Journal of Monetary Economics 25 (1990) 145-150. North-Holland

A TEST OF A KEYNESIAN ALTERNATIVE TO HERCOWITZ’S AGGREGATE SUPPLY THEORY*

John McCALLUM

McGill University, Montreal, Quebec, Canada H3A 2T7

Received September 1988, final version received April 1989

Hercowitz (1986) estimates aggregate output and hours equations which are derived from a new classical model of aggregate supply. Under an alternative interpretation, his aggregate supply equation is in fact an IS curve with one omitted variable, while his hours equation is best interpreted as an Okun’s Law relationship. When the Hercowitz equations are nested in more general formulations which incorporate these alternatives, the new classical restrictions are rejected resoundingly.

1. Introduction

In terms of business cycle theory, the main achievement of Hercowitz (1986) is to develop an equilibrium model that generates co-movements of macroeco- nomic variables of the kind that we actually observe. In particular, the model is consistent with positive co-movements of employment, capital utilization, labour productivity, output, investment, and consumption. A key point that drives these results is the hypothesis that aggregate supply depends negatively on the real interest rate, rather than positively as in Lucas and Rapping (1969). Empirically, the author avoids the usual simultaneity problems by estimating his aggregate supply function with Canadian data and by invoking the not unreasonable assumption that Canada is a small open economy facing an exogenous real interest rate that is set in the United States. Since the empirical results appear to support the model, one might conclude that there is both theoretical and empirical support for the equilibrium approach to the business cycle.

Hercowitz’s aggregate supply equation 20’ can be written:

y, = a, + air, + azgt + a3kt + a,r + elt, 0)

a, < 0, a2 > 0, a3 >< 0, a,>(),

where y, is the logarithm of per capita constant dollar GDP; ‘; is an estimate of the U.S. real interest rate; g, is the logarithm of per capita constant dollar

*This research was supported by Fonds FCAR. The author thanks John Galbraith, Jagdish Handa, David Laidler, and the editors for their useful comments.

0304-3932/90/$3.500 1990. Elsevier Science Publishers B.V. (North-Holland)

146 J. McCallum, Alternative to Hercowitz’s uggregute supp!v theocv

current government expenditure on goods and services; k, is the logarithm of the beginning-of-year per capita net stock of fixed nonresidential capital; t is a time trend; and e,, is an error term.

Although Hercowitz interprets this equation as an aggregate supply relation- ship, it certainly looks like an IS curve: output depends negatively on the real interest rate, positively on government spending, ambiguously on the stock of fixed nonresidential capital,’ and positively on a trend (which could reflect the accumulation of assets other than physical capital). On the other hand, particularly for a relatively small open economy like Canada, foreign output is a key determinant of exports, and hence aggregate demand. Consequently, if we wish to interpret (1) as an IS curve, we should add a foreign or U.S. output

variable ( _yr* ) as follows:

y, = (Y() + air, + (big, + cx3kl + a4t + a5jt* + e2,,

a1 co. a2 ’ 0, a,><& a,‘O, q > 0,

where yr* will in fact be defined as a U.S. output variable.

(2)

Because of a possible simultaneity problem, it is important to note that it is the estimated value of U.S. output ( jt*) rather than the actual value (_t+*) that is entered in eq. (2). Suppose that a major component of the North American business cycle is driven by technology or supply shocks that are common to both the United States and Canada. In this case a positive coefficient on y,* could reflect the influence of continental supply shocks as well as or instead of the influence of Canadian export demand. Hercowitz’s interpretation of eq. (2) as an aggregate supply relation could then be maintained even with (Ye > 0. For this reason, eq. (2) will be estimated by two-stage least squares with demand-side variables used as instruments for y,*. (A similar argument can be advanced in favor of treating r, as endogenous.2)

Our main test of the Hercowitz model, then, is to estimate eq. (2) and see whether the coefficient (Ye is significantly greater than zero. If it is, we conclude that the Hercowitz model is rejected and that his estimated output equation is best interpreted as an IS curve with one omitted variable. I now present the results of this test and then turn to the hours equation.

2. Test results

In estimating eq. (2) by two-stage least squares, the government spending variable g,, the beginning-of-period capital stock k,, and the time trend are

‘With eq. (1) interpreted as an IS curve, the capital stock is likely to have a positive effect on consumption and a negative effect on investment, with the result that the overall effect on aggregate demand is ambiguous.

‘If technology shocks that are common to both countries have a major influence on both U.S. output and U.S. real interest rates, then each of these variables will be correlated with the error term of eq. (2).

J. McCullum, Alternative to Hercowitz’s aggregate supply theoty 141

Table 1

Estimates of as based on eq. (2) (with t-statistics in parentheses).”

Level specification

First-difference specification

1955-84 F,,,,, 0.60 (3.8) 0.73 (3.8) ,. ‘,/I, 0.63 (4.3) 0.71 (4.2)

1955-80 ?,>,,, 0.55 (3.7) 0.60 (3.1) (.I,! 0.59 (4.1) 0.63 (3.3)

1960-84 ?,,,,, 0.64 (5.4) 0.78 (6.1) t.*, 0.64 (5.4) 0.77 (6.2)

.‘Estimates of as are reported for both level and first-difference specifications, for three alternative sample periods, and for estimates of the real interest rate based on the producer price index (i,,,,,) and the consumer price index (i,,,,,). The sample period for the first-difference specihcatron begins in 1956 rather than 1955.

treated as exogenous, while yI* and r, are treated as endogenous. Instruments for rr and y,* include r,_l, rr_2, y,T1, ytt2, DMR,*_l, DMR,*_,, g:, and g;“_,, where DMR: is a Barro-type money shock term for the United States and g(* is the logarithm of U.S. constant dollar federal purchases. Note that the contemporaneous money shock term is not treated as exogenous. Following Hercowitz, the regressions are estimated with an AR(l) term, although more general specifications for the error term are presented later in the paper. Fair (1970) has shown that under such conditions, consistent estimates require the inclusion of the lagged dependent and independent variables as instruments. Hence the full list of instruments includes the variables just mentioned, plus a constant term, t, Y,-~, g,, g,_l, k,, and k,_,. Eq. (2) was also estimated in first-difference form, and in this case the instruments were also first-dif- ferenced.

Most of the data used in the tests are taken directly from Hercowitz’s tables 3a, 3b, and 3c. The exceptions are y,* and g:, which are taken from the U.S. National Income and Product Accounts (and deflated by Hercowitz’s popula- tion variable), and DMR:, which is taken from McCallum (1989). All regres- sions were estimated with Micro TSP (version 6.0). Table 1 reports estimates of the key coefficient (Ye for a variety of specifications and sample periods. It can be seen that the estimates of (us lie between 0.55 and 0.78, with t-statistics between 3 and 6. Hence the Hercowitz model is rejected.

Table 2 reports four regressions. Columns 1 and 2 report the original specification in level and first difference form. However, further tests indicated that this specification was too restrictive. As indicated in columns 3 and 4, the standard errors of the regressions fall considerably when the lagged dependent variable and an AR(2) term are added and when the government spending variable is entered with a one-period lag. It can be seen from these last two regressions that the estimated coefficients for the level and first-difference

148 J. McCullum, Alternative to Hercowitz’s aggregate supp!v theory

Table 2

Regression results (with t-statistics in parentheses).”

Original specification Amended specification

Level First difference Level First difference

(1) (2) (3) (4)

Constant 7.08 0.0311 2.17 0.0062

(5.5) (1.8) (6.1) (2.2)

t 0.051 0.0051

(3.8) (3.8)

.r; I 0.44 0.39

(5.4) (3.6)

5 -0.18 ~ 0.22 -0.33 -0.33

(1.1) (1.3) (4.3) (2.5)

Ri 0.17 0.01

(1.8) (0.0)

gi I 0.16 0.17 (3.3) (2.4)

k, - 1.54 - 0.65

(2.9) (1.1)

.r;,* 0.60 0.73 0.43 0.40

(3.8) (3.8) (8.7) (3.5)

AR(l) 0.12 0.30 - 0.30 ~ 0.39

(X.1) (1.5) (1.6) (2.0)

AR(2) - - 0.33 -- 0.40

(2.6) (3.5)

MA(l) - 0.49

(1.6)

Period 1955-84 1956-84 1956-84 1957-84 SE 0.0150 0.0178 0.011 0.012

Q(4) 0.96 2.96 2.17 2.70 jj? 0.994 0.55 0.998 0.76

_

“The variables are defined as follows: J, is logarithm of per capita constant dollar GDP: r, is estimate of U.S. real interest rate based on producer price index; g, is logarithm of constant dollar per capita current government spending on goods and services; k, is the logarithm of the beginning-of-year net stock of nonresidential capital: t is a time trend; and );* is the logarithm of per capita constant dollar U.S. GNP. The regressions were estimated by two-stage least squares. and the instruments are reported in the text.

J. McCallum, Alternative to Hercowitz’s aggregate supply theory 149

specifications are very similar and of reasonable magnitude.3 In particular, the elasticity of Canadian output with respect to U.S. output is about 0.4 in the short run and 0.7 in the long run.

3. Equation for hours worked

The tests that were just reported for Hercowitz’s output equation can in principle be applied to his hours equation. One simply appends the foreign output variable to his hours equation 18’ to obtain

(3)

and then tests the hypothesis that & = 0. This exercise was carried out, and the hypothesis was always rejected.

However, whereas there is an excellent rationale for the inclusion of foreign output in an IS equation, there seems no particular reason why foreign output should have a direct effect on domestic employment. The natural interpreta- tion of (3) is simply that the demand for labour is a derived demand and that foreign output influences n, indirectly via its influence on y,. Indeed it might also be true that rt, g,, and k, influence n, only via their effects on y,. If this were so, then (3) could be written in the more compact form

e n, = c0 + cly, + c,t + edr. (4)

In fact, the restrictions implied by the ‘Okun’s Law’ equation (4) could not be rejected at conventional significance levels. The equations estimated over the period 1955-84 in both level and first-difference form were as follows (with t-statistics in parentheses):

n I = 2.77 + O&J, - O.O09t, (7.8) (5.2) (5.2)

AR(l) = 0.53 (2.8), SE = 0.012, DW= 1.62, R2 = 0.77,

Dn, = -0.011 +0.56DJt, (3.8) (6.1)

SE = 0.013, D W = 1.94, E2 = 0.58.

These regressions confirm the well-known empirical fact that productivity is procyclical. As Hercowitz observes (p. 140), this means that the empirical results are not consistent with textbook Keynesian models in which firms are

31t might be observed that the estimated coefficient on the trend in the level specification (0.0051) corresponds to the constant term in the first-difference specification (0.0062). Also, equivalence of the two specifications would imply an MA(l) coefficient of -1.0 for the first- difference regression, as compared with the reported estimate of - 0.49 (SE = 0.30).

150 J. McCullum, Allernative lo Hercowitz’s aggregate supply theo?

always on their standard labour demand curves and higher output is associ- ated with a lower marginal product of labour. However, the results are consistent with the disequilibrium approach introduced by Barro and

Grossman (1971), and also with the class of models introduced by Oi (1962) that incorporate costs of adjusting labour input.

4. Conclusions

Hercowitz’s new classical model of aggregate supply is rejected resound- ingly, and it appears that his aggregate supply relation is best interpreted as an IS curve, while his hours equation is best interpreted as an Okun’s Law relation.

References

Barro, Robert J. and H.I. Grossman, 1971. A general disequilibrium model of income and employment, American Economic Review 61, 82-93.

Fair, Ray C.. 1970 The estimation of simultaneous equation models with lagged endogenous variables and first order serially correlated errors, Econometrica 38, 507-516.

Hercowitz, Zvi, 1986, The real interest rate and aggregate supply, Journal of Monetary Economics 18, 121-145.

Lucas, R.E. and L.A. Rapping, 1969, Real wages, employment and inflation, Journal of Political Economy 11, 721-154.

McCallum, John, 1989, Credit rationing and the monetary transmission mechanism, Mimeo. (Department of Economics, McGill University, Montreal).

Oi, Walter, 1962. Labour as a quasi-fixed factor, Journal of Political Economy 70. 538-555.