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STO CH A STIC TRENDS AND EC O N O M IC
FLUCTUATIONS
Robert G. King, Charles I. Plosser,
James H. Stock and Mark W. Watson
Working Paper Series
Macro Economic Issues
Research Department
Federal Reserve Bank of Chicago
February, 1991 (WP-91-4)
Stochastic Trends and Economic Fluctuations
by
Robert G. King
Department of Economics
University of Rochester
Rochester, NY 14627
Charles I. Plosser
William E. Simon Graduate School of Business Administration
University of Rochester
Rochester, NY 14627
James H. Stock
Department of Economics
University of California, Berkeley
Berkeley, CA 94720
and
Mark W. Watson
Department of Economics
Northwestern University
Evanston, IL 60208
and Chicago Federal Reserve Bank
Revised: January 1991
The authors thank Ben Bernanke, C.W.J. Granger, Robert Hall, Gary Hansen, Bennett
McCallum, Thomas Sargent, John Taylor, James Wilcox, and two referees for helpful discussions
and comments, and Craig Burnside and Gustavo Gonzaga for valuable research assistance. This
research was supported in part by the National Science Foundation, the Sloan Foundation and
the John M. Olin Foundation at the University of Rochester.
A bstract
Are business cycles mainly the result of permanent shocks to productivity? This paper uses
a long-run restriction implied by a large class of real business cycle models -- identifying
permanent productivity shocks as shocks to the common stochastic trend in output, consumption
and investment -- to provide new evidence on this question. Econometric tests indicate that
this common stochastic trend/cointegration implication is consistent with postwar U.S. data.
However, in systems with nominal variables, the estimates of this common stochastic trend
indicate that permanent productivity shocks typically explain less than half of the business cycle
variability in output, consumption and investment. (JEL 131,211)
A central, surprising and controversial result of some current research on real business
cycles is the claim that a common stochastic trend -- the cumulative effect of permanent shocks
to productivity -- underlies the bulk of economic fluctuations. If confirmed, this finding
would imply that many other forces have been relatively unimportant over historical business
cycles, including the monetary and fiscal policy shocks stressed in traditional macroeconomic
analysis. This paper shows that the hypothesis of a common stochastic productivity trend has a
set of econometric implications that allows us to test for its presence, measure its importance
and extract estimates of its realized value. Applying these procedures to consumption,
investment and output for the postwar U.S., we find results that both support and contradict
this claim in the real business cycle literature. The U.S. data a re consistent with the presence
of a common stochastic productivity trend. Such a trend is capable of explaining important
components of fluctuations in consumption, investment and output in a three variable reduced
form system. But the common trend’s explanatory power drops off sharply when measures of
money, the price level and the nominal interest rate are added to the system. The key
implication of the standard real business cycle model - that permanent productivity shocks are
the dominant source of economic fluctuations -- is not supported by these data. Moreover, our
empirical results cast doubt on other explanations of the business cycle: estimates of permanent
nominal shocks, which are constrained to be neutral in the long-run, explain little real activity.
Our econometric methodology can determine the importance of productivity shocks within a
wide class of real business cycle (RBC) models with permanent productivity disturbances. To
explain why this is so, we begin by discussing three features of the research on which our
analysis builds. First, there is a long tradition of empirical support for balanced growth in
which output, investment and consumption all display positive trend growth but the
consumption-output and investment-output "great ratios” do not (see, for example, Robert
Kosobud and Lawrence Klein (1961)). Second, in large part because of this ratio stability, most
-1 -
RBC models are one sector models which restrict preferences and production possibilities so that
"balanced growth" occurs asymptotically when there is a constant rate of technological progress.
Third, these RBC models imply that permanent shifts in productivity will induce (i) long-run
equiproportionate shifts in the paths of output, consumption and investment; and (ii) dynamic
adjustments with differential movements in consumption, investment and output.
The econom etric procedures developed here use the models’ long-run balanced growth
implication to isolate the permanent shocks in productivity, and then to trace out the short-run
effects of these shocks. These econometric procedures rely on the fact that balanced growth
under uncertainty implies that consumption, investment and output are cointegrated in the sense
of Robert Engle and Clive Granger (1987). In turn, this means that a cointegrated vector
autoregression (V A R ) nests log-linear approximations of all RBC models that generate long-run
balanced growth. Our empirical analysis is based on such a cointegrated V A R (or vector error
correction model), which is otherwise unrestricted by preferences or technology. Thus, our
conclusions can be interpreted as casting doubt on the strong claims emerging from an entire
class of real business cycle models.
The empirical analysis is structured around three questions. First, what are the
cointegration properties of postwar U.S. data, and are these properties consistent with the
predictions of balanced growth? Second, how much of the cyclical variation in the data can be
attributed to innovations in the common stochastic trends? Third, a natural alternative to RBC
models is one in which nominal variables play an important role. Do innovations associated
with nominal variables explain important cyclical movements in the real variables?
The empirical results provide robust answers to these questions. First, cointegration tests
and estimated cointegrating vectors indicate that the data are consistent with the balanced
growth hypothesis. Second, in a three variable model incorporating output, consumption and
investment, the balanced growth shock explains 60-75% of the variation of output at business
cycle horizons (4 to 20 quarters). Moreover, the estimated response of the real variables to the
-2-
balanced growth shock is similar to the dynamic multipliers implied by simple RBC models
driven by random walk productivity. Third, these results change significantly when nominal
variables are added to the empirical model. When money, prices, and interest rates are added,
the balanced growth shock explains less of the fluctuations in output, 35% to 44%, depending on
the particular specification used. Permanent nominal shocks, identified by imposing long-run
neutrality for output, explain little of the variability in the real variables. Much of the shortrun variability in output and investment is associated with a shock that has a persistent effect
on real interest rates. These results suggest that models that rely solely on permanent
productivity and/or long-run neutral nominal shocks are not capable of capturing important
features of the postwar U.S. experience.
The paper is organized as follows. Section II provides theoretical background and reviews
recent work on real business cycles. Section III outlines the empirical model and discusses
identification. Sections IV and V present the empirical results. Section VI concludes.
II. Growth and Fluctuations: Theoretical Background
To fix some ideas and notation, this section outlines a simple real business cycle model with
permanent productivity shocks. The model is of the general class put forward by Fynn
Kydland and Edward Prescott (1982) and is detailed in Robert King, Charles Plosser, and Sergio
Rebelo (1988). Output, Y t, is produced via a constant returns to scale Cobb-Douglas
production function:
(2.1)
Yt = AtKj-*N?
where Kt is the capital stock and N t represents labor input. Total factor productivity, At,
follows a logarithmic random walk:
-3-
(2.2)
l°g(*t) =
+ log(At-l) +
where the innovations, {£t}, are independent and identically distributed with a mean of 0 and a
2
variance of a . The parameter
represents the average rate of growth in productivity;
represents deviations of actual growth from this average.
Within the basic neoclassical model with deterministic trends, it is familiar - from Robert
Solow (1970) - that per capita consumption, investment and output all grow at the rate /x^/0 in
steady state.
1
The common deterministic trend implies that the "great ratios" of investment and
consumption to output are constant along the steady state growth path. When uncertainty is
added, realizations of
change the forecast of trend productivity equally at all future dates:
Ejlog(At+s)=Et j(At+s)+£t* A positive productivity shock raises the expected long-run growth
path: there is a common stochastic trend in the logarithms of consumption, investment and
output. The stochastic trend is log(At)/0 and its growth rate is (/x^+£t)/0, the analogue of the
deterministic model’s common growth rate restriction, jjl^ /0. With common stochastic trends the
great ratios C ^ /Y ^ and I^/Y^. become stationary stochastic processes.
These theoretical results have a natural interpretation in terms of cointegration. Let X t be a
vector of the logarithms of output, consumption and investment at date t, denoted by yt, ct and
it. Each component of X t is integrated of order one (1(1) or, loosely speaking, "nonstationary")
because of the random walk nature of productivity. Yet, the balanced growth implication of
the theory implies that the difference between any two elements of X t is integrated of order
zero (1(0) or "stationary"). In Engle and Granger’s (1987) terminology, the two linearly
independent cointegrating vectors, aj=(-l, 1 , 0 )’ and
>0 , 1 )’ isolate stationary linear
combinations of X t corresponding to the logarithms of the balanced growth great ratios.
In this basic one-sector model and variants of it, the precise dynamic adjustment process to
a permanent productivity shock depends on the details of preferences and technology. For
-4-
example, recent RBC research has studied alterations in the investment technology (time-tobuild, adjustment costs, inventories), the production technology (variable capacity utilization,
labor indivisibilities, employment adjustment costs), preferences (nonseparabilities in leisure,
durable consumption goods) and serial correlation in the productivity growth process. Two
general properties emerge from these investigations. First, the productivity shock sets off
transitional dynamics, as capital is accumulated and the economy moves towards a new steady
state. During this transition, work effort and the great ratios change temporarily. Second,
there is a common stochastic trend in consumption, investment and output arising from
9
productivity growth. These two properties motivate the econometric theory and empirical
research described in the next sections.
In systems that incorporate both real and nominal variables, additional cointegrating relations
may plausibly arise. Two are relevant for our empirical analysis. The first is the money
demand relation:
(2.3)
mt - pt = /3yyt - % R t + vt
where mt-pt is logarithm of real balances, R t is the nominal interest rate and
is the money
demand disturbance. The second is the conventional Fisher relation:
(2.4)
R t = rt + E tApt+ 1
where rt is the ex-an te real rate of interest, pt is the logarithm of the price level and E tApt+^
denotes the expected rate of inflation between t and t+1. If real balances, output and interest
rates are 1(1), while the money demand disturbance in (2.3) is 1(0), then real balances, output
and nominal interest rates are cointegrated. If the real rate is 1(0) and the inflation rate is 1(1),
then (2.4) implies that nominal interest rates and inflation are cointegrated. The empirical
-5-
analysis investigates these cointegrating relations and isolates the common stochastic trends that
they imply.
III. Econometric Methodology
This section provides an overview of the econometric techniques used to answer the
questions posed in the introduction. The first question, concerning the integration and
cointegration properties of the data, can be addressed using techniques that are now familiar.
This section therefore focuses on the specification of an econometric model in which the trends
and their impulse response functions can be identified and estimated.
Let X t denote an nxl vector of time series. The individual series are assumed to be 1(1) (so
that they must be differenced before they are stationary) and to have the Wold representation:
(3.1)
AXt = n + C(L)et
where et is the vector of one-step-ahead forecast errors in X t given information on lagged
values of X t- The et’s are serially uncorrelated with a mean of zero and covariance matrix
Equation (3.1) is a reduced form relation, and except for purposes of forecasting, is of little
inherent interest. What is of interest is the set of structural relations that leads to (3.1), and the
primary purposes of this section are to discuss (i) how the balanced growth and other
cointegration restrictions outlined in the last section restrict this set of structural relations, and
(ii) how these restrictions can be exploited to draw inferences about structural relations from
consistent estimators of C(L) and S £.
To be specific, consider a structural model of the form:
(3.2)
AXt = n + r(L)r,t
-6-
where rjt is a nxl vector of serially uncorrelated structural disturbances with a mean of zero and
a covariance matrix E . The reduced form of (3.2) will be of the form (3.1) with e^=TQri^ and
C(L)=r(L)ro1 3
The identification problem can now be stated as follows: under what conditions is it possible
to deduce the structural disturbances rjt and matrix of lag polynomials r(L) from the reduced
form errors e t and matrix of lag polynomials C(L)? In the classic literature on simultaneous
equation models, the identification problem is solved by postulating that certain blocks of T(L)
are zero, so that some of the X’s are exogenous or predetermined. In linear rational
expectations models, the identification problem is solved by imposing cross-equation restrictions
on the various elements of r(L), as described, for example in Kenneth Wallis (1980) and Lars
Hansen and Thomas Sargent (1980). The literature on vector autoregressive models addressed
the identification problem by imposing restrictions on the covariance matrix E^ and the matrix
of structural impact multipliers, Tq. For example, in his classic paper on vector autoregressions,
Christopher Sims (1980) assumes that E^ is diagonal and Tq is lower triangular, assumptions
analogous to a Wold causal chain; Olivier Blanchard and Mark Watson (1986) modify Sims’
original procedure by imposing restrictions on Tq analogous to those appearing in static
simultaneous equation models.
In this paper, identification is achieved through two sets of restrictions. First, the
cointegration restrictions impose constraints on the matrix of long-run multipliers r(l)
(=Xi°=Qri) in (3.2). This identifies the permanent components. Second, the innovations in the
permanent components are assumed to be uncorrelated with the innovations to the remaining
transitory components. This identifies the dynamic response of the economic variables to the
permanent innovations. A concise algebraic summary of the identification scheme is given in
the appendix, with a more extensive discussion in King, Plosser, James Stock and Watson
(1987); here we outline the major ideas and relate our procedure to other recent work.
-7-
Consider the three variable model with X t=(yt,ct,it)- Because there are two cointegrating
vectors, there is only one permanent innovation, the balanced growth innovation r?J. This
shock corresponds to
3
r)t
2
in the neoclassical model of Section II. The other two shocks, r?t and
have only transitory effects on X {. Thus, the first identification restriction (the balanced
growth cointegrating vectors) implies that the matrix of long-run multipliers is
(3.3)
1 0
1 0
1 0
r(i)
0
0
0
where the values of the coefficients in the first column of r(l) are normalized to 1 to fix the
scale of r;J. Equation (3.3) serves to identify the balanced growth shock as the common longrun component in X t, since the innovation in the long-run forecast of X t is (11 l)’7?J=C(l)et,
which can be calculated directly from the reduced form (3.1). The second restriction - that
1
2
3
1
»7t is uncorrelated with r;t and r;t -- is used to determine the dynamic effect of r?t on
X t, that is, to identify the first column of r(L). The reason this assumption is needed is clear:
the impulse responses given by the first column of r(L) are the partial derivatives of AXt+^
with respect to
The second restriction specifies what is being held constant in computing
these second derivatives.
Another way to motivate these identifying restrictions is to rewrite the model in terms of
the stationary variables Z^Ay^Cj-y^ij-yj)’. The productivity shock, i
has a long-run
effect on yt, but no long-run effect on the stationary ratios, ct-yt and it*yt- Thus, rjJ can be
identified as the innovation in the long-run component of the first element of Z t. The other
2
3
two disturbances, r;t and
have temporary effects on yt and the ratios.
Olivier Blanchard and Danny Quah (1989) used a special case of this identification scheme
to analyze Z^=<Ayt,ut), where ut was the unemployment rate, which they assumed to be 1 (0 ).
1
2
Their disturbances were f/t, a "supply shock," and rjt, a "demand shock." These shocks
were restricted to be uncorrelated, and only the supply shock, fjj, was allowed to have a
-8-
long-run effect on yt. Thus, except for the fact that their system is bivariate and ours is
trivariate, the identifying restrictions are identical. Indeed, if we eliminate one of the ratios, so
that our model is bivariate, our identifying restrictions are formally equivalent to those used by
Blanchard and Quah.
This equivalence highlights what we consider to be two practical advantages of the empirical
specification employed here. First, work in the tradition of Milton Friedman (1957) links
consumption to permanent income. This suggests that the emphasis on real flow variables,
rather than on unemployment and output as in Blanchard and Quah, arguably will result in
better estimates of the trend components of output and the parameters of the structural model.
4
Second, our application is to multivariate systems rather than bivariate systems. This has two
advantages: first, more macroeconomic variables are used to estimate the common trends, and
second, by allowing for a wider range of shocks, a richer set of alternative models is
considered.
To introduce nominal shocks, the three-variable real model is augmented by real balances,
nominal interest rates and inflation. The resulting six variable model has three common
stochastic trends, and this makes identification more complicated since the individual permanent
innovations must be sorted out. We use a version of Sims’ (1980) procedure for this purpose.
The general indentification problem can be described as follows. Suppose that there are k
common stochastic trends driving the nxl vector X^. Partition the vector of structural
1 2
disturbances rjt into two components, ri^=(rj^
r^’)’, where rj1t contains the disturbances
2
that have permanent effects on the components of X t and rj t are disturbances that have only
temporary effects. (In our six variable model k=3, and
is a 3x1 vector containing the
balanced growth shock, a long-run neutral inflation shock, and a real interest rate shock.)
Partition T(l) conformably with r?t as r(l)=[A 0], where A is the nxk matrix of long-run
multipliers for 77J and 0 is an nx(n-k) matrix of zeros corresponding to the long-run
2
multipliers for rj^ The matrix of long-run multipliers is determined by the condition that its
-9-
columns are orthogonal to the cointegrating vectors, and A»jJ represents the innovations in the
long-run components of X t.
Identification of the individual elements of
becomes more complicated when there is
more than one permanent innovation because the unique influence of each permanent
component needs to be isolated. Formally, while the cointegration restrictions identify the
permanent innovations ArjJ, they fail to identify
because Ar?j=(APXP ^»?J) for any
nonsingular matrix P. The following restrictions are used to identify the model. First, as in
1
the model with k=l, we assume that r^ and
2
are uncorrelated. Second, the permanent
shocks, rjJ, are assumed to be mutually uncorrelated. Third, A is assumed to be lower
triangular, which permits writing A=An, where A is a matrix with no unknown parameters
(analogous to the vector of l ’s in the k=l model) and n is a kxk lower triangular matrix. As
will become clear in the next section, A can be chosen in a way that associates each shock
with a familiar econom ic mechanism: the first disturbance is interpreted as a balanced growth
shock, the second is a long-run neutral inflation shock and the third is a permanent real interest
rate shock. Finally, the constrained reduced form is estimated as a V A R with error correction
terms, i.e., a vector error correction model (VECM).
IV. Empirical Results
IV.a The Data
The data are quarterly U.S. observations on real aggregate national income account flow
variables, the money supply, inflation and a short term interest rate. The three aggregate real
flow variables are the logarithms of per capita real consumption expenditures (c), per capita
gross private domestic fixed investment (i) and per capita "private" gross national product (y),
defined as total gross national product less real total government purchases of goods and
services. The measure of the money supply used is M2 (per capita in logarithms, m). The
-10-
price level is measured by the implicit price deflator for our measure of private GNP (in
logarithms, p) and the interest rate (R) is the three-month U.S. Treasury bill rate. Regressions
were run over 1949:1 through 1988:4 for statistical procedures that involve only real flows. To
avoid observations that occurred during periods of price controls, the Korean War, and the
Treasury-Fed accord, the regressions were run over 1954:1 to 1988:4 when money, interest rates,
or prices are involved. Data prior to 1949:1 (respectively 1954:1) were used as initial
observations in regressions that contain lags.^
Because the national product measure is not the standard one we have graphed the logarithm
of the real variables (y, c, i and m-p) in Figure la. These plots show the familiar growth and
cyclical characteristics of the data. Output, consumption and investment display strong upward
trends. Investment is the most volatile component, followed by output and then consumption.
Real balances (m-p) also display an upward trend. Figure lb plots the logarithm of the
consumption output ratio (c-y) and the logarithm of the investment ratio (i-y). Over the
postwar period, these ratios display the stability reported by prior researchers; it is easy to view
them as fluctuating around a constant mean. This suggests that the growth evident in Figure 1
occurs in a manner that is "balanced" between investment and consumption.
IV .b In te g ra tio n a n d C o in te g ra tio n P r o p e r tie s o f the D a ta
Univariate analysis of these six variables indicates that the real flow variables, y, c and i can
be characterized as 1(1) processes with positive drift, and that R, price inflation (Ap) and
nominal money growth (Am) can be characterized as 1(1) processes without drift. These results
are consistent with the large literature on the "unit root" properties of U.S. macroeconomic time
. 6
series.
The balanced growth conditions also appear consistent with the data: we can reject the
presence of unit root components in the great ratios.
Augmented Dickey-Fuller t-statistics
(f , with 5 lags) testing for a unit autoregressive root in c-y and i-y have values of -4.21 and
- 11 -
-3.99 respectively; both are significant at the 1% level, suggesting that (c,y) and (i,y) are
cointegrated. The log of real balances, m-p, appears to be an 1(1) process with drift, even
though both m and p can be characterized as 1(2) processes. (The f r statistic for A(m-p) is
significant at the 1% level). The best characterization of the real interest rate is unclear. The
e x -p o st
real rate, R-Ap, has sample AR(1) coefficient equal to .86, suggesting stationary
behavior, but the augmented Dickey-Fuller t-statistic (f #, with 5 lags) is only -1.8, which is
consistent with one unit root in the real rate.
Tables 1-3 present a variety of statistics calculated from multivariate systems, starting with
the results for the three variable (y,c,i) model. Panel A of Table 1 shows the largest
eigenvalues from the companion matrix of an estimated VAR(6). The one common stochastic
trend (balanced growth) model implies that the companion matrix should have one unit
eigenvalue, corresponding to the common trend, and all the other eigenvalues should be less
than one in modulus. The point estimates accord with this prediction. Panel B presents formal
tests for cointegration using procedures developed by Soren Johansen (1988) and Stock and
Watson (1988). Both procedures take as their null hypothesis that the data are integrated but
not cointegrated, so that there are three unit roots in the companion matrix. The first two rows
of Panel B test this against an alternative of at most 2 unit roots, while the third row tests the
null against the sharper alternative of at most 1 unit root. The results are consistent with the
one unit root (one common trend) specification.
7
The final panel in Table 1 presents maximum likelihood estimates of the cointegrating
vectors, conditional on the presence of one unit root in the VAR, computed using the dynamic
OLS procedure of Stock and Watson (1989). The point estimates are close to (1,0,-1) and (0,1,1), the values that imply balanced growth in output, consumption and investment. These
balanced growth restrictions impose two constraints on the cointegrating vectors. In Table 1,
these restrictions are tested using a Wald statistic based on the dynamic OLS point estimates and
standard errors; under the null hypothesis, this statistic has an asymptotic chi-squared
-12-
distribution with two degrees of freedom. Although the restriction is rejected at the 10% (but
not the 5%) level, the estimated cointegrating vector is broadly consistent with the balanced
g
growth prediction.
Table 2 explores two sets of cointegrating relations suggested by the nonstationarity of the
nominal and real interest rates. The top panel of the table reports an estimated cointegrating
9
relation between real balances, output and nominal interest rates.
The estimate of the long-
run income elasticity is close to unity (although statistically significantly larger than 1); the
estimated interest rate semi-elasticity is small, but statistically significantly less than zero.
The second issue examined in Table 2 is the possibility that the consumption/output and
investment/output ratios might exhibit permanent shifts resulting from permanent shifts in real
rates. Estimated bivariate cointegrating relations (c-y)=0j(R-Ap) and (i-y)=<^2 (R-Ap) are
shown in panel B of Table 2. As predicted by the long-run theory of the growth model, for
example, a higher real interest rate lowers the share of product going into investment and,
symmetrically, raises the share of consumption. However, the long-run effects are imprecisely
estimated and small: a permanent increase in the annual real rate of one percentage point is
associated with an increase in the consumption-output ratio of 0.3 percentage points.
The cointegration properties of the six variable system (y,c,i,m-p,R,Ap) are investigated in
Table 3. The theoretical analysis suggests 3 stochastic trends in the system - a balanced
growth trend, an inflation/money growth trend and, possibly, a real interest rate stochastic
trend. Equivalently, three cointegrating relations should be present in the system —two
(interest rate adjusted) balanced growth relations and a long-run money demand relation. The
f
q (6,3) statistic reported at the bottom of Table 3, panel A provides some evidence for the
three-trend specification, rejecting six unit roots against three with a p-value of 11%.
The Wald tests in panel B of Table 3 investigate various hypotheses about the cointegrating
vectors, under the maintained hypothesis that the number of cointegrating vectors is correctly
specified. The first hypothesis (panel B, row 1) is that the cointegrating vectors are the
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balanced growth and money demand cointegrating vectors. This hypothesis is rejected at the
10%, but not the 5% level. Despite this formal rejection at the 10% level, we interpret the
balanced growth/money demand cointegrating restrictions as providing a good qualitative
description of the cointegrating vectors for the system. The remaining lines of Table 3, panel B
investigate alternative cointegration restrictions. There is strong evidence against a fourth
cointegrating vector implying stationary real rates (line 2) and against the stationary velocity
model, even permitting cointegration between the great ratios and real rates (line 4). The
evidence is weakest against the hypothesis that the great ratios and real rates are cointegrated,
combined with the money demand cointegrating vector (line 3).
10
Taken together, these results suggest that a money demand cointegrating relation is
consistent with the observed behavior of the time series. There is some evidence that the shares
of consumption and investment move with permanent shifts in the real rate. Yet, this effect is
negligibly small in the long-run, and the hypothesis of "balanced growth" also appears generally
consistent with the data.
IV .c A T h re e V a ria b le S y ste m o f R eal F lo w V a ria b les
The results for the three-variable system are based on an estimated VECM using eight lags
of the first differences of y, c and i with an intercept and the two theoretical error correction
terms, y-c and y-i. The only identifying assumption needed to analyze the dynamics of the
system is that the permanent shock is uncorrelated with the transitory shocks. The impulse
response of y, c and i to a one standard deviation innovation in the common trend is plotted in
11
Figure 2, together with one standard deviation confidence bands. . (The standard deviation of
the balanced growth shock is 0.7%, and, as discussed in Section III, the system is normalized so
that a one unit innovation eventually leads to a one unit increase in y, c, and i.) In response to
a shock that leads to a one percent permanent increase in y, c and i, output and investment
increase by more than one percent in the near term (one to two years), while consumption
-14-
moves less. Most of the adjustment is completed within four years. The results for c and i are
consistent with the simple theoretical model discussed in Section II where the capital stock
rapidly increases at the short-run cost of consumption.
Are these responses large enough to explain a substantial fraction of the short-run variation
in the data? This key question is addressed in Table 4, which shows the fraction of the
variance of the forecast error variance attributed to innovations in the common stochastic trend,
at horizons of one to twenty-four quarters. These variance decompositions suggest that
innovations in the permanent component appear to play a dominant role in the variation in GNP
and consumption. At the one to four quarter horizon, the point estimates suggest that 45% to
58% of the fluctuations in private GNP can be attributed to the permanent component. This
increases to 68% at the two year horizon and to 81% at the six year horizon. The results for
consumption are broadly similar. Notably, the permanent component explains a much smaller
fraction of the movements in investment: only 31% at the one year horizon, increasing to 47% at
the six year horizon.
This evidence suggests the existence of a persistent, potentially permanent, component which
shifts the composition of real output between consumption and investment. (If there were
temporary components with negligible effect on forecast errors after three or more years, then
the population counterparts of the variance ratios in Table 4 would increase more sharply at the
longer horizons.) Thus, the results motivate us to investigate the possibility of additional
permanent components.
IV ,d S ix -V a ria b le S y ste m s w ith N o m in a l V a ria b les
Augmenting output, consumption and investment by real balances, nominal interest rates and
inflation yields a six variable system. The results of Section IV.b suggest that a reasonable
specification incorporates three cointegrating relations (and thus three common trends) among
the six variables. We have estimated a variety of six variable models, with different numbers
-15-
of trends, different cointegrating relations and different ordering of the shocks. The detailed
results for a benchmark model are reported in this subsection, and the results for the other
models are summarized in the next subsection. The benchmark model incorporates the
cointegrating relations (c-y)=<£j(R-Ap), (i-y)=$ 2 (R-Ap) and m-p=y9yy-/9j^R. The first two
relations link variation in the real ratios to permanent shifts in the real interest rate, although
the estimates reported above suggest that this effect is small. The third implies that "money
demand" disturbances are 1(0). The estimates of <f>^, ^
Py
and p ^ are the restricted dynamic
OLS estimates given in Table 2.
The permanent components and their impulse responses are identified by specifying a
structure for the matrix of long-run multipliers. In the notation of Section III, with
X=(y,c,i,m-p,R,Ap), the particular structure adopted is:
0
1
0
1
0
0
h
1
0
0
*21
1
0
*31
*32
i-l
(4.1)
1
<f>2
A
h
'^R
'^R
0
1
1
0
1
0
The 6x3 matrix A is the matrix of long-run multipliers from the three permanent shocks. In
the notation of Section III, the two matrices on the right hand side of (4.1) are A and n
respectively.
Our interpretation of the shocks follows from the structure placed on the long-run
multipliers in (4.1). The first shock is a real balanced growth shock, since it leads to a unit
long-run increase in y, c and i. Through the money demand relation, it also leads to a /?y
increase in real balances. The second shock is a neutral inflation shock. It has no long-run
effect on y, c, or i, and has a unit long-run effect on inflation and nominal interest rates.
Further, the unit increase in nominal interest rates arising from this shock leads to reduction of
-16-
real balances of
The final permanent shock is a real interest rate shock. A one unit
increase in this shock leads to a change of <j>^ and <j>2 in c-y and i-y. There is also a one unit
increase in nominal interest rates and a decrease of
in real balances. The coefficients in n
are determined by the requirement that the permanent innovations are mutually uncorrelated.
In the standard V A R terminology, following Sims (1980), the balanced growth disturbance is
ordered first, the inflation disturbance is second and the real rate shock is third.
The model is estimated using a VECM with eight lags and the three error correction terms
implied by the cointegrating relations. Table 5 presents the variance decompositions of the
forecast errors from the benchmark model. Four aspects of this table are of particular interest.
First, relative to the three variable model, the real or "balanced growth" shock is less important
for output and consumption, especially at the one to four quarter horizon. At the three to five
year horizon, however, this shock has important explanatory power: roughly one half the
variation in these forecast errors is attributable to the first permanent component. Second,
including the additional shocks in this expanded model does not enable the first permanent
component to explain the short-run variations in investment. Third, the component associated
with permanent shifts in the rate of inflation explains a considerable amount of the variation in
inflation, but little else. Fourth, the component associated with permanent movements in the
real interest rate explains most of the forecast errors in the nominal rate. It also is important
for real activity: it explains substantial amounts of the output and investment forecast errors,
particularly at the short horizons.
Figure 3 illustrates the roles played by the different shocks by plotting the forecast error at
the three year horizon and the portion attributable to each stochastic trend for y, c and i.
These plots highlight the negligible role of the inflation shock and the substantial role played by
the balanced growth shock and the real interest rate shock. Looking at specific episodes in this
figure, one finds that the balanced growth shock has particular explanatory power for the
sustained growth of the 1960s. On the other hand, the real rate shock seems particularly
important in the contraction of 1974 and the 1981-82 recession.
-17-
Figure 4 shows the responses of the variables to one standard deviation impulses in the
balanced growth shock, the inflation shock and the real interest rate shock. The estimated
standard deviations of these underlying shocks are 0.7%, 0.08% and 0.12% per quarter. The
response of output to the balanced growth shock is negligible over the first few quarters, while
consumption increases slightly and investment declines. After one year, major increases in
output, consumption and investment are present. While these responses are smaller than those
in the three variable model, they conform to how one might think a system would respond to
news about technological developments. The inflation shock has very little impact on output
and consumption. Investment, on the other hand, shows a large positive response to this shock
for the first three quarters.
We have already noted that the real interest rate shock plays an important role in explaining
the short-run behavior of output and investment. The impulse response functions make
interpreting this shock as a permanent change in the real rate of interest somewhat difficult.
A ll three of the real flow variables have an initial response to this shock that is strongly
positive, before turning negative after two to three quarters. While there may be economic
models that predict such responses to a permanent change in the real rate, standard ones do not.
We draw four conclusions from this analysis. First, permanent innovations account for a
substantial fraction of transitory economic fluctuations. Second, the balanced growth factor
retains a significant role in explaining movements at horizons greater than 2 years, although it
has considerably less explanatory power in the six variable system than in the three variable
system. Third, a large fraction of the short-run (0-2 year) variability in output and investment
is explained by a factor that is related to persistent movements in the real rate of interest.
Fourth, the impulse response functions appear consistent with the interpretation of the first
shock as a real or balanced growth shock, but lead us to be uncertain about the interpretation of
the third, real rate shock, at least within the context of standard macroeconomic models.
-18-
IV .e S e n sitiv ity A n a ly sis
It is important to explore the sensitivity of these main conclusions to changes in
cointegrating vectors and changes in the ordering of the permanent components: we do this by
estimating a variety of five- and six-variable models. To save space, we focus on a key
measure, the fraction of the variance of the three-year-ahead forecast error in each variable
explained by the balanced growth permanent innovation. The results, summarized in Table 6,
suggest four conclusions. First, looking across specifications, a substantial fraction of the
forecast errors in output and consumption is explained by the balanced growth innovation; the
point estimates range from one-third to two-thirds. Second, in systems including nominal
variables, the fraction of the forecast error variance of investment explained by the balanced
growth real permanent component is never large (at most 27% in model M.l.) Third, little
changes when balanced growth is imposed by setting <j>^ and ^ equal to zero. Fourth, changing
the ordering of the shocks (for example, putting the balanced growth shock last in the Wold
causal ordering, as in model M.3) does not change the main qualitative features of the results.
In short, the sensitivity analysis indicates that the principle results for the base six-variable
model are robust to a wide variety of changes in the identifying restrictions.
12
V. Analysis of Trend Components of Private GNP
In the neoclassical growth framework of Section II, the common long-run movements in
aggregate variables arise from changes in productivity. Is there any evidence that productivity
movements are related to innovations in the balanced growth component of GNP? We
investigate this by comparing these estimated innovations to a popular estimate of the change in
total factor productivity in the economy, Solow’s (1957) residual. If the economy can be
described by a Cobb-Douglas production function -- as in the theoretical model of Section 2 the Solow residual has the convenient interpretation of being exactly
-19-
in (2.2). We use two
measures of this productivity residual: Robert Hall’s (1988, Table 1) for total manufacturing
and that produced by Prescott (1986).^
The time path of the Solow residual and the change in the balanced growth trend component
of private GNP from the 6 variable model are plotted in Figure 5a for Hall’s measure and in
Figure 5b for Prescott’s measure. The graphs suggest a very modest relation between Hall’s
Solow residual and our estimated balanced growth shock (the correlation is .19) and a stronger
relationship between Prescott’s Solow residual and the estimated shock (the correlation is .48).
The Solow residual is an imperfect measure of technical change. For example, Prescott
(1986) points to errors in measuring the variables used in its construction and Hall (1988)
suggests that this measure of productivity misrepresents true technological progress in
noncompetitive environments where price exceeds marginal cost. Nonetheless, these results
suggest some link between the real permanent shocks from the model and the two measures of
the Solow residual. These comparisons thus lend some credence to the interpretation in Section
IV of the permanent real shocks as measuring economy-wide shifts in productivity.
The focus so far has been to use the empirical model to evaluate a class of real business
cycle models. However, the empirical model also provides a solution to a classic problem in
descriptive econom ic statistics: how to decompose an economic time series into a "trend" and
"cyclical" component. A natural definition of the trend is the long-run forecast of the variables
(see Harvey (1989), chapter 6), and some simple algebra (see the appendix) shows that changes
in this trend are just linear combinations of the permanent innovations. Thus, the empirical
model can be used to form a multivariate generalization of the trend-cycle decomposition
proposed by Stephen Beveridge and Charles Nelson (1981).
The implied trend component of output computed using the six-variable model is shown in
Figure 6 along with Edward Denison’s (1985) estimate of real potential GNP per capita.^
Despite the very different approaches used to construct the two trend estimates they are broadly
similar. The three major differences between the two series are the prolonged growth of the
1960s, the 1974 contraction and the slowdown of the late 1970s.
-20-
VI. Conclusion
In this paper we analyzed the stochastic trend properties of postwar U.S. macroeconomic
data to evaluate the empirical relevance of standard RBC models with permanent productivity
shocks. Several aspects of these results are consistent with the central proposition of most real
business cycle models. Real per capita output, consumption and investment (as well as real
balances and interest rates) appear to share common stochastic trends. The cointegrating
relations among the real flow variables are consistent with balanced growth; in addition, money,
prices, output and interest rates are consistent with a long-run money demand cointegrating
relation. In a three variable real model, innovations in the balanced growth component account
for more than two-thirds of the unpredictable variation in output over the two- to five-year
horizon.
Yet much evidence is at odds with one-sector RBC models in which permanent productivity
changes play a major role. Even in the three-variable model, the balanced growth innovation
accounts for less than two-fifths of the movements of investment at horizons up to six years.
The explanatory power of the balanced growth innovation for output is reduced to under 45%
by introducing nominal variables. Moreover, the explanatory power arises mainly from some
specific episodes, notably the sustained growth of the 1960’s. The balanced growth innovation
sheds little light on other important episodes, such as 1974-75 and 1981-82. Thus, we are led
to conclude that the U.S. data are not consistent with the view that a single real permanent
shock is the dominant source of business cycle fluctuations.
What are the omitted sources of the business cycle. From a monetarist perspective, it is
surprising that such a small role is played by the inflation shock. Accelerations and
decelerations in money growth and inflation, which are assumed to have no long-run effect on
real flow variables and real interest rates, explain a trivial fraction of the variability in output
-21-
and consumption, and a small fraction of the variability in investment. The results point
toward an additional permanent (or at least highly persistent) component associated with real
interest rates, which has large effects on investment.
-22-
Appendix
This appendix presents a discussion of identification and estimation. The definitions in the
text are:
(A.1) Reduced Form: AXt = m + C(L)£t.
(A.2) Structural Model: AXt = fi + r(L)r;t.
The identifying restrictions are:
(A.3)
where Tq exists.
(A.4) r(l) = [A n 0],
where A is a known nxk matrix with full column rank, n is a kxk lower triangular matrix
with full rank and l ’s on the diagonal, and 0 is a nx(n-k) matrix of 0’s. The covariance matrix
of the structural disturbances is assumed to be:
(A.5)
- E(r?tr?t')
V
0
',2-1
1 2
1
2
where E^ is partitioned conformably with T7t=(r7t’ rj t’)’, where rjt is kxl and rjt is
(n-k)xl, and where
is diagonal.
Equations (A.l) and (A.2) are the definitions of the reduced form and structural models
given in (3.1) and (3.2) in the text. Assumption (A.3) says that the structural disturbances are
in the space spanned by current and lagged values of X t, and that there are no singularities in
the structural model. Assumption (A.4) is discussed in the text explicitly for the six-variable
model. It also applies to the three-variable model also by defining A to be a vector of l ’s.
In assumption (A.4) the diagonal elements of n are normalized to unity without loss of
generality, since the variances of j, in (A.5), are unrestricted.
-23-
The permanent innovations, »jj, can be determined from the reduced form, (A.1), as
follows. From (A.1)-(A.3), C(L)=r(L)rQ^, so that C(l)=r(l)rQ\ Let D be any solution of
C(1)=AD (for example, D =(A ’A)’^A’C(1)). Thus A D e^ A n ^ J so, since
1 1
E tj^ - E ^ , DE£D —IIE^n’. Let n* be the unique lower triangular square root of DE£D ’,
h.
h.
and let n and E ^ be the unique solutions to IIE^^n*, where n and E ^ satisfy (A.4) and
-1
-1
(A.5). Then A =A n, and the first k rows of Tq are given by G=n D. Since D is unique
up to premultiplication by a nonsingular matrix, G is unique. Finally, »7t=rQ^et implies that
1= r
Vt
Gev
The dynamic multipliers associated with
These can be calculated as follows. First, write
are given by the first k columns of r(L).
T q = (H
J), where
H
is nxk and J is nx(n-k).
Since r(L)=C(L)rQ, the first k columns of r(L) are given by C(L)H. Finally, €t=rQ»7t implies
-1
-1
1
E I q —rQ E£, so that from (A.5) H -E f?jGE£; thus, the dynamic multipliers for
are
C(L)E£G’SJ7'J.
Both the structural and reduced form lead naturally to the multivariate version of the
Beveridge-Nelson (1981) decomposition used to estimated trend output and plotted in Figure 6.
The structural form can be expressed, X t=XQ+/it+^_jr(L)r?s or, setting X q=0,
t
1
*
*
00
X t= jtt+ r(l)^ =1»?s+r (L )rjv where Tj = -£ i= j+1rj. Let
Xt=xP+X®, where
X \= T
t
1
then this becomes,
(L)r?t is the stationary component of Xt and
XP=Mt+ r (l)^ =1^ = Mt+ A rt is the permanent component of X {. By construction, X^
satisifies the natural notion of a trend as the infinitely long-run forecast of X, based on
information through time t.
The only restrictions that the structural model places on the reduced form are the
cointegration restrictions. This implies that efficient estimates of the structural model can be
calculated in two steps: first, the reduced form is estimated imposing only the cointegration
restrictions, and second, this estimated reduced form is transformed into the structural model
using the relations given in the last two paragraphs. In all models reported in this paper, the
-24-
reduced form was parameterized as a VECM (a cointegrated VAR). The estimated VECM was
inverted to yield an estimate of the moving average representation of the reduced form in
(A.l).
-25-
References
Beveridge, Stephen and Nelson, Charles R., "A New Approach to Decom position of Econom ic
Time Series into Permanent and Transitory Components with Particular A ttention to
Measurement of the ’Business Cycle’," Journal of Monetary Economics 7 (1981), 151-74.
Blanchard, Olivier J. and Quah, Danny, "The Dynamic Effects of Aggregate Supply and
Demand Disturbances," American Economic Review. 79, (1989), 655-73.
Blanchard, Olivier J. and Watson, Mark W., "Are Business Cycles A ll Alike?" in R.J. Gordon,
ed., The American Business Cycle: Continuity and Change. National Bureau of Economic
Research Studies in Business Cycles, Vol. 25, Chicago and London: U niversity of
Chicago Press, (1986), 123-82.
Cochrane, John H., "Univariate vs. Multivariate Forecasts of GNP Growth and Stock Returns:
Evidence and Implications for the Persistence of Shocks, Detrending Methods, and Tests
of the Permanent Income Hypothesis," NBER Working Paper 3427, September 1990.
Cochrane, John H. and Sbordone, Argra M., "Multivariate Estimates of the Permanent
Components of GNP and Stock Prices," Journal of Economic Dynamics and Control. 12,
no. 2/3 (1988), 255-96.
Dickey, David A. and Fuller, Wayne A., "Distribution of the Estimators for Autoregressive
Time Series With a Unit Root,” Journal of the American Statistical Association 74, no.
366 (1979), 427-431.
Denison, Edward, Trends in American Economic Growth. 1929-1982. Washington, DC: The
Brookings Institute, (1985).
Doan, Thomas A. and Litterman, Robert B., RATS User’s Manual, version 2.00 (1986).
Engle, Robert F. and Granger, Clive W.J., "Cointegration and Error Correction:
Representation, Estimation, and Testing," Econometrica 55 (1987), 251-276.
Fama, Eugene F., "Transitory Variation in Investment and Output," Manuscript, University of
Chicago.
-26-
Friedman, Benjamin M. and Kuttner, Kenneth N., "Money, Income, Prices and Interest Rates
After the 1980’s," Federal Reserve Bank of Chicago Working Paper 90-11, (1990).
Friedman, Milton, A Theory of the Consumption Function. Princeton University Press:
Princeton, NJ, (1957).
Hall, Robert E., "The Relationship Between Price and Marginal Cost in U.S. Industry," Journal
of Political Economy. Vol. 96, no. 3 (October 1988), 921-47.
Hansen, Gary, "Technical Progress and Aggregate Fluctuations," U.C.L.A. working paper,
(1988).
Hansen, Lars P. and Sargent, Thomas J., "Formulating and Estimating Dynamic Linear Rational
Expectations Models," Journal of Economic Dynamics and Control. Vol. 2, (1980).
Harvey, Andrew C., Forecasting. Structural Time Series Models and the Kalman Filter.
Cambridge University Press: New York, NY, (1989).
___________________and J.H. Stock, "Continuous Time Autoregressions with Common Stochastic
Trends,” Journal of Economic Dynamics and Control. 12, no. 2/3 (1988), 365-84.
Hoffman, Dennis R. and Rasche, Robert, "Long-Run Income and Interest Elasticities on Money
Demand in the United States," NBER Discussion Paper No. 2949 (1989).
Johansen, Soren, "Statistical Analysis of Cointegration Vectors," Journal of Econom ic Dynamics
and Control 12, no. 2/3 (1988), 231-54.
King, Robert G., Plosser, Charles I. and Rebelo, Sergio P., "Production, Growth, and Business
Cycles: II. New Directions," Journal of Monetary Economics. 21:2/3, (1988), 195-232.
King, Robert G., Plosser, Charles I. Stock, James H. Stock and Watson, Mark W., "Stochastic
Trends and Economic Fluctuations," NBER Discussion Paper #2229, April 1987.
Kosobud, Robert and Klein, Lawrence, "Some Econometrics of Growth: Great Ratios of
Economics," Quarterly Journal of Economics. 25 (May 1961), 173-98.
Kydland, Fynn and Prescott, Edward C., "Time to Build and Aggregate Fluctuations,"
Econometrica 50 (1982), 1345-70.
-27-
Long, John B. and Plosser, Charles L "Real Business Cycles," Journal of Political E conom y 91
(1983), 39-69.
Lucas, Robert E., "Money Demand in the United States: A Quantitative Review," CarnegieRochester Conference on Public Policy. 29 (1988), 137-68.
Prescott, Edward C., "Theory Ahead of Business Cycle Measurement," Carnegie-Rochester
Conference on Public Policy. 25 (1986), 11-66.
Sims, Christopher A., "Macroeconomics and Reality," Econometrica 48, no. 1 (January 1980),
1-48.
_______________________ , Stock, James H. and Watson, Mark W., "Inference in Linear Time
Series Models with Some Unit Roots," Econometrica. Vol. 58, no.l (1990), 113-44.
Solow, Robert M., Growth Theory: An Exposition. Oxford: Clarendon Press, (1970).
___________________ , "Technical Change and the Aggregate Production Function." Review
of
Econom ics and Statistics 39 (1957), 312-20.
Stock, James H. and Watson, Mark W., "Testing for Common Trends," Journal of the American
Statistical Association 83, (1988), 1097-1107.
________________________________________ , "A Simple MLE of Cointegrating Vectors in Higher
Order Integrated Systems," NBER Technical Working Paper, 83, December 1989.
Wallis, Kenneth F., "Econometric Implications of the Rational Expectations Hypothesis,"
Econometrica. Vol. 48, no. 2 (1980).
-28-
Endnotes
1. This follows directly from the economy’s commodity resource constraint Y t=Ct+It, its
investment technology, Kt+^=(l-<5)Kt+It, with <5 being the rate of depreciation, and the fact that
the econom y’s allocation of time between work and leisure must be constant in steady state.
2. As one example of how an extension of the basic model preserves the stochastic trend
implication, consider the time-to-build investment technology of Kydland and Prescott (1982).
All of the stages of investment in their model inherit the common stochastic trend. Similar
conclusions hold for the other examples in the text. There are two important categories of RBC
models that need not display a single common stochastic trend when there are permanent
productivity shocks. Multi-sector models can have separate productivity trends in each sector,
as in John Long and Plosser (1983). Models of stochastic endogenous growth such as those
constructed by King and Rebelo (1988) generate a stochastic trend in the level of productivity
when shocks are stationary; with endogenous growth, permanent changes in taxes or in the level
of exogenous productivity lead to permanent changes in the growth rates.
3. This assumes that the structural disturbances lie in the space spanned by current and lagged
values of X^.
4. This notion, that consumption might provide a good measure of permanent income, has been
recently exploited by John Cochrane and Argia Sbordone (1988), Andrew Harvey and James
Stock (1988), Eugene Fama (1990), and Cochrane (1990).
5. All data were obtained from Citibase. Using the Citibase mnemonics for the series, the
precise definitions of the variables are GC82 (consumption), GIF82 (investment) and (GNP82GGE82) (real private output). The Citibase M2 series (FM2) was used for 1959:1-85:4; the
earlier M2 data was formed by splicing the M2 series reported in Banking and Monetary
Statistics, 1941-1970, Board of Governors of the Federal Reserve System to the Citibase data in
January 1959. The monthly data were averaged to obtain the quarterly observations. The price
deflator was obtained as the ratio of nominal private GNP (the difference between Citibase
series GNP and GGE) to real private GNP (the difference between Citibase series GNP82 and
- 29-
GGE82). The interest rate is FYGM3. It is measured as an annual percentage (a typical value
is 10.0%). Price inflation was also measured as an annual percentage (400*ln(Pt/P t ^)).
Output, consumption, investment and money are on per capita basis using total civilian
noninstitutional population (P16).
6. Because the techniques and results are now familiar, they are omitted here; interested
readers are referred to an earlier version of this paper (King, Plosser, Stock and Watson (1987))
for details.
f
7. The multiviarate unit root tests in Tables 1-3 are based on the J^(r), JT(r), q^(k,m), and
f
qT(k,m) statistics. The J(r) statistics are Johansen’s (1988) test of the null of r cointegrating
f
vectors against >r cointegrating vectors, and the q (k,m) statistics are Stock and Watson’s (1988)
test of the null of k unit roots in the multivariate system against the alternative of m (m <k) unit
roots; V and V ' subscripts respectively denote the tests computed using demeaned data and
data that have been both demeaned and linearly detrended. (Detrending is appropriate if the
f
f
series have a nonzero drift under the null.) Asymptotic critical values for q^ and qr are
taken from Stock and Watson (1988). Asymptotic p-values for
and Jf differ from those
tabulated in Johansen (1988) because of the demeaning/detrending. It is straightforward to
derive and to compute the asymptotic null distribution of J^ and Jr using the results in Sims,
Stock and Watson (1990). We have done this, and the p-values shown in the table are based on
these asymptotic distributions. All multivariate tests were computed using six lagged levels,
which these procedures parameterize as five lags in first differences and a single lagged level.
8. The cointegrating vectors reported in Tables 1-3, and the estimated cointegrating vectors
used as the basis of the V A R analysis in Tables 4-6, were estimated using the Stock-Watson
(1989) dynamic OLS procedure, which is asymptotically equivalent to the Gaussian maximum
likelihood procedure for a triangular error correction system. If there are r cointegrating
vectors, then there are r regression equations; each equation has n-r regressors in levels (where
n is the number of variables), a constant, and m leads, m lags, and the contemporaneous values
of the differences of right-hand side levels variables. Standard errors, calculated using a
VAR(4) estimator of the spectral density matrix of the errors in these r equations, are given in
-30-
parentheses in Tables 1-3. All results are based on m=5; to allow for the leads, the dynamic
OLS regressions end at 87:3 (all other regressions go through 88:4). Wald statistics computed
using the estimated covariance matrix have the usual large-sample x
2
distributions.
The log likelihoods provided in this and subsequent tables are provided as a guide for
readers interested in exploring the shape of the likelihood surface. It should be stressed that,
because of the hypothesized unit roots, the usual chi-squared inference does not always apply to
likelihood ratio statistics computed from the reported values.
9. See Robert Lucas (1988), Dennis Hoffman and Robert Rasche (1989), and Benjamin
Friedman and Kenneth Kuttner (1990) for recent empirical investigations of the stability of
long run money demand.
10. To check robustness, the cointegrating vectors in Tables 1, 2, 3, and 6 were also estimated
using Johansen’s (1988) MLE for a VECM with 5 lagged differences, one lagged level, and a
constant. The Johansen MLE point estimates (available from the authors upon request) are
similar to the dynamic OLS point estimates. For example, the Johansen MLE of the money
demand equation is m-p = 1.134y - .0093R (cf. Table 2, model (1)).
11. The standard errors for the impulse response functions and variance decompositions were
approximated using 500 simulations as discussed by Thomas Doan and Robert Litterman (1986),
page 19-4.
12. Additional sensitivity analyses were performed: substituting short-term private and long
term public interest rates for the short term public rate, dropping interest rates entirely and
changing the number of lags. The results, available from the authors on request, are consistent
with the summary conclusions in this and subsequent sections.
13. Because Hall’s series is annual, Prescott’s quarterly series was aggregated to an annual
level.
14. Denison computed his measure of potential output by adjusting actual output using an
Okun’s law relationship, by adjusting for capacity utilization, and by making other adjustments
such as for labor disputes, the weather, and the size of the armed forces. Source: Denison
(1985), Tables 2-4.
-31-
T ab le 1
C oin tegration S ta tis tic s :
3 V a r ia b le M od el, (y , c , i ) , 1 9 4 9 :1
- 88:4
A. Results from Unrestricted Levels Vector Autoregressions
Largest Eigenvalues of Estimated Companion Matrix
VAR(6) with Constant
Real
1.00
0.83
0.83
-0.62
-0.62
0.50
Modulus
1.00
0.85
0.85
0.77
0.77
0.71
Imaginary
0.00
0.19
-0.19
0.46
-0.46
0.50
Log Likelihood:
VAR(6) with Constant and Trend
Real
0.97
0.83
0.83
-0.62
-0.62
0.49
Imaginary
0.00
0.18
-0.18
0.46
-0.46
0.49
2199.66
Modulus
0.97
0.85
0.85
0.77
0.77
0.69
2200.95
B . Multivariate Unit Root Statistics
Statistic
Value
Nul1/Alternat ive
35.4
(0.13)
3 unit roots/at most 2 unit roots
qf(3.2)
-29.4
(0.21)
3 unit roots/at most 2 unit roots
qf(3,l)
-29.4
(<.01)
3 unit roots/at most 1 unit root
JT(0)
Log
Log
Log
Log
(P-Value)
Likelihood
Likelihood
Likelihood
Likelihood
(3
(2
(1
(0
Unit
Unit
Unit
Unit
Roots):
Roots):
Roots):
Roots):
2183.27
2193.11
2199.43
2200.95
C. Estimated Cointegrating Vectors
Null
Variable
Hypothesis
Estimates
C*2
ai
a2
c
i
0
1.00a
o.ooa
i
0
1
0.00a
1.00a
y
-1
-1
-1.058
(0.026)
-1.004
(0.038)
Wald test of balanced growth restrictions:
x\ - 4.96 (0.08)
Notes to Table 1: Values in parentheses are p-values (for the test statistics) or
standard errors (for the estimators). The roots and likelihoods in panel A
correspond to an unrestricted VAR(6) in levels. The multivariate unit root tests in
panel B, which are described in detail in footnote 7, were computed from VAR(6)'s in
levels. Jr(r) is Johansen's (1988) tes£ of the null of r cointegrating vectors
against >r cointegrating vectors, and q^(k,m) is Stock and Watson's (1988) test of
the null of k unit roots in the multivariate system against the alternative of m
(m<k) unit roots, where V
denotes linear detrending. The log likelihoods in Panel
B are for models that include a constant and linear time trend. Panel C reports the
cointegrating vectors for (c,i,y), estimated by Stock and Watson's (1989) dynamic OLS
(with a constant, 5 leads and 5 lags) procedure. t-statistics formed using the
standard errors have asymptotic normal distributions. The Wald statistic, which
tests that the cointegrating vectors lie in the hypothesized subspace, is computed
using the dynamic OLS estimates and standard errors described in footnote 8.
T ab le 2
Estimated Cointegrating Vectors
A.
(1)
Money Demand. 1954:1 - 88:4
m - p = 1.197 y -0.013 R
(0.062) (0.004)
q; ( 3 , 2) = - 20.6
(0.54)
Wald test of velocity restriction (/? =1 and y0^=O) : X 2 = 12.7
B.
(2)
(3)
(<0.01)
Real Ratios and Real Interest. 1954:1-1988:4
c - y =0.0033 (R - Ap)
(0.0022)
i - y = -0.0028 (R - Ap)
(0.0050)
qf(2,l) - -70.0
(<0.01)
J (0) = 19.3
(0.17)
q^(2,l) - -62.2
(C0.01)
J (0) = 15.6
(0.11)
qf(2,l) = -73.8
(C0.01)
q„(2,l) = -65.1
M
(<0.01)
Notes to Table 2:
Jr(0) - 42.6
(0.03)
J (0) = 26.9
(0.02)
J (0) = 24.8
^
(0.01)
Values in parentheses are p-values (for the test
statistics) or standard errors (for the estimators).
The cointegrating
vectors (l)-(3) were estimated by dynamic OLS (with 5 leads and 5 lags),
£
including a constant in the regression, equation-by-equation. The qfl and J[L
tests are computed using demeaned data (see footnote 7). The Wald statistic
is described in the notes to Table 1 and in footnote 8.
T ab le 3
C oin tegration S t a t is t ic s
6 V a r ia b le M od el, ( y ,c ,i ,m - p ,R ,A p ) , 1 9 5 4 :1
- 88:4
A. Estimated Cointegrating Vectors
Variable
di
d3
“2
c
1.00a
0.00a
0.00a
i
0.00a
1.00a
0.00a
m-p
0.00a
0.00a
1.00a
y
-1.118
(0.050)
-1.120
(0.083)
-1.152
(0.063)
R
0.004
(0.003)
0.002
(0.005)
0.009
(0.004)
Ap
0.004
(0.003)
0.006
(0.004)
0.002
(0.003)
q*(6,3)- -27.5
(p-value =0.11)
Log Likelihood = 2826.54
B. Tests of Restrictions on Cointegrating Vectors
Null Hypothesis
d,f.
(c-y), (i-y), m-p-j8yy+/3RR
7
12.6 (0.08)
(c-y), (i-y), m-p-0yy+£RR, R-Ap
6
40.1 (<0.01)
(c-y)-^1(R-Ap) , (i-y)-<£2(R-Ap) , m-p-/3yy+£RR
5
7.6 (0.18)
(c-y)-<^1(R-Ap) , (i-y)-<02(R-Ap) , m-p-y
7
42.0 (C0.01)
Wald test
Notes to Table 3: Values in parentheses are p-values (for the test
statistics) or standard errors (for the estimators). Panel A reports dynamic
OLS estimates for the 3-equation system. Panel B reports tests of whether the
cointegrating vectors fall in the hypothesized subspace, conditional on the
number of cointegrating vectors. The Wald statistics are described in the
notes to Table 1 and in footnote 8.
anormalized.
T ab le 4
F o r e c a st E rror V arian ce D eco m p o sitio n s:
3 v a r ia b le r e a l m odel (y, c, i ) , 1949:2 - 88:4
Fraction of the forecast error variance
atributed to the real permanent shock
i
Horizon
y
i.
0.45
(0.28)
0.88
(0.21)
0.12
(0.18)
4.
0.58
(0.27)
0.89
(0.19)
0.31
(0.23)
8.
0.68
(0.22)
0.83
(0.18)
0.40
(0.18)
12.
0.73
(0.19)
0.83
(0.18)
0.43
(0.17)
16.
0.77
(0.17)
0.85
(0.16)
0.44
(0.16)
20.
0.79
(0.16)
0.87
(0.15)
0.46
(0.16)
24.
0.81
(0.16)
0.89
(0.13)
0.47
(0.16)
1.00
1.00
1.00
oo
C
Note: Based on an estimated vector error correction model of X t * (yt_,ct
with 8 lags of AXt , one lag each of the error correction terms c-y and iand a constant. Approximate standard errors, shown in parentheses, were
computed by Monte Carlo simulation using 500 replications.
A. Fraction of the forecast error variance
attributed to balanced growth shock
1
.
4.
8.
r—H
16.
o
CM
24.
0 .00
(0..13)
0 .05
(0.•14)
0 .22
(0..13)
0..44
(0..14)
0 .54
(0,•15)
0 .59
(0..15)
0 .62
(0.•14)
.
.
.
,
,
.
1
.
o
o
00
y
c
0 .02
(0..09)
0 .15
(0..13)
0..31
(0..18)
0 .48
(0.•21)
0..59
(0.■21)
0 .63
(0..19)
0 .65
(0,■17)
.
.
.
.
.
0
.92
.
i
0 .11
(0..16)
0 .06
(0. 11)
0 .14
(0.•11)
0 .27
(0,.16)
0..32
(0..17)
0 .33
(0.•17)
0 .33
(0.•16)
.
.
.
.
,
.
0
.97
.
m-p
0 .79
(0..23)
0 ,76
(0. 23)
0 ,70
(0. 24)
0 .72
(0..25)
0.,74
(0. 24)
0 ,75
(0.,22)
0 ,77
(0.,22)
.
.
.
.
.
.
0
00
Horizon
CM
.
R
0 ,14
(0. 19)
0..11
(0. 19)
0 .11
(0..20)
0..11
(0..20)
0..11
(0..20)
0 .12
(0..21)
0 .14
(0..22)
.
.
.
.
0
.23
.
Ap
0 .30
(0..13)
0..22
(0..08)
0 .20
(0..07)
0 .17
(0..06)
0..16
(0. 07)
0 .15
(0..07)
0 .14
(0..08)
.
.
.
.
.
0
,04
.
B. Fraction of the forecast error variance
attributable to inflation shock
Horizon
y
0..00
(0..12)
0..04
(0.■14)
0..04
(0..12)
0..03
(0..10)
0..02
(0..10)
0..02
(0..10)
0,.02
(0,.10)
0..02
(0..11)
0..01
(0..10)
0..01
(0..11)
0..01
(0..12)
0..01
(0..12)
0..02
(0..12)
0..02
(0..11)
i
0..08
(0..16)
0,.23
(0,•19)
0,.20
(0.•15)
0..12
(0,.12)
0..10
(0.■ID
0..10
(0. 11)
0,.09
(0,.11)
m-p
0..01
(0..13)
0..04
(0..14)
0.,01
(0. 14)
0..01
(0. 14)
0..01
(0..13)
0..01
(0. 13)
0..01
(0. 13)
o
o
0,
o
o
0.
CM
0.
c
O
00
O
o
T ab le 5
F o r e c a st E rror V arian ce D eco m p o sitio n s:
6 V a ria b le M odel ( 4 .1 ) , 54:1 - 8 8 :4
0.
R
Ap
0..03
(0..14)
0..04
(0,.15)
0..02
(0..16)
0..02
(0..15)
0..03
(0..15)
0..03
(0..15)
0..02
(0.•14)
0.,43
(0. 19)
0.,37
(0.,13)
0.,36
(0. 12)
0..45
(0. 11)
0..49
(0.•ID
0..53
(0. 12)
0..55
(0. 13)
0..01
0..96
Table 5
(Continued)
C. Fraction of the forecast error variance
attributable to real interest rate shock
Horizon
i.
4.
8.
12.
16.
20.
24.
00
y
c
i
m-p
R
Ap
0..67
(0..19)
0..74
(0..20)
0..55
(0..16)
0..39
(0.■11)
0..32
(0..10)
0..28
(0..09)
0..25
(0,.08)
0.,35
(0..22)
0..24
(0..18)
0..12
(0..10)
0..11
(0..09)
0..09
(0..09)
0..09
(0..08)
0..10
(0..08)
0,.44
(0,.20)
0,.50
(0,.20)
0..37
(0 ■14)
0,.36
(0,.12)
0 .37
(0,•12)
0 .34
(0,•12)
0,.34
(0,.11)
0 .03
(0..13)
0 .07
(0 ■14)
0 .20
<0 •17)
0 .21
(0..18)
0 .20
(0,•17)
0,.18
(0 .16)
0 .16
(0 .15)
0,.63
(0,■21)
0..72
(0,.21)
0,.77
(0..22)
0 .78
(0,.22)
0,.78
(0,.22)
0,.78
(0..22)
0 .77
(0..22)
0..00
(0.,11)
0. 10
(0.,08)
0.,16
(0..09)
0..14
(0..08)
0..13
(0.•07)
0..12
(0. 07)
0..11
(0..07)
0..00
0..06
0,.03
0 .21
0 .77
0,.00
Note: Based on an estimated vector error correction model of
= (y,c,i,mp,R,Ap) with 8 lags of AXt , one lag each of the error correction terms c-y<^(R-Ap) , i-y-^2 (R-Ap) , and m-p-|0 y-^R, and a constant. Approximate standard
errors, shown in parentheses, were computed by Monte Carlo simulation using
500 replications.
T ab le 6
3 Y ear Ahead F o r e c a st E rror V arian ce D eco m p o sitio n s:
Sum m ary o f R e s u l t s o f V a r io u s M o d els
Model Est. Per.
Tests of Restrictions
on Cointegrating Vectors
Fraction of forecast error
variance attributed to
the permanent real shock
d.f.
y
c
i
Wald test
Log Lik.
m-p
R
AP
R.l
49:2-88:4
2
4.96 (0.08)
2196.67
.73
.83
.43
M.l
54:1-88:4
5
7.60 (0.18)
2816.06
.44
00
.27
.72
.ii
.17
M. 2
54:1-88:4
7
12.60 (0.08)
2814.64
.42
.52
.25
.68
.07
.16
M. 3
54:1-88:4
.35
.30
.12
.26
.02
.16
M.4
54:1-88:4
6
.37
.40
.15
.56
.01
.18
M.5
54:1-88:4
4
.42
.47
.23
.64
.06
M.6
54:1-88:4
.42
.36
.19
.46
.01
—
same as M.l ......
40.10 (<0.01) 2820.48
3.04 (0.55)
—
2812.52
same as M.5 ......
Model Description
Model R.l:
Three variable (y,c,i) model with cointegrating relations c-y and
i-y.
Model M.l:
Six variable (y,c ,i,m-p,R,Ap) baseline model of Table 5.
Model M.2:
Identical to M.l, except that the coefficients <t>^ and <
j>^ are set
to zero in the cointegrating vectors and the A matrix, i.e.,
cointegration of shares and the real interest rate is dropped.
Model M.3:
Identical to M.l, except that the stochastic trend innovations are
reordered to place the inflation shock first, the real interest
rate shock second and the balanced growth trend third.
Model M.4:
A two stochastic trend model for (y,c ,i,m-p,R,Ap), obtained by
assuming that the real interest rate is stationary. The
cointegrating relations are c-y, i-y, (ni*p)’l y + ^ R and R-Ap, and
A - [A-^ A^], where A^ = (1 1 1 p 0 0)' (balanced growth
shock) and ^ = (0 0 0 -/?R 1 1)' (neutral inflation shock).
Model M.5:
A five variable system (y,c,i, m-p,R) with cointegrating relations
c-y, i-y and (m-p)-/3 y+/?RR, and A * [A^ A,], where Aj - (1 1 1
B 0) ' (balanced growth shock) and A^ * (0 0 0 -/3R 1)' (neutral
interest rate shock).
Model M.6: Identical to M.5, except that the ordering of stochastic trend
innovations is reversed, so A = [A^ A^].
Notes to Table 6: The estimation period denotes the sample used to estimate
the VECM, with earlier data used for initial conditions for the lags. The
Wald statistics test the hypothesis that the true cointegrating subspace is
spanned by the hypothesized cointegrating vectors, or equivalently is
orthogonal to the A matrix, and are described in the notes to Table 1 and
footnote 8.
Figure 1
0.0
0.2
0.4
0.6
0.8
CO
51
54 57 60 63 66 69 72 75 78 81 84 87
A, Logarithms of Private Output (y). Consumption (c). Investment (I)
90
r
i r 11 r-i | i i | ■i t i > i | ■
»i r
i "i r » j i"■i r ».i ..r i.i 'i . ■ i ■ ■
0.0
0.1
0.2
0.3
0.4
0.5
and Real M o ney B a la n c e s ( m - p )
i
1,1 L, 1 .1 X- i - J -U.1—
1 ,1 L-l. J. 1 -1 ,1.1 1 I I I I i J- L-l L i■I ..1■I I I I I
51 54 57 60 63 66 69 72 75 78 81 84 87 90
B. Logarithms of the Consumption-Output (c-y)
and Investment-Output (i-y) Ratios
Figure 2
Responses in 3 —variable Model to a One Std. Dev. Shock
in the Real Permanent Component
I
1
3
5
7
9
11
13
15
17
19
21
23
25
15
17
19
21
23
25
15
17
19
21
23
25
<«) y
I
1
3
5
7
9
11
13
0>) C
I
1
3
5
7
9
11
13
(c) i
3
Historical F o re c a s t Decomposition
Six Variable Model
Consumption
Output
Investment
an
o
d
Balanced
Q
Growth
9
O
Component
an
o
V 58
62
66
70
74
78
82
86
90
TO
an
o
o
Inflation
o
o
Component 6
an
o
V 58
62
66
70
74
78
82
86
90
TO
Real I nte res t
Rate
Component
V 58
62
66
70
74
78
82
86
90
Total F o r e c a s t E r r o r
P e r m a n e n t C o m po ne n t
C\|
Fig ure 4
Selected Impulse R e sp o n se s fo r the
Six Variable Model
Balanced Growth Shock
0
3
6
9
12
15
18
21
24
27
0
3
6
9
12
15
18
21
24
27
0
3
6
9
12
15
18
21
24
27
1.6
-
0.8
-
0.0
0.8
1.6
-
1.6
-
0.8
-
0.0
0.8
1.6
-
1.6
-
0.8
-
0.0
0.8
Inflation Shock
1 0
3
6
9
12
15
18
21
24
27
Figure 5a
------------ Hall’s Solow Residual
----------
Balanced Growth Shock from the 6-variable Benchmark Model
Figure 5b
Precott’s Solow Residual
Balanced Growth Shock from the 6—variable Benchmark Model
Figure 6
5.8
-5.6
-5.4
Estimates of Annual Trend Output
Dennison (1985. Table 2-2)
Permanent Component of y from the 6—variable
Benchmark Model
@FedFRASER