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Cycles in Lending
Standards?
John A. Weinberg
T
he lending activity of commercial banks has long received considerable
attention as an important contributor to the performance of the economy.
This attention has, perhaps, become sharper in the wake of the difficulties experienced by the banking industry in the 1980s. In recent years, the public
perception of bank lending seems to have swung through a cycle. In the early
1990s, the prevailing view was that the bank loan market was experiencing a
credit crunch in which banks set unreasonably high credit standards, denying
credit to qualified borrowers.1 By late 1994, with growth in bank loans picking
up, some voiced concerns that banks were possibly becoming too loose in their
standards for acceptable credit risks. These concerns appeared in the pages of
the American Banker and other professional journals and in speeches by the
Chairman of the Federal Reserve Board and the Comptroller of the Currency.2
Do swings from tightness to laxity in credit standards constitute an inherent
part of bank lending activity? Some observers have suggested that such cycles
can be caused by an imperfection in bank credit markets that results in a systematic tendency for banks to overextend themselves during general expansions
of lending. In some expressions of this view, the imperfection is the result of
government intervention in banking markets, while in others it results from
the very nature of credit markets. In any case, the implied consequence is a
cycle in lending behavior that is distinct from and may exert an independent
influence on the general business cycle.
The existence of a systematic cycle in lending standards could have important public policy implications. If lending displays a bias toward too much
The author thanks Tom Humphrey, Mary Finn, Tony Kuprianov, and John Walter for their
comments on an earlier draft. The views expressed in this article are the author’s and do
not necessarily represent the views of the Federal Reserve Bank of Richmond or the Federal
Reserve System.
1 For general discussions of credit crunches and the experience of the early 1990s, see
Bernanke and Lown (1991) and Owens and Schreft (1995).
2 See, for instance, Dunaief (1995), Connor (1995), and Stevenson (1994).
Federal Reserve Bank of Richmond Economic Quarterly Volume 81/3 Summer 1995
1
2
Federal Reserve Bank of Richmond Economic Quarterly
risk during expansions, resulting in increased risks of bank failures and losses
to bank insurance funds, then bank regulation might justifiably seek to smooth
out lending cycles. Placing greater scrutiny and restrictions on bank lending
during periods of loan growth would slow the expansions but limit ensuing
losses.
Justifying a regulatory policy aimed at eliminating or reducing cyclical
swings in credit standards and lending activity requires that expansions of credit
be inherently excessive. Excessive, however, is a relative term. A relevant question in this regard concerns the behavior of an “ideal” credit market, in which
there is no source of market failure or government-induced distortion. Would
such a market produce cycles in lending standards? This is the question that
this article explores. First, Section 1 describes the notion of cycles in standards
in a bit more detail. The following sections examine a stylized model of credit
market behavior. This model suggests that lending standards might naturally be
expected to change with market conditions. In fact, the model serves to make
the point that in a well-functioning credit market, an expansion of lending
almost necessarily implies an easing of standards and the extension of credit
to “riskier borrowers.” This is true when borrower riskiness is defined in terms
of borrower-specific characteristics drawn, for instance, from a commercial
borrower’s recent income statement or balance sheet. These characteristics tell
only part of the story of the true expected profitability of a loan to a particular
borrower. Also important are aggregate conditions that affect the demand for
the borrower’s product or the supply of its inputs. These factors are typically
not well captured by borrower-specific indicators of credit quality.
1.
CYCLES IN CREDIT STANDARDS
In discussions of bank lending activity, the notion of cycles in lending standards
typically begins with expansions; standards fall with heightening competition
in expansions and rise in contractions as banks respond to their own capital
shortfalls or the constraints of regulators. On the downside, this view is related
to the notion of a credit crunch, which has received much attention in recent
years. On the upside, the view often seems to embody the belief that expansions
of credit are inherently excessive. In March of 1995, for instance, the American Banker reported on growing concerns about easing credit standards and on
warnings from the loan officers’ professional association advising lenders to
resist competitive pressures and maintain credit quality.3 Such warnings come
very close to the caution voiced by Federal Reserve Chairman Alan Greenspan
in speeches given in late 1994.
If, indeed, there is a natural tendency for expansions of credit to push down
lending standards, then it would naturally follow that expansions would lead,
3 See
Mathews (1995).
J. A. Weinberg: Cycles in Lending Standards?
3
at least sometimes, to significant increases in losses on loans. Further, the contraction phase of the credit cycle could be worsened as banks find themselves
with bad assets on their books. Hence, under this view the primary driving
force of cycles in the credit markets is the propensity of lenders to succumb
to an unrealistic optimism in good times, creating lending booms that sow the
seeds of their own demise. Just such a description of the cycle appears in some
discussions by credit professionals.4 In this sort of description the expansion
of lending could, itself, be the impulse that drives the cycle, as a spontaneous
wave of optimism hits the lending community. Alternatively, the expansion
could be an overreaction to other shocks to the economy that “legitimately”
shift the supply of or demand for credit.
The view of cycles outlined above is one of fluctuations in a number of
variables. In this view, the cycle in lending standards coincides with the cycle in
the amount of lending, while movements in the amount of lending are followed
by movements in loan losses. In particular, an increase in lending activity is
followed by an increase in losses. Does this pattern appear in the data? Figure
1 displays the behavior of the growth rate of total loans and loan charge-offs as
a fraction of total loans at insured banks and thrifts in the United States from
1950 to 1992.
While the relationship is not striking, there do appear to be periods in which
unusually strong loan growth preceded rising charge-offs. For instance, in 1972
and 1973, annual loan growth was about 18 percent compared to around 10
percent in 1971 and 1974. From 1973 through 1976, charge-offs, as a percent
of total loans, grew from 0.33 percent in 1973 to 0.77 percent in 1976. It is
important to note that accounting standards allow banks some discretion as to
when to write off a nonperforming loan.5 Consequently, charge-offs resulting
from an episode of poor credit quality can be spread out over time, delaying
and smoothing the apparent response of losses to an expansion of lending. The
last several years of data in Figure 1 seem to reflect the downside of the cycle:
an extended period of rising losses followed by a period of declining loan
growth, culminating in two years of declines in the level of lending activity.
While not conclusive, the behavior of loan growth and charge-offs is not
inconsistent with the notion of a cycle in lending standards. Alternatively, one
might seek more direct evidence on lending standards. The Federal Reserve
Board’s Senior Loan Officer Opinion Survey on Bank Lending Practices contains explicit questions on this topic. Schreft and Owens (1991) provide a
detailed description of this survey evidence. Broadly stated, they find that loan
officers’ self-professed tendency to tighten lending standards follows a cyclical
pattern that tends to peak (attain the greatest tightening of lending standards)
4 See,
5 This
for instance, Mueller and Olson (1981) and Stevenson (1994).
issue is discussed in Darin and Walter (1994).
4
Federal Reserve Bank of Richmond Economic Quarterly
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
-0.1
+
Charge-Off Rate
(Right Scale)
0.015
0.010
Growth Rate of Total Loans
(Left Scale)
Percent
Percent
Figure 1 Loan Growth and Charge-Offs
0.005
0.000
50
55
60
65
70
75
80
85
90
just prior to or during general economic downturns. They find similar behavior
in responses to questions about loan officers’ general willingness and unwillingness to make loans. Further, peaks in the loan growth series in Figure 1
tend to occur at or around the troughs in the lending standards series studied
by Schreft and Owens. In other words, the survey data suggest that low selfreported standards coincide with high growth in actual lending, while bankers
report tightest credit around periods when lending growth is slow.
An additional source of information on lending standards could come from
examining the characteristics of banks’ borrowers. If borrowers’ average financial conditions become weaker, then one might be able to conclude that
standards have eased, especially if one can control for the general state of the
economy. Cunningham and Rose (1994) perform such a study. They examine
evidence on the financial conditions of small and mid-sized commercial borrowers. This evidence appears to be consistent with a general easing of standards
from 1978 to 1988 and a tightening from 1988 to 1991. Except for 1978 itself,
1978 to 1988 was a period of fairly steady loan growth, with a few relatively
strong years. By contrast, 1988 to 1991 was a period of generally declining
loan growth. Hence, this study serves to confirm the general coincidence of
changes in standards with movements in total lending activity.
The evidence discussed above is broadly consistent with the notion of
cycles in lending standards. More difficult is the question of whether expansions
represent a systematic tendency of banks to become too easy in the extension
J. A. Weinberg: Cycles in Lending Standards?
5
of credit. There are at least two views as to what might drive lenders to display
an excessively tolerant attitude toward credit risk. One view is that there is a
fundamental imperfection in financial markets and, in particular, markets for
bank credit. Under this view, the source of the imperfection seems to be that
the credit quality of borrowers is difficult and costly to observe. Banks spend
resources gathering information on borrower characteristics, but it is difficult
for outside observers to verify the information obtained by the bank. Such limits
to the flow of information form the basis of much of the recent work in the
theory of banking. Limited information, however, does not necessarily imply a
bias toward accepting greater risks. Indeed, if providers of funds feel that they
are at an informational disadvantage, the cost of funds could be higher than
in the case of perfect information. This could have the effect of making banks
less willing to accept risks than they otherwise might be.
In some discussions of cycles in lending the supposed market imperfection
simply seems to be a general failure of lenders to make good credit decisions.
Sometimes this failure takes the form of basing decisions on the decisions of
other lenders, rather than on an independent evaluation of market conditions.
A model of banking markets that encompasses this view is presented by Rajan
(1994). In that model, bankers are driven by a concern for their reputations,
which could suffer if they fail to expand credit while others are doing so.
Related to this view is the belief that competition drives lenders to ease their
standards.6 Hence, one might refer to this type of imperfection as the “herd
mentality” problem.
The second, and perhaps more widely advanced view on the source of
excessive risk tolerance by banks, is deposit insurance. It is well understood
by now that federal deposit insurance has the potential of distorting banks’
attitudes toward credit risk, and indeed there is an extensive literature on this
subject.7
There is nothing inherently cyclical in the distortion caused by deposit
insurance. In fact, by placing a limit on the losses a bank can incur, deposit
insurance is likely to have its greatest effect on incentives when the overall
financial condition of banks is weak. This seems to run counter to the view of
a lending cycle in which banks overextend in good times, making themselves
vulnerable to adverse shocks. It is possible that the interaction of deposit insurance and the behavior of bank regulators could produce the type of cyclical
behavior described. Such behavior could arise if the scrutiny of and restrictions
on bank lending applied by examiners and regulators varied, after the fact, with
observed performance of banks. While times are good, regulators might impose
little interference, intervening only after significant losses have been incurred.
6 This
7A
view is expressed in the discussion described by Randall (1994).
notable statement of the problem is given by Kareken (1983).
6
Federal Reserve Bank of Richmond Economic Quarterly
Some have argued that the “prompt corrective action” requirements for regulators in the Federal Deposit Insurance Corporation Improvement Act of 1991
(FDICIA) embody just such a procyclical effect. On the other hand, recent,
pre-FDICIA experience might be better characterized by a different pattern of
regulatory response to changes in banks’ conditions. In particular, it is possible
that regulatory forbearance during the 1980s resulted in too few restrictions on
the activities of troubled banks.
Much of the research and writing on cycles in bank lending activity often
has been driven by the observation that expansions in credit-risk exposure tend
to be concentrated in a particular type of lending; lending for commercial real
estate development in New England in the 1980s is a typical example.8 From
the point of view of an individual bank, loan concentration implies a loan portfolio that is, perhaps, not as well diversified as it might be. In an environment
characterized by deposit insurance and bank regulation, concentrations could
present a problem to regulatory agencies. The description of cycles driven
by concentrated expansions of lending, however, is very similar to the more
general description mentioned above; lenders take on excessive risks as the
market is swept up in a lending euphoria that skews individuals’ evaluations
of credit quality.
Each of the various views on market or regulatory failures driving cycles in lending behavior embodies some theory of the behavior of banks and
financial markets. While each may have the ability to explain some aspect of
observed behavior, the question remains: Compared to what? The following
sections explore the implications of a benchmark model, in the absence of
market imperfections or government intervention.
2.
A MODEL OF A CREDIT MARKET
An explicit model of equilibrium in a market for loans is useful for interpreting
observed lending patterns. The model presented below is one without many of
the transaction costs and other frictions that are often thought to be important
to banking and credit markets. While such frictions are probably important
for an explanation of the institutional structure of these markets, they are not
necessarily essential to every aspect of observed market behavior. Exploration
of an “ideal” or frictionless model will help to uncover what aspects of observed
behavior result from market frictions and what aspects arise simply from the
competitive allocation of credit among heterogeneous borrowers.
First, it is useful to think of the activity in the model economy as taking place over two time periods. These two time periods might be thought
of as a component of a more explicitly dynamic model, in which aggregate
8 Randall
(1994) summarizes the proceedings of a conference on this subject.
J. A. Weinberg: Cycles in Lending Standards?
7
conditions evolve over time. A dynamic model would, ultimately, be better
suited to closely matching the notion of cycles in lending standards. In the
simple environment considered here, the forces that might drive cycles show
up clearly in the comparative statics of the two-period version of the model.
Since, in the model, lending standards arise as a characteristic of equilibrium,
one can examine directly how changes in exogenous (to the decisions of market
participants) market conditions change the equilibrium standards.
The demand side in the model economy’s credit market is composed of
a large number, N, of potential borrowers. Suppose that these are all business
borrowers. Each borrower’s business is particularly simple. A business requires
a fixed amount of resources in the first period and, if successful, produces a
fixed amount of output in the second period. If unsuccessful, the business
produces nothing. Both input and output should be thought of as measured in
monetary units, and, for simplicity, it is useful to set the value of the fixed
amount of inputs required equal to one. The output of a successful business
enterprise is denoted by Y > 1. This output, or revenue, should be thought of
as being net of the opportunity cost of the business owner’s time.
Borrowers’ businesses vary in their likelihood of success. In general, probability of success might be thought of as depending on an array of borrower
characteristics such as education, past business experience, and the nature of the
product or service being produced. A detailed list of characteristics is beyond
the scope of the model. Suppose that all of the relevant information about a
borrower can be reduced to a single summary statistic, or “score,” that can
be expressed as a probability. Hence, a borrower’s type is φ, 0 ≤ φ ≤ 1.
An important market characteristic, then, is the distribution of individual borrower types, represented by a cumulative distribution function F(φ). That is,
F(φ) denotes the fraction of the population with type no greater than φ. This
fraction increases with φ. Since φ is a probability, F(0) = 0 and F(1) = 1.
This distribution function has an associated density, denoted by f (φ) where
f (φ) ≡ F (φ).
In addition to their individual types, businesses’ prospects may depend on
aggregate conditions. One way of introducing aggregate conditions into the
model is to assume that the economy is subject to an aggregate technology
shock that affects the output of successful firms. In this regard, Y should be
thought of as a random variable, the realization of which is not known until
firms observe their productive outcomes. Accordingly, a firm’s output is the
product of two random variables, its own success or failure and the aggregate
shock to technology. The expected output for a firm of type φ is φEY, where
EY is the expected value of Y.
An important ingredient of any model of resource allocation among
heterogeneous users is its information structure. The ability of market mechanisms to assign resources to their most productive users can depend on whether
8
Federal Reserve Bank of Richmond Economic Quarterly
individual productive capabilities are public or private information.9 Similarly,
if it is difficult for outsiders to observe a business’s productive outcome, then
the business may not be able to commit to payments contingent on that outcome. This limit on payment schemes can, in turn, limit the opportunities for
gains from trade. In what follows, these complications are assumed away. This
simplifying assumption is not based on a belief that informational frictions are
not an important characteristic of financial market transactions. Indeed, such
frictions are probably essential for understanding why financial institutions
and contracts look the way they do. The assumption of perfect information
allows the model to focus more directly on the implications of diversity among
borrowers for the market allocation of credit.
Suppose that business owners have no funds of their own. Since there is an
interval of time between the employment of inputs and the realization of output,
providing a business with the funds to acquire inputs necessarily involves an
extension of credit. If there are limited funds available for acquiring resources,
or if there are alternative uses of funds that provide sufficient returns, then some
businesses will operate and others will not. The credit market in this model
economy allocates savers’ funds to ultimate business borrowers and thereby
determines which firms operate.
Business borrowers are assumed to be risk-neutral, caring only about the
expected value of their profits. A business is willing and able to borrow funds
if, after paying for the loan, it expects to cover at least the opportunity cost of
its owner-manager’s time. Since the resolution of uncertainty occurs between
the extension and the repayment of the loan, the measure of the cost of credit
relevant to the borrower’s decision is the expected payment. This expected
payment, or expected return from the lender’s point of view, plays the role of
the price in this market. That is, in order to attract funds, a borrower must be
able to offer payments that have an expected value equal to the market return,
denoted r.
Given a market return, r, any borrower whose expected output, φEY, is at
least equal to r will be willing and able to profitably take a loan. Any such
borrower can fashion a feasible repayment schedule that yields an expected return of r to lenders. Repayment schedules, here, are particularly simple. Since
an unsuccessful business has no proceeds, no payment is made in that event. A
successful business can make a repayment up to its realized output Y. Since Y is
a random variable (common to all borrowers) that is realized before repayments
are made, the repayment by a successful borrower can be contingent on Y as
well as on the borrower’s type, φ. Hence, the repayment made by a successful
borrower will be denoted by ρ(φ, Y, r), where r is added as an argument to
indicate dependence on the required expected return.
9 See
Lacker and Weinberg (1993) for a discussion of private information in a model very
similar to that in the present discussion.
J. A. Weinberg: Cycles in Lending Standards?
9
The expected repayment from a type φ borrower is φEY [ρ(φ, Y, r)], where
the notation EY indicates expectation with respect to the aggregate random variable Y. An “acceptable” payment schedule is one that, given the borrower’s
type, meets the market expected-return requirement. That is, to be acceptable,
a schedule must satisfy
φEY [ρ(φ, Y, r)] ≥ r.
(1)
Recall that each loan is assumed to be one unit of funds. Therefore, given
the required expected return, r, the demand for funds is simply given by the
portion of the population of borrowers that can structure a payment schedule
that yields an expected payment of at least r. The most a successful borrower
can pay is the entire realized output, Y. A borrower who agrees to pay this maximum for all realizations of Y will be just indifferent between borrowing and
remaining idle. The lowest-type borrower for whom obtaining a loan that the
market finds acceptable will be worthwhile is the borrower for whom φEY = r,
or φ = r/EY. Hence, the demand for funds is the number of borrowers with
φ ≥ r/EY. Accordingly, demand can be expressed as a function of the required
return and the expected value of the aggregate technology shock:
D(r, EY) = [1 − F(r/EY)]N.
(2)
The right-hand side of this equation is simply the size of the population, N,
times the fraction of borrowers with probabilities of success above the cutoff value, r/EY. Notice that this function has the usual property of a demand
function: demand is decreasing in the cost of funds, r. In addition, all businesses
are willing to borrow when r = 0, and none will be willing when r = EY.
Given a cost of funds, the cut-off value of the individual probability of
success resembles a credit standard. This probability of success is assumed to
be a function of the observable characteristics of the borrower. Hence, setting
a minimum level for φ amounts to establishing a standard based on borrowers’
characteristics.
In addition to borrowers, the economy is populated by other individuals
who are endowed with funds in the first period, but may seek to save some of
those funds for consumption purposes in the second period. Depending on the
existence of alternatives, some or all of these funds may be saved in the form
of loans to the business borrowers described above.
The behavior of savers will not be as carefully described as that of borrowers. For purposes at hand, it is sufficient to specify a function, S(r), that
determines the aggregate savings available to business borrowers. This function
could take a variety of forms. One possibility is that S(r) is a constant value
(perfectly inelastic). This would occur if loans to businesses constituted the
only outlet for savings and if savers’ preferences were such that their desired
10
Federal Reserve Bank of Richmond Economic Quarterly
savings were independent of the rate of return.10 Alternatively, if the business
borrowers represent only a small portion of the economy’s demanders of funds,
then the supply of funds to this small sector can be treated as perfectly elastic;
as much funding as is demanded will be forthcoming so long as borrowers are
able to pay an expected return at least as great as that available elsewhere in
the economy.
A last possibility for the behavior of savers arises if, for instance, they
use first-period funds for both first- and second-period consumption, and if
the only means of saving is through loans to businesses. In this case, savings
increase with r. The key maintained assumptions about S(r) are that S(0) < N
and S(EY) > 0. The first of these two assumptions assures that if credit is free,
demand exceeds supply. The second implies that there are at least some rates
of return for which supply exceeds demand.
In addition to the expected return, savings may depend on other aggregate
conditions. For instance, one might imagine variation in savers’ first-period
endowment. This endowment might be, in part, the result of an aggregate
shock in the first period that may, in turn, affect people’s expectations about
the second-period technology shock. Hence, one can imagine shifts in aggregate conditions that cause shifts in both the supply of funds by savers and the
demand of borrowers.
To this point, there has been no mention of the role of intermediaries in
the credit market. The model’s specification is such that intermediaries are not
necessary to allocate effectively the economy’s resources among borrowers. In
fact, borrowers could sell securities, offering prorated shares of the payment
schedules, ρ(φ, Y, r). In equilibrium, a saver might own shares of the securities
issued by a variety of borrowers. The ability of borrowers to contract directly
with the savers arises from the absence of informational frictions in the model.
Although intermediaries are not necessary, one can imagine credit in this
economy flowing through institutions that take the funds of savers, in exchange
for some promised return, and distribute those funds to borrowers. Borrowers,
then, make the payments ρ(φ, Y, r) to the intermediaries, who use those payments to pay the savers. If the activity of intermediating is costless and if
there is free entry into this activity, then, in equilibrium, intermediaries will
have zero profits and the allocation of resources will be the same as in the
case of the direct securities market. While the discussion that follows will
adopt this interpretation of an intermediated credit market, note that the model
applies more generally to the competitive allocation of credit, with or without
intermediation.
It is assumed that all market participants—borrowers, savers, and intermediaries—take the expected return, r, as given in making their economic
10 For
instance, if savers care only about consumption in the second period, then they will
save all of their resources, regardless of the rate of return.
J. A. Weinberg: Cycles in Lending Standards?
11
decisions. Under the intermediary interpretation, intermediaries attract total
savings of S(r), accept loan applications from borrowers, and adopt a lending standard. A lending standard, here, is simply a rule stating the minimum
acceptable probability of success. A type φ borrower takes a loan from the
intermediary that offers the lowest expected repayment, EY [ρ(φ, Y, r)], among
those that are willing to accept a type φ credit risk.
3.
EQUILIBRIUM
The equilibrium expected return equates savers’ supply of funds to the demand
of the borrowers. Hence, equilibrium can be represented by a standard supply
and demand diagram, as in Figure 2.11 The market-clearing return, denoted r∗ ,
in turn determines the minimum acceptable probability of success. This lending
standard, φ̂, satisfies
(3)
φ̂EY = r∗ .
Given this cut-off, the aggregate amount of credit extended is N[1 − F(r∗ /EY)].
Competition assures that each loan makes zero expected profits for the lender.
This zero-profit condition implies that repayment schedules for borrowers of
type φ are such that φEY [ρ(φ, Y, r∗ )] = r∗ . Note that all loans are risky in the
sense that a borrower only makes payments when successful (with probability
φ). Hence, for all but the best borrower (type φ = 1) there is a positive markup
between the average contractual repayment rate, EY [ρ(φ, Y, r∗ )], and the cost
of funds, r∗ . Notice, also, that this markup gets larger as the probability of
success, φ, gets smaller.
All lending brings with it an expectation of losses. A loss will be said
to occur when no payment is made. This amounts to defining default as arising only from the borrower-specific failure to produce, not from variations in
payments arising from changes in the aggregate shock. That is, the loss on a
failed borrower is equal to the payment that would have been expected had
that borrower succeeded. For a type φ borrower, then, the expected loss to the
lender is weighted by the probability of failure:
L(φ, r∗ ) ≡ (1 − φ)EY [ρ(φ, Y, r∗ )] =
(1 − φ) ∗
r .
φ
(4)
Note that L(φ) is decreasing in φ, as one would expect.
Equation (4) gives a narrow specification of losses. A loan to a given
borrower suffers a loss only if that borrower fails to produce a positive output.
A broader specification of losses might include shortfalls of payments caused
by bad realizations of the aggregate shock. For instance, one might define a
loss as occurring any time the realized payment is less than EY [ρ(φ, Y, r∗ )].
11 Note,
however, that in Figure 2, price is on the horizontal axis.
12
Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Supply and Demand in the Credit Market
Loan Volume
N
S(r)
ˆ
N[1 - F()]
D(r, EY) = [1- F(r/EY)]N
0
EY
r*
+
r
This broader specification would complicate the analysis without changing the
essential fact that expected losses rise as the probability of success falls.
The narrower specification of losses, L(φ), treats as losses only those shortfalls resulting from borrower-specific performance. With the narrow specification, changes in the equilibrium lending standard will affect expected aggregate
losses primarily through the change in the riskiness of loans made. Since this
interaction is the intended focus of this article, the narrow specification is
sufficient. Expected aggregate losses, per loan made, are given by
1
L̄(r ) ≡
[1 − F(r∗ /EY)]
∗
1
L(φ, r∗ )f (φ)dφ.
(5)
r∗ /EY
The right-hand side of equation (5) is the expected value of the function
L(φ, r∗ ), given that φ is greater than the threshold value φ̂. For each type,
L(φ, r∗ )f (φ) gives the expected losses weighted by that type’s weight in the
J. A. Weinberg: Cycles in Lending Standards?
13
population’s distribution of types. The integral, then, can be viewed as adding
up these weighted expected losses across all borrowers that receive loans.
Dividing by the number of borrowers receiving credit gives average losses.
The competitive equilibrium described above achieves an efficient allocation of funds.12 Given the expected technology shock, the market extends
credit to businesses from the top down, until the supply of funds has been
exhausted. In other words, it is impossible to find two businesses, one funded
and one unfunded, such that the unfunded firm has a higher probability of
success. There is no “credit rationing” in the sense that the term is often used.
All borrowers who are willing and able to pay the required return (in expected
value) receive loans.
4.
COMPARATIVE STATICS
As presented above, the key exogenous variable in the model is EY, the
expected value of the aggregate shock. Accordingly, all of the endogenous variables determined in equilibrium are functions of EY. Expectations, of course,
are not truly exogenous variables. Rather, economic decisionmakers form expectations by observing current conditions and making assumptions about the
relationship between current conditions and future conditions. Recall that the
model’s description specifies a two-period time horizon. The market for funds
allocates credit in the first period, based on expectations of conditions in the
second period. The expectation EY, then, may be a function of some condition
observed in the first period. Since that condition would be exogenous to the
decisions made by borrowers and lenders, including the additional variable
would contribute little to the current analysis. This section, therefore, simply
treats EY as an exogenous variable and examines how the endogenous variables
respond to changes in EY.13
In terms of Figure 2, an increase in EY brings about an increase in demand, as represented by an outward shift of the demand curve. For any required
return, the set of borrowers who can profitably meet that requirement grows
as EY rises. Specifically, the marginal borrower is that for whom φEY = r ;
this is the borrower who, if promising payments with expected value r, has
expected earnings net of payments just equal to zero. An increase in EY lowers
the marginal type. Hence, as expected aggregate conditions improve, riskier
borrowers become acceptable for any given required return.
12 A deeper discussion of the efficiency of equilibrium would have to include a more explicit
treatment of the economic decisions made by savers; as long as those decisions are such that an
equilibrium exists and there are no externalities, equilibrium will be efficient (Pareto-optimal).
13 More precisely, the distribution of Y is assumed to be subject to exogenous changes.
Market participants are aware of the true distribution when they make their first-period decisions
and, hence, form expectations rationally.
14
Federal Reserve Bank of Richmond Economic Quarterly
Of course, since more businesses are able to borrow profitably at any given
return, market forces will cause the equilibrium required return, r∗ , to rise if,
as in Figure 2, the supply of funds depends on the return to saving. The overall
change in the lending standard, φ̂, is
∗
d
r∗
dr∗ 1
r
dφ̂
=
−
=
.
(6)
dEY
dEY EY
dEY EY (EY)2
The right-hand side of this expression includes a direct and an indirect effect.
The direct effect is given by the second term; for a fixed required return, r∗ , φ̂
falls as EY rises. The first term is the indirect effect that comes from the change
in r∗ . If the increase in demand causes r∗ to rise, then the rising cost of funds
will dampen the decline in the lending standard. If the supply of funds is not
perfectly inelastic, the equilibrium lending standard falls, and credit is extended
to a wider range of borrowers as EY rises. If supply is perfectly elastic (for
instance, if this credit market is a small portion of broader financial markets),
then the first term on the right side of equation (6) is zero and only the direct
effect on φ̂ is present.
Since all loans have some risk of loss, the expansion of lending when
expected conditions improve causes expected losses on loans to rise. Expected
losses, however, also rise as a percent of total loans. That is, L̄(r∗ ) rises as EY
rises. This can occur for two reasons. First, as seen in equation (4), the expected
loss on a loan to any given borrower rises if r∗ rises. Second, expansion of
lending comes at the bottom of the range of acceptable credit risks. Hence,
added borrowers bring greater-than-average expected losses.
Closely related to loan losses is the average markup of repayments by
businesses that do not fail over the cost of funds, r. This markup for a type
φ borrower is E[ρ(φ, Y, r∗ )]/r∗ . Since, in equilibrium, φE[ρ(φ, Y, r∗ )] = r∗
for all acceptable borrowers, the markup for a type φ borrower is simply
1/φ. Therefore, for any given borrower, the average markup of repayment
(conditional on successful production) over the lender’s opportunity cost of
funds is independent of aggregate conditions. Accordingly, the average markup
across borrowers rises as the lending standard, τ̂ , falls with a rising EY. Additional borrowers have a higher risk of failure and, therefore, must make greater
average payments, conditional on success.
5.
DISCUSSION
Can the model presented above yield insights into cycles in lending standards? Recall that in a fully dynamic model, aggregate conditions would evolve
according to a process that would allow market participants to form the expectation, EY, using observations of current and, perhaps, past conditions. More
precisely, EY would be a function of at least the most recently observed productive (or profit) experience of successful business firms.
J. A. Weinberg: Cycles in Lending Standards?
15
If there is persistence in the aggregate technology shock that determines Y,
then expected output is high when current output is high. Hence, loan growth
and falling standards occur during good economic times. Since a lower standard
means the extension of credit to borrowers with a higher risk of default, it seems
as though movements in the lending standard can have the effect of widening
the swings in the economy, especially in downturns. If, when expectations were
high, a downturn occurs because of a low realization of Y, output falls not just
because of the low aggregate shock, but also because of the higher number of
failures of marginal borrowers. On the other hand, if the economy experiences
a string of good aggregate performance, it is possible to have a period of rising
losses while total loans and economic activity continue to grow. This possibility
might be reminiscent of much of the expansion of the U.S. economy in the
1980s.
In the model outlined above, changes in lending and lending standards
come entirely from the demand side of the market. It may be reasonable to
suppose that changes in aggregate conditions also affect the supply of funds.
In particular, if a high current value of the output of successful firms means
that savers’ total resources are correspondingly high, then the supply of funds
may expand together with the demand. This would tend to reinforce the effect
of aggregate shocks on the expansion of credit while making the effect on
expected returns ambiguous.
Popular discussions of changes in lending standards often implicitly treat
these changes as originating from the supply side. When standards rise, for instance, there are often references and anecdotes concerning borrowers who are
being rationed out of the market. Such references seem to suggest that demand
has not changed, but that borrowers who are willing and able to take loans
under prevailing market conditions are being denied credit. Casual evidence,
however, is difficult to interpret, particularly when credit flows from suppliers
to demanders through intermediaries. One can interpret the model presented
above as one of an intermediated market. Under this interpretation, for the
market to achieve its equilibrium allocation it is not necessary for borrowers
to be aware of aggregate conditions. If every borrower submits applications
to one or more banks, then the banks, knowing the aggregate condition, will
make credit decisions according to the equilibrium described above. Hence, the
perceptions of borrowers who get screened out as standards rise (as EY falls)
can be misleading.
Another aspect of popular discussions of variations in lending standards
is the degree of competition among intermediaries. As in late 1994 and early
1995, observers often point to rising competition as a cause of weakening bank
standards. The model presented in this article, as a model of perfect competition, cannot capture rising competition. Presumably, if competitiveness is to
vary, competition must be imperfect. Imperfect competition might arise from
technological or legal barriers to entry. Procyclical competitiveness might arise
16
Federal Reserve Bank of Richmond Economic Quarterly
if lenders in an imperfectly competitive market find it more difficult to refrain
from aggressive competition during expansions. This sort of varying competition would reinforce the countercyclical lending standards of this article’s
model.
The model clearly does not imply that a goal of supervision and regulation
of financial intermediaries should be to encourage “smooth” lending standards
that do not vary with aggregate conditions in the economy. Rather, the model
implies that in times of strong and improving economic activity, businesses
(and households) whose individual characteristics make them look risky may
become acceptable borrowers. A policy aimed at smoothing credit standards
would limit the ability of participants in the economy to adapt to changing economic conditions and to take productive risks when those risks are warranted.
Of course, the existence of regulatory oversight of bank activities seems to
presume a bias toward excessive risk taking on the part of banks. The source of
that presumption is deposit insurance. It is not clear why the adverse incentive
created by deposit insurance should be greatest during periods of strong economic performance and expanding credit. If anything, deposit insurance should
have its greatest effect on incentives at the other end of the cycle in credit
market conditions. The argument in the literature on this topic is that insured
lenders may have an incentive to “bet the bank” when they are in weak financial condition. This suggests that regulatory scrutiny of bank lending behavior
should be greatest during a period of low returns (profitability) for banks.
6.
CONCLUSION
Lending standards are usually thought of in terms of requirements placed on the
characteristics of individual loans and borrowers. A central point of this article
is that there is a natural tendency for standards to vary inversely with the level
of activity in the credit markets. There is, of course, a sense in which lending
standards do not vary at all in this article’s model. The marginal borrower is
always that borrower who can just afford repayment terms that just cover (in
expected value) the opportunity cost of savers’ funds. That the quality of the
marginal borrower varies is the result of the interaction of individual and aggregate conditions in determining the payoff to extending credit. As (expected)
aggregate conditions improve, a borrower who did not look creditworthy yesterday may now be deserving of credit.
The frictionless model examined herein is missing many of the ingredients
that are perhaps important to the character of modern credit markets and institutions. Even with further complications, however, the fundamental role of the
credit market would be the same. Given perceptions of the general condition
of the economy, the credit market sets a threshold for acceptable risks. If the
market functions well, subject to whatever imperfections may be present in
J. A. Weinberg: Cycles in Lending Standards?
17
the economic environment, then an improvement in participants’ perceptions
of aggregate conditions will lower the threshold.
The concerns about credit quality that are often expressed in times of
expanding credit are typically driven by very current news. A more “global”
perspective would recognize the interaction of individual and aggregate conditions. The model examined in this article suggests that a policy goal of credit
standards that are constant over time is not only unwarranted, but could be
counterproductive.
REFERENCES
Bernanke, Ben S., and Cara S. Lown. “The Credit Crunch,” Brookings Papers
on Economic Activity, 2:1991, pp. 205–47.
Connor, John. “Comptroller Unveils Credit Committee, Citing Slippage in
Banks’ Loan Standards,” The Wall Street Journal, April 10, 1995, p. 2.
Cunningham, Donald F., and John T. Rose. “Bank Risk Taking and Changes in
Financial Condition of Banks’ Small and Midsized Commercial Customers,
1978–1988 and 1988–1991,” Journal of Financial Services Research, vol.
8 (December 1994), pp. 301–10.
Darin, Robert M., and John R. Walter. “Were Bank Examiners Too Strict with
New England and California Banks?” Federal Reserve Bank of Richmond
Economic Quarterly, vol. 80 (Fall 1994), pp. 25–47.
Dunaief, Daniel. “Bankers Agree: Credit Standards Have Been Sliding,”
American Banker, April 11, 1995, pp. 1 and 24.
Kareken, John H. “Deposit Insurance Reform or Deregulation Is the Cart, Not
the Horse,” Federal Reserve Bank of Minneapolis Quarterly Review, vol.
7 (Spring 1983), pp. 1–9.
Lacker, Jeffrey M., and John A. Weinberg. “Coalition Proof Equilibrium in a
Private Information Credit Economy,” Economic Theory, vol. 3 (Spring
1993), pp. 279–96.
Mathews, Gordon. “Worries Grow that Downturn in Economy Could Punish
Banks for Easing Credit,” American Banker, March 27, 1995, pp. 1 and
28.
Mueller, Henry P., and Leif Olson. “Credit and the Business Cycle,” in Herbert
V. Prochnow, ed., Bank Credit. New York: Harper and Row, 1981, pp.
78–91.
Owens, Raymond E., and Stacey L. Schreft. “Identifying Credit Crunches,”
Contemporary Economic Policy, vol. 13 (April 1995), pp. 63–76.
18
Federal Reserve Bank of Richmond Economic Quarterly
Rajan, Raghuram G. “Why Bank Credit Policies Fluctuate: A Theory and
Some Evidence,” Quarterly Journal of Economics, vol. 109 (May 1994),
pp. 399–441.
Randall, Richard E. “Safeguarding the Banking System in an Environment
of Financial Cycles: An Overview,” New England Economic Review,
March/April 1994, pp. 1–13.
Schreft, Stacey L., and Raymond E. Owens. “Survey Evidence of Tighter Credit
Conditions: What Does it Mean?” Federal Reserve Bank of Richmond
Economic Review, vol. 77 (March/April 1991), pp. 29–34.
Stevenson, Bruce G. “Research Report . . . Capital Flows and Loan Losses
in Commercial Banking,” Journal of Commercial Lending, vol. 77
(September 1994), pp. 18–26.
Errors in Variables and
Lending Discrimination
Jed L. DeVaro and Jeffrey M. Lacker
D
o banks discriminate against minority loan applicants? One approach
to answering this question is to estimate a model of bank lending
decisions in which the probability of being denied a loan is a function
of a set of creditworthiness variables and a dummy variable for the applicant’s
race (z = 1 for minorities, z = 0 for whites). A positive coefficient on the
race dummy is taken as evidence that minority applicants are less likely to be
granted loans than white applicants with similar qualifications. This approach
is employed in many empirical studies of lending discrimination (Schill and
Wachter 1994; Munnell et al. 1992), in U.S. Department of Justice lending
discrimination suits (Seiberg 1994), and in regulatory examination procedures
(Bauer and Cromwell 1994; Cummins 1994).
One weakness of this approach is that an estimate of the discrimination
coefficient may be biased when measures of creditworthiness are fallible. In
such situations, distinguishing racial discrimination from unmeasured racial
disparities in creditworthiness can be difficult. If true creditworthiness is lower
on average for minority applicants, the model may indicate that race adversely
affects the probability of denial, even if race plays no direct causal role.
There are good reasons to believe that measures of creditworthiness are
fallible. First, regulatory field examiners report difficulty finding matched pairs
of loan files to corroborate discrimination identified by regression models. An
applicant’s file often yields a picture of creditworthiness different from the one
given by model variables. Second, including more borrower financial characteristics generally reduces discrimination estimates, sometimes to zero (Schill and
Wachter 1994). Third, studies of default data find that minority borrowers are
more likely than white borrowers to default, even after controlling for income,
The authors thank Mary Finn and Peter Ireland for helpful conversations during this research
and Tom Humphrey, Tony Kuprianov, and Stacey Schreft for helpful comments on an earlier
draft. The authors are solely responsible for the contents of this article. The views expressed
do not necessarily reflect those of the Federal Reserve Bank of Richmond or the Federal
Reserve System.
Federal Reserve Bank of Richmond Economic Quarterly Volume 81/3 Summer 1995
19
20
Federal Reserve Bank of Richmond Economic Quarterly
wealth, and other borrower characteristics related to creditworthiness (Berkovec
et al. 1994). This finding suggests that there are race-related discrepancies between the true determinants of creditworthiness and the measures available to
econometricians.
Our objective is to develop a method for assessing the sensitivity of lending
discrimination estimates to measurement error. In particular, we study the classical errors-in-variables model, in which the components of a vector x of observed
measures of creditworthiness are, one for one, fallible measures of those in a
vector of true qualifications x∗ .1 The implications of errors in variables in the
standard linear regression model are well known (Klepper and Leamer 1984;
Goldberger 1984).2 We briefly review these implications in Section 1. Models
of lending discrimination generally specify a nonlinear regression model, such
as the logit model, because the dependent variable is dichotomous (y = 1 if the
loan application is denied; y = 0 if it is accepted). In this article we extend the
results for the linear case to cover the nonlinear logit regression model widely
used in lending discrimination studies.
Linear errors-in-variables models are underidentified because variation in
true qualifications cannot be distinguished from error variance. Assuming that
the errors are normally distributed with known parameters, however, the linear
model is just-identified, allowing estimation of model parameters depending
on the assumed error-variance parameters. Assuming zero error variance yields
the standard linear regression model as a special case. By estimating under a
range of error-variance assumptions, one can trace out the potential effect of
measurement error on model parameter estimates. Note that since the errorvariance assumptions make the model just-identified, no one assumption about
the error-variance parameters is more likely than any other; that is, estimates of
model parameters under alternative error-variance assumptions are all equally
consistent with the data. Also note that in the case of normally distributed regressors in the linear model, parameter estimates for alternative error-variance
1 The classical errors-in-variables model is not the only one in which observed variables,
taken together, are fallible measures of true creditworthiness. Alternatives include “multipleindicator” models in which observed variables are fallible measures of a single index of creditworthiness, and “omitted-variable” models in which some determinants of creditworthiness are
unobservable. All are alike in that a component of the true model is unobserved by the econometrician; thus, all are latent-variable models. Because errors in variables is one of the simplest
and most widely studied models of fallible regressors, it is a useful starting point in examining
fallibility in empirical models of lending discrimination.
2 Interest in the errors-in-variables problem has surged since 1970. As Hausman and colleagues (1995) stated, “During the formative period of econometrics in the 1930’s, considerable
attention was given to the errors-in-variable[s] problem. However, with the subsequent emphasis
on aggregate time series research, the errors-in-variables problem decreased in importance in most
econometric research. In the past decade as econometric research on micro data has increased
dramatically, the errors-in-variables problem has once again moved to the forefront of econometric
research” (p. 206).
J. L. DeVaro and J. M. Lacker: Errors in Variables
21
assumptions can be obtained through an algebraic correction to the ordinary
least squares estimates.
In Section 2 we examine the logit model under errors in variables and
show how estimators depend on assumptions about error variance. Adjusting
estimators for error variance is no longer an algebraic correction as it is in the
linear setup; the model must be reestimated for each error-variance assumption.
For the case in which the independent variables are continuous-valued, we show
how to estimate the logit model under various assumptions about error variance.
Because of the nonlinearity, the logit model is in some cases identified without
error-variance assumptions. In practice, however, the logit model is quite close
to underidentified, and little information can be obtained from the data about
error-variance parameters. Therefore, we advocate estimating models under a
range of error-variance assumptions to check the sensitivity of estimates to
measurement error.
In Section 3 we demonstrate our method using artificial data. We show how
estimates of a discrimination parameter can be biased when a relatively modest
amount of measurement error is present. The magnitude of the bias depends on
the model’s fundamental parameters. By estimating the model under different
assumptions about measurement error variance, we can gauge the sensitivity of
the estimators to errors in variables. Section 4 concludes and offers directions
for further research.
Bauer and Cromwell (1994) have also studied the properties of logit regression models of lending discrimination, focusing on the small-sample properties
of a misspecified model using simulated data. They found that tests for lending
discrimination were sensitive to sample size. Our work focuses on the effect
of errors in variables on the large-sample properties of otherwise correctly
specified logit models of lending discrimination.
1.
ERRORS IN VARIABLES
The implications of errors in variables are easiest to see in a linear setup such as
the following simple model of salary discrimination.3 Suppose that an earnings
variable (y) is determined according to the following equations:
y = βx∗ + αz + v,
(1a)
x∗ = x0 + µz + u,
(1b)
3 The exposition in this section is based on Goldberger (1984). This model of salary
discrimination has a close parallel in the permanent income theory. Friedman (1957) discusses
how racial differences in unobserved permanent income (the counterpart of qualifications in the
salary model and creditworthiness in the lending model) bias estimates of racial differences in
the consumption function intercept.
22
Federal Reserve Bank of Richmond Economic Quarterly
x = x∗ + e,
(1c)
where the scalar x∗ = true qualification, x = measured qualification, and z is
a race dummy (z = 1 for minorities, z = 0 for whites). We take v, u, and e to
be mutually independent random variables with zero means and variances σv2 ,
σu2 , and σe2 , all independent of z. The earnings variable in (1a) is a stochastic
function of the true qualifications and race. The parameter α represents the independent effect of race on salary, and α < 0 represents discrimination against
minorities. If better-qualified applicants obtain higher salaries, then β > 0. In
(1b) qualification is allowed to be correlated with race; the expectation of x∗
is x0 for whites and x0 + µ for minorities. The empirically relevant case has
µ < 0. Observed qualification in (1c) is contaminated by measurement error e.
Consider a regression of y on the observed variables x and z. This estimates
E[y | x, z] = bx + az.
Since the variances and covariances are the same for both white and minority applicants, we can use conditional covariances to calculate the regression
slopes. We focus on relationships in a population and thus ignore sampling
variability. The least squares estimators are
b = cov(x, y | z)/v(x | z) = cov(x∗ , y | z)/v(x | z) = (1 − δ)β
and
a = E[y | z = 1] − E[y | z = 0] − b{E[x | z = 1] − E[x | z = 0]}
= α + βµ − bµ
= α + δβµ,
where
δ ≡ σe2 /(σu2 + σe2 ).
When there is measurement error (σe2 > 0), the regression estimator of β is
biased toward zero. To see why, substitute for x∗ in (1a) using (1c) to obtain
y = βx + αz + (v − βe). The “error” v − βe in the regression of y on x and
z is correlated with x via (1c). Thus a key assumption of the classical linear
regression model is violated, and the coefficients are no longer unbiased.
In our case (β > 0, µ < 0), the estimator of α is biased downward as well.
Bias creeps in because z is informative about x∗ , given x;
E[x∗ | x, z] = (1 − δ)x + δ(x0 + µz).
Given observed qualification x, race can help “predict” true qualification x∗ .
Race can then help “explain” earnings, even in the absence of discrimination
(α = 0), because race is correlated with true qualifications.
The model (1) is underidentified (Kapteyn and Wansbeek 1983). A regression of x on z recovers the nuisance parameters x0 and µ, along with
J. L. DeVaro and J. M. Lacker: Errors in Variables
23
v(x | z) = σu2 + σe2 . Other population moments provide us with a and b, but
these are not sufficient to identify α, β, and δ. No sample can provide us with
enough information to divide v(x | z) between the variance in true qualifications
σu2 and the variance in measurement error σe2 . Under the assumptions β > 0
and µ < 0, any value of α > a, including the no-discrimination case α = 0, is
consistent with the data for some β and σe2 .
If σe2 were known independently, then we would know δ = σe2 /(σu2 + σe2 )
and could calculate the unbiased estimators α̂ and β̂ by correcting the ordinary
least squares estimators as follows:
β̂ = b/(1 − δ)
(2a)
α̂ = a − δbµ/(1 − δ).
(2b)
One could use (2) to study the implications of alternative assumptions about the
variance of measurement error; different values of σe2 would trace out different
estimates of α.
In (1) the direction of bias in a is known when the sign of βµ is known.
Matters are different when x is a vector of characteristics affecting qualifications. Consider a multivariate model:
y = β x∗ + αz + v,
(3a)
x∗ = x0 + µz + u,
(3b)
x = x∗ + e,
(3c)
∗
where x and x are now k × 1 random vectors and β, µ, and x0 are k × 1
parameter vectors. We take u and e to be normally distributed random vectors,
independent of v, z, and each other, with zero means and covariance matrices
Σ∗ and D. The classical assumption is that measurement errors are mutually
independent, so D is diagonal.
The least squares estimators are now
b = (Σ∗ + D)−1 Σ∗ β
(4a)
a = α + (β − b) µ.
(4b)
and
The direction of bias is now uncertain, even under the usual assumption that
measurement errors are independent (D is diagonal). To see why, suppose that
k = 2, Σ∗ has ρ as the off-diagonal element, and Σ∗ + D has ones on the
diagonal (a normalization of units). Then (4b) becomes
a = α + [(D11 β1 − ρD22 β2 )µ1 + (D22 β2 − ρD11 β1 )µ2 ]/(1 − ρ2 ).
The bias in a could be positive or negative, depending on parameter values.
For example, suppose only one component of x is subject to measurement
24
Federal Reserve Bank of Richmond Economic Quarterly
error, say, x1 (D11 > 0 and D22 = 0). By itself this would bias b1 downward,
resulting in an upward bias in a. But b2 = ρβ1 D11 /(1 − ρ2 ) + β2 is now biased
as well, and this would induce downward bias in a if ρβµ > 0. The overall
direction of bias is indeterminate (Rao 1973; Hashimoto and Kochin 1980).
But again, if the measurement error parameters D were known, then the least
squares estimators a and b could be corrected by a simple transformation of
(4) (using Σ∗ = Σ − D, where Σ = v(x | z)). Each alternative measurement
error assumption would imply a different estimator.4
2.
ERRORS IN VARIABLES IN A
LOGIT MODEL OF DISCRIMINATION
In model (3) the dependent variable is a linear function of the explanatory variables. In models of lending decisions the dependent variable is dichotomous:
y = 1 if the applicant is denied a loan, and y = 0 if the applicant is accepted.
In this case the linear formulation in (3) is unattractive (Maddala 1983). A
common alternative is the logit model, shown here without errors in variables:
Pr[y = 1 | x, z] = G(β x + αz),
(5a)
1
,
(5b)
1 + e−t
where x is a vector of characteristics influencing creditworthiness. The empirically relevant case has β < 0, so applicants who are more creditworthy are
less likely to be denied loans. A value of α > 0 would indicate discrimination
against minorities: a minority applicant is approximately α(1 − G) times more
likely than an identical white applicant to be denied a loan.5
The parameters α and β can be estimated by the method of maximum
likelihood. The log likelihood function for a sample of n observations
{yi , xi , zi , i = 1, . . . , n} is
G(t) =
log L =
n
i=1
log Pr(yi , xi , zi ) =
n
i=1
log Pr(yi | xi , zi ) +
n
log Pr(xi , zi ),
(6)
i=1
where
Pr(yi | xi , zi ) = G(β xi + αzi )yi [1 − G(β xi + αzi )](1−yi ) .
Estimators are found by choosing parameter values that maximize log L. The
likelihood depends on the parameters of the conditional distribution in (5) as
4 Klepper and Leamer (1984) and Klepper (1988b) show how to find bounds and other
diagnostics for the linear errors-in-variables model.
5 The elasticity of G with respect to z is αG /G = α(1 + e−t )e−t /(1 + e−t )2 = αe−t /(1 +
−t
e ) = α(1 − G), where G is evaluated at β x + αz.
J. L. DeVaro and J. M. Lacker: Errors in Variables
25
well as on the “nuisance parameters” governing the unconditional distribution
of (x, z). Since the nuisance parameters appear only in the second sum in (6),
while α and β appear only in the first sum, α and β can be estimated in this
case without estimating the nuisance parameters.
Under errors in variables, (5a) is replaced with
Pr[y = 1 | x∗ , z] = G(β x∗ + αz),
(7)
where x∗ is the vector of true characteristics. The resulting log likelihood function is
log L =
=
n
i=1
n
log Pr(yi , xi , zi )
(8)
Pr(yi |
log
xi∗ , zi )Pr(xi
|
xi∗ )Pr(xi∗ , zi )dxi∗ .
i=1
The likelihood function now depends on Pr(x | x∗ ), the probability that x is
observed if the vector of true characteristics is x∗ . Since x − x∗ is the vector
of measurement errors, Pr(x | x∗ ) is the probability distribution governing the
measurement error. In the linear model (3) the least squares estimators could be
corrected algebraically for measurement error of known variance. In the logit
model, however, there is no simple way to adjust maximum likelihood estimators for errors in variables, since the regression function is nonlinear. Instead,
we must estimate α and β for each distinct assumption about Pr(x | x∗ ).
Unlike the one in (6), the log likelihood function in (8) is not separable
in the nuisance parameters of the distribution Pr(x∗ , z). Even if we posit an
error distribution Pr(x | x∗ ), estimating α and β requires estimating the parameters of Pr(x∗ , z) as well. The estimation of these nuisance parameters will be
sidestepped here by maximizing the conditional likelihood function
log L̃ =
=
n
i=1
n
log Pr(yi | xi , zi )
(9)
log
Pr(yi |
xi∗ , zi )Pr(xi∗
|
xi , zi )dxi∗ .
i=1
We will assume that Pr(x∗ | x, z), the distribution of true characteristics conditional on observed characteristics and race, is known.
Our model is completed by adding specific assumptions about the distributions Pr(x | x∗ ) and Pr(x∗ | z), which will allow us to derive Pr(x∗ | x, z). We
will maintain the assumptions embodied in (3b) and (3c):
x∗ = x0 + µz + u,
(10a)
x = x∗ + e,
(10b)
26
Federal Reserve Bank of Richmond Economic Quarterly
where β, µ, and x0 are k × 1 parameter vectors and where u and e are normally
distributed random vectors, independent of v, z, and each other, with zero
means and covariance matrices Σ∗ and D. Given x and z, x∗ is then normally
distributed with mean vector m∗ and covariance matrix S∗ , where
m∗ = DΣ−1 µz + (I − DΣ−1 )x,
(11a)
S∗ = (I − DΣ−1 )D.
(11b)
With this result in hand, we find that, conditional on x and z, the argument of
G is normally distributed with mean β m∗ + αz and variance β S∗ β. Therefore,
the likelihood in (9) can be written as
Pr(y | x, z) = G(m + σs)(2π)−1/2 exp(−s2 /2)ds,
(12)
where
m = β (I − DΣ−1 )x + (α + β DΣ−1 µ)z,
σ = [β (I − DΣ−1 )Dβ]1/2 .
When D = 0, m collapses to β x + αz and σ = 0, which is the error-free
model.6
Because of the nonlinearity of G, the logit model can potentially be identified without error-variance assumptions, unlike the linear model in Section
1. Thus, in principle, the error-variance parameters could be estimated rather
than imposed. In practice, however, the model is so close to linear that the
error-variance parameters cannot be estimated; even large samples are uninformative about D. We therefore recommend estimating the model under a range
of alternative error-variance assumptions.
To summarize the procedure, first calculate least squares estimators for the
parameters x0 , µ, and Σ. These parameters are treated as fixed and combined
with an assumed D to obtain the distribution Pr(x∗ | x, z), which is used in (12)
and (9) to obtain maximum likelihood estimates of α and β. This procedure
treats the error variance D as known, just as the error-free model treats D
6 The joint normality of x and x∗ given z implies that given x and z, x∗ is normal with
parameters that can be derived algebraically from the parameters of Pr(x | x∗ ) and Pr(x∗ | z).
Other distributional assumptions on x and x∗ are far less convenient. For example, when x∗ takes
∗
on discrete values, a more general approach is required to derive Pr(x
| x, z).∗Given∗ a distribution
∗
of the observables Pr(x, z), recover Pr(x | z) using Pr(x | z) = Pr(x | x )Pr(x | z)dx∗ , and
then use Bayes’s rule to obtain Pr(x∗ | x, z) = Pr(x | x∗ )Pr(x∗ | z)/Pr(x | z). The first of these
steps involves inverting a very large matrix.
J. L. DeVaro and J. M. Lacker: Errors in Variables
27
as identically zero. Estimates of α can then be traced out under alternative
assumptions on D.7
Our procedure will misstate the uncertainty about parameter estimates,
even conditioning on D. By implicitly assuming that the estimated parameters
x0 , µ, and Σ are known, we are neglecting their sampling variability. These
parameters appear in (12) and thus influence estimates of α and β. Our procedure therefore misstates their sampling variability as well. When D = 0, the
nuisance parameters disappear from (12), and this problem does not arise.8
3.
EXAMPLES
In the examples in this section, we apply our procedure in a logit model of
discrimination to show how the technique is capable of detecting the sensitivity
of parameter estimates to errors in variables. We find it convenient to use artificially generated data sets to illustrate our results. Artificial data allow us to
isolate important features of the errors-in-variables model for a wide array of
cases. Observations are randomly generated under a given, true error variance,
and the model is then estimated under various hypothesized error variances.
In the simplest case there is only one explanatory variable besides race
(k = 1). We assume α = 0, β = −1, µ = −2, and Σ = 1. (We focus on
the no-discrimination case, α = 0, solely for convenience.) In this case, if a
is significantly different from zero, then it is also significantly greater than α,
and the usual t-statistic on a will also show whether a is significantly biased.
The sample was assumed to be half white (z = 0) and half minority (z = 1).
Using these values and an assumed true error variance D, we generated 10,000
random observations on x∗ , x, and y using equations (7) and (10). We then
estimated the model using maximum likelihood, assuming that the true values
of µ and Σ were known and making an assumption about D̃ (not necessarily
the same as D). The results are displayed in Table 1. The sample size of 10,000
was chosen to reduce sampling variance.
For the estimates shown in Panel A of Table 1, the true variance of the
measurement error is D = 0.1. This represents one-tenth of the total variance
in observed x, a relatively modest amount. The first line reports estimation
under the (incorrect) assumption that the error variance is zero. As expected,
the estimate b is biased toward zero. Consequently, a is biased upward, toward
showing discrimination, and is significant.
7 In related work, Klepper (1988a) extended the diagnostic results of Klepper and Leamer
(1984) and Klepper (1988b) to a linear regression model with dichotomous independent variables.
These earlier approaches attempted to characterize the set of parameters that maximize the likelihood function. Levine (1986) extended the results of Klepper and Leamer (1984) to the probit
model.
8 Specifically, the hessian of the log likelihood function is then block diagonal across (α, β)
and (x0 , µ, Σ).
28
Federal Reserve Bank of Richmond Economic Quarterly
Table 1 Coefficient Estimates for
Alternative Error-Variance Assumptions, k = 1
µ = −2, Σ = 1, n = 10, 000.
a
A.
True parameters α = 0, β = −1, and D = 0.1:
Assumed D̃
0.0
0.05
0.1
B.
b
0.1446
(2.4380)
0.0482
(0.7780)
−0.0607
(−0.9308)
−0.9208
(−32.3477)
−0.9775
(−31.8322)
−1.0418
(−31.2626)
True parameters α = 0.1, β = −0.9, and D = 0.0:
Assumed D̃
0.0
0.05
0.1
0.1609
(2.7101)
0.0640
(1.0315)
−0.0456
(−0.6986)
−0.9260
(−32.4378)
−0.9832
(−31.9159)
−1.0480
(−31.3393)
Notes: t-statistics are shown in parentheses beneath the coefficient estimates. For each panel, we
drew a set of 10,000 random realizations for (y, x): 5,000 with z = 0 and 5,000 with z = 1.
Within each panel, estimation was performed on the same data set with different assumptions
about the error variance D̃.
The last two lines in Panel A show estimates assuming positive error variance. For larger values of D̃, b is closer to one and a is closer to zero, the true
value. The discrimination parameter is not significantly different from zero
when estimated assuming D is 0.05 or 0.1. In this case, then, our procedure
successfully detects the sensitivity of parameter estimates to errors in variables.
In Panel B we examine the case in which no measurement error is present
and the true discrimination parameter is positive. The (correct) assumption of
no measurement error now yields estimates that are unbiased; they differ from
the true parameters only because of sampling error. Imposing the (incorrect)
assumption of positive measurement error variance “undoes” a nonexistant bias,
resulting in a near zero and a larger negative b.
Table 2 shows how the magnitude of the bias varies with the correlation
between components of x when k = 2. Σ has diagonal elements equal to
one and off-diagonal elements equal to a scalar ρ, where −1 < ρ < 1. D
has diagonal elements all equal to 0.1; the independent variables other than
race suffer from measurement error of the same variance. We maintain α = 0,
J. L. DeVaro and J. M. Lacker: Errors in Variables
29
Table 2 Coefficient Estimates for Alternative Correlation and
Error-Variance Assumptions, k = 2
α = 0, β =
−1
,
−1
µ=
−2
,
−2
Σ=
a
A. ρ = 0:
Assumed d̃
0.0
0.1
B. ρ = 0.5:
Assumed d̃
0.0
0.1
C. ρ = −0.5:
Assumed d̃
0.0
0.1
1 ρ
ρ 1
and
D=
0.1 0
,
0 0.1
b1
D̃ =
d̃ 0
,
0 d̃
n = 10, 000.
b2
0.4299
(4.2028)
0.0394
(0.3483)
−0.8340
(−25.2057)
−0.9557
(−24.3926)
−0.8663
(−25.7966)
−0.9924
(−24.9389)
0.2975
(3.4439)
0.0419
(0.4506)
−0.8797
(−22.8422)
−0.9726
(−20.2974)
−0.8705
(−22.5769)
−0.9597
(−20.0418)
0.7672
(5.5997)
−0.0457
(−0.2720)
−0.7816
(−21.8801)
−1.0084
(−21.4374)
−0.7714
(−21.5103)
−0.9969
(−21.1531)
Notes: t-statistics are shown in parentheses beneath the coefficient estimates. For each panel, we drew a
set of 10,000 random realizations for (y, x): 5,000 with z = 0 and 5,000 with z = 1. Within each panel,
estimation was performed on the same data set.
β = (−1, −1), and µ = (−2, −2). Panel A shows that when the components of
x are uncorrelated, the bias is larger than in the comparable k = 1 model: 0.43
versus 0.14. When the components of x are positively correlated (ρ = 0.5), the
bias is smaller by almost a third but is still significant. When the components of
x are negatively correlated (ρ = −0.5), the bias is substantially larger. Thus the
bias in a varies negatively with ρ, just as the linear case suggested. A positive
value of ρ implies that measurement error in x1 biases the coefficient on x2
away from zero, counteracting the effect of measurement error in x2 . Although
bi is biased toward zero by measurement error in xi , the bias is somewhat offset
by the effects of measurement error in other components of x.
When k = 1, the direction of bias is determined entirely by the sign of βµ.
When k > 1, the direction of bias depends on Σ and D, even when β µ can be
signed. Table 3 illustrates this fact for k = 2, showing a set of parameters for
which a is biased against finding discrimination. Both x1 and x2 are plagued
by measurement error, but with a strong positive correlation between the two,
30
Federal Reserve Bank of Richmond Economic Quarterly
Table 3 Coefficient Estimates for
Alternative Error-Variance Assumptions, k = 2
α = 0, β =
D=
−0.1
,
−1
0.1 0
,
0 0.1
µ=
D̃ =
−2
,
−0.1
0.1
1 0.75
0.75 1
and
d̃ 0
,
0 d̃
n = 10, 000.
a
Assumed d̃
0.0
Σ=
−0.2445
(−3.4442)
0.0312
(0.2888)
b1
−0.2352
(−7.0602)
−0.0887
(−1.6430)
b2
−0.7703
(−21.9616)
−0.9962
(−17.3444)
Notes: t-statistics are shown in parentheses beneath the coefficient estimates. For each panel, we
drew a set of 10,000 random realizations for (y, x): 5,000 with z = 0 and 5,000 with z = 1.
Within each panel, estimation was performed on the same data set.
each has a dampening effect on the bias in the coefficient of the other variable.
The net bias in b2 is toward zero, but b1 is biased away from zero. Since x1 is
more strongly correlated with z, the net effect is a negative bias in a. With the
correct error-variance assumption, the model detects the lack of discrimination.
In Table 4 we display results for a model with k = 10, a size that is more
like that of the data sets encountered in actual practice. With ρ = 0, we see in
Panel A that with more correlates plagued by measurement error, the bias in
a is larger. With ρ = 0.5, the various measurement errors partially offset each
other, but a remains significantly biased. Once again, our technique faithfully
compensates for known measurement error.
4.
SUMMARY
We have described a method for estimating logit models of discrimination under
a range of assumptions about the magnitude of errors in variables. Using artificially generated data, we showed how the bias in the discrimination coefficient
varies with measurement error and other basic model parameters. Our method
successfully corrects for known measurement error, and can gauge the sensitivity of parameter estimates to errors in variables. Our method can be applied
to the studies of lending discrimination cited in the introduction. It can also
be applied to the empirical models employed in lending discrimination suits
and regulatory examinations. Since the stakes are high in such applications, the
models ought to be routinely tested for sensitivity to errors in variables.
Further extensions of our method would be worthwhile. Although we allow
for errors only in continuous-valued independent variables, studies of lending
J. L. DeVaro and J. M. Lacker: Errors in Variables
31
Table 4 Race Coefficient Estimates for Alternative Correlation and
Error-Variance Assumptions, k = 10
α = 0, β is a k × 1 vector of −1s, µ is a k × 1 vector of 1s, Σ is a k × k matrix
with 1s on the diagonal and off-diagonal elements equal to ρ, D is a k × k matrix with
0.1s on the diagonal and off-diagonal elements equal to 0, D̃ is a k × k matrix with
elements d̃ on the diagonal and off-diagonal elements equal to 0, and n = 10, 000.
a
A.
True parameter ρ = 0:
Assumed d̃
0.0
0.1
B.
True parameter ρ = 0.5:
Assumed d̃
0.0
0.1
1.0033
(3.3154)
−0.0339
(−0.1006)
0.2266
(3.4658)
0.0645
(0.5988)
Notes: t-statistics are shown in parentheses beneath the coefficient estimate. For each panel, we
drew a set of 10,000 random realizations for (y, x): 5,000 with z = 0 and 5,000 with z = 1. Within
each panel, estimation was performed on the same data set.
discrimination often include discrete variables that are likely to be fallible as
well. It would be worthwhile to allow for errors in the discrete variables, as
Klepper (1988a) does for the linear regression model. In addition, it would be
useful to allow for uncertainty about the nuisance distributional parameters that
our method treats as known.
REFERENCES
Bauer, Paul W., and Brian A. Cromwell. “A Monte Carlo Examination of
Bias Tests in Mortgage Lending,” Federal Reserve Bank of Cleveland
Economic Review, vol. 30 (July/August/September 1994), pp. 27–44.
Berkovec, James, Glenn Canner, Stuart Gabriel, and Timothy Hannan. “Race,
Redlining, and Residential Mortgage Loan Performance,” Journal of Real
Estate Finance and Economics, vol. 9 (November 1994), pp. 263–94.
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Federal Reserve Bank of Richmond Economic Quarterly
Cummins, Claudia. “Fed Using New Statistical Tool to Detect Bias,” American
Banker, June 8, 1994.
Friedman, Milton. A Theory of the Consumption Function. Princeton, N.J.:
Princeton University Press, 1957.
Goldberger, Arthur S. “Reverse Regression and Salary Discrimination,”
Journal of Human Resources, vol. 19 (Summer 1984), pp. 293–318.
Hashimoto, Masanori, and Levis Kochin. “A Bias in the Statistical Estimation
of the Effects of Discrimination,” Economic Inquiry, vol. 18 (July 1980),
pp. 478–86.
Hausman, J. A., W. K. Newey, and J. L. Powell. “Nonlinear Errors in
Variables: Estimation of Some Engel Curves,” Journal of Econometrics,
vol. 65 (January 1995), pp. 205–33.
Kapteyn, Arie, and Tom Wansbeek. “Identification in the Linear Errors in
Variables Model,” Econometrica, vol. 51 (November 1983), pp. 1847–49.
Klepper, Steven. “Bounding the Effects of Measurement Error in Regressions
Involving Dichotomous Variables,” Journal of Econometrics, vol. 37
(March 1988a), pp. 343–59.
. “Regressor Diagnostics for the Classical Errors-in-Variables
Model,” Journal of Econometrics, vol. 37 (February 1988b), pp. 225–50.
, and Edward E. Leamer. “Consistent Sets of Estimates for
Regressions with Errors in All Variables,” Econometrica, vol. 52 (January
1984), pp. 163–83.
Levine, David K. “Reverse Regressions for Latent-Variable Models,” Journal
of Econometrics, vol. 32 (July 1986), pp. 291–92.
Maddala, G. S. Limited-Dependent and Qualitative Variables in Econometrics.
Cambridge: Cambridge University Press, 1983.
Munnell, Alicia H., Lynn E. Browne, James McEneaney, and Geoffrey M.
B. Tootell. “Mortgage Lending in Boston: Interpreting the HMDA Data,”
Working Paper Series No. 92. Boston: Federal Reserve Bank of Boston,
1992.
Rao, Potluri. “Some Notes on the Errors-in-Variables Model,” American
Statistician, vol. 27 (December 1973), pp. 217–28.
Schill, Michael H., and Susan M. Wachter. “Borrower and Neighborhood
Racial and Income Characteristics and Financial Institution Mortgage
Application Screening,” Journal of Real Estate Finance and Economics,
vol. 9 (November 1994), pp. 223–39.
Seiberg, Jaret. “When Justice Department Fights Bias by the Numbers, They’re
His Numbers,” American Banker, September 14, 1994.
Some Key Empirical
Determinants of
Short-Term Nominal
Interest Rates
Yash P. Mehra
M
any previous studies cannot account for the high level of interest
rates in the early 1980s. For example, Clarida and Friedman (1983,
1984) demonstrate that relative to the predictions of either a structural or an astructural model, interest rates in the early 1980s were too high.
This article suggests that the customary empirical measures that gauge the
short-run impact of monetary policy on interest rates have become increasingly
noisy in the 1980s and that this factor may have been partly responsible for
deterioration in the predictive ability of these interest rate equations.
In most previous studies, the short-run impact of monetary policy on rates
has been captured by detrended measures of the real money supply defined
either as real M1 or real M2. Such empirical measures of the money supply
do not provide a consistent basis of comparison over time when deposit rate
ceilings are removed and new liquid financial claims are introduced. These
and other financial developments of the 1980s also have altered the underlying
relationships between the public’s demand for these assets and their traditional
economic determinants, including the nominal rate (Hetzel and Mehra 1989;
Feinman and Porter 1992). As a result, many of the published reduced forms
for nominal rates that gauge the short-run impact of monetary policy by the
The views expressed are those of the author and do not necessarily represent those of the
Federal Reserve Bank of Richmond or the Federal Reserve System.
Federal Reserve Bank of Richmond Economic Quarterly Volume 81/3 Summer 1995
33
34
Federal Reserve Bank of Richmond Economic Quarterly
real money supply perform poorly in predicting the actual behavior of rates in
the 1980s.1
In this article, I present an interest rate equation in which the short-run impact of monetary policy on the real component of rates is captured by changes
in the real federal funds rate rather than in the real money supply.2 In addition,
I distinguish between the short- and long-run empirical determinants of rates
using cointegration and error-correction methodology. In previous short-rate
studies, stationarity properties of the data have largely been ignored, thereby
muddling the important distinction between the short- and long-run determinants of rates.3
The empirical work presented here focuses on the behavior of one-year
Treasury bill rates and finds that inflation is the main long-run economic determinant of the level of the nominal rate. Conversely, several customary empirical
measures of fiscal policy and a measure capturing foreign capital inflows are
not significant when included in the long-run (cointegrating) interest-inflation
regression. Also, in the short run, changes in the nominal rate depend largely
upon changes in inflation, real output, and the real federal funds rate. Changes
in fiscal policy measures and foreign capital inflows do not affect the nominal
rate even in the short run.
The short-run interest rate equation estimated here does not exhibit any
simultaneous equation bias. The real funds rate is therefore exogenous in the
1 Other
studies that have addressed this issue are those of Peek and Wilcox (1987) and Hendershott and Peek (1992). Peek and Wilcox continue to employ the real money supply measure
and attribute the high level of rates in the early 1980s to a less accommodative monetary policy.
They, however, use dummy variables to capture differences in the average tightness of policy
during different Fed chairman regimes. Hendershott and Peek, on the other hand, abandon the
money supply measure and use instead innovations in the slope of the term structure (the ratio of
the 6- to 60-month Treasury rates) to measure the short-run impact of monetary policy on the real
component of short-term rates. Given this proxy, monetary policy is highly relevant in explaining
the behavior of rates in the early 1980s.
2 Unlike the slope of the term structure used in Hendershott and Peek (1992), the nominal
federal funds rate has been the main instrument of monetary policy during most of the sample
period studied here. I use the real funds rate because in the short run the Fed influences the
nominal rate mainly by affecting its real component. Moreover, the real funds rate is correlated
with the nominal funds rate in the short run. The simple correlation between these two variables
is 0.46. Recently, Goodfriend (1993) has used the federal funds rate to measure the short-run
impact of monetary policy on the real component of the long rate.
3 This distinction is important in describing the effect of monetary policy on the nominal rate.
Many analysts believe that monetary policy can control the level of the nominal rate in the long
run only through its control over inflation, even though in the short run it could have substantial
effects. This distinction between short- and long-run effects can be made more precise using cointegration and error-correction methodology. Thus, the statement that monetary policy determines
the level of the nominal rate in the long run through its control over inflation can be interpreted to
mean that measures of monetary policy are not at the source of long-run, stochastic movements
in the level of the nominal rate, but inflation is. That is, the nominal rate is cointegrated with
inflation, but not with measures of monetary policy. However, short-run stationary movements in
the nominal rate could still be correlated with measures of monetary policy, indicating that in the
short run, monetary policy also influences the nominal rate.
Y. P. Mehra: Short-Term Nominal Interest Rates
35
short-run equation, which means that the real funds rate is not correlated with
contemporaneous shocks to the short rate. Thus, the Federal Reserve does influence the market rate in the short run. In addition, the nominal rate equation
that captures the impact of monetary policy by the real funds rate explains
reasonably well the nominal rate in the 1980s, although it does not completely
solve the puzzle of high rates during the early 1980s. The equation, for example, significantly underpredicts the level of the short rate in 1981. Finally, the
results with other measures of short- and medium-term nominal rates indicate
that in the short run, nominal yields on the short end of the U.S. Treasury term
structure are determined largely by movements in inflation, the real funds rate,
and real GDP. The short-run influences of these economic factors, however,
decline with the term to maturity, suggesting that the yield curve at the short
end is affected most by the outlook for Fed policy and the state of the economy.
The plan of this article is as follows. Section 1 briefly describes the model
and the method used in estimating the nominal rate equation. Section 2 presents
empirical results, and Section 3 contains concluding observations.
1.
THE MODEL AND THE METHOD
An Economic Specification of the Interest Rate Equation
The nominal interest rate equation that underlies the empirical work presented
here is based on a variant of a loanable funds model employed by Sargent
(1969), among others. According to that model, the nominal rate depends upon
anticipated inflation, changes in the real money supply and income, the deficit,
and the level of income.4 The version examined here measures the impact of
monetary policy on the nominal rate by changes in the real federal funds rate
rather than in the real money supply. In addition, I examine the role of other fiscal policy measures such as government purchases, net taxes, and foreign capital
inflows in determining the nominal rate. Since this model has already been described in an earlier paper (Mehra 1994), I report below just the relevant econometric specifications that are used to investigate the behavior of the short rate.
The short-run interest rate equation is estimated using cointegration and
error-correction modeling. If the nominal rate and empirical measures of its
potential economic determinants are nonstationary, then tests for cointegration
provide inferences about the existence of a long-run, equilibrium relationship
between the nominal rate and its potential determinants. The error-correction
equation then explains short-run changes in the nominal rate.
4 The
nominal rate responds positively to anticipated inflation, the deficit, and real growth
and responds negatively to increases in the real money supply. A rise in the level of real income,
however, generates a larger volume of savings and hence depresses the equilibrium real rate
(Sargent 1969).
36
Federal Reserve Bank of Richmond Economic Quarterly
The nominal rate equation estimated here has two parts: a long-run part
and a short-run part. The long-run part that specifies the potential long-run
determinants of the level of the nominal rate is given in (1):
Rt = a0 + a1 ṗet + a2 RFRt + a3 FPt − a4 ln ryt + a5 ∆ ln ryt + Ut ,
(1)
where R is the nominal interest rate, ṗe is anticipated inflation, RFR is the
real federal funds rate, FP is a fiscal policy variable, ln ry is the logarithm of
real income, and U is the disturbance term. Equation (1) describes the longrun response of the nominal rate to anticipated inflation, the real funds rate,
a fiscal policy variable, changes in real income, and the level of real income.
The coefficients ai , i = 1 to 5, measure the long-run responses in the sense
that they are the sums of coefficients that appear on current and past values
of the relevant economic determinants. The term a1 ṗet in (1) captures the inflation premium in the nominal rate, whereas the remaining terms capture the
influence of other variables on the equilibrium real component of the short
rate. If the nominal rate and anticipated inflation variables are nonstationary
but cointegrated as in Engle and Granger (1987), then the other remaining
long-run impact coefficients in (1) may all be zero.
Equation (1) may not do well in explaining short-run movements in the
nominal rate for a number of reasons. First, it ignores the short-run effects of
economic factors. Some economic factors, including those measuring monetary
policy actions, may be important in explaining short-run changes in the nominal rate, even though they may have no long-run effects. Second, it completely
ignores short-run dynamics. Hence, in order to explain short-run changes in
the nominal rate, consider the following error-correction model of the nominal
rate:
∆Rt = c0 + c1 ∆ṗet + c2 ∆RFRt + c3 ∆FPt + c4 ∆ ln ryt
+c5 ∆2 ln ryt +
n
c6s ∆Rt−s + c7 Ut−1 + 1t ,
(2)
s=1
where Ut−1 is the lagged residual from the long-run nominal equation (1), ∆2
is the second-difference operator, and other variables are as defined earlier.
Equation (2) is the short-run interest rate equation, and the coefficients ci ,
i = 1 to 5, capture the short-run responses of the interest rate to economic
determinants suggested here.
Estimation Issues: OLS Works If Empirical Measures of Economic
Determinants Are Nonstationary or Exogenous
Both the long- and short-run equations (1) and (2) contain contemporaneous
values of economic fundamentals. Those values are likely to be correlated
with contemporary shocks to the nominal rate. In particular, if the Federal
Reserve contemporaneously adjusts its short-run funds objective with respect
Y. P. Mehra: Short-Term Nominal Interest Rates
37
to movements in short rates, then the real federal funds rate is likely to be
correlated with the disturbance term in the regression.5 Hence, these equations
cannot be consistently estimated by ordinary least squares unless some special
assumptions hold.
The long-run equation (1) can be consistently estimated by ordinary least
squares if empirical measures of the economic factors included in (1) are nonstationary but cointegrated as in Engle and Granger (1987). Tests of hypotheses
on coefficients that appear in (1) can then be carried out by estimating Stock
and Watson’s (1993) dynamic OLS regressions. I, therefore, test first for nonstationarity and cointegration. The empirical work here examines the stationarity
properties of the data using unit root and mean stationarity tests. The test for
cointegration used is the one proposed in Johansen and Juselius (1990).6
The economic variables that appear in the short-run equation (2) are stationary. This equation can be consistently estimated by ordinary least squares
if contemporaneous right-hand side explanatory variables are uncorrelated with
the disturbance term. That condition can be tested by performing the test for
exogeneity given in Hausman (1978). To implement the test, consider the
following VAR representation of these contemporaneous right-hand side explanatory variables. (For simplicity, I am ignoring fiscal policy and some other
variables.)
∆ṗet = d0 +
k
d1s ∆ṗet−s +
k
s=1
+
k
d3s ∆ ln ryt−s +
s=1
∆ ln ryt = e0 +
k
d4s ∆Rt−s + 2t
(3)
s=1
k
s=1
+
d2s ∆RFRt−s
s=1
k
s=1
e1s ∆ṗet−s +
k
e2s ∆RFRt−s
s=1
e3s ∆ ln ryt−s +
k
e4s ∆Rt−s + 3t
(4)
s=1
5 To illustrate, consider a scenario in which the incoming new data indicates that in the
current quarter real growth or inflation is going to be higher than what the market expected based
on the past information. If the market believes such information, short rates could rise because
accelerations in real growth or inflation are generally associated with higher rates. If the Fed
also reacts contemporaneously to such new information and the resulting rise in short rates, then
changes in the funds rate would be correlated with the disturbance term. Such correlation will be
absent, however, if the Fed does not react or reacts with a lag.
6 These tests are described in detail in Mehra (1994).
38
Federal Reserve Bank of Richmond Economic Quarterly
∆RFRt = f0 +
k
f1s ∆ṗet−s
+
k
s=1
+
k
f3s ∆ ln ryt−s +
s=1
f2s ∆RFRt−s
s=1
k
f4s ∆Rt−s + 4t
(5)
s=1
This VAR includes only past values of the economic factors that appear in the
economic model used here and hence can be consistently estimated by ordinary
least squares. Then consider an expanded version of equation (2) given below:
∆Rt = d0 + d1 ∆ṗet + d2 ∆RFRt + d3 ∆ ln ryt +
n
d4 ∆Rt−s
s=1
+d5 Ut−1 + d6 ˆ2t + d7 ˆ3t + d8 ˆ4t + 1t ,
(6)
where ˆ2 , ˆ3 , and ˆ4 are residuals from the VAR. If d6 = d7 = d8 = 0, then
∆ṗet , ∆RFRt , and ∆ ln ry are uncorrelated with the disturbance term and hence
exogenous in this equation. The hypothesis d6 = d7 = d8 = 0 can be tested
using the F-test.7
Data and Definition of Variables
The empirical work uses quarterly data from 1955:1 to 1994:3. The shortterm nominal rate, R1, is the nominal yield on one-year U.S. Treasury bills. I
consider two proxies for anticipated inflation. The first one uses actual inflation
as measured by the behavior of the consumer price index ( ṗ). The second one
uses one-year-ahead inflation rates from the Livingston survey ( ṗe ). The real
federal funds rate, RFR, is the nominal federal funds rate minus the actual,
annualized quarterly inflation rate. Nominal interest rate data are observations
from the last month of the quarter, and inflation ( ṗ) is calculated as the change
in the log form from the last month of the previous quarter price level to that
of the current. In some specifications in which the real money supply is used
to measure the impact of monetary policy actions on the real component of
the nominal rate, the measure of money used is M2 scaled by the real GDP
deflator. Real income, ry, is real GDP. The real deficit scaled by real GDP
(DEF/y),8 real government purchases ( fg), and real government tax (net of
transfers) receipts (tx) are alternatively used to measure the impact of fiscal
policy on the real component of the short rate. I also consider the impact of
7 The hypothesis that the right-hand side contemporary regressors in (2) are independent
of the disturbance term also can be tested by comparing ordinary least squares and instrumental
variables estimates of the equation. Under the null hypothesis that there is no simultaneity equation
bias, OLS estimates (β̂OLS ) should not be statistically different from IV estimates (β̂IV ). Hausman
(1978) shows that the statistic that tests the null hypothesis β̂OLS = β̂IV is distributed Chi-squared
with a degree of freedom parameter equal to the number of parameters estimated in the equation.
See Maddala (1988) for a simple description of these test procedures.
8 This specification reflects the assumption that in a growing economy higher deficits result
in higher rates only if the deficit rises relative to GDP.
Y. P. Mehra: Short-Term Nominal Interest Rates
39
foreign capital inflows measured as the ratio of U.S. Treasury securities held by
foreigners to the total of U.S. securities held by domestic and foreign residents
( fh).9
2.
ESTIMATION RESULTS
On the Long-Run Determinants of the Nominal Rate
I first present test results that help determine which economic determinants
suggested in the long-run equation (1) are relevant.
Table 1 presents test results for determining whether empirical measures of
potential determinants such as R1, ṗ, ṗe , DEF/y, ln fg, ln tx, fh, ln rM2, ln ry, and
RFR have a unit root or are mean stationary. As can be seen, the t-statistic
(tp̂ ) that tests the null hypothesis that a particular variable has a unit root
is small for all these series. On the other hand, the test statistic (n̂u ) that
tests the null hypothesis that a particular variable is mean stationary is large
for all these variables with the exception of RFR. These results indicate that
R1, ṗ, ṗe , DEF/y, ln fg, fh, ln tx, ln rM2t , and ln ry have a unit root and thus are
nonstationary in levels.10 The results are inconclusive for the real funds rate
RFR. Together, these results indicate that most empirical measures of the potential determinants suggested here are nonstationary and hence could be the
source of long-run stochastic movements in the nominal rate.
Table 2 presents test statistics for determining whether the short-term
nominal rate (R1) is cointegrated with any of these nonstationary measures
of inflation, fiscal and monetary policies, and foreign capital inflows. Trace
and maximum eigenvalue statistics, which test the null hypothesis that there
is no cointegrating vector, are large for systems (R1, ṗ), (R1, ṗe ), (R1, ln ry),
(R1, DEF/y), and (R1, ln rM2), but are very small for systems (R1, RFR),
(R1, ln fg), (R1, ln tx), and (R1, fh). These results indicate that the nominal rate
is cointegrated with inflation (actual or anticipated), the real money supply,
the level of income, and the deficit, but not with the real federal funds rate,
government purchases, net taxes, and foreign capital inflows. The evidence
continues to favor the presence of at least one cointegrating vector even in
expanded systems that include inflation and fiscal and monetary policy variables
9 The data on the Livingston survey are provided by the Philadelphia Fed. The data used
in measuring capital inflows are from the Federal Reserve Board’s flow of funds data. All other
data series are from the Citibank database.
10 The t-statistic that tests the null hypothesis that first differences of a series have a unit
root takes values −5.2, −4.9, −6.4, −15.9, −4.2, −6.2, −5.2, −5.6, −4.7, and −7.4 for
∆R1, ∆ṗ, ∆ṗe , ∆RFR, ∆ ln rM2, ∆DEF/y, ∆ ln fg, ∆ ln tx, ∆fh, and ∆ ln ry, respectively.
These t-values are large, indicating that first differences of these series are stationary. The 5
percent critical value taken from Fuller (1976) is −2.9.
40
Federal Reserve Bank of Richmond Economic Quarterly
Table 1 Tests for Unit Roots and Mean Stationarity
ρρ̂
Panel A
Tests for Unit Tools
tρρ̂
k
Panel B
Tests for
Mean Stationarity
n̂u
0.94
0.84
0.98
0.85
0.99
0.97
0.92
0.98
0.96
0.98
−1.93
−2.83
−1.80
−2.51
−1.87
−1.80
−2.54
−1.63
−1.54
−1.58
2
7
2
2
1
1
1
3
1
5
0.84∗
0.49∗
0.98∗
0.37
1.80∗
0.33
1.42∗
0.91∗
1.54∗
1.19∗
Series X
R1
ṗ
ṗe
RFR
ln rM2
ln ry
DEF/y
ln fg
ln tx
fh
∗
Significant at the 5 percent level.
Notes: R1 is the one-year Treasury bill rate; ṗ is the annualized quarterly inflation rate measured
by the consumer price index; ṗe is the Livingston survey measure of one-year-ahead expected
inflation; RFR is the real federal funds rate; rM2 is the real money supply; ry is real GDP;
DEF/y is the ratio of federal government deficits to nominal GDP; fg is real federal government
purchases; tx is real federal government tax (net of transfers) receipts; and fh is the ratio of U.S.
Treasury securities held by foreigners to total of U.S. Treasury securities held by domestic and
foreign residents. The sample period studied is 1955:1 to 1994:3. The values for ρ and t-statistics
(tρ̂ ) for ρ = 1 in Panel A above are from the Augmented Dickey-Fuller regressions of the form
Xt = a0 + ρXt−1 +
k
as ∆Xt−s ,
(a)
s=1
where X is the pertinent series. The number of lagged first differences (k) included in these
regressions are chosen using the procedure given in Hall (1990). The procedure starts with some
upper bound on k, say k max, chosen a priori (eight quarters here). Estimate (a) above with k set
at k max. If the last included lag is significant, select k = k max. If not, reduce the order of the
autoregression by one until the coefficient on the last included lag is significant. The test statistic
n̂u in Panel B above is the statistic that tests the null hypothesis that the pertinent series is mean
stationary. The 5 percent critical value for n̂u given in Kwiatkowski et al. (1992) is 0.463.
[see systems (R1, ṗ, DEF/y, ln rM2, ln ry), (R1, ṗe , DEF/y, ln rM2, ln ry), (R1,
ṗ, DEF/y, ln ry), and (R1, ṗe , DEF/y, ln ry)].
Panels A and B in Table 3 help determine which variables included in the
cointegrating regression11 are statistically significant. It presents the dynamic
OLS estimates of the potential cointegrating regressions with and without the
real money supply. As can be seen, inflation (actual or anticipated) is the
only variable that enters significantly in these cointegrating regressions. Other
11 In this article, I focus on a single cointegrating regression that is normalized on the shortterm nominal rate. The analysis here thus ignores the possibility that in larger systems there may
be multiple cointegrating vectors.
Y. P. Mehra: Short-Term Nominal Interest Rates
41
Table 2 Cointegration Test Results
System
(R1,
(R1,
(R1,
(R1,
(R1,
(R1,
(R1,
(R1,
(R1,
(R1,
(R1,
(R1,
(R1,
ṗ)
ṗe )
RFR)
ln rM2)
ln fg)
ln tx)
DEF/y)
fh)
ln ry)
ṗ, DEF/y, ln ry)
ṗe , DEF/y, ln ry)
ṗ, DEF/y, ln rM2, ln ry)
ṗe , DEF/y, ln rM2, ln ry)
Trace Test
Maximum Eigenvalue
Test
k
22.2∗
22.1∗
17.6
20.3∗
10.1
12.6
30.7∗
14.7
48.6∗
97.1∗
86.1∗
148.8∗
146.4∗
17.9∗
18.9∗
14.3
16.9∗
5.3
8.5
27.5∗
10.7
43.4∗
42.9∗
34.9∗
65.2∗
64.9∗
8
2
8
8
2
4
8
8
2
4
4
2
2
∗
Significant at the 5 percent level.
Notes: Trace and maximum eigenvalue tests are tests of the null hypothesis that there is no
cointegrating vector in the system. The lag length in the relevant VAR system is k and is chosen
using the likelihood ratio test given in Sims (1980). In particular, the VAR model initially was
estimated with k set equal to a maximum number of eight quarters. This unrestricted model was
then tested against a restricted model, where k is reduced by one, using the likelihood ratio test.
The lag length finally selected is the one that results in the rejection of the restricted model.
nonstationary variables such as the real deficit and the real money supply are
not significant. Real GDP is significant in some regressions and not in others
(see Panels A and B, Table 3). These results thus indicate that inflation is the
main long-run economic determinant of the short-term nominal rate.
Panel C in Table 3 presents the cointegrating regressions that include only
the inflation variable. The cointegrating regression is estimated with and without the restriction that the nominal rate adjusts one for one with inflation in
the long run. The χ2 statistic that tests the validity of the full Fisher-effect
restriction is not large, indicating that this restriction is consistent with data.
These results also indicate that the real rate of interest on one-year Treasury
bills is mean stationary. The estimate of this mean falls in a 2.3 to 3.2 percent
range (see the constant term in Panel C regressions).
On the Short-Run Determinants of the Nominal Rate
The short-run equation (2) is estimated here jointly with its long-run part and
hence includes levels as well as first differences of the relevant economic
42
Federal Reserve Bank of Richmond Economic Quarterly
Table 3 Cointegrating Regressions; Dynamic OLS
(Leads, Lags)
Panel A: With Real Money Supply
(−4, 4)
R1t = 0.7ṗt − 17.6 ln rM2t + 2.05 ln ryt + 0.22(DEF/y)t
(3.9)
(0.9)
(1.0)
(0.5)
(−4, 4)
R1t = 1.1ṗet + 6.1 ln rM2t + 6.8 ln ryt + 0.02(DEF/y)t
(5.9) (0.4)
(0.4)
(0.1)
Panel B: Without Real Money Supply
(−4, 4)
R1t = 0.9ṗt − 1.0 ln ryt + 0.18(DEF/y)
(21.8) (2.2)
(1.5)
(−4, 4)
R1t = 1.1ṗet − 1.5 ln ryt − 0.10(DEF/y)t
(23.1) (12.9)
(0.7)
Panel C: With Inflation Only
(−4, 4)
R1 = 3.2 + 0.8ṗt ; R1 = 2.3 + 1.0ṗt ; χ2 (1) = 2.8(0.10)
(−4, 4)
R1 = 2.5 + 1.1ṗet ; R1 = 2.6 + 1.0ṗet ; χ2 (1) = 0.32(0.57)
Notes: All regressions are estimated by the dynamic OLS procedure given in Stock and Watson
(1993), using leads and lags of first differences of the relevant right-hand side explanatory variables. Parentheses contain t-values corrected for the presence of moving average serial correlation.
χ2 (1) is the χ2 statistic with one degree of freedom (significance levels in parentheses); it tests
the hypothesis that the coefficient on ṗ or ṗe is unity.
determinants.12 A preliminary specification search indicated that in the short
run, changes in the nominal rate depend largely upon contemporaneous changes
in inflation, the real funds rate, and real GDP. The lagged level of the funds rate
is also significant. The empirical measures of fiscal policy and foreign capital
inflows, however, did not enter the short-run equation. (I formally test these
restrictions later on.)
Table 4 presents ordinary least squares as well as instrumental variables
estimates of the pertinent short-run equation. The instruments chosen are basically the lagged values of the right-hand side explanatory variables that appear
12 The short-run equation (2) of the text includes a one-period lagged value of the residual
from the long-run equilibrium equation. In joint estimation, the lagged residual is replaced by
lagged levels of the variables that enter the long-run equilibrium equation. To see it, assume for
the sake of explanation that the long-run cointegrating regression is given in (b) below:
R1t = a0 + a1 ṗet + Ut .
(b)
If we solve (b) for Ut−1 and substitute for Ut−1 into (2), then the short-run equation (2) will
include the lagged level of the nominal rate (R1) and the inflation rate ( ṗe ).
Y. P. Mehra: Short-Term Nominal Interest Rates
43
Table 4 Short-Run Nominal Interest Rate Equations, 1957:1 to 1994:3
Explanatory
Variables
constant
∆ṗt
∆ṗet
∆RFRt
∆ ln ryt
R1t−1
ṗt−1
ṗet−1
RFRt−1
∆R1t−1
∆R1t−2
∆R1t−3
∆R1t−4
SER
Q(36)
Q(8)
Q(4)
Sargan’s χ2
F
Instrumental Variables
(A.1)
(A.2)
−0.22(1.4)
0.71(8.2)
0.61(6.5)
0.12(2.6)
−0.37(3.4)
0.37(3.4)
0.34(2.7)
−0.23(6.2)
0.07(1.2)
−0.04(0.5)
0.15(2.9)
1.0 (2.9)
0.9 (3.2)
0.11(1.2)
−0.02(0.3)
−0.55(4.9)
0.55(4.9)
0.28(3.3)
0.547
33.8
6.9
2.8
0.98
27.7
5.9
3.9
12.5(0.19)
1.4(0.24)
6.5(0.36)
1.0(0.37)
Ordinary Least Squares
(B.1)
(B.2)
0.06( 0.8)
0.72(12.8)
0.64(11.3)
0.06( 3.9)
−0.33( 4.1)
0.33( 4.1)
0.30(
−0.21(
0.04(
−0.06(
0.14(
3.3)
5.0)
0.9)
0.9)
2.9)
0.52
37.1
9.2
2.2
0.6 (3.8)
0.8 (3.3)
0.17(3.5)
0.06(2.1)
−0.53(6.1)
0.53(6.1)
0.30(4.7)
0.94
38.9
11.9
8.5
∗
Significant at the 5 percent level.
Notes: See notes in Table 1 for definition of variables. Parentheses following coefficients contain
t-values. SER is the standard error of estimate, and Q(36), Q(8), and Q(4) are the Ljung-Box
Q-statistics based on 36, 8, and 4 autocorrelations of the residuals.
Sargan’s χ2 tests the independence of the instruments and the disturbance term (parentheses
contain the significance level of the test). F is the F-statistic that tests the null hypothesis that the
residuals from the reduced-form regressions of real growth, inflation, and the real federal funds
rate are not jointly significant when included in the interest rate equation (parentheses contain the
significance level of the test). See footnote 13 of the text for a description of instruments used.
in (2).13,14 The null hypothesis that contemporary values of changes in inflation, the real funds rate, and real growth are jointly exogenous with respect to
13 When
actual inflation is the proxy for anticipated inflation, the instruments used for estimating the short-run equation are a constant, one-period lagged value of the short rate, inflation,
and the real funds rate, two-period lagged values of the change in the short rate, and four-period
lagged values of changes in inflation, real income, and the real funds rate. When the Livingston
survey is the proxy for anticipated inflation, I use similar instruments, but I treat the Livingston
survey as exogenous in the short equation. Hence, first differences and one-period lagged values
of the Livingston survey (∆ṗet , ṗet−1 ) are used as instruments.
14 Two considerations are important in the choice of instruments. First, the instruments chosen should be uncorrelated with the disturbance term. Second, they should be highly correlated
44
Federal Reserve Bank of Richmond Economic Quarterly
short-run impact coefficients is not rejected (the relevant Hausman F-statistics
reported in Table 4 are small).15 Sargan’s (1964) χ2 specification test statistic
presented in Table 4 is also small, indicating that the instruments chosen are
independent of the disturbance term. A casual look at the estimates reported in
Table 4 indicates that instrumental variables estimates of short-run impact coefficients are not strikingly different from OLS estimates.16 These results suggest
that in the short-run equation, changes in inflation, the funds rate, and real
income are not correlated with contemporary shocks to the short rate. Hence,
the estimates of short-run impact coefficients reported here are consistent.17
The short-run equation is reported for the full sample 1955:1 to 1994:3.18
As can be seen, the relevant explanatory variables have coefficients that are
with the contemporaneous values of endogenous variables. In the short-run equation estimated
here, lagged endogenous variables are valid as instruments if the equation does not exhibit any
serial correlation. The Ljung-Box Q-statistics reported in Table 4 indicate that serial correlation is
not a problem in the regressions reported there. As regards the second point, lagged endogenous
variables are good instruments because they are likely to be highly correlated with the contemporaneous endogenous variables. However, these variables may not be strictly exogenous in the
sense that they are completely independent of past shocks to economic variables, including the
nominal rate. Thus, changes in the real funds rate may be uncorrelated with the disturbance term
in the regression but may not be independent of the past behavior of economic fundamentals,
including the short rate.
15 The individual t-statistics that appear on the residuals from the reduced-form regressions
of inflation, the funds rate, and real growth (the t-statistics for d6 = 0, d7 = 0, or d8 = 0 in (6))
are not large in the short-run interest rate equation either. In contrast, the null hypothesis that in
the real funds rate equation contemporaneous values of changes in inflation, real GDP, and the
nominal rate are jointly exogenous is usually rejected by the F-test.
16 The null hypothesis that instrumental variables estimates of the short-run equation (2) of
the text are not jointly different from OLS estimates is not rejected by the Hausman test described
in footnote 7. For equations A.1 and A.2 of Table 4, the relevant χ2 statistics are 2.1 and 2.6,
respectively. Both these statistics are small.
17 If one begins with a structural equation in which the nominal rate depends upon contemporaneous anticipated values of economic variables, then lagged values are valid as instruments
for unobservables if the order of lag in instruments chosen exceed the order of lag in the serial
correlation of the disturbance term. The empirical work here does not begin with any particular
structural equation. It does, however, assume that the disturbance term in the short-run equation
does not have any serial correlation. Nevertheless, in order to check the robustness of results
to the order of lag in instruments, I also reestimated the short-run equations using successively
more than one-period lagged values of the right-hand side explanatory variables as instruments,
going as far back as five- through eight-period lags. The point-estimates of the short-run impact
coefficients that appear on inflation, real GDP, and the funds rate move around somewhat, but
they continue to have expected signs and are generally significant. As expected, standard errors
of the estimated coefficients increase as the order of lag in instruments chosen increases.
18 The instrumental variables estimates of the short-run equations for the subsample 1955:1
to 1979:3 are given below, and those estimates look very similar to the ones for the whole period.
∆R1t = 0.06 + 0.76∆ṗt + 0.67∆RFRt + 0.00∆ ln ryt − 0.16∆R1t−1 − 0.15∆R1t−2
(8.7)
(6.8)
(0.0)
(2.2)
(2.2)
− 0.13R1t−1 + 0.13ṗt−1 + 0.12RFRt−1
(1.7)
(1.7)
(1.5)
Y. P. Mehra: Short-Term Nominal Interest Rates
45
of expected signs and statistically significant. Thus, the nominal rate responds
positively to short-run increases in inflation, real GDP, and the real funds rate.
The coefficients that appear on contemporary values of these variables range
from 0.7 to 0.9 for inflation, 0.1 to 0.6 for the real funds rate, and 0.0 to 0.12
for real growth. Thus, a one percentage point rise in inflation raises the short
rate between 70 and 90 basis points, whereas a similar increase in the real
funds rate raises it by 10 to 60 basis points in the short run. A one percentage
point rise in the growth rate of real GDP raises the short rate by about 12 basis
points in the regression that uses actual inflation data.19
The short-run equations reported in Table 4 embody the long-run relationship among the levels of economic determinants. The null hypothesis that
coefficients appearing on one-period lagged levels of inflation and the nominal
rate sum to zero is not rejected, indicating that the nominal rate adjusts one
for one with inflation in the long run. The coefficient that appears on the
lagged level of the real funds rate is large and remains statistically significant,
indicating that (stationary) movements in the real funds rate have substantial
short-run effects on the real component of the short rate.
The short-run equations reported in Table 4 do not include any fiscal policy measure. Nor do they allow for the effect of foreign capital inflows. At
this point I formally test the hypothesis that these variables have no significant
effects on the nominal rate. Table 5 presents Lagrange multiplier tests for omitted variables.20 Those test results indicate that the real deficit, real government
∆R1t = 1.0 + 1.0∆ṗet + 0.29∆RFRt − 0.06∆ ln ryt + 0.17∆R1t−1
(3.4)
(2.0)
(0.1)
(1.4)
− 0.54R1t−1 + 0.54ṗet−1 + 0.38RFRt−1
(3.3)
(3.3)
(2.5)
Parentheses contain t-values.
19 The empirical work here uses actual values of the real funds rate, in contrast with previous studies in which the short-run impact of monetary policy on rates is captured by employing
innovations in the pertinent money supply or term-structure measure. The latter approach reflects
two assumptions. First, anticipated values of these variables affect the short rate by altering its
expected-inflation component. In the short run, only unanticipated changes have an effect on the
real component of the short rate. Second, it is possible to decompose the pertinent monetary
policy measure into its anticipated and unanticipated components. Some of these assumptions are
questionable. Nevertheless, I also estimate the short-run equation using residuals from the shortrun real funds rate equation like (5) as a measure of unanticipated monetary policy actions. The
coefficient that appears on this measure of policy remains positive and is statistically significant.
The estimated coefficient is 0.46 (t-value = 8.6) when the actual inflation data is used, whereas
it is 0.20 (t-value = 3.2) when the expected inflation measure is used. These results indicate that
monetary policy effects when using the actual funds rate are not spurious.
20 A Lagrange multiplier test for a set of p-omitted variables is constructed by regressing
the model’s residuals on both the set of original regressors and on the set of omitted variables.
If the omitted variables do not belong in the equation, then multiplying the R2 statistic from this
regression by the number of observations will produce a statistic distributed as χ2 with p degrees
of freedom (Engle 1984; Breusch and Pagan 1980).
46
Federal Reserve Bank of Richmond Economic Quarterly
Table 5 χ 2 Tests for Omitted Variables
Candidate
Variable X
Lags
(0 to k)
∆(DEF/y)
(0,0)
(0,1)
(0,4)
(0,0)
(0,1)
(0,4)
(0,0)
(0,1)
(0,4)
(0,0)
(0,1)
(0,4)
∆ ln fg
∆ ln tx
∆fh
Equation
B.1
Equation
B.2
0.0
0.7
6.1
0.0
1.8
3.9
0.2
2.4
4.2
1.2
3.2
6.1
1.5
1.5
4.6
0.9
1.2
1.8
2.2
4.7
7.3
0.1
0.4
1.6
(0.96)
(0.68)
(0.30)
(0.80)
(0.40)
(0.54)
(0.28)
(0.30)
(0.51)
(0.27)
(0.20)
(0.30)
(0.21)
(0.47)
(0.47)
(0.33)
(0.55)
(0.87)
(0.14)
(0.10)
(0.20)
(0.71)
(0.81)
(0.89)
Notes: See notes in Table 1 for definition of variables. The statistics reported are the Wald test
of the null hypothesis that the pertinent variable is not an omitted variable from the relevant
regression. If this statistic is large for some variable, then the variable should be included in the
regression. Parentheses contain significance levels of the test.
purchases, net taxes, and foreign capital inflows do not enter the short-run interest equation. Overall, these results assign no significant role to fiscal policy
measures and foreign capital inflows in explaining short-run movements in the
nominal rate.
Predicting the Behavior of the Nominal Rate in the 1980s
I now examine whether the short-run equation estimated here can predict the
actual behavior of the nominal rate during the 1980s. The predicted values
used are the out-of-sample, one-year-ahead dynamic forecasts that cover the
subperiod from 1979 to 1993. I focus on the equation that uses actual inflation.
Table 6 presents predicted values generated using the interest rate regression presented in Table 4. Actual values and prediction errors are also reported
there. As can be seen, this equation predicts reasonably well the actual behavior
of the nominal rate during this period. The mean error is small, only 3 basis
points, and the root mean squared error is 0.4 percentage points. This regression
outperforms a purely eight-order autoregressive model of the short rate. For the
time series model, the mean prediction error is 51 basis points and the root
mean squared error is 1.3 percentage points.21
21 The interest rate regression with the real funds rate also outperforms the version in which
changes in the real funds rate are replaced by changes in the real money supply. The mean error
is 7 basis points and the root mean squared error is 1.22 percentage points.
Y. P. Mehra: Short-Term Nominal Interest Rates
47
Table 6 Predictive Performance
Year
Actual (A)
Predicted (P)
Error
Panel A: Actual and Predicted One-Year Treasury Bill Rate, 1979 to 1993
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
10.7
12.6
14.5
11.9
9.7
10.9
8.3
6.3
6.9
7.8
8.5
7.8
5.7
3.9
3.5
11.2
12.2
13.5
12.2
9.9
10.8
8.9
6.7
6.4
7.9
9.0
7.5
5.4
−0.1
3.2
Mean Error
RMSE
−0.5
0.4
1.0∗
−0.3
−0.2
0.1
−0.6
−0.3
0.5
−0.1
−0.6
0.3
0.3
0.3
0.2
0.03[−0.051]
0.43[ 1.31 ]
Panel B: At = d0 + d1 Pt + et
Model
Interest Rate Equation
(A.1, Table 4)
AR(8)
d0
d1
F
0.05
0.48
1.0
0.89
0.1
3.1
∗
The predictor error is twice the root mean squared error.
Notes: The predicted values reported above are generated using the regression A.1 of Table 4.
AR(8) is an eight-order autoregressive process for explaining changes in the one-year Treasury
bill rate. RMSE is the root mean squared error. Brackets contain the mean error and the RMSE
value generated using the time series model. F is the F-statistic that tests the null hypothesis that
d0 = 0 and d1 = 1.
I evaluate further the predictive performance of the interest rate equation
with the real funds rate from 1979:1 to 1993:4 by estimating regressions of
the form
At = d0 + d1 Pt ,
where A is the quarterly value of the short rate and P is the value predicted by
the short-rate regression. If d0 = 0 and d1 = 1, then regression forecasts are
unbiased. As can be seen from Table 6, the coefficients d0 and d1 take values
48
Federal Reserve Bank of Richmond Economic Quarterly
0.05 and 1.0, respectively, for the interest rate equation (A.1 of Table 4) and
0.48 and 0.89, respectively, for the time series model. The hypothesis d0 = 0
and d1 = 1 is rejected for the time series model, but not for the interest rate
equation.
Despite this overall good predictive performance, the interest rate equation
with the real funds rate does not completely solve the puzzle of high rates
during the early 1980s. The equation significantly underpredicts the level of the
nominal rate in 1981. It predicts very well, however, the declines in nominal
rates that have occurred since 1990.22
The Short End of the Term Structure Is Dominated by the
Outlook for Inflation, Fed Policy, and the State of the Economy
The empirical analysis of one-year Treasury bills summarized in Tables 4 and
5 indicate that in the short run, changes in the nominal rate depend largely
upon changes in inflation, real GDP, and the real funds rate. I now argue that
the same economic factors pretty much determine the behavior of the short end
of the U.S. Treasury term structure. Table 7 presents short-run coefficients that
appear on these economic determinants when alternative measures of shortto medium-term interest rates are used in the regression (2). As can be seen,
those estimates indicate that changes in inflation, real GDP, and the real funds
rate influence most the short end of the term structure. The short-run impact
coefficients that appear on these variables steadily decline in size with the term
to maturity. These results indicate that the yield curve at the short end of the
term structure is dominated by the outlook for inflation, Fed policy, and the
state of the economy.
3.
CONCLUDING OBSERVATIONS
It is a widely held view both in financial press and academic circles that the
Federal Reserve influences short-term nominal interest rates. The empirical
work presented here provides one perspective on the potential role of the
Federal Reserve in determining short-term rates. The results indicate that it
is inflation, not the real federal funds rate, that is at the source of long-run
stochastic movements in the level of the nominal rate. Therefore, only through
its control over inflation can the Federal Reserve exercise control over the level
of the nominal rate in the long run. In other words, the Federal Reserve cannot
permanently lower the nominal rate by affecting its real component.
22 In contrast, the interest rate regression with the real money supply performs very poorly
in predicting the behavior of the nominal rate during the early 1980s. Nor does it predict well the
declines in nominal rates that have occurred since 1990.
Y. P. Mehra: Short-Term Nominal Interest Rates
49
Table 7 Term Structure Effects
Dependent
Variable
R3M
R1
R3
R5
R10
Coefficients (t-values) on Contemporary Values of
∆ṗt
∆RFRt
∆ ln ryt
0.73
0.71
0.51
0.41
0.29
(8.9)
(8.2)
(6.9)
(5.9)
(5.0)
0.57
0.62
0.43
0.33
0.22
(7.5)
(6.4)
(4.5)
(3.6)
(2.8)
0.14
0.11
0.07
0.05
0.03
(3.5)
(2.6)
(1.4)
(1.2)
(0.9)
Notes: R3M is the three-month Treasury bill rate; R1, R3, R5, and R10 are the nominal yields
on one-year, three-year, five-year, and ten-year Treasury bills. The regression equation used is
equation A.1 reported in Table 4. In all of these regressions, the long-run coefficient that appears
on the level of the inflation rate (ṗ) is constrained to be unity, so that in the long run the nominal
rate adjusts one for one with inflation. The regressions are estimated by instrumental variables
over the sample period 1957:1 to 1994:3. Parentheses above contain t-values.
The results also indicate that in the short run, changes in the real funds rate
have considerable effects on the nominal rate. Moreover, the real funds rate
variable is exogenous in the short-run interest rate equations, indicating that
short-run changes in the real federal funds rate do not respond to contemporaneous movements in the nominal rate. This finding, however, is quite consistent
with the possibility that in reduced-form regressions like (5), changes in the
real funds rate are highly correlated with lagged values of economic variables,
including the nominal rate. Together, these results indicate that in the short run,
the Federal Reserve influences the market as well as may be influenced by it.
Fiscal policy measures such as the deficit, government purchases, net taxes,
and foreign capital inflows do not affect the short rate, once one controls for
the effects of inflation, the real funds rate, and real growth. Overall, the results indicate that in the short run, the behavior of short-term nominal rates
is dominated by the outlook for inflation, the funds rate, and the state of the
economy.
The interest rate equation predicts reasonably well the nominal rate during
the 1980s. It does not, however, completely solve the puzzle of high rates
during the early 1980s, particularly in 1981. The analysis here indicates that a
significant part of the rise in rates in the early 1980s can be attributed to the
behavior of inflation and the real federal funds rate. Since over long periods
these variables are endogenously determined, a complete explanation of the
behavior of short rates must include an explanation of the behavior of these
two variables.
50
Federal Reserve Bank of Richmond Economic Quarterly
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39 (July 1984), pp. 671–82.
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Papers on Economic Activity, 2:1983, pp. 553–78.
Engle, Robert F. “Wald, Likelihood Ratio, and Lagrange Multiplier Tests
in Econometrics,” in Zvi Griliches and Michael D. Intriligator, eds.,
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, and C. W. J. Granger. “Co-Integration and Error Correction:
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Feinman, Joshua, and Richard D. Porter. “The Continuing Weakness in M2,”
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(November 1978), pp. 1251–71.
Hendershott, Patric H., and Joe Peek. “Treasury Bill Rates in the 1970s and
1980s,” Journal of Money, Credit, and Banking, vol. 24 (May 1992), pp.
195–214.
Hetzel, Robert L., and Yash P. Mehra. “The Behavior of Money Demand in
the 1980s,” Journal of Money, Credit, and Banking, vol. 21 (November
1989), pp. 455–63.
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and Inference on Cointegration—With Applications to the Demand for
Money,” Oxford Bulletin of Economics and Statistics, vol. 52 (May 1990),
pp. 169–210.
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Kwiatkowski, Denis, Peter C. B. Phillips, Peter Schmidt, and Yongcheol Shin.
“Testing the Null Hypothesis of Stationarity Against the Alternative of a
Unit Root: How Sure Are We That Economic Time Series Have a Unit
Root?” Journal of Econometrics, vol. 54 (October-December 1992), pp.
159–78.
Maddala, G. S. Introduction to Econometrics. New York: Macmillan, 1988.
Mehra, Yash P. “An Error-Correction Model of the Long-Term Bond Rate,”
Federal Reserve Bank of Richmond Economic Quarterly, vol. 80 (Fall
1994), pp. 49–68.
Peek, Joe and James A. Wilcox. “Monetary Policy Regimes and the Reduced
Form for Interest Rates,” Journal of Money, Credit, and Banking, vol. 19
(August 1987), pp. 273–91.
Sargan, J. D. “Wages and Prices in the United Kingdom: A Study in
Econometric Methodology” in P. E. Hart, G. Mills, and J. K. Whitaker,
eds., Econometric Analysis for National Economic Planning. London:
Butterworth Co., 1964.
Sargent, Thomas J. “Commodity Price Expectations and the Interest Rate,”
Quarterly Journal of Economics, vol. 83 (February 1969), pp. 127–40.
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Vectors in Higher Order Integrated Systems,” Econometrica, vol. 61 (July
1993), pp. 783–820.
Quantitative Theory
and Econometrics
Robert G. King
Q
uantitative theory uses simple, abstract economic models together with
a small amount of economic data to highlight major economic mechanisms. To illustrate the methods of quantitative theory, we review
studies of the production function by Paul Douglas, Robert Solow, and Edward
Prescott. Consideration of these studies takes an important research area from
its earliest days through contemporary real business cycle analysis. In these
quantitative theoretical studies, economic models are employed in two ways.
First, they are used to organize economic data in a new and suggestive manner.
Second, models are combined with economic data to display successes and
failures of particular theoretical mechanisms. Each of these features is present
in each of the three studies, but to varying degrees, as we shall see.
These quantitative theoretical investigations changed how economists
thought about the aggregate production function, i.e., about an equation describing how the total output of many firms is related to the total quantities
of inputs, in particular labor and capital inputs. Douglas taught economists
that the production function could be an important applied tool, as well as a
theoretical device, by skillfully combining indexes of output with indexes of
capital and labor input. Solow taught economists that the production function
could not be used to explain long-term growth, absent a residual factor that
he labeled technical progress. Prescott taught economists that Solow’s residual
was sufficiently strongly procyclical that it might serve as a source of economic
The author is A.W. Robertson Professor of Economics at the University of Virginia, consultant to the research department of the Federal Reserve Bank of Richmond, and a research
associate of the National Bureau of Economic Research. This article was originally prepared
as an invited lecture on economic theory at the July 1992 meeting of the Australasian Econometric Society held at Monash University, Melbourne, Australia. Comments from Mary Finn,
Michael Dotsey, Tom Humphrey, Peter Ireland, and Sergio Rebelo substantially improved this
article from its original lecture form. The views expressed are those of the author and do not
necessarily reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve
System.
Federal Reserve Bank of Richmond Economic Quarterly Volume 81/3 Summer 1995
53
54
Federal Reserve Bank of Richmond Economic Quarterly
fluctuations. More specifically, he showed that a real business cycle model
driven by Solow’s residuals produced fluctuations in consumption, investment,
and output that broadly resembled actual U.S. business cycle experience.
In working through three key studies by Douglas, Solow, and Prescott, we
focus on their design, their interrelationship, and the way in which they illustrate how economists learn from studies in quantitative theory. This learning
process is of considerable importance to ongoing developments in macroeconomics, since the quantitative theory approach is now the dominant research
paradigm being used by economists incorporating rational expectations and
dynamic choice into small-scale macroeconomic models.
Quantitative theory is thus necessarily akin to applied econometric research, but its methods are very different, at least at first appearance. Indeed,
practitioners of quantitative theory—notably Prescott (1986) and Kydland and
Prescott (1991)—have repeatedly clashed with practitioners of econometrics.
Essentially, advocates of quantitative theory have suggested that little is learned
from econometric investigations, while proponents of econometrics have suggested that little tested knowledge of business cycle mechanisms is uncovered
by studies in quantitative economic theory.
This article reviews and critically evaluates recent developments in quantitative theory and econometrics. To define quantitative theory more precisely,
Section 1 begins by considering alternative styles of economic theory. Subsequently, Section 2 considers the three examples of quantitative theory in the
area of the production function, reviewing the work of Douglas, Solow, and
Prescott. With these examples in hand, Section 3 then considers how economists
learn from exercises in quantitative theory.
One notable difference between the practice of quantitative theory and of
econometrics is the manner in which the behavioral parameters of economic
models are selected. In quantitative theoretical models of business cycles, for
example, most behavioral parameters are chosen from sources other than the
time series fluctuations in the macroeconomic data that are to be explained in
the investigation. This practice has come to be called calibration. In modern
macroeconometrics, the textbook procedure is to estimate parameters from the
time series that are under study. Thus, this clash of methodologies is frequently
described as “calibration versus estimation.”
After considering how a methodological controversy between quantitative
theory and econometrics inevitably grew out of the rational expectations revolution in Section 4 and describing the rise of quantitative theory as a methodology
in Section 5, this article then argues that the ongoing controversy cannot really
be about “calibration versus estimation.” It demonstrates that classic calibration
studies estimate some of their key parameters and classic estimation studies are
frequently forced to restrict some of their parameters so as to yield manageable
computational problems, i.e., to calibrate them. Instead, in Section 6, the article
argues that the key practical issue is styles of “model evaluation,” i.e., about
R. G. King: Quantitative Theory and Econometrics
55
the manner in which economists determine the dimensions along which models
succeed or fail.
In terms of the practice of model evaluation, there are two key differences
between standard practice in quantitative theory and econometrics. One key difference is indeed whether there are discernible differences between the activities
of parameter selection and model evaluation. In quantitative theory, parameter
selection is typically undertaken as an initial activity, with model evaluation
being a separate secondary stage. By contrast, in the dominant dynamic macroeconometric approach, that of Hansen and Sargent (1981), parameter selection
and model evaluation are undertaken in an essentially simultaneous manner:
most parameters are selected to maximize the overall fit of the dynamic model,
and a measure of this fit is also used as the primary diagnostic for evaluation
of the theory. Another key difference lies in the breadth of model implications
utilized, as well as the manner in which they are explored and evaluated.
Quantitative theorists look at a narrow set of model implications; they conduct
an informal evaluation of the discrepancies between these implications and
analogous features of a real-world economy. Econometricians typically look at
a broad set of implications and use specific statistical methods to evaluate these
discrepancies.
By and large, this article takes the perspective of the quantitative theorist. It
argues that there is a great benefit to choosing parameters in an initial stage of
an investigation, so that other researchers can readily understand and criticize
the attributes of the data that give rise to such parameter estimates. It also argues that there is a substantial benefit to limiting the scope of inquiry in model
evaluation, i.e., to focusing on a set of model implications taken to display
central and novel features of the operation of a theoretical model economy.
This limitation of focus seems appropriate to the current stage of research in
macroeconomics, where we are still working with macroeconomic models that
are extreme simplifications of macroeconomic reality.
Yet quantitative theory is not without its difficulties. To illustrate three of
its limitations, Section 7 of the article reconsiders the standard real business
cycle model, which is sometimes described as capturing a dominant component of postwar U.S. business cycles (for example, by Kydland and Prescott
[1991] and Plosser [1989]). The first limitation is one stressed by Eichenbaum (1991): since it ignores uncertainty in estimated parameters, a study
in quantitative theory cannot give any indication of the statistical confidence
that should be placed in its findings. The second limitation is that quantitative
theory may direct one’s attention to model implications that do not provide
much information about the endogenous mechanisms contained in the model.
In the discussion of these two limitations, the focus is on a “variance ratio” that
has been used, by Kydland and Prescott (1991) among others, to suggest that a
real business cycle arising from technology shocks accounts for three-quarters
of postwar U.S. business cycle fluctuations in output. In discussing the practical
56
Federal Reserve Bank of Richmond Economic Quarterly
importance of the first limitation, Eichenbaum concluded that there is “enormous” uncertainty about this variance ratio, which he suggested arises because
of estimation uncertainty about the values of parameters of the exogenous driving process for technology. In terms of the second limitation, the article shows
that a naive model—in which output is driven only by production function
residuals without any endogenous response of factors of production—performs
nearly as well as the standard quantitative theoretical model according to the
“variance ratio.” The third limitation is that the essential focus of quantitative
theory on a small number of model implications may easily mean that it misses
crucial failures (or successes) of an economic model. This point is made by
Watson’s (1993) recent work that showed that the standard real business cycle
model badly misses capturing the “typical spectral shape of growth rates” for
real macroeconomic variables, including real output. That is, by focusing on
only a small number of low-order autocovariances, prior investigations such
as those of Kydland and Prescott (1982) and King, Plosser, and Rebelo (1988)
simply overlooked the fact that there is an important predictable output growth
at business cycle frequencies.
However, while there are shortcomings in the methodology of quantitative
theory, its practice has grown at the expense of econometrics for a good reason:
it provides a workable vehicle for the systematic development of macroeconomic models. In particular, it is a method that can be used to make systematic
progress in the current circumstances of macroeconomics, when the models
being developed are still relatively incomplete descriptions of the economy.
Notably, macroeconomists have used quantitative theory in recent years to
learn how the business cycle implications of the basic neoclassical model are
altered by a wide range of economic factors, including fiscal policies, international trade, monopolistic competition, financial market frictions, and gradual
adjustment of wages and prices.
The main challenge for econometric theory is thus to design procedures that
can be used to make similar progress in the development of macroeconomic
models. One particular aspect of this challenge is that the econometric methods
must be suitable for situations in which we know before looking at the data
that the model or models under study are badly incomplete, as we will know
in most situations for some time to come. Section 8 of the article discusses a
general framework of model-building activity within which quantitative theory
and traditional macroeconometric approaches are each included. On this basis, it
then considers some initial efforts aimed at developing econometric methods to
capture the strong points of the quantitative theory approach while providing the
key additional benefits associated with econometric work. Chief among these
benefits are (1) the potential for replication of the outcomes of an empirical
evaluation of a model or models and (2) an explicit statement of the statistical
reliability of the results of such an evaluation.
R. G. King: Quantitative Theory and Econometrics
57
In addition to providing challenges to econometrics, Section 9 of the article
shows how the methods of quantitative theory also provide new opportunities
for applied econometrics, using Friedman’s (1957) permanent income theory
of consumption as a basis for constructing two more detailed examples. The
first of these illustrates how an applied econometrician may use the approach
of quantitative theory to find a powerful estimator of a parameter of interest.
The second of these illustrates how quantitative theory can aid in the design of
informative descriptive empirical investigations.
In macroeconometric analysis, issues of identification have long played
a central role in theoretical and applied work, since most macroeconomists
believe that business fluctuations are the result of a myriad of causal factors.
Quantitative theories, by contrast, typically are designed to highlight the role
of basic mechanisms and typically identify individual causal factors. Section
10 considers the challenges that issues of identification raise for the approach
of quantitative theory and the recent econometric developments that share its
model evaluation strategy. It suggests that the natural way of proceeding is to
compare the predictions of a model or models to characteristics of economic
data that are isolated with a symmetric empirical identification.
The final section of the article offers a brief summary as well as some concluding comments on the relationship between quantitative theory and econometrics in the future of macroeconomic research.
1.
STYLES OF ECONOMIC THEORY
The role of economic theory is to articulate the mechanisms by which economic
causes are translated into economic consequences. By requiring that theorizing
is conducted in a formal mathematical way, economists have assured a rigor of
argument that would be difficult to attain in any other manner. Minimally, the
process of undertaking a mathematical proof lays bare the essential linkages
between assumptions and conclusions. Further, and importantly, mathematical
model-building also has forced economists to make sharp abstractions: as model
economies become more complex, there is a rapidly rising cost to establishing
formal propositions. Articulation of key mechanisms and abstraction from less
important ones are essential functions of theory in any discipline, and the speed
at which economic analysis has adopted the mathematical paradigm has led it to
advance at a much greater rate than its sister disciplines in the social sciences.
If one reviews the history of economics over the course of this century,
the accomplishments of formal economic theory have been major. Our profession developed a comprehensive theory of consumer and producer choice, first
working out static models with known circumstances and then extending it
to dynamics, uncertainty, and incomplete information. Using these developments, it established core propositions about the nature and efficiency of
general equilibrium with interacting consumers and producers. Taken together,
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Federal Reserve Bank of Richmond Economic Quarterly
the accomplishments of formal economic theory have had profound effects on
applied fields, not only in the macroeconomic research that will be the focal
point of this article but also in international economics, public finance, and
many other areas.
The developments in economic theory have been nothing short of remarkable, matched within the social sciences perhaps only by the rise of econometrics, in which statistical methods applicable to economic analysis have been
developed. For macroeconomics, the major accomplishment of econometrics
has been the development of statistical procedures for the estimation of parameters and testing of hypotheses in a context where a vector of economic
variables is dynamically interrelated. For example, macroeconomists now think
about the measurement of business cycles and the testing of business cycle
theories using an entirely different statistical conceptual framework from that
available to Mitchell (1927) and his contemporaries.1
When economists discuss economic theory, most of us naturally focus on
formal theory, i.e., the construction of a model economy—which naturally is a
simplified version of the real world—and the establishment of general propositions about its operation. Yet, there is another important kind of economic
theory, which is the use of much more simplified model economies to organize economic facts in ways that change the focus of applied research and
the development of formal theory. Quantitative theory, in the terminology of
Kydland and Prescott (1991), involves taking a more detailed stand on how economic causes are translated into economic consequences. Quantitative theory,
of course, embodies all the simplifications of abstract models of formal theory.
In addition, it involves making (1) judgments about the quantitative importance
of various economic mechanisms and (2) decisions about how to selectively
compare the implications of a model to features of real-world economies. By its
very nature, quantitative theory thus stands as an intermediate activity to formal
theory and the application of econometric methods to evaluation of economic
models.
A decade ago, many economists thought of quantitative theory as simply
the natural first step in a progression of research activities from formal theory
to econometrics, but there has been a hardening of viewpoints in recent years.
Some argue that standard econometric methods are not necessary or are, in fact,
unhelpful; quantitative theory is sufficient. Others argue that one can learn little
from quantitative theory and that the only source of knowledge about important
economic mechanisms is obtained through econometrics. For those of us that
honor the traditions of both quantitative theory and econometrics, not only did
the onset of this controversy come as a surprise, but its depth and persistence
1 In particular, we now think of an observed time series as the outcome of a stochastic
process, while Mitchell and his contemporaries struggled with how to best handle the evident
serial dependence that was so inconsistent with the statistical theory that they had at hand.
R. G. King: Quantitative Theory and Econometrics
59
also were unexpected. Accordingly, the twin objectives of this paper are, first, to
explore why the events of recent years have led to tensions between practitioners of quantitative theory and econometrics and, second, to suggest dimensions
along which the recent controversy can lead to better methods and practice.
2.
EXAMPLES OF QUANTITATIVE THEORY
This section discusses three related research topics that take quantitative theory
from its earliest stages to the present day. The topics all concern the production
function, i.e., the link between output and factor inputs.2
The Production Function and Distribution Theory
The production function is a powerful tool of economic analysis, which every
first-year graduate student learns to manipulate. Indeed, the first example that
most economists encounter is the functional form of Cobb and Douglas (1928),
which is also the first example studied here. For contemporary economists, it is
difficult to imagine that there once was a time when the notion of the production
function was controversial. But, 50 years after his pioneering investigation, Paul
Douglas (1976) reminisced:
Critics of the production function analysis such as Horst Mendershausen and
his mentor, Ragnar Frisch, . . . urged that so few observations were involved
that any mathematical relationship was purely accidental and not causal. They
sincerely believed that the analysis should be abandoned and, in the words
of Mendershausen, that all past work should be torn up and consigned to the
wastepaper basket. This was also the general sentiment among senior American economists, and nowhere was it held more strongly than among my senior
colleagues at the University of Chicago. I must admit that I was discouraged
by this criticism and thought of giving up the effort, but there was something
which told me I should hold on.
(P. 905)
The design of the investigation by Douglas was as follows. First, he enlisted
the assistance of a mathematician, Cobb, to develop a production function with
specified properties.3 Second, he constructed indexes of physical capital and
labor input in U.S. manufacturing for 1899–1922. Third, Cobb and Douglas
estimated the production function
2 A replication diskette available from the author contains computer programs used to produce the four figures that summarize the Douglas, Solow, and Prescott studies in this section,
as well as to produce additional figures discussed below. That diskette also contains detailed
background information on the data.
3 But the mathematics was not the major element of the investigation. Indeed, it was not
even novel, as Tom Humphrey has pointed out to the author, having been previously derived in
the realm of pure theory by Wicksell (1934, pp. 101–28).
60
Federal Reserve Bank of Richmond Economic Quarterly
Yt = A Ntα Kt1−α .
(1)
In this specification, Yt is the date t index of manufacturing output, Nt is the
date t index of employed workers, and Kt is the date t index of the capital
stock. The least squares estimates for 1899–1922 were  = 1.01 and α̂ = 0.73.
Fourth, Cobb and Douglas performed a variety of checks of the implications
of their specification. These included comparing their estimated α̂ to measures
of labor’s share of income, which earlier work had shown to be reasonably
constant through time. They also examined the extent to which the production
function held for deviations from trend rather than levels. Finally, they examined the relationship between the model’s implied marginal product of labor
(αY/N) and a measure of real wages that Douglas (1926) had constructed in
earlier work.
The results of the Cobb-Douglas quantitative theoretical investigation are
displayed in Figure 1. Panel A provides a plot of the data on output, labor,
and capital from 1899 to 1922. All series are benchmarked at 100 in 1899, and
it is notable that capital grows dramatically over the sample period. Panel B
displays the fitted production function, Ŷ = ÂN α̂ K 1−α̂ , graphed as a dashed
line and manufacturing output, Y, graphed as a solid line. As organized by
the production function, variations in the factors N and K clearly capture the
upward trend in output.4
With the Cobb-Douglas study, the production function moved from the
realm of pure theory—where its properties had been discussed by Clark (1889)
and others—to that of quantitative theory. In the hands of Cobb and Douglas,
the production function displayed an ability to link (1) measures of physical
output to measures of factor inputs (capital and labor) and (2) measures of real
wages to measures of average products. It thus became an engine of analysis
for applied research.
But the authors took care to indicate that their quantitative production
function was not to be viewed as an exact model of economic activity for
three reasons. Cobb and Douglas recognized that they had neglected technical
progress, but they were uncertain about its magnitude or measurability. They
also viewed their measure of labor input only as a first step, taken because
there was relatively poor data on hours per worker. Finally, and importantly,
they found it unsurprising that errors in the production function were related
4 The
Cobb-Douglas methodology was also applied to cross-sections of industries in followup investigations (as reviewed in Douglas [1948]). Many of these subsequent cross-section
investigations used the high-quality New South Wales and Victoria data from Australia. These
cross-section studies provided further buttressing of Douglas’s perspective that the production
function was a useful applied tool, although from a modern perspective they impose too much
homogeneity of production technique across industries.
R. G. King: Quantitative Theory and Econometrics
61
Figure 1
A. Cobb-Douglas Data
450
Capital
400
Index: 1899 = 100
350
300
250
Output
200
Labor
150
100
50
1900
5
05
10
15
20
B. Cobb-Douglas Result
250
Output
Index: 1899 = 100
200
Production
Function
150
100
50
1900
+
5
05
10
15
20
62
Federal Reserve Bank of Richmond Economic Quarterly
to business cycles. They pointed out that during slack years the production
function overpredicted output because reductions in hours and capital utilization were not measured appropriately. Correspondingly, years of prosperity were
underpredicted.
The Production Function and Technical Progress
Based on the Cobb and Douglas (1928) investigations, it was reasonable to think
that (1) movements in capital and labor accounted for movements in output,
both secularly and over shorter time periods and (2) wages were equated with
marginal products computed from the Cobb-Douglas production function.
Solow’s (1957) exercise in quantitative theory provided a sharp contradiction of the first conclusion from the Cobb-Douglas studies, namely, that output
movements were largely determined by movements in factor inputs. Taking
the marginal productivity theory of wages to be true, Solow used the implied
value of labor’s share to decompose the growth in output per man-hour into
components attributable to capital per man-hour and a residual:
yt − nt = sk (kt − nt ) + at ,
(2)
where y is the growth rate of output; n is the growth rate of labor input; k
is the growth rate of capital input; and a is the “Solow residual,” taken to be
a measure of growth in total factor productivity. The share is given by the
competitive theory, i.e., sk is the share of capital income. Solow allowed the
share weight in (2) to vary at each date, but this feature leads to essentially the
same outcomes as simply imposing the average of the sk . Hence, a constant
value of sk or α = (1 − sk ) is used in the construction of the figures to maintain
comparability with the work of Cobb and Douglas. Solow emphasized the trend
(low frequency) implications of the production function by looking at the longterm average contribution of growth in capital per worker (k − n) to growth
in output per worker (y − n). Over his 50-year sample period, output roughly
doubled. But only one-eighth of the increase was due to capital; the remaining
seven-eighths were due to technical progress.
Another way of looking at this decomposition is to consider the trends
in the series. Figure 2 displays the nature of Solow’s quantitative theoretical
investigation in a manner comparable to Figure 1’s presentation of the CobbDouglas data and results. Panel A is a plot of the aggregate data on output,
capital, and labor during 1909–49. For comparability with Figure 1, the indexes
are all scaled to 100 in the earliest year. The striking difference between Panel
A of the two figures is that capital grows more slowly than output in Solow’s
data while it grows more rapidly than output in the Cobb-Douglas data. In turn,
Panel B of Figure 2 displays the production function Ŷt = A Ntα Kt1−α , with α
chosen to match data on the average labor share and A chosen so that the fit is
correct in the initial period. In contrast to Panel B of Figure 1, the production
R. G. King: Quantitative Theory and Econometrics
63
Figure 2
A. Solow Data
350
Index: 1909 = 100
300
Output
250
200
Capital
150
Labor
100
50
1910
15
20
25
30
35
45
B. Solow Version of Cobb-Douglas Plot
350
300
Index: 1909 = 100
40
Output
250
200
150
Production
Function
100
50
1910
+
15
20
25
30
35
40
45
64
Federal Reserve Bank of Richmond Economic Quarterly
function, which is the dashed line, does not capture much of the trend variation
in output, Y.5
It is likely that these results were unexpected to Solow, who had just
completed development of a formal theoretical model that stressed capital
deepening in his 1956 classic, “A Contribution to the Theory of Economic
Growth.” Instead of displaying the importance of capital deepening, as had
the Cobb-Douglas investigations, Solow’s adventure into quantitative theory
pointed toward the importance of a new feature, namely, total-factoraugmenting technical progress. There had been hints of this result in other
research just before Solow’s, but none of it had the simplicity, transparency,
or direct connection to formal economic theory that marked Solow’s work.6
Cyclical Implications of Movements in Productivity
By using the production function as an applied tool in business cycle research,
Prescott (1986) stepped into an area that both Douglas and Solow had avoided.
Earlier work by Kydland and Prescott (1982) and Long and Plosser (1983) had
suggested that business cycle phenomena could derive from an interaction of
tastes and technologies, particularly if there were major shocks to technology.
But Prescott’s (1986) investigation was notable in its use of Solow’s (1957)
procedure to measure the extent of variations in technology. With such a measurement of technology or productivity shocks in hand, Prescott explored how
a neoclassical model behaved when driven by these shocks, focusing on the
nature of business cycles that arose in the model.
Figure 3 displays the business cycle behavior of postwar U.S. real gross
national product and a related productivity measure,
log At = log Yt − α log Nt + (1 − α) log Kt .
(3)
This figure is constructed by first following a version of the Solow (1957)
procedure to extract a productivity residual and then filtering the outcomes
with the procedure of Hodrick and Prescott (1980). These “business cycle
components” are very close to deviations from a lengthy centered moving
5 It is an interesting question as to why Douglas’s production function results differed so
much from Solow’s, particularly since the latter’s have largely been sustained in later work. To
begin, Douglas’s work looked only at 23 years, from 1899 to 1922, and the study was restricted
to manufacturing. During this interval, there was substantial growth in manufacturing capital (as
displayed in Figure 1). Ultimately, it is this growth in capital that underlies the different conclusions from the Solow investigation. The capital series itself was created by Cobb and Douglas
using interpolation between U.S. census estimates in 1899, 1904, and 1922.
6 The construction of Panel B of Figure 2 actually departs somewhat from the procedure
used by Solow. That is, the indexes of capital and labor from Panel A are used directly, while
Solow corrected capital for utilization by multiplying the capital stock by the unemployment rate.
For the trend properties of Figure 2, this difference is of little importance, but for the business
cycle issue to which we will now turn, it is likely of greater importance.
R. G. King: Quantitative Theory and Econometrics
65
Figure 3
A. Output
Percent
5
0
-5
1955
60
65
70
75
80
85
90
85
90
85
90
B. Solow Residual and Output
Percent
5
0
-5
1955
60
65
70
75
80
C. Labor Component and Output
Percent
5
0
-5
1955
60
65
70
75
80
D. Capital Component and Output
Percent
5
0
-5
1955
60
65
70
75
80
85
90
+
Notes: Panel A displays the behavior of output relative to trend, measured as discussed in the
text. Panels B, C, and D show the decomposition of cyclical output into components attributable
to the Solow residual, labor input, and capital input (the solid lines). In each of these panels,
cyclical output (the dotted line) is also displayed to provide the reader with a reference point.
66
Federal Reserve Bank of Richmond Economic Quarterly
average; they also closely correspond to the results if a band-pass filter is
applied to each time series.7 Panel A displays the business cycle variation in
output, which has a sample standard deviation of 1.69 percent. This output
measure is also the solid line in the remainder of the panels of Figure 3. It
is clear from Panel B of this figure that the productivity residual is strongly
procyclical: the correlation between the series in Panel B is 0.88. The labor
component of output—the filtered version of α log Nt —is also strongly procyclical: the correlation between the series in Panel C is 0.85. Finally, Panel
D shows there is little cyclical variation in capital input, the filtered series
(1 − α) log Kt : its correlation with output is −0.12.
Unlike the earlier studies, real business cycle (RBC) exercises in quantitative theory are explicitly general equilibrium: they take measurements of
individual causes and trace the consequences for a number of macroeconomic
variables. The simplest RBC model is essentially the growth model of Solow
(1956), modified to include optimal choice of consumption and labor input over
time. In the Solow model, long-term growth in output per capita must come
about mainly from productivity growth. Thus, RBC models may be viewed
as exploring the business cycle implications of a feature widely agreed to be
important for lower frequency phenomena.
In his quantitative theoretical investigation, Prescott thus assumed that productivity in a model economy was generated by a stochastic process fit to
observations on log At . He then explored how consumption, investment, output, and input would evolve in such an economy.8 There were two striking
outcomes of Prescott’s experiment, which have been frequently highlighted by
adherents of real business cycle theory.
The first, much-stressed finding is that a large fraction of variation in output
is explained by such a model, according to a statistical measure that will be
explained in more detail below. Figure 4 shows that an RBC model’s output
(the dotted line) is closely related to actual output (the solid line), albeit with
7 For some additional discussion of the Hodrick and Prescott filter, see King and Rebelo
(1993). Recent related work, Baxter and King (1995), showed that Hodrick-Prescott filtered time
series on gross national product resembles (1) the deviation from a centered five-year (equalweight) moving average and (2) a particular approximate high-pass filter, which is designed to
eliminate slow-moving components and to leave intact components with periodicities of eight
years or less.
In terms of the discussion in Section 7 below, it is useful to have a version of “cyclical
variation in productivity” present in Figure 3. However, for Figure 4 below, the original time
series on log At was fed into the dynamic model, not the filtered one. Since optimal dynamic
responses imply that “shocks” at one frequency are generally translated to all others, it would be
hard to justify using filtered log At as a forcing process.
8 To explore the consequences of such “technology shocks,” the current discussion will
assume that the model economy is driven by the actual sequence of log At , which is the strategy
used in Plosser’s (1989) version of this quantitative theoretical experiment. Prescott (1986) instead assumed that the technology shocks, εt , were drawn from a random-number generator and
explored the properties of simulated economies.
R. G. King: Quantitative Theory and Econometrics
67
Figure 4
A. Actual and Model-Generated Output
Percent
5
Model-Generated
Actual
0
-5
1955
60
65
70
75
80
85
90
B. Model-Generated Consumption and Output
Percent
5
0
-5
1955
60
65
70
75
80
85
90
C. Model-Generated Labor and Output
Percent
5
0
-5
1955
60
65
70
75
80
85
90
D. Model-Generated Investment and Output
10
Percent
5
0
-5
-10
1955
60
65
70
75
80
85
90
+
Notes: Panel A displays U.S. cyclical output (the solid line, also previously reported in Panel
A of Figure 3) as well as model cyclical output (the dotted line). The model cyclical output
was generated by simulating a basic real business cycle, with the U.S. productivity residual as
a driving process. Panels B, C, and D show the comparable simulation results for consumption,
labor input, and investment (the solid lines). In each of these panels, model cyclical output (the
dotted line) is also displayed to provide the reader with a reference point.
68
Federal Reserve Bank of Richmond Economic Quarterly
somewhat smaller amplitude.9 In particular, the variance of output in the model
is 2.25 percent with the variance of actual output being 2.87 percent, so that
the ratio of variances is 0.78. Further, the correlation between the actual and
model output series is 0.90. In this sense, the RBC model captures a major part
of economic fluctuations.
The second, much-stressed finding is that the model economy captures
other features of observed business cycles, notably the fact that consumption is
smoother than output and that investment is more volatile than output, as shown
in the additional panels of Figure 4.10 Moments from the model economy that
measure consumption’s relative volatility—the ratio of its standard deviation to
that of output—are about as large as the comparable ratio for postwar U.S. data.
The design of Prescott’s (1986) investigation was solidly in the tradition of
Douglas and Solow: a very simple theoretical model was constructed and it was
shown to have very surprising empirical properties. Prior to Prescott’s study,
it was possible to dismiss the RBC interpretation of postwar U.S. economic
fluctuations out of hand. One needed only to make one of two arguments that
were widely accepted at the time. The first common argument was that there
was no evidence of large, procyclical productivity shocks. The second was that
equilibrium macroeconomic models, even with productivity shocks, were evidently inconsistent with major features of the U.S. macroeconomic time series,
such as the procyclicality of labor input and the high volatility of investment.
After Prescott’s investigation, as Rogoff (1986) pointed out, it was no longer
possible to do this: it was necessary to undertake much more subtle critiques
of the RBC interpretation of economic fluctuations.
3.
HOW WE LEARN FROM QUANTITATIVE THEORY
How, in general, do economists learn from quantitative theory? Taken together,
the preceding examples help answer this question.
Theory as Abstraction
An essential characteristic of formal economic theory is that it involves abstraction. In constructing a theoretical model, we focus on one set of mechanisms
and variables; we neglect factors that are hypothesized to be secondary or are
9 The model economy is the one discussed in King, Plosser, and Rebelo (1988), with parameter choices detailed there. The productivity process is assumed to be a random walk, and the
changes in the Solow residual (relative to the mean change) are taken to be productivity shocks.
The dynamic model is then used to produce a time series on output, consumption, investment,
and labor input; the resulting simulated outcomes are then run through the Hodrick-Prescott filter.
10 Less realistically, in the specific simple version of the model underlying Figure 4, labor is
less volatile than output. However, as Prescott (1986) discussed, there are several ways to increase
labor volatility to roughly the same level as output volatility without major changes in the other
features of the model.
R. G. King: Quantitative Theory and Econometrics
69
simply not the focus of the investigation. Quantitative theory similarly involves
the process of making sharp abstractions, but it also involves using the theory
as an empirical vehicle. In particular, we ask whether an abstraction provides
a compelling organization of key facts in an area of economic inquiry.
For Douglas, there were two key questions. First, was there a systematic
empirical relationship between factor inputs and outputs of the type suggested
by the formal economic theories of Clark and others? Second, was this relationship also consistent with the competitive theory of distribution, i.e., did
real wages resemble marginal products constructed from the production function? Prior to Douglas’s work, there was little reason to think that indexes
of production, employment, and capital would be linked together in the way
suggested by the theoretical production function. After Douglas’s work, it was
hard to think that there were not empirical laws of production to be discovered
by economists. Further, Douglas’s work also indicated a strong empirical relationship between real wages and average products, i.e., a rough constancy of
payments to labor as a fraction of the value of output. Thus, Douglas’s work
made the theory of production and the theory of competitive determination of
factor income payments operational in ways that were crucial for the development of economics. To be sure, as Douglas (1976) pointed out, his simple
theory did not work exactly, but it suggested that there was sufficient empirical
content to warrant substantial additional work.
One measure of Douglas’s success on both fronts is the character of Solow’s
quantitative theoretical investigation. Solow simply took as given that many
economists would accept both abstractions as useful organizing principles: he
assumed both that the aggregate production function was an organizing principle for data and that the marginal productivity theory of wages was appropriate.
But he then proceeded to challenge one of the substantive conclusions of Douglas’s analysis, namely, that most of the movement in output can be explained
by movements in factors of production. In particular, after reading Solow’s article, one finds it hard not to believe that other factors besides physical capital
formation are behind the secular rise in wage rates.11 Solow’s reorganization of
11 That is, as it did for Douglas’s indexes, input from physical capital would need to grow
much faster than final output if it is to be responsible for the secular rise in wage rates. If capital
input is proportional to capital stock, the capital stock series that Solow used would have to be
very badly measured for this to be the case. However, an important recent exercise in quantitative
theory by Greenwood, Hercovitz, and Krusell (1992) suggested that, for the United States over
the postwar period, there has been a major mismeasurement of capital input deriving from lack of
incorporation of quality change. Remarkably, when Greenwood, Hercovitz, and Krusell corrected
investment flows for the amount of quality change suggested by Gordon’s (1990) study, capital
input did grow substantially faster than output, sufficiently so that it accounted for about twothirds of growth in output per man-hour. However, one interpretation of Greenwood, Hercovitz,
and Krusell’s results is that these authors produced a measurement of technical progress, albeit a
more direct one than that of Solow. Thus, the net effect of Greenwood, Hercovitz, and Krusell’s
study is likely to be that it enhances the perceived importance of technical progress.
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Federal Reserve Bank of Richmond Economic Quarterly
the production function facts spurred the development of the growth accounting
literature—as summarized by Maddison (1981)—and continues to provide major challenges to economists’ thinking about growth and development problems.
Prescott’s (1986) analysis of the role of productivity fluctuations in business
cycles builds directly on the prior investigations of Douglas and Solow, yielding
two key findings. The first finding is that there is important procyclical variation
in Solow residuals. The second finding is that cyclical variations in productivity can explain cyclical variations in other macroeconomic quantities. The first
component of this investigation in quantitative theory has had sufficient impact
that even new Keynesian macroeconomists like Blanchard and Fischer (1989)
list procyclical productivity as one of the key stylized facts of macroeconomics
in the opening chapter of their textbook. The second has stimulated a major
research program into the causes and consequences of cyclical variations in
productivity.
In each of these cases, the investigation in quantitative theory had a surprising outcome: prior to each investigation, there was little reason to think that
a specific economic mechanism was of substantial empirical importance. Prior
to Douglas’s work, there was little reason to look for empirical connections
between outputs and quantities of factor inputs. Prior to Solow’s, there was
little reason to think that technical progress was an important contributor to
economic growth. Prior to Prescott’s, there was little reason to think that procyclical variations in productivity were important for the business cycle. Each
investigation substantially changed the views of economists about the nature
of economic mechanisms.
Challenges to Formal Theory
Quantitative theory issues important challenges to formal theory; each of the
three examples contains such a challenge.
Flexible and Tractable Function Forms: The Cobb-Douglas investigations
led to a search for alternative functional forms that could be used in such investigations, but that did not require the researcher to impose all of the restrictions
on substitutability, etc., associated with the Cobb-Douglas specification.
Factor Income Payments and Technical Progress: Solow’s investigation
reinforced the need for theories of distribution of income to different factor
inputs in the presence of technical progress. Initially, this was satisfied by the
examination of different forms of technical progress. But more recently, there
has been much attention given to the implications of technical progress for the
competitive paradigm (notably in Romer [1986, 1987]).
Cyclical Variation in Productivity: Prescott’s investigation showed dramatically how macroeconomic theories without productivity variation typically have
important counterfactual productivity implications (i.e., they imply that output
per man-hour is countercyclical). Recent work has explored a range of theories
R. G. King: Quantitative Theory and Econometrics
71
designed to generate procyclical output per man-hour, including (1) imperfect
competition, (2) external effects, and (3) time-varying capacity utilization.
4.
HOW MACROECONOMETRICS FAILED (TWICE)
In recent years, the methods of quantitative theory have become an increasingly
used tool in applied research in macroeconomics. This growth results from
two very different historical episodes during which macroeconometrics failed,
though for very different reasons. In the first of these episodes, econometric
practice was subject to too little discipline from economic and econometric
theory in the development of large-scale macroeconometric models. While
the behavioral equations were sometimes motivated by a prevailing economic
theory, in practice generous incorporation of lags and dummy variables gave
rise to essentially unrestricted empirical specifications. As a result, most
models fit the historical data very well, even though they had much more modest
success in short-term forecasting. In the second of these episodes, econometric
practice was subject to too much discipline from economic and econometric
theory: rational expectations econometrics produced tightly restricted dynamic
macroeconomic models and concluded that no models fit the data very well.
Keynesian Econometrics: The Promise
The research program of Koopmans and his coworkers at the Cowles Foundation, as reported in Hood and Koopmans (1953), provided a formal structure
for Keynesian macroeconometrics, the dynamic simultaneous equations model.
This econometric structure utilized economic theory in a very important manner: it stressed that theory could deliver the exclusion restrictions that were
necessary for identification of the behavioral parameters. Initially, the Keynesian macroeconometric models were of sufficiently small scale that they could
be readily studied by other researchers, like those of Klein (1950) and Klein
and Goldberger (1955). Klein’s (1950) work is particularly notable in terms of
its insistence on developing the relevant behavioral theory for each structural
equation. The promise of Keynesian macroeconometrics was that empirical
macroeconomic models were to be a research laboratory, the basis for systematic refinement of theoretical and empirical specifications. They were also to
be a device for concrete discussion of appropriate policy actions and rules.
Keynesian Econometrics: The Failure
By the mid-1970s, the working Keynesian macroeconometric models had
evolved into extremely large systems; this growth occurred so that their builders
could readily answer a very wide range of questions posed by business and
government policymakers. In the process of this evolution, they had strayed
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Federal Reserve Bank of Richmond Economic Quarterly
far from the promise that early developments had suggested: they could not be
readily studied by an individual researcher nor was it possible to determine the
components of the model that led to its operating characteristics. For example,
in response to most policy and other disturbances, most models displayed outcomes that cycled for many years, but it proved difficult to understand why this
was the case. The consequent approach was for individual academic researchers
to concentrate on refinement of a particular structural equation—such as the
money demand function or the consumption function—and to abstain from
analysis of complete models. But this strategy made it difficult to discuss many
central issues in macroeconomics, which necessarily involved the operation of
a full macroeconomic system.
Then, in the mid-1970s, two major events occurred. There was worldwide
“stagflation,” with a coexistence of high inflation and high unemployment.
Major macroeconometric models simply got it very wrong in terms of predicting this pattern of events. In addition, Lucas’s (1976) famous critique of
econometric policy evaluation highlighted the necessity of producing macroeconomic models with dynamic choice and rational expectations. Taken together with the prior inherent difficulties with macroeconometric models, these
two events meant that interest in large-scale macroeconometric models essentially evaporated.
Lucas’s (1976) critique of macroeconometric models had two major components. First, Lucas noted that many behavioral relations—investment, consumption, labor supply, etc.—depended in a central way on expectations when
derived from relevant dynamic theory. Typical macroeconomic models either
omitted expectations or treated them as essentially static. Second, Lucas argued
that expectations would likely be rational—at least with respect to sustained
policy changes and possibly for others—and that there were quantitatively major consequences of introducing rational expectations into macroeconometric
models. The victory of Lucas’s ideas was swift. For example, in a second-year
graduate macro class at Brown in 1975, Poole gave a clear message when
reviewing Lucas’s critique in working-paper form: the stage was set for a
complete overhaul of econometric models.12 Better theoretical foundations and
rational expectations were to be the centerpieces of the new research.
Rational Expectations Econometrics: The Promise
As a result of the rational expectations revolution, there was a high demand for
new methods in two areas: (1) algorithms to solve dynamic rational expectations
models, and (2) econometric methods. The high ground was rapidly taken by
the linear systems approach best articulated by Hansen and Sargent (1981):
dynamic linear rational expectations models could be solved easily and had
12 See
also Poole (1976).
R. G. King: Quantitative Theory and Econometrics
73
heavily constrained vector autoregressions as their reduced form. As Sargent
(1981) stressed, the general equilibrium nature of rational expectations models
meant that parameters traditionally viewed as important for one “behavioral
equation” in a traditional macroeconomic model would also be important for
others. For example, parameters of the investment technology would have implications for other quantities (for instance, consumption and labor supply)
because changes in investment dynamics would alter the optimal choices of
consumption and labor supply by influencing the rational expectations of future wages and interest rates implied by the model. Thus, complicated and
time-consuming systems methods of estimation and testing were employed in
the Hansen-Sargent program.
The promise of rational expectations econometrics was that “fully articulated” model economies were to be constructed, their reduced forms determined,
and the resulting systems compared with unrestricted dynamic models. The
“deep parameters” of preferences and technologies were to be estimated via
maximum likelihood, and the fully articulated economies were to be used to
evaluate alternative macroeconomic policies in a manner consistent with the
requirements of Lucas (1976).
Rational Expectations Econometrics: The Failure
With a decade of hindsight, though, it is clear that the Hansen-Sargent program
was overambitious, in ways that are perhaps best illustrated by discussing the
typical study using this technology in the mid-1980s. The author would (1)
construct a rich dynamic macroeconomic model, (2) estimate most of its parameters using maximum likelihood, and (3) perform a likelihood ratio test to
evaluate the model, i.e., compare its fit to an unrestricted vector autoregression.
Since the estimation of the model had been very time-consuming, the author
would have been able to produce only a small number of experiments with
alternative specifications of preferences, technologies, and forcing processes.
Further, it would be difficult for the author and the audience to interpret the
results of the study. Typically, at least some of the parameter estimates would be
very strange, such as implausible discount factors or utility functions lacking
concavity properties. Since only limited experimentation with the economic
structure was feasible, the author would consequently struggle to explain what
features of the data led to these aberrant outcomes. Further, the model would
be badly rejected and again the author would have difficulty explaining which
features of the macroeconomic time series led to this rejection. Overall, the
author would be hard pressed to defend the specific model or, indeed, why he
had spent his time conducting the investigation. This experience produced a
general reaction that the Hansen-Sargent program had not produced a workable
vehicle for systematic development of macroeconometric models.
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Federal Reserve Bank of Richmond Economic Quarterly
Applied Macroeconomic Research After the Fall
There were three basic reactions to this double failure of econometrics.
First, some researchers sought to use limited-information methods to estimate the parameters of a single behavioral equation in ways consistent with
rational expectations (as in McCallum [1976], Kennan [1979], and Hansen and
Singleton [1982]). While this work was valuable, it did not aim at the objective
of constructing and evaluating complete models of macroeconomic activity.
Second, some researchers virtually abandoned the development of dynamic
rational expectations models as part of their applied work. This rejectionist
approach took hold most strongly in Cambridge, Massachusetts. One of the
most sophisticated early applications of rational expectations methods is
Blanchard (1983), but this researcher’s applied work moved from routinely
using the dynamic rational expectations models to the polar alternative, direct
behavioral specifications that typically lack any expectational elements.13 The
final approach, quantitative theory, is the topic considered next.
5.
THE RISE OF QUANTITATIVE THEORY
In the two seminal theoretical papers on real business cycles, Kydland and Prescott’s (1982) “Time to Build and Aggregate Fluctuations” and Long and
Plosser’s (1983) “Real Business Cycles,” each pair of authors faced the following problem. They had constructed rich dynamic macroeconomic models
driven by “technology shocks” and wanted to illustrate the implications of their
models for the nature of economic fluctuations. Each set of authors sought to
use parameters drawn from other sources: data on the input/output structure
of industries in Long and Plosser and data on various shares and elasticities
in Kydland and Prescott, with the latter pair of authors particularly seeking
to utilize information from microeconomic studies.14 One motivation for this
strategy was to make clear that the models were not being “rigged” to generate
fluctuations, for example, by fitting parameters to best match the business cycle
components of macroeconomic time series.
13 Recently, a small group of researchers has moved to a modified usage and interpretation
of the Hansen-Sargent methodology. An early example along these lines is Christiano (1988). In
recent work, Leeper and Sims (1994) use maximum likelihood methods to estimate parameters,
but supplement likelihood ratio tests with many other model diagnostics. Potentially, promising
technical developments like those in Chow (1993) may make it possible to execute the HansenSargent program for interesting models in the future.
14 In one case, in which they were least sure about the parameter value—a parameter indicating the extent of time nonseparability in utility flows from work effort, which determines the
sensitivity of labor supply to temporary wage changes—Kydland and Prescott explored a range
of values and traced the consequences for model implications.
R. G. King: Quantitative Theory and Econometrics
75
Calibration of Parameters
This process is now called the “calibration” approach to model parameter
selection.15 In line with the definition above, these papers were “quantitative
theory”: they provided a strong case for the general mechanisms stressed by the
authors. That is, they showed that the theory—restricted in plausible ways—
could produce outcomes that appeared empirically relevant. These models were
notably interesting precisely because they were evidently “post-Lucas critique”
models: expectations were determined rationally about the future productivity.
While they did not contain policy rules or policy disturbances, it was clear that
these extensions would be feasible, and similar models have since incorporated
policy anticipations, particularly on the fiscal side.
In terms of parameter selection, Lucas (1980) set forth a cogent argument
for the use of estimates from microeconomic data within his articulation of the
quantitative theory approach in his “Methods and Problems in Business Cycle
Theory,” albeit in a slightly different context than that so far discussed:
In the case of the equilibrium account of wage and employment determination,
parameters describing the degree of intertemporal substitutability do the job
(in an empirical sense) of the parameter describing auctioneer behavior in the
Phillips curve model. On these parameters, we have a wealth of inexpensively
available data from census cohort information, from panel data describing the
reactions of individual households to a variety of changing market conditions,
and so forth. In principle (and perhaps before too long in practice . . .) these
crucial parameters can be estimated independently from individual as well as
aggregate data. If so, we will know what the aggregate parameters mean, we
will understand them in a sense that disequilibrium adjustment parameters will
never be understood. This is exactly why we care about the “microeconomic
foundations” of aggregate theories.
(P. 712)
Thus, one feature of the calibration of a model is that it brings to bear available
microeconomic evidence. But Lucas also indicated the value of comparing estimates obtained from microeconomic studies and aggregate time series evidence,
so that calibration from such sources is simply one of several useful approaches
to parameter selection.
Evaluating a Calibrated Model
The evaluation of the calibrated models by Kydland and Prescott (1982) and
Long and Plosser (1983) was based on whether the models could capture some
15 The calibration approach is now mainly associated with work on real business cycles,
but it was simply a common strategy when used by Kydland and Prescott (1982) and Long and
Plosser (1983). For example, Blanchard’s (1980) work on dynamic rational expectations models
with nominal rigidities also utilized the calibration approach to explore the implications of a very
different class of models.
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key features of economic fluctuations that the authors were seeking to explain.
This evaluation was conducted outside of an explicit econometric framework.
In justifying this choice, Kydland and Prescott argued:
We choose not to test our model against the less restrictive vector autoregressive model. This would most likely have resulted in the model being rejected
given the measurement problems and the abstract nature of the model. Our
approach is to focus on certain statistics for which the noise introduced by
approximation and measurement errors is likely to be small relative to the
statistic. Failure of the theory to mimic the behavior of the post-war U.S.
economy with respect to these stable statistics with high signal to noise ratios
would be grounds for its rejection.
(P. 1360)
Their argument, then, was essentially that a model needs to be able to capture
some first-order features of the macroeconomic data before it can be taken seriously as a theory of business fluctuations. Their conclusions incorporated three
further clarifications of the role of quantitative theory. First, they concluded that
their model had some success: the “results indicate a surprisingly good fit in
light of the model’s simplicity” (p. 1368). Second, they articulated a program
of modifications of the theory that were important, many of which have been
the subject of their subsequent research activities, including introduction of a
variable workweek for capital. Third, they argued that applications of currently
prevailing econometric methods were inappropriate for the current stage of
model development: “in spite of the considerable recent advances made by
Hansen and Sargent, further advances are necessary before formal econometric
methods can fruitfully be applied to testing this theory of aggregate fluctuations”
(p. 1369).
In Long and Plosser’s (1983) study of the response of a multi-sector
macroeconomic model to fluctuations in productivity, the focus was similarly
on a subset of the model’s empirical implications. Indeed, since their explicit
dynamic equilibrium solution meant that the aggregate and industry allocations
of labor input were constant over time, these authors did not focus on the
interaction of productivity and labor input, which had been a central concern
of Kydland and Prescott. Instead, Long and Plosser highlighted the role of
output interrelationships arising from produced inputs for generating sectoral
comovement, a topic about which the highly aggregated Kydland and Prescott
theory had been silent.
Formal Econometric Analysis of RBC Models
Variants of the basic RBC model were evaluated by Altug (1989) and Christiano (1988) using formal econometric techniques that were closely related to
those of Hansen and Sargent (1981). There were three major outcomes of these
analyses. First, predictably, the economic models performed poorly: at least
some key parameter estimates typically strayed far from the values employed
R. G. King: Quantitative Theory and Econometrics
77
in calibration studies and the models were decisively rejected as constrained
vector autoregressions. Second, the Altug and Christiano studies did not naturally lead to new research, using either the methods of quantitative theory or
econometrics, that aimed at resolving specific puzzles that arose in their work.
Third, the rejectionist position of Kydland and Prescott hardened.
In particular, Kydland and Prescott called into question all econometric
evidence. Prescott (1986) argued that “we do not follow the [econometric]
approach and treat the [leisure share] as a free parameter because it would
violate the principle that parameters cannot be specific to the phenomena being
studied. What sort of a science would economics be if micro studies used
one share parameter and aggregate studies another?” (p. 25) More recently,
Kydland and Prescott (1991) argued that current econometrics is not faithful
to the objectives of its originators (notably Frisch), which they interpret as
being the construction of calibrated models. In the conclusion to their paper,
they argued “econometrics is by definition quantitative economic theory—that
is, economic analysis that provides quantitative answers to clear-cut questions”
(p. 176). Thus, macroeconomic research has bifurcated, with a growing number
of researchers using calibrated economic models and a very small number using
the methods of Hansen and Sargent (1981).
6.
COMPARING THE METHODOLOGIES
By its very nature, quantitative theory strays onto the turf of theoretical and applied econometrics since it seeks to use economic models to organize economic
data. Thus there has been substantial—at times heated—controversy about the
methods and conclusions of each area. Some of this controversy is a natural
part of the way that economists learn. In this regard, it is frequently the case
that applied econometrics poses challenges to quantitative theory: for example,
McCallum (1989) marshals the results of various applied econometric studies
to challenge RBC interpretations of the Solow residual as a technology shock.
However, controversy over methods can sometimes interfere with accumulation of knowledge. For example, adherents of RBC models have sometimes
suggested that any study based on formal econometric methods is unlikely
to yield useful information about the business cycle. Econometricians have
similarly suggested that one learns little from investigations using the methods of quantitative theory. For this reason, it is important to look critically at
the differences that separate studies using the methods of quantitative theory
from those that use the more familiar methods of econometrics. As we
shall see, a key strength of the quantitative theory approach is that it permits
the researcher to focus the evaluation of a model on a specific subset of its
empirical implications.
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Parameter Selection and Model Evaluation
To look critically at the methodological differences, we will find it useful to
break the activities of econometricians into two general topics: selection of parameters (estimation) and evaluation of economic models (including testing of
hypotheses and computation of measures of fit). Theoretical econometricians
work to devise procedures for conducting each of these activities. Applied
econometricians utilize these procedures in the context of specific economic
problems and interpret the results. For concreteness in the discussion below,
let the vector of parameters be β and the vector of model implications be µ.
Solving a model involves constructing a function,
µ = g(β),
(4)
that indicates how model implications are related to parameters. The business
of econometrics, then, is to determine ways of estimating the parameters β and
evaluating whether the implications µ are reliably close to some empirical counterparts m. In the approach of Hansen and Sargent (1981), the economic model
implications computed are a set of coefficients in a reduced-form vector autoregression. There are typically many more of these than there are parameters of
the model (elements of β), so that the theoretical model is heavily overidentified
in the sense of Hood and Koopmans (1953). The empirical counterparts m are
the coefficients of an unconstrained vector autoregression. The estimation of
the β parameters thus involves choosing the model parameters so as to maximize the fit of a constrained vector autoregression; model evaluation involves
a comparison of this fit with that of an unconstrained time series model.
In the studies in quantitative theory reviewed above, there were also analogous parameter selection and model evaluation activities. Taking the first study
as an example, in Douglas’s production function, Y = AN α K 1−α , there were
the parameters A and α, which he estimated via least squares. The model
implications that he explored included the time series observations on Y and
the values of α derived from earlier studies of factor income shares.
Quantitative theory, then, involves selection of values of the parameters
β and the comparison of some model implications µ with some empirical
counterparts m, which results in an evaluation of the model. Thus, it shares a
formal structure with econometric research. This identification of a common
structure is important for three reasons. First, while frequently suggested to
be dramatic and irreconcilable, the differences in the two methods must be,
at least on some level, ones of degree rather than kind. Second, this common
structure also indicates why the two methods have been substitutes in research
activity. Third, the common structure indicates the potential for approaches that
combine the best attributes of quantitative theory and econometrics.
However, it is notable that quantitative theory, as an abstraction of reality,
nearly always delivers a probability model that is remarkably detailed in its
R. G. King: Quantitative Theory and Econometrics
79
implications and, hence, is too simple for direct econometric implementation.
In the Douglas-Solow setting, the production function is essentially free of
error terms as it is written down: it is equally applicable at seasonal, cyclical,
and secular frequencies. But it is clear that Douglas and Solow did not view
the production function as exact. Instead, they had strong views about what
components of the data were likely to be most informative about aspects of the
production function; in his own way, each concentrated on the trend behavior
of output and factor inputs. The model driven solely by a productivity shock
displays another form of exactness, which is sometimes called a “stochastic
singularity.” There are not enough error terms for the joint probability distribution of the model’s outcomes to be nondegenerate. Since the model is dynamic,
this does not mean that all the variables are perfectly correlated, but it does
mean that there is (1) a singular variance-covariance matrix for the innovations
in the model’s reduced form and (2) a unit coherence between variables in the
frequency domain.
The key implication of this essential simplicity is that there is always some
way to reject econometrically a quantitative theory with certainty. Quantitative
theorists have had to come to terms with this simplicity on a number of fronts.
First, when we learn from a quantitative theory, it is not because it is literally
true but because there is a surprising magnitude of the gap between model implications µ and empirical counterparts m. Sometimes the gap is surprisingly
small, as in the work of Douglas and Prescott, and at others it is surprisingly
large, as in the work of Solow. Second, when we investigate the implications
of a quantitative theory, we must select a subset of implications of particular
interest. Thus, for example, Douglas and Solow focused mainly on asking
whether the production function explained the trend variation in output. By
contrast, Prescott’s concern was with business cycle variations, that is, with
deviations from trend.
Calibration Versus Estimation
The central tension between practitioners of quantitative theory and econometrics is sometimes described as the choice between whether parameters are to
be estimated or calibrated. However, returning to the three examples, we see
clearly that this cannot be the case. Douglas estimated his parameter α from
observations on Y, K, and N: he then compared this estimated value to averages of real labor cost as a fraction of manufacturing output, i.e., labor’s share.
Solow “calibrated” his labor share parameter from data on labor income as a
fraction of total income.16 Prescott’s construction of an RBC model involved
16 As noted above, Solow’s calibration is at each point in the sample, but this article’s version
of it uses the average value of labor’s share over the entire sample. The difference between these
two procedures is quantitatively unimportant.
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estimating the parameters of the stochastic productivity process from aggregate
time series data and calibrating other parameters of the model.
Further, applications of the Hansen-Sargent methodology to dynamic
macroeconomic models typically involved “fixing” some of the parameters at
specified values, rather than estimating all of them. For example, the discount
factor in dynamic models proved notoriously hard to estimate so that it was
typically set at a reasonable (calibrated) value. More generally, in such studies,
maximizing the likelihood function in even small-scale dynamic models proved
to be a painfully slow procedure because of the system nature of the estimation.
Many researchers resorted to the procedure of fixing as many parameters as
possible to increase the pace of improvement and the time to convergence.
Thus, almost all studies in quantitative theory and macroeconometrics involve a mixture of estimation and calibration as methods of parameter selection.
The main issue, consequently, is: How should we evaluate models that we
know are major simplifications of reality? In this context, it is not enough to
compare the likelihoods of a heavily restricted linear time series model (for
example, an RBC model with some or all of its parameters estimated) to an
unrestricted time series model. For some time, the outcome of this procedure
will be known in advance of the test: the probability that the model is true is
zero for stochastically singular models and nearly zero for all other models of
interest.
Model Evaluation in Quantitative Theory
Implicitly, the three exercises in quantitative theory reviewed above each directed our attention to a small number of features of reality, i.e., a small vector
of empirical features m that are counterparts to a subset of the model’s implications µ = g(β). In Douglas’s and Prescott’s case, the quantitative theory was
judged informative when there was a small gap between the predictions µ and
the empirical features m. In Solow’s case, there was a strikingly large gap for
a production function without technical progress, so we were led to reject that
model in favor of an alternative that incorporates technical progress.
Looking at the studies in more detail clearly shows that for Douglas, there
were two key findings. First, it was instructive that variation in factors of
production, when combined into Ŷ, explained so much of the evolution of
output Y. Second, it was striking that the time series of real wages was close
to output per unit of labor input multiplied by α̂ (i.e., that the value of labor’s
share obtained was close to the estimate α̂ obtained from the time series).
This latter finding is an example of Lucas’s reconciliation of aggregate and
micro evidence: we understand the parameter α better because it represents
a consistent pattern of behavioral response in micro and aggregate settings.
For Solow, it was instructive that the variation in factors of production explained so little of the evolution of output; thus, there was a large amount
of growth attributed to technical progress. Kydland and Prescott (1991) noted
R. G. King: Quantitative Theory and Econometrics
81
that “contrary to what virtually everyone thought, including the authors, . . .
technology shocks were found to be an important contributor to business cycle
fluctuations in the U.S. postwar period” (p. 176). Prescott (1986) provided a
more detailed listing of successes: “Standard theory . . . correctly predicts the
amplitude of these fluctuations, their serial correlation properties, and the fact
that the investment component of output is about six times as volatile as the
consumption component” (p. 11).
For these researchers and for many others, the key idea is that one learns
from an investigation in quantitative theory when there are relatively small or
large discrepancies between a set of empirical features m and model implications µ = g(β). A key strength of the quantitative theory approach is that it
permits the researcher to specify the elements of m, focusing the evaluation of
the model on a specific subset of its empirical implications.
Limitations of the Quantitative Theory Approach
However, there are also three important limitations of the method of model
evaluation in quantitative theory, which may be illustrated by posing some
questions about the outcomes of an exercise in quantitative theory.
First, as Singleton (1988) observed, quantitative theory does not provide
information about the confidence that one should have in the outcomes of an
investigation. For example, when we look at a specific implication µ1 and its
empirical counterpart m1 , how likely is it that a small value of m1 − µ1 would
occur given the amount of uncertainty that we have about our estimates of β
and m1 ? When we are looking at a vector of discrepancies m − µ, the problem
is compounded. It is then important to be explicit about the joint uncertainty
concerning β and µ, and it is also important to specify how to weight these
different discrepancies in evaluating the theory.17
Second, each of the quantitative theory investigations discussed above
spawned a major area of research. In the follow-up research, many investigations adopted the quantitative theory methods of the original investigation.
This process of model refinement thus raises an additional question: When
can we say with confidence that model A represents an improvement over
model B?
17 Particularly in the area of real business cycles, some investigations in quantitative theory
follow Kydland and Prescott (1982) by providing “standard deviations” of model moments. These
statistics are computed by simulating the calibrated model over a specified sample period (say,
160 quarters) so as to determine the approximate finite sample distribution of the model moments
under the assumption that the model is exactly true. Such statistics cannot be used to answer the
question posed in the text since they do not take into account the joint distribution of the parameters β and the empirical implications m. Instead, these statistics indicate how much uncertainty
in sample moments is introduced by sample size if the theory were literally true, including use
of the exact parameter values.
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Third, if a strength of the quantitative theory approach is that one looks
at a subset of model implications of the model economy, is this not also a
limitation? In particular, when we look at a subset of model implications, this
leaves open the possibility that there are also other, interesting implications of
the model that are not examined.
Operating independently of econometrics, quantitative theory gives us no
ability even to pose, much less answer, these three key questions.
7.
REEVALUATING THE STANDARD RBC MODEL
The issues of model evaluation raised above are not debating points: absent
some systematic procedures for evaluating models, one may regularly miss
core features of reality because the “tests” are uninformative. To document this
claim, we will look critically at two features of RBC models, which many—
beginning with Prescott (1986) and Kydland and Prescott (1991)—have argued
are supportive of the theory. These two features are summarized by Prescott
(1986) as follows: “Standard theory . . . correctly predicts the amplitude of
. . . fluctuations [and] their serial correlation properties” (p. 11).
The first of these features is typically documented in many RBC studies
by computing the “variance ratio”:
λ=
var(log Ym )
.
var(log Yd )
(5)
In this expression, var(log Ym ) is the variance of the logarithm of the model’s
output measure, and var(log Yd ) is the variance of the corresponding empirical
measure.18 In terms of the series displayed in Figure 4, in particular, the variance of log Ym is 2.25 and the variance of log Yd is 2.87, so that the implied
value of λ = 0.78. That is, Kydland and Prescott (1991) would say that the
baseline RBC model explains 78 percent of business fluctuations.
The second of these features is typically documented by looking at a list
of autocorrelations of model time series and corresponding time series from
the U.S. economy. Most typically, researchers focus on a small number of loworder autocorrelations as, for example, in Kydland and Prescott (1982) and
King, Plosser, and Rebelo (1988).
Parameter Uncertainty and the Variance Ratio
In a provocative recent contribution, Eichenbaum (1991) argued that economists
know essentially nothing about the value of λ because of parameter uncertainty.
18 In
particular, these constructs are 100 times the Hodrick-Prescott filtered logarithms of
model and data output, so that they are interpretable as percentage deviations from trend.
R. G. King: Quantitative Theory and Econometrics
83
He focused attention on uncertainty about the driving process for technology and, in particular, about the estimated values of the two parameters that
are taken to describe it by Prescott (1986) and others. The core elements of
Eichenbaum’s argument were as follows. First, many RBC studies estimate the
parameters of a low-order autoregression for the technology-driving process,
specifying it as log At = ρ log At−1 + εt and estimating the parameters (ρ, σε2 )
by ordinary least squares. Second, following the standard quantitative theory
approach, Eichenbaum solved the RBC model using a parameter vector β̂ that
contains a set of calibrated values β1 and the two estimates, β̂2 = [ρ̂ σ̂ε2 ].
Using this model solution, Eichenbaum determined that the population value
of λ(β̂) is 0.78 (which is identical to the sample estimate taken from Figure
4). Third, Eichenbaum noted that estimation uncertainty concerning β̂2 means
there is substantial uncertainty about the implications that the model has for
λ: the standard error of λ from these two sources alone is huge, about 0.64.
Further, he computed the 95 percent confidence interval for λ as covering the
range from 0.05 to 2.19 Eichenbaum’s conclusion was that there is a great deal
of uncertainty over sources of business cycles, which is not displayed in the
large number of quantitative theory studies that compute the λ measure.
The Importance of Comparing Models
There is an old saying that “it takes a model to beat a model.”20 Overall, one
reason that quantitative theory has grown at the expense of econometrics is
that it offers a way to systematically develop models: one starts with a benchmark approach and then seeks to evaluate the quantitative importance of a new
wrinkle. However, when that approach is applied to the basic RBC model’s
variance ratio, it casts some doubt on the standard, optimistic interpretation of
that measure.
To see why, recall that the derivation of the productivity residual implies
log Ydt = log Adt + α log Ndt + (1 − α) log Kdt ,
where the subscript d indicates that this is the version of the expression
applicable to the actual data. In the model economy, the comparable construction is
log Ymt = log Amt + α log Nmt + (1 − α) log Kmt ,
19 There are some subtle statistical issues associated with the computation of this confidence
interval, as Mary Finn and Adrian Pagan have pointed out to the author. In particular, the point
estimate in Eichenbaum [1991] for ρ is 0.986, thus suggesting that the finite sample approximate
confidence interval on ρ̂ is both wide and asymmetric because of considerations familiar from the
analysis of “near unit root” behavior. Pagan indicates that taking careful account of this would
shrink Eichenbaum’s confidence interval so that it had a lower bound for λ of 0.40 rather than
0.05.
20 Both Thomas Sargent and Robert Barro have attributed this saying to Zvi Griliches.
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Federal Reserve Bank of Richmond Economic Quarterly
where the subscript m indicates that this is the model version. In the simulations
underlying Figure 4, the model and data versions of the technology shocks are
set equal. Thus, the difference of output in the data from that in the model is
log Ydt − log Ymt = α(log Ndt − log Nmt ) + (1 − α)(log Kdt − log Kmt ).
(6)
That is, by construction, deviations between the data and the model arise only
when the model’s measures of labor input or capital input depart from the
actual measures.
In terms of the variance ratio λ, this means that we are giving the model
credit for log At in terms of explaining output. This is a very substantial asset:
in Panel A of Figure 3, the Solow residual is highly procyclical (strongly
correlated with output) and has substantial volatility.
Now, to take an extreme stand, suppose that one’s benchmark model was
simply that labor and capital were constant over the business cycle: log Ymt =
log At . This naive model has a value of λ = 0.49: the “Solow residual” alone
explains about half the variability of output. Thus, the marginal contribution of
variation in the endogenous mechanisms of the model—labor and capital—to
the value of λ = 0.78 is only λ − λ = 0.29. From this standpoint, the success
of the basic RBC model looks much less dramatic, and it does so precisely
because two models are compared.
This finding also illustrates the concerns, expressed by McCallum (1989)
and others, that RBC models may be mistakenly attributing to technology
shocks variations in output that arise for other reasons. It is indeed the case
that RBC models explain output very well precisely because they utilize output (productivity) shifts as an explanatory variable, rather than inducing large
responses to small productivity shifts.
From this standpoint, the major success of RBC models cannot be that
they produce high λ values; instead, it is that they provide a good account
of the relative cyclical amplitude of consumption and investment. Perhaps
paradoxically, this more modest statement of the accomplishments of RBC
analysis suggests that studies like those of Prescott (1986) and Plosser (1989)
are more rather than less important for research in macroeconomics. That is,
these studies suggest that the neoclassical mechanisms governing consumption
and investment in RBC economies are likely to be important for any theory of
the cycle independent of the initiating mechanisms.
Looking Broadly at Model Implications
Many RBC studies including those of Prescott (1986) and King, Plosser, and
Rebelo (1988) report information on the volatility and serial correlation properties of real output in models and in U.S. data. Typically, the authors of these
studies report significant success in the ability of the model to match the empirical time series properties of real output. However, a recent study by Watson
R. G. King: Quantitative Theory and Econometrics
85
(1993), which is an interesting blend of quantitative theory and econometrics,
questions this conclusion in a powerful manner.
On the econometric side, Watson estimated the time series of productivity
shocks, working with two key assumptions about a baseline RBC model drawn
from King, Plosser, and Rebelo (1988). First, he assumed that the model correctly specifies the form of the driving process, which is taken to be a random
walk with given drift and innovation variance. Second, he assumed that the
realizations of productivity shocks are chosen to maximize the fit of the model
to the U.S. data. By doing so, he gave the basic RBC model the best possible
chance to match the main features of the business cycle.21 The model did relatively poorly when Watson placed all weight on explaining output variability:
the model could explain at most 48 percent of variance in output growth and
57 percent of variance in Hodrick-Prescott filtered data.
On the quantitative theory side, Watson traced this conclusion to a simple
fact about the spectrum of output growth in the basic RBC model and in the
U.S. data. A version of Watson’s result is illustrated in Figure 5, which displays
two important pieces of information. To begin to interpret this figure, recall that
the power spectrum provides a decomposition of variance by frequency: it is
based on dividing up the growth in output into periodic components that are
mutually uncorrelated. For example, the fact that the data’s power spectrum in
Figure 5 has greater height at eight cycles per period than at four cycles per
period means that there is greater variation in the part of output growth that
involves two-year cycles than one-year cycles. The general shape of the power
spectrum in Figure 5 thus indicates that there is a great deal of variability
in output growth at the business cycle frequencies defined in the tradition of
Burns and Mitchell (1946). These cyclical components, with durations between
eight years and eighteen months, lie between the vertical lines in the figure.
In addition, since the power spectrum is the Fourier transform of the autocovariance generating function, it displays the same set of information as the full
set of autocorrelations. In terms of the autocovariances, the key point is that
the overall shape of the power spectrum suggests that there is substantial predictability to output growth.22 Further, the empirical power spectrum’s shape for
21 This fitting exercise is one that is easy to undertake because Watson poses it as a simple
minimization problem in the frequency domain.
22 Previously, the empirical power spectrum’s shape, or, equivalently, the pattern of autocorrelations of output growth, has been important for the empirical literature on the importance
of stochastic trends, as in Cochrane (1988) and Watson (1986). This literature has stressed that
it would be necessary to have a relatively lengthy univariate autoregression in order to fit the
spectrum’s shape well. Such a long autoregression could thus capture the shape with one or more
positive coefficients at low lags (to capture the initial increase in the power spectrum as one
moves from high to medium frequencies) and then many negative ones (to permit the subsequent
decline in the spectrum as one moves from medium to very low frequencies). However, short
autoregressions would likely include only the positive coefficients.
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 5
Contribution to Variance (Power)
A. Basic Neoclassical Model
Data
Model
0
32
6
2
Cycles per Period
Contribution to Variance (Power)
B. Time-to-Build Model
Data
Model
0
32
6
4
2
Cycles per Period
+
Notes: Panels A and B contain the estimated power spectrum of the growth rate of quarterly
U.S. gross national product. The power spectrum summarizes the decomposition of this growth
rate series into periodic components of varying duration. Business cycle components (defined as
cycles of length between 6 and 32 quarters) lie between the vertical lines in the figure. Since the
total variance of the growth rate is proportional to the area under the power spectrum, this shape
of the estimate indicates that a great deal of variability in growth rates occurs at the business
cycle frequencies.
Panel A also contains the power spectrum of output growth in a basic real business cycle
model if it is driven by random-walk shocks to productivity. Panel B displays the power spectrum
for a model that contains the time-to-build investment technology.
R. G. King: Quantitative Theory and Econometrics
87
output is also a “typical spectral shape for growth rates” of quarterly U.S.
time series. King and Watson (1994) showed that a similar shape is displayed
by consumption, investment, output, man-hours, money, nominal prices, and
nominal wages; it is thus a major stylized fact of business cycles.
Yet, as shown in Panel A of Figure 5, the basic RBC model does not come
close to capturing the shape of the power spectrum of output growth, i.e., it
does not generate much predictable output growth. When driven by randomwalk productivity shocks, the basic model’s spectrum involves no peak at the
business cycle frequencies; it has the counterfactual implication that there is
greater variability at very low frequencies than at business cycle frequencies.23
A natural immediate reaction is that the model studied in King, Plosser, and
Rebelo (1988) is just too simple: it is a single state variable version of the RBC
model in which the physical capital stock is the only propagation mechanism.
One might naturally conjecture that this simplicity is central for the discrepancy
highlighted in Panel A of Figure 5. However, Panel B of Figure 5 shows that the
introduction of time-to-build investment technology as in Kydland and Prescott
(1982) does not alter the result much. There is a slight “bump” in the model’s
spectrum at four quarters since there is a four-quarter time-to-build, but little
change in the general nature of the discrepancies between model and data.
Overall, there is an evident challenge of Watson’s (1993) plot: by expressing
the deficiencies of the standard RBC theory in a simple and transparent way,
this exercise in quantitative theory and econometrics invites development of
a new class of models that can capture the “typical spectral shape of growth
rates.”24
8.
CHALLENGES TO ECONOMETRICS
The growth of business cycle analysis using the quantitative theory approach
has arisen because there is a set of generally agreed-upon procedures that makes
feasible an “adaptive modeling strategy.” That is, a researcher can look at a
set of existing models, understand how and why they work, determine a new
line of inquiry to be pursued, and evaluate new results. The main challenge to
econometrics is to devise ways to mimic quantitative theory’s power in systematic model development, while adding the discipline of statistical inference
and of replicability.
23 There is also a sense in which the overall height of the spectrum is too low, which may
be altered by raising or lowering the variance of the technology shock in the specification. Such
a perturbation simply shifts the overall height of the model’s spectrum by the same proportion at
every frequency.
24 Rotemberg and Woodford (1994) demonstrated that a multivariate time series model displays a substantial ability to predict output growth and also argued that the basic RBC model
cannot capture this set of facts. Their demonstration was thus the time-domain analogue to Watson’s findings.
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Indeed, development of such new econometric methods is essential for
modern macroeconomic analysis. At present, it is the last stage of the quantitative theory approach that is frequently most controversial, i.e., the evaluation
of new results. This controversy arises for two reasons: (1) lack of generally
agreed-upon criteria and (2) lack of information on the statistical significance
of new results. On the former front, it seems impossible to produce a uniform
practice, and it is perhaps undesirable to try to do so: the criteria by which
results are evaluated has always been part of the “art” of econometrics. But on
the latter, we can make progress. Indeed, short of such developments, research
in modern business cycle analysis is likely to follow the path of the NBER
research that utilized the Burns and Mitchell (1946) strategy of quantitative
business cycle analysis: it will become increasingly judgmental—leading to difficulties in replication and communication of results—and increasingly isolated.
As suggested earlier, the main challenge for econometric theory is to derive
procedures that can be used when we know ex ante that the model or models
under study are badly incomplete. There are some promising initial efforts
under way that will be referred to here as the “Northwestern approach,” whose
core features will now be summarized. There are two stages in this procedure:
parameter estimation and model evaluation.
Parameter Estimation
The strength of the quantitative theory approach in terms of the selection of
parameters is that it is transparent: it is relatively easy to determine which
features of the real-world data are important for determining the value of a
parameter used in constructing the model economy.
The estimation strategy of the Northwestern approach is similar to quantitative theory in this regard and in sharp contrast to the Hansen and Sargent (1981)
approach. Rather than relying on the complete model to select parameters, it
advocates using a subset of the model’s empirical implications to estimate
parameters. It consequently makes transparent which features of the data are
responsible for the resulting parameter estimates.
Essentially, these chosen features of the model are used for the measurement of parameters, in a manner broadly consistent with the quantitative theory
approach. However, and importantly, the Northwestern approach provides an
estimate of the variance-covariance matrix of parameter estimates, so that
there is information on the extent of parameter uncertainty. To take a specific example, in building dynamic macroeconomic models incorporating an
aggregate production function, many researchers use an estimate of labor’s
share to determine the value of α in the production function, appealing to
the work of Solow. This estimate is sometimes described as a first-moment
estimate, since it depends simply on the sample average labor’s share. Further,
since observations on period-by-period labor’s share are serially correlated,
R. G. King: Quantitative Theory and Econometrics
89
obtaining an estimate of the amount of uncertainty about α requires that one
adopt a “generalized least squares” procedure such as Hansen’s (1982) generalized method of moments. The procedure is remarkably transparent: the
definitional and statistical characteristics of labor income and national income
dictate the estimate of the parameter α.
In a core paper in the Northwestern approach, Christiano and Eichenbaum
(1992) demonstrated how a subset of model relations may be used to estimate a vector of model parameters with Hansen’s (1982) method in a manner
like that used for α in the discussion above.25 In this study, some of the parameters are determined from steady-state relations, so that they are naturally
first-moment estimators, like the labor share and other long-run parameters
which are “calibrated” in Prescott’s (1986) study. Other parameters describe
aspects of the model’s exogenous dynamics, including driving processes for
productivity and government purchases. These parameter selections necessarily
involve consideration of second moments, as in Eichenbaum’s (1991) estimate
of the productivity process discussed previously.
More generally, second-moment estimates could also be used for internal
elements of the model. For example, an alternative transparent approach to
estimating the production function parameter α is to use the joint behavior of
output, capital, and labor as in Cobb and Douglas (1928). With this alternative
method, aspects of the trends in output, capital, and labor would be the core
features that determined the estimate of α, as stressed above. However, given
the recent focus on technology shocks and the endogenous response of capital
and labor to these disturbances, one would presumably not follow Douglas’s
ordinary least squares approach. Instead, one would employ a set of instrumental variables suggested by the structure of the model economy under study.
For example, in Christiano and Eichenbaum’s setting, one would likely employ
government purchases as an instrumental variable for the production function
estimation. In this case, the estimate of α would depend on the comovements
of government purchases with output, capital, and labor.
Thus, in the case of α, a researcher has latitude to determine the core features of the model that are used to estimate the parameter of interest: either firstor second-moment estimators are available. However, with some alternative
structural elements of dynamic macroeconomic models, only second-moment
estimators are available since the parameters govern intrinsically dynamic
elements of the model. For example, absent information on the distribution of
investment expenditure over the life of investment projects, there is simply no
way to determine the parameters of the “time-to-build” technology of Kydland
and Prescott (1982) from steady-state information. However, second-moment
25 In
an early appraisal of the relationship between quantitative theory and econometrics,
Singleton (1988) forecasted some of the developments in Christiano and Eichenbaum (1992).
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Federal Reserve Bank of Richmond Economic Quarterly
estimates of a dynamic investment function could readily recover the necessary
time-to-build parameters.
Overall, the parameter selection component of the Northwestern approach
is best viewed as an application of the instrumental variables methods of McCallum (1976) and Hansen and Singleton (1982). A key feature, shared with
quantitative theory and in contrast to the typical application of the HansenSargent (1981) strategy, is that the complete model is not used for parameter
estimation. Rather, since a carefully chosen subset of the model’s relations is
used, it is relatively easy to understand which features of the data are important for results of parameter estimation. Yet, in contrast to standard techniques
in quantitative theory, there are explicit measures of the extent of parameter
uncertainty provided by the Northwestern method.
From this perspective, we know from prior theoretical and applied econometric work that instrumental variables estimates of structural parameters from
a subset of the model’s equations are no panacea. Minimally, one loses efficiency of parameter estimation relative to procedures like those of Hansen
and Sargent (1981), if the model is well specified. More generally, there will
always be potential for problems with poor instruments and inappropriate selection of the subset of relations that is chosen for estimation. However, at
the present stage of macroeconomic research, economists are far from having
a well-specified macroeconomic model. It seems best to opt for simple and
transparent procedures in the selection of model parameters.
Model Evaluation
A notable feature of quantitative theory is that a selected subset of model
implications are compared to their empirical counterparts. This method has its
benefits and costs. On the positive side, it permits the researcher to specify
a subset of implications that are viewed as first-order for the investigation.
This allows an individual researcher to focus on a manageable problem, which
is essential for research progress in any discipline. On the negative side, this
freedom may mean that the researcher may study model implications that are
not too informative about the economic model or models under study. However,
once methods are in place that allow for replication and criticism of the results
of studies, it is likely that competition among researchers will provide for
extensive exploration of the sensitivity of the results of the model evaluation
stage of research.
Like quantitative theory, the Northwestern model evaluation approach
permits the researcher to focus on a subset of model implications in evaluating
small-scale dynamic models. However, since it utilizes standard econometric
methodology, it provides diagnostic information about the extent of uncertainty that one has about gaps between model and empirical implications. Any
such evaluation of the discrepancies between a model’s implications and the
corresponding empirical features involves taking a stand on a penalty function
R. G. King: Quantitative Theory and Econometrics
91
to be applied to the discrepancies, i.e., on a function ∆ that assigns a scalar loss
to the vector of discrepancies (m − µ). If this penalty function is assumed to
be quadratic, then one must simply specify a matrix L that is used to penalize
discrepancies between the model’s implication and the corresponding empirical
features. That is, models may be evaluated using a discrepancy function like
∆(m, β) = [m − g(β)]L[m − g(β)] ,
(7)
whose statistical properties will depend on the choice of L.
Christiano and Eichenbaum (1992) took the null hypothesis to be that the
model is correctly specified and choose L to be the inverse of the variancecovariance matrix of [m − g(β)]. This permitted them to perform powerful
chi-squared tests of a subset of the model’s implications. Essentially, Christiano
and Eichenbaum (1992) asked: Can we reject the hypothesis that discrepancies
between the model’s implication and the corresponding empirical features are
due to sampling error?26 If the answer is yes, then the model is viewed as
deficient. If the answer is no, then the model is viewed as having successfully
produced the specified empirical features. While their paper focused on implications for selected second moments (variances and covariances of output,
etc.), their procedure could readily be applied to other implications such as
spectra or impulse responses.
King and Watson (1995) alternatively assumed the null hypothesis that
models are not correctly specified and require, instead, that the researcher
specify the discrepancy function L. In contrast to the procedures of Christiano and Eichenbaum (1992), their tests are thus of unknown power. While
they discussed how to use this assumed discrepancy function to construct tests
of the adequacy of an individual model, most of their attention was directed
to devising tests of the relative adequacy of two models. Suppose, for sake
of argument, that model A has a smaller discrepancy measure than model B.
Essentially, the question that they asked was: Can we reject the hypothesis
that a difference in discrepancy measures between two models A and B is
due to sampling error? If the answer is yes, then King and Watson would
say that model A better captures the specified list of empirical features. While
their paper focused on implications for selected impulse responses (comparative
dynamics), the procedure could also be applied to other implications such as
selected moments or spectra.
26 The
phrase sampling error in this question is a short-hand for the following. In a time
series context with a sample of length T, features of the macroeconomic data (such as the variance
of output, the covariance of output and labor input, etc.) are estimates of a population counterpart
and, hence, are subject to sampling uncertainty. Model parameters β estimated by generalized
method of moments are consistent estimates, if the model is correctly specified in terms of the
equations that are used to obtain these estimates, but these estimates are also subject to some
sampling error. Evaluation of discrepancies ∆ under the null hypothesis that the model is true
takes into account the variance-covariance matrix of m and β.
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Thus, the Northwestern approach involves two stages. In the first, the parameters are estimated using a subset of model relations. In the second, a set
of features of interest is specified; the magnitude of discrepancies between
the model’s implications and the corresponding empirical features is evaluated.
The approach broadly parallels that of quantitative theory, but it also provides
statistical information about the reliability of estimates of parameters and on
the congruence of the model economy with the actual one being studied.
It is too early to tell whether the Northwestern approach will prove broadly
useful in actual applications. With either strategy for model evaluation, there
is the potential for one of the pitfalls discussed in the previous section. For
example, by concentrating on a limited set of empirical features (a small set
of low-order autocovariances), a researcher would likely miss the main discrepancies between model and data arising from the “typical spectral shape of
growth rates.” Other approaches—perhaps along the lines of White (1982)—
may eventually dominate both. But the Northwestern approach at least provides
some econometric procedures that move in the direction of the challenges raised
by quantitative theory. The alternative versions of this approach promise to
permit systematic development of macroeconomic models, as in quantitative
theory, with the additional potential for replication and critique of the parameter
selection and model evaluation stages of research.
9.
OPPORTUNITIES FOR ECONOMETRICS
The approach of quantitative economic theory also provides opportunities for
econometrics: these are for learning about (1) the features of the data that are
most informative about particular parameters, especially dynamic ones, and (2)
how to organize data in exploratory empirical investigations. This section provides one example of each opportunity, drawing on the powerful intuition of the
permanent income theory of consumption (Friedman 1957). For our purposes,
the core features of Friedman’s theory of consumption are as follows. First,
consumption depends importantly on a measure of wealth, which Friedman
suggested contains expectations of future labor income as its dominant empirical component. Second, Friedman argued that it is the changes in expectations
about relatively permanent components of income that exert important wealth
effects and that it is these wealth effects that produce the main variations in
the level of an individual’s consumption path.
Identifying and Estimating Dynamic Parameters
A common problem in applied econometrics is that a researcher may have little
a priori idea about which features of the data are most likely to be informative
about certain parameters, particularly parameters that describe certain dynamic
responses. Quantitative economic theories can provide an important means of
R. G. King: Quantitative Theory and Econometrics
93
learning about this dependence: they can indicate whether economic responses
change sharply in response to parameter variation, which is necessary for precise estimation.
To illustrate this idea, we return to consideration of productivity and the
business cycle. As previously discussed, Eichenbaum (1991) suggested that
there is substantial uncertainty concerning the parameters that describe the
persistence of productivity shocks, for example, the parameter ρ in log At =
ρ log At−1 + . In his study, Eichenbaum reported a point estimate of ρ = 0.986
and a standard error of 0.026, so that his Solow residual is estimated to display
highly persistent behavior. Further, Eichenbaum concluded that this “substantial
uncertainty” about ρ translates into “enormous uncertainty” about the variance
ratio λ.
Looking at these issues from the standpoint of the permanent income
theory of consumption, we know that the ratio of the response of consumption
to income innovations changes very dramatically for values of ρ near unity. (For
example, the line of argument in Goodfriend [1992] shows that the response
of consumption to an income innovation is unity when ρ = 1, since actual and
permanent income change by the same amount in this case, and is about onehalf when ρ = 0.95).27 That is, it matters a great deal for individual economic
agents where the parameter ρ is located. The sensitivity of individual behavior
to the value of ρ carries over to general equilibrium settings like Eichenbaum’s
RBC model in a slightly modified form: the response of consumption to the
productivity shock t increases rapidly as ρ gets near unity. In fact, there are
sharp changes in individual behavior well inside the conventional confidence
band on Eichenbaum’s productivity estimate of ρ. The economic mechanism is
that stressed by the permanent income theory: wealth effects are much larger
when disturbances are more permanent than when they are transitory. Thus, the
parameter ρ cannot move too much without dramatically changing the model’s
implication for the comovement of consumption and output. More specifically,
the key point of this sensitivity is that a moment condition relating consumption
to technology shocks is plausibly a very useful basis for sharply estimating the
parameter ρ if the true ρ is near unity. Presumably, sharper estimates of ρ
would also moderate Eichenbaum’s conclusion that there is enormous uncertainty about the variance ratio λ.
27 The analytics are as follows: If income is a first-order autoregression with parameter ρ,
then the change in permanent income is proportional to the innovation in income with a coefficient:
r/(r + 1 − ρ). Thus, if ρ = 1, an innovation in income has a unit effect, and correspondingly,
if ρ = 0, then the effect of a change in income is simply the annuity factor r/(r + 1), which is
much closer to zero than to one. Even fairly persistent changes in income have much less than a
one-for-one effect: with r = 0.05 and ρ = 0.95, then it follows that the adjustment coefficient is
0.5.
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Organizing Economic Data
Another problem for applied researchers is to determine interesting dimensions
along which to organize descriptive empirical investigations. For example, there
have been many recent papers devoted to looking at how countries behave differentially in terms of their national business cycles, but relatively few of these
investigations provide much information on characteristics of countries that are
correlated with differences in national business cycles. There are some interesting and natural decompositions: for example, Baxter and Stockman (1989)
asked how national business cycles are related to a country’s exchange rate
regime and Kouparitsas (1994) asked how national business cycles are linked
to industrial structure and the composition of trade. But, beyond these immediate decompositions, how is a researcher to determine revealing ways of
organizing data? In the discussion that follows, we show how quantitative
theories can provide a vehicle for determining those dimensions along which
revealing organizations of economic data may be based.
To continue on this international theme, open economy RBC models such
as those of Backus, Kehoe, and Kydland (1992) and Baxter and Crucini (1993)
have the implication that there should be a near-perfect cross-country correlation of national consumptions. This striking implication is an artifact of two
features of these models. First, they incorporate the desire for consumption
smoothing on the part of the representative agent (as in Friedman’s permanent
income theory). Second, they allow for rich international asset trade, so that
agents in different countries can in fact purchase a great deal of risk sharing,
i.e., they are complete-markets models.
But cross-national correlations of consumption are in fact lower than crossnational correlations of output.28 A plausible conjecture is that incomplete asset
markets, particularly those for national human capital, are somehow responsible
for this gap between theory and reality. Thus, a natural research strategy would
be to try to measure the extent of access to the world capital market in various
countries and then to look cross-sectionally to see if this measure is related to
the extent of a country’s correlation with world consumption.
Quantitative theory suggests that this strategy, while natural, would miss
a central part of the problem. It thus suggests the importance of an alternative organization of the international economic data. In particular, Baxter and
Crucini (1995) investigated a two-country RBC model in which there can be
trade in bonds but no trade in contingent claims to physical and human capital.
They thus produced a two-country general equilibrium version of the permanent
income hypothesis suitable for restricted asset markets. Further, as suggested
by Hall’s (1978) random-walk theory of consumption, the restriction on asset
28 That
is, there is a smaller correlation of Hodrick-Prescott filtered output than of HodrickPrescott filtered consumption, as shown in Backus and Kehoe (1992).
R. G. King: Quantitative Theory and Econometrics
95
markets leads to a random-walk component of each nation’s economic activity,
which arises because shocks redistribute wealth between countries. When there
is low persistence of productivity shocks in a country, Baxter and Crucini
(1995) found that the restricted asset two-country model does not behave very
differently from its complete-markets counterpart. That is, trade in bonds offers
the country a great ability to smooth out income fluctuations; the wealth effects
associated with transitory shocks are small influences on consumption. Yet, if
the persistence of productivity shocks in a country is high, then there are major
departures from the complete-markets model and it is relatively easy for crossnational consumption correlations to be small (or even negative).
In this international business cycle example, quantitative theory thus suggests that a two-way classification of countries is important. One needs to
stratify both by a measure of the persistence of shocks and by the natural measure of access to the international capital market. Further, it also makes some
detailed predictions about how the results of this two-way classification should
occur, if restricted access is indeed an important determinant of the magnitude
of international consumption correlations. More generally, quantitative theory
can aid the applied researcher in determining those dimensions along which
descriptive empirical investigations can usefully be organized.29
10.
THE CHALLENGE OF IDENTIFICATION
To this point, our discussion has focused on evaluating a model in terms of its
implications for variability and comovement of prices and quantities; this has
come to be the standard practice in quantitative theory in macroeconomics. For
example, in Prescott’s (1986) analysis of productivity shocks, measured as in
Solow (1957), the basic neoclassical model successfully generated outcomes
that “looked like” actual business cycles, in terms of the observed cyclical
amplitude of consumption, investment, and labor input as well as the observed
comovement of these series with aggregate output. In making these comparisons, Prescott measured cyclical amplitude using the standard deviation of an
individual series, such as consumption, and cyclical comovement by the correlation of this series with output. That is, he evaluated the basic neoclassical
model in terms of its implications for selected second moments of consumption, investment, labor, and output. Further, in Watson’s (1993) analysis of
the same model reviewed above, the number of second moments examined
was expanded to include all of the autocorrelations of these four variables, by
plotting the spectra. However, in these discussions, there was relatively little
explicit discussion of the link between the driving variable—productivity—and
29 Prescott (1986) previously suggested that theory should guide measurement in macroeconomics. The example in the current section is thus a specific application of this suggestion,
applied to the organization of international economic data.
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the four endogenously determined variables of the model. In this section, we
take up some issues that arise when this alternative strategy is adopted, with
specific emphasis on evaluating models of business fluctuations.
Identifying Causes of Economic Fluctuations
Identification involves spelling out a detailed set of linkages between causes
and consequences. In particular, the researcher takes a stand on those economic
variables that are taken to exert a causal influence on economic activity; he
then studies the consequences that these “exogenous variables” have for the
dynamic evolution of the economy.
To implement this strategy empirically, the researcher must detail the measurement of causal variables. This might be as simple as providing a list of
economic variables assumed to be causal, such as productivity, money supply,
taxes or government deficits, and so on. But it might also be more elaborate,
involving some method of extracting causal variables from observed series.
For example, researchers examining the effects of government budget deficits
on macroeconomic activity have long adjusted the published deficit figures to
eliminate components that are clearly dependent on economic activity, creating
a “cyclically adjusted” or “full employment” deficit measure. Modern versions
of this identification strategy involve treating unobserved variations in productivity as the common stochastic trend in real consumption, investment, and
output (as in King, Plosser, Stock, and Watson [1991]) or extracting the unobserved policy-induced component of the money supply as that derived from
unpredictable shifts in the federal funds rate (as in Sims [1989]). This stage
of the analysis is necessarily open to debate; researchers may adopt alternative
identifications of causal variables.
Once these causal variables are determined, then they may be put to three
uses. First, one can employ them as the basis for estimation of behavioral
parameters, i.e., they may be used as instrumental variables. Second, one can
investigate how an actual economy reacts dynamically to shocks in these causal
variables, estimating “dynamic multipliers” rather than behavioral parameters.
Such estimates may serve as the basis for model evaluation, as discussed further
below. Third, one can consider the practical consequences of altering the historical evolution of these causal variables, conducting counterfactual experiments
that determine how the evolution of the economy would have been altered in
response to changes in the time path of one or more of the causal variables.
Identification in Quantitative Theory
Since they involve simple, abstract models and a small amount of economic
data, exercises in quantitative theory typically involve a very strong identification of causal factors. It is useful to begin by considering the identifications
implicit in the previously reviewed studies.
R. G. King: Quantitative Theory and Econometrics
97
For Douglas, the working hypothesis behind the idea of a “practical”
production function was that the factors of production, capital and labor, represented causal influences on the quantity of production: if a firm had more of
either or both, then output should increase. Estimates of production functions
along these lines are now a standard example in introductory econometrics
textbooks. Operating prior to the development of econometric tools for studying simultaneous-equations systems, Douglas simply used capital and labor as
independent variables to estimate the parameters of the production function. He
was not naive: as discussed above, he was concerned that his assumption of a
pure causal influence might not be correct, so he gave more weight to the results
suggested by the “trends” in capital, labor, and output than he did to results
suggested by shorter-term variation. A central objective of his investigation
was to be able to make “what if” statements about the practical implications of
varying the quantities of these inputs. Thus, he employed his identification for
precisely the first and third purposes discussed above. Viewed with more than
50 years of hindsight, many would now be led to question his identification: if
there is technical progress, then capital and labor will typically respond to this
factor. However, Douglas’s transparent identification led to striking results.
In Solow’s (1957) procedure, particularly as it has been used in RBC modeling, the nature of the identification is also transparent: output is transformed,
using data on factor shares and factor inputs, to reveal an unobserved component, productivity, to which a causal role may be assigned.30 In his investigation,
Solow focused mainly on asking the hypothetical question: What if there had
been capital growth, but no productivity growth, in the United States?
In the RBC analysis of Prescott (1986), the Solow productivity process
was used as the driving variable, with the objective of describing the nature of
macroeconomic fluctuations that arise from this causal factor. In line with our
third use of an identified causal disturbance, the type of historical simulation
produced in Section 2 above provides an implicit answer to the question: How
large would fluctuations in macroeconomic activity have been without fluctuations in productivity, as identified by the Solow procedure? The confidence
placed in that answer, of course, depends on the extent to which one believes
that there is an accurate identification of causal disturbances.
30 There are essentially two equations of Solow’s investigation. In logarithms, the production function is y − k = α(n − k) + a, and the competitive marginal productivity condition is
w = (y − n) + log α + ε, where time-dating of variables is surpressed for simplicity. In these
expressions, the notation is the same as in the main text, except that w is the log real wage and
ε is a disturbance to the marginal productivity condition. To estimate α, Solow assumed that
the ε are a set of mean-zero discrepancies. He made no assumptions about the a process. Thus,
Solow’s identification of a was conditioned on some additional assumptions about the natures of
departures from the competitive theory; it is precisely along these lines that it was challenged in
the work of Hall (1988) and others.
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Identification and Moment Implications
Consideration of identification leads one squarely to a set of potential difficulties with the model evaluation strategy used in quantitative RBC theory,
reviewed in Section 2 above, and also in the econometric strategies described
in Section 8 above. To see the nature of this tension, consider the following
simple two-equation model:
y − k = α(n − k) + a,
(8)
n − k = βa + θb.
(9)
The first of these specifications is the Cobb-Douglas production function in logarithmic form, with y = log Y, etc. The second may be viewed as a behavioral
rule for setting hours n as a function of productivity a and some other causal
factor b, which is taken for simplicity not to affect output except via labor
input. In the case in which a and b are not correlated, these two expressions
imply that the covariance of y − k and n − k is:
cov( y − k, n − k) = (1 + αβ)βσa2 + αθ2 σb2 ,
(10)
where σa2 is the variance of a and σb2 is the variance of b. This covariance
is, of course, the numerator of the correlation that is frequently examined to
explore the comovement of output and labor input. Expression (10) shows that
the covariance depends on how labor and output respond to productivity, i.e.,
on α and β; on how output and labor respond to the other causal factor (θ);
and on the extent of variability in the two causal factors (σa2 , σb2 ).
In models with multiple causal factors, moments of endogenous variables
like y−k and n−k are combinations of behavioral responses determined by the
model (α, β, θ), with these responses weighted by the variability of the causal
factors. If, as most macroeconomists believe, business cycles are the result of
a myriad of factors, it follows that covariances and other second moments of
endogenous variables will not be the best way to determine how well a model
economy describes response to a particular causal factor.
Implications for Model Evaluation
These difficulties have led to some work on alternative model evaluation strategies; a core reference in this line is Sims (1989). The main idea is to evaluate
models based on how well they describe the dynamic responses to changes in
identified causal factors.31 The basic ideas of this research can be discussed
within the context of the preceding two-equation model.
31 Sims
(1989) studied how several models match up to multiple identified shocks. Singleshock analyses include Rotemberg and Woodford’s (1992) analysis of government-purchase disturbances. King and Watson (1995) considered evaluation and comparison of models using individual
and multiple shocks.
R. G. King: Quantitative Theory and Econometrics
99
To begin, consider the relationship between the causal factor a, productivity, and the endogenous variables y − k and n − k. One way to summarize this
linkage is to consider the coefficient that relates y to a: call this a “response
coefficient” and write it as the coefficient π in y−k = πya a, with the subscripts
indicating a linkage from a to y. In the model above, this response coefficient is
m
= (1 + β), with the superscript indicating that this is the model’s response:
πya
productivity exerts an effect on output directly and through the labor input
effect β. In the preceding model, the comparable response coefficient that links
m
= β.32
labor and productivity is πna
An identified series of empirical productivity disturbances, a, can be used
d
d
and πna
, where the
to estimate comparable objects empirically, producing πya
superscript indicates that these are “data” versions of the response coefficients.
The natural question then is: Are the model response coefficients close to those
estimated in the data? Proceeding along the lines of Section 8 above, we can
m
m
πna
] and
conduct model evaluation and model comparison by treating [πya
d
d
[πya πna ] as the model features to be explored. The results of model evaluations and model comparisons will certainly be dependent on the plausibility
of the initial identification of causal factors. Faced with the inevitable major
discrepancies between models and data, a researcher will likely be forced to
examine both identification and model structure.
Explicit evaluation of models along these lines subtly changes the question
that a researcher is asking. It changes the question from “Do we have a good (or
better) model of business cycles?” to “Do we have a good (or better) model of
how fluctuations in x lead to business cycles?” Given that it is essentially certain
that business cycles originate from a multitude of causes, it seems essential to
ask the latter question as well as the former.
11.
SUMMARY AND CONCLUSIONS
In studies in quantitative theory, economists develop simple and abstract models
that focus attention on key features of actual economies. In this article, quantitative theory is illustrated with examples of work on the production function
by Douglas, Solow, and Prescott.
It is typical to view econometrics as providing challenges to quantitative
theory. In particular, in the years after each of the aforementioned studies,
applied researchers using econometric tools indicated that there were elements
missing from each theoretical framework.
32 An
alternative procedure would be to focus on a covariance, such as cov(y−k, a), with the
difference in the current context simply reflecting whether one scales by the variance of a. In the
dynamic settings that are envisioned as the main focal point of this research, one can alternatively
investigate impulse responses at various horizons or cross-correlations between endogenous and
exogenous variables. These features summarize the same information but present it in slightly
different ways.
100
Federal Reserve Bank of Richmond Economic Quarterly
However, particularly in terms of the development of dynamic macroeconomic models, there are also important challenges and opportunities that
quantitative theory provides to econometrics. To begin, it is not an accident
that there has been substantial recent growth in dynamic macroeconomic research using the methods of quantitative theory and little recent work building
such models using standard econometric approaches. The article argues that
the style of econometrics needed is one that is consistent with the ongoing
development of simple dynamic macroeconomic models. It must thus aid in
understanding the dimensions along which simple theories capture the interactions in the macroeconomic data and those along which they do not. In this
sense, it must become like the methods of quantitative theory. But it can be
superior to the current methods of quantitative theory because econometrics
can provide discipline to model development by adding precision to statements
about the success and failure of competing models.
Quantitative theory, however, is also not without its difficulties. To provide
a concrete example of some of these problems, the article uses quantitative
theory and some recent econometric work to evaluate recent claims, by Kydland and Prescott (1991) and others, that the basic RBC model explains most
of economic fluctuations. This claim is frequently documented by showing that
there is a high ratio of the variance of a model’s predicted output series to the
actual variance of output. (It is also the basis for the sometimes-expressed view
that business cycle research is close to a “closed field.”) The article’s contrary
conclusion is that it is hard to argue convincingly that the standard RBC model
explains most of economic fluctuations. First, a simple exercise in quantitative
theory indicates that most of the variance “explained” by a basic version of the
RBC theory comes from the direct effects of productivity residuals, not from
the endogenous response of factors of production. Second, recent econometric
research indicates that the basic RBC model misses badly the nature of the
business cycle variation in output growth (as indicated by comparison of the
power spectrum of output growth in the theory and the U.S. data). Thus, the
more general conclusion is that business cycle research is far from a closed
field. The speed at which a successful blending of the methods of quantitative
theory and econometrics is achieved will have a major effect on the pace at
which we develop tested knowledge of business cycles.
R. G. King: Quantitative Theory and Econometrics
101
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