THE TIME-VARYING PERFORMANCE OF THE LONG-RUN DEMAND FOR MONEY IN THE UNITED STATES.

HONDROYIANNIS, GEORGE ; SWAMY, P.A.V.B. ; TAVLAS, GEORGE S. 等

GEORGE S. TAVLAS [*]

This article investigates the issues of the stability and predictability and interestsensitivity of money demand over 1870-1997 Two different estimation methodologies are used-random coefficient (RC) modeling and vector error correction (VEC) modeling. The former procedure allows the profiles of the coefficients to be traced over time and relaxes several restrictions routinely imposed in applied work. The results indicate that different estimation methodologies using different data periods and frequencies yield estimates of some of the coefficients of the long-run demand for money that fall within a fairly narrow range. The results also suggest that specification errors have had an important influence on the time profile of the interest elasticity of money demand and that there is a tendency for the interest elasticity to decline in absolute value as interest rates decline. (JEL C20, E47)

I. INTRODUCTION

This article reinvestigates the issues of the stability, predictability, and interestsensitivity of the long-run demand for money in the United States. [1] The model used is closely based on that used by Friedman and Schwartz (1982). The data are also those of Friedman and Schwartz--extended by Bordo et al. (1997) and us to include more recent observations--while the data frequency is annual rather than averaged over business cycle phases (as used by Friedman and Schwartz). [2] Two very different methodologies are used to examine the issue of money-demand behavior--vector error correction (VEC) modeling and random coefficient (RC) modeling. The former approach is aimed at addressing problems of spurious correlation produced by integrated variables and dynamic misspecification due to inadequate lag structures and attempts to integrate short-run dynamics with departures from long-run equilibrium relationships. The underlying philosophy of this approach is the general-to-specific methodology popularized by Hend ry and his associates (e.g., Hendry and Ericsson [1991]; Ericsson et al. [1998]). The RC approach, developed by Swamy and Tinsley (1980), Swamy and Tavlas (1992, 1995, 2000), and Christou et al. (1996, 1998), is aimed at dealing with four major specification problems (discussed in section II) that often arise in econometric estimation. We use it to shed light on such issues as the stability and predictability of money demand and whether the demand for money was more interest sensitive during the 1930s Great Depression than in other periods (and, indeed, whether a liquidity trap existed in the 1930s).

The remainder of this article is divided into three sections. Section II discusses the basic money-demand specification, the data, and the VEC and RC estimation procedures. Section III presents the estimation results. Both the VEC and RC models are estimated using annual data covering 1870-1989 and postsample forecasts are generated over 1990-1997. The models are then reestimated over earlier periods to test how well they forecast in a variety of circumstances, including the Great Depression years. For as Goldfeld (1992, 623) put it in his survey of the money-demand literature: "Ultimately, of course, such models need to stand the forward test of time; that is, they need to continue to hold outside the period of estimation." [3] An aim of the article is to inquire whether two different empirical methodologies can provide a stable long-run demand for money function that can perform well in prediction. If so, this should provide some reassurance about the empirical properties of that function. Section IV concl udes.

II. THE MODEL, DATA, AND ESTIMATION PROCEDURES

A Basic Friedman-Schwartz Specification

Friedman and Schwartz (1982) posited the following money-demand function:

(1) ln([m.sub.t]) = [[alpha].sub.0] + [[alpha].sub.1][r.sub.t] + [[alpha].sub.2]ln([y.sub.t]) + [[alpha].sub.3][g.sub.yt] + [u.sub.t],

where [m.sub.t] is the stock of real money balances at time t, [r.sub.t] is an interest rate, [y.sub.t] is real income, [g.sub.yt] is the growth in nominal income, and [u.sub.t] is an error term. Money (M2) consists of the sums of currency held by the public plus adjusted deposits at all commercial banks, less large negotiable certificates of deposit since 1961. [4] Income is net national income. Both the money series and the income series are deflated by the implicit price deflator for net national product and are expressed in per-capita terms. [5] In what follows, two interest rate series are used (both used by Friedman and Schwartz)--the commercial paper rate on four- to six-months' bills to represent the short rate and the basic yield (ten years' maturity) on corporate bonds for the long rate; each of these interest rates is used in (two) alternate specifications of equation (1). Except for these two specifications, no other variables are entered in equation (1). One rationale for using [g.sub.y] is that, because it is "the sum of the rate of change in prices and the rate of change in output, and the rate of change of output is an estimate ... of the real yield [on physical assets], the rate of change of nominal income can be regarded as a better proxy than the rate of change of prices alone for the total nominal yield on physical assets" (Friedman and Schwartz (1982, 276, original italics). Another rationale for this variable is that in periods of regulated interest rates it captures the opportunity cost of holding money balances better than do interest rates, which, in such circumstances, may be subject to ceilings. Indeed, in periods of rising inflation with controlled interest rates, real assets are often viewed as desirable alternatives to holding money balances. Friedman and Schwartz also used several dummy variables (e.g., for depression and war years) that are not used in this paper, as the RC approach allows all the coefficients of equation (1) to change in every period. Friedman and Schwartz constru cted a series representing the own rate of return on money and subtracted it from the opportunity cost variables to derive the interest differentials; we did not perform this adjustment because an own rate series was not available to us. Their preferred specification used the differential between the commercial paper rate and the own rate. Their estimation period was 1867-1975.

VEC Estimation

An objective of VEC is to test for the existence of a long-run equilibrium, or cointegrating, relationship among the levels of the variables in equation (1). Some authors (e.g., Hafer and Jansen [1991]; Hoffman and Rasche [1991, 1996]; Hoffman et al. [1995]) have used cointegration to examine long-run elasticities in a money-demand relation, without great concern for the short-run dynamics. Often, however, if such a relationship is found to exist, it is embellished with lagged and unlagged differences of these variables and other stationary variables that economic theory may suggest as belonging in equation (1) in an attempt to capture the short-run dynamics of the long-run relationship. Standard methodology employs a three-step procedure (see, for example, Enders [1995, chapter 6]). First, the variables are tested for stationarity using Augmented Dickey-Fuller (ADF) tests. Of the five variables--real per-capita money balances, real per-capita income, the commercial paper rate, the yield on corporate bonds, and the rate of change in nominal income--the first four were found to contain a unit root, but became stationary on first differencing. The rate of change of nominal income was found to be stationary. The results of the ADF tests are not reported but are available from the authors. The ADF results are interpreted to imply that, in testing for cointegration, we should alternately use (1) [m.sub.t], [y.sub.t], and the commercial paper rate; and (2) [m.sub.t], [y.sub.t], and the yield on corporate bonds.

The second step in the standard procedure is to test for cointegration among the integrated variables. To do so, the Johansen (1991) maximum likelihood procedure is used. The Johansen procedure purportedly has several advantages over other methods (such as the Engle-Granger procedure). First, it tests for all the cointegrating vectors among the variables. Two test statistics are used to evaluate the number of cointegrating relationships: the trace test and the maximal eigenvalue test. With three variables, the Johansen procedure yields at most two cointegrating vectors. Second, it treats all the variables included in equation (1) as endogenous, thus avoiding an arbitrary assumption of exogeneity. Third, it provides a unified approach for estimating and testing cointegrating relations within the framework of a VEC model. Providing one or more cointegrating relationships exist, the third step involves the estimation of a VEC specification containing the cointegrating relationship(s), current and lagged first d ifferences of the variables in the cointegrating relationship(s), and any stationary variables thought to influence money demand (in this case, [g.sub.y]).

RC Estimation

Standard estimation procedures often impose a number of restrictions when applied to equations such as equation (1), including the following: (i) [[alpha].sub.0] [[alpha].sub.1], [[alpha].sub.2], and [[alpha].sub.3] are constants; (ii) excluded explanatory variables are proxied through the use of an error term, and, therefore, these excluded variables are assumed to have means equal to zero and to be mean independent of the included explanatory variables; (iii) the true functional form is known (whether linear or nonlinear); and (iv) the variables are not subject to measurement error.

Swamy and Tavlas (1995, 2000) and Chang et al. (2000) define (I) any variable or value that is not mismeasured is true and (II) any economic relationship with the correct functional form, without any omitted explanatory variable and without mismeasured variables is true. Using these definitions, we can specify a class of functions which is wide enough to cover the true money demand function (in the sense of definition [II]) as a member. To rewrite this class in a form that has the same explanatory variables as equation (1), we assume that explanatory variables that are in the true money demand function but excluded from equation (1) are related linearly or nonlinearly to the explanatory variables included in equation (1). This assumption is reasonable, given that economic variables are rarely if ever uncorrelated and may not be linearly related to each other. To account for measurement errors, we assume that each variable in equation (1) is the sum of the underlying true value and the appropriate measurement error. These assumptions imply that equation (1) does not correspond to the true money demand function unless it is changed to

(2) ln([m.sub.t]) = [[gamma].sub.0t] + [[gamma].sub.1t][r.sub.t] + [[gamma].sub.2t]ln([y.sub.t]) + [[gamma].sub.3t][g.sub.yt], t = 1, 2, . . . , T,

where the real-world interpretations of the coefficients follow from the derivation of equation (2): [[gamma].sub.0t] is the sum of three parts: (a) the intercept of the true money demand model, (b) the joint effect on the true value of ln([m.sub.t]) of the portions of excluded variables remaining after the effects of the true values of the included explanatory variables have been removed; and (c) the measurement error in ln([m.sub.t]). The coefficient [[gamma].sub.1t]( [[gamma].sub.2t] or [[gamma].sub.3t]) is also the sum of three parts: (a) a direct effect of the true value of [r.sub.t](ln[[y.sub.t]] or [g.sub.yt]) on the true value of ln([m.sub.t]), (b) a term capturing omitted-variables bias, and (c) a mismeasurement effect due to mismeasuring [r.sub.t](ln[[y.sub.t]] or [g.sub.yt],) (see Chang et al. [2000]). The direct effects provide economic explanations. An implication of these interpretations is that the explanatory variables of equation (2) are correlated with their coefficients. With these correlat ions, none of the explanatory variables is exogenous. The effects of such dynamic factors as technical change in the payments system and excluded lagged explanatory variables are captured in the omitted-variables bias component of each of the coefficients of equation (2). Consequently, equation (2) is a dynamic specification.

One question that needs to be answered before estimating equation (2) is that of parametrization: which features of equation (2) ought to be treated as constant parameters? Inconsistencies arise if this parametrization is not consistent with the real-world interpretations of [gamma]'S. To achieve consistency, the [gamma]'S are estimated using concomitants. A formal definition of concomitants is provided in Chang et al. (2000) and Swamy and Tavlas (2000). Intuitively, these may be viewed as variables that are not included in the equation used to estimate money demand, but help deal with the correlations between the [gamma]'S and the included explanatory variables ([r.sub.t], ln[[y.sub.t]], and [g.sub.yt]).

Assumption I. The coefficients of equation (2) are linear functions of p variables, called concomitants, including a constant term with added error terms, which may be contemporaneously and serially correlated. The error terms are mean independent of the concomitants.

Assumption II. The explanatory variables of equation (2) are independent of their coefficients' error terms, given any values of the concomitants.

Assumption II captures the idea that the explanatory variables of equation (2) can be independent of their coefficients conditional on the given values of concomitants even though they are not unconditionally independent of their coefficients. This property provides a useful procedure for consistently estimating the direct effects contained in the coefficients of equation (2). Under Assumptions I and II, equation (2) can be written as

(3) ln([m.sub.t]) = [[pi].sub.00][z.sub.0t] + [[[sigma].sup.p-1].sub.j=1][[pi].sub.0j][z.sup.jt] + [[pi].sub.10][r.sub.t][z.sub.0t] + [[[sigma].sup.p-1].sub.j=1][[pi].sub.1j][z.sup.jt][p.sub.t] + [[pi].sub.20]ln([y.sub.t])[z.sub.0t] + [[[sigma].sup.p-1].sub.j=1][[pi].sub.2j][z.sup.jt]ln([y.sub.t]) + [[pi].sub.30][g.sub.yt][z.sub.0t] + [[[sigma].sup.p-1].sub.j=1][[pi].sub.3j][z.sup.jt][g.sub.yt] + [[epsilon].sub.0t] + [[epsilon].sub.1t][r.sub.t] + [[epsilon].sub.2t]ln([y.sub.t]) + [[epsilon].sub.3t][g.sub.yt],

where the z's denote concomitants and the [epsilon]'S denote the error terms of the coefficients of equation (2). In our empirical work we set p = 3, [z.sub.0t] = 1 for all t, [z.sub.1t] = the short-term or long-term interest rate, and [z.sub.2t] = the inflation rate. This means that we use three concomitants to estimate the [gamma]'S. In equation (2) with [r.sub.t] = the long-term interest rate, the concomitants are [z.sub.0t], the short-term rate, and the inflation rate. In equation (2) with [r.sub.t] = the short-term interest rate, we use [z.sub.0t] the long rate, and the inflation rate as concomitants. We are attempting, therefore, to capture the direct effect contained in [[gamma].sub.1t] in, say, the equation that uses the short rate as a regressor, by using a linear function ([[pi].sub.10] + [[pi].sub.11][Z.sub.1t] of the long rate. The indirect and mismeasurement effects are captured by using a function ([[pi].sub.12][Z.sub.2t] + [[epsilon].sub.1t]) of the inflation rate and [[epsilon].sub.1t]. The me asures of direct effects contained in [[gamma].sub.2t] and [[gamma].sub.3t] are [[epsilon].sub.20] + [[epsilon].sub.21][Z.sub.1t] and [[epsilon].sub.30] + [[epsilon].sub.31][Z.sub.1t], respectively, and those of indirect and mismeasurement effects contained in [[gamma].sub.2t] and [[gamma].sub.3t] are [[pi].sub.22][Z.sub.2t] + [[epsilon].sub.2t] and, [[pi].sub.32][Z.sub.2t] + [[epsilon].sub.3t], respectively. The components of the coefficients of equation (2) can take different values in different phases of the business cycle. The demand for money may be lower in periods of contraction than in periods of expansion. Consequently, changes in the values of the included explanatory variables that occur during the peak of a business cycle may exhibit very different effects on money demand than the same changes that occur during the trough of a business cycle. If so, more accurate results can be obtained by taking changing conditions into account. For this reason, we use the inflation rate as a proxy for these chan ging conditions.

Note that equation (3) has four error terms, three of which are the products of [epsilon]'s and the included explanatory variables of equation (1). The sum of these four terms is both heteroscedastic and serially correlated. Under Assumptions I and II, the right-hand side of equation (3) with the last four terms suppressed gives the conditional expectation of the left-hand side variable as a nonlinear function of the conditioning variables. This conditional expectation is different from the right-hand side of equation (1) with [u.sub.t] suppressed. This result shows that the addition of a single error term to a mathematical equation and the exclusion of the interaction terms on the right-hand side equation (3) introduce inconsistencies in the usual situations where measurement errors and omitted-variable biases are present and the true functional forms are unknown. A computer program developed by Chang et al. (2000) is used to estimate equation (3).

The RC methodology is applicable to situations involving nonstationary data. A stochastic process is said to be stationary if the distribution of variables underlying the process is the same when displaced in time. The distributions implied by RC models, however, are nonstationary. This can be seen by noting that the conditional means and the conditional variances and covariances of the dependent variable implied by RC models vary over time (Swamy and Tavlas [2000]). The important point to observe is that equation (3) deals with those nonstationarities that are relevant to equation (2), which, for certain variations in the [gamma]'s, coincides with an actual economic relationship.

III. EMPIRICAL RESULTS

Table 1 summarizes the results of cointegration analysis among the three variables, the stock of real money balances, either of the two interest rates, and real income. Specifications VEC1 and VEC2 correspond to the use of the long-term interest rate and the short-term rate, respectively. To determine the lag lengths of the vector autoregressive (VAR) models of the two sets of three variables, three versions of system were initially estimated involving four lags, three lags, and two lags, respectively. Then, an Akaike Information Criterion (AIC), a Schwarz Bayesian Criterion, and a likelihood ratio test (Sims' test) were used to test the hypothesis that all three specifications are equivalent. The AIC test suggested VAR = 3 and the other two tests suggested VAR = 2. VAR = 2 is used in the estimation procedure of cointegration to avoid overparameterization (see Pesaran and Pesaran [1997, 293]) of VAR models and because with two lags the residuals of the individual equations suggest that serial correlation is not present.

As noted, to test for cointegration we use the Johansen maximum likelihood approach employing both the maximum eigenvalue and trace statistics, The two test statistics both provide evidence to reject the null of zero cointegrating vectors in favor of one cointegrating vector at the 5% level of significance. On the basis of the empirical results, the long-run money demand (equation [1] with [[alpha].sub.3] = 0) finds statistical support over the estimation period. [6] Likelihood ratio tests (described in Johansen [1992] and Johansen and Juselius [1992]) indicate that in each specification the long-run coefficients in the cointegrating relationships are statistically significant. Having determined that the variables are cointegrated, VEC models can be applied. The VEC specifications include the [g.sub.y] term, which was found to be stationary. VEC models are useful as a further test of the cointegration hypothesis, measuring through the error term the size of the deviation in an equilibrium relationship. The r estricted error-correction models pass a series of diagnostic tests, including serial correlation based on the inspection of the residuals as well as the Lagrange multiplier test. The coefficient of the error correction term is negative and statistically significant in both specifications. [7]

Our interest is in the long-run demand for money. Accordingly, Table 2 reports (1) the coefficients of the regressors in each of the specifications estimated--including those in the cointegrating vectors--for the period 1870-1989. Specification RC1 uses the long-term interest rate and RC2 uses the short-term rate under the RC procedure. We stress that the coefficients in the RC specifications represent total effects. Separate estimates of direct and total effects are presented below. The difference between these two estimates gives an estimation of the sum of omitted-variable biases and mismeasurement effects contained in the coefficients of equation (2). Both the RC and VEC models produce estimates of the elasticities of income and the interest rate that are within the range of estimates yielded in previous empirical studies of money demand (e.g., see Thompson [1993]), with the VEC specification giving higher income elasticities than the RC specification and the latter providing lower interest rate elasticit ies (-0.04 for the short rate and -0.07 for the long rate). [8] For purposes of comparison, the bottom three rows of Table 2 reproduce: (1) Friedman and Schwartz's final specification (1982, 282, Table 6.14) covering the period 1869-75, using data averaged over the business cycle and using dummy variables for World Wars I and II, postwar readjustments, and the Great Depression; (2) long-run (implied) elasticities obtained by Hafer and Jansen (1991) using a cointegrating equilibrium specification for the demand for M2 over the period 1915-88, using quarterly observations and the commercial paper rate as the opportunity cost variable; and (3) long-run elasticities obtained by Hafer and Jansen using the same procedure and estimation period but with the corporate bond rate as the opportunity cost variable.

As noted, the interest rate variable preferred by Friedman and Schwartz is the differential between the commercial paper rate (used in this article) and the own rate of return on money. The elasticity of this variable should be larger in absolute value than the elasticity of the commercial paper rate alone obtained in equations RC2 and VEC2 because the commercial paper rate, to the extent that it captures the roles of both own rate and opportunity cost, picks up both the positive effects of the own rate and the negative effects of the opportunity cost. Not surprisingly, therefore, Friedman and Schwartz's interest rate elasticity (-0.32) is higher than those obtained for RC2 (-0.04) and VEC2 (-0.28), and their income elasticity (1.15) is also higher than those obtained in RC1 (1.08) and RC2 (1.06); their elasticity of the growth of nominal income is -0.02 compared with -0.25 in RC2. Hafer and Jansen obtained an implied elasticity of -0.12 for the commercial paper rate. Their implied income elasticity was 1.08 . For their specification using the corporate bond rate, their interest rate and income elasticities were -0.19 and 1.07, respectively. [9] These income elasticities are remarkably close to those of RC1 and RC2. The results reported in Table 2 indicate that several different estimation methodologies--one using RC estimation on annual data over 1870-1989, another using the same data but based on the use of variables integrated of order one (and, therefore, excluding [g.sub.y]), which (in this case) cointegrate into a single equilibrium relationship, another using cointegration on quarterly data over 1915-88, and still another using ordinary least squares on phase-averaged data over 1869-1975 with dummy variables to capture shifts in behavior--yield estimates of some of the coefficients of the long-run money demand functions that fall within a fairly narrow range.

The RC and VEC specifications were used to forecast over successive decades beginning with the 1920s, and ending with the 1990s (i.e., 1990-97). The decades encompass a variety of conditions. For example, the 1920s, 1930s and 1940s correspond to decades of (1) relative prosperity; (2) the Great Depression; and (3) World War II, postwar readjustment, and the Fed's policy of pegging interest rates, respectively. To generate these forecasts, each specification was reestimated over the within sample period using a set amount of data (40 years) prior to the forecast period. Thus, forecasts for the 1930s, for instance, are based on the RC and VEC specifications estimated over 1890-1929. To generate the VEC forecasts, the conditional expectations implied by complete VEC models (i.e., with lagged terms) were used. These conditional expectations do not minimize the mean-square error of ln([m.sub.t]) if the VEC specifications with the growth rate of [m.sub.t] as their dependent variable imply that this mean square err or is infinite, as shown by Christou et al. (1998). Optimal forecasting methods for RC specifications are described in Swamy and Tinsley (1980). These results are reported in Table 3. Each of the specifications produces fairly uniform root mean square errors (RMSE), except for the decade of the 1940s, which has relatively high RMSEs. The 1940s encompassed World War II and the Fed's policy of pegging interest rates. Interestingly, there is no marked deterioration in the RMSEs during the 1990s, a period often considered to be characterized by unstable velocity.

As reported in Table 3, the VEC models provide low RMSEs in the 1990s, a period in which many authors have found that the M2 relation breaks down. Specifically, the years 1990-93 were marked by falling interest rates and a sharp slowdown in the pace of economic activity (including a recession in 1990-91). Other things equal, the fall in interest rates should have caused an increase in the demand for M2. Instead, M2 shifted downward as commercial banks and thrifts failed in large numbers. The VEC models were able to account for the fall in M2 because the high estimates of the income elasticities (around 1.7) yielded by these models and the weak economic activity combined to offset the effect of lower interest rates on M2 demand. When income declined, the high income elasticity produced an even larger proportionate decline in M2. Thus, the VEC models appear to have captured the decline in M2 for reasons apart from the failed commercial banks and thrifts.

Because the interest rate coefficients are so small in the RC models, these variables were dropped and the resulting specifications were used to forecast. These specifications are denoted RC3 and RC4 in Table 3, where the former includes the short rate and the inflation rate as concomitants (as in RC1) and the latter includes the long rate and the inflation rate as concomitants (as in RC2). Comparing the RC specification with and without interest rates, the results provide support for Friedman's (1959) finding that interest rates are not critical for forecasting real money balances. RC3 performs better than RC1 in four of eight cases; RC4 performs better than RC2 in six of eight cases.

To examine the sensitivity of the results to changes in the specifications, the following equations were estimated: (1) RC1 and RC2 with the log of the interest rate variables--instead of the levels of the variables; (2) RC1 and RC2 without the g variable; (3) VEC1 and VEC2 with the (stationary) [g.sub.y] variable included in the cointegrating relationship. The results are reported in Table 4. The largest changes occur in RC2 when the [g.sub.y] variable is dropped; the income elasticity falls to 0.69 and the interest rate elasticity declines (in absolute value) to -0.02. The other sepcification changes lead to smaller changes in elasticities.

Table 5 reports the averages of the total and direct effects of the coefficients for specifications RC1 and RC2 over the period 1870-1989. There are some differences between the average total and direct effects. The direct-effect components of income and interest rate elasticities and the direct-effect components of the coefficients on [g.sub.y] are somewhat higher in absolute value than the total effects. Therefore, based on the two assumptions (Assumptions I and II) used to derive equation (3) from equation (2), measurement error and omitted-variable biases would appear, on average, to have some effect on the RC direct-effect estimates.

Using specifications RC1 and RC2, Figures 1 and 2 present the time profiles of the coefficients (both direct and total effects) on the long-term interest rate and short-term interest rate, respectively. Several points are worth noting. First, although the direct and total effects tend to move together, the pattern of the direct effect displays much less volatility than that of the total effect, indicating that the impact of specification errors on the time profile (as opposed to the average values of the coefficients) of the interest rate coefficients have been important. Second, the interest elasticity tended to decline during the economic contraction of 1920-21 and the Depression years 1929-33. Using the total effects shown in Figures 1 and 2, in neither case did these changes approach anything like a liquidity trap. [10] The elasticity of the short rate reached minimum values of about -0.17 in the early 1920s and about -0.08 in the early 1930s, whereas the elasticity of the long rate reached -0.19 in the early 1920s and about -0.11 in the early 1930s. For the direct effects, the declines in elasticities were even less. Third, beginning around 1933, both interest elasticities show a tendency to rise (decrease in absolute value), despite a continuing decline in interest rates. The year 1933 marked the beginning of the New Deal and the enactment of three kinds of legislative measures to deal with the banking panic of that year: emergency measures designed to reopen closed banks and to strengthen banks permitted to open; the introduction of federal deposit insurance and other measures that affected a lasting alteration in the banking structure; and measures that altered the structure and powers of the Federal Reserve System (Friedman and Schwartz [1963, 420-92]). A tentative hypothesis is that such changes increased confidence in the banking system and led to a decline in the interest-rate sensitivity of holding money. Fourth, beginning in the early 1940s the interest elasticity moved either close to zero or into positive territory and remained there until the early 1950s. These were years dominated by the Fed's pegging of interest rates to help the Treasury in its funding operations; the period 1935-51 represents the years of the lowest interest rates in our sample. Yet the interest elasticity was positive (or near zero)--quite the antithesis of the liquidity-trap prediction. Fifth, after the Fed-Treasury Accord of 1952, interest-rate elasticity (particularly the direct effect elasticity) increased in absolute value (became more negative). The post-1952 years involved the proliferation of money substitutes, which made money demand more interest-sensitive.

Friedman (1959), using the period 1869-1957 and treating each business cycle as a single observation, found that the demand for money was not very interest-sensitive. Figure 1 suggests that Friedman's results were not spurious. A casual inspection of Figure 1 indicates that during 1881-1957 the interest rate elasticity of the demand for money was, on average, not far from zero. Accordingly, we reestimated the RC1 and RC2 specifications over two time periods--1870-1957 and 1958-1989. Using the coefficients for the direct effects, the average elasticities are -0.04 and -0.10 for the earlier and later periods, respectively. For the long rate, during the earlier period the elasticity is -0.11; and for the later period, the elasticity is -0.25. These results suggest that work subsequent to Friedman's (1959) study, using extended data samples, represented not a refutation of Friedman's results but a confirmation that the demand for money was becoming increasingly interest-sensitive over time in association with th e processes of financial deregulation and liberalization.

IV. CONCLUSIONS

The main results may be summarized as follows. (1) Using a Friedman and Schwartz money-demand model over 1870-1989 with annual data frequency, two very different empirical methodologies produce estimates of the coefficients in a long-run M2 demand function that are within the range yielded by earlier studies and can be reconciled with those obtained by Friedman and Schwartz, who used phase-averaged data and a different empirical methodology and time horizon. (2) Separate estimates of the coefficients that do not correct for four main specification errors and that do correct for such errors suggest that such specification errors have been important over time. (3) The interest sensitivity of money-demand has been well below unity. The time profile of the interest rate coefficient does not approach anything like a liquidity trap in the 1930s or, for that matter, in other decades. Indeed, there is a tendency for the interest rate elasticity to increase (i.e., become less negative) as interest rates decrease. (4) Until the late 1950s, the average interest elasticity of the demand for money was small. It appears to have increased in absolute value since the late 1950s in light of the increased competition in the financial services sector and the payment of interest on components included in the definition of money (M2).

Hondroyiannis: Economist, Bank of Greece, 21 E. Venizelos Avenue, GR 102 50 Athens, Greece, and Assistant Professor, Harokopio University, Athens, Greece. Phone 011-301-3202429, Fax 011301-3233025, E-mail ghondr@hua.gr

Tavias: Chief, General Resources Division, IMF, and Director, Bank of Greece, 21 E. Venizelos Avenue, GR 102 50 Athens, Greece. Phone 011-301-3237224, Fax 011-301-3233025, E-mail gtavlas@otenet.gr

Swamy: Financial Economist, U.S. Comptroller of the Currency, 250 E Street, S.W., Mail Stop No. 2-3, Washington, D.C., 20219. Phone 1-202-874-4751, Fax 1-202-874-5394, E-mail swamy.paravastu@occ.treas.gov

(*.) The views expressed are our own and should not be interpreted as those of our respective institutions or the U.S. Department of the Treasury. We are grateful to Dennis Jansen and two referees for constructive comments and to Michael Bordo for providing us with data.

(1.) The standard reference of the money-demand literature is Laidler (1993). Other good surveys include Goldfeld (1992) and Thompson (1993).

(2.) Using data with a high degree of time aggregation has its costs because it reduces degrees of freedom and averages random errors out of the data, which might shed light on other systematic influences on money demand. Hendry and Ericsson (1991) criticized Friedman and Schwartz (1982) for using highly aggregated data.

(3.) For similar views, see Friedman and Schwartz (1991) and Thomas (1997, 361). Christou et al. (1998) and Swamy and Tavlas (2000) provide a formal analysis of the use of out-of-sample predictability as a criterion for model validation.

(4.) The data for deposits were constructed by Friedman and Schwartz for the period 1869 through 1946. The data for currency were constructed by Friedman and Schwartz for 1869 through 1942. Thereafter, Federal Reserve estimates are used. The definition of M2 has changed over time. In 1980, M2 was redefined to include overnight repurchase agreements issued by commercial banks and certain overnight Euro-dollars held by nonbank U.S. residents, Negotiable Order of Withdrawal (NOW) and Automatic Transfer Service (ATS) accounts, money market mutual fund shares and savings, and small-denomination time deposits at all depository institutions. RC estimation is suited to this situation. The deviation of M2 from a particular definition of M2 shows up as the measurement errors in M2, which are taken into account in RC estimation.

(5.) Per-capita values for money and income were used by Friedman and Schwartz (1982, 39-40) in line with their view that the demand for money should be estimated at the level of the individual wealth holder.

(6.) Previous studies of M2 demand cointegration using quarterly, post-World War II data have yielded mixed results. Miyao (1996) provides an overview of these studies. Miyao uses several tests of M2 demand cointegration in the postwar period and finds that although cointegration may possibly exist prior to 1990, there is virtually no support for cointegration when using 1990s data. Hafer and Jansen (1991) find that cointegration exists for M2 over the periods 1915-88 and 1953-88.

(7.) Because we are concerned with the long-run demand for money, we do not present these results in this paper. They are available from us on request.

(8.) As Thompson (1993, 72) puts it in his survey of the literature, "elasticity values with respect to the scale variable differ considerably between studies, and range from less than one-half to well above unity, with the most common value either just above or slightly below unity. Broader definitions of money have, if anything, tended to produce higher long-run elasticity values; a finding that is consistent with the link between narrow definitions of money and transactions cost-based theories of the demand for money which, in their basic form at least, predict economies of scale in money holding."

(9.) Hafer and Jansen also report elasticities over the shorter time horizon of 1953-88. For example, using the commercial paper rate, these elasticities are -0.03 (interest rate) and 1.08 (income).

(10.) According to the liquidity-trap hypothesis, at low levels of the rate of interest the demand for money becomes highly elastic with respect to that variable. Thus, at low levels of the rate of interest any increase in the supply of money will be absorbed without any fall in interest rates so that monetary policy is impotent. Although some previous empirical work found support for this hypothesis, most empirical studies have cast doubt on its validity. Thompson (1993, 70-71) reviews some of the relevant literature. All of these studies used indirect estimation procedures; none used time-varying estimation. Recently, some writers (e.g., McKinnon and Ohno [1997]; Krugman [1998]) have argued that the Japanese economy is mired in a liquidity-trap situation.

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ABBREVIATIONS

ADF: Augmented Dickey-Fuller

AIC: Akaike Information Criterion

M2: Money Supply

RC: Random Coefficient

RMSE: Root Mean Square Error

VAR: Vector Autoregressive

VEC: Vector Error Correction

 Johansen and Juselius Cointegration Test
 Long-Run Demand for Money, 1870-1989
 VAR = 2, Model: VEC1
 Critical Values
Null Alternative Eigenvalue 95% 90%
Maximum Eigenvalues
r = 0 r = 1 31.18 22.04 19.86
r [less than] 1 r = 2 11.32 15.87 13.81
 Critical Values
Null Alternative Trace 95% 90%
Trace Statistic
r = 0 r [greater than] 1 47.78 34.87 31.93
r [less than] 1 r = 2 16.60 20.18 17.88
 VAR = 2, Model: VEC2
 Critical Values
Null Alternative Eigenvalue 95% 90%
Maximum Eigenvalues
r = 0 r = 1 24.83 22.04 19.86
r [less than] 1 r = 2 10.42 15.87 13.81
 Critical Values
Null Alternative Trace 95% 90%
Trace Statistic
r = 0 r [greater than] 1 39.53 34.87 31.93
r [less than] 1 r = 2 14.70 20.18 17.88

Notes: VAR denotes vector autoregression and r indicates the number of cointegrating relationships. Maximum eigenvalue and trace test statistics are compared with the critical values from Johansen and Juselius (1990).

 Long-Run Results (Elasticities) [a]
 Coefficients On
Model Constant Implied Elasticity (r) In(y) ([g.sub.y])
VEC1 2.42 -0.24 1.78 no
 (15.0) (6.8) (11.8)
VEC2 1.84 -0.28 1.64 no
 (8.4) (13.0) (15.3)
RC1 -0.46 -0.07 1.08 -0.31
 (-1.5) (-0.9) (11.3) (-0.8)
RC2 -0.55 -0.04 1.06 -0.25
 (-1.9) (-2.3) (10.2) (-2.4)
F-S [b] 1.53 -0.32 1.15 -0.02
 (9.4) (-4.5) (50.7) (-3.5)
H-J [c] - -0.19 1.07
 (-) (-)
H-J [d] - -0.12 1.08
 (-) (-)

Notes: (a.) Estimation period for the RC models is 1870-1989. The coefficients for the RC models are averages over the estimation period. Figures in parentheses are t-ratios. The figures in parentheses for the VEC models are distributed as Chi-square with one degree of freedom testing the hypotheses that the corresponding variable enters the cointegrating vector at a statistically significant level.

(b.) F-S refers to Friedman and Schwartz (1982). Estimation period is 1867-1975 using data averaged over the business cycle. The interest rate variable is the commercial paper rate minus the own rate of return on money; the elasticity of this variable was derived by multiplying the coefficient on its level by the average value of the interest rate (.037) over the estimation period. The elasticity of the [g.sub.y] variable was derived by multiplying its coefficient by its average value (.05) over the estimation period. Source: Table 6.14 of Friedman and Schwartz (1982, 282).

(c.) H-J refers to Hafer and Jansen (1991) cointegrating equation using corporate bond rate. H-J do not report t-ratios.

(d.) Hafer and Jansen cointegrating equation using the commercial paper rate.

 Postsample Forecasts (RMSEs)
Model 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s
VEC1 0.304 0.113 0.473 0.080 0.172 0.144 0.054 0.020
VEC2 0.345 0.122 0.432 0.077 0.189 0.194 0.066 0.033
RC1 0.093 0.152 0.358 0.075 0.124 0.069 0.161 0.188
RC2 0.273 0.126 0.197 0.077 0.085 0.064 0.205 0.200
RC3 0.199 0.112 0.378 0.080 0.128 0.040 0.050 0.157
RC4 0.103 0.098 0.203 0.062 0.083 0.095 0.102 0.148
Note: Forecasts are based on equations estimated over the 40 years prior to
the forecast interval. For example, forecasts for the 1920s are based on
estimates made over the period 1880-1919.
 Comparisons of Alternative Specifications
 Coefficients On
Model Constant Elasticity (r) ln(y) [g.sub.y] RMSE (1990s)
Using levels
 VEC1 2.96 -0.38 1.89 yes 0.246
 VEC2 2.52 -0.36 1.79 yes 0.255
 RC1 -0.26 -0.14 1.28 0.191
 RC2 -1.93 -0.02 0.69 0.071
Using logs
 RC1 -0.60 -0.05 1.05 -0.29 0.144
 RC2 -0.60 -0.10 1.06 -0.30 0.155
 RC Estimates of Direct and Total Effects
 Coefficients On
 Elasticity (r) ln(y) ([g.sub.y]
Model Total Effect Direct Effect Total Effect Direct Effect Total Effect
RC1 -0.07 -0.15 1.08 1.09 -0.31
 (-0.9) (-1.6) (11.3) (11.2) (-0.8)
RC2 -0.04 -0.06 1.06 1.09 -0.25
 (-2.3) (-2.2) (10.2) (10.5) (-2.4)
Model Direct Effect
RC1 -0.45
 (-1.2)
RC2 -0.59
 (-6.0)
Notes: Numbers in parenthesis are t-ratios. Estimation period is 1870-1989.