money demand: the effects of inflation and alternative adjustment mechanisms

Money Demand: The Effects of Inflation and Alternative Adjustment MechanismsAuthor(s): Stephen M. Goldfeld and Daniel E. SichelSource: The Review of Economics and Statistics, Vol. 69, No. 3 (Aug., 1987), pp. 511-515Published by: The MIT PressStable URL: http://www.jstor.org/stable/1925540 .

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NOTES

MONEY DEMAND: THE EFFECTS OF INFLATION AND ALTERNATIVE ADJUSTMENT MECHANISMS

Stephen M. Goldfeld and Daniel E. Sichel*

Abstract-The paper first reconciles a variety of specification tests for partial adjustment money demand models and points out a fundamental identification problem which makes it impossible to distinguish between the real and nominal partial adjustment models if inflation has an independent effect on the long-run demand for money. The paper also finds that empirical estimates of simple partial adjustment models have some undesirable properties and then considers the short- and long-run effects of inflation in a more general distributed lag model.

I. Introduction

In the last decade, episodes of apparently errant behavior have spawned voluminous research on the proper specification and estimation of money demand functions. The dynamics of short-run money demand have received particular attention. Although it is not without its shortcomings, the partial adjustment mechanism (PAM) has often served as the workhorse to capture the underlying dynamics. Indeed, two alternative PAMs, the so-called real and nominal models, have been utilized. The primary difference between these two specifications is the implication of the real model that there is an instantaneous adjustment of money balances to changes in the price level whereas the nominal model posits that this adjustment is subject to a distributed lag. Given this difference, it is not surprising that there has been considerable interest in formal comparisons of the two models as evidenced, for example, by the work of Milbourne (1983), Spencer (1985), Hwang (1985) and Hafer and Thornton (1986). At the same time, there has been interest in examining the role of the rate of inflation in money demand. What does not seem to have been fully appreciated, however, is that this question is intertwined with the nature of the adjustment mechanism.

The outline of the paper is as follows. In section II we derive the real and nominal partial adjustment models from a general cost-minimizing framework, allowing for the possibility that inflation affects the desired stock of real balances. We then spell out a fundamental identification problem in sorting out the role of inflation and

the adjustment mechanism. In so doing, we clarify the potential pitfalls with existing econometric tests of the real vs. nominal models. Empirical results are presented that illustrate these points. In section III we use a model more general than the real or nominal PAM to explore further the relationship between money demand and inflation.

II. Partial Adjustment Models and Identification

The real partial adjustment mechanism (RPAM) is given by

ln mt-lnmt_l = (In mt*- - _1) + u, 1

where m, is the actual stock of real money balances and m * is the "desired" stock, and u, is a random disturbance. The nominal partial adjustment model (NPAM) is given by

ln M - ln MA_1 = X(ln M,*-ln MAi ) + ut (2)

where M, = m Pt, Mt* = m *Pt and P, is the price level. The difference between (1) and (2) can be seen by rewriting (1) as

ln Mt = X(ln m* - ln mt l) + A ln Pt + u,.

Hence, an implication of the RPAM is that there is an instantaneous adjustment to changes in the price level. To convert (1) or (2) into a form suitable for estimation, requires a specification for m*. Before doing this, however, it will be helpful if we comment briefly on the derivation of the PAM.

Partial adjustment is typically motivated by cost- minimizing behavior wherein the costs of disequilibrium are balanced against adjustment costs. Following Hwang (1985), we specify a quadratic cost function of the form

C = a [ln Mt* -ln Mt]2 + a2 [(ln M.-ln M-1 )

+8(ln P, - ln t_1)12.

(3)

The first and second terms of (3) correspond to the disequilibrium and adjustment costs, respectively. For 8 = 0, (3) posits that the adjustment term is solely a

Received for publication August 22, 1986. Revision accepted for publication December 3, 1986.

*Princeton University.

Copyright ? 1987 [ 511 1



512 THE REVIEW OF ECONOMICS AND STATISTICS

function of nominal magnitudes, while for 8 = 1 the second term in (3) takes the form a2(In m, - ln mt_1)2. Intermediate values of 8 correspond to hybrid models which combine elements of both the RPAM and the NPAM.1

Minimizing costs with respect to Mt yields

In M - In M,-, == A(ln M* - In M,-,)

+ -y(In Pt- In Pt-,1) (4)

where X a1/(a1 + a2) and y = 3a2/(a1 + a2) =

8(1 - X). We thus see that when 8 = 1, (4) reduces to the RPAM given by (1), while when 8 = 0, (4) collapses to the NPAM, (2).

To complete the story we adopt a standard specification for m * given by

ln m* = 00 + Olln yt + 021n rt + ?03 t (5)

where yt is a transactions variable such as real GNP, rt represents one (or more) interest rates, and s7t is the rate of inflation measured by 7rt = ln(Pt/Pt-1) = APt/IPt-. Combining (4) and (5) and rearranging terms yields

lnm, = X0O + XO1ln yt + X021n rt

+ (1- )lnmt-, + fln(PtI/Pt1) (6)

where

3 = [-(1-X) + X03 + ?(1-X)]. (7)

The nature of the identification problem is readily revealed by examining (6) and (7). Estimation of (6) will provide estimates of X and 8, X and /, but from (7) we see that these estimates are consistent with an infinite set of underlying values of 03 and 8. In general, it is impossible to identify the underlying values of 03 and 8 and to perform hypothesis tests on these parameters individually. Even if one only permits values of 8 corresponding to the RPAM and NPAM, 8 = 1 and 8 = 0, respectively, one is still faced with two possible values for 03: for 8 = 1, 03 = 8/A; for 8 = 0, 03 =

( + 1-X)/X. What this says is that to test whether inflation affects

desired real balances (03 * 0), we must assume an underlying adjustment model (i.e., a value for 8). Alterna- tively, to test the real vs. nominal adjustment models, we must assume a value for 03, typically zero. We cannot, however, do both at once.

Noting this identification problem, we can readily interpret recently proposed tests of the RPAM and

NPAM and their limitations. Milbourne (1983) following Goldfeld (1973), observed that the restriction (1 -

A) = -,B would reduce (6) to a conventional NPAM and proposed testing this restriction. Milbourne's primary interest was to use this as a test of the role of inflation, although he loosely suggested that this served as a test of the real vs. nominal models. This bit of schizophrenia is precisely what is indicated by (7) from which we see that restriction (1 - A) = -,B implies that X03 + 8(1 - A) = 0. Thus, the test of the restriction (1 - A) = -/3 does not shed individual light on 03 or 8. If, however, we are willing to posit a value for one of these, we can test hypotheses about the other. For example, if we assume 03 = 0, the restriction (1 - A) = - / implies 8(1 - A) = 0, providing an indirect test of 8 = 0. This is, in essence, the approach taken by Spencer (1985) and Hwang (1985). Alternatively with 8 = 0 (NPAM), the restriction (1 - A) = -,8 implies A03 = 0, providing an indirect test of 03 = 0. This is the primary approach adopted by Hafer and Thornton.2

While these results mean that it is inappropriate to perform the tests done by many researchers, equation (6) is not uninformative. Within the framework of the simple partial adjustment model with serially correlated errors, equation (6) is the reduced form which sum- marizes all information in the data. Equation (6) can be used to see which conclusions are consistent with alternative priors for 8 and 03.

To obtain conditional estimates and standard errors for 8 and 03, we estimated a conventional money demand function for M1. The scale variable, yt, is real GNP and P, was taken to be the implicit GNP deflator.3 Two interest rates were used: the commercial paper rate, RCP, and the commercial bank passbook rate, RCBP. Table 1 reports the results of estimating equation (6) over two sample periods, 1952:3-1973:4 and 1952:3-1978:4, by maximum likelihood allowing for

'See Hwang (1985) for a more detailed discussion of (3). As Hwang also notes, it makes no difference whether one writes the first term in (3) as a function of nominal or real magnitudes.

2Hafer and Thornton criticize the indirect nature of Milbourne's test and propose directly testing 03 = 0. However, except for unusual circumstances, the Milbourne test of (1 - X) = -,8, a test of 03 = 0 conditional on 8 = 0, and a test of 8 = 0, conditional on 03 = 0 should yield comparable test statistics, a finding confirmed by our empirical results. Hafer and Thornton report more disparate results for the first two tests but, as they indicate, their tests are suspect because X is essentially zero in their specification. Although Hafer and Thornton carry out their tests they seem to be aware of some of the difficulties of interpretation implied by (6) and (7). Milbourne (1986) contains other criticisms of the Hafer-Thorn- ton approach.

3The GNP data reflect the major 1985 revisions of the national income accounts. The M1 data, in quarterly average form, utilize the revised data available as of mid-1986. Since the latest conceptual revisions of M1 are only available begin- ning in 1959, we spliced in the early data using the 1959:1 data for the old and new series. The results are quite insensitive to alternative ways of doing this.



NOTES 513

first-order autocorrelation of the disturbance.4 The second sample period includes the episode of the "missing money" and, as is well established, equation (6) fails a stability test over the complete sample. To correct for this we initially tried to follow Hafer and Hein (1982) who included an intercept dummy variable, D, which takes on the value of 0 prior to 1974:2 and 1 otherwise. They suggested this was sufficient but our further tests indicated that a preferred specification included slope dummies for y and RCP and excluded the intercept dummy.' Thus, the equation estimated through 1978 reported in table 1 includes the variables D -ln y and D -ln RCP.6

Armed with the estimates in table 1, and the underlying variance-covariance matrices, we can readily derive estimates and standard errors for 03 conditional on 8 and vice versa.7 Table 2 reports some sample calcula- tions. Among other possibilities, we see that we can explain the data either with the nominal model (8 = 0) with 03 = 0 or with the real model (8 = 1) with a significantly negative 03.

To summarize the results of this section, estimates of equation (6) are open to varying interpretations. Never- theless, the estimates clearly say that inflation will affect real balances independently of the Fisher effect operat- ing through nominal interest rates. The identification

problem merely means that we cannot sort out whether this result comes about because vT, matters in m because of the adjustment mechanism, or for both reasons. However, some researchers have indicated a pref- erence for the NPAM for informal reasons, suggesting that theory does not readily provide an explanation other than the adjustment mechanism for an inflation effect independent of the Fisher effect. The point, of course, is that if we have strong priors about 8 or 03, we can make inferences conditional on these priors.8

Ill. General Distributed Lag Models

While the methodological point of the previous section should put to rest certain kinds of tests and may lead to abandoning distinctions between RPAM and NPAM in the presence of inflation effects, it does not mean we are forced to be agnostic about whether inflation affects real balances. For this purpose, however, equation (6) is a rather restrictive vehicle. It does not allow a distinction between short- and long-run inflation effects and requires that real money balances adjust with the same geometric lag to each right-hand side variable. A detailed look at the estimates of (6) provides additional reasons for questioning its adequacy.

TABLE 1.-A CONVENTIONAL MONEY DEMAND MODEL

Sample Period Intercept y RCP RCBP mt-1 Pt/Pt- P

1952:3- 0.400 0.134 -0.016 -0.032 0.782 -0.707 0.491 1973:4 (1.5)a (4.6) (5.2) (3.1) (10.8) (6.9) (4.5)

1952:3- - 0.012 0.082 - 0.015 - 0.015 0.909 - 0.784 0.368 1978:4b (0.1) (6.5) (6.0) (3.1) (29.0) (8.8) (3.9)

aCoefficients divided by asymptotic standard errors are given in parentheses.

bAlso includes slope dummies for y and RCP with coefficients -0.0043 (2.9) and 0.011 (2.0), respectively.

TABLE 2.-CONDITIONAL VALUES OF 83 AND 8

1952:3-1973:4 1952:3-1978:4

Parameter t-statistic Parameter t-statistic

93 given 8 = 0 0.341 0.71 1.366 1.27 93 given 8 = 1/2 - 1.448 - 2.17 - 3.616 -2.20 03 given 8 = 1 - 3.239 -2.40 -8.599 - 2.56 8 given 93 = 0 0.096 0.81 0.137 1.46 8 given 03 = - 2 0.654 2.71 0.338 2.82

4To estimate an equation of the form m, = ?x, + Xm1-I + e, with sample size T where e, = pe,_1 + u, maximum likelihood amounts to minimizing au/(j - p2)1/T where

T

(T -1)2 = E [(m, - pm,-,) - f(x, - px,-) t=3

-X(Mt_-I PMt2 12)

+(1- P2)(M2- iX2 - Xm)2.

The optimization was done using the package GQOPT. Virtu- ally identical estimates were obtained with nonlinear least squares or with the much-maligned Cochrane-Orcutt tech- nique.

'We did a sequential likelihood ratio test procedure begin- ning with an intercept dummy and a full set of slope dummies (for RCP, RCBP, PI/P,-, y, m,-1). This suggested the seri- ous candidates were D and the slope dummies for RCP and y. A likelihood ratio test for excluding the intercept dummy from this set of three yielded a 42-statistic of 2.62 where the corresponding 5% critical value is 3.84. The 4,2-statistic for excluding the,slope dummies was 6.56 with a critical value of 5.99.

6See footnote 15 below for an explanation of why we ended our sample period in 1978:4. Of course, the methodological point of this section is unaffected by this.

7For example, to obtain estimates for 03 we can rewrite (7) as

03 = [? + (1 - X)(1 - 8)I/X

and plug in the assumed value of 8 along with the estimates of ,B and X. This same expression can be used to derive the standard error of 03, based on the underlying variance-covariance matrix of the estimates.

8The data also support the unconditional hypothesis that /3 = X - 1 (Milbourne's test). From (7) we know that 8 = 0 implies 03 = (/3 + 1 - X)/X and 8 = 1 implies 03 = A8/X. Thus, when /8 = X - 1 we have 03 = 0 for the nominal model and 03 = (A - 1)/k for the real model. An advocate of the nominal model is likely to regard the latter as a peculiar restriction.



514 THE REVIEW OF ECONOMICS AND STATISTICS

First, a closer examination of the residuals of (6) over the longer sample period revealed that there was significant fourth order serial correlation, suggesting an in- complete dynamic specification. Second, there appears to be a problem with the implicit assumption of homogeneity with respect to the price level that is em- bodied in (6). Homogeneity of money demand, at least in the long run, is generally presumed to be a feature of any well-specified money demand function. Conse- quently, rejection of homogeneity is typically taken to be evidence of misspecification. Tests for homogeneity in the context of the NPAM and RPAM are typically accomplished by including In P, in an equation like (6) and doing a t-test to see if the coefficient is significantly different from zero. Generally speaking, at least for sample periods that avoid the missing money, such tests have accepted homogeneity for both NPAM and RPAM.9 It was somewhat to our surprise, therefore, that when we reestimated (6) through 1973 including the variable ln Pt, we found ln P, had a t-statistic of about 2.5. Extending this sample period to 1978, and including the same dummy variables as in table 2, also led to a rejection of homogeneity. It thus appears that the revision of the GNP data released in 1985 has further served to call into doubt the adequacy of the partial adjustment model.'0

Overall, there are sufficient reasons to consider a model more general than equation (6)." One conveni- ent generalization that retains the spirit of (6)-and, of course, some of its defects-that has found favor in recent years is given by

nl n2

ln m, = g + E a,ln yt-i + . biln rt-i i=O i=O

n3 n4

+ c,ln mt-i + E e,ln Pt-i + t. (8) i=l i=O

Note that in the context of (8), homogeneity of real balances with respect to prices is equivalent to the restriction that Ee, = 0. Moreover, if this restriction is

satisfied (8) can be written in the form of (9). nl n2

In m, = g + Y. a,ln yt-i + E, biln rt-, i=O i=O

n3 n4-1

+ E ciln m,_i + E d,a,t-, + Et (9) i=l i=O

where d. = EJ=O e, Equations (8) and (9) resolve many of the shortcom-

ings of equation (6). Real balances are no longer con- strained to adjust with the same geometric distributed lag to each independent variable. Further, it is possible to explicitly test for short- and long-run inflation effects. It is worth emphasizing that equations (8) and (9) are not motivated by an underlying cost function as are the NPAM and RPAM and therefore are not intended to shed light on the preferred adjustment process.

To use (8) and (9) to assess whether any inflation effect is present, we can test the restriction that ei = 0 for all i in (8) or di = 0 for all i in (9). If the homogeneity restriction held exactly, test statistics for these two tests would be identical. In practice, if homogeneity is accepted, these tests should be quite close. The hypothesis that d, = 0 for all i amounts to testing whether real money balances adjust without lag to changes in prices.

A second less restrictive hypothesis is that there is a short-run effect of inflation on real balances, but that

Ithere is no long-run effect. In terms of (9), this hypothesis is simply ,n4- l d, = 0. Given homogeneity, the corresponding test in terms of (8) is ,40 i = 0. Again, if homogeneity held exactly the tests would be identical.'2 In equation (6), this distinction between short- and long-run inflation effects is not possible.'3

To estimate (8) or (9) we first need to choose the relevant lag lengths. Examination of results in the litera- ture and modest experimentation suggested that nl = n 2 = n 3 = n 4 = 4 was adequate.'4 Results using this

9This is the conclusion in Goldfeld (1973) and Spencer (1985). Hwang (1985) accepts the hypothesis for the NPAM but rejects it for RPAM, although his sample extends through 1974, part of the troublesome period.

l?Using the latest revisions of the money data with the unrevised GNP data does not lead to a rejection of homogeneity over the sample period through 1973.

1 One bit of evidence to the contrary is provided by estimating a general alternative with just enough lags of each independent variable to nest (6) with an AR(1) disturbance. We carried out the likelihood ratio test for the restrictions required to go from the more general model to (6). We could not reject (6) at conventional levels of significance for either sample period. See Harvey (1983), ch. 5, for a more detailed discussion of the approach.

"2For example, if n4 = 3, we have 3

E e,ln P,ti = eoln i, + (eo + el)ln7Tt_l i=o

+(eO + el + e2)lnsT,_2

+ (eo + el + e2 + e3 )ln Pt,3.

The test of no long-run inflation under homogeneity (i.e., E e, = 0), is thus

Ed, = 3eo + 2e, + e2 = 0.

13 With one coefficient for iT,, as in (6), there is no difference between the hypothesis that the coefficient is zero and the hypothesis that the sum of the coefficients is zero.

14Spencer (1985) uses a common lag length of 4. However, he does not include lagged money but rather captures the dynamics via an AR(2) process for the disturbance. Including lagged dependent variables is a less restrictive way of modeling the dynamics.



NOTES 515

TABLE 3.-A GENERAL MoNEY DEMAND MODEL

Sample Period y RCP RCBP m,1_ Pt P,/Pt-

1953:1-- 0.114 a -0.010 -0.024 0.825 -0.019 1973:4 (2.2)b (2.3) (1.9) (8.8) (1.3) -

0.061 -0.007 -0.013 0.914 - -0.638 (2.0) (1.9) (1.4) (14.1) - (2.4)

1953:1- 0.118 -0.012 -0.023 0.824 -0.020 -

1978:4C (2.5) (2.8) (2.0) (9.5) (1.4) -

0.064 -0.009 -0.013 0.911 - -0.632 (2.3) (2.4) (1.4) (14.8) - (2.4)

aSums of the distributed lag coefficients. bt-statistics in parentheses are for the test that the sums of the coefficients

in the distributed lag equal zero. 'In the specification with P, the sums of the distributed lags on RCP and

its slope dummy and y and its slope dummy are -0.004 and 0.114, respectively. For the PIP,-. specification, these sums are -0.004 and 0.062, respectively.

TABLE 4.-TESTS OF HOMOGENEITY AND INFLATION EFFECTS IN THE GENERAL MODELI

Sample Period

Restriction 1953:1-1973:4 1953:1-1978:4

4

(1) L ef = 0 .200 .162 i=O

(2) e, = 0, i 0,...,4 .000 .000 4

(3) Zie4_ =0 .060 .064 1=1

(4) Joint test of (1) and (3)b .030 .024 3

(5) d,= 0 .020 .019 , =o

(6) d, = 0, i=0...,3 .000 .000

aEntries are marginal significance levels of the F-test for rejecting the restrictions.

bThis tests the same hypothesis as line (5).

lag length are reported in the first parts of tables 3 and 4 for the sample period ending 1973. Table 3 displays the sums of the distributed lag coefficients for each of the variables while table 4 gives the marginal significance level for rejecting the various hypotheses of interest. The results are as follows: (1) homogeneity is not rejected at conventional levels of significance; (2) the no-inflation effects model is overwhelmingly rejected; and (3) there is a long-run significant negative effect of inflation on real money balances.

As before, to extend the results to 1978, it was necessary to include dummy variables to achieve stability.15 Some limited experimentation confirmed the ap-

propriateness of the choice of the slope dummies for RCP and y utilized in table 2 which were included at every lag of RCP and y in (8) or (9). With these emendations, the estimates for the extended sample are extremely similar to those for the shorter period. As a consequence, the various hypothesis tests in table 4 yield identical conclusions for the longer and shorter sample periods. Moreover, the results for the extended sample period display no evidence of high-order serial correlation, providing yet another reason for preferring

(8) to (6).

IV. Conclusion

We first characterized the underlying identification problem that confounds interpreting RPAM and NPAM in the presence of an inflation effect on the desired stock of real balances. This aside, the empirical results for the PAM did indicate that inflation played a significantly negative role in the demand for money. We then noted the a priori restrictiveness of the PAM specification and also uncovered some problems with serial correlation and homogeneity. Estimation of a distributed lag model provided a more general way for testing homogeneity and for examining the role of inflation on money demand. Unlike the PAM specification, this more general model passed the homogeneity test at the same time that it confirmed the role for inflation. As noted, however, without an explicit cost-based derivation of the general model, the precise source of the inflation effect remains an open question.

15We attempted to extend the sample period beyond 1978 but parameter constancy could not be obtained without a substantial modification of the specification. At a minimum, credit control dummies for 1980 are needed to take the sample through 1981 and beyond that more drastic surgery is required.

REFERENCES

Goldfeld, Stephen M., "The Demand for Money Revisited," Brookings Papers on Economic Activity (3, 1973), 577-638.

Hafer, R. W., and Scott E. Hein, "The Shift in Money De- mand: What Really Happened?" Review, Federal Re- serve Bank of St. Louis (Feb. 1982), 11-16.

Hafer, R. W., and Daniel L. Thornton, "Price Expectations and the Demand for Money: A Comment," this REVIEW

68 (Aug. 1986), 539-542. Harvey, Andrew, The Econometric Analysis of Time Series

(Oxford: Phillip Allan Publishers, 1983). Hwang, Hae-shin, "Test of the Adjustment Process and Linear

Homogeneity in a Stock Adjustment Model of Money Demand," this REVIEW 67 (Nov. 1985), 689-692.

Milbourne, Ross, "Price Expectations and the Demand for Money: Resolution of a Paradox," this REVIEW 63 (Nov. 1983), 633-638.

_ "Price Expectations and the Demand for Money: Reply," this REVIEW 68 (Aug. 1986), 543-544.

Rose, Andrew, "An Alternative Approach to the American Demand for Money," Journal of Money, Credit and Banking (Nov. 1985, pt. I), 439-455.

Spencer, David E., "Money Demand and the Price Level," this REVIEW 67 (Aug. 1985), 490-496.



money demand: the effects of inflation and alternative adjustment mechanisms

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