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The long-run and short-run demand for money: Additional evidence
DANIEL L THORNTON Resetwe Bank of St. L.ouis
Federal
The Long-Run and Short-Run Demand for Money: Additional Evidence* This paper presents estimates of Chow’s money-demand equations using Chow’s data. The equations are adjusted for autocorrelation using two autocorrelation transformations, the standard Cochrane-Orcutt transformation that deletes “initial” observations, and one that does not delete these observations. The estimates focus on the question of the asset versus the transactions specifications. The results reaffirm Chow’s original conclusion which supported the asset motive and, thereby, reversed nearly all of the findings recently reported by Lieberman (1986). The paper concludes that one should be wary about using the Cochrane-Orcutt transformation, especially when the ratio of the number of deleted observations to total observations is large.
1. Introduction Over the past several decades there has been a great deal of empirical research on the demand for money. While various studies have addressed a number of theoretical and empirical issues, much of this research has directly or indirectly focused on two important questions: the role of interest rates and the appropriateness of current income vs. some measure of wealth as a scale variable. While the particular results vary, most of the empirical research supports wealth as the appropriate scale variable and indicates a significant interest elasticity of the demand for money. These findings were strongly supported in a classic study by Chow (1966).’ Recently, Lieberman (1980) has presented results that seriously question these conclusions. Reestimating Chow’s long-run and short-run demands for money, adjusting for autocorrelation, and using Chow’s data, Lieberman finds strong support for the transactions demand rather than the asset demand, as Chow reported. *I would like to thank an anonymous referee of this paper. ‘Other well-known and respected studies Laidler (1966a, 1966b) and Meltzer (1963).
for helpful
comments
on an earlier
draft
Journal Copyright
of
Macroeconomics, 0
1982
by
Wayne
Summer 1982, State University
which
Vol.
support
4, No. Press.
3, pp.
this
conclusion
are:
325-338
325
Daniel L. Thornton He found that neither permanent income nor wealth were statistically significant when included with current GNP if the equations are adjusted for first-order serial correlation and estimated over the period 1934-58. Lieberman argues that there was a structural shift in the demand for money in 1933, and that the absence of a wealth effect in the demand for money is evidence against monetarist third-level crowding out of fiscal policy.’ He also finds long-run interest elasticities that are much smaller than those reported by Chow and others, and he notes that in some cases they are only “marginally” significant. Lieberman credits his results to institutional changes which took place in 1933 and to the serial-correlation adjustment. These results stand in stark contrast to those obtained by Chow and, taken together, give strong support to the transactions motive over the asset motive for holding money. Unfortunately, Lieberman’s results were obtained using an estimation procedure which, though commonly used, requires that some observations be deleted. This accommodation to convenience is particularly damaging when the Chow data are used, especially for the 1934-58 period. The purpose of this paper is to demonstrate that, when the long-run and short-run demands for money are estimated with a technique that does not require this accommodation, results are obtained which differ significantly from Lieberman’s. Current GNP continues to perform well; however, the case against permanent income and wealth is much less clear. The estimates of the interest elasticity are generally larger than those obtained using the more common estimation technique, and are always statistically significant. Other elasticity estimates differ markedly as well. Since the transformation employed here is not widely used, a comparison of this transformation with the more common one used by Lieberman is made in Section 2. Sections 3 and 4 contain estimates of the long-run and short-run demands, respectively. Ordinary Least Squares (OLS) estimates are presented along with estimates using both autocorrelation transformations. The results are summarized in Section 5. 2. Adjusting for Autocorrelation with Missing Observations The problem with adjusting for autocorrelation using the Chow data is that there are breaks in the data for the war years. When
Two 326
‘There have been useful references
a number of theoretical are: Blinder and Solow
studies (1973)
on third-level and Hayakawa
crowding (1979).
out.
Long-Run,
Short-Run
Demand
for
Money
traditional methods of estimation are employed, the first observation and the initial observation after each break in the time series are deleted. Although it has not been widely discussed in the econometric literature, there is a transformation which does not require one to delete these observations. The transformation can be compared with the one employed by Lieberman in terms of the general model with 9 consecutive missing time-series observations, given by Equation (1).
(1) where Y, is a T, x 1 vector of observations on the dependent variable prior to the gap, and Y, is a T2 x 1 vector of observations on the dependent variable after the gap, X is a T( =T, + TJ X k matrix of observations on the fared regressor variables, and U is a T X 1 vector of random disturbances. Both X and U are partitioned to conform with Y. It is assumed that the tth element of U, u,, is generated by: u,
where ai V (1) is ploys (1) is
=
put-1
+ et;
IPI <
1 >
(2)
E, - NZD (0, I$). G iven these assumptions, the E(U U’) = where V is a symmetric matrix involving p. When Equation adjusted for the autocorrelation given in (2), one usually ema transformation matrix, M, such that M’M = V-‘. Equation then transformed by premultiplying by M to obtain:
Y*=MY=MXP+MU=X*P+U*.
(3)
Parameter estimates are obtained from Equation (3) using one of several techniques. 3 The problem is that, while there is only one transformation matrix M such that M’M = V-‘, there are several transformations which yield V-’ approximately.4 However, only the approximate transformation which Lieberman employed will be considered here. It is easy to show that the exact transformation can be obtained with the T x T matrix M below. 3Two of the ‘M is unique
most common are Hildreth-Lu and up to an orthogonal transformation.
Cochrane-Orcutt
327
Daniel L. Thornton (1 -
py
0
-p
0
, . .
0
lO...O
0
-pl...O
0
0
0
0
oo...oo
0
0
. , .
o...
0
0
0
0
Tl
M=
0
oo...-p
0
0
0
oo...o
1
0 . . .
0
-gpq+’ 0
oo...oo
g
0 . . .
0
0
1
0
0
-p
. . .
T2
0
oo...
0
0
oo...-pl
where g” = (1 - p’)/(l - pz(g”‘). The approximate transformation is the same as M except that the first and T, + 1 rows are deleted. Thus, transforming Equation (1) with the approximate transformation is tantamount to dropping the initial observation and the first observation after a break in the data. The two transformations produce estimators with identical asymptotic properties; however, the exact transformation will be more efficient than the approximate transformation on average. Indeed, Kadiyala (1968) has shown that estimates obtained with approximate transformation may be less efficient than OLS. Moreover, since the deleted observations may be important, especially when the sample size is small, estimates obtained using the exact transformation will be preferred.
3. The Long-Run
Demand
for Money
The estimates presented in this and the following section are forms of Chow’s well-known long-run and short-run demand functions that employ Chow’s data. Chow’s data consists of annual observations over the period 1897-1958.’ All equations were specified in log-linear form, and all dollar-denominated variables are in real 51 am grateful 328
to Charles
Lieberman
for
providing
me
with
the
Chow
data.
Long-Run,
Short-Run
Demand
for Money
terms. Only equations that include GNP and permanent income or GNP and wealth were estimated, since they are the relevant equations for determining the appropriate scale variable. Three estimates of each equation are presented. The first estimates are OLS (denoted by a one). The second are Hildreth-Lu (HL) estimates which employ the approximate transformation (denoted by a two). The third set of estimates (denoted by a three) are maximumlikelihood (ML) est imates which use the transformation matrix M. The ML estimates were obtained using a search technique identical to that employed to obtain the HL estimates.6 The OLS, HL, and ML estimates for the period 1897-1958 are presented in Table 1. The t-values appear in parentheses below each coefficient estimate. Equation A.1 presents OLS estimates of Chow’s equation with income and permanent income as scale variables. This equation indicates that permanent income is the appropriate scale variable. The Durbin-Watson statistic of 0.65, however, indicates the presence of first-order autocorrelation. When this equation is reestimated using the HL technique, the basic result holds up, but there are important differences. The interest and permanent-income elasticities are cut in half. The permanent income elasticity of 0.495 indicates substantial economies of scale in holding money. When this equation is reestimated using a maximum-likelihood technique that employs the exact transformation, the estimate of permanent-income elasticity is 1.171, which is not significantly different horn one at the 0.05 level. The estimate of the interest elasticity is also larger than in A.2. In Equation B wealth replaces permanent income as the scale variable. In this case the results were not greatly affected by the estimation technique employed. Both income and wealth remain statistically significant, and the estimates of the interest and income ‘The
ML
estimates
are
obtained RSS (p)/[(l
where search
RSS (p) denotes was carried out
the residual to an accuracy
by
minimizing
- pS)2/(l sum of of 0.01.
- p)qQ+‘)]l’r
)
squares conditional In all cases, the
on p. The grid likelihood surfaces
appear to have only one extremum. In some cases, the search was restricted to positive values of p since nearly all previous estimates of p for the demand for money are positive. ‘The standard Durbin-Watson statistic is valid even when the fact that there are missing observations has been ignored, but its power is reduced. See Savin and White (1978) for details. The standard errors and R’s between Equations (2) and (3) are not directly comparable because they are based on different sample sizes.
B.3
B.2
B.l
A.3
A.2
-0.645
-1.624
(-4.522) -1.361 (-3.269) -2.503 (-4.481)
1.051
0.042 (0.319) 0.116 (1.056) -0.037 (-0.303) 0.408 (3.947) 0.255 (3.545) 0.289 (3.028) (7.231) 0.495 (2.191) 1.171 (8.438)
YP
Demand for
Y
of the Long-Run
(-2.784) 4.502 (2.410) -1.234 (-2.975)
A.1
Estimates
Constant
1.
Equation
TABLE
0.626 (5.652) 0.732 (9.368) 0.790 (7.452)
W R2
-0.720 0.992 (-13.482) -0.355 0.997 (-3.757) -0.586 0.985 (-7.743) -0.628 0.990 (-11.390) -0.527 0.997 (-7.473) -0.557 0.982 (-6.614)
Bond
Money: 1897-1958
0.045
0.034
1.79
2.00
0.61
1.54
0.042 0.071
1.67
0.65
D.W.
0.039
0.064
S.E.
0.68 (6.62) 0.73 (7.85)
(21.72) 0.70 (7.20)
0.95
P
s
.Y
Long-Run,
Short-Run
Demand
for Money
elasticities are nearly identical. Thus, the estimates over the 1897-1958 period support the asset motive for holding money, regardless of the estimation technique employed. Lieberman argues that when the equations are estimated over the 193458 period this conclusion is reversed. The estimates for this period appear in Table 2. Again, the OLS estimates (Equation C.l) indicate that permanent income is the appropriate scale variable. When this equation is reestimated adjusting for autocorrelation using the approximate transformation, income becomes significant and permanent income becomes insignifkant. Also, the interest elasticity is cut by more than half and is only “marginally” significant. When this equation is estimated using the ML technique, Lieberman’s finding is reversed. Measured income is insignificant and permanent income is significant at the 0.05 level. Furthermore, the ML estimate of the interest elasticity of the demand for money is much closer to that obtained using OLS. Wealth replaces permanent income in Equation D. OLS estimates of this equation suggest wealth is the appropriate constraint. Adjusting for autocorrelation using either the HL or the ML technique reverses this conclusion. However, the estimate of the long-run interest elasticity from Equation D.3 is substantially larger than that of Equation D.2. When the results of Equations C.3 and D.3 are compared with those in C.l and D. 1, a case much less damaging to the asset motive appears. Measured income performs well only when included with wealth. Furthermore, permanent income cannot be rejected as a determinant of money demand, and the estimated long-run interest elasticities are much larger than Lieberman’s and are clearly statistically significant. Thus, Chow’s previously reported large interest elasticities are not the result of autocorrelation as Lieberman suggests.’
4. The Short-Run
Demand
for Money
Chow’s short-run demand for money is obtained by assuming that individuals adjust their actual money holdings to their desired ‘It could be argued that neither autocorrelation adjustment is correct since both implicitly assume that the autocorrelation process continued uninterrupted over the war years. Thus, the long-run demand for the 193458 period was estimated using a transformation which assumes that the autocorrelation process stopped during the war years and started again in 1946. See Thornton (1982). These estimates did not change the conclusions reached here. These results will be provided by the author upon written request. 331
D.3
D.2
D.l
c.3
c.2
-0.204 (-0.110)
0.289 (1.031)
0.662
(1.181)
0.217
(1.789)
0.861
W
(-3.673)
-0.517
(-2.167)
-0.268
(-5.333)
-0.959
(-6.104)
-0.564
(-1.707)
-0.223
(-7.140)
-0.601
Bond
for Money: 193458
(3.154)
0.597
(3.659)
1.303
(1.010)
0.200
(0.510)
(-1.092)
1.075 (3.872)
0.015
(0.296)
0.089
(5.240)
1.444
YP
Demand
(0.065)
-0.742 (-0.989) -1.960
0.680
(3.033)
2.199
(1.465)
-0.287
(-1.259)
-1.308
(-2.231)
c.1
Y
of the Long-Run
Constant
Estimates
Equation
TABLE 2.
0.997
0.995
0.973
0.999
0.995
0.988
23’
0.042
0.028
0.070
0.037
0.029
0.047
S.E.
1.87
1.64
1.12
1.68
1.81
1.19
D.W.
0.84 (6.92)
(4.67)
0.74
0.58 (3.18)
(4.96)
0.76
P
3 $
2
i
F.3
F.2
F.l
E.3
E.2
-0.290 (-1.430) -0.449 (-1.525) -0.341 (-1.395)
(0.590)
0.281 (1.694) 0.204 (0.798) 0.150
Constant
Equation
E.l
Estimates
TABLE 3.
0.162 (1.916) 0.126 (1.186) 0.156 (1.527) 0.154 (2.951) 0.194 (2.918) 0.173 (2.927)
Y,
-0.365 (-4.168) -0.323 (-8.043) -0.395 (-5.701) -0.341 (-7.015)
(-3.801)
-0.395
(-4.875)
-0.298
Bond,
for Money:
0.279 (4.412) 0.356 (3.302) 0.293 (3.820)
Demand
0.216 (1.668) 0.363 (1.760) 0.345 (1.763)
YPt
of the Short-Run
(7.964)
0.607 (8.669) 0.511 (3.712) 0.499 (4.342) 0.551 (10.294) 0.435 (3.961) 0.519
M,-,
1897-1958
0.998
0.998
0.998
0.999
0.997
0.997
R2
0.030
0.032
0.032
0.033
0.035
0.037
S.E.
0.51
-0.47
1.56
0.07
-0.28
2.76
h
0.31 (1.76) 0.22 (1.37)
(2.55)
0.40 (2.W 0.44
P
Daniel L. Thornton levels via the standard partial-adjustment mechanism. The short-run demand for money differs from the long-run demand in that the lagged dependent variable appears on the R.H.S. of the equation. The presence of the lagged dependent variable in the short-run demand function complicates the estimation. OLS estimates of short-run demand are neither consistent nor efficient; however, the HL and ML estimates presented here are consistent and the ML estimates are asymptotically efficient.g Thus, the short-run demand for money was estimated using OLS, HL and ML as before. Chow’s short-run demand for money was estimated using the same data as in the previous section. The results for the periods 1897-1958 and 193458 appear in Tables 3 and 4, respectively. Since the Durbin-Watson statistic is biased toward accepting the null hypothesis when a lagged dependent variable is present on the R. H. S., Durbin’s (1970) h-statistic replaces the Durbin-Watson statistic in these tables.” The estimates of the short-run demand for money over the 1897-1958 period are relatively insensitive to the estimation technique employed. Perhaps the most significant finding, as far as the asset motive for holding money is concerned, is that income becomes insignificant when the equation is estimated adjusting for autocorrelation. Thus, the results presented in Table 3 support the asset motive for holding money, as Chow suggested. The estimates of the short-run demand for money for the period 193458 are presented in Table 4. OLS estimates indicate that income, permanent income and wealth are all insignificant at the 0.05 level. When the equation is reestimated using the HL tech@Actually, Lieberman presents estimates of the short-run demand for money based on several estimation techniques. First, he presents results based on an iterative Cochrane-Orcutt procedure. Second, Lieberman estimated the equation including wealth with a combination of instrumental variables and a HL scan. Since the instrumental-variable estimator will be less efficient than the ML estimator unless the optimal instruments are chosen, only the ML estimates are presented along with OLS. Moreover, the HL-type estimates presented in Equation (H.2) do not differ greatly from Lieberman’s combined instrumental variables-H1 estimates. “‘The reader is cautioned that Durbin’s h-statistic may not be directly applicable to the case of missing data points. Savin and White’s (1978) conclusion with respect to the Durbin-Watson statistic, however, may apply to Durbin’s h-statistic as well (see Footnote 7 above). Also, the estimated standard errors of the coefficients obtained directly from the autocorrelation adjustment are biased downward when a lagged dependent variable is present. The t-values presented in Tables 3 and 4 are based on asymptotically unbiased estimates of the coefficient variances using a transformation suggested by Cooper (1972). Thus, the t-ratios for the HL estimates differ from those reported by Lieberman. 334
H.3
H.2
H.l
G.3
G.2
G.l
Equation
1.237 (1.527) 2.494 (1.168) 1.186 (1.622) 0.622 (0.584) 1.389 (0.856) 0.690 (0.781)
Constant y’pt
for
0.126 (0.425) 0.300 (1.257) 0.115 (0.491)
w
Demand
0.408 -0.108 (1.618) (-0.237) 0.663 0.268 (2.610) (OJW 0.402 -0.054 (1.838) (0.135) 0.265 (1.150) 0.672 (2.978) 0.295 (1.612)
Y,
TABLE 4. Estimates of the Short-Run M,-,
0.585 (3.385) -0.192 (-0.481) 0.541 (3.609) 0.535 (5.239) -0.190 (-0.560) 0.509 (6.046)
193458
-0.238 (-1.755) -0.233 (-1.587) -0.251 (-2.093) -0.294 (-2.092) -0.267 (-1.968) -0.296 (-2.445)
Bond,
Money: S.E.
0.999 0.029
0.86
N.A.
1.30
0.992 0.035 0.994 0.030
1.03
N.A.
1.68
h
0.999 0.029
0.991 0.032
0.993 0.035
Ii2
0.75 (4.71) 0.25 (1.02)
0.79 (4.06) 0.25 (0.89)
P
s s cc f u 3
g z "5.
Daniel L. Thornton nique, income becomes significant while permanent income and wealth remain insignificant. Moreover, the interest-rate coefficient is insignificant in the permanent-income equation, and the adjustment coefficient is not significantly different from one in either equation. When these equations are reestimated using the exact transformation, most of these findings are reversed. In fact, the ML estimates do not differ greatly from the OLS estimates.” The only scale variable that is significant is income, and then only in the permanent-income equation. Furthermore, the interest rate is significant in both equations, and the estimated speed of adjustment is reduced markedly. l2 Thus, the ML estimates give slight support to the transactions motive in that income is the only scale variable that is significant. Given the relatively poor performance of the variables in these equations, however, one should be very cautious in interpreting the results.13
5. Conclusions The first and perhaps most significant conclusion that can be drawn from the above is that the method one uses to adjust for serial correlation is very important. If the standard transformation which requires the researcher to delete observations from consideration is used, the results obtained may differ significantly from those obtained using a method that does not require that these observations be deleted. This is likely to be more important the more breaks in the data, and the larger the ratio of deleted to total available observations (in this case at least 5.5 percent). Thus, researchers should be very careful when employing techniques that use the standard autocorrelation transformation. Perhaps econometrics software vendors should modify their procedures to employ
“This is expected since the ML estimates of p are not signillcantly different zero. “The adjustment coefficient is one minus the coefficient on the lagged dependent variable. It is interesting to note that the adjustment coefficient cannot be less than one. Thus, the negative coefficients on Equations (G.2) and (H.2) are particularly disturbing. [For a theoretical discussion of the stock adjustment model, see Griliches (1967).] “It should be noted that the null hypothesis of no first-order serial correlation cannot be rejected using Durbin’s (1970) h-test at the 0.05 level for the OLS equations. The ML estimates of p are consistent with the pre-test results, while the HL estimates are not.
from
336
Long-Run,
Short-Run
Demand
for Money
the exact autocorrelation transformation, even when there are no breaks in the data. While many of the differences reported here were due to significantly smaller standard errors obtained when the exact transformation was used, there were several striking differences in the magnitude of certain parameter estimates as well. Second, Lieberman’s finding that the transactions motive appears to be stronger than the asset motive is not substantiated by these results. In particular, Lieberman’s conclusion that “wealth and permanent income are statistically insignificant in all cases where current income enters as a measure of the volume of transactions, ” is shown to be false. Third, Lieberman’s contention that money-demand equations in particular may provide misleading coefficient estimates unless they are adjusted to remove serial correlation is not supported by these results. In most instances, ML estimates did not change the conclusions concerning the role of interest rates and the relevant scale variables from the OLS results; however, many of the parameter estimates differed markedly. Fourth, the results give only modest support to Lieberman’s contention that there was a structural shift in the demand for money in 1933. Parameter estimates between the two periods only change dramatically for the long-run demand for money which includes wealth, or if the approximate transformation is used. Finally, there is evidence of a wealth effect in the demand for money. Thus, there is no evidence against possible third-level crowding out, which has been so widely discussed in the theoretical literature. Received: October, 1980 Final oersion received: January,
1982
References Blinder, A. S. and R. M. Solow. “Does Fiscal Policy Matter?’ Journal of Public Economics 2 (November 1973): 31937. Chow, G.C. “On the Long-Run and Short-Run Demand for Money.” Journal of Political Economy 74 (April 1966): 11131. Cooper, J.P. “Asymptotic Covariance Matrix of Procedures for Linear Regression in the Presence of First-Order Autoregressive Disturbances.” Econometrica 40 (March 1972): 305-10. Durbin, J. “Testing for Serial Correlation in Least-Squares Regression When Some of Regressors are Lagged Dependent Vari337
Daniel L. Thornton ables.” Econometrica 38 (May 1972): 410-21. Friedman, M.S. “The Demand for Money: Some Theoretical and Empirical Results. ” Journal of Political Economy 67 (August 1959): 32751. Griliches, Z. “Distributed Lags: A Survey. ” Econometrica 35 (January 1967): 16-49. Hayakawa, H. “Does Fiscal Policy Really Matter in the Context of Variable Prices?” JournaE of Macroeconomics 1 (Fall 1979): 321-46. Kadiyala, K.R. “A Transformation Used to Circumvent the Problem of Auto-Correlation. ” Econometrica 36 (January 1968): 93-96. Laidler, D. “Some Evidence on the Demand for Money.” Journal of Political Economy 64 (February 1966): 55-68. -. “The Rate of Interest and the Demand for Money-Some Empirical Evidence. ” Journal of Political Economy 64 (December 1966): 543-55. Lieberman, C. “The Long-Run and Short-Run Demand for Money, Revisited. ” Journal of Money, Credit and Banking 12 (February 1980): 4357. Meltzer, A. “The Demand for Money: The Evidence from the Time Series.” Journal of Political Economy 61 (June 1963): 219-46. Savin, N.E. and K.J. White. “Testing for Autocorrelation with Missing Gbservations. ” Econometrica 46 (January 1978): 59-67. Thornton, D. L. “The Appropriate Autocorrelation Transformation when the Autocorrelation Process has a Finite Past. ” Federal Reserve Bank of St. Louis Working Paper 82-002, 1982.
Federal
The Long-Run and Short-Run Demand for Money: Additional Evidence* This paper presents estimates of Chow’s money-demand equations using Chow’s data. The equations are adjusted for autocorrelation using two autocorrelation transformations, the standard Cochrane-Orcutt transformation that deletes “initial” observations, and one that does not delete these observations. The estimates focus on the question of the asset versus the transactions specifications. The results reaffirm Chow’s original conclusion which supported the asset motive and, thereby, reversed nearly all of the findings recently reported by Lieberman (1986). The paper concludes that one should be wary about using the Cochrane-Orcutt transformation, especially when the ratio of the number of deleted observations to total observations is large.
1. Introduction Over the past several decades there has been a great deal of empirical research on the demand for money. While various studies have addressed a number of theoretical and empirical issues, much of this research has directly or indirectly focused on two important questions: the role of interest rates and the appropriateness of current income vs. some measure of wealth as a scale variable. While the particular results vary, most of the empirical research supports wealth as the appropriate scale variable and indicates a significant interest elasticity of the demand for money. These findings were strongly supported in a classic study by Chow (1966).’ Recently, Lieberman (1980) has presented results that seriously question these conclusions. Reestimating Chow’s long-run and short-run demands for money, adjusting for autocorrelation, and using Chow’s data, Lieberman finds strong support for the transactions demand rather than the asset demand, as Chow reported. *I would like to thank an anonymous referee of this paper. ‘Other well-known and respected studies Laidler (1966a, 1966b) and Meltzer (1963).
for helpful
comments
on an earlier
draft
Journal Copyright
of
Macroeconomics, 0
1982
by
Wayne
Summer 1982, State University
which
Vol.
support
4, No. Press.
3, pp.
this
conclusion
are:
325-338
325
Daniel L. Thornton He found that neither permanent income nor wealth were statistically significant when included with current GNP if the equations are adjusted for first-order serial correlation and estimated over the period 1934-58. Lieberman argues that there was a structural shift in the demand for money in 1933, and that the absence of a wealth effect in the demand for money is evidence against monetarist third-level crowding out of fiscal policy.’ He also finds long-run interest elasticities that are much smaller than those reported by Chow and others, and he notes that in some cases they are only “marginally” significant. Lieberman credits his results to institutional changes which took place in 1933 and to the serial-correlation adjustment. These results stand in stark contrast to those obtained by Chow and, taken together, give strong support to the transactions motive over the asset motive for holding money. Unfortunately, Lieberman’s results were obtained using an estimation procedure which, though commonly used, requires that some observations be deleted. This accommodation to convenience is particularly damaging when the Chow data are used, especially for the 1934-58 period. The purpose of this paper is to demonstrate that, when the long-run and short-run demands for money are estimated with a technique that does not require this accommodation, results are obtained which differ significantly from Lieberman’s. Current GNP continues to perform well; however, the case against permanent income and wealth is much less clear. The estimates of the interest elasticity are generally larger than those obtained using the more common estimation technique, and are always statistically significant. Other elasticity estimates differ markedly as well. Since the transformation employed here is not widely used, a comparison of this transformation with the more common one used by Lieberman is made in Section 2. Sections 3 and 4 contain estimates of the long-run and short-run demands, respectively. Ordinary Least Squares (OLS) estimates are presented along with estimates using both autocorrelation transformations. The results are summarized in Section 5. 2. Adjusting for Autocorrelation with Missing Observations The problem with adjusting for autocorrelation using the Chow data is that there are breaks in the data for the war years. When
Two 326
‘There have been useful references
a number of theoretical are: Blinder and Solow
studies (1973)
on third-level and Hayakawa
crowding (1979).
out.
Long-Run,
Short-Run
Demand
for
Money
traditional methods of estimation are employed, the first observation and the initial observation after each break in the time series are deleted. Although it has not been widely discussed in the econometric literature, there is a transformation which does not require one to delete these observations. The transformation can be compared with the one employed by Lieberman in terms of the general model with 9 consecutive missing time-series observations, given by Equation (1).
(1) where Y, is a T, x 1 vector of observations on the dependent variable prior to the gap, and Y, is a T2 x 1 vector of observations on the dependent variable after the gap, X is a T( =T, + TJ X k matrix of observations on the fared regressor variables, and U is a T X 1 vector of random disturbances. Both X and U are partitioned to conform with Y. It is assumed that the tth element of U, u,, is generated by: u,
where ai V (1) is ploys (1) is
=
put-1
+ et;
IPI <
1 >
(2)
E, - NZD (0, I$). G iven these assumptions, the E(U U’) = where V is a symmetric matrix involving p. When Equation adjusted for the autocorrelation given in (2), one usually ema transformation matrix, M, such that M’M = V-‘. Equation then transformed by premultiplying by M to obtain:
Y*=MY=MXP+MU=X*P+U*.
(3)
Parameter estimates are obtained from Equation (3) using one of several techniques. 3 The problem is that, while there is only one transformation matrix M such that M’M = V-‘, there are several transformations which yield V-’ approximately.4 However, only the approximate transformation which Lieberman employed will be considered here. It is easy to show that the exact transformation can be obtained with the T x T matrix M below. 3Two of the ‘M is unique
most common are Hildreth-Lu and up to an orthogonal transformation.
Cochrane-Orcutt
327
Daniel L. Thornton (1 -
py
0
-p
0
, . .
0
lO...O
0
-pl...O
0
0
0
0
oo...oo
0
0
. , .
o...
0
0
0
0
Tl
M=
0
oo...-p
0
0
0
oo...o
1
0 . . .
0
-gpq+’ 0
oo...oo
g
0 . . .
0
0
1
0
0
-p
. . .
T2
0
oo...
0
0
oo...-pl
where g” = (1 - p’)/(l - pz(g”‘). The approximate transformation is the same as M except that the first and T, + 1 rows are deleted. Thus, transforming Equation (1) with the approximate transformation is tantamount to dropping the initial observation and the first observation after a break in the data. The two transformations produce estimators with identical asymptotic properties; however, the exact transformation will be more efficient than the approximate transformation on average. Indeed, Kadiyala (1968) has shown that estimates obtained with approximate transformation may be less efficient than OLS. Moreover, since the deleted observations may be important, especially when the sample size is small, estimates obtained using the exact transformation will be preferred.
3. The Long-Run
Demand
for Money
The estimates presented in this and the following section are forms of Chow’s well-known long-run and short-run demand functions that employ Chow’s data. Chow’s data consists of annual observations over the period 1897-1958.’ All equations were specified in log-linear form, and all dollar-denominated variables are in real 51 am grateful 328
to Charles
Lieberman
for
providing
me
with
the
Chow
data.
Long-Run,
Short-Run
Demand
for Money
terms. Only equations that include GNP and permanent income or GNP and wealth were estimated, since they are the relevant equations for determining the appropriate scale variable. Three estimates of each equation are presented. The first estimates are OLS (denoted by a one). The second are Hildreth-Lu (HL) estimates which employ the approximate transformation (denoted by a two). The third set of estimates (denoted by a three) are maximumlikelihood (ML) est imates which use the transformation matrix M. The ML estimates were obtained using a search technique identical to that employed to obtain the HL estimates.6 The OLS, HL, and ML estimates for the period 1897-1958 are presented in Table 1. The t-values appear in parentheses below each coefficient estimate. Equation A.1 presents OLS estimates of Chow’s equation with income and permanent income as scale variables. This equation indicates that permanent income is the appropriate scale variable. The Durbin-Watson statistic of 0.65, however, indicates the presence of first-order autocorrelation. When this equation is reestimated using the HL technique, the basic result holds up, but there are important differences. The interest and permanent-income elasticities are cut in half. The permanent income elasticity of 0.495 indicates substantial economies of scale in holding money. When this equation is reestimated using a maximum-likelihood technique that employs the exact transformation, the estimate of permanent-income elasticity is 1.171, which is not significantly different horn one at the 0.05 level. The estimate of the interest elasticity is also larger than in A.2. In Equation B wealth replaces permanent income as the scale variable. In this case the results were not greatly affected by the estimation technique employed. Both income and wealth remain statistically significant, and the estimates of the interest and income ‘The
ML
estimates
are
obtained RSS (p)/[(l
where search
RSS (p) denotes was carried out
the residual to an accuracy
by
minimizing
- pS)2/(l sum of of 0.01.
- p)qQ+‘)]l’r
)
squares conditional In all cases, the
on p. The grid likelihood surfaces
appear to have only one extremum. In some cases, the search was restricted to positive values of p since nearly all previous estimates of p for the demand for money are positive. ‘The standard Durbin-Watson statistic is valid even when the fact that there are missing observations has been ignored, but its power is reduced. See Savin and White (1978) for details. The standard errors and R’s between Equations (2) and (3) are not directly comparable because they are based on different sample sizes.
B.3
B.2
B.l
A.3
A.2
-0.645
-1.624
(-4.522) -1.361 (-3.269) -2.503 (-4.481)
1.051
0.042 (0.319) 0.116 (1.056) -0.037 (-0.303) 0.408 (3.947) 0.255 (3.545) 0.289 (3.028) (7.231) 0.495 (2.191) 1.171 (8.438)
YP
Demand for
Y
of the Long-Run
(-2.784) 4.502 (2.410) -1.234 (-2.975)
A.1
Estimates
Constant
1.
Equation
TABLE
0.626 (5.652) 0.732 (9.368) 0.790 (7.452)
W R2
-0.720 0.992 (-13.482) -0.355 0.997 (-3.757) -0.586 0.985 (-7.743) -0.628 0.990 (-11.390) -0.527 0.997 (-7.473) -0.557 0.982 (-6.614)
Bond
Money: 1897-1958
0.045
0.034
1.79
2.00
0.61
1.54
0.042 0.071
1.67
0.65
D.W.
0.039
0.064
S.E.
0.68 (6.62) 0.73 (7.85)
(21.72) 0.70 (7.20)
0.95
P
s
.Y
Long-Run,
Short-Run
Demand
for Money
elasticities are nearly identical. Thus, the estimates over the 1897-1958 period support the asset motive for holding money, regardless of the estimation technique employed. Lieberman argues that when the equations are estimated over the 193458 period this conclusion is reversed. The estimates for this period appear in Table 2. Again, the OLS estimates (Equation C.l) indicate that permanent income is the appropriate scale variable. When this equation is reestimated adjusting for autocorrelation using the approximate transformation, income becomes significant and permanent income becomes insignifkant. Also, the interest elasticity is cut by more than half and is only “marginally” significant. When this equation is estimated using the ML technique, Lieberman’s finding is reversed. Measured income is insignificant and permanent income is significant at the 0.05 level. Furthermore, the ML estimate of the interest elasticity of the demand for money is much closer to that obtained using OLS. Wealth replaces permanent income in Equation D. OLS estimates of this equation suggest wealth is the appropriate constraint. Adjusting for autocorrelation using either the HL or the ML technique reverses this conclusion. However, the estimate of the long-run interest elasticity from Equation D.3 is substantially larger than that of Equation D.2. When the results of Equations C.3 and D.3 are compared with those in C.l and D. 1, a case much less damaging to the asset motive appears. Measured income performs well only when included with wealth. Furthermore, permanent income cannot be rejected as a determinant of money demand, and the estimated long-run interest elasticities are much larger than Lieberman’s and are clearly statistically significant. Thus, Chow’s previously reported large interest elasticities are not the result of autocorrelation as Lieberman suggests.’
4. The Short-Run
Demand
for Money
Chow’s short-run demand for money is obtained by assuming that individuals adjust their actual money holdings to their desired ‘It could be argued that neither autocorrelation adjustment is correct since both implicitly assume that the autocorrelation process continued uninterrupted over the war years. Thus, the long-run demand for the 193458 period was estimated using a transformation which assumes that the autocorrelation process stopped during the war years and started again in 1946. See Thornton (1982). These estimates did not change the conclusions reached here. These results will be provided by the author upon written request. 331
D.3
D.2
D.l
c.3
c.2
-0.204 (-0.110)
0.289 (1.031)
0.662
(1.181)
0.217
(1.789)
0.861
W
(-3.673)
-0.517
(-2.167)
-0.268
(-5.333)
-0.959
(-6.104)
-0.564
(-1.707)
-0.223
(-7.140)
-0.601
Bond
for Money: 193458
(3.154)
0.597
(3.659)
1.303
(1.010)
0.200
(0.510)
(-1.092)
1.075 (3.872)
0.015
(0.296)
0.089
(5.240)
1.444
YP
Demand
(0.065)
-0.742 (-0.989) -1.960
0.680
(3.033)
2.199
(1.465)
-0.287
(-1.259)
-1.308
(-2.231)
c.1
Y
of the Long-Run
Constant
Estimates
Equation
TABLE 2.
0.997
0.995
0.973
0.999
0.995
0.988
23’
0.042
0.028
0.070
0.037
0.029
0.047
S.E.
1.87
1.64
1.12
1.68
1.81
1.19
D.W.
0.84 (6.92)
(4.67)
0.74
0.58 (3.18)
(4.96)
0.76
P
3 $
2
i
F.3
F.2
F.l
E.3
E.2
-0.290 (-1.430) -0.449 (-1.525) -0.341 (-1.395)
(0.590)
0.281 (1.694) 0.204 (0.798) 0.150
Constant
Equation
E.l
Estimates
TABLE 3.
0.162 (1.916) 0.126 (1.186) 0.156 (1.527) 0.154 (2.951) 0.194 (2.918) 0.173 (2.927)
Y,
-0.365 (-4.168) -0.323 (-8.043) -0.395 (-5.701) -0.341 (-7.015)
(-3.801)
-0.395
(-4.875)
-0.298
Bond,
for Money:
0.279 (4.412) 0.356 (3.302) 0.293 (3.820)
Demand
0.216 (1.668) 0.363 (1.760) 0.345 (1.763)
YPt
of the Short-Run
(7.964)
0.607 (8.669) 0.511 (3.712) 0.499 (4.342) 0.551 (10.294) 0.435 (3.961) 0.519
M,-,
1897-1958
0.998
0.998
0.998
0.999
0.997
0.997
R2
0.030
0.032
0.032
0.033
0.035
0.037
S.E.
0.51
-0.47
1.56
0.07
-0.28
2.76
h
0.31 (1.76) 0.22 (1.37)
(2.55)
0.40 (2.W 0.44
P
Daniel L. Thornton levels via the standard partial-adjustment mechanism. The short-run demand for money differs from the long-run demand in that the lagged dependent variable appears on the R.H.S. of the equation. The presence of the lagged dependent variable in the short-run demand function complicates the estimation. OLS estimates of short-run demand are neither consistent nor efficient; however, the HL and ML estimates presented here are consistent and the ML estimates are asymptotically efficient.g Thus, the short-run demand for money was estimated using OLS, HL and ML as before. Chow’s short-run demand for money was estimated using the same data as in the previous section. The results for the periods 1897-1958 and 193458 appear in Tables 3 and 4, respectively. Since the Durbin-Watson statistic is biased toward accepting the null hypothesis when a lagged dependent variable is present on the R. H. S., Durbin’s (1970) h-statistic replaces the Durbin-Watson statistic in these tables.” The estimates of the short-run demand for money over the 1897-1958 period are relatively insensitive to the estimation technique employed. Perhaps the most significant finding, as far as the asset motive for holding money is concerned, is that income becomes insignificant when the equation is estimated adjusting for autocorrelation. Thus, the results presented in Table 3 support the asset motive for holding money, as Chow suggested. The estimates of the short-run demand for money for the period 193458 are presented in Table 4. OLS estimates indicate that income, permanent income and wealth are all insignificant at the 0.05 level. When the equation is reestimated using the HL tech@Actually, Lieberman presents estimates of the short-run demand for money based on several estimation techniques. First, he presents results based on an iterative Cochrane-Orcutt procedure. Second, Lieberman estimated the equation including wealth with a combination of instrumental variables and a HL scan. Since the instrumental-variable estimator will be less efficient than the ML estimator unless the optimal instruments are chosen, only the ML estimates are presented along with OLS. Moreover, the HL-type estimates presented in Equation (H.2) do not differ greatly from Lieberman’s combined instrumental variables-H1 estimates. “‘The reader is cautioned that Durbin’s h-statistic may not be directly applicable to the case of missing data points. Savin and White’s (1978) conclusion with respect to the Durbin-Watson statistic, however, may apply to Durbin’s h-statistic as well (see Footnote 7 above). Also, the estimated standard errors of the coefficients obtained directly from the autocorrelation adjustment are biased downward when a lagged dependent variable is present. The t-values presented in Tables 3 and 4 are based on asymptotically unbiased estimates of the coefficient variances using a transformation suggested by Cooper (1972). Thus, the t-ratios for the HL estimates differ from those reported by Lieberman. 334
H.3
H.2
H.l
G.3
G.2
G.l
Equation
1.237 (1.527) 2.494 (1.168) 1.186 (1.622) 0.622 (0.584) 1.389 (0.856) 0.690 (0.781)
Constant y’pt
for
0.126 (0.425) 0.300 (1.257) 0.115 (0.491)
w
Demand
0.408 -0.108 (1.618) (-0.237) 0.663 0.268 (2.610) (OJW 0.402 -0.054 (1.838) (0.135) 0.265 (1.150) 0.672 (2.978) 0.295 (1.612)
Y,
TABLE 4. Estimates of the Short-Run M,-,
0.585 (3.385) -0.192 (-0.481) 0.541 (3.609) 0.535 (5.239) -0.190 (-0.560) 0.509 (6.046)
193458
-0.238 (-1.755) -0.233 (-1.587) -0.251 (-2.093) -0.294 (-2.092) -0.267 (-1.968) -0.296 (-2.445)
Bond,
Money: S.E.
0.999 0.029
0.86
N.A.
1.30
0.992 0.035 0.994 0.030
1.03
N.A.
1.68
h
0.999 0.029
0.991 0.032
0.993 0.035
Ii2
0.75 (4.71) 0.25 (1.02)
0.79 (4.06) 0.25 (0.89)
P
s s cc f u 3
g z "5.
Daniel L. Thornton nique, income becomes significant while permanent income and wealth remain insignificant. Moreover, the interest-rate coefficient is insignificant in the permanent-income equation, and the adjustment coefficient is not significantly different from one in either equation. When these equations are reestimated using the exact transformation, most of these findings are reversed. In fact, the ML estimates do not differ greatly from the OLS estimates.” The only scale variable that is significant is income, and then only in the permanent-income equation. Furthermore, the interest rate is significant in both equations, and the estimated speed of adjustment is reduced markedly. l2 Thus, the ML estimates give slight support to the transactions motive in that income is the only scale variable that is significant. Given the relatively poor performance of the variables in these equations, however, one should be very cautious in interpreting the results.13
5. Conclusions The first and perhaps most significant conclusion that can be drawn from the above is that the method one uses to adjust for serial correlation is very important. If the standard transformation which requires the researcher to delete observations from consideration is used, the results obtained may differ significantly from those obtained using a method that does not require that these observations be deleted. This is likely to be more important the more breaks in the data, and the larger the ratio of deleted to total available observations (in this case at least 5.5 percent). Thus, researchers should be very careful when employing techniques that use the standard autocorrelation transformation. Perhaps econometrics software vendors should modify their procedures to employ
“This is expected since the ML estimates of p are not signillcantly different zero. “The adjustment coefficient is one minus the coefficient on the lagged dependent variable. It is interesting to note that the adjustment coefficient cannot be less than one. Thus, the negative coefficients on Equations (G.2) and (H.2) are particularly disturbing. [For a theoretical discussion of the stock adjustment model, see Griliches (1967).] “It should be noted that the null hypothesis of no first-order serial correlation cannot be rejected using Durbin’s (1970) h-test at the 0.05 level for the OLS equations. The ML estimates of p are consistent with the pre-test results, while the HL estimates are not.
from
336
Long-Run,
Short-Run
Demand
for Money
the exact autocorrelation transformation, even when there are no breaks in the data. While many of the differences reported here were due to significantly smaller standard errors obtained when the exact transformation was used, there were several striking differences in the magnitude of certain parameter estimates as well. Second, Lieberman’s finding that the transactions motive appears to be stronger than the asset motive is not substantiated by these results. In particular, Lieberman’s conclusion that “wealth and permanent income are statistically insignificant in all cases where current income enters as a measure of the volume of transactions, ” is shown to be false. Third, Lieberman’s contention that money-demand equations in particular may provide misleading coefficient estimates unless they are adjusted to remove serial correlation is not supported by these results. In most instances, ML estimates did not change the conclusions concerning the role of interest rates and the relevant scale variables from the OLS results; however, many of the parameter estimates differed markedly. Fourth, the results give only modest support to Lieberman’s contention that there was a structural shift in the demand for money in 1933. Parameter estimates between the two periods only change dramatically for the long-run demand for money which includes wealth, or if the approximate transformation is used. Finally, there is evidence of a wealth effect in the demand for money. Thus, there is no evidence against possible third-level crowding out, which has been so widely discussed in the theoretical literature. Received: October, 1980 Final oersion received: January,
1982
References Blinder, A. S. and R. M. Solow. “Does Fiscal Policy Matter?’ Journal of Public Economics 2 (November 1973): 31937. Chow, G.C. “On the Long-Run and Short-Run Demand for Money.” Journal of Political Economy 74 (April 1966): 11131. Cooper, J.P. “Asymptotic Covariance Matrix of Procedures for Linear Regression in the Presence of First-Order Autoregressive Disturbances.” Econometrica 40 (March 1972): 305-10. Durbin, J. “Testing for Serial Correlation in Least-Squares Regression When Some of Regressors are Lagged Dependent Vari337
Daniel L. Thornton ables.” Econometrica 38 (May 1972): 410-21. Friedman, M.S. “The Demand for Money: Some Theoretical and Empirical Results. ” Journal of Political Economy 67 (August 1959): 32751. Griliches, Z. “Distributed Lags: A Survey. ” Econometrica 35 (January 1967): 16-49. Hayakawa, H. “Does Fiscal Policy Really Matter in the Context of Variable Prices?” JournaE of Macroeconomics 1 (Fall 1979): 321-46. Kadiyala, K.R. “A Transformation Used to Circumvent the Problem of Auto-Correlation. ” Econometrica 36 (January 1968): 93-96. Laidler, D. “Some Evidence on the Demand for Money.” Journal of Political Economy 64 (February 1966): 55-68. -. “The Rate of Interest and the Demand for Money-Some Empirical Evidence. ” Journal of Political Economy 64 (December 1966): 543-55. Lieberman, C. “The Long-Run and Short-Run Demand for Money, Revisited. ” Journal of Money, Credit and Banking 12 (February 1980): 4357. Meltzer, A. “The Demand for Money: The Evidence from the Time Series.” Journal of Political Economy 61 (June 1963): 219-46. Savin, N.E. and K.J. White. “Testing for Autocorrelation with Missing Gbservations. ” Econometrica 46 (January 1978): 59-67. Thornton, D. L. “The Appropriate Autocorrelation Transformation when the Autocorrelation Process has a Finite Past. ” Federal Reserve Bank of St. Louis Working Paper 82-002, 1982.