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Interest

Rate Expectations

and the Slope of

the Money Market Yield Curve
Timothy Cook and Thomas Hahn

What determines the relationship between yield
and maturity (the yield curve) in the money market?
A resurgence of interest in this question in recent
years has resulted in a substantial body of new
research. The focus of much of the research has been
on tests of the “expectations theory.” According to
the theory, changes in the slope of the yield curve
should depend on interest rate expectations: the more
market participants expect rates to rise, the more
positive should be the slope of the current yield
curve. The expectations theory suggests that variation in the slope of the yield curve should be
systematically related to the subsequent movement
in interest rates. Much of the recent research has
focused on whether this prediction of the theory is
supported by the data. A surprising finding is that
parts of the yield curve have been useful in forecasting
interest rates while other parts have not.
A novel and interesting aspect of some of the recent literature is its emphasis on the possible role
of monetary policy in explaining the behavior of the
yield curve. This literature views the Federal
Reserve’ policy instrument as the federal funds rate,
s
and it posits that money markets rates at different
maturities are strongly influenced by current and
expected levels of the funds rate. In this view, explaining the behavior of the yield curve requires
understanding how the Federal Reserve moves the
funds rate over time. A key paper in this area
(Ma&w and Miron [ 1986]), for example, argues that
the pexuhvzce of changes in the federal funds rate
engineered by the Federal Reserve helps explain why
the yield curve from three to six months has had
negligible forecasting power.
* Timothy Cook is an economist at the Federal Reserve Bank
of Richmond and Thomas Hahn is a consultant with TKH
Associates, San Francisco, California. The authors benefitted
from comments by Marvin Goodfriend, Ward McCarthy, Yash
Mehra, and Roy Webb. They are also grateful to J. Huston
McCulloch for providing interest rate data used in the paper and
to the publishers of the Washington
Bond d MoneyMarketReport
for permitting use of their survey data on interest rate
expectations.
FEDERAL

RESERVE

l

This paper surveys the recent literature on the
determinants of the yield curve. It begins by reviewing the expectations theory and recent empirical tests
of the theory. It discusses two general explanations
for the lack of support for the theory from these tests.
Finally, the paper discusses in more detail the
behavior of market participants that might influence
the yield curve, and the role that monetary policy
might play in explaining this behavior.

I.
THE EXPECTATIONS THEORY
Concepts
Two concepts central to the tests of the expectations theory reviewed below are the “forward rate
premium” and the “term premium.” Suppose an investor can purchase a six-month Treasury bill now
or purchase a three-month bill now and reinvest his
funds three months from now in another three-month
bill. The forward rate is the hypothetical rate on the
three-month bill three months in the future that
equalizes the rate of return from the two options,
given the current three- and six-month rates.’ The
forward rate calculated from the current six-month
rate (R6) and the current three-month rate (R3),
which we denote F(6,3), is defined as:
(1 + R6) = (1 + R3)(1 + F(6,3)), or
F(6,3)

= [(l +R6)/(1 +R3)]

(1)

- 1

where the yields are simple unannualized

yields.

Virtually all of the studies surveyed in this paper
use continuously compounded yields, which enable
the forward rate to be expressed as an additive
(rather than a multiplicative) function of the current
six- and three-month
rates. Using continuously
1 The intuition behind the term “forward rate” is that a market
participant who can borrow and lend at currently quoted threeand six-month rates can fix the rate at which he borrows or
lends funds three months forward by an appropriate set of
current transactions. See Shiller 11987, pp. 6-71.
BANK

OF RICHMOND

3

rate premium is 2 percentage points. The forward
rate premium can be decomposed into an expected
change in the three-month rate of 1 Yz percentage
points and an expected term premium of ‘ percen/2
tage point.

compounded annualized yields (denoted here by
lower case letters) the forward rate becomes?
f(6,3) = 2r6 - r3

(2)

The “forward rate premium” is defined as the difference between the forward rate and the current
short-term spot rate:
f(6,3) - r3 = (2r6-r3)

- r3 = Z(r6 -r3)

An equivalent decomposition of the forward rate
premium used in some papers employs the concept
of “holding period yield,” which is the return earned
on a security sold prior to maturity. The forward rate
premium can be divided into (1) the expected change
in the three-month rate and (2) the difference between the expected holding period yield earned by
investing in a six-month bill and selling it when it
is a three-month bill three months in the future,
Eh(6,3:t +3), and the return from investing in a threemonth bill?

(3)

When the maturity of the long-term rate is twice the
maturity of the short-term rate, as in this case, the
forward rate premium is simply twice the spread between the long- and short-term rates.
The “term premium (0)” is generally defined as the
difference between the forward rate and the corresponding expected spot rate:

f(6,3) - r3 = [Er(3:t +3) - r3]
+ [Eh(6,3:t+3)
- r3]

(4)

8 = f(6,3) - Er(3:t+3),

In the above example, the forward rate premium
of 2 percentage points can be decomposed into an
expected change in the short-term rate of 1 Yz percentage points and an expected excess return of ‘
/2
percentage point for holding six-month bills for three
months rather than investing in three-month bills.

where r(3:t +3) denotes the three-month rate three
months in the future and E denotes the current
expectation of that rate. Spot and forward rates not
followed by a colon are measured as of time “t”. Quation (4) can be rewritten in terms of the forward rate
premium by rearranging terms and subtracting r3
from both sides:
f(6,3) - r3 = [Er(3:t+3)

1 r3] + 8

Assumptions
The “expectations theory” is based on two assumptions about the behavior of participants in the money
market. The first is that the term premium that
market participants demand for investing in one
maturity rather than another (and issuers are willing
to pay to issue that maturity) is constant over time.4
Under this assumption equation (5) becomes:

(5)

This expression now decomposes the forward rate
premium into the expected change in interest rates
and a term premium.
To illustrate these concepts, suppose the current
three-month rate is 6 percent, the current six-month
rate is 7 percent, and the expected three-month rate
three months in the future is 7 ‘ percent. Then the
/z
implied forward rate on a three-month security three
months in the future is 8 percent and the forward

Er(3:t+3)

- r3 = -c

+ [f(6,3) - r31

r6 = %c + %[r3 + Er(3:t+3)],

ECONOMIC

REVIEW,

(7)

where c is now a constant term premium. Note that
equation (7) can be rewritten using equation (3) as:

2 The relationship between a simple yield (R) and the corresponding continuously compounded yield (r) is:
(l+R)
= exp(r).
Hence, using continuously compounded yields, equation (1) in
the text can be rewritten:
exp(r6) = exp(r3)exp(f(6,3)),
which taking logarithms of both sides becomes:
f(6,3) = r6 -r3.
If we now let the lower case letters stand for annualized continuously compounded yields, the expression for the forward rate
becomes:
l/qf(6,3) = %r6 - Gr3, or
f(6,3) = 2r6 -r3.
4

(6)

(8)

which says that under the expectations hypothesis
the long-term rate is equal to an average of the current and expected short-term rates plus a constant
which reflects the term premium.
3 Fama (1986, pp. 180-1821 and Fama and Bliss (1987, pp.
681-6823 derive this decomposition.
4 Some papers
assumption of
the hypothesis
of the theory.

SEPTEMBER/OCTOBER

equate the term “expectations theory” with the
a constant term premium, while others include
of rational expectations (discussed below) as part
In this paper we follow the latter procedure.
1990

Equation (7) is the focus of most of the recent
empirical work testing the expectations hypothesis.
Researchers using equation (7) to test the expectations hypothesis do not know the values of
Er(3:t +3). The procedure generally used to get these
values is to assume that interest rate expectations are
formed “rationally,” so that:
(9)

r(3:t +3) = Er(3:t +3) + e:t+3,

where e:t + 3 is a forecast error that has an expected
value of zero and is assumed to be uncorrelated with
any information available at time t. The ideas behind
the rational expectations assumption are that (1) there
is a stable economic environment, (2) market participants understand this environment, (3) therefore,
they should not systematically over- or under-forecast
future interest rates, and (4) they should not ignore
any readily available information that could improve
their forecasts. This assumption specifically requires
that forecast errors are not correlated with the forward rate premium at time t or its two components,
the expected change in interest rates and the expected term premium. Substituting (9) into (7) yields
the following regression equation:
r(3:t+3)

- r3 = a + b[f(6,3)
+ u:t+3

used by Fama [1984a, 1986) replaces the,change in
the three-month rate in equation ( 10) with the holding
period premium:
h(6,3:t+3)

- r3]
(11)

The estimates of the coefficients of equation (11)
provide the same information as the estimates of
equation (10) because the dependent variables in the
two equations sum to the common independent
variable (as indicated by equation 6). Hence, b plus
bl equal one, and the sum of the constants in the
two equations equals zero.’ A value of bl greater
than zero is evidence that the current yield curve has
forecasting power for the excess return earned by investing in six-month bills for three months over the
return from investing in three-month bills. Given the
rational expectations assumption, a value of b 1 equal
to one would indicate that all variation in the yield
curve is due to variation in expected excess returns
(i.e. the term premium) and none due to variation
in the expected change in rates.
II.

REGRESSION ESTIMATES

- r3]
(10)

Under the rational expectations assumption the
error term in equation (10) is uncorrelated with the
right-hand side variable so that the coefficient b can
be estimated consistently. The theory predicts that
b should not differ significantly from one. A significantly different value would contradict either the
assumption of a fixed term premium or the rational
expectations assumption.5 An estimated coefficient
of zero would be evidence that the forward rate
premium has no forecasting power for the subsequent
behavior of the three-month rate.
While equation (10) is the most common regression estimated in this literature, a number of other
specifications have also been used.6 An alternative
5 We discuss in detail in Sections III and IV and in Appendix
II the expected effect on the estimate of “b” if either of these
assumptions is not valid.
6 Campbell and Shiller 119891 derive and estimate two other
specifications to test the expectations theory using a short
m-period rate and a longer n-period rate. In the fust the difference
between the yield on an n -m period bond m periods ahead and
the current yield on an n-period is regressed on the spread
between the current n-period and m-period rates, where the
spread is weighted by m/(n-m).
In the second a weightedaverage change of the m-period rate over (n - m)/m periods is
regressed on the current spread between the n-period and
m-period rates.
FEDERAL

- r3 = al + bl[f(6,3)
+ u:t+3

RESERVE

Three major sets of postwar monthly interest rate
data have been used by the studies surveyed in this
article to estimate equations (10) and (11): (1)
Treasury yields from the Center for Research in
Security Prices (CRSP) at the University of Chicago,
(2) Yield series constructed from Treasury rate data
by a cubic spline curve-fitting
technique
by
McCulloch [1987] and (3) Yields for Treasury and
private sector securities from Salomon Brothers’ An
Analytcal Record of Yields and Yield Spmads. In addition, Hardouvelis
[1989] uses weekly data on
Treasury bills obtained from the quotation sheets of
the Federal Reserve Bank of New York, and Mankiw
and Miron [1986] construct a quarterly series on
three- and six-month loan rates at New York banks
from 1890 through 1958. The regression results we
report in this paper use the McCulloch data for
Treasury rates and the Salomon Brothers data for
private sector rates. We also used Treasury rates from
the CRSP data and the Salomon Brothers data and
found little difference in the results. All interest rates
used in the paper are converted to continuously cornpounded annual rates as described in the Appendix I.
7 This statement is correct if the long maturity (n) is equal to
twice the short maturity (m). If n is not equal to Zm, then the
statement is still true if the dependent variable is multiplied by
an appropriate constant.
BANK OF RICHMOND

5

Estimates

of the Standard

Regression

The standard test of the expectations theory uses
a long-term rate with a maturity equal to twice that
of the short rate. Numerous studies have used the
three- and six-month rates to calculate a three-month
forward rate three months in the future and estimate
the coefficients of equation (10) or a comparable
equation using data over the postwar period. These
include Hamburger and Platt [ 197.51, Mankiw and
Miron [ 19861, Mankiw and Summers [ 19843, and
Shiller, Campbell and Schoenholtz [ 19831. All these
studies report coefficients for the forward rate
premium that are not significantly different from zero,
indicating that the yield curve from three to six
months has had negligible power to forecast the
changes in the three-month rate. Fama [ 1986) finds
that the Treasury bill yield curve from six to twelve
months has had no forecasting power for the subsequent six-month rate, although he does find some
forecasting power for the CD yield curve from six
to twelve months.8
The lack of support for the expectations theory
using postwar Treasury bill rates at the three-, six-,
and twelve-month maturities is shown in the top of
Table I, which reports regression results using the
McCulloch data.9 Table I also shows little support
for the theory using private security rates. The
coefficients in these regressions all are positive, but
only one is significant at the five percent level, and
the explanatory power of the regressions is negligible. The results for the private rates are similar to
those reported by Fama [ 19861, except that his
dependent variable is the holding period premium
so that his coefficients are roughly 1 minus the
coefficients reported in Table I.
Mankiw and Miron [ 19861 estimate equation (10)
from 1890 to 1914, prior to the founding of the
* Also, Hendershott [ 19841 finds forecasting power for the bill
yield curve from six to twelve months after adding unexpected
changes in inflation and unexpected changes in other variables
to his estimated equation.
9 The forecast horizon in these regressions is generally longer
than the monthly period between observations. As a result there
will likely be serial correlation in the error term of the regressions. For example, a regression of the three-month change
in the three-month rate on the forward rate premium using
monthly three- and six-month rates will likely generate a
moving average error term of order 2 because the forecasts in
months two and three are made before the error from month
one’ forecast is known. The standard errors provided in the
s
tables are calculated using the consistent variance-covariance
estimate from Hansen 119821 with the modification by Newey
and West 119873. For discussion of this procedure see Mishkin
11988,
pp. 307-3091
6

ECONOMIC

REVIEW.

Federal Reserve, and over four subperiods from 19 14
through 1979. They find that the spread between
the six- and three-month
rates had substantial
forecasting power for the three-month rate only in
the period prior to 1914. In fact, the estimated slope
coefficient in this period is only slightly below the
value predicted by the expectations theory. We
discuss this interesting result in more detail below.
Estimates

of Non-Standard

Regressions

A number of recent studies also report regression
results for sections of the yield curve over which the
maturity of the long-term rate is not equal to twice
that of the short rate. One type of regression
measures the “cumulative” predictive power of the
slope of the yield curve between a one-period rate
and longer-term rates at various maturities. For example, we can estimate the predictive power of the
yield curve from one to six months with the
regression:
r(l:t+S)

- rl

= a + b[(f6,5)
+ u:t+s

- rl]
(12)

The dependent variable in this regression is the
change in the one-month rate over the following five
months. The independent variable is the difference
between the forward rate for a one-month bill five
months in the future and the current one-month spot
rate. The forward rate on a one-month bill five
months in the future can be calculated from the current five- and six-month yields - hence, the notation f(6,.5).i” A coefficient of 1 for b in this regression supports the expectations hypothesis, and a
coefficient less than 1 but significantly greater than
zero provides evidence that the yield curve over this
range has forecasting power for the subsequent movement in rates.
A second type of non-standard regression estimates
the “marginal” ability of small sections of the yield
curve to forecast the subsequent movements in rates
over a corresponding future period. For example, the
predictive power of the yield curve for the change
in rates from four to five months in the future can
be estimated with the regression:
r(l:t +5) - r(l:t +4) = a + b[f(6,5) - f(.5,4)]
+ u:t+s
(13)
where the dependent variable is the change in the
one-month rate from four to five months in the future
10The formula used to calculate the forward rate on an n-m
month bill m months in the future is:
f(n,m) = [l/(n -m)][nr(n)
- mr(m)j.

SEPTEMBER/OCTOBER

1990

Table I

ESTIMATES OF THE STANDARD REGRESSION (n =2rd*
r(m:t+m)
Dependent

Variable

Treasury

-

rm = a + b[f(n,m)

Bills

r(3:t+3)

-

-a

- rml + u:t+m

r3

-R2

b

0.10

Period

-0.15
(0.19)

(0.09)

0.00

52: l-86:8

0.00

52: l-86:8

r(6:t+6)

-

r6

0.04
(0.17)

0.04
(0.30)

r(3:t+3)

-

r3

0.13
(0.15)

-0.20
(0.22)

0.01

66: 12-86:8

r(6:t+6)

-

r6

0.04
(0.25)

-0.01
(0.32)

0.00

66.12-86:8

Certificates

of Deposit

r(3:t+3)

-

r3

- 0.05
(0.17)

0.36
(0.19)

0.02

66: 12-86:8

r(6:t+6)

-

r6

0.07
(0.32)

0.52
(0.26)

0.06

71: lo-86:8

-

r3

-0.06
(0.20)

0.38
(0.25)

0.02

66: 12-86:8

- 0.02
(0.16)

0.40
(0.22)

0.03

66: 12-86:8

Eurodollars
r(3:t+3)

Commercial
r(3:t+3)

Paper
-

r3

* Standard errors are in parentheses and are calculated as described in footnote
annual rates in percentage points. “m” and “n” refer to maturity in months.

and the independent variable is the difference between the one-month forward rate five months in the
future (calculated from the current five- and six-month
rates) and the one-month forward rate four months
in the future (calculated from the current four- and
five-month rates). A coefficient not significantly different from one supports the expectations hypothesis,
and a coefficient less than one but significantly greater
than zero provides evidence that the yield curve from
four to six months has predictive power for the movement in the one-month rate four to five months in
the future.
Estimates for the non-standard regressions using
the McCulloch data are shown in Table II. The
estimates of the cumulative regressions in the top
of the table show positive and steadily declining coefficients over the money market yield curve out to
six months, although only the coefficient in the first
regression is significant at the 5 percent level. The
FEDERAL

RESERVE

9. Interest

rates are continuously

compounded

results of the marginal predictive power regressions
show that virtually all of the forecasting power of the
bill yield curve is in the spread between the onemonth ahead one-month forward rate and the current one-month spot rate.
Fama [1984a] estimates cumulative and marginal
predictive power regressions using Treasury bill
rates with maturities up to six months from the CRSP
data from 1959 through 1982, and Mishkin [1988]
repeats Fama’ regressions using the same data set
s
extended through 1986. Both studies report full
sample results for the cumulative predictive power
regressions roughly similar to those reported in the
top of Table II. One difference is that Fama finds
coefficients significant at the five percent level in his
regressions covering the cumulative change in rates
one, two, and three months in the future, and Mishkin
finds significant coefficients in regressions covering
the cumulative change in rates one and two months
BANK OF RICHMOND

7

Table

II

ESTIMATES OF NON-STANDARD REGRESSIONS*
A. Cumulative
Dependent

Regressions:

r(l:f+n-l)-rl

= a + b[f(n,n-U-r11

Variable

+ u:t+n-1
b

-a

-R2

r(l:t+ 1) -rl

-0.18
(0.04)

0.50
(0.12)

0.09

r(l:t+2)-rl

-0.19
(0.11)

0.36
(0.20)

0.03

r(l:t+3) -rl

-0.21
(0.14)

0.33
(0.21)

0.03

r(l:t+4)

-0.04
(0.10)’

0.09
(0.15)

0.00

0.03
(0.12)

0.02
(0.14)

0.00

-rl

r(l:f+5)-rl

B. Marginal
Dependent

Regressions:

r(l:t+n-l)-r(l:t+n-2)

Variable

= a + b[f(n,n-l)-f(n-l,n-2)l

+ u:t+n-1

b

-R2

-0.18
(0.04)

0.50
(0.12)

0.09

-0.01
(0.06)

0.12
(0.21)

0.00

r(l:f+3)ir(l:t+2)

0.02
(0.04)

-0.07
(0.14)

0.00

r(l:t+4)-r(l:t+3)

- 0.00
(0.04)

0.09
(0.20)

0.00

r(l:t+5)-r(l:t+4)

-0.03
(0.04)

0.62
(0.34)

0.02

r(l:t+

a

U-r1

r(l:t+2) -r(l:t+

1)

*Standard errors are in parentheses.
Interest rates are continuously
in months. Estimation period is 1952:l to 1986:8.

in the future. As with the McCulloch data, however,
the full sample marginal predictive power regressions
reported by Fama and Mishkin have significant coefficients only in the regression for the change in rates
one month ahead, r( 1:t + 1) - rl, confirming that
virtually all of the forecasting power of the bill yield
curve is in the shortest maturities.
Fama estimates subperiod regressions for 1959 to
1964, 1964 to 1969, and 1969 to 1982, and Mishkin
reports regressions for these subperiods and also for
1982 to 1986. They find that in each subperiod the
difference between the one-month ahead one-month
forward rate and the current spot rate had forecasting
power for the movement in the one-month rate over
the following month. They also find that in some
8

ECONOMIC

REVIEW.

compounded

annual

rates in percentage

points.

“n”

refers to maturity

subperiods-notably
those in the 1960s-the
difference between the two-month ahead forward rate and
the one-month ahead forward rate had significant
forecasting power for the change in rates one to two
months in the future.
Hardouvelis [ 19881 uses weekly data on Treasury
bill rates from 1972 through 198.5 to calculate twoweek forward rates at one week intervals from one
to twenty-four weeks in the future. Hardouvelis
estimates coefficients for cumulative and marginal
forecasting regression equations over three periods
corresponding
to three Federal Reserve policy
regimes from 1972 through October 1979, October
1979 through October 1982, and October 1982
through November 1985. In the first period the yield
SEPTEMBER/OCTOBER

1990

curve has forecasting power for only one week, while
in the latter two periods the marginal forecasting
power of the yield curve lasts eight or nine weeks.
These results are roughly consistent with those of
Fama and Mishkin, who also find that the forecasting
power of the money market yield curve was weakest
in the 1970s.” A striking feature of Hardouvelis’
results is that the coefficient in the regression for the
one week ahead change in rates is close to 1 in each
of the three periods, which suggests that in these
periods the shortest end of the yield curve behaved
closely in accordance with the expectations theory.

Table III

FORECASTING POWER OF YIELD CURVE
FROM ONE TO FIVE YEARS*

Dependent

ii In a related paper Simon [ 19901 tests the forecasting power
of the spread between the three-month Treasury bill rate and
the overnight federal funds rate for the average funds rate over
the following three months. His full sample covers the period
from 1972 to 1987, and his three subperiods correspond to those
in Hardouvelis’ paper. Simon [p. 574, Table III finds that the
s
spread has forecasting power in the latter two subperiods but
not in the 1970s.
FEDERAL

RESERVE

Variable

a

-

rll

1) -

rl

-

0.15

0.38

-

rl

rl

-R2

(0.27)

0.25

0.73

0.02.

(0.52)

0.17

1.28

(0.55)
r(l:t+4)

-

rl

+ u:t+n-1

-b

(0.55)

r(l:t+2)

The Forecasting Power of the Yield Curve
from One to Five Years

Campbell and Shiller [1989] use the McCulloch
data to test a different specification of the expectations theory in which the current spread between an
n-period maturity rate (such as a five-year rate) and
a shorter m-period maturity (one-year) rate forecasts
a weighted average change of the m-period rate over
the next n - 1 periods (4 years). They regress the
weighted average change of the m-period rate on the
current spread and get results similar to those of
Fama and Bliss. Specifically, they find that the spread
between the 4-year and l-year rates and the spread
between the S-year and l-year rates have significant
forecasting power for the weighted average change
in the one-year rate over the next 3 or 4 years.

rl = a + b[f(n,n-1)

(0.25)

r(l:t+

r(l:t+3)

A final set of regression results that we briefly
review relate to the forecasting power of the yield
curve from one to five years. Fama and Bliss [ 19871
find that the yield curve from one to five years has
had substantial forecasting power for the change in
rates over the following three or four years. For example, they find that the difference between the forward rate on a one-year Treasury security four years
in the future (calculated from the current four- and
five-year rates) and the current one-year rate explains
48 percent of the variance of the 4-year change in
the one-year rate. Table III reports these regressions
using the McCulloch data. The results are generally
similar to those reported by Fama, although the explanatory power of the four-year rate change regression is smaller.

-

r(l:f+n-1)

1.53
(0.33)

0.23

(0.31)

0.10

0.08

(0.51)

0.29

Standard errors are in parentheses.
Interest rates are continuously compounded annual rates in percentage points. “n” refers to maturity in years.
Estimation period is 1952:l to 1983:2.
l

III.

EVIDENCEOFAVARIABLETERMPREMIUM
The studies surveyed in the previous section
strongly reject the expectations theory, especially
when the theory is tested with the standard regression using three- and six-month or six- and twelvemonth rates. The rejection of the theory implies that
either (1) the term premium is not constant, (2) the
rational expectations assumption is not valid, or
(3) both. We discuss evidence regarding the variable
term premium in this section and evidence regarding
the rational expectations assumption in the following section.
Most explanations of the lack of empirical support
for the expectations theory have focused on the
possibility that the expected term premium is not
constant, as assumed by the theory, but varies
substantially over time. If the term premium is
variable, the estimate of b in equation (10) will
differ from the value of one predicted by the expectations theory. A number of papers have discussed
the determinants of the estimated coefficient and
derived expressions for the probability limit of the
coefficient when the variance of,the term premium
is positive. (See Hardouvelis [ 1988, pp. 342-3431 and
Mankiw and Miron [ 1986, pp. ‘
218~2201.) The
derivation of one of these expressions is shown in
Appendix II. One conclusion of these papers is that,
generally, the greater the fraction of the variance in
the spread between the forward and spot rates due
to the variance in the expected term premium-and
the smaller the fraction due to the variance of the
BANK OF RICHMOND

9

forecasting equations using data that was available
to market participants at the time of their forecasts.
Startz [1982] regresses the current interest rate, r,
on lagged values of spot and forward rates. He then
uses the standard error of this equation as a maximum estimate or “upper bound” of the standard
deviation of the market’ forecast error, assuming that
s
the set of variables used in the regression represents
a minimum set of information available to market participants to forecast rates.

expected change in rates-the
lower will be the
coefficient below the value of one predicted by the
expectations theory. ia If the variance of the expected
change in rates is equal to the variance of the expected term premium, then the estimate of the
coefficient converges to one-half.
From this perspective the relevant questions are
(1) does the expected term premium vary and (2)
how much does it vary relative to the expected
change in rates. Evidence from a variety of sources
suggests that the expected term premium does vary
substantially over time and, moreover, that the
magnitude of the variance is comparable to the
variance in the expected change in rates.

Startz then decomposes the spread between the
forward rate and the subsequent matching spot rate
(which he labels the “forward deviation”) into the expected term premium (P) and the forecast error (e):
f-r:t+3

Evidence from Holding Period
Premium Regressions

+ (Er:t+3 -r:t+3)
+
e

(15)

The variance of (f -r:t +3) is:

As discussed in Section I, an alternative and compiementary way to estimate the standard regression
is to make the dependent variable the holding period
premium rather than the expected change in rates:
h(6,3:t +3) - r3 = al + bl[f(6,3)
+ u:t+3

= (f-Er:t+3)
=
P

var(f -r:t +3) = var(P) + var(e) +Zcov(P,e)

(16)

The covariance of P and e is zero under the rational
expectations assumption, however, because P is
known at the time of the forecast and should not be
correlated with forecast errors. Hence,

- r3]
(14)

var(P) = var(f-r:t+3)

A value of bl greater than zero is evidence that the
forward rate premium has had forecasting power
for the excess return from holding six-month versus
three-month securities over a three-month period.
A value of bl equal to one would be evidence that
virtually all variation in the yield curve is due to variation in expected returns. (This conclusion, of course,
depends on the rational expectations assumption.)

- var(e)

(17)

From equation (17) we can see that if v%(e)-the
standard error of the regression squared-is
an upper bound estimate for the true variance of the
market’ forecast error, then var(f -r:t +3) - v%(e)
s
is a lower-bound estimate of the true variance of the
term premium.

A few papers have tried to measure the variance
of the term premium by estimating interest rate

Startz calculates lower bound estimates over the
period from 1953 through 1971 of the proportion
of the variance of the spread between the forward
rate for a one-month bill and the subsequent matching spot rate that was due to variation in the term
premium. These estimates range from one-third to
two-thirds over horizons from one to twelve months.
This conclusion implies that lower bound estimates
of the ratio of the variance of the premium to the
variance of the forecast error ranged from one-half
to two. Of course, this is a lower bound estimate of
the ratio of the variance of the premium to the
variance of the forecast error, not to the variance of
the expected change in rates. Nevertheless, these
results suggest that the variation in the premium is
substantial. l3

‘ Specifically, these papers find that if the correlation coeffi2
cient between the expected change in rates and the expected
term premium is zero or greater than zero, the probability limit
of the estimated coefficient in equation (10) is a strictly increasing function of the ratio of the variance of the expected change
in rates to the variance of the expected term premium.

13 Moreover,
in the interest rate survey data discussed in the
following section, the variance of the expected change in rates
is less than the variance of forecast errors, in which case onehalf to two would also be a lower bound estimate for the ratio
of the variance of the premium to the variance.of the expected
change in rates.

Fama [ 19861 estimates equation (14) using oneand three-month rates, three- and six-month rates,
and six- and twelve-month rates. He reports values
of b 1 that are close to one. for bills and average a
little over one-half for CDs and commercial paper.
These results indicate that variation in the slope of
the yield curve provides systematic information about
expected excess returns. As Fama [1984a, p. 5121
emphasizes, this is evidence that the current slope
of the money market yield curve is influenced by
expected term premiums that change over time.
Evidence

10

from Lower

Bound Estimates

ECONOMIC

REVIEW,

SEPTEMBER/OCTOBER

1990

DeGennaro and Moser [ 19893 employ essentially
the same procedure as Startz to calculate lower bound
estimates over the period from 1970 through 1982
of the proportion of the variance of the spreads between the forward rates for four- and eight-week bills
and the subsequent matching spot rates that was due
to variation in the term premium. Their estimates
range from one-fifth to three-fifths for horizons from
one to 49 weeks.
IV.

EVIDENCEONTHE
RATIONALEXPECTATIONSASSUMPTION
The previous section presented evidence that a
variable term premium contributes to the rejection
of the expectations hypothesis in the tests reported
in Section II. The remaining question is whether
violation of the rational expectations assumption also
contributes to the regression results. A way to
evaluate this question is to use survey data to get
an estimate of the market’ interest rate expectations.
s
For instance, suppose ESr(3:t +3) is the mean
response from survey participants of the expected
level of the three-month rate three months in the
future. Then the coefficients of the standard equation can be estimated with the regression:
ESr(3:t +3)-r3

= a + b[f(6,3) -r3]

+ u

(18)

The variables in equation (18) are all measured at
time t, the time of the survey. Consequently, the
expected coefficient estimates do not depend on the
rational expectations assumption. That is, if the term
premium is constant, then the estimated coefficient
of the forward rate premium in equation (18) should
be close to 1 regardless of how expectations are
formed.
A small number of studies, including Friedman
[ 19791 and Froot [ 19891, have used the “GoldsmithNagan” survey data to estimate versions of equation
(18). The survey data are based on a quarterly survey
of 25 to 45 market participants on the interest rates
they expect three and six months in the future. The
survey was originally conducted by the GO&F&Nagan Bond and Money Market Letter and is now
published in the newsletter Washington Bond &f’
Money
Market Report. The survey collects forecasts of the
three-month bill rate, the twelve-month bill rate, and
a private sector three-month rate, along with forecasts
of a number of long-term interest rates. Through the
March 1978 survey the private rate was the threemonth Eurodollar rate. Since then, the private rate
has been the three-month commercial paper rate.
FEDERAL

RESERVE

There is typically about a two-week period between the time the survey forms are mailed to the
respondents and the latest market close prior to
publication of the responses. The average timing of
the latest close prior to publication is the end of
the quarter, and in estimating equation (18) we
matched the survey data with the end-of-quarter data
on Treasury bill rates from McCulloch and the endof-quarter data on Eurodollar and commercial paper
rates from Salomon Brothers.r4 We also used the sixand nine-month
Treasury
bill rates from the
McCulloch data to calculate the six-month ahead forward rate for a three-month bill, f(9,6), and estimated
the coefficients of an equation with the survey expected change in the three-month bill rate six months
in the future as the dependent variable:
ESr(3:t+6)

-r3

= a + b[f(9,6) -r3]

+ u. (19)

Equation (19) can not be estimated for private sector rates because Salomon Brothers does not have
the nine-month rates needed to calculate f(9,6).
The top part of Table IV shows the regression
results for equations (18) and (19) using the
Goldsmith-Nagan survey data. The coefficients of
the forward rate premium in these regressions are
all positive and significant. The low Durbin-Watson
statistics, however, suggest that serial correlation is
a serious problem, and inspection of the regression
residuals indicated that they fall sharply in recessions.
Consequently, we reestimated the regressions with
a dummy variable set equal to one for all the survey
dates that occurred in recessions.r5 The coefficients
of the forward rate premium in these regressions
range from 0.45 to 0.59 and are significant at
the one percent level in the Treasury bill rate and
r4 On average over the period covered by the survey regressions
the latest market close prior to publication of the survey results
falls on the last day of the quarter. The average absolute difference between the latest close and the last day of the quarter
is four days. We know of no reason to expect that the differences
between the timing of the survey and the timing of the calculation of the forward rate premium would bias the estimate of b
in equations (18) and (19). Froot [1989, p. 285, footnote 91
experiments with data sets one week and two weeks before the
end of the quarter and finds that the regression results are the
same as with end-of-quarter data.
1s The dummv variable equals 1 from the fourth quarter of 1969
through the third quarter-of 1970, the fourth quarter of 1973
throueh the fourth Quarter of 1974. and the first auarter of 1980
throuih the third quarter of 1982. ‘
The latter period covers two
recessions that are separated by three quarters.
BANK

OF RICHMOND

11

Table

IV

TEST OF THE EXPECTATIONS THEORY USING SURVEY DATA*

A.

Dependent

Variable:
+ n -m)

ESr(m:t

Survey Expected

Change

. rm = a + b[f(n,n

in Rates

-m)

-

rml + cD + U
Es$en;~n

a

Treasury

c
-

-Ra

-DW

0.19

0.43

69:3-86:2

0.70

‘
1.24

69:3-86:2

0.31

0.59

69:3-86:2

0.68

1.27

69:3-86:2

0.42

0.67

69:3-78: 1

0.56

1.17

69:3-78: 1

0.35

b

1.35

78:2-86:2

0.58

2.11

78:2-86:2

(quarter)

Bills

ESr(3:t+3)

-

r3

-0.33
(0.11)

0.44
(0.14)

ESr(3:t+3)

-

r3

-0.11
(0.07)

0.54
(0.11)

ESr(3:t+6)

-

r3

-0.43
(0.13)

0.53
(0.11)

ESr(3:t+6)

-

r3

-0.08
(0.10)

0.50
(0.08)

ESr(3:t+3)

-

r3

-0.67
(0.12)

0.75
(0.19)

ESr(3:t+3)

-

r3

-0.37
(0.10)

0.45
(0.18)

ESr(3:t+3)

-

r3

-0.12
(0.10)

0.90
(0.19)

ESr(3:t+3)

-

r3

0.14
(0.08)

0.59
(0.15)

-0.96
(0.10)

- 1.09
(0.15)

Eurodollars

Commercial

-0.70
(0.23)

Paper

-0.86
(0.23)

* Standard errors are in parentheses.
Interest rates are continuously compounded annual rates in percentage points. “n” and “m” refer to maturity in months.
D is a dummy variable set equal to 1 from the fourth quarter of 1969 through the third quarter of 1970, the fourth quarter of 1973 through the fourth quarter
of 1974, and the first quarter of 1980 through the third quarter of 1982.

B. Dependent

Variable:

r(m:t+n-m)

-

Actual

Change

in Rates

rm = a + b[f(n,n-m)
a

Treasury

-

rml

+ u:t+m
-Ra

-DW

Estimation
Period
(quarter)

-0.35
(0.42)

0.02

2.56

69:3-86:2

0.14

0.00

1.61

69:3-86:2

b

Bills

r(3:t+3)

-

r3

r(3:t+6)

-

r3

0.20
(0.26)
-0.16

(0.35)

(0.24)

r3

-0.31
(0.28)

0.62
(0.36)

0.07

1.57

69:3-78: 1

r3

-0.04
(0.42)

0.23
(0.82)

0.00

2.65

78:2-86:2

Eurodollars
r(3:t+3)

Commercial
r(3:t+3)

12

-

Paper
-

ECONOMIC

REVIEW,

SEPTEMBER/OCTOBER

1990

commercial paper rate regressions.i6 The coefficients
of the dummy variable are all negative and significantly different from zero. Moreover, the DurbinWatson statistics for these regressions rise sharply,
although they still indicate some serial correlation in
three of the four regressions. A plausible explanation for the significance of the dummy variable
coefficient is that the term premium rises in recessions. We discuss this possibility in Section VI.
For comparison with the survey regression results,
the bottom part of Table IV shows estimates of the
regressions over the same sample period and the
same quarterly observations but with the actual
change in rates as the dependent variable. The
negligible explanatory power of these regressions is
in sharp contrast to the survey regressions.
We can derive an estimate of the term premium
implied by the survey results by subtracting the
expected change in rates from the forward rate
premium at the time of the survey. This estimate
can be used to calculate an estimate of the relative
magnitude of the variances of the premium and the
expected change in rates. These variances are summarized in Table V for both Treasury bills and private
securities at the three-month forecast horizon. The
ratio of the variance of the premium to the variance
of the expected change in rates is 1.11 for bills and
about 0.65 for private securities. These numbers
appear roughly consistent with the evidence from the
studies reviewed in Section III that used forecasting
equations to generate lower bound estimates of the
variance of the premium.
The survey regression results suggest that the
rational expectations assumption used in the studies
surveyed in Section II is not valid for the time period
covered by the survey data. To see this, note that
the actual change in the three-month rate used as
the dependent variable in these studies can be
decomposed into the expected change in the rate plus
a forecast error. If the actual change in interest rates
r6 Froot [ 1989, ~293, Table III] reports estimates of equations
for the three-month ahead expected changes in the three-month
bill ,rate, the twelve-month bill rate, and the three-month
Eurodollar rate, and six-month ahead expected changes in the
three-month bill rate and the twelve-month bill rate. He also
finds a strong positive correlation between forward rate premiums
and the survey expectations. The major difference between his
results and those reported here is that he reports a negative coefficient of -0.05 in the regression for the six-month ahead forecast
of the change in the three-month bill rate. Hamburger and Platt
11975. o. 191. footnote 51 find the correlation between forward
rates and the survey’ expected rates to be so strong that they
s
cite it as evidence that forward rates are the market’ expectas
tions of future spot rates.
FEDERAL

RESERVE

Table

V

VARIANCE OF SURVEY EXPECTED CHANGE
IN RATES AND SURVEY PREMIUM
Treasury Bills (69:3-86:2)
Variance

of Premium

Variance

of Expected

Ratio

0.42
Change

in Rates

0.38

Change

in Rates

0.46

in Rates

0.62

= 1.11

Eurodollars

f69:3-78:l)

Variance

of Expected

0.29

of Premium

Variance
Ratio

= 0.63

Commerciai

Paper

(78:2-86~2)

Variance

of Premium

Variance

of Expected

Ratio

0.41
Change

= 0.66

is uncorrelated with the forward rate premiumas indicated by the regression results reported in
Section II-but the expected change is positively
correlated with the forward rate premium, then the
survey forecast error must be negatively correlated
with the forward rate premium. This is a violation
of the rational expectations assumption specified by
equation (9).
As shown in Appendix II, a negative correlation
between forecast errors and the forward rate premium
reduces the coefficient of the forward rate premium
in tests of the expectations theory (estimated with
actual changes in rates) below the value of 1 predicted
by the theory. Following Froot [ 1989, pp. 290-94,
we can use the survey data to get estimates of the
effects of the variable term premium and forecast
errors on the coefficient of the forward rate premium.
The probability limit of the coefficient of the forward
rate premium can be written as one minus a deviation due to the variable term premium plus a deviation due to systematic expectational errors:

where FP refers to the forward rate premium, 8 refers
to the term premium, and e refers to expectational
errors. The survey data can be used to derive
estimates of the terms on the right-hand side of equation (20). According to these estimates, shown in
Table VI, roughly half the deviation from 1.O of the
BANK OF RICHMOND

13

Table

DECOMPOSITION

VI

OF THE COEFFICIENT OF THE FORWARD RATE PREMIUM
IN TESTS OF THE EXPECTATIONS THEORY*
(1)
Forecast
Horizon

Instrument

3-Month
Treasury

3 Months

-0.35

(2)
Component
Attributable
to
Term Premium

(3)
Component
Attributable
to
Forecast Errors

0.56

-0.79

Bill

3-Month
Treasury

Regression
Coefficient

6 Months

0.14

0.47

-0.39

3 Months

0.62

0.25

-0.13

3 Months

0.23

0.10

-0.67

Bill

3-Month
Eurodollar
3-Month
Commercial
Paper

* The construction of this table follows Froot f1989,
p. 291, Table III. The regression coefficients are from Part B of
Table IV. Columns (2) and (3) are calculated using the Goldsmith-Nagan
survey data from the third quarter of 1969
through the second quarter of 1986. Column (1) equals 1.0 minus column (2) plus column (3).

coefficient of the forward rate premium is due to a
variable term premium and half results from the correlation of forecast errors with the forward rate
premium.
The survey regression results suggest that market
participants build their expectations into the yield
curve, but their forecasts have been so poor at the
three- and six-month horizons that the yield curve
has had negligible forecasting power for the subsequent three-month and six-month rates. l7 Of course,
this interpretation depends critically on the assumption that the mean of the survey forecasts is an
unbiased estimate of the market forecast, and that
the survey expectations can be interpreted as determining the current slope of the yield curve. One can
imagine circumstances under which this might not
be true. For example, the forecasters used in the
survey might be influenced by the current shape of
the yield curve in determining their interest rate
forecasts, in which case the regression results would
be spurious. Or they might systematically differ in
I7 The poor forecasting of market participants at the three- and
six-month horizons is documented by Hafer and Hein 11989,
p. 37, Table 11, who evaluate the forecasting power of both the
Goldsmith-Nagan
survey data and the Treasury bill futures
market. They find that naive forecasts of no change in the threemonth bill rate over the following three and six months do about
as well as the changes forecast by the Goldsmith-Nagan survey
or by the futures market. Similarly, Belongia [1987, p. 13, Table
l] finds that a forecast of no change in rates over six months
does as well as the consensus forecast of a group of economists
surveyed regularly by the Wal/ Streer Journal.
14

ECONOMIC

REVIEW,

their forecasts from the market in general for other
reasons, perhaps because their forecasts are made
public or because they are not actively involved in
buying and selling securities.ra We know of no evidence that either of these possibilities is true.19
A final point to make here is that there is a distinction between the specialized form of the rational
expectations
hypothesis used in the literature
surveyed in this article-indicated
by equation (9)and the general principle of rational expectations,
which is that market participants use available information efficiently in forming their expectations. Webb
119873 discusses a number of reasons why rational
market participants might not behave over a given
time period according to the specialized form of the
hypothesis. A general point is that it is difficult to
say anything definite about whether market participants have formed expectations rationally without a
clear understanding of the process determining
interest rate movements.
r* Kane [ 1983, pp. 117-l 181 emphasizes that survey respondents
should be decision-makers with the authority and willingness to
commit funds in support of their forecasts. He finds [p. 1191
that in his survey the response of “bosses” (i.e., decision-makers)
differs from the response of non-bosses. Many of the respondents
in the Goldsmith-Nagan
survey are senior officials in their
respective organizations and would seem to fit the label of
“decision-maker.” We are not aware, however, of any general
classification of the Goldsmith-Nagan respondents along these
lines.
19More detailed discussions and evaluations of the GoldsmithNagan survey data are found in Prell 119733, Throop [1981],
Friedman 119801, and Hafer and Hein [1989].

SEPTEMBER/OCTOBER

1990

To illustrate this point, consider the behavior of
the Goldsmith-Nagan forecasts over the period from
late 1979 through mid-1982. In reaction to rising
inflation, the Federal Reserve at the beginning of the
period unexpectedly raised short-term interest rates
sharply and then kept them at an unusually high level
for most of the following 2 ‘ years. The Fed’ policy
/2
s
over this period was generally not anticipated by
market participants. As a result, following the initial
increase in rates the survey participants forecasted
large declines in rates for several quarters in a row.
(See Chart 1 in the following section.) It seems
reasonable that in this episode the expectations of
market participants at the three- and six-month
horizons would be influenced by their judgment that
monetary policy actions had driven short-term rates
to a level that could not be sustained. Moreover, the
expectation that rates were going to fall sharply
eventually proved correct. Yet ex post the expected
declines in rates at the three- and six-month horizons
over this period were accompanied by large positive
forecast errors. This contributed to a negative correlation over the estimation period between the expected change in rates and forecast errors, but it
does not seem accurate to say that market participants formed expectations irrationally over this
period.
V.
FEDERAL FUNDS RATE EXPECTATIONS AND

The regression results reported in Section II indicate that the slope of the yield curve from three to
twelve months has had no forecasting power for
three- and six-month rates. A puzzling aspect of the
strong rejection of the expectations theory from this
type of test is that it seems at odds with the standard view among money market participants that
money market’
rates are largely determined by the
current and expected levels of the shortest-term rate,
the federal funds rate. A second puzzling aspect of
the regression results is that the yield curve from one
week to three months and from one year to five years
has had forecasting power, even though the yield
curve from three to twelve months has had no
forecasting power. This section discusses possible
explanations for these two puzzles.
Federal Funds Rate Expectations
the Mankiw-Miron Hypothesis

and

Market participants view the federal funds rate as
the instrument used by the Federal Reserve to carry
out its policy decisions. In forming expectations of
FEDERAL

RESERVE

the future level of the funds rate they attempt to
identify Federal Reserve actions signaling changes
in the funds rate target, and they attempt to forecast values of macroeconomic variables they believe
influence the Fed’ decisions.*O Many studies over
s
the past decade have found that Treasury bill rates
respond to monetary policy announcements or actions that influence funds rate expectations. Similarly,
many studies have found that bill rates respond to
incoming news on variables-such
as the money
supply-that
market participants believe are likely
to influence policy actions. If money market rates
are so sensitive to funds rate expectations, as these
studies suggest, why do tests of the expectations
theory using rates from three to twelve months fail
so badly?
A possible answer focuses on the way market participants form expectations of the future behavior of
the federal funds rate. Mankiw and Miron [ 1986, p.
22.5] suggest that at each point in time the Federal
Reserve sets the short rate (i.e., the federal funds
rate) at a level that it expects to maintain given the
information affecting its policy decisions. They
hypothesize that market participants understand this
behavior and therefore expect changes in the short
rate at any point in time to be zero: “Under this
characterization of policy, while the Fed might change
the short rate in response to new information, it
always (rationally) expected to maintain the short rate
at its current level.” If this view is correct, then the
whole spectrum of money market rates would
adjust up and down in response to changes in the
funds rate targeted by the Fed, but the dope of the
yield curve would be unchanged. Hence, expected
changes in interest rates would be negligible, and the
variance of expected changes in rates would be small.
This expectations behavior coupled with a variable
term premium could explain the regression results
in Section II. The paradox according to this explanation is that tests of the expectations theory using
three- and six-month rates provide little support
for the theory, even though rates at these maturities
are, in fact, responding strongly to funds rate
expectations.
Mankiw and Miron provide support for this argument by testing the expectations theory using threeand six-month interest rates over the Z-year period
prior to the founding of the Fed and over four periods
z” See McCarthy 119873 for a description of “Fed-watching”
behavior bv market oarticioants and Goodfriend 119901 for a
description’ of the Gederai Reserve’ interest rate targeting
s
procedures.
BANK OF RICHMOND

15

since. They find virtually no support .for the expectations theory in any of the latter periods. In the
period prior to the founding of the Fed, however,
they find that the yield curve from three to six months
had substantial forecasting power and that the slope
coefficient for this time period is only modestly below
the value predicted by the expectations theory.
Mankiw and Miron also present evidence that after
the founding of the Fed there was a sharp deterioration in the ability of time series forecasting equations to forecast changes in interest rates three
months in the future. In light of this evidence they
conclude that the ability of market participants to
forecast changes in short-term rates fell sharply after
the founding of the Fed, resulting in a sharp rise in
the ratio of the variance of the premium to the
variance of the expected change in interest rates.
Cook and Hahn 11989, p. 345) catalogue the reactions of the three.-month, six-month, and twelvemonth Treasury bill rates to events changing federal
funds rate expectations and find these reactions to
be broadly consistent with the Mankiw-Miron
hypothesis. These reactions are summarized in
Table VII, which shows the estimated coefficients
of regression equations of the form:
ARTBi

= a + bAXj + u,

(21)

Where ARTBi is the change in the Treasury bill rate
at matuiity i and AXj is the change in a variable j
that influences the market’ funds rate expectations.
s
The top of the table shows the reaction of bill rates
to changes in the Federal funds rate target over the
period from September 1974 to September 1979.21
The middle shows the reaction of bill rates to discount rate announcements with policy content in the
1973-1985 period (i.e., announcements indicating
the discount rate is being changed for reasons other
than to simply realign it with market rates).Z2 The
bottom of the table shows the reaction of bill rates
to announcements
of unexpected changes in the
money supply. Under the “policy anticipations
hypothesis”-which
is the most widely accepted
explanation for this phenomenon-this
reaction
occurs because the Fed is expected to raise or lower
the funds rate .in response to deviations of money
from its target path.
z1 This period is unique in that the
so closely that market participants
in the funds rate target on the day
by the Fed. See Cook and Hahn

Fed controlled the funds rate
could identify most changes
they were first implemented
(1989, pp. 332-3381.

z2 Cook and Hahn 119881 find that throughout this period the
Fed systematically used discount rate announcements with policy
content to signal persistent changes in the federal funds rate.
16

ECONOMIC

REVIEW,

A striking aspect of the regression coefficients in
Table VII is the relative stability from the threemonth through the twelve-month maturities. This
suggests that new information influencing expectations of the future level of the funds rate-even
though it has a strong effect on bill rates at all
maturities-has
little effect on the slope of the
Treasury yield curve from three to twelve months.
In light of this evidence it seems plausible that the
variance of the yield curve over this range has been
dominated by movement in the term premium, as
suggested by Mankiw and Miron.
A More General Monetary Policy
Explanation for the Regression Results
While the Mankiw-Miron hypothesis can help
explain the absence of forecasting power of the yield
curve from three to twelve months, it is inconsistent
with the evidence that the yield curve up to three
months and from one to five years has had forecasting
power. One can pose a more general version of the
monetary policy explanation that is consistent with
this evidence, and, we believe, mdre in line with the
way market participants actually view monetary
policy.
The Mankiw-Miron hypothesis assumes that the
Fed reacts continuously to new information affecting
its policy decisions, whereas in practice Fed policy
changes are of a more discontinuous nature. That
is, changes in the Fed’ target for the funds rate
s
typically occur infrequently after they are triggered
by the cumulative weight of ‘
new information on
economic activity and inflation. Consequently,
at
times there is a gap between the release of new
information influencing policy expectations and when
policy actually changes. This information could take
the form of a policy announcement-such
as a discount rate announcement-which
signals an upcoming change in the funds rate target. Or it could take
the form of news on an economic variable-such
as
the money supply or employment-that
is viewed
by market participants as likely to influence the Fed’
s
target for the funds rate.
If policy and news announcements affect expectations of changes in the funds rate over a relatively
short term, then the slope of the bill yield curve out
to three months will vary more in response to changing interest rate expectations than will the slope from
three to twelve months. In this case the reaction of
market participants to such announcements c6uld
generate a pattern of funds rate expectations that is
consistent with the regression results. For example,

SEPTEMBER/OCTOBER

1990

Table VII

THE REACTION OF TREASURY BILL RATES BY MATURITY TO
EVENTS CHANGING FEDERAL FUNDS RATE EXPECTATIONS*
(Coefficients

of Treasury

Bill Rate Regressions)

3-month

Federal

6-month

12-month

funds rate target changes:

Discount
Jan.

1974-Oct.

0.554

0.541

0.500

(8.10)

Sept.

(10.25)

(9.61)

1979

rate announcements:
0.26

0.32

0.30

(2.66)

1973-O&.

(3.54)

(3.15)

1979

0.73

0.61

0.59

(7.38)

Oct. 1979-Dec.

(7.61)

(7.54)

1985

Money announcements:
Sept.

0.072

0.072

(3.11)

1977-O&.

(4.73)

1979

Oct.

1982-Sept.

0.364

0.338

(6.58)

Oct. 1979-O&

(7.59)

1982

1984

0.190

0.216

(5.77)

(5.62)

* The funds rate target regression coefficients
and the discount rate announcement
regression coefficients
are from
Cook and Hahn [1988. 19891. The monev announcement
regression coefficients are from Gavin and Karamouzis f19841.
.
t-statistics are in par&the&.

suppose a discount rate announcement generates expectations of a 50 basis point change in the funds
rate the following week, after which no further change
in the rate is expected. Under the expectations
theory the effect on the slope of the yield curve out
to one or two months would be considerable, but
the effect on the slope from three to six months and
six to twelve months would be negligible.

changes are persistent (i.e., not quickly reversed)
but not permanent. 23 If so-and if market participants expected this type of funds rate behaviorthen increases in the funds rate target would be
associated with decreases in the slope of the yield
curve between short-term rates and rates on longer
maturities of five to ten years, and changes in this
slope would have some forecasting accuracy.

Hegde and McDonald [ 19863 find that Treasury
bill futures rates have substantially outperformed a
no-change forecast from one to four weeks prior to
delivery, even though they have not been superior
to a no-change forecast from five to thirteen weeks
prior to delivery. This evidence is consistent with
the hypothesis that market participants are at times
able to forecast rate changes over the near-term and
build these expectations into the yield curve.

A number of recent papers have suggested that the
forecasting power of the spread between long- and
short-term rates is at least partially a reflection of
monetary policy. (See Bernanke and Blinder [ 19891,
Laurent [ 19893, and Stock and Watson [ 1990, pp.
2.5261.) The basic reasoning is that monetary policy
has a strong influence over short-term rates but that

A second modification one could make to the
Mankiw-Miron hypothesis notes that funds rate target
FEDERAL

RESERVE

23 Fama and Bliss [ 19871 find that the one-year Treasury rate
is highly autocorrelated but slowly mean-reverting, which is
consistent with the view that changes in the funds rate are highly
persistent but not permanent.
BANK

OF RICHMOND

17

this influence diminishes at longer maturities. Hence,
if short-term rates are high relative to long-term rates
that is an indication that monetary policy is contractionary and a decline in inflation and interest rates
is likely in the future.

Chart 1

EXPECTED
CHANGE IN AND
LEVELOFFEDERALFUNDSRATE
Percentage

This explanation for the forecasting power of the
yield curve between long- and short-term rates seems
especially relevant for the periods from late 1969 to
late 1970, mid-1973 to mid-1974, and late 1979 to
mid-1982. Near the beginning of each of these
periods, the Fed raised the funds rate sharply because
of its concern over accelerating inflation, and shortterm rates rose well above long-term rates. (See
Laurent (1989, Figure 2, p. 261.) In each period the
rise in the funds rate and the downward-sloping yield
curve were eventually followed by a recession and
falling interest rates. 24 As illustrated in Chart 1,
the Goldsmith-Nagan survey participants expected
large declines in the funds rate throughout these
periods.25 These expectations
had considerable
accuracy at longer horizons of two to four years, even
though they were not very accurate at the three- and
six-month horizons.z6
If the above adjustments to the Mankiw-Miron
hypothesis are correct, then one would expect to see
the slope of the yield curve out to three months and
the slope from one to five years vary more than the
slope from three to twelve months in response to
policy actions or announcements signaling changes
in the funds rate.27 Numerous studies have provided evidence that the response of interest rates to
24 For more discussion of these episodes see Romer and Romer
[ 19893, who also suggest that the sharp rise in interest rates in
these periods resulted from monetary policy actions intended
to lower the rate of inflation.,
2s The funds rate used in Chart 1 is the average rate for the week
at the end of the quarter, as determined by the weekly rate that
had the greatest overlap with the last five trading days of the
quarter. Special factors at the end of 1985 and 1986 caused the
year-end weekly average rate to rise sharply above its level over
the surrounding weeks. In these two cases Chart 1 uses the
average rate for the previous week.
26 Of course, the evidence from the survey data that market participants expected large declines in interest rates three and six
months in the future in these episodes is inconsistent with the
Mankiw-Miron
hypothesis that market participants always
forecast small changes in rates at the three- and six-month
horizons. These episodes constitute a relatively small part of
the period covered by the survey data, however, and they may
be unique to this era. It may be that over the longer period
studied by Mankiw and Miron the generalization that expected
changes in interest rates at the three- and six-month horizons
were generally small is an accurate one.
27 In the case of a&&changes
in the funds rate target, however,
one would expect very short maturity rates to vary as much as
three- and six-month rates, since in this case the level of the
funds rate rises immediately.

18

ECONOMIC

REVIEW,

Points

Percent

20

1

18

0

16
-1
14
-2

12
10

-3

8
-4

6

-5

4
1970

72

74

76

78

80

82

84 86

88

policy actions and announcements influencing policy
expectations gradually declines at maturities greater
than a year. For example, Cook and Hahn [1989]
find that the reaction of Treasury rates to funds rate
target changes falls from 0.50 at the one-year maturity
to 0.29 at the five-year maturity and 0.13 at the tenyear maturity. Likewise, several papers including
Hardouvelis (19841 and Gavin and Karamouzis
[ 1984) have reported that the reaction of Treasury
rates to money announcements declines at longer
maturities.
The evidence at the short-end of the yield curve
is more limited and somewhat more ambiguous. The
reaction of interest rates to money announcements
has been studied by many authors, but only a few
have looked at the reaction of rates with shorter
maturities than three months. These studies have
found that the reaction of the one-month rate to
money announcements is smaller than the reaction
of longer-term money market rates, which is consistent with the notion that the yield curve out to
three months varies more in response to changing
policy anticipations than the curve from three months
to a year. Husted and Kitchen [1985, p. 4601 find
that the reaction of Eurodollar rates to announcements of unexpected increases in the money supplyas determined by the coefficients of a regression
similar to equation (2 1) above-rose from 0.28 at the
one-month maturity to 0.46 at the three-month
maturity and 0.44 at the six-month maturity. Hardouvelis [ 19841 finds that the reaction of the onemonth bill rate (0.24) was smaller than the reaction
of the one- to two-month forward rate (0.45), the

SEPTEMBER/OCTOBER

1990

two- to three-month forward rate (0.40) and the
three- to six-month forward rate (0.35). (The
sample period for the studies cited in this and the
following paragraph is from late 1979 or early 1980
to late 1982.).
Surprisingly, hotiever, the money announcement
literature indicates that the reaction of the one-day
funds rate and the one-week bill ‘
rate to money
announcements is’
not smaller than the reaction of
longer-term money market rates. Hardouvelis finds
a coefficient of 0.38 for the one-day funds rate, and
Roley and Walsh [ 19851 find .a coefficient of 0.43
for’the one-day funds. rate, 0.37 for the one-week
bill rate, and 0.36 for the three-month bill rate. A
possible explanation for the relatively large response
of the one-day and one-week rates is that under the
lagged reserve accounting system prevailing prior to
February 1984 the weekly money announcement
provided information on the current.statement week’
s
aggregate demand for reserves that influenced the
expected average funds rate for the statement
bill rateweek-and,
hence, the one-week
independent of any policy anticipations effect.28
VI.
BFHAVIOR OF ;I‘ TERM PREMIUM
HE
The evidence presented in Section III suggests that
a variable term premium plays an important role in
explaining the negligible forecasting power of ‘
the
yield curve from three to twelve months. This conclusion raises a final set of questions. First, how does
the term premium behave on average and at different
maturities? Second, what causes the term premium
to’change over time? The literature in this areaespecially
regarding .the second question-is
voluminous, yet largely inconclusive. Our purpose
here is simply to provide a brief review of this
literature and the difficulties researchers have faced
in trying to measure the term premium..

most common practice is to use the one-month rate
for the,benchmark (m = 1). The literature in this area
has found ,a positive average term premium in the
Treasury bill market at all. maturities. The average
term premium rises sharply for the first .couple of
months, increases at a decreasing rate out to around
five or six months,, and then flattens out. This
behavior is illustrated in Chart 2 which shows
estimates of the average. premium at different
maturities using the McCulloch data.29 McCulloch
[ 19871 shows. that the CRSP data provide a similar
picture. of the relationship between the average
premium and maturity in the bill market.30 Researchers have.found no evidence of a significant term
premium, on average, in the markets for private
money market securities such as commercial paper,
CDs, and Eurodollar- CDs. Fama [ 1986; p. ,178,
Table 11; for example, finds that- average term
29 For maturities up to six months, the estimate of the average
term premium in Chart 2 is the average of the annualized.returns
earned by holding a Treasury bill at a given maturity for one
month less the returns on a one-month bill. For maturities of
nine and twelve months, the estimate of the average premium
is the average return from holding a nine- or twelve-month bill
for three months less the return on a one-month bill. (In the
latter two cases three months is the shortest holding period yield
that can be calculated .using the McCulloch data.),
30 Fama (1984bl provides evidence using the.CRSI? data.that
the premium’
declines between nine and ten months; McCulloch
[ 19871, however, shows that this evidence results from the’
small
bid-ask spread on nine-month Treasury bills in the period.from
1964 through 1972 when the Treasurv was issuing new bills at
that maturib. He concludes that the. description ‘
ihat best ‘
fits
the CRSP data is that the premium rises monotonically to about
five months and has no further significant change.
‘ :
3’
,
; :. .
Chart 2

,,

::
-.

‘
3.

Pkrcentage
Points

(Treasury

.’

Bill&j

,,

‘
,

ESTIMATES OF ,AVERAGE
TERM PREMIUM

.’

The Average.
Term Premium in the
Money Market
Researchers have generally estimated the average
term premium in the money market .by calculating
over long periods of time the average excess returns
from holding n-month securities for m months versus the return from holding m-month securities. The
28 Along these lines, Strongin and Tarhan [1990, pp. 151-1521
conclude that “The response of the Fed funds rate [to money
announcements] cannot be,explained by either [policy anticipations or.expected inflation], but instead by the peculiar way
money shocks are transmitted to the reserve market under lagged
reserves accounting.”
FEDERAL

RESERVE

I

I

I

I

I

I

9

123456

.12

Maturity
BANK OF RICHMOND

19

premiums for privately issued securities over. his
whole sample period from January 1967 to January
198.5 are.close to zero.,Fama, however, also divides
his sample into .months when the yield curve was
monotonically upward sloping and months when the
yield curve was “humped” (i.e initially rising and then
falling). He finds ‘
that in months when the private
yield curve was upward sloping, the average term
premium was positive and rose with maturity,- and
in months when the private yield curve was humpedi
the average term premium initially was positive’
but
then.bedame negative at the longer maturities.
One type of, explanation for the positive average
premium in the bill market focuses on the preferences
of individual investors. Hicks [1946, pp. 144-Z)
argues that investors have a preference for shorter;
term securities because of the greater price volatility of long-term securities when interest rates
change. In contrast, he reasons that many borrowers
have a preference for long-term borrowing. Hence,
there is a “constitutional weakness” on the long side
of the inark& such that in equilibrium investors have
tb be offered a premium to invest iti longer-term
securities. In a similar vein, Kessel [1965, p. 451
argots that the ‘
maiket has a preference for shorterterm securities because of their greater liquidity: “The
shorter’
the t&m to maturity of a security, the smaller
is its vtilnerability’ to’ capital loss, and hence the
greater its liquidity and the smaller the yield differential ,between that security and money.“31
:

More recent papers have analyzed the term
premium in ,the &ext
of individuals who maximize
the expected utility of their lifetime consumption.J*
An idea that comes out of this literature is that the
term premium is likely to be positive if unexpected
capital losses (i.e. ‘
pdsitive future interest rate surprises) are generally positively correlated with
negative consumption surprises. In other words, investors are likely to demand a higher yield on longterm securities if they are likely to experience unexpected capital losses when times are unexpectedly
bad and their marginal utility of consumption is
relatively high.
A second explanation for the positive average
premium in the bill market, suggested by Rowe,
31There was a huge amount of literature on the expectations
theory and the term premium in the 1960s and early 1970s. For
a review of this literature see Van Horne [ 1984, Chapter 41 and
Malkiel [ 19701.

Lawler, and Cook [1986, pp. 9-101 and Toevs and
Mond [ 19881, focuses on the unique characteristics
of the market. Treasury bills can be used to satisfy
numerous institutional and regulatory requirements,
such as serving as collateral for tax and loan accounts
at commercial banks and satisfying margin ,requirem&s’ on futures contracts. To the extent that the
holding period for these purpdses tends to be short,
investors might prefer to minimize capital risk by
ho!ding short-term bills to satisfy them. Moreover,
the Treasury is not sensitive to interest rates at
different maturities in its supply of bills, so there is
no pressure from the supply &de to equalize the
expected cost of issuing bills at different maturities.
In contrast, banks might be expected, to issue more
three-month CDs if the expecttid cost of raising funds
this way were systematically lower than the.cost of
raising funds by issuing six-month CDs, and this
behavior would raise the three-month rate relative
to the six-month rate and reduce the premium.
Measuring the Behavior of:the
Term Premium over Time
A number of approaches have been taken in the
literature to measure the behavior of the term
premium over time. The simplest approach is to
assume that the forward rate premium is an accurate
representation of the, term premium. Suppose the
expected, change in rates at, any point in time is
negligible so that the forward.rate premium is completely dominated by variation in the term premium.
Then, as Fama, [ 1986, p. isi] suggests, the forward
rate premium can “provide a direct picture of the
behavior of the expected term...premium.”
As
discussed earlier, ‘
however, the Goldsmith-Nagan
survey data suggest that at times, market participants
have expected large changes in rates. If so, then in
these periods the forward rate premium provides an
inaccurate picture of the term premium.
A second approach to measuring the term premium
is to subtract the expected interest rate level from
the Goldsmith-Nagan survey from the comparable
forward rate at the time of the survey. Chart 3 shows
(a) the difference between the forward rate on threemonth bills three months ahead, and the expected
three-month bill rate three months ahead and (b) the
difference between the forward rate on threeymonth,
bills six months ahead.and the expected three-month
bill rate six months ahead.s3 The chart shows a clear
33The vertical lines in Charts 3,4, and 5 show quarterly business
cycle peaks and troughs. Peaks are the fourth-quarte; of 1969,
fourth auarter of. 1973. first auarter of 1980. and third Quarter
of 198 1. Troughs are the fourth quarter of 1970, the first quarter.
of 1975, third quarter of 1980, and the fourth quarter of 1982.
L

L

32 For an example of this approach see Sargent
102-1051: Abken [1990) discuSses this literature.
20

ECONOMIC

[1987, pp.

REVIEW,

SEPTEMBER/OCTOBER

1990

Chart 3

Chart 4

SURVEY TREASURY BILL
TERM PREMIUMS

FORWARD RATE PREMIUM AND
SURVEY TERM PREMIUM

Percentage
Points

Percentage
Points

4
3

Forward
- Rate
Premium

2
1
0
-1

I

-1
1970 72

74

76

78

80

82

84

Note: The dashed line is the difference between the forward and expected three-month
rates six months
ahead. The solid line is the difference between the
forward and expected three-month rates three months
ahead. The shaded areas represent recessions (peak to
trough).

tendency for the survey term premiums to be relatively high in recessions and low in expansions. (This
tendency was captured in the regression results
reported in Section IV by the signficant negative
coefficient of the recession dummy variable.)
Chart 4 shows that the survey term premium and
the forward premium generally move together, but
large differences occasionally occur when the survey
indicates large expected declines in rates. The most
striking difference in the two estimates of the term
premium is in the period from late 1979 though
mid-1982 when interest rates were unusually high
and were expected to fall by the survey participants.
In this situation the survey term premium rose well
above the forward rate premium. Chart 5 compares
the survey premiums for bills and private money
market securities. The private premium generally
follows the same pattern as the bill premium-rising
in recessions and falling in expansions-although
occasionally there are significant differences in the
two premiums.
A third estimate of the term premium is the forward deviation, i.e. the difference between the forward rate and the subsequently realized spot rate.
As discussed in Section III, the forward deviation can
be decomposed into an expected term premium and
an, interest rate forecast error. Both the GoldsmithNagan survey data and futures market data indicate
that market participants .have had little ability to
forecast rates at the three- and six-month horizons.
FEDERAL

1970 72

86

RESERVE

74

I

76

I

1

78

1 82

84

86

Note: The forward rate premium is the forward rate on
three-month Treasury bills three months ahead minus
the current three-month spot rate. The survey term
premium is the forward rate minus the expected threemonth spot rate three months ahead. Shaded areas
represent recessions.

As a result, the forward deviation is an extremely
volatile measure dominated by interest rate forecast
errors.
A final approach used to estimate the term
premium is to employ regression methods to generate
“expected” interest rates with data available to market
participants at the time of the forecast., These

Chart 5

SURVEY TERM PREMIUMS ON
TREASURY BILLS AND
PRIVATE ISSUES
Percentage
Points

4
3
'2
1
0
-1
:
1970 72

74

76

78. 80

82

84

86

Note: The term premiums are the difference between
the forward and expected three-month spot rates three
months ahead. Shaded areas represent recessions.
BANK OF RICHMOND

21

estimates can then be used along with contemporaneous forward rates to calculate estimates of the term
premium. Since these forecasting equations have little
power to predict changes in interest rates, one might
expect this approach to provide estimates of the term
premium that are similar to the forward rate premium.
We are not aware, however, of any studies that have
made this comparison.
Determinants of the
Variable Term Premium
The estimates of the term premium shown in
Charts 3-5 suggest that term premiums in the money
market tend to be low in periods of economic expansions and high in periods of weakness. This is
consistent with a recent conclusion by Fama and
French [1989, p. 431 that, term premiums “move
opposite to business conditions,” This is not a
universally accepted description of the behavior of
term premiums, however. Numerous variables are
correlated with economic conditions, and the charts
might be capturing a correlation of the premium with
some other variable such as the level of interest
rates.34 Moreover, even if one, accepts the description that term premiums move opposite to business
conditions as accurate, there is still no generally accepted explanation for why term premiums rise
around recessions and fall in expansions.
Numerous
papers have attempted
to make
judgments about the determinants
of the. term
premium by regressing one of the measures of the
premium described above on various possible explanatory variables .3s Two explanatory variables often
included in these regressions are the volatility of
interest rates and the,level of interest rates. Hicks
[ 19461 reasons.that.the term premium is compensation for the capital risk resulting from interest rate
movements and, therefore, increases in interest rate
34 For example, on the basis of the Goldsmith-Nagan survey data
and a chart similar to Chart 3, Froot [1989, p. 299, Figure l]
concludes that “the surveys suggest that term premia rose
substantially during periods of high interest rate volatility [p.
3001.” He also concludes that the survey premia “are highly
positively correlated with nominal, interest rates ,and inflation
[p. 303].“.Friedman 11979, p. 9721 on the basis of regressions
using the Goldsmith-Nagan data from September 1969 through
March 1977 concludes that “the results make clear that the basic
relation is between the term premium and interest rate levels,
not economic activity . . .“.
3s For example Friedman I19791 uses the premium calculated
from the survey data as the dependent variable in his regressions, Kessel 119651 uses the forward deviation in his regressions, and Pesando [1975] estimates interest rate forecasting
equations to calculate an estimate of the term premium to use
as the dependent variable in his regressions.

22

ECONOMIC

REVIEW,

volatility should increase the premium demanded by
investors. The argument that the level of rates should
be a determinant of the term premium is generally
associated with Kessel [ 1965, pp. 25-261. He argues
that short-term bills are better money substitutes than
long-term bills, and since an increase in interest rates
increases the cost of holding money, the yield on
short-term bills should fall relative to the yield on
long-term bills when rates rise.
Papers that find the volatility of interest .rates to
be a significant determinant of the premium include
Fama [ 19761, Heuson [ 19881, and Lauterbach
[ 19891 .s6 Papers that find the level of rates to be a
determinant of the premium include Kessel [ 19651,
Pesando [1975], and Friedman [1979]. Other explanatory variables that have been used in studies
of the premium include the relative supplies of
securities at different maturities, the unemployment
rate, industrial production, and the spread between
yields on high- and low-risk securities. As ,Shiller
[ 1987, pp. 56-571 concludes in his survey article on
the term structure of interest rates: “It is difficult to
produce a useful summary of [the] conflicting results”
from the empirical studies of the term premium. The
main conclusion is that no consensus has emerged
in the literature on what macroeconomic variable the
term premium is most closely related to or on why
the term premium varies so much.3’
VII.

CONCLUSION
The studies surveyed in this paper find that over
long periods of time the yield curve from three to
twelve months has had negligible power to forecast
interest rates three and six months in the future. The
36 Fama 119761 assumes that the expected real return on a onemonth Treasury bill is constant over his sample period and
therefore concludes that his measure of the volatility of interest
rates is capturing the positive effect of inflation uncertainty on
expected term premiums on multimonth bills.
37 One possibility that we do not discuss here is that the variable
term premium results from factors related to specific Treasury
bill issues and maturities. For example, Park and Reinganum
[ 19861 find that Treasury bill yields maturing at the end of the
month and especially atthe end of the year-have lower yields
than surroundine maturities. and Nelson and Siegel 119871 find
evidence of both maturity-specific and issue-spe&id effects on
bill yields. Also, it is also widely believed in the financial markets
that a shortage or abundance of a particular bill issue can cause
that issue’ yield to differ significantly from the yields on surs
rounding maturities. The McCulloch data used in this paper,
however, are constructed from a curve-fitting technique and
therefore should generally not be affected by such factors.
Moreover, the evidence presented earlier in the paper suggests
the variable term premium is pervasive throughout the money
market and not just due to special factors operating in the bill
market.

SEPTEMBER/OCTOBER

1990

yield curve out to three months has had forecasting
power for the one-month ahead rate, however, and
the yield curve from one to five years has had
forecasting power for the one-year rate over ‘
the
following three or four years.
The research in this area has suggested two broad
reasons for the poor forecasting poker of the yield
curve from three to twelve months:The first is that
the variation in the term premium at the three- and
six-month horizons has been substantial relative to
the variation in the expected change in rates:The
second is that even when market participants have
forecasted significant changes in interest rates at the
three- and six-month horizons; their forecasts have
been poor rit these horizons.
,’
An understanding of how market participants form
monetary policy expectations may provide insight
into some of the results in this literature. A monetary
policy explanation for the poor forecasting power of
the yield curve from three to twelve months is that
market participants expect changes in the Fed’
s
federal.funds rate target to be persistent. According
to this explanation, three-, six-, and twelve-month
rates tend to move the same amount in reaction to
changes in the funds rate target and, therefore,
changes in the slope of the yield curve over this range

are dominated by movement in the term premium.
The forecasting power of the yield curve out to three
months may reflect the ability of market participants
to forecast over short horizons the reaction of the
Fed to new information influencing its policy decisions. And the forecasting power of the yield curve
from one to five years may partially reflect the belief
of market participants that over longer periods of time
‘
changes in the funds rate target are likely to be
reversed, especially after the Fed has raised the funds
rate sharply in reaction to rising ‘
inflation.
The evidence cited in this paper in favor of a
monetary policy explanation for the regression results
is limited, however, and the explanation has not been
*
uruversally, or’
even widely, accepted. There’ also
is
no general agreement on why the term premium
varies so much,, although the Goldsmith-Nagan
survey data strongly suggest that the premium rises
when economic conditions deteriorate. A brief assessment of the literature surveyed in this paper is that
it has done a good job of documenting the forecasting
power of various parts of the yield curve, and it has
suggested some plausible and interesting answers to
some of the major questions in this area. A comprehensive explanation for these questions, however,
‘
awaits further research.

APPENDIXI
INTERESTRATECONVERSIONS
All interest rates in the paper are continuously compounded annual rates. No conversion is necessary
for the, McCulloch .Treasury bill rate data, which
come in this form. Three-month Treasury bill rate
forecasts from the‘
Goldsmith-Nagan
survey are on
a 360-day discount basis, however, as are all commercial paper rates used in the paper. Eurodollar,
CD, and federal funds rates are quoted on a simple
interest 360-day basis. Prices per $1 of return are
calculated from the quoted yields, Q, u,sing the
formulas:
P=l

- [(Q/lOO)~(t/360)1

,:

FEDERAL

(1)

RESERVE

for bills and commercial paper and
P = l/[(Q/lOO)(t/360)

+ 11

(2)

for Eurodollars, CDs, and federal funds rates. “t” is
the. days from settlement to maturity: .30, 90, and
180 days. for- commercial
paper, CDs, and
Eurodollars; 91 days for Treasury bills; and 1 day
for federal funds; Prices are converted to continuously
compounded yields using the formula:
.
r = 1(365/t)lnP
'(3)
where 1nP is the natural logarithm of P.

BANK OF RICHMOND

23

APPEN~IxII
THECOEFFICIENT.OFTHEFORWARD&ATEPREMIUM
INTHESTANDARDRE,GRESSION
:
The standard regression equation .is:
r(3:t+3)

- r3 = a + b[f(6,3)
+ u:t+3

- r3]
(1)

Denote the ‘
variance of x as d(x), .the standard
deviation as a(x), and the correlation coefficient
between x and y as p. Recall that cov(x,y) =
pa(x
Then equation (7) can be written:

To simplify the notation rewrite this as:
Ar = a + b(FP)

(2)

= B[E(Ar)]

FP .r E(Ar) + 8

(3)

Ar = E(Ar) + e

0

The probability limit (abbreviated as plt) of the
ordinary least squares estimate of b in equation (2) is:
cov(FP, Ar)

(5)

var(FP)
Substituting
plt b =

d[E(Ar)]

+ d(0)

+ Zcov[E(Ar),B]

c?[E(Ar)] + pc@ja[E(Ar)]

where Ar is the rate change and FP is the forward
rate premium. Recall also that the forward rate
premium can be decomposed into the.expected rate
change and the expected term premium, 8, and
the actual change in the interest rate can be decomposed into the expected change and a forecasi
error, e:

plt b =

oa[E(Ar)] + cov[e,E(Ar)]

plt b =

+ u:t+3

+ &J)

+ Zpa[E(Ar)]@3)

(8)

This is the,expression in Hardouvelis [ 1988, p.
3421. It is, also similar to the expression in Mankiw
and Miron [ 1986, ,p. 2191, except that the term
premium in their framework is equal to one-half the
premium.above.
Note that the probability limit of
b is one if the premium is a constant and one-half
if the standard deviation of the term premium equals
the standard deviation of the expected change in
rates.
Now .substitute
plt b =

equation (4) into (5) to get:

cov(FP,

E(Ar) + e)

var(FP)

(3) and (4) into (5) yields:

= cov(FP,

E(A)) + cov(FP,e)
var(FP)

cov]E(Ar) + 8, E(Ar) + ej
var[E(Ar) +0]

Substituting

(9)

(3) into (9) yields:

= cov(E(Ar) +8, E(Ar)]
pit b _ cov(FP, FP-8)

var [E(Ar) + 01

+ cov(FP,e)

var(FP)

+ cov[E(Ar) +8, e]

(6)

‘ var(FP)
=

var[E(Ar) +e]

- cov(FP,B)

+ cov(FP,e)

var(FP)
Suppose ‘
the rational expectations assumption is
valid. Then the forecast error, e, is not correlated
with information available at the time of the forecast, which includes the expected change in rates and
the expected premium. Then the second term on
the right-hand side of equation (6) equals 0 and we
get the expression:

pit b = cov[WAd +e, Wdl

(7)

var(E(Ar) +t?]

24

ECONOMIC

REVIEW,

= l-

cov(FP,B) + cov(FP,e)
var(FP)

(1 o)

var(FP)

Equation (10) says that a positive correlation of the
term premium with the forward rate premium or a
negative correlation of forecast errors with the forward rate premium will reduce the coefficient of the
forward rate premium below the value of one
predicted by the expectations theory.

SEPTEMBER/OCTOBER

1990

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Fama, Eugene F., and Kenneth R. French. “Business Conditions and Expected Returns on Stocks and Bonds.” Journal
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Heuson, Andrea J. “The Term Premia Relationship Implicit
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Hicks, J. R. Valueand Capita/, second edition. London: Oxford
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Husted, Steven, and John Kitchen. “Some Evidence on the
International Transmission of the U.S. Money Supply
Announcement
Effects.” Journal of Money, Credit and
Banking 17 (November 1985, Part I): 456-466.
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Friedman, Benjamin M. “Interest Rate Expectations Versus
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Th Joumaf of Finance 34 (September 1979): 965-973.

Kessel; Reuben A. T/reCyckcalBehavior of the Term Structureof
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Rationality’ of Interest
Rate Expectations.” Jounral of Monetary
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1980): 453-465.

Laurent, Robert D. “Testing the Spread.” Federal Reserve Bank
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Froot, Kenneth A. “New Hope for the Expectations Hypothesis
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Lauterbach, Beni. “Consumption Volatility, Production Volatility, Spot-Rate Volatility, and the Returns on Treasury
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BANK OF RICHMOND

25

Mankiw, N. Gregory, and Jeffrey A. Miron. “The Changing
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Rowe, Timothy D., Thomas A. Lawler, and Timothy Cook.
“Treasury Bill Versus Private Money Market Yield Curves.”
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(July/August 1986): 3~12.
Sargent, Thomas J.. DynamicMacmeconomic
Thoty. Cambridge,
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McCarthy, F. Ward. “Basics of Fed Watching” in Frank J.
Fabozzi (ed.) Th Handbookof TredcurySecUrties. Chicago:
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McCulloch, J. Huston. “The Monotonicity of the Term
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. “U.S. Term Structure Data.” Appendix II in
Robert J. Shiller “The Term Structure of Interest Rates.”
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Mishkin, Frederic S. “The Information in the Term Structure:
Some Further Results.” Joumai of Appked Ecotaotnetks3
(October-December
1988): 307-314.

Shiller, Robert J., John Y. Campbell, and Kermit L. Schoenholtz.
“Forward Rates and.Future Policy: Interpreting the Term
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Acriwiity(1:1983): 173-217.
Simon, ‘
David P. “Expectations and the Treasury Bill-Federal
Funds Rate Spread over Recent Monetary Policy Regimes.”
Th Journal of Finance 45 (June 1990): 567-577.
Startz, Richard. “Do Forecast Errors or Term Premia Really
Make the Difference Between Long and Short Rates?”
JoumalofFinan&~&mon+r 10 (November 1982): 323-329.

Nelson, Charles R., and Andrew F. Siegel. “Parsimonious
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Policy.” Journa/ of ,Money, Credit and Banking 22 (May
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Park, Sang Yong, and Marc R. Reinganum. “The Puzzling
Price Behavior of Treasury Bills that Mature at the Turn
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Toevs, Alden L., and David J. Mond. “Capturing Liquidity
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Pesando, James E. “Determinants of Term Premiums in the
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ReGw (September-October
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100 (Supplement 1985): 101 l-1039.

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Economic Research, May 1989.

26

ECONOMIC

REVIEW,

SEPTEMBER/OCTOBER

1990

The Macroeconomic Effects of Government Spending
C/ring-Shng Mao *

I.
INTRODUCTION
Peacetime
government
spending
has risen
steadily from less than 10 percent of GNP in the
1920s to about 30 percent of GNP today.’ The larger
role of government has generated increasing interest
in the macroeconomic effects of government spending.-This paper examines the effects of government
spending in a simple macromodel. A small-scale
neoclassical model is used for analyzing a classical
problem in the literature, namely, the effects of temporary and persistent changes in government spending under a balanced budget. It is found that under
a simple lump-sum tax financing scheme, persistent
changes in government spending have much larger
effects .on economic aggregates (such as consumption, output, labor, and investment) than do temporary changes. This result replicates the findings
of recent studies by King (1989) and Aiyagari,
Christiano, and Eichenbaum (1990).
The second purpose of this paper is to analyze the
effects of government spending under different tax
financing regimes. For simplicity or technical reasons,
the above studies assume that government purchases
are financed by lump-sum taxes.* This assumption
severely limits the applicability of the theory because
most taxes are distortionary. The current paper extends the existing literature to the important case of
income tax financing. The results ‘
stemming from this
extension are fundamentally different from those of
lump-sum tax financing. For example, an increase
in government spending that is financed by a lumpsum tax under a balanced budget will increase labor
effort and real output because of the dominating income effect. Under income tax financing, however,
both labor effort and output decline instead of rise
in response to an increase in government spending.
’ I would like to thank Tim Cook, Michael Dotsey, Marvin
Goodfriend, Robert Hetzel, and Tom Humphrey for helpful
comments. All errors are my own.
r For a statistical review of government

spending,

This paper is organized as follows: Section II
describes a model economy that will be used for
analyzing the effects of government spending. Section III analyzes the consumer’ problem. Section IV
s
then calibrates the model economy and considers a
specific example. The effects of temporary and persistent changes in government spending, under both
the lump-sum tax and the income tax regime, are
discussed in Section V. Section VI concludes the
paper and points out possible extensions for future
studies.

II.
THE ECONOMY
The hypothetical
economy is assumed to be
populated by a large,number of identical and infinitely
lived consumers. Since consumers are all alike by
assumption, their behavior can be represented in
terms of a single representative agent. At each date
t, the representative agent values services from consumption of a single commodity ct and leisure It. It
is assumed that both leisure and the consumption
goods are normal in the sense that more is always
desired to less and that the utility function u(ct,lt)
satisfies the usual restrictions, namely, it is strictly
increasing, concave, and twice differentiable.
The consumer derives his income from three different sources. First, at time t the consumer provides
labor s,ervices nt (hours of work) to the market and
earns wage income wtnt, where wt is the marketdetermined hourly real wage rate expressed in consumption units. Labor hours are constrained by the
total time endowment, which is normalized to one.
Thus, It + nt = 1. The second source of income
derives from the holding of a single asset called
capital. At the beginning of each period, the consumer rents to firms the amount of capital kt carried
from the previous period and collects capital income
rtkt, where rt is the market-determined
rental rate
expressed in consumption units. In each period,
the government imposes a uniform tax rate rt on
wage income and capital income so that the net-oftax earned income for the consumer is (1- -rt)(wtnt
+ rtkt).3 The final source of income is the lump-sum

see Barro

(1984).

2 A notable exception is Baxter and King (1990) who considered
the case of a proportional tax. Barro (1984) also discussed the
implications of income tax financing.
FEDERAL

RESERVE

3 For simplicity, wage income and’
capital income are assumed
to be taxed at the ‘
same rate. This assumption may not represent the actual tax scheme in the U.S. where capital income (e.g.,
capital gains) is usually taxed at a lower rate than is wage income.
BANK OF RICHMOND

27

transfer vt from the government. Depending upon
the budget constraint of the government, the lumpsum payment may be negative, in which case there
is a lump-sum tax imposed on the consumer. The
total disposable income for the consumer at time t
is therefore (1 - 7Jwtnt + (1 - Tt)rtkt + vt, which will
be allocated between consumption and investment.
In short, the budget constraint for the consumer at
time t is:
ct + it = (1 - Tt)wtnt + (1 -Tt)rtkt

+ vt,

(1)

where it = kt + I- (1 - 6)kt is gross investment4 arid
6 is the depreciation rate of capital (0 < 6 < 1).
While the capital stock will always be positive, gross
investment is allowed to become negative. That is,
investment is reversible in the sense that the consumer may acttially eat some existing capital stock
if he decides to do so.5
The consumer’ problem is to choose a sequence
s
of contingent plans for consumptitin and labor supply,
taking prices as given, so as to maximize his discounted expected lifetime utility subject to the budget
constraint. .Formally, the consumer solves the following maximization problem:

mar%[t~081u(r,,lt~‘
,0
subject to

< P <

ct + it = (l.-~t)(w*nt

1,

+ rtkt) + vt,

and
It + nt = 1,

for all t,

where fl is the time preference discounting factor and
Eo is the conditional expectation operator. Expectations are taken conditional on the future course
of government spending, which will be discussed
shortly. The optimal solution of the consumer’
s
problem will be characterized in the next section.
As in the c,ase of consumers, there are a large
number of identical firms in the economy; each firm
accesses. a constant returns to scale technologjr
represented by the production function F(kt,nt).
During each period, the firm chooses inputs in order
to maximize the current profit (or output) at the
market-determined wage rate and rental rate. Let yt
denote output’ at time t; then the firm solves the
following problem:

max

[yt - wtnt - r,kJ

subject to

yt = F(kt,nt).

Note that the firm’ problem is much simpler than
s
that of consumers; it does not involve any intertemporal trade-off as in the consumer’ problem. Since
s
the market is assumed to be competitive, the zero
profit condition dictates that capital and labor will
be employed up to the point where the rental rate
rt and the wage rate wt equal the marginal product
of capital and’
labor, r&pectively. That is:

wt = F&t,nt) and rt = F&m)

(2)

where F, and Fk are the marginal product of labor
and capital, respectively.6 To focus on government
fiscal shocks, it has been assumed that there is no
uncertainty in the firm’ production process. Incors
porating such uncertainty into the model is easy, but
unnecessary. Also, for simplicity, it is assumed that
the firm’ income or profit is not taxed.’
s
The role of the govern&&t in this ,hypothetical
economy is a simple one. It collects taxes and consumes portions of real output each period. It is assumed that government spending is not utility- or
production-enhancing;
the resources claimed by the
government are simply “thrown into the ocean”
and vanish. This’ assumption may not be the most
interesting way to model tlie function of the government, but it serves as a useful point of departure.
Thus, let gt, be the percentage of output that the
government claims each period. .Then government
purchases at time t are gtyt. In order to finance its
purchases, the government collects taxes Tt(wtnt +
rtkt), which are equal to 7tyt in. view of the constant
returns to scale technology. As noted before, the
variable Tt is the income tax rate. The budget constraint of the government at time t, expressed in per
capita terms, is:

gtyt + vt = 7tyt.

(3)

In short; equation (3) states that the sum of government purchases g,y, and lump-sum transfers vt must
equal tax revenues 7tyt. I rule out the possibility of
debt financing and money creation as alternative
means to finance government purchases. That is, the

4 The gross investment it is the sum of the net investment
(kt $I- kt), and the replacement investment 6kt.

6 Throughout the paper, the notation F,,(.) wiIl be used to denote
the partial derivative of the function F with respect to the
argument nt, which is the marginal product of labor. Similar
quantities are defined accordingly.

5 Later on, the shock I choose turns out to generate negative
gross investment at the time of impact, but not later.

’ It should be mentioned, however, that the personal income
tax in the hypothetical economy is equivalent to a production tax.

28

ECONOMIC

REVIEW.

SEPTEMBER/OCTOBER

1990

government fiscal policy will be conducted
balanced budget constraint.

III.
THE EQUILIBRIUM

under a

The variable gt is an exogenous policy instrument
that is assumed to be a random variable: Ideally, the
government would make gt contingent on certain
variables in the economy such as output, and labor
hours. However, a simplistic view-will be taken regarding the policy process {gt):Specifically, gt.is assumed to follow a first-order Markov process with
a given transition probability that is known to all
agents in the economy. For the bulk of the analysis,
the transition probability will be further structured
so that it gives rise to the following autoregressive
representation:

The equilibrium of the model economy requires
that the commodity market clear at each date and
that consumers and firms solve their maximization
problems at the given market prices. A formal definition of the equilibrium is discussed in the appendix.
Here we focus on characterizing the firm’ and cqns
sumer’ equilibrium.
s
As noted before, the firm’ problem is straightfors
ward. It requires, as stated in equation (Z), that the
rental rate and the real wage rate equal the marginal
product of capital and labor, respectively. The consumer’ problem requires that the following two firsts
order necessary conditions be satisfied in equilibrium:

Eigt + 1ktl = (1 -p)g’ + P&t, :

W(Ct,wJch~t)

0 SP < 1.

(4)

In this representation,, the conditional mean of gt + 1
depends only on its immediate past plus a constant
term (1 -p)g*. The quantity g’ is the steady state
or long-run level of the government share of GNP.
The autoregressive parameter p, assumed to be nonnegative and less than one, will determine the persistence of government spending. The larger p is,
the more lasting will be the displacement of gt. If
p = 0, then changes ‘ government spending will
in
be completely temporary.
Although the government is not allowed to print
money or issue debts to finance its purchases,‘ still
it
has some latitude in choosing different tax schemes.
Two idealized tax systems will be considered in this
paper: (1) rt = 0 .and (2) rt = gt. In the first case,
the government finances all its purchases by a lumpsum tax. That is, the transfer vt is negative and equals
gtyt in absolute value. In the second case, all government purchases are financed by an income tax and
the lump-sum transfer will be zero (i.e., vt = 0). This
policy exerts the greatest distortion on the behavior
of consumers.
It is not difficult to conceive that the effects of
government spending will depend upon the way it
is financed. For instance, if the spending is financed
by an income tax, there will be substitution effects
that will distort market outcomes. Even’ the case
in
of a lump-sum tax, market prices will still have to
adjust in response to changes in quantities that are
induced by income effects. It is impossible to assess
the impact of government spending without explicitly
considering’ the market equilibrium.
FEDERAL

RESERVE

=

(1 - 7th.

(5)

UcWt) = P Et[uch + dt + 1)
[1 + (1 -7t+drt+l-611.

(6)

Equation (5) states that the rate of substitution of
consumption for leisure (i.e., the ratio of their
marginal utilities) should equal the opportunity cost
of leisure, which is the after-tax wage rate. Equation
(6) states that the utility-denominated price of current consumption (i.e., marginal utility of consumption) should equal the discounted expected
return on saving, which is the expected value of
the product of the after-tax return to investment
[ 1 + (1 -rt + i)rt + i-61 and the next periods
marginal utility of consumption discounted at the rate
fl.8 This condition implies that in equilibrium the
consumer is indifferent between consuming one
extra unit of output today and investing it in the form
of capital and consuming tomorrow. Equations (5)
and (6) together with the ,budget constraint (1) and
the time .constraint It + nt = 1 completely
characterize the. consumer’ equilibrium.
s
Figure 1 ‘
presents a two-quadrant’
diagram to illustrate the determination of the consumer’ equilibrium.
s
For this purpose, we assume that there is no uncertainty in the economy and that the utility function
is homothetic9 The right-hand quadrant depicts the
‘
.
a Note ‘
that in a deterministic context, the gross return to
investment will be equal to’
one plus the real interest rate, which
is the ratio-of marginal utilities of consumption between time
t and time t+l.
9 A utility function is called homothetic if the marginal rate of
substitution depends only on the consumption-leisure
ratio. A
homothetic utility function has the property that the slope of
the indifference curve is constant along a given ray from the
origin.
BANK

OF RICHMOND

29

Figiire 1

.those implied..by the-intratemporal-equilibrium point
E.10 Points E and F jointly characterize .the consumer’ equilibrium. Other quantities such as leisure
s
(labor hours) and time t + 1 consumption can be easily
derived once the equilibrium point is determined.
I

CONSUMER’S EQUILIBRIUM
Taking the real interest rate. the after-tax wage rate,

and the after-tax rental rate as given
’

5

1

-[(l-~t+,)rt+,+

Slope

=

A

The. appendix sketches a. numerical procedure
which’
permits computation of the equilibrium and
quantitative assessment of the effects of government
spending. This approach requires one&to take an explicit stand on -the parameter structure of the
economy. The’
.rest of the paper’ therefore focuses
-on a specific example and works out the equilibrium
implications. of changes in government spending.

l-4

IV.
CALIBRATING THE MODEL
Intertemporal
Equilibrium

Intratemporal
Equilibrium

The example considered
rithmic utility function:
uict, it) F @ In ct + (T-0)

trade-off between consumption (measured along the
vertical axis) and leisure (measured along the horizontal axis) for a given wage rate and tax rate at time
t. The budget line in the right-hand quadrant for the
consumer at time t has two .components: the vertical segment corresponds to the nonwage income
which is fixed at the beginning of the period and
equals [ 1 + (1 - 7t)rt - 61kt + vt; the sloping segment
corresponds to labor income (1 - Tt)wtnt and has the
slope -(l.-7t)wt.
From equation .(5), the.slope df
-the indifference curve must equal the after-tax wage
rate in equilibrium. Since the utility function is
homothetic, this condition determines an equilibrium
consumption-leisure ratio that is represented by the
ray OA extended from the origin. .
Equation (5) alone cannot pin down. the e&ilibrium
point, however. To Jocate the equilibrium, ‘
one must
determine saving from equation (6). Consider the
.point E along the. ray QA..There is an indifference
curve tangent at E-.with slope equal to -(l-T 7t)wt.
The total income associated with this point, OB, is
divided between consumption-and investment. If invested,, the income available at time t + 1 is OC,
which is measured from right to left along the
horizontal,axis in quadrant 2. The absolute, value of
the Slope of the bud&t line BC is the after-tax rate
of return to capital. [i.e., 1 f (1 -.+rt+‘ + 1 -S].
l)rt
According to equation (6), the intertemporal equilibrium will be achieved at point F, where the indifference curve is tgngent to the budget line BC. The
pdint F determines the optimal saving (i.e., kt+ 1)
BD and time t consumption OD which coincide with
30

ECONOMIC

REVIEW,

and a Cobb-Douglas
. .
F&t,

nt)

=

kt” nt

here involves a loga-

In !t,

production
1--,

,

0 <’ 0 <

1,

function:

0 < a,<

1.

This specification is widely used’
& the literature
because it generates dynamics that rodghly match
several imp&&
features of business cycles’ the
in
U.S. (see, for example, King, Plosser, and Rebel0
(1988)). Our experiment assumes the following
values: ai = 0:3, 19=’0.3, p = 0.96, and 6 = 0.05.
In addition to pieferendes and technology, one
needs explicitly to ‘
spell out .the process of .governmerit’ spending. As mentioned before, the variable
gt, i.e., the ratio qf government spending to real outptit, is assumed to follow a first-order Mtirkov process., In what follows,’the autoregressive parameter
p is assigned kither a value of 0 in the case of’ tems
porary government spending oi a value of 0.9 in the
case of ‘ more persistent goveriment spending.‘
a
Further, the random variable gt is assumed to possess
a binomial distribution with probabilities concentrated
on’
five distinct points o&r a bounded interva!. The
mean and variance..of gt are ,ta&en to be 9.3.. and
0.005, respectititiljr. These figures imply that gt will
fluctuate around 30 ,percenF. (i.e., g’ = 0.3),
ranging .approximately from 15, percent to 45
.

._’

;O.if the intratemporal
equilibrium and the intertemporal
equilibrium do not imply the same consumption and saving
decisions, then andther point along the ray OA must be chosen
until the two equilibria are consistent:

SEPTEMBER/OCTOBER

1990

percent. Given this specification, the transition probability of gt, needed to numerically solve the model,
is constructed using the method proposed by Rebel0
and Rouwenhorst (1989).
V.

DYNAMIC ‘EFFECTS OF
GOVERNMENT
SPENDING
Consider the following scenario: Suppose, initi:
ally, that the economy has settled at its steady state
equilibrium, and that government spending has
reached its long-run level relative to the economy’
s
real output such that 30 percent of real output is
claimed by the government. At date 1 the government raises taxes and increases spending. Thereafter,
the ratio of government spending to real output
follows a time path prescribed by the autoregressive
process and gradually returns to its steady state. The
left- and right-hand sides of Figure 2 plot the mean
path of gt, measured as percentage deviations from
the steady state, for p = 0 and p’ = 0.9, respectively. These hypothetical paths are generated by
taking an average of 5000 random realizations of gt,
conditional on the given change at the initial date:
Notice that the case of p = 0.9 yields a more lasting
displacement of gt.
,.
Given the displacement of government spending,
what would be the dynamic response of quantities
and prices in the pure lump-sum versus pure income
tax regime? To answer this question, one needs to
understand the forces that govern individual behavior.
It is instructive to consider a simpler case in which
the increase in government spending is financed by.
a lump-sum tax. Figure 3 shows the shift in the consumer’ equilibrium for this case. As in Figure 1, the
s
points E and F represent the initial equilibrium prior
to the occurrence of shocks to government spending.
As government spending rises, the budget line shifts
downward by an amount equal to the increment of
government spending, i.e., -Avt = A&y,). With
lump-sum tax financing, the slope of the budget line.
or the after-tax wage rate remains unchanged. As a
result, the new equilibrium will still lie on the rays
OA and OB (recall that the utility function is
homothetic). Given the new budget constraint, the
intratemporal and intertemporal equilibrium will be
achieved at point E ’and F ‘ respectively.’ Since there
,
is only an adverse income effect, represented by the
downward and parallel shift of the budget line, the
new equilibrium displays less consumption for both
periods and greater work effort. The individual is willing to work harder because leisure is a normal good
and the individual is poorer than before due to tax
FEDERAL

RESERVE

increases. Because both income and consumption are
lower, the effect on saving is indeterminate. In other
words, at the initial interest rate, saving or investment may rise or fall, so it appears that, the
equilibrium interest rate may go either way. In the
simulation below, however, we will see that it rises.
How might results differ with income tax financing? Now, substitution effects of changes in the aftertax wage rate and rental rate become potentially important. A change in the income tax rate will induce
not only a substitution between consumption and
leisure at a given date, but also a substitution of consumption over time. In order to assess the impact
of government spending, it is necessary to trace out
the equilibrium paths of quantities and prices.
The dynamic responses of the system are displayed
below the dotted line in Figure 2. These response
functions are calculated by taking an average from
5000 random realizations of the system, conditional
on the initial displacement 9f government spending.
To contrast the effects under different tax regimes,
each figure contains two transition paths of the same
variable; the solid line traces out the dynamic
response under lump-sum tax financing; the dotted
line traces out the dynamic response under income
tax financing. Since the steady state is different for
the two tax regimes, these responses are expressed
in terms of percentage deviations from the steady
state. The following discusses the different implications under the two tax financing schemes.
Lump-Sum Tax vs. Income Tax Financing
(Temporary Case)
Consider first the case of a temporary increase in
government spending in which gt jumps from 30
percent to above 40 percent at date 1. Since the
shock is temporary, it lasts for about one period (see
Figure 2, left-hand side). As the left-hand side of
Figure 2 shows, both lump%um tax financing and
income tax financing have ‘
negative effects on capital,
consumption, and investment. The magnitudes are
quite different, however. In the case of lump-sum
tax financing, capital falls by 3 percent on impact,
while consumption and investment decrease by 2
percent and 70 percent, respectively. The negative
effects are much more severe under income tax financing; capital falls by over 9 percent while consumption and investment drop by more than 5 percent and 180 percent, respectively. Two reasons are
responsible ‘
for these results. First, a rise in the
income tax rate decreases the after-tax marginal
product of capital. In addition, a decrease in labor
BANK OF RICHMOND

31

5
8

Figure 3

The lower wage rate implies that leisure
is less expensive relative to consumption
and as a result, consumers are more willing
to take leisure instead of consumotion.
Finally, there is an interest rate effect.
According to Figure 2, the real interest
rate rises on impact, which largely reflects
the increase of aggregate demand associated
with an increase in government spending.
The rise in the interest rate encourages consumers to work harder due to a higher rate
of return. Under lump-sum tax financing,
- Av,
the wage effect is dominated by the income
effect and the interest rate effect, resulting
= N%Y,)
in greater labor effort. Since the capital stock
is fixed at the beginning of the period, output also increases. Although the interest rate
rises even higher in the case of income tax
financing, this rise together with the income
effect is not sufficient to outweigh the
hours, which results also from a lower after-tax wage
wage effect so that both labor hours and real output
rate, pushes the marginal product of capital even
decrease.
lower.” As the productivity of capital falls, agents
The initial response of the interest rate and outhave less incentive to accumulate capital so that the
put can be analyzed using the traditional aggregate
decrease in investment is larger under income tax
demand and aggregate supply paradigm. Figure 4a
financing. Finally, the lower productivity of capital
depicts the equilibrium shift in the goods market
and labor represents an additional loss of income
when a lump-sum tax is used to finance government
which makes agents poorer than in the lump-sum tax
spending. The real interest rate and output are
case. Therefore, the decrease in consumption is also
measured on the vertical and horizontal axis, respeclarger under income tax financing. In order to induce
tively. The point E is the initial equilibrium point.
agents to consume less, the real interest rate will go
As government spending rises, the aggregate demand
up to maintain equilibrium in the goods market.
schedule shifts to the right because of the increase
The most visible difference between lump-sum tax
in goods demanded by the government. The aggrefinancing and income tax financing shows up in their
gate supply schedule also shifts to the right because,
effects on labor effort and real output. In the case
as explained above, labor supply increases. However,
of lump-sum tax financing, both labor effort and real
since the increase in government spending is temoutput rise on impact by about 2 percent, while inporary, the shift in aggregate supply will be relacome tax financing causes them to decrease by more
tively small due to the negligible income effect. As
than 24 percent and 17 percent, respectively. Three
a result, there is an excess demand at the initial
forces determine the response of labor supply. First,
interest rate r , which must rise in order to restore
an increase in government spending leads to the use
equilibrium in the goods market. As the real interest
of real resources and makes agents poorer. This
rate rises, aggregate supply (labor effort) increases
adverse income effect motivates consumers to work
while aggregate demand (consumption and investharder. However, since the disturbances are temment) decreases and the new equilibrium is reached
porary, this effect is relatively small. Second, there
at point F. Comparing points E and F reveals that
is a wage effect. As Figure 2 shows, the after-tax wage
both output.and the real interest rate are higher.
:
rate falls by more than 13 percent in the income tax
case, as opposed to a tiny 0.6 percent drop in the
The case of income tax financing can be analyzed
lump-sum tax case. The larger decrease in the wage
in a similar fashion (see Figure 4b). The principal
rate tends to dampen the response of labor supply.
difference here is that the aggregate supply schedule
will now shift. to the left because of the decrease in
labor supply. The ,shift in aggregate supply will of
I1 Since capital and labor are complements in production, a
decrease in labor input lowers the productivity of capital.
course depend,on the extent to which the marginal
CONSUMER’
S
EQUILIBRIUM:
EFFECTS
OF AN INCREASE
IN LUMP-SUM
TAX

l

FEDERAL

RESERVE

BANK

OF RICHMOND

33

EFFECTS
OF TEMPORARY
INCREASES
IN GOVERNMENT
SPENDING
Figure 4a
With Lump-Sum Tax

Y: (Aggregate

\

Supply)

0
Figure

4b

With Income Tax

Y: (Aggregate

Lump-Sum Tax vs. Income Tax Financhg
(Persistent Case)

Supply)

Suppose n&v that the increase in government
spending is more persistent (i.e., p = 0.9j. The right
panel shows that the responses are very similar to
those of a temporary increase in government spend&The
principal difference is the implied wealth
effect. Because the shock is expected to persist for
a longer period of time, the wealth effect will now
play a more-important role in the response of quantities and prices.

r’

‘
II-I

I’

1

I

i

(Aggregate
Demand)

i

C

;

.,

product of labor is reduced.
case under consideration the
outweighs that of aggregate
decreases while the interest

It turns out that in the
shift of aggregate supply
demand .so that output
rate rises.

The analysis up. to this point has focused on the
short-run effects of an increase in government spending. Consider now the transition dynamics of the
system after the initial. impact. Since the capital stock
is lower at date. 1, the .marginal product of capital
34

increases.‘ As a result, agents begin to accumulate
2
more capital after date 1. As the .capital stock or
investment increases, the real interest ‘
rate (or the
marginal product of capital) falls and consumption
begins to rise. Consumption rises over the transition
period because current consumption becomes less
expensive relative to future consumption as the
interest rate declines over time.13 This response
applies to the lump-sum tax financing as well as the
income tax financing. Figure 2 shows, however, that
the transition path of real output and labor effort will
depend on the tax regimes. In the lump-sum tax case,
both. labor hours and real output decrease over time
because the real interest rate falls (recall that a lower
interest rate implies a lower labor effort). In the case
of income tax, the rising wage rate, due to a decrease
in the income tax rate and an increase in the capital
stock, becomes an overriding force that pushes labor
hours up over the .transition period. As can be seen
from the figure, labor supply will temporarily overshoot the steady state and then decline to the initial
equilibrium. As labor supply and the capital-stock
rise, output also increases until the steady state is
reached.

ECONOMIC

REVIEW,

Consider the case of lump-sum tax financing.
Figure 2 shows that labor hours rise by 13 percent
and consumption falls by 10 percent on impact.
These responses are more than five times the
responses in the ‘
temporary case. These results
occur because consumers are poorer than in the case
of a temporary shock. To induce agents to consume
less and work harder, the real interest rate will also
r; Since labor hours rise under lump-sum tax financing, iipushes
the marginal product of capital even higher. Under income tax
financing, labor effort decreases, but the decrease outweighs that
of capital (see Figure. 2, left-hand side) and the capital-labor
ratio is lower at date 1, implying a higher marginal product of
capital.
I3 The negative correlation between current consumption and
the real interest rate is sometimes called the effect of intertemporal substitution.
SEPTEMBER/OCTOBER

1990

increase by a larger magnitude. Again, since capital
is predetermined,
real output rises with labor
supply. Perhaps the most interesting difference here
is that investment does not go down as much as in
the temporary case. The principal reason for this
result is that the increase in labor hours occurs over
a more extended time period and pushes up the
marginal product of capital both now and in the
future, thus raising the rate of return to investment.
It should be noted, however, that investment will
still go down on impact as consumers try to smooth
out consumption by holding less capital.
The adverse income effect works in a similar
fashion under the income tax regime. In particular,
consumption drops by more than 10 percent, as
opposed to a 5 percent decrease in the temporary
case. Because of the income effect, the decrease in
labor hours, which is caused by a lower after-tax wage
rate, is smaller than that in the temporary case.
Consequently, the decrease of real output is also
smaller. Because the decrease of labor effort is
smaller, the marginal product of capital does not go
down as much as in the temporary case, leading to
a smaller decrease in investment.
Although the initial effects of a persistent increase
in the income tax rate are not as large as those in
the temporary case (except consumption), major
variables such as output and investment will stay
below their steady state for a long period of time.
In fact, the shock is so persistent that agents will eat
up some existing capital for one period before consumption (and capital) begins to rise over the transition period. This is the case of .a severe recession.
The reason for this result is that the marginal product of capital is so low in the future that agents have
very little incentive to accumulate capital.
A surprising feature of the income tax regime is
that the real interest rate declines in response to a
persistent increase in government spending. Again,
this result can be attributed to the income effect. As
noted before, output supply will decline, but the
decrease will not be as much as that in the temporary
case because the income effect motivates agents
to work harder. On the demand side, the income
effect and the lower productivity of capital in the
future decrease both consumption and investment
at the initial real interest rate. The decrease of consumption and investment may reach the point at
which it outweighs the increase of government purchases, leading to a decrease of aggregate demand.
The extent to which aggregate demand decreases will
depend on how long the shock persists. It turns out
FEDERAL

RESERVE

that in the case under consideration, .the decrease
in aggregate demand is quite sizable so that at the
initial real interest rate there is an excess supply,
resulting in a lower interest rate. Clearly, this argument hinges on the persistence of the shock and the
intensity of the income effect. If the government
spending shock is less persistent, then the interest
rate will decline by a smaller amount or even increase
as in the pure temporary case.
VI.

CONCLUSIONSANDEXTENSIONS
This paper examines the balanced budget effects
of government spending under different tax financing schemes. The results suggest that, in the case
of lump-sum tax financing, persistent changes, in
government spending have larger effects on prices
and quantities except investment. This result, due
to larger income effect and interest rate effect, is
consistent with the findings of King (1989) and
others. In general, an increase in government spending under lump-sum tax financing will reduce consumption and investment but raise ‘
labor effort and
real output. This result is driven by the income and
interest rate effects that encourage individuals to,work
harder. Under income tax financing, however, some
of the above results are reversed. In particular,
regardless of the persistence of spending shocks, both
output and labor effort now decline in response to
an increase in government spending. This result
occurs because the decline in the wage rate dominates
the income and interest rate effects.
There are several features of the model that are
oversimplified and can be improved upon. Most
notably, the government budget is assumed to be
balanced in each period. This assumption prevents
one from seriously considering the implications of
deficit or debt financing. It is relatively easy to
introduce such a financing scheme into the model.
Extension along this line will probably yield fruitful
results if government debts coexist with some types
of distortionary tax such as the income tax considered
in this paper. The most important implication of
debt financing is that it allows the tax burden to be
smoothed out over time. This mechanism reduces
the distortionary effect on labor supply, particularly when the increase in government spending is
temporary. In this case, real output and labor hours
may no longer decline as in the case of a balanced
budget.
Another extension worth undertaking concerns the
function of government spending. The current paper
assumes that government spending is a waste of
BANK

OF RICHMOND

35

resources and is not utility- or production-enhancing.
This assumption is inappropriate for some types of
government spending that may either substitute for
private consumption or increase the economy’ pros
ductivity. These features could be introduced into

the model by specifying a more general utility function or production function, such as those employed
by Barro (1984). Such refinements would nullify or
even reverse some of the negative effects associated
with income tax financing.

APPENDIX
This appendix presents a definition of the
equilibrium discussed in the text and outlines a
numerical method to construct the equilibrium. Formally, the general equilibrium for the model economy
consists of a sequence of quantities {ct,kt + l,nt,lt}
and prices {wt,rt) that satisfy the following two conditions: (1) the sequence {ct,kt + l,nt,lt) solves the
maximization problems of consumers and firms for
a given sequence of prices {wt,rt) and (2) the commodity market clears at each date t such that aggregate demand equals aggregate supply:

ct + it + gtyt = yt.

(Al)

Equation (Al) states that the total of consumption,
investment, and government purchases must exhaust
total output. The government budget constraint,
which must also be satisfied in equilibrium, is implied by the market-clearing condition (Al) and the
individual budget constraint (1) in the text.
To further characterize the equilibrium one must
solve the maximization problems of consumers and
firms. The firm’ problem is straightforward. It res
quires, as stated in equation (Z), that the rental rate
and the wage rate be equal to the marginal product
of capital and labor, respectively. This condition
defines the equilibrium prices that will clear the labor
market and the rental market for the existing capital
stock. As discussed in the text, the consumer’
s
equilibrium is characterized by the budget constraint
(1) and the time constraint It + nt = 1 together with
two first-order necessary conditions, which are rewritten as follows:
u1hlt)~u,(ct,1t)
uchld

=

=
P Et[uch

(1 - 7t)Wt.

(AZ)

+ l,lt + I)

11 + (1-7t+drt+~-4].

(A3)

The meaning of (AZ) and (A3) is discussed.in
text.
36

ECONOMIC

the

REVIEW.

The approach used to determine the equilibrium
of the model economy is as follows. First, substitute
the time constraint and equations (1) - (3) and (Al)
into (AZ) and (A3) to obtain

-al

ul[(l -gt)F(kt,nt)+(l-Qkt-kt+l,l

u,Kl -gt)F(kt,d

+(I -W

-kt+ 1~1-4

= (1 - dF&,nt),

(A4)

and

u&l

-gdF(kt,nt)+(1-6)kt-kt+1,1-ntl

P Et {udl

=

-gt + dF(kt + m + 1)

+ (1-6)kt+l-kt+2,1-nt+ll
x [(1-7t+l)Fk(kt+l,nt+l)

+ (1-N).

645)

Note that equations (A4) and (A5) are alternative
versions of the consumer’ equilibrium with quans
tities and prices replaced by the market-clearing
condition and the firm’ marginal conditions. These
s
two equations jointly determine the equilibrium level
of capital kt + 1 and labor nt,14 which can be used to
determine consumption,
investment, output and
equilibrium prices. Note that given the beginning of
period capital kt, a decision rule for kt + 1 is equivalent
for a saving decision made at time t.
In general, an analytical solution to equations (A4)
and (A5) does not exist except for a very few special
cases. Numerical methods are therefore required to
obtain an approximate solution. The following briefly
describes an iterative procedure used to solve the
model. Technical details of this method can be found
in Coleman (1989) and will not be presented here.
Basically, the solution to equations (A4) and (A5)
comprises a pair of decision rules for capital kt + 1 and
labor nt that can be expressed as functions of kt and
I4 Note that kt + 2 and nt + I are “integrated out” when (A4) and
(A5) are solved.
SEPTEMBER/OCTOBER

1990

gt (i.e., the state of the system). The numerical procedure involves approximation of these decision rules
over a finite number of discrete points on the space
of kt and gt. Starting from an arbitrary capital rule
(usually, a zero function), the procedure first solves
the labor rule from equation (A4) and then iterates
on equation (AS) until the capital rule converges to
a stationary point, that is, until capital as a function
of kt and gt does not change over consecutive iterations. The resulting stationary function is the
equilibrium solution for capital and labor.
By construction, the above procedure yields solutions that satisfy both (A4) and (AS) for all contingencies of government spending. These solutions
imply three imputed or shadow prices that are consistent with the market equilibrium. Specifically, the
equilibrium wage rate wt and rental rate rt can be
computed from the firm’ marginal condition (Z), and
s
the real interest rate r4, by definition, is the ratio of
the marginal utilities of consumption’
between
time
t and time t + 1, i.e., u,(ct,lt)/[PEtuC(ct + r,lt + I)]. In
a deterministic equilibrium, the gross real interest rate
r: is equal to (1 - 6) plus the capital rental rate rt + I,
as can be seen from equations (A3) and (AS). This
is the price that will clear the commodity market.

FEDERAL

RESERVE

References
Aiyagari, S. R., L. J. Christiano, and M. Eichenbaum. “The
Output, Employment, and Interest Rate Effects of Government Consumption,” Discussion Paper 25, Institute for
Empirical Macroeconomics,
Federal Reserve Bank of
Minneapolis, March 1990.
Barro, R. J. Mameconomics, 2nd edition.
Wiley & Sons, 1984.

New York: John

Baxter, M. and R. G. King. “The Equilibrium Approach to
Fiscal Policy: Equilibrium Analysis and Review,” Manuscript, University of Rochester, May 1990.
Coleman, W. J. “Equilibrium in an Economy with Capital and
Taxes on Production,” Manuscript, Federal Reserve Board,
May 1989.
King, R. G. “Value and Capital in the Equilibrium Business
Cycle Program,” Working Paper, University of Rochester,
February 1989.
King, R. G., C. Plosser, and S. Rebelo. “Production, Growth
and Business Cycles I. The Basic Neoclassical Model,”
Journal of MonetaryEconomics 2 1, (March/May 1988).
Rebelo, S. and G. Rouwenhorst. “Linear Quadratic Approximations Versus Discrete State Space Methods: A Numerical Evaluation,” Manuscript,
University of Rochester,
March 1989.

BANK

OF RICHMOND

37

Why Do Estimates of Bank Scale Economies Differ?*

A number of policy issues turn on whether or not
large commercial banks, merely because of their size,
are more efficient than small banks. Such scale
economies, where average cost declines as bank output rises, would result from spreading fixed costs over
a greater volume of output. Scale economies are an
important policy consideration in interstate bank
branching.
Interstate branching was long prohibited on the
grounds that (1) industry concentration and monopoly
power would result, and (2) local areas may be Iess
well served by giant banks having We interest in
these localities, as more profitable uses for funds
would likely be found elsewhere. Cost savings
associated with laree scale economies. however.
might overcome these negatives. As weli, interstat;
branching would allow banks to diversify their portfolios g&graphically, strengthening the indusuy.
Consumer and business bank customers would Likely
benefit from lower prices and reduced banking risks
which could follow.
~

~
-

-

-

-

In contrast, if scale economies were small, fears
of concentration might outweigh any perceived
benefits of expansion. It would then be more politically tenable to limit the size and geographical
distribution of banks. While there still could be loan
risk diversification, this benefit by itself might not
justify the concentration of economic power in uuly
giant banking organizations.
T h e level of bank scale economies is an empirical
question, but one where widely differing results have
made it difticult to form a clear and unambiguous conclusion. Fortunately, there are now enough studies
to attempt to sort out why past results have differed.
Such a sorting out is useful in its own right and for
the implications it has for policy decisions that depend on scale economies in banking. It also illustrates
the benefits a detailed analysis could have for other
areas of economics where empirical findings diier
and can cloud proper policy formation (such as in
the appropriate defiiition of the money supply).

sciences, researchers use the same experimental technique to generate new and independent data and then look for consistency in the results.
In contrast, economists generally use. similar data but
vary the experimental technique-that is, the particular specification and definition of variables, functional form, and time period used. Thus robust results
are less frequent. If enough studies are performed,
however, a pattern to the results may emerge suggesting why they differ. Then we can compare the
relative advantages of different experimental techniques. Instead of a single scale economy conclusion
that applies in all cases, we obtain a set of different
results h a t illustrate how sensitive our measures are
to the research design chosen. From this and from
some additiqnal thought on how we best measure
scale economies, we develop a general conclusion
on the size and significance of scale economies in
banking.

n.

COMMON
DI~ERENCES
AMONG STUDIES
Graphically, bank scale economies appear as the
slope of an average cost curve indicating how costs
vary with output. An example is shown in Figure 1.
A series of short-run average cost curves (solid lines)
for three different-sized banks, each producing different levels of bank output, trace out an implied longrun average cost curve (dotted Line). A downwardsloping long-run average cost curve reflects scale
economies. An upward slope reflects diseconomies,
since higher average costs are incurred when more
output is produced. The assumption is that a crossFigure 1

..IllustrativeBank Average Cost Curves

I

Series of Short-Run Cost Curves

Implied Long-Run Cost Curve
Comments by Mike Docsq. Bob Graboyes, Tom Humphrey,
and Dave Mengle are appreciated, although the opinions expressed are those of the author alone. Able research assistance
was provided by Bill Whelpley.

38

(low)

EWNOMH: REVIEW, SEPTEMBEWOCTOBER 1990

(medium)

Bank Output

(high)

section of different-sized banks at a point in time will
reveal the appropriate long-run curve; from this is
derived a measure of scale economies. Thus as
smaller banks expand their output in the future, their
costs are likely to “look like” the costs of larger banks
today.
The cost curve itself (and the implied scale
economies reflected in it) is actually derived from an
equation similar to (l), below, where costs (C) are
regressed on the level of bank output (Q) and other
variables which affect costs but need to be held constant in the cross-section data set:
(1) C = f(Q, other variables).
Other variables, such as the prices of labor and capital
factor inputs, need to be held constant in a crosssection in order to statistically separate movements
al’ the cost curve (due to changes in output) from
ong
s/$?.r in the cost curve (due to influences on bank
costs which are essentially unrelated to output).
With this background, we now outline the most
common differences observed in bank scale economy
studies and assess how these differences have affected
the results derived from them. More specifically, our
purpose is to critically review the literature on bank
scale economies, to select a preferred method for
estimating these economies, and thereby to determine which empirical result is the most appropriate
for policy purposes, as well as defensible on
theoretical grounds. The most common research
design differences among studies of bank scale
economies concern the following:
(1) Cost definition (operating
cost);

cost versus total

(2) Bank output definition (numbers of accounts
versus dollars in these accounts);
(3) Functional form used (linear versus quadratic);
(4) Scale economy evaluation (single office versus banking firm);
(5) Time period used (high versus low interest rate
period);
(6) Commingling scale with scope (single versus
multiple output); and
(7) Bank efficiency differences (assume all observations are efficient versus only those on the efficient
frontier).
In the following sections, each of these differences
is discussed in conjunction with one or more published studies. Some other differences occur and,
when appropriate, they too are noted.
FEDERAL

RESERVE

III.

OPERATINGVERSUSTOTALBANKCOSTS
This section concerns
how the dependent
variable-cost
(C)-is defined in equation (1). Many
studies relate only operating costs to bank output
levels in estimating scale economies (Langer, 1980;
Nelson, 1985; Hunter and Timme, 1986; Evanoff,
Israilevich, and Merris, 1989). Operating costs
include wages, fringe benefits, physical capital,
occupancy, and materials cost, along with management fees and data processing expenses paid to the
holding company and other entities. On average,
operating costs only comprise slightly over 2.5 percent of total costs. Most other studies have used total
costs, which are obtained by adding interest expenses
on purchased funds and core deposits to operating
costs.’ The two interest cost categories are large
and each exceed operating costs since they comprise
around 3.5 and 40 percent, respectively, of total costs.
Clearly, it makes a difference which definition of cost
is used to derive an estimate of scale economies.
The difference in cost definitions-operating
versus total costs-would not be an issue if all banks
had the same percentage composition of interest and
operating expenses regardless of their size. This is
because interest expenses typically have little or no
economies associated with them. Therefore, adding
these roughly constant cost expenses to operating
costs (giving total costs) means that any scale
economies or diseconomies found using operating
costs alone would only be attenuated, rather than
reversed, if the ratio of interest to operating costs
were the same across banks. But this ratio is not even
close to being stable across banks. The proportion
of assets funded with purchased funds rises substantially as banks get larger so that the proportion of
purchased funds interest expense in total cost rises
while the proportion of core deposit interest expense
and operating cost falls.
For example, at small branching banks (those with
$50 to $75. million in assets in 1984), purchased funds
were 12 percent of the value of core deposits plus
purchased money. For medium-sized banks (with
$300 to $500 million in assets), the purchased funds
proportion rises to 19 percent. And for large banks
(with $2 to $5 billion and then over $10 billion in
r Purchased funds are purchased federal funds, CDs of $100
thousand or above, and foreign deposits (which are almost alwavs
over $100 thousand). Core-or produced deposits are demand
deoosits and small denomination (i.e., less than $100 thousand)
time and savings deposits. The costs of equity and subordinated
notes and debentures are small and are almost always excluded
from bank cost studies.
BANK

OF RICHMOND

39

assets), the proportion rises further to 36 and finally
60 percent. At unit state banks for the same four size
groupings, the purchased funds proportions are 16,
3 1, 61, and 78 percent. Thus the percentage composition of interest and operating expenses varies
considerably across banks and is closely related to
bank size, which is the key to the problem which
arises when operating costs are used.
Purchased funds have very low operating expenses
per dollar raised; their only significant cost is interest
expense. In contrast, core or produced deposits
generate the major portion (49 percent) of all
operating (capital, labor, materials) expenses. Since
purchased funds are a strong substitute for core
deposits, the interest expense of purchased funds is
also a substitute for the operating and interest expenses of core deposits. To accurately gauge how
bank costs really change with size thus requires that
purchased funds and core deposit interest expenses
be included with operating costs. Taken together,
these components allow one to determine the average
cost actually faced by a bank even as its funding mix
is altered. In this way, changes in the funding mix
do not bias the results.
This point is illustrated by comparing the actual
average operating cost (operating expenses divided by
total assets) for 1984 with the average torah cost
(operating plus interest expenses divided by total
assets) for the same year across 13 size classes of
banks (see Figure 2). The branching state bank comparison is shown in Panel A with the unit state bank
comparison in Panel B.2 Operating cost per dollar
of assets is seen to fall more rapidly than total costs
per dollar of assets. Thus if only operating costs are
used in a statistical analysis of bank scale economies,
as some investigators have done, greater scale
economies (or lower diseconomies) will typically be
measured when an equation like (1) is estimated and
a curve is fitted to these raw data points.3
Hunter
when they
operating
estimates

and Timme, 1986, obtained this result
alternatively used operating costs and then
plus interest costs in their statistical
of scale economies for 91 large bank

a The top line in each comparison is the mean
curve (solid line). To make this comparison
for average operating costs-right
side of the
shifted up so that the two curves will appear
same point for the first size class. The scale
costs is on the left side.

average total cost
clearer, the scale
figure-has been
to start from the
for average total

3 The same sort of bias toward finding scale economies when
only operating costs are used also exists for thrift institutions.
This can be seen in the raw data presented in Verbrugge,
McNulty, and Rochester, 1990, Table 1.

40

ECONOMIC

REVIEW,

holding companies over 11 years (1972-82). They
found significant operating cost scale economies
(using only operating costs) but no significant total
cost scale economies (when interest expenses were
included). Their study covered large banks separately
and did not include any small or medium-sized
institutions.
While operating costs are of some interest in
themselves, it would be misleading to conclude that
reductions in the ratio of operating costs to assets
accurately reflects inherent bank scale economies.
If this were true then a bank with a wholesale orientation (large purchased funds, small core deposits)
would always experience lower costs solely because
of lower operating costs per dollar of assets. But lower
operating costs per dollar of assetSare typically offset by having greater interest costs per dollar of assets
through more intensive reliance on purchased funds
instead of core deposits. Thus the proper comparison
of costs, and measurement of scale economies, must
rely on total costs rather than only on operating costs
by themselves. When this is done, then differences
in a bank’ funding mix will not bias the results.4
s

IV.
BANKOUTPUTMEASUREMENT:
NUMBEROF
ACCOUNTS VERSUS
DOLLARSINTHE
ACCOUNTS
Another important difference in published studies
concerns the definition of bank output (Q), a key
independent variable in equation (1). In most other
industries, the measurement
of output is not a
problem. Output is a flow concept measured in
physical terms, either because the physical unit is
homogeneous and can be easily observed or because
there is a convenient index of the value of the
output flow which can be deflated by an appropriate
output price index. In banking, neither of these alternatives exists and data availability dictates how bank
output is defined. Output flow information is not
available for each individual bank so information on
the stock of output is used instead. Generally, researchers assume that the unobserved output flow
is proportional to the observed output stock. Thus
use of stock information in statistical analyses is
presumed to give results similar to those obtainable
using flow data.
4 If the U.S. banking system were considerably more consolidated, as could occur if full interstate branching were permitted, then the importance of purchased funds would of course
be reduced. Once this occurs, looking at operating cost per dollar
of assets could be more revealing. There would be less substitution of purchased funds for produced deposits and the funding
mix bias that exists in current studies using only operating cost
would be attenuated.

SEPTEMBER/OCTOBER

1990

Figure

2

Comparing Actual Average Operating and Average Total Cost
(1984 data points)

.ll

Panel A:

-

Branching

State Banks
- .045

verage Total Cost (@ per $)
- .04
\\
\\
\\
\\,-d----

i

I

I

-c-c /----N5w

,-‘
----------ww
0’
--H /’
z

I

I

I

--a

**

-\

- .035
-x \

Average Operating Cost (Q per $) “
N,
- .03
\\
(right scale)
\
I

I

I

I

I

- .025
I

.ll
Panel B:

Unit State Banks

.105
.04
.lO

.095
.09

-\

-.

-\

-\

\

.025
\

Average Operating Cost (C per $) \
\
(right scale)
\
\
‘
7

.02
s.015

I

I

I

I

I

I

I

I

I

I

I

1

Asset Size Classes
(millions = M)
(billions = B)

FEDERAL

RESERVE

BANK

OF RICHMOND

41

Data on the number of deposit and loan accounts,
an output stock measure, also are not available for all
banks. Nevertheless, some information is given in
the Federal Reserve’ annual Functional Cost ArzaLysis
s
(FCA) survey. This survey covers 400 to 600 banks
but typically excludes the very largest (those with
more than $1 billion in assets). Also, the same banks
are not in the sample each year.s Alternatively, the
value of dollars in the various deposit and loan accounts, another output stock measure, is publicly
available for each individual bank in every year in
the Report of Condition and Income (Call Report).
Some researchers have made a strong argument
for using the number of accounts as an indicator of
bank output (Benston and Smith, 1976). Fortunately,
it turns out that the scale economy results are
reasonably robust to the use of either the number
of accounts or the dollar value in the accounts. That
is, using both of these alternative representations of
bank output in the same model for the same year
leads to similar scale economy results (Benston,
Hanweck, and Humphrey, 1982; Berger, Hanweck,
and Humphrey, 1987). This occurs because these
two approximations to bank output, while numerically
quite different, are highly correlated, both in the U.S.
and elsewhere (see Berg, Forsund, and Jansen, 1990).
A preferable measure for bank output would
measure the flow of some physical aspects of bank
output rather than just the stock of accounts serviced or their dollar values. While the Bureau of
Labor Statistics compiles such a measure annually,
it applies only to the aggregate of all banks in the
U.S. (BLS, 1989). This aggregate flow measure is
a specially weighted index of the number of checks
processed (for demand deposit output), the number
of savings account deposits and withdrawals (for
savings and small-denomination time deposit output),
the number and type of new loans made (for various
loan outputs), and the number of trust accounts serviced (for trust output).‘
j

ALINEARVERSUSA
QUADRATIC
FWNC~I-IONALFORM

5 The sample has varied by as much as 15 to 20 percent each
year. Also, credit unions and thrift institutions (such as MSBs)
can and do particioate in the FCA survev. In 1984. the oarticipation raie of &rift and credit unions was almost l? percent
of the total sample.
6 The FCA data also provide.physical flow information, similar
to that used by the BLS, but these data are available only for
banks in the survey, not for all banks.
ECONOMIC

REVIEW,

A related issue, often noted in the literature, concerns the similarity of the survey bank data from the
FCA versus that for the population of all banks in
the Call Report. The only published study addressing this issue concluded that while there were
statistically significant differences between the FCA
sample and the Call Report population data (in terms
of portfolio composition, capital/asset ratio, and total
cost/asset ratio), these differences were quantitatively
small. In fact, FCA banks in 1970 experienced mean
average costs which were 6 percent lower than the
average costs for the mean of the non-FCA bank sizematched sample (Heggestad and Mingo, 1978).
Updating this comparison for 1984, but using all
banks, we find that the mean difference is now only
3 percent, and most of this arises for banks with the
highest costs. Thus, FCA data should not lead to
markedly different scale economy results compared
to use of data on all banks, or on only large banks
not covered in the FCA sample.
V.

Over a recent lo-year period (1977-86), the BLS
aggregate measure of bank output rose by 40.4 percent. Over the same period, a cost share-weighted
index of the vahe of demand deposits, savings and
small time deposits, real estate, installment, and com-

42

mercial and industrial loans (all deflated by the GNP
deflator) rose by 43.8 percent (Humphrey, forthcoming). These 5 output stock categories accounted
for around 7.5 percent of bank value-added during
the 1980s and so clearly reflect the majority of services produced by banks (in a flow sense). Importantly, the similar growth rates indicate, at the aggregate level at least, that the flow and stock measures
of bank output closely correspond to one another.
This suggests that use of a stock measure of bank
output (the only one available at the individual bank
level for all banks) may be a reasonable approximation of the unobserved flow measure for recent time
periods. Thus it would seem that little bias has been
introduced in past scale economy studies when a
stock of output measure is used in place of a flow
measure. Also, either the stock of accounts or the
stock of dollars in those accounts seems to give
qualitatively similar scale economy results (when
properly used in the same model).

Historically, bank scale economies were typically
estimated using a linear functional form for equation
(l), such as the log-linear Cobb-Douglas form.7 Such
forms were commonly used in cost or production
analyses in areas where the research emphasis was
’ Greenbaum, 1967, is an important exception as he used a
simple quadratic equation and, as a result, found a U-shaped
average cost curve (in contrast to studies using a Cobb-Douglas
form).

SEPTEMBER/OCTOBER

1990

on factor shares in the distribution of income and on
estimating the various sources of output growth over
time. Unfortunately, one property of the log-linear
Cobb-Douglas form is that the same cost economies
or diseconomies will be measured for ail banks in
the sample regardless of their size. Put differently,
all banks will either have scale economies, scale
diseconomies,
or constant costs. A U-shaped
long-run cost curve, similar to that illustrated in
Figure 1, cannot be estimated when only Q enters
the regression equation (1). What is needed is a
specification that includes Q and Q2, making (1) a
quadratic equation.
Earlier studies, such as the comprehensive analyses
of Benston, 1965 and 1972, and Bell and Murphy,
1968, used a Cobb-Douglas form and found that scale
economies existed in many banking services.*
Overall, these economies were relatively small. The
average scale economy value was .92.9 This means
that for each 10 percent increase in bank output, costs
rise by only 9.2 percent, so average costs would be
estimated to fall as a bank gets larger. A scale
economy value greater than one-say
1.05-would
have suggested a 10.5 percent rise in costs for each
10 percent increase in output (thus reflecting scale
diseconomies).
Recently, more flexible functional forms have been
developed and used. One of the most common is
the translog form, which is .a quadratic form. That
is, the translog has linear output terms, like the CobbDouglas, but also squared output terms. As a result,
the translog form can estimate a U-shaped cost curve
if one exists in the data. If a U-shaped cost curve
were in fact estimated, it would show scale economies
at smaller banks and diseconomies at larger ones, like
that illustrated in Figure 1. Unlike the Cobb-Douglas
form, quadratic forms capture variations of scale
economies across different sizes of banks.
Studies using the translog form, such as Gilligan,
Smirlock, and Marshall, 1984, Lawrence and Shay,
1986, or Benston, Hanweck, and Humphrey, 1982,
generally find that bank cost curves are weakly
s Squared terms of some independent variables were used in
Benston’ regressions but only rarely applied to the output
s
variables. Thus U-shaped cost curves could not, except in these
infrequent cases, be estimated.
9 Simple averages of Benston’ 1965, direct and indirect expense
s,
scale economies were .87 and .98, respectively (Table 26,
p.544). As indirect expenses were 43 percent of coral operating
expenses, this yielded a weibhhted average scale economy of
.87(.57) + .98(.43) = .92. Bell and Murphy obtained an overall
scale economy of .93 (Table 4, p.8).
FELiERAL

RESERVE

U-shaped. Scale economies exist in banking but seem
to be limited to the relatively smaller banks. Either
constant costs (for banks in branching states) or some
scale diseconomies (for those in unit banking states)
seems to apply to larger institutions. Since under certain restrictions the translog reduces to the CobbDouglas form, it is possible to ‘
see if these restrictions significantly reduce the ability of the model to
fit the underlying data. In these tests, the CobbDouglas has been rejected in favor of the more
general translog form. That is, the restrictions the
Cobb-Douglas form places on the translog model
(equal scale economies for all sizes of banks and all
elasticities of factor input substitution equal to 1.0)
are rejected.
Use of the translog instead of the Cobb-Douglas
is one way these restrictions can be relaxed. Another
way is through a specialized adjustment (called a
Box-Cox adjustment) to the Cobb-Douglas model,
as applied by Clark, 1984, and Lawrence, 1989. With
such an adjustment, Clark finds only scale economies
in his small and medium-sized unit bank data set (the
largest bank had only $425 million in assets). In contrast, Kilbride, McDonald, and Miller, 1986, find
scale economies at small unit banks but diseconomies
at large ones using the same technique as Clark. Since
the Kilbride, et al. study differs in two respects-it
covered a later time period (1979-83 versus Clark’
s
1972-77) and added large unit banks up to $10 billion
in assets to the unit bank sample-it is not clear which
change led to the reversal in Clark’ results: the
s
different time period covered, the inclusion of large
banks, or both.
Recently, Lawrence, 1989, generalized the BoxCox adjustment of the Cobb-Douglas model by
adding the possibility of multiple outputs-either
multiple classes of loans or loans plus certain types
of deposits. Both the Clark and the Kilbride, et al.,
studies had used a single composite measure of bank
output. With this adjustment, both the multiple output translog and the single output Cobb-Douglas
forms can be tested to see which form best fits the
data. The single output Cobb-Douglas form, even
with a Box-Cox adjustment, was rejected in favor of
the multiple output translog. Thus it appears that
both the possibility of U-shaped cost curves and cost
complementarities among different bank outputs are
important generalizations of the single output CobbDouglas form (which cannot reflect either of these
more flexible specifications). In sum, a functional
form that permits the estimated average cost curve
to be U-shaped, rather than monotonic, is preferred.
Thus a quadratic form dominates a linear form when
BANK

OF RICHMOND

43

measuring bank scale economies and typically yields
different scale economy conclusions as well.
Closely related to the choice of a proper functional
form is the assumed constancy of the estimated relationship for all sizes of banks. More precisely, all
banks in a particular sample are presumed to lie on
the same average cost curve. While some studies
estimate scale economies for only large banks and
others estimate these economies for small and
medium-sized institutions, few have systematically
tested to see if all banks lie on the same curve, and
therefore face the same technology. This hypothesis
has been rejected statistically (Lawrence, 1989),
likely due to the large samples which produce a very
peaked sampling distribution. However, contrasts of
published results for large and small banks separately suggest that scale economy values may not
differ much in an economic sense. That is, the
relatively flat U-shaped cost curves identified using
all banks are replicated when only large banks are
used separately (e.g., Noulas, Ray, and Miller, 1990).
In either case, it is clear that on average the very
largest banks do not appear to have a significant cost
advantage due to scale economies compared to most
smaller institutions.

VI.
SCALEECONOMIESATTHEOFFICEOR
BANKINGFIRMLEVEL
When only bank-incurred costs are being minimized, scale economies for the average banking
office and the average banking firm-both
derived
from equation (1)-should
be the same. But when
costs include both the production and the delivery
of output to the customer, as occurs in banking, these
two measures can differ. In effect banks minimize
both bank and customer-incurred costs together, but
only the bank portion is observed. Some banks will
find it profitable to do more delivery-branchingthan others. These banks will save customers’
transportation and transaction costs (Nelson, 1985,
Evanoff, 1988) but will add to bank costs, and so
look to be less efficient compared to others which
provide less delivery. As customer costs are unobserved, differences in delivery strategies can give the
appearance of higher than minimum bank costs, even
though profits may be maximized in either case. In
this situation, scale economies can be measured at
the office level (as seen in the results of Lawrence
and Shay, 1986, who only measure office economies)
while diseconomies can be measured at the firm level
(as found in Hunter and Timme, 1986, and Berger,
Hanweck, and Humphrey, 1987).

44

ECONOMIC

REVIEW,

Some insight into resolving this difficulty, however,
may be obtained by observing how banks behave
when they have virtually no branches. Here the
office is the firm. This is the result when scale
economies are estimated for banks in unit banking
states.10 Scale diseconomies are regularly observed
for the larger unit banks. Because these banks have
(except in rare instances) no branch network to provide “convenience” to customers, these diseconomies
must therefore be related to production inefficiencies alone, not to the extra expense of providing consumer convenience. In contrast, banks operating in
branching states and hence providing customer convenience through a branching network have lower
scale diseconomies at the firm level and slight
economies at the office level (for all sizes of banks).
Thus it appears that permitting a bank to branch will
itself lower costs for the larger banks. The implication is that branching, far from being an extra cost
of customer convenience, actually Ibwm both bank
and customer costs. Branching permits a banking fum
to lower costs by producing services in more optimally sized “plants”‘ offices rather than producing
or
virtually all of the output at a single office, as occurs
in unit banking states. I1 Thus the customer convenience aspect of branching would appear to be
largely a side effect of a bank’ desire to lower scale
s
diseconomies by choosing a more optimal configuration of production facilities.
For banks in branching states, which in 1988
included all but Colorado, Illinois, Montana, and
Wyoming, the average number of accounts per banking firm rises steadily with bank size, while the
average number of accounts per office remains steady
after a certain minimum is reached. This fact implies
that branching banks can add output (deposits and
loans) in either of two different ways: by adding
additional offices in new market areas (which attract
new accounts and balances) or by adding new accounts and balances to existing offices. The data
indicate that the former method of output expansion,
which includes internal growth as well as mergers,
dominates the latter (Benston, Hanweck, and Humphrey, 1982, Table 1).
10Early on, published studies lumped banks in unit banking and
branching states together. This is inappropriate since more
recent studies have shown that these two classes of banks are
significantly different from one another in terms of how costs
vary with size. It should be noted that banks in unit banking
states do at times have a limited number of branches while unit
banks--those
with no branches-exist
in branching states.
*I Two studies which contrast unit and branching bank scale
economies are Benston, Hanweck, and Humphrey, 1982, and
Berger, Hanweck, and Humphrey, 1987. Other studies generally
parallel these results for banks in these two different regulatory
environments.

SEPTEMBER/OCTOBER

1990

To determine economies at the average banking
office, the number of branches is included as an explanatory variable in equation (1) and scale economies
at the office level are obtained from a partial derivative of the estimated total cost equation with respect
to scale (or output) alone. For economies at the
average banking firm, the same model is estimated
but the total derivative of the equation with respect
to both scale and number of branches is used.
Equivalently, the variable measuring the number of
branch offices can be deleted from (1) to obtain scale
economies at the firm level. The results typically
indicate that the average office still has some
realizable scale economies whereas for the firm, these
economies have either disappeared or have turned
into slight diseconomies.
Researchers have in the past estimated scale
economies for the average banking office and then
conclude that large banks (banking,firms) have lower
costs. They do so without realizing there can be a
difference between the office and firm results. In fact,
most of the early studies of bank scale economies
are deficient in this regard because they typically
specified the number of branches as an independent
variable in their estimating equation and then proceeded to derive scale economies as the partial
derivative of costs with respect to output. But this
derivation only gives scale economies when the
number of banking offices is held constant and thus
reflects only one of the two ways that bank output
expansion can affect costs. A better approach is to
compute scale economies both ways, and be clear
about what concept is being measured, or to compute only those economies which apply to the banking firm as a whole-the
relevant concept for policy
purposes. That is, most policy issues in banking,
whether relating to interstate banking, foreign bank
competition, or bank costs faced by users, are a function of the relation between costs and firm size, not
costs and the size of the average office. The prices
of banking services necessarily reflect all banking
costs, so the former, not the latter, is the appropriate
point for scale economy evaluation.

Each of the three major components of average
cost-purchased
funds interest cost, core deposit
interest cost, and the prices of factor inputs which
comprise operating cost-are
influenced by the
interest rate cycle in cross-section data sets, but by
differing amounts and with different lags. For example, average operating cost rises, with a lag, with
the rate of inflation while the average cost of purchased funds rises immediately and fully reflects the
level of,market interest rates. In contrast, the average
interest cost of core deposits almost always rises by
less than the rise in market rates and usually with
a lag. Since larger banks rely more on purchased
funds, it is easy to see that large banks will necessarily have.higher average costs than smaller banks
when interest rates are high. This holds even if equal
average costs would prevail across all banks when
interest rates are at their “normal” level. Similarly,
the reverse can hold if interest rates are at an exceptionally low level. I2
Simply put, the slope of the average cost curve and
estimates of bank scale economies can differ when
they are based on single year cross-section data
simply because the level of the market interest rate
varies over time. Since the vast majority of scale
economy estimates are in fact derived from singleyear cross-section studies, interest rate variations can
be an important consideration in explaining why some
studies show more or less scale economies than
others. Such variations are especially important when
studies conducted in the 1960s and early 197Os,
periods of relatively low interest rates, are contrasted
with studies of the late 1970s and early 1980s
periods of unusually high rates. But even over
1980-84 when rates were high there was enough
variation in the market interest rate to alter the slope
of the average cost curve, shifting around the large
banks so that small scale economies became small
diseconomies (Humphrey, 1987, Figures 4a and 4b).
To abstract from this problem, time-series studies
are needed since they can control for the year-to-

VII.
TIMEPERIODSWITHHIGHVERSUS
L~WINTERE~TRATES
The time period chosen for a cross-section study
of scale economies can affect the estimated slope of
the average cost curve. The reason is that total
costs-the
appropriate cost concept to use when
measuring scale economies-will
vary over the
FEDERAL

interest rate cycle and alter the slope of the estimated
cost curve.

RESERVE

12 If core deposits could be easily and rapidly substituted for
purchased funds when market rates were relatively high, and
vice versa when these rates were low, then the slope of the
average cost curve would not be dependent on the interest rate
cycle in the manner just described. But since such substitution
is quite limited in practice, and because core deposits are typically
treated as quasi-fmed inputs to the banking fnm (Flannery, 1982),
the effects of the interest rate cycle on cross-section scale
economy estimation are operative.
BANK

OF RICHMOND

45

year variation in the level of interest rates.13 It turns
out that those few time-series studies that do exist
show constant costs for large banks-a flat average
cost curve-when evaluated using the average interest
rate over the sample period (Hunter and Timme,
1986). When a broader sample of banks are used over
time, slight economies are measured for small banks
(around .95) and slight diseconomies for the largest
banks (around 1.05). l4 Overall, these time-series
results are quite similar to many, but not all, of the
studies that used cross-section data for a single year.15
Thus, while the time period can affect the slope of
the average cost curve and therefore the estimate of
the associated scale economy, in practice the bias
appears to have been relatively small. In any event,
the safest course is to rely on generalizations of a
number of single year cross-section results (as Mester,
1987, and Clark, 1988, have done) rather than
generalize from only a single one. The close correspondence between many cross-section studies and
the few time-series studies which exist supports this
conclusion.
VIII.

SINGLEVERSUSMULTIPLE
BANKOUTPUTS
Until quite recently, scale economy estimates were
based on how costs varied with changes in a single,
aggregate (stock) measure of bank output. That is,
Q rather than the separate and different bank outputs (Qi) that make up Q were specified in equation
(1). A problem with this approach is that there are
at least two quite different reasons why costs may
vary with an aggregate measure of output and only
one of them reflects scale economies. The other
reflects economies of scope, or cost changes related
to the number and joint production nature of the
different outputs produced. Scope economies occur
when costs fall as product mix is expanded, allowing fixed costs to be spread over a larger number of
different outputs.
l3 Making the average interest rate an independent variable in
equation (1) will control for the small variation in this rate across
banks in a cross-section analysis but will not control for the bias
introduced if the level of interest rates are atypically high or low
for the time period studied.
r4 These results are from unpublished work by the author using
a panel of almost 700 banks over 1977-88 that accounted for
$2 trillion out the $3 trillion in total U.S. banking assets.
r5 A large number of cross-section studies are summarized in
the comprehensive surveys of bank scale economies done by
Mester, 1987, and Clark, 1988. Their conclusions are similar
to those here in that scale economies seem to exist for small
banks while constant costs or slight diseconomies are measured
at the largest.
46

ECONOMIC

REVIEW,

In single-output studies, there is the possibility that
economies associated with output levels have been
confounded with economies associated with joint production. One may avoid this problem by specifying
a multiproduct estimating framework (using a number
of different Qis), rather than relying on an aggregate
index of the different outputs (where Q is a weighted
sum of the Qis). In this way, the two separate
influences-scale
and scope-can
be separated.16
A number of studies have tested the (functional
separability) conditions needed to justify a single
index of bank output and have rejected them statistically (Kim, 1986). Even so, as often happens,
statistical rejection has not led to economic rejection:
the scale economy results from single output studies
are quite similar to those found in multiproduct
analyses. That is, slight but significant economies are
measured at the office level (.96 to .98) for all sizes
of banks whereas the average cost curve describes
a relatively flat U-shape at the level of the banking
firm, this shape indicating significant economies at
small banks (around .94) but significant diseconomies
at the largest (around 1.06).17 As a result, biases
that could be due to commingling scope economies
with scale economies appear in practice to be slight.
Banks produce very similar product mixes, on
average, so that the importance of measured scope
economies using current observed production is
apparently small enough not to bias the scale
economy results obtained specifying single versus
multiple outputs. l8 In sum, there are strong theoretical reasons to (1) reject studies of scale economies
that have aggregated all bank outputs into a single
index and (2) use an explicit multiproduct specification in its place. In practice, however, the overall
I6 Strictly speaking, the relationship between scale and scope
economies is SIJ = (W Sr + (1 -W) Sz)/(l -SC) where St,2
is the measure of overall economies of scale (in a two-output
situation), Sr and Sz are the product-specific scale economies
of the two outputs, S, is the scope economy measure, and W
is a weight which is similar to the share of variable costs in total
cost for output 1 (See Bailey and Friedlaender,
1982, pp.
1031-32). Thus, the measure of overall economies of scale is
related to scope economies in the usual aggregate (single) output situation. Even if Sr and Sz show constant costs, the overall
scale measure (Sr.2) can falsely reflect economies or diseconomies
depending on the value of scope economies (S,).
I7 These results hold for both banks in unit banking and
branching states, with the exception that the results noted in
the text for the firm also apply to the average office in unit states
(Berger and Humphrey, 1990).
**This result refers to the small expansion path subadditivitv
results in Hunter, Timme, and yang, 1988, and Berger’
,
Hanweck, and Humphrey, 1987. Scope economies are a special
case of subadditivity and the complete specialization needed to
reflect the scope concept is rarely seen in banking.

SEPTEMBER/OCTOBER

1990

measure of scale economies is little affected by this
adjustment. l9
IX.

ALLBANKS ARE EFFICIENTVERSUS
ONLY
THOSEONTHEFRONTIER
A final source of bias in the estimation of bank scale
economies is the possibility that the economies exhibited by the set of most efficient or “best practice”
banks can differ from those exhibited by all banks,
efficient and inefficient. The potential for such bias
exists because scale economies measured using all
banks may be affected by other inefficiencies,
unrelated to scale. These other factors would give
a distorted picture of the true scale effects obtainable
if all banks were as well managed and efficiently
organized as those best practice banks with the lowest
average costs.
This possibility arises because substantial cost
differences, likely reflecting inefficiencies, seem to
exist in banking (Humphrey, 1987). When all banks
are stratified by size and then divided up into quartiles based on their levels of average costs for various
years during the 198Os, the mean variation in average
cost between the highest and lowest average cost
quartiles of banks is 34 (31) percent for branching
(unit) state banks. Since the mean variation in average
cost across size classes was only 8 (12) percent, the
variation between quartiles is seen to be 4 (2) times
the variation across size classes. This pattern indicates that relative efficiency differences between
similarly sized banks far exceed those obtainable by
only altering bank size.rO
To put these results differently, if a $500 million
asset bank experienced a drop in its average cost from
19One benefit of a multiproduct specification, however, is that
scale economies for each output can be determined separately
and contrasted. The scope economy results derived from a
multiproduct specification have, however, been disappointing
as there has been a lack of consistency in the value of scope
economies estimated. It has been shown that one reason for the
markedly different scope economy results in different studies
is a limitation in the translog functional form itself (virtually the
only form used today in banking studies). When a form that better
fits the data is used instead, consistent values for scope
economies result regardless of the point of evaluation (Pulley
and Humphrey, 1990).
20 These differences are not due to chance occurrences of high
or low costs among banks as they exist for the same banks
during different time periods, when chance variations would be
expected to average out. As well, low-cost banks consistently
have higher profits (and vice versa). Thus whatever is happening on the cost side rolls over to the revenue side as well, rather
than being the result of high-cost banks producing a different
output which is offset by higher revenues (Rerger and Humphrey,
forthcoming).
FEDERAL

RESERVE

the mean of the highest to the mean of the lowest
average cost quartile, costs would have fallen by 3 1
to 34 percent. Such a cost reduction would be
equivalent to a scale economy value of .69 to .66.
Since this figure far exceeds most estimates attributable to scale economies (e.g., .95), it is seen that
even the existence of substantial scale economies at
higher cost banks will not enable them to become
competitive
with smaller OT larger banks that
happen to be in the lowest cost quartile. Thus the
competitive implications of scale economies at large
banks are qualified by the existence of offsetting
differences in cost levels or relative efficiency for all
sizes of banks.2’
Surprisingly, given the large differences in average
costs between low- and high-cost banks, the scale
economy results for banks in the lowest cost quartile (and therefore on the efficient cost frontier) are
very similar to those -obtained when all banks are
pooled together (Berger and Humphrey, 1990). Thus
while there are considerable differences in cost efficiency across banks, these differences do not
significantly affect the scale economy results or conclusions of the previous section. Frontier analyses,
which focus on low-cost or efficient banks, give the
same results as the more traditional studies which
estimate scale economies for all banks in a sample.
X.

S~MMARYANDCONCLUSIONS
There are important economic and political issues
related to the size of scale economies in banking.
Measurement of these economies is an empirical
issue and, when many studies exist, it is possible to
sort out the likely reasons for seemingly conflicting
results. Such an understanding of the data and the
results of different research designs permits the
derivation of a consensus position useful for policy
purposes.
Seven common differences in existing bank scale
economy studies have been identified and discussed.
These are summarized in Table I. Of the seven, only
three (numbers 1, 3, and 4) led to problems sufficiently serious to warrant discounting the conclusions
of studies incorporating them. Analyses which relate
operating costs-not total costs-to variations in bank
output contain a bias due to differences in the funding mix across banks. As a result, these analyses are
typically biased toward finding scale economies when
zr Similar conclusions apply to thrift institutions
McNuIty, and Rochester, 1990).
BANK

OF RICHMOND

(Verbrugge,

47

Table

Summary of Differences
Common

I

Among Bank Scale Economy Studies
Bias Found:

Differences:

1. Cost Definition
(operating versus total cost)

Use of operating
scale economies.

2.

Either output

Output measurement
(number of accounts

cost gives bias toward finding

measure

gives similar

results.

versus dollars in the accounts)

3.

Functional form
(linear versus quadratic)

Linear (Cobb-Douglas)
scale economies.

form gives bias toward finding

4.

Point of scale economy evaluation
(single office versus banking firm)

Evaluation for average
policy purposes.

banking

5.

Time period used
(high versus low interest

Bias exists but is minor.

not relevant

for

rates)

6.

Commingling
scale with scope
(single versus multiple outputs)

Similar scale economy
multiple outputs.

7.

Efficiency differences
(average bank versus those on frontier)

No effect

none may exist after proper account is taken of all
costs associated with producing bank outputs. Thus,
believable scale economy estimates should be based
on models using total costs, not just operating costs.
As well, a quadratic functional form such as the
translog that permits a U-shaped cost curve to be
estimated if it exists in the data, is always favored
over a linear function such as the Cobb-Douglas. This
eliminates the majority of the earlier studies in which
the (log linear) Cobb-Douglas form was used and
scale economies were regularly (mis)identified.
Lastly, only those scale economies evaluated at the
level of the banking firm are pertinent to the policy
issues at hand since it is the size of the banking firm,
not the size of the average office, which captures the
full cost efficiency associated with the two ways that
bank output can be expanded. While some problems
are encountered in using different measures of bank
output, selecting different time periods for estimation, commingling scale with scope economies, and
pooling efficient with inefficient banks, the resulting
scale estimates obtained in these four cases are
reasonably robust to these different treatments.
Overall, a consensus conclusion of the preferred
studies on bank scale economies suggests that the
average cost curve in banking reflects a relatively flat
U-shape at the firm level, with significant economies
at small banks (around .94) but small and significant
48

office

results with either

on scale economy

single or

results.

diseconomies at the largest (around 1.06). This
relatively flat U-shape also holds even when large
banks are viewed separately. The implication is that
the slight diseconomies identified for all large banks
together represents an average for some of the smaller
large banks possessing economies and the very largest
which seem to possess diseconomies.
From these results, some practical conclusions may
be inferred. First, there would seem to be little
benefit of a cost-reducing nature from a marked
increase in bank size alone, although significant
benefits from loan diversification would exist for giant
nationwide banks. Second, the measured scale or cost
economies are small in comparison to existing differences in cost levels between similarly sized banks.
This finding implies that even if cost economies were
pervasive, which they are not, they would have a
much smaller competitive impact than has been
heretofore presumed. The large and persistent cost
differences between banks of a similar size and
product mix suggest that greater competition within
the banking industry would be beneficial but that this
need not be associated with bank size. One way to
enhance competition is to permit easier entry into
and exit from the industry. A step in this direction
will come with full interstate banking during the next
decade when geographical restrictions on entry are
to be removed.

ECONOMIC REVIEW, SEPTEMBER/OCTOBER

1990

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OF RICHMOND

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50

ECONOMIC

REVIEW,

SEPTEMBER/OCTOBER

1990


Federal Reserve Bank of St. Louis, One Federal Reserve Bank Plaza, St. Louis, MO 63102