# The value of time and recent U.S. money demand instability.

I. Introduction

Much controversy has surrounded the U.S. money demand relationship, or more precisely, its stability. This is rather disappointing in light of the central importance of money demand estimates for the formulation and implementation of effective monetary policy and for almost all theories of aggregate economic activity. To date, the vast majority of alternative money demand specifications have generally failed to produce a stable money demand function throughout the 1970s and 1980s.

A common weakness in most previous studies in this area is their total neglect of the "value-of-time" hypothesis when modelling money demand. Yet, some early work due to Dutton and Gramm |20~, and Karni |49; 50~ have provided some empirical evidence that a variable representing the value of time (like the wage rate) should be included as an additional argument in the money demand function. Dutton and Gramm |20, 652~ note "if the use of money saves transactions time, it increases the amount of leisure. This suggests an additional determinant of the demand for money, the consumer's valuation of time, i.e., the wage rate."(1) Indeed, the underlying thesis of Baumol's famed money demand approach is that the holding of money saves transaction time, the cost of which varies with the value of time. As Laidler |52~ and Dowd |19~ pointed out, failure to allow for the role of the wage rate in the "standard" transactions models of money demand could result in serious misspecification. It is, of course, conceivable that such misspecification may have caused instability in the U.S. money demand function. Ironically, none of the previous studies that did examine the effect of the time factor on money demand, including Dowd for the U.K., pays any attention to its implication for the stability property of the estimated function. Such issue, however, forms the basis of this study. Besides the wage rate as a proxy for the value of time, the U.S. money demand model estimated here further allows for the potential effects of financial uncertainty as measured by the volatility of money growth and the variability of interest rates. Previous theorizing |e.g., Friedman |29~ and Tobin |76~~ has suggested that both variables may exert important effects on money demand. Moreover, the paper addresses the co-integratedness of the variables and estimates the implied error-correction model. The rest of the paper is organized as follows. Section II briefly discusses the issues involved, while section III presents the model. Section IV reports the empirical results and section V analyzes their implications for the stability debate. Section VI offers some concluding remarks.

II. Some Issues

Dowd presents a lucid theoretical rationale for the value-of-time hypothesis based on utility maximization and thus needs not be repeated here. As a result of the optimization process, Dowd estimates for the U.K. economy a money demand equation comprising the wage rate, interest rates, and consumers expenditure as regressors. His estimates show that the wage coefficient is positive as expected and statistically significant with an elasticity of about one half. Earlier, Dutton and Gramm |20~ and Karni |49; 50~ reported similar findings for the U.S. using annual data. Further empirical support for the effect of the wage rate on the U.S. money demand has also come from the work of Diewert |17~, Philps |60~, and Dotsey |18~. As we mentioned earlier, despite this apparent strong support (both theoretical and empirical) for the value-of-time hypothesis, the implication of this hypothesis for the recent instability controversy surrounding the U.S. money demand has been surprisingly overlooked. However, a thoroughly examined hypothesis is the Friedman |29~ proposition that the U.S. money demand instability of the 1980s is caused by higher volatility of money growth in those years. Friedman theorizes that the increased volatility of money growth--following the 1979 change in the Federal Reserve operating strategy--has increased the degree of perceived uncertainty, leading to decreased money velocity since 1982. The empirical evidence on the Friedman hypothesis has been mixed. For example, Hall and Noble |41~, Fisher |27~, and Fisher and Serlitis |28~ have reported results in favor of the Friedman contention. On the other hand, Brocato and Smith |9~, and Mehra |55; 56; 57~ found results inconsistent with that contention. It should be noted that all these empirical tests of the Friedman hypothesis appear plagued with possible omission-of-variables bias for they use the restrictive bivariate framework in which money growth volatility is the only regressor in the demand for money (velocity) equation.(2) Clearly, a more credible test of the Friedman hypothesis should take into account the effects of other determinants of money demand. In the words of Mehra |56, 265~, "it may be necessary to reexamine the role of volatility in a more general framework that controls (for) the influence of other factors on velocity." On the other hand, drawing on Tobin's |76~ theoretical model, studies by Slovin and Sushka |69~, and Garner |31~ have argued that interest-rate variability can influence money demand. They contend that risk-averse asset holders would seek greater liquidity in response to heightened interest-rate risk.

III. The Model

Based on the preceding discussion, the money demand equation may take the following form:

ln |m*.sub.t~ = ||Alpha~.sub.0~ + b |center dot~ ln |w.sub.t~ + c |center dot~ v|(M).sub.t~ + d |center dot~ v|(R).sub.t~ + |Z.sub.t~|Omega~, (1)

where |m*.sub.t~ denotes desired real money balances, |w.sub.t~ is the real wage rate representing the value of time,(3) v|(M).sub.t~ is the volatility of money growth, v(R) is the variability of interest rates, |Omega~ is a column vector of parameters associated with a row vector of variables, Z, comprising three traditional regressors (real consumers expenditure (x), the interest rate spread defined as market interest rates minus the own rate of return on money (r - rm), and the inflation rate (|Pi~)).(4) Because the underlying theory is micro in nature and as such refers to individual behavior, the aggregate variables (money and consumers expenditures) are expressed as per capita figures. Note that v |(M).sub.t~ and v(R) are not expressed in natural logarithms because they assume negative values over some quarters for which the logarithms are undefined. To make equation (1) estimatable, we need to replace desired (unobservable) money demand with actual (observable) levels. This is done by allowing for a less-than-immediate adjustment to desired money demand. A procedure that has gained popularity particularly in money demand literature is the simple Koyck-lag mechanism whereby the entire adjustment is represented by adding a lagged dependent variable (nominal or real) as a regressor. However, several researchers have criticized this Koyck-lag structure due to its several restrictive assumptions.(5) Consequently, recent studies have increasingly employed other more flexible lag structures when estimating money demand equations.

Therefore, equation (1) can be rewritten as:

ln |Delta~|m.sub.t~ = ||Alpha~.sub.0~ + |summation of~ |b.sub.i~ |Delta~ ln |w.sub.t-i~ where i = 0 to |n.sub.1~ + |summation of~ |c.sub.i~|Delta~v|(M).sub.t-i~ where i = 0 to |n.sub.2~ + |summation of~ |d.sub.i~|Delta~v|(R).sub.t-i~ where i = 0 to |n.sub.3~

+ |summation of~ |f.sub.i~|Delta~ ln |x.sub.t-i~ where i = 0 to |n.sub.4~ + |summation of~ |g.sub.i~|Delta~ ln |(r - rm).sub.t-i~ where i = 0 to |n.sub.5~ + |summation of~ |h.sub.i~|Delta~||Pi~.sub.t-i~ + |e.sub.t~ |where~ i = 0 to |n.sub.6~ (2)

where |n.sub.j~ (j = 1,2,...6) are the various lags on the regressors; m is the money stock deflated by the implicit GDP deflator, and the resultant real money stock variable is divided by population to obtain the corresponding per capita figures; w is real wages measured by the index of average hourly earnings deflated by the implicit GDP deflator; v(M) is the volatility of M1 stock measured, as in Mehra |55~ and others, by the eight-quarter moving average of the standard deviation of quarterly M1 growth; v(R) is the variability of interest rates defined, as in Evans |24~, by the eight-quarter moving average of the standard deviation of Moody's AAA corporate rate; x is per capita consumers' expenditure; (r - rm) is the interest rate spread where r is the 4-6 month commercial paper rate and rm is the own rate of return on money defined, as in Hetzel |44~, by the weighted-average of the explicit rate of interest paid on the components of money; |Pi~ is the inflation rate measured by the percentage change in the implicit GDP deflator; and e is a white-noise error term assumed as in the usual fashion to be serially-uncorrelated with zero-mean and a constant variance. The money demand equation is estimated over the quarterly period 1963:1-1991:4 (after adjustment for all lags). All time series (with one exception)(6) are obtained from the Citibank data tape. The underlying money demand theory suggests the following a priori sign restrictions for the summed coefficients:

|summation of~ |b.sub.i~, |summation of~ |c.sub.i~, |summation of~ |d.sub.i~, |summation of~ |f.sub.i~ |is greater than~ 0; and |summation of~ |g.sub.i~, |summation of~ |h.sub.i~ |is less than~ 0.

Before turning to the empirical analysis, three comments pertaining to equation (2) bear emphasis. First, the distributed-lag on each variable is estimated by means of the Almon procedure which has been extensively employed in applied literature.(7) Following Schmidt and Waud |65~, Sims |68~, and Thornton and Batten |75~, we used the Almon procedure here without imposing the endpoint constraints since they lack any theoretical basis and, moreover, could yield biased results. In addition, we do not impose the usual implicit restriction that the underlying lag weights are all non-negative. This latter additional restriction is often imposed when researchers |e.g., Goldfeld |34~, and Berman |31~~ choose the lag length on each explanatory variable using a priori expectations regarding the signs of the variables. Clearly, this practice presupposes that the weights are all positive. Following Schmidt and Waud |65~, and Giles and Smith |33~, the lag structure on each variable (lag length and degree of polynomial) is determined by Theil's minimum residual-variance criterion.

The second comment relates to the particular definition of money stock used in the empirical analysis. Compared to alternative aggregates, a growing number of studies have paid increasing attention to M2 as a more appropriate monetary target |e.g., Wenninger |78~, Hetzel |44~, Mehra |55; 56; 57; 58~, Roberds |61~, Hafer and Jansen |39~, Hallman, Porter and Small |42~, and Thornton |74~~. Indeed, since the mid 1980s, the Fed has dropped M1 from its list of intermediate policy aggregates, making M2 the primary policy target. The Fed has not set target ranges for the M1 aggregate in recent years, "largely because movements in this aggregate no longer track movements in income and prices very well" (Furlong and Trehan |30, 1~). Recent work |e.g., Mehra |55; 57~~ has also suggested that the M1 demand relationship is inherently unstable casting doubt on its usefulness for policy analysis. A further rationale for our focus on M2 can be distilled from the work of Mehra |55~ and Cox and Rosenbloom |13~. Their analyses imply that broader monetary aggregates like M2 provide a more credible test of the Friedman hypothesis since M2 can internalize deregulation-induced substitutions among the aggregate components. All in all, then, the behavior of the broad M2 aggregate appears worthy of special attention. Thirdly, most recent studies of money demand have employed first-differences to avoid the spurious regression phenomenon (see Granger and Newbold |37~). However, Engle and Granger |22~ have shown that a model estimated using differenced data will be misspecified if the variables are cointegrated and the cointegrating relationship is ignored. Indeed, as Boughton |7~ pointed out, overlooking the cointegrating relationship may explain some of the apparent instability TABULAR DATA OMITTED of money demand equations. Thus, testing for cointegration among the variables seems to be necessary if possible biases were to be avoided.

The cointegration tests can be implemented using the two-step procedure described in Engle and Yoo |23~. In the first step, we test for nonstationarity (presence of unit roots) in each of the variables using levels (of natural logarithms) and first-differences. Results from the Dickey-Fuller (DF) and the augmented DF (ADF) tests are displayed in Table I.

For the levels of all series, both tests could not reject the null hypothesis of non-stationarity at the 5 percent level of significance. However, with first-differences, each series (with one minor exception) clearly indicates rejection of the null hypothesis at the 5 percent level or better. The only exception is that for m (real M2). While the DF and the ADF (with 2 lags) reject non-stationarity for the first-difference of real M2 at the 5 percent level, the ADF test (with 4 lags) does so only at the 10 percent level (test statistic = -3.30). Nonetheless, first-differences seem adequate for real M2 as well since second-differencing the variable appears to entail over-differencing.(8) These unit-root tests consistently suggest that all series in the real M2 demand equation are stationary if used in first-differences. Hence, all variables in equation (2) are cast in first-differences to achieve stationarity. The second step in the Engle-Yoo methodology is to examine whether the non-stationary variables (in levels) are cointegrated. If they are, the use of first-differences without due adjustments might introduce biases since it would filter out low-frequency (long-run) information |see Hendry |43~, Engle and Granger |22~, and Miller |59~~. Thus, cointegrating regressions of (non-stationary) variables are estimated and their residuals are tested for the presence of unit roots.

When estimating cointegrating regressions, a choice must be made regarding which series is used as the left-hand-side conditioning variable.(9) Since cointegrating parameters may not be unique, we follow Hall |40~ and Miller |59~ and examine all possible cointegrating regressions and choose that which yields the highest adjusted |R.sup.2~. We found that the logarithm of real M2 displays the highest adjusted |R.sup.2~ (=0.9711). Consequently, the logarithm of real M2 was chosen as the left-hand-side conditioning variable.

The residuals from the cointegrating regression are then tested for non-stationarity. Both DF and ADF tests reject the null hypothesis of non-cointegration (non-stationarity) in the real M2 regression. Specifically, the test statistics are -6.60 for the DF test, -4.32 for the ADF test (with 2 lags); and -3.29 for the ADF test (with 4 lags). Except for the ADF test with 4 lags, the other two tests soundly reject the null hypothesis at least at the 5 percent level. Higher lag orders in the ADF confirm this conclusion. For example, test statistics for 8 and 10 lags are respectively -4.06 and -3.71, both of which are significant at least at the 10 percent level. This finding of cointegration in the real M2 regression accords with the evidence reported recently by Hafer and Jansen |39~, Mehra |58~, and Miller |59~ among others. An attractive alternative to the above Engle-Yoo (single equation) approach for testing cointegrating relationships is the Johansen |45~ multivariate technique which has gained a lot of popularity in recent applied literature. As Dickey, Jansen, and Thornton |16~ pointed out, the Johansen approach is particularly promising for it is based on the well-established likelihood ratio principle. Furthermore, the Johansen method is not subject to biases due to arbitrary normalization choices since all of the variables are jointly endogenized in the testing process. Monte Carlo evidence reported by Gonzalo |36~ also supports the relative power of Johansen's methodology over other alternative (single- and multivariate) techniques.

The Johansen test utilizes two likelihood ratio test statistics for the number of cointegrating vectors; namely the trace and the maximum eigenvalue statistics.(10) In our case, both statistics indicate the presence of one cointegrating relationship. Specifically, for the trace test, the hypothesis that the number of cointegrating vectors (k) is less than or equal to 1, 2...6 cannot be rejected at the 5 percent level of significance. However, the hypothesis that k = 0 is easily rejected (test statistic = 173.22, compared to the 5 percent critical value of 131.70). This finding of one cointegrating relationship is, perhaps more clearly confirmed,(11) by the maximum eigenvalue test which suggests rejection of the null hypothesis k = 0 in favor of the explicit alternative hypothesis that k = 1 (test statistic = 48.24, compared to the 5 percent critical value of 46.45).(12)

In sum, both the Engle-Yoo and the Johansen procedures suggest that real M2 has a long-run relationship with the determinants proposed in the model. To avoid the loss of potentially relevant (low-frequency) information, the error-correction (one-lagged error) term generated from the Johansen multivariate procedure is included as another regressor in the money demand equation (2). The resultant error-correction model integrates the short-run dynamics in the long-run M2 demand relationship.

IV. Empirical Results

To achieve stationarity and to reintroduce the low-frequency (long-run) information into the M2 real money demand equation, the basic equation (2) underwent two adjustments: all variables were cast in first-differences, and the error-correction term was added as another regressor. Over the quarterly period 1963:1-1991:4, the unconstrained Almon/Theil procedure yielded the empirical estimates displayed in Table II.(13) According to the relatively high value of adjusted |R.sup.2~ (=0.79), the model fits the data quite well, considering that the dependent variable is cast in first-differences. Plots of actual and predicted real M2 demand (not shown here) indicate that the error-correction model of U.S. M2 demand adequately traces the behavior of real money demand and its turning points. As is clear from the table, all of the diagnostic tests support the statistical appropriateness of the estimated M2 demand equation. In particular, for examining serial correlation of the residuals, we applied the following tests: the Durbin-Watson and the Durbin-m tests for first-order autocorrelation;(14) the Breusch-Godfrey test of different autoregressive and moving-average order processes;(15) and the Geary nonparametric test for a general (unspecified) autocorrelation process. These tests uniformly fail to reject the null hypothesis of no autocorrelation in the residuals. Moreover, heteroscedasticity does not seem to be a problem according to the Glejser test. The Plosser-Schwert-White Differencing test could not reject the hypothesis that the estimated equation is correctly specified, and the Ramsey RESET test concurs with that verdict and reveals no serious omission of variables. Furthermore, the White test indicates absence of significant simultaneity bias in the estimates. Having provided some evidence supporting the adequacy of the estimated error-correction model of U.S. M2 demand equation, the following observations can be made regarding the obtained parameter estimates.

We should first note that the one-lagged error-correction term appears with a statistically significant coefficient and displays the appropriate (negative) sign, a finding that accords well with some previous studies |59~. This implies that overlooking the cointegratedness of the variables would have introduced misspecification in the underlying dynamic structure. It should also be noted that literature on cointegrated systems suggests that only one-lagged EC is needed to represent the cointegrating scheme. Indeed, higher powers of EC were included but proved statistically insignificant.

Secondly, and perhaps more importantly, Table II shows that the summed coefficient of real wages (long-run elasticity) is positive as hypothesized by the value-of-time theory and is statistically significant at better than the one percent level. Our estimate of 0.93 for the real wage elasticity is within the range of estimates reported earlier for the U.S. by Dutton and Gramm |20~ of about 1.02; by Karni |49~ of 1.00; and by Dotsey |18~ of about 0.72. Therefore, our empirical evidence corroborates previous findings supportive of the value-of-time hypothesis. Further discussion of this important finding is provided below.

Thirdly, the empirical results do not lend strong support to the Friedman hypothesis that money growth volatility is a major culprit behind changes in the demand for M2. Thus, similar to the findings of Mehra |55; 56; 57~, our results reveal no significant impact of the volatility of money growth on real M2 demand at the conventional significance levels.(16)

TABULAR DATA OMITTED

Fourthly, the coefficient on interest rate variability is statistically insignificant at the 5 percent level, a finding that accords well with the evidence reported by Garner |31~ who discussed various reasons for the negligible effect of interest rate variability on money demand. Interestingly, the sign of the summed coefficient of interest-rate variability is negative. One explanation for this negative (albeit insignificant) impact may be found in Evan's |24~ and Tatom's |72~ argument that increased interest-rate risk exerts an adverse effect on aggregate production which in terms decreases the transaction demand for money. Note also that v(R) may instead reflect inflation uncertainty.(17) Higher inflation uncertainty makes financial assets (e.g., M2) riskier to hold, in turn inducing some investors to reallocate their portfolio away from them in favor of real assets.(18) This view is consistent with Klein's |51~ choice-theoretic (utility maximization) approach. In Klein's model, the value of money is primarily derived from the flow of services it renders the asset holders. Inflation uncertainty can thus be viewed as a negative technological change. Empirical support for Klein's theoretical proposition has come, for example, from Blejer |4~ and Smirlock |70~ who report some evidence showing a negative relationship between inflation uncertainty and money demand. Finally, the results in Table II indicate that the conventional variables of real consumers expenditures, interest rate spread and inflation are all correctly signed, with highly significant summed coefficients. The long-run elasticity estimates are also within the range of recent U.S. estimates reported, for example, by Roley |62~, Garner |31~, Mehra |56~, and Emery |21~.

Taken together, the model of Table II fits the U.S. quarterly data quite well, and the estimated real M2 demand equation appears correctly specified. Most importantly, the empirical results lend strong support to the value-of-time hypothesis in that real wages exert significant positive impact upon U.S. real M2 demand. Further, and perhaps more substantive, the evidence indicates that real wages must be included in the analysis if the estimated money demand equation is to exhibit the desired property of structural stability.

Before turning to the stability implications of the estimated model, a comment about the lag structure and method of estimation appears in order. It seems prudent to check whether the results reported in Table II are invariant to both of these important aspects of the estimation. A popular procedure employed in a growing number of studies is to reestimate the basic model by means of the Hendry general-to-specific modelling strategy which eliminates lags with insignificant parameters |32~. Among others, the virtue of this approach is that it uses unconstrained Ordinary Least-Squares (OLS) and, as such, can assess the robustness of the Almon results. Moreover, by eliminating insignificant lagged coefficients (at the 10 percent significance level), the approach is parsimonious guarding against possible over-parameterization. Money demand estimates from the Hendry strategy are displayed in Table III. As seen from the table, the coefficients estimates are very similar to those reported in Table II which are based on the Almon procedure.(19) It is, of course, encouraging that the results are not particularly sensitive to the method of estimation nor are they to the problem of over-parameterization. Observe, however, that the Hendry estimates of Table III seem to suffer from significant autocorrelation according to the various diagnostic tests.(20) Owing to this possible misspecification, and in light of the various difficulties with Hendry's strategy discussed recently by Boughton and Tavlas |8~, our analysis of the stability issue of real M2 demand equation will primarily focus on the Almon estimates reported in Table II.(21)

TABULAR DATA OMITTED

V. Stability Implications

Structural stability of the estimated money demand equation is vital for drawing meaningful policy inferences. As is well-known, the money demand equation represents the link between monetary policy and the rest of the macroeconomy. In order to adequately predict the impact of a given change in money stock on key macro variables, the underlying money demand equation must remain stable over time.

The task of this section is two-fold. We examine first the stability property of the basic equation (2) using several stability tests. As Boughton |6~ pointed out, when testing for money demand instability, several testing procedures should be applied since each procedure examines a particular type of structural instability. Thus, the following three testing procedures are used: the Chow test |10~; the Farley-Hinich test |25~; and the Ashley |1~ Stabilogram test. The second task of this section is to investigate the contribution of real wage to the stability property of the estimated real M2 demand equation. The Chow test is perhaps the most widely used procedure to assess the structural stability of estimated equations. However, as Toyoda |77~, and Schmidt and Sickles |64~ demonstrated, the Chow test presupposes homoscedastic error terms. Thus, it is advisable that heteroscedasticity be tested before applying the Chow test. As Table II shows, the Glejser test did not suggest the presence of significant heteroscedasticity which then permits the application of the Chow test. Unlike the Chow test, the alternative Farley-Hinich procedure does not require splitting the sample period at a particular date. As Farley, Hinich and McGuire |26~ noted, the Farley-Hinich procedure is a robust test for a gradual (continuous) drift in the parameters, in contrast to the single-point shift tested by the Chow procedure. Finally, the Ashley Stabilogram (stab) procedure is a flexible and perhaps more rigorous test against parameter instability of various types. Ashley |1~ reported Monte Carlo simulation evidence supporting the empirical power of the Stab test over a number of standard stability techniques, including the Chow test. Before applying the Chow test, we need to determine the appropriate breaking point(s) at which the money demand equation may have hypothetically shifted. Extensive empirical literature that appeared in the mid-1970s and early 1980s have suggested the possibility of a structural shift in the U.S. money demand relationship around the start of 1973.(22) Another break could have occurred as the result of the well-known October 1979 change in the Federal Reserve operating procedure.(23) Finally, money demand may have shifted as a result of the nation-wide introduction of interest-bearing checkable deposits and other financial deregulations in the early 1980.(24) Consequently, three breaking dates seem appropriate for testing money demand stability; namely, 1973:1; 1979:4; and 1980:4.

Table IV-A reports the results from the alternative stability tests of equation (2). The evidence from all three tests is remarkably consistent and uniformly suggests that the estimated U.S. real M2 demand equation of Table II is structurally stable throughout the estimation period (1963:1-1991:4).

An important issue relates to the contribution of the time-value variable (real wages) to the stability property of the estimated money demand equation. To measure that, the same equation was reestimated and tested after dropping the wage variable. Table IV-B reports the obtained statistics of the three stability tests. The Chow test persistently indicates that the money demand equation is rendered structurally unstable at least at the five percent level, a verdict that is corroborated by the Farley-Hinich test which rejects stability at the stronger one percent level. The Ashley test too reinforces the above conclusion and shows that, once the wage variable is omitted from the equation, it becomes structurally unstable. Except for the interest-rate spread variable, the Ashley test suggests that all other determinants of U.S. real M2 demand display significant unstable behavior at least at the ten percent level. Further support for the importance of including real wages into the M2 demand equation comes from the Durbin-m and the Breusch-Godfrey TABULAR DATA OMITTED tests. These tests reveal significant autocorrelation in the absence of real wages (Durbin-m = 2.23; BG(8) = 16.14; BG(10) = 17.14; BG(12) = 19.15). Of course, significant autocorrelation may be the result of an omission of important variables.

These various tests then convincingly show that without real wages, the U.S. real M2 demand equation exhibits serious problems and becomes structurally unstable. Indeed, Mehra |57~, Baum and Furno |2~ and Emery |21~ have reported instability evidence for conventional U.S. real M2 demand equations, and Judd and Trehan |48~ have also expressed similar doubts. Interestingly, none of these recent studies of the U.S. M2 demand equation considered the role of the time factor (wages) in the instability debate. Yet, results reported in this paper consistently suggest that some of the instabilities and statistical difficulties that have apparently plagued the U.S. real M2 demand equation may have resulted from ignoring the value of time in the underlying portfolio behavior. VI. Concluding Remarks

The primary objective of this paper has been to explore the empirical validity of the value-of-time hypothesis for the U.S. M2 demand relationship. Also examined is the empirical importance of including money-growth volatility (Friedman) and interest-rate variability (Tobin) as additional regressors in the U.S. M2 demand equation. In so doing, the emphasis has been on the stability implication for the estimated money demand equation.

The empirical results from the quarterly U.S. data (1963:1-1991:4) strongly support the value-of-time hypothesis. The results show that the wage variable, as a proxy for the value of time, exerts a highly significant positive impact upon money demand. Perhaps more critically, the overall money demand equation requires the inclusion of the wage variable in order to exhibit the desired property of structural stability. Without such a variable, the equation appears seriously misspecified and structurally unstable according to a battery of alternative tests. On the other hand, this paper finds little empirical evidence that money growth volatility and interest rate variability have significantly influenced M2 demand.

It can thus be argued that policy-makers should take into account the value of time, e.g., movements in real wages, when setting their targets for the M2 aggregate. In contrast, money growth volatility and interest rate variability may be safely ignored in the M2 targeting process.(25)

1. As Dutton and Gramm |20~ pointed out, the wage rate could measure leisure since in equilibrium the marginal valuation of one unit of leisure is equal to the wage rate. A similar view can be found in Karni |49~.

2. A similar model was tested with international data by Chowdhury |11~, and Shams |66~.

3. Evidence reported by Deacon and Sonstelie |14~ supports the use of the wage rate as a proxy for the value of time.

4. The value of time hypothesis requires the use of consumers' expenditure as the scale variable instead of the more traditional real GNP measure. Support for this view comes from Mankiw and Summers |54~ and Dowd |19~.

5. On this, see Boughton |6~, and Maddala |53~. Among the many restrictions imposed by the Koyck-lag process is the assumption of identical adjustment elasticities; geometrically declining paths of reactions; and non-negative lag weights. As Griliches |38~ demonstrated, none of these assumptions has any basis in theory.

6. The exception is data on the own rate of return on money which came from Hetzel |44~ for the period 1963:1-1989:2. Data over 1989:3-1991:4 were extracted using Hetzel's methodology.

7. The Almon procedure is of course not without drawbacks as Thomas |73~ pointed out. Several alternative lag schemes have been proposed and used in applied econometric literature, including the Jorgenson (rational) distributed lag and the Akaike FPE criterion.

8. The coefficient of the lagged dependent variable (in second-differences) significantly exceeds minus one in absolute value (= -1.73, t = 2.43). Miller |59~ found similar evidence.

9. This "normalization" choice in fact represents one major problem with the Engle-Yoo approach. As Dickey, Jansen and Thornton |16~ noted, the test results are known to be highly sensitive to such choice.

10. For a concise but informative account of the Johansen approach, see Johansen and Juselius |46~ and Dickey, Jansen, and Thornton |16~.

11. According to Johansen and Juselius |46~, relative to the trace test we should expect "the maximum eigenvalue test to produce more clear cut results," page 19.

12. Detailed test results from the Johansen procedure are not shown here to conserve space, but available from the authors upon request.

13. Results from the White |79~ test suggest absence of significant simultaneity bias (F = 1.01). Consequently, and following Cooley and LeRoy |12~ and Maddala |53~, we use the OLS procedure to estimate the model. Indeed, as an indirect evidence for the appropriateness of the OLS, an instrumental variables approach produced very similar estimates which are available upon request. 14. Dezhbakhsh |15~ reports evidence supporting the use of Durbin-m over alternative testing procedures in dynamic linear models.

15. On this, see Johnston |47~.

16. The variable, however, appears with the correct (positive) sign.

17. Note that since inflation enters the money demand equation, it can be argued that interest rates (or their spread) reflect real rates.

18. As Garner |31~ noted, such a view is not unanimously held. Some theoretical and empirical studies |e.g., Boonekamp |51~ have suggested a positive impact of inflation uncertainty on money demand.

19. The only notable difference in the two sets of coefficient estimates relates to the size of the impact of money growth volatility.

20. The Durbin-m test indicates significant autocorrelation at the 5 percent level, while the Breusch-Godfrey procedure (at various orders) suggests significant autocorrelation at the 10 percent level.

21. We have also reestimated the lag structure of the basic equation by means of the Akaike FPE criterion. Although the implied lag structure is generally shorter, the results are nevertheless very consistent with those reported in Table II particularly in regards to the signs obtained for the summed coefficients and their statistical significance (or insignificance).

22. This has been dubbed by Goldfeld |35~ as "The Case of the Missing Money." In tact, breaking the sample at 1973 for stability testing has become standard practice. See Miller |59~.

23. Some work, e.g., Mehra |55~, implies that the October 1979 shift in the Fed operating strategy began to exert pronounced effects only by late 1979 and early 1980. Thus, the fourth (rather than the third) quarter of 1979 appears a more appropriate date for testing parameter stability.

24. Several researchers, e.g., Roth |63~, Mehra |55~, and Siklos |67~, have pointed out the importance of these dates for testing the instability of U.S. money demand. It should be noted that other alternative breaking dates yielded very similar stability results.

25. The results further imply that M2 demand may not work as a channel transmitting the effects--if any--of money growth volatility and interest rate variability to the real side of the economy and/or inflation. Evans |24~ and Tatom |71; 72~ have argued that these factors, perhaps through other channels, may still have important effects on real output and prices. See Garner |31~.

References

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16. Dickey, David A., David W. Jansen, and Daniel L. Thornton, "A Primer on Cointegration with an Application to Money and Income." Federal Reserve Bank of St. Louis, Review, March/April 1991, 58-78.

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29. Friedman, Milton, "Lessons from the 1979-1982 Monetary Policy Experiments." American Economic Review, Papers and Proceedings, May 1984, 397-400. 30. Furlong, Fredrick T. and Bharat Trehan, "Interpreting Recent Money Growth." Federal Reserve Bank of San Francisco, Weekly Letter, September 28, 1990, 1-3. 31. Garner, C. Alan, "Does Interest Rates Volatility Affect Money Demand?" Federal Reserve Bank of Kansas City, Economic Review, January 1986, 25-37.

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42. Hallman, Jeffrey J., Richard D. Porter, and David H. Small, "Is the Price Level Tied to the M2 Monetary Aggregate in the Long Run?" American Economic Review, September 1991, 841-58.

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45. Johansen, Soren, "Statistical Analysis of Cointegration Vectors." Journal of Economic Dynamics and Control, June/September 1988, 231-54.

46. ----- and Katrina Juselius, "Maximum Likelihood Estimation and Inference on Cointegration-with Application to the Demand for Money." Oxford Bulletin of Economics and Statistics, May 1990, 169-210.

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56. -----, "Velocity and the Variability of Money Growth: Evidence from Granger Causality Tests." Journal of Money, Credit and Banking, May 1989 a, 262-66. 57. -----, "Some Further Results on the Source of Shift in MI Demand in the 1980's." Federal Reserve Bank of Richmond, Economic Review, September/October 1989 b, 3-13.

58. -----, "An Error-Correction Model of U.S. M2 Demand." Federal Reserve Bank of Richmond, Economic Review, May/June 1991, 3-12.

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@ @

Much controversy has surrounded the U.S. money demand relationship, or more precisely, its stability. This is rather disappointing in light of the central importance of money demand estimates for the formulation and implementation of effective monetary policy and for almost all theories of aggregate economic activity. To date, the vast majority of alternative money demand specifications have generally failed to produce a stable money demand function throughout the 1970s and 1980s.

A common weakness in most previous studies in this area is their total neglect of the "value-of-time" hypothesis when modelling money demand. Yet, some early work due to Dutton and Gramm |20~, and Karni |49; 50~ have provided some empirical evidence that a variable representing the value of time (like the wage rate) should be included as an additional argument in the money demand function. Dutton and Gramm |20, 652~ note "if the use of money saves transactions time, it increases the amount of leisure. This suggests an additional determinant of the demand for money, the consumer's valuation of time, i.e., the wage rate."(1) Indeed, the underlying thesis of Baumol's famed money demand approach is that the holding of money saves transaction time, the cost of which varies with the value of time. As Laidler |52~ and Dowd |19~ pointed out, failure to allow for the role of the wage rate in the "standard" transactions models of money demand could result in serious misspecification. It is, of course, conceivable that such misspecification may have caused instability in the U.S. money demand function. Ironically, none of the previous studies that did examine the effect of the time factor on money demand, including Dowd for the U.K., pays any attention to its implication for the stability property of the estimated function. Such issue, however, forms the basis of this study. Besides the wage rate as a proxy for the value of time, the U.S. money demand model estimated here further allows for the potential effects of financial uncertainty as measured by the volatility of money growth and the variability of interest rates. Previous theorizing |e.g., Friedman |29~ and Tobin |76~~ has suggested that both variables may exert important effects on money demand. Moreover, the paper addresses the co-integratedness of the variables and estimates the implied error-correction model. The rest of the paper is organized as follows. Section II briefly discusses the issues involved, while section III presents the model. Section IV reports the empirical results and section V analyzes their implications for the stability debate. Section VI offers some concluding remarks.

II. Some Issues

Dowd presents a lucid theoretical rationale for the value-of-time hypothesis based on utility maximization and thus needs not be repeated here. As a result of the optimization process, Dowd estimates for the U.K. economy a money demand equation comprising the wage rate, interest rates, and consumers expenditure as regressors. His estimates show that the wage coefficient is positive as expected and statistically significant with an elasticity of about one half. Earlier, Dutton and Gramm |20~ and Karni |49; 50~ reported similar findings for the U.S. using annual data. Further empirical support for the effect of the wage rate on the U.S. money demand has also come from the work of Diewert |17~, Philps |60~, and Dotsey |18~. As we mentioned earlier, despite this apparent strong support (both theoretical and empirical) for the value-of-time hypothesis, the implication of this hypothesis for the recent instability controversy surrounding the U.S. money demand has been surprisingly overlooked. However, a thoroughly examined hypothesis is the Friedman |29~ proposition that the U.S. money demand instability of the 1980s is caused by higher volatility of money growth in those years. Friedman theorizes that the increased volatility of money growth--following the 1979 change in the Federal Reserve operating strategy--has increased the degree of perceived uncertainty, leading to decreased money velocity since 1982. The empirical evidence on the Friedman hypothesis has been mixed. For example, Hall and Noble |41~, Fisher |27~, and Fisher and Serlitis |28~ have reported results in favor of the Friedman contention. On the other hand, Brocato and Smith |9~, and Mehra |55; 56; 57~ found results inconsistent with that contention. It should be noted that all these empirical tests of the Friedman hypothesis appear plagued with possible omission-of-variables bias for they use the restrictive bivariate framework in which money growth volatility is the only regressor in the demand for money (velocity) equation.(2) Clearly, a more credible test of the Friedman hypothesis should take into account the effects of other determinants of money demand. In the words of Mehra |56, 265~, "it may be necessary to reexamine the role of volatility in a more general framework that controls (for) the influence of other factors on velocity." On the other hand, drawing on Tobin's |76~ theoretical model, studies by Slovin and Sushka |69~, and Garner |31~ have argued that interest-rate variability can influence money demand. They contend that risk-averse asset holders would seek greater liquidity in response to heightened interest-rate risk.

III. The Model

Based on the preceding discussion, the money demand equation may take the following form:

ln |m*.sub.t~ = ||Alpha~.sub.0~ + b |center dot~ ln |w.sub.t~ + c |center dot~ v|(M).sub.t~ + d |center dot~ v|(R).sub.t~ + |Z.sub.t~|Omega~, (1)

where |m*.sub.t~ denotes desired real money balances, |w.sub.t~ is the real wage rate representing the value of time,(3) v|(M).sub.t~ is the volatility of money growth, v(R) is the variability of interest rates, |Omega~ is a column vector of parameters associated with a row vector of variables, Z, comprising three traditional regressors (real consumers expenditure (x), the interest rate spread defined as market interest rates minus the own rate of return on money (r - rm), and the inflation rate (|Pi~)).(4) Because the underlying theory is micro in nature and as such refers to individual behavior, the aggregate variables (money and consumers expenditures) are expressed as per capita figures. Note that v |(M).sub.t~ and v(R) are not expressed in natural logarithms because they assume negative values over some quarters for which the logarithms are undefined. To make equation (1) estimatable, we need to replace desired (unobservable) money demand with actual (observable) levels. This is done by allowing for a less-than-immediate adjustment to desired money demand. A procedure that has gained popularity particularly in money demand literature is the simple Koyck-lag mechanism whereby the entire adjustment is represented by adding a lagged dependent variable (nominal or real) as a regressor. However, several researchers have criticized this Koyck-lag structure due to its several restrictive assumptions.(5) Consequently, recent studies have increasingly employed other more flexible lag structures when estimating money demand equations.

Therefore, equation (1) can be rewritten as:

ln |Delta~|m.sub.t~ = ||Alpha~.sub.0~ + |summation of~ |b.sub.i~ |Delta~ ln |w.sub.t-i~ where i = 0 to |n.sub.1~ + |summation of~ |c.sub.i~|Delta~v|(M).sub.t-i~ where i = 0 to |n.sub.2~ + |summation of~ |d.sub.i~|Delta~v|(R).sub.t-i~ where i = 0 to |n.sub.3~

+ |summation of~ |f.sub.i~|Delta~ ln |x.sub.t-i~ where i = 0 to |n.sub.4~ + |summation of~ |g.sub.i~|Delta~ ln |(r - rm).sub.t-i~ where i = 0 to |n.sub.5~ + |summation of~ |h.sub.i~|Delta~||Pi~.sub.t-i~ + |e.sub.t~ |where~ i = 0 to |n.sub.6~ (2)

where |n.sub.j~ (j = 1,2,...6) are the various lags on the regressors; m is the money stock deflated by the implicit GDP deflator, and the resultant real money stock variable is divided by population to obtain the corresponding per capita figures; w is real wages measured by the index of average hourly earnings deflated by the implicit GDP deflator; v(M) is the volatility of M1 stock measured, as in Mehra |55~ and others, by the eight-quarter moving average of the standard deviation of quarterly M1 growth; v(R) is the variability of interest rates defined, as in Evans |24~, by the eight-quarter moving average of the standard deviation of Moody's AAA corporate rate; x is per capita consumers' expenditure; (r - rm) is the interest rate spread where r is the 4-6 month commercial paper rate and rm is the own rate of return on money defined, as in Hetzel |44~, by the weighted-average of the explicit rate of interest paid on the components of money; |Pi~ is the inflation rate measured by the percentage change in the implicit GDP deflator; and e is a white-noise error term assumed as in the usual fashion to be serially-uncorrelated with zero-mean and a constant variance. The money demand equation is estimated over the quarterly period 1963:1-1991:4 (after adjustment for all lags). All time series (with one exception)(6) are obtained from the Citibank data tape. The underlying money demand theory suggests the following a priori sign restrictions for the summed coefficients:

|summation of~ |b.sub.i~, |summation of~ |c.sub.i~, |summation of~ |d.sub.i~, |summation of~ |f.sub.i~ |is greater than~ 0; and |summation of~ |g.sub.i~, |summation of~ |h.sub.i~ |is less than~ 0.

Before turning to the empirical analysis, three comments pertaining to equation (2) bear emphasis. First, the distributed-lag on each variable is estimated by means of the Almon procedure which has been extensively employed in applied literature.(7) Following Schmidt and Waud |65~, Sims |68~, and Thornton and Batten |75~, we used the Almon procedure here without imposing the endpoint constraints since they lack any theoretical basis and, moreover, could yield biased results. In addition, we do not impose the usual implicit restriction that the underlying lag weights are all non-negative. This latter additional restriction is often imposed when researchers |e.g., Goldfeld |34~, and Berman |31~~ choose the lag length on each explanatory variable using a priori expectations regarding the signs of the variables. Clearly, this practice presupposes that the weights are all positive. Following Schmidt and Waud |65~, and Giles and Smith |33~, the lag structure on each variable (lag length and degree of polynomial) is determined by Theil's minimum residual-variance criterion.

The second comment relates to the particular definition of money stock used in the empirical analysis. Compared to alternative aggregates, a growing number of studies have paid increasing attention to M2 as a more appropriate monetary target |e.g., Wenninger |78~, Hetzel |44~, Mehra |55; 56; 57; 58~, Roberds |61~, Hafer and Jansen |39~, Hallman, Porter and Small |42~, and Thornton |74~~. Indeed, since the mid 1980s, the Fed has dropped M1 from its list of intermediate policy aggregates, making M2 the primary policy target. The Fed has not set target ranges for the M1 aggregate in recent years, "largely because movements in this aggregate no longer track movements in income and prices very well" (Furlong and Trehan |30, 1~). Recent work |e.g., Mehra |55; 57~~ has also suggested that the M1 demand relationship is inherently unstable casting doubt on its usefulness for policy analysis. A further rationale for our focus on M2 can be distilled from the work of Mehra |55~ and Cox and Rosenbloom |13~. Their analyses imply that broader monetary aggregates like M2 provide a more credible test of the Friedman hypothesis since M2 can internalize deregulation-induced substitutions among the aggregate components. All in all, then, the behavior of the broad M2 aggregate appears worthy of special attention. Thirdly, most recent studies of money demand have employed first-differences to avoid the spurious regression phenomenon (see Granger and Newbold |37~). However, Engle and Granger |22~ have shown that a model estimated using differenced data will be misspecified if the variables are cointegrated and the cointegrating relationship is ignored. Indeed, as Boughton |7~ pointed out, overlooking the cointegrating relationship may explain some of the apparent instability TABULAR DATA OMITTED of money demand equations. Thus, testing for cointegration among the variables seems to be necessary if possible biases were to be avoided.

The cointegration tests can be implemented using the two-step procedure described in Engle and Yoo |23~. In the first step, we test for nonstationarity (presence of unit roots) in each of the variables using levels (of natural logarithms) and first-differences. Results from the Dickey-Fuller (DF) and the augmented DF (ADF) tests are displayed in Table I.

For the levels of all series, both tests could not reject the null hypothesis of non-stationarity at the 5 percent level of significance. However, with first-differences, each series (with one minor exception) clearly indicates rejection of the null hypothesis at the 5 percent level or better. The only exception is that for m (real M2). While the DF and the ADF (with 2 lags) reject non-stationarity for the first-difference of real M2 at the 5 percent level, the ADF test (with 4 lags) does so only at the 10 percent level (test statistic = -3.30). Nonetheless, first-differences seem adequate for real M2 as well since second-differencing the variable appears to entail over-differencing.(8) These unit-root tests consistently suggest that all series in the real M2 demand equation are stationary if used in first-differences. Hence, all variables in equation (2) are cast in first-differences to achieve stationarity. The second step in the Engle-Yoo methodology is to examine whether the non-stationary variables (in levels) are cointegrated. If they are, the use of first-differences without due adjustments might introduce biases since it would filter out low-frequency (long-run) information |see Hendry |43~, Engle and Granger |22~, and Miller |59~~. Thus, cointegrating regressions of (non-stationary) variables are estimated and their residuals are tested for the presence of unit roots.

When estimating cointegrating regressions, a choice must be made regarding which series is used as the left-hand-side conditioning variable.(9) Since cointegrating parameters may not be unique, we follow Hall |40~ and Miller |59~ and examine all possible cointegrating regressions and choose that which yields the highest adjusted |R.sup.2~. We found that the logarithm of real M2 displays the highest adjusted |R.sup.2~ (=0.9711). Consequently, the logarithm of real M2 was chosen as the left-hand-side conditioning variable.

The residuals from the cointegrating regression are then tested for non-stationarity. Both DF and ADF tests reject the null hypothesis of non-cointegration (non-stationarity) in the real M2 regression. Specifically, the test statistics are -6.60 for the DF test, -4.32 for the ADF test (with 2 lags); and -3.29 for the ADF test (with 4 lags). Except for the ADF test with 4 lags, the other two tests soundly reject the null hypothesis at least at the 5 percent level. Higher lag orders in the ADF confirm this conclusion. For example, test statistics for 8 and 10 lags are respectively -4.06 and -3.71, both of which are significant at least at the 10 percent level. This finding of cointegration in the real M2 regression accords with the evidence reported recently by Hafer and Jansen |39~, Mehra |58~, and Miller |59~ among others. An attractive alternative to the above Engle-Yoo (single equation) approach for testing cointegrating relationships is the Johansen |45~ multivariate technique which has gained a lot of popularity in recent applied literature. As Dickey, Jansen, and Thornton |16~ pointed out, the Johansen approach is particularly promising for it is based on the well-established likelihood ratio principle. Furthermore, the Johansen method is not subject to biases due to arbitrary normalization choices since all of the variables are jointly endogenized in the testing process. Monte Carlo evidence reported by Gonzalo |36~ also supports the relative power of Johansen's methodology over other alternative (single- and multivariate) techniques.

The Johansen test utilizes two likelihood ratio test statistics for the number of cointegrating vectors; namely the trace and the maximum eigenvalue statistics.(10) In our case, both statistics indicate the presence of one cointegrating relationship. Specifically, for the trace test, the hypothesis that the number of cointegrating vectors (k) is less than or equal to 1, 2...6 cannot be rejected at the 5 percent level of significance. However, the hypothesis that k = 0 is easily rejected (test statistic = 173.22, compared to the 5 percent critical value of 131.70). This finding of one cointegrating relationship is, perhaps more clearly confirmed,(11) by the maximum eigenvalue test which suggests rejection of the null hypothesis k = 0 in favor of the explicit alternative hypothesis that k = 1 (test statistic = 48.24, compared to the 5 percent critical value of 46.45).(12)

In sum, both the Engle-Yoo and the Johansen procedures suggest that real M2 has a long-run relationship with the determinants proposed in the model. To avoid the loss of potentially relevant (low-frequency) information, the error-correction (one-lagged error) term generated from the Johansen multivariate procedure is included as another regressor in the money demand equation (2). The resultant error-correction model integrates the short-run dynamics in the long-run M2 demand relationship.

IV. Empirical Results

To achieve stationarity and to reintroduce the low-frequency (long-run) information into the M2 real money demand equation, the basic equation (2) underwent two adjustments: all variables were cast in first-differences, and the error-correction term was added as another regressor. Over the quarterly period 1963:1-1991:4, the unconstrained Almon/Theil procedure yielded the empirical estimates displayed in Table II.(13) According to the relatively high value of adjusted |R.sup.2~ (=0.79), the model fits the data quite well, considering that the dependent variable is cast in first-differences. Plots of actual and predicted real M2 demand (not shown here) indicate that the error-correction model of U.S. M2 demand adequately traces the behavior of real money demand and its turning points. As is clear from the table, all of the diagnostic tests support the statistical appropriateness of the estimated M2 demand equation. In particular, for examining serial correlation of the residuals, we applied the following tests: the Durbin-Watson and the Durbin-m tests for first-order autocorrelation;(14) the Breusch-Godfrey test of different autoregressive and moving-average order processes;(15) and the Geary nonparametric test for a general (unspecified) autocorrelation process. These tests uniformly fail to reject the null hypothesis of no autocorrelation in the residuals. Moreover, heteroscedasticity does not seem to be a problem according to the Glejser test. The Plosser-Schwert-White Differencing test could not reject the hypothesis that the estimated equation is correctly specified, and the Ramsey RESET test concurs with that verdict and reveals no serious omission of variables. Furthermore, the White test indicates absence of significant simultaneity bias in the estimates. Having provided some evidence supporting the adequacy of the estimated error-correction model of U.S. M2 demand equation, the following observations can be made regarding the obtained parameter estimates.

We should first note that the one-lagged error-correction term appears with a statistically significant coefficient and displays the appropriate (negative) sign, a finding that accords well with some previous studies |59~. This implies that overlooking the cointegratedness of the variables would have introduced misspecification in the underlying dynamic structure. It should also be noted that literature on cointegrated systems suggests that only one-lagged EC is needed to represent the cointegrating scheme. Indeed, higher powers of EC were included but proved statistically insignificant.

Secondly, and perhaps more importantly, Table II shows that the summed coefficient of real wages (long-run elasticity) is positive as hypothesized by the value-of-time theory and is statistically significant at better than the one percent level. Our estimate of 0.93 for the real wage elasticity is within the range of estimates reported earlier for the U.S. by Dutton and Gramm |20~ of about 1.02; by Karni |49~ of 1.00; and by Dotsey |18~ of about 0.72. Therefore, our empirical evidence corroborates previous findings supportive of the value-of-time hypothesis. Further discussion of this important finding is provided below.

Thirdly, the empirical results do not lend strong support to the Friedman hypothesis that money growth volatility is a major culprit behind changes in the demand for M2. Thus, similar to the findings of Mehra |55; 56; 57~, our results reveal no significant impact of the volatility of money growth on real M2 demand at the conventional significance levels.(16)

TABULAR DATA OMITTED

Fourthly, the coefficient on interest rate variability is statistically insignificant at the 5 percent level, a finding that accords well with the evidence reported by Garner |31~ who discussed various reasons for the negligible effect of interest rate variability on money demand. Interestingly, the sign of the summed coefficient of interest-rate variability is negative. One explanation for this negative (albeit insignificant) impact may be found in Evan's |24~ and Tatom's |72~ argument that increased interest-rate risk exerts an adverse effect on aggregate production which in terms decreases the transaction demand for money. Note also that v(R) may instead reflect inflation uncertainty.(17) Higher inflation uncertainty makes financial assets (e.g., M2) riskier to hold, in turn inducing some investors to reallocate their portfolio away from them in favor of real assets.(18) This view is consistent with Klein's |51~ choice-theoretic (utility maximization) approach. In Klein's model, the value of money is primarily derived from the flow of services it renders the asset holders. Inflation uncertainty can thus be viewed as a negative technological change. Empirical support for Klein's theoretical proposition has come, for example, from Blejer |4~ and Smirlock |70~ who report some evidence showing a negative relationship between inflation uncertainty and money demand. Finally, the results in Table II indicate that the conventional variables of real consumers expenditures, interest rate spread and inflation are all correctly signed, with highly significant summed coefficients. The long-run elasticity estimates are also within the range of recent U.S. estimates reported, for example, by Roley |62~, Garner |31~, Mehra |56~, and Emery |21~.

Taken together, the model of Table II fits the U.S. quarterly data quite well, and the estimated real M2 demand equation appears correctly specified. Most importantly, the empirical results lend strong support to the value-of-time hypothesis in that real wages exert significant positive impact upon U.S. real M2 demand. Further, and perhaps more substantive, the evidence indicates that real wages must be included in the analysis if the estimated money demand equation is to exhibit the desired property of structural stability.

Before turning to the stability implications of the estimated model, a comment about the lag structure and method of estimation appears in order. It seems prudent to check whether the results reported in Table II are invariant to both of these important aspects of the estimation. A popular procedure employed in a growing number of studies is to reestimate the basic model by means of the Hendry general-to-specific modelling strategy which eliminates lags with insignificant parameters |32~. Among others, the virtue of this approach is that it uses unconstrained Ordinary Least-Squares (OLS) and, as such, can assess the robustness of the Almon results. Moreover, by eliminating insignificant lagged coefficients (at the 10 percent significance level), the approach is parsimonious guarding against possible over-parameterization. Money demand estimates from the Hendry strategy are displayed in Table III. As seen from the table, the coefficients estimates are very similar to those reported in Table II which are based on the Almon procedure.(19) It is, of course, encouraging that the results are not particularly sensitive to the method of estimation nor are they to the problem of over-parameterization. Observe, however, that the Hendry estimates of Table III seem to suffer from significant autocorrelation according to the various diagnostic tests.(20) Owing to this possible misspecification, and in light of the various difficulties with Hendry's strategy discussed recently by Boughton and Tavlas |8~, our analysis of the stability issue of real M2 demand equation will primarily focus on the Almon estimates reported in Table II.(21)

TABULAR DATA OMITTED

V. Stability Implications

Structural stability of the estimated money demand equation is vital for drawing meaningful policy inferences. As is well-known, the money demand equation represents the link between monetary policy and the rest of the macroeconomy. In order to adequately predict the impact of a given change in money stock on key macro variables, the underlying money demand equation must remain stable over time.

The task of this section is two-fold. We examine first the stability property of the basic equation (2) using several stability tests. As Boughton |6~ pointed out, when testing for money demand instability, several testing procedures should be applied since each procedure examines a particular type of structural instability. Thus, the following three testing procedures are used: the Chow test |10~; the Farley-Hinich test |25~; and the Ashley |1~ Stabilogram test. The second task of this section is to investigate the contribution of real wage to the stability property of the estimated real M2 demand equation. The Chow test is perhaps the most widely used procedure to assess the structural stability of estimated equations. However, as Toyoda |77~, and Schmidt and Sickles |64~ demonstrated, the Chow test presupposes homoscedastic error terms. Thus, it is advisable that heteroscedasticity be tested before applying the Chow test. As Table II shows, the Glejser test did not suggest the presence of significant heteroscedasticity which then permits the application of the Chow test. Unlike the Chow test, the alternative Farley-Hinich procedure does not require splitting the sample period at a particular date. As Farley, Hinich and McGuire |26~ noted, the Farley-Hinich procedure is a robust test for a gradual (continuous) drift in the parameters, in contrast to the single-point shift tested by the Chow procedure. Finally, the Ashley Stabilogram (stab) procedure is a flexible and perhaps more rigorous test against parameter instability of various types. Ashley |1~ reported Monte Carlo simulation evidence supporting the empirical power of the Stab test over a number of standard stability techniques, including the Chow test. Before applying the Chow test, we need to determine the appropriate breaking point(s) at which the money demand equation may have hypothetically shifted. Extensive empirical literature that appeared in the mid-1970s and early 1980s have suggested the possibility of a structural shift in the U.S. money demand relationship around the start of 1973.(22) Another break could have occurred as the result of the well-known October 1979 change in the Federal Reserve operating procedure.(23) Finally, money demand may have shifted as a result of the nation-wide introduction of interest-bearing checkable deposits and other financial deregulations in the early 1980.(24) Consequently, three breaking dates seem appropriate for testing money demand stability; namely, 1973:1; 1979:4; and 1980:4.

Table IV-A reports the results from the alternative stability tests of equation (2). The evidence from all three tests is remarkably consistent and uniformly suggests that the estimated U.S. real M2 demand equation of Table II is structurally stable throughout the estimation period (1963:1-1991:4).

An important issue relates to the contribution of the time-value variable (real wages) to the stability property of the estimated money demand equation. To measure that, the same equation was reestimated and tested after dropping the wage variable. Table IV-B reports the obtained statistics of the three stability tests. The Chow test persistently indicates that the money demand equation is rendered structurally unstable at least at the five percent level, a verdict that is corroborated by the Farley-Hinich test which rejects stability at the stronger one percent level. The Ashley test too reinforces the above conclusion and shows that, once the wage variable is omitted from the equation, it becomes structurally unstable. Except for the interest-rate spread variable, the Ashley test suggests that all other determinants of U.S. real M2 demand display significant unstable behavior at least at the ten percent level. Further support for the importance of including real wages into the M2 demand equation comes from the Durbin-m and the Breusch-Godfrey TABULAR DATA OMITTED tests. These tests reveal significant autocorrelation in the absence of real wages (Durbin-m = 2.23; BG(8) = 16.14; BG(10) = 17.14; BG(12) = 19.15). Of course, significant autocorrelation may be the result of an omission of important variables.

These various tests then convincingly show that without real wages, the U.S. real M2 demand equation exhibits serious problems and becomes structurally unstable. Indeed, Mehra |57~, Baum and Furno |2~ and Emery |21~ have reported instability evidence for conventional U.S. real M2 demand equations, and Judd and Trehan |48~ have also expressed similar doubts. Interestingly, none of these recent studies of the U.S. M2 demand equation considered the role of the time factor (wages) in the instability debate. Yet, results reported in this paper consistently suggest that some of the instabilities and statistical difficulties that have apparently plagued the U.S. real M2 demand equation may have resulted from ignoring the value of time in the underlying portfolio behavior. VI. Concluding Remarks

The primary objective of this paper has been to explore the empirical validity of the value-of-time hypothesis for the U.S. M2 demand relationship. Also examined is the empirical importance of including money-growth volatility (Friedman) and interest-rate variability (Tobin) as additional regressors in the U.S. M2 demand equation. In so doing, the emphasis has been on the stability implication for the estimated money demand equation.

The empirical results from the quarterly U.S. data (1963:1-1991:4) strongly support the value-of-time hypothesis. The results show that the wage variable, as a proxy for the value of time, exerts a highly significant positive impact upon money demand. Perhaps more critically, the overall money demand equation requires the inclusion of the wage variable in order to exhibit the desired property of structural stability. Without such a variable, the equation appears seriously misspecified and structurally unstable according to a battery of alternative tests. On the other hand, this paper finds little empirical evidence that money growth volatility and interest rate variability have significantly influenced M2 demand.

It can thus be argued that policy-makers should take into account the value of time, e.g., movements in real wages, when setting their targets for the M2 aggregate. In contrast, money growth volatility and interest rate variability may be safely ignored in the M2 targeting process.(25)

1. As Dutton and Gramm |20~ pointed out, the wage rate could measure leisure since in equilibrium the marginal valuation of one unit of leisure is equal to the wage rate. A similar view can be found in Karni |49~.

2. A similar model was tested with international data by Chowdhury |11~, and Shams |66~.

3. Evidence reported by Deacon and Sonstelie |14~ supports the use of the wage rate as a proxy for the value of time.

4. The value of time hypothesis requires the use of consumers' expenditure as the scale variable instead of the more traditional real GNP measure. Support for this view comes from Mankiw and Summers |54~ and Dowd |19~.

5. On this, see Boughton |6~, and Maddala |53~. Among the many restrictions imposed by the Koyck-lag process is the assumption of identical adjustment elasticities; geometrically declining paths of reactions; and non-negative lag weights. As Griliches |38~ demonstrated, none of these assumptions has any basis in theory.

6. The exception is data on the own rate of return on money which came from Hetzel |44~ for the period 1963:1-1989:2. Data over 1989:3-1991:4 were extracted using Hetzel's methodology.

7. The Almon procedure is of course not without drawbacks as Thomas |73~ pointed out. Several alternative lag schemes have been proposed and used in applied econometric literature, including the Jorgenson (rational) distributed lag and the Akaike FPE criterion.

8. The coefficient of the lagged dependent variable (in second-differences) significantly exceeds minus one in absolute value (= -1.73, t = 2.43). Miller |59~ found similar evidence.

9. This "normalization" choice in fact represents one major problem with the Engle-Yoo approach. As Dickey, Jansen and Thornton |16~ noted, the test results are known to be highly sensitive to such choice.

10. For a concise but informative account of the Johansen approach, see Johansen and Juselius |46~ and Dickey, Jansen, and Thornton |16~.

11. According to Johansen and Juselius |46~, relative to the trace test we should expect "the maximum eigenvalue test to produce more clear cut results," page 19.

12. Detailed test results from the Johansen procedure are not shown here to conserve space, but available from the authors upon request.

13. Results from the White |79~ test suggest absence of significant simultaneity bias (F = 1.01). Consequently, and following Cooley and LeRoy |12~ and Maddala |53~, we use the OLS procedure to estimate the model. Indeed, as an indirect evidence for the appropriateness of the OLS, an instrumental variables approach produced very similar estimates which are available upon request. 14. Dezhbakhsh |15~ reports evidence supporting the use of Durbin-m over alternative testing procedures in dynamic linear models.

15. On this, see Johnston |47~.

16. The variable, however, appears with the correct (positive) sign.

17. Note that since inflation enters the money demand equation, it can be argued that interest rates (or their spread) reflect real rates.

18. As Garner |31~ noted, such a view is not unanimously held. Some theoretical and empirical studies |e.g., Boonekamp |51~ have suggested a positive impact of inflation uncertainty on money demand.

19. The only notable difference in the two sets of coefficient estimates relates to the size of the impact of money growth volatility.

20. The Durbin-m test indicates significant autocorrelation at the 5 percent level, while the Breusch-Godfrey procedure (at various orders) suggests significant autocorrelation at the 10 percent level.

21. We have also reestimated the lag structure of the basic equation by means of the Akaike FPE criterion. Although the implied lag structure is generally shorter, the results are nevertheless very consistent with those reported in Table II particularly in regards to the signs obtained for the summed coefficients and their statistical significance (or insignificance).

22. This has been dubbed by Goldfeld |35~ as "The Case of the Missing Money." In tact, breaking the sample at 1973 for stability testing has become standard practice. See Miller |59~.

23. Some work, e.g., Mehra |55~, implies that the October 1979 shift in the Fed operating strategy began to exert pronounced effects only by late 1979 and early 1980. Thus, the fourth (rather than the third) quarter of 1979 appears a more appropriate date for testing parameter stability.

24. Several researchers, e.g., Roth |63~, Mehra |55~, and Siklos |67~, have pointed out the importance of these dates for testing the instability of U.S. money demand. It should be noted that other alternative breaking dates yielded very similar stability results.

25. The results further imply that M2 demand may not work as a channel transmitting the effects--if any--of money growth volatility and interest rate variability to the real side of the economy and/or inflation. Evans |24~ and Tatom |71; 72~ have argued that these factors, perhaps through other channels, may still have important effects on real output and prices. See Garner |31~.

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Publication: | Southern Economic Journal |

Date: | Jan 1, 1994 |

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