Exogeneity within the M2 demand function: evidence from a large macroeconomic system.

Schmidt, Martin B.

A large body of literature investigates whether a stable and predictable long-run association between money and its arguments exists. One point of variation between models is whether to include an interest rate measure directly within the long-run relationship. Several recent studies indicate that empirical findings are sensitive to the choice. Therefore, the present article reexamines the empirical significance of the interest rate within a four-equation macroeconomic system. The results suggest that the interest rate (1) may be excluded from the M2 demand function, (2) is strongly exogenous to most of the system's remaining variables, and (3) may represent a common trend. (JEL E41, E52, C32)

I. INTRODUCTION

Much of recent macroeconomic literature has been dedicated to investigating whether long-run relationships exist between macroeconomic variables. (1) Such preoccupation is understandable given the myriad models in which these relationships play a dominant role. A significant portion of this literature has focused on whether a long-run relationship exists between money demand and its arguments. (2) Of course, the close association between the variables is an integral part of theories that suggest that the monetary authority may qualitatively influence the direction of the aggregate economy.

The estimation of long-run money demand models have differed in their methodological approach and in countless other ways. One specific way in which they may differ in whether the investigator(s) incorporates an interest rate measure directly within the long-run relationship. For example, although Fischer and Nicholetti (1993) and Baba et al. (1992) incorporate various interest rate measures to identify the long-run money demand function, Hendry and Mizon (1998) and Hendry and Ericsson (1991) estimate the long-run money demand function without an interest rate measure.

Although the theoretical importance of the interest rate within the money demand function predates Pigou's (1917) classic treatment and has been discussed at length elsewhere, the importance empirically has recently been high-lighted by Sims (1980). In examining whether money Granger causes output, Sims's results suggest that the tests are sensitive to whether a short-term interest rate measure is included. Campbell and Perron (1991) further argue that Sims's findings are an outgrowth of whether the trivariate model of money, prices and output are cointegrated, that is, I(0), or whether reduction to stationarity requires the introduction of a fourth variable, the interest rate. Reduction to stationarity is important because estimating Granger causality with nonstationary residuals may yield biased estimates due to nonstandard residual distributions.

The present article reexamines the significance of the interest rate within the long-run money demand function. Specifically, I examine two aspects of interest rate-money behavior. The initial examination is to test whether the interest rate measure (three-month Treasury bill rate) can be directly excluded from the long-run M2 demand function. Following the methodology advanced by Hendry and Mizon (1993), Johansen and Juselius (1992, 1994), and Ericsson et al. (1990), the test involves estimating the long-run relationship with the interest rate measure and then impose the overidentifying zero restriction. The associated log-likelihood ratio statistics are distributed [chi square] (n) where (n) represents the number of overidentifying restrictions.

This approach, however, only examines whether the interest rate may be removed from the long-run money demand vector. A second area of interest is concerned with the short-run responses of the interest rate. More specifically, the short-run responses provide a test for Granger causality within the cointegrated variables. Following Ericsson et al. (1998), Granger causality is estimated where one or more cointegrated variables reject strong exogeneity. Strong exogeneity requires that the variable(s) fail to respond to either (1) deviations from the defined long-run equilibrium, that is, the so-called speeds of adjustment; or (2) movements in the associated vector autoregressive (VAR) lags.

Rather than estimate these within the usual single equation setting, the present study embeds the M2 demand function within a larger macroeconomic system of equations. Recent theoretical work by Johansen (1992a) and Phillips (1991) has demonstrated that the omission of relevant variables in an analysis of cointegration may produce biased and inefficient estimates of both the number of cointegrating relationships and of the cointegrating coefficients. Given the fact that most (if not all) economic variables and/or relationships are not determined in isolation, estimating within a single equation format may subject the results to the concerns raised by Johansen and Phillips. In addition, Engle and Hendry (1992), Hendry and Ericsson (1991), and Hendry (1988) raise the concern that weak exogeneity may be an outgrowth of model misspecification. Finally, studies by King et al. (1991) and Cutler et al. (1997) have found empirical support for the efficiency gains associated with embedding a single macroeconomic equation within the framework of a larger macroeconomic system.

The present system of equations largely follows the model proposed by King et al. (1991). The authors combine real gross national product (GNP), consumption, investment, real money balances, the three-month Treasury bill rate, and the inflation rate into cointegrating vectors that represent money, consumption, and investment markets. However, their model is augmented in two ways. The first is to separate real money balances into its two components, M2 and prices. Recently, Fisher and Nicholetti (1993) and Cutler et al. (1997) have found greater success in estimating money demand relationships once both nominal money and prices are introduced. (3)

The second modification is to include a money supply function alongside its demand counterpart. Specifically, the estimation of the money demand function usually involves the use of a final monetary aggregate as the dependent variable whose value is most likely supply determined or at the very least simultaneously determined. Although the use of this aggregate creates few difficulties in estimating the long-run relationship, as both the demand for and the supply of money are thought to be in equilibrium, the short-run properties are more susceptible to this error in specification. (4) For example, a change in the money supply process may exogenously alter the level of the endogenous money variable, which in turn may produce short-run changes in the levels of the price, income, and/or interest rate variables. (5) Such feedback from the hypothesized endogenous variable to the hypothesized exogenous variables would constitute a specification error and may produce biased results.

To address these concerns, the present paper modifies the King et al. (1991) three-equation system to incorporate the money supply relationship. Specifically, the supply relationship is introduced to account for short-run responses associated with returning the long-run money supply relationship. A relatively simple long-run money supply representation is offered by Baghestani and Mott (1997), McCallum (1989), and Gordon (1984). Each maintains that the long-run behavior of the money supply is well represented by the interaction of an interest rate measure, a final monetary aggregate, and the monetary base.

In general, the results of the article suggest that both interest rate and consumption variables are largely determined outside the system's equations. The evidence from unit root, trace, and Johansen and Juselius's overidentifying tests indicate that the system of equations produces results consistent with expectation. Furthermore, weak exogeneity tests indicate that the system of equations yields many of the expected short-run adjustments.

With respect to the interest rate, the results indicate that the measure may be excluded from the four equations. There is, however, marginal evidence of significance within the money supply vector. The short-run results further suggest the exogeneity of the interest rate to the system of equations. In general, the interest rate fails to respond to deviations from the estimated long-run relationships or the VAR lags. The one exception is that the interest rate did respond to the price lags. However, this response is not unexpected because the incorporated interest rate is a nominal value and one might expect inflation to impact nominal values. In the end, following Harbo et al. (1998), the result may suggest that the interest rate represents a common trend. Finally, although there is evidence to suggest that output Granger causes money, there is little evidence to suggest the reverse.

The outline of the article is as follows. Section II offers a brief summary the common and rudimentary macroeconomic model introduced. Section III presents the empirical approach and economic data, as well as detailing the model's estimated long- and short-run properties. Finally, section IV concludes.

II. A MACROECONOMIC MODEL

To more closely examine the role of the interest rate within the money demand function, the present article examines the following four-equation macroeconomic model:

(1) [c.sub.t] - [[beta].sub.1][y.sub.t] - [[beta].sub.2][r.sub.t] = [[member of].sub.ct],

(2) [i.sub.t] - [[gamma].sub.1][y.sub.t] - [[gamma].sub.2][r.sub.t] = [[member of].sub.it],

(3) [M.sup.S.sub.t] - [[alpha].sub.1][B.sub.t] - [[alpha].sub.2][R.sub.t] = [[member of].sub.MSt],

(4) [M.sup.D.sub.t] - [[delta].sub.3][P.sub.t] - [[delta].sub.1][y.sub.t] - [[delta].sub.2][R.sub.t] = [[member of].sub.MDt].

Here [c.sub.t] represents real consumption, [i.sub.t] represents real investment, [M.sup.s.sub.t] represents the supply of nominal money balances, and [M.sup.D.sub.t] represents the demand for nominal money balances. In addition, [y.sub.t] represents real output, [r.sub.t] the ex post real interest rate, [R.sub.t], the nominal rate of return, [P.sub.t] the aggregate price level, and [B.sub.t] represents the monetary base. Finally, the [[member of].sub.ct], [[member of].sub.it] and [[member of].sub.MSt], [[member of].sub.MDt] are error terms associated with consumption, investment, money supply, and money demand, respectively.

The four-equation model is a variant of those commonly used in the literature. (6) The seven variables imply four cointegrating vectors: equation (1) yields a consumption vector, (2) an investment vector, (3) a money supply vector, and (4) a money demand vector. The four vectors represent logically distinct but clearly interrelated vectors. Each equation further represents a long-run equilibrium relation with the error terms capturing disequilibria within each vector that are hypothesized to contain unique stationary processes.

The first two equations are hypothesized to reflect the real side of the economy where the "great ratio" assumptions predict that both [[beta].sub.1] and [[gamma].sub.1] equal 1. Though the sign of [[beta].sub.2] is ambiguous, theory predicts that [[gamma].sub.2] < 0. The money market equations are hypothesized to capture nominal sector behavior. Therefore, a great ratio between the base and the monetary aggregate, that is, [[gamma].sub.1] = 1, is expected. A great ratio between the money supply and the monetary base is hypothesized due to the presumption of long-run stationarity within the money multiplier. In this case, a percentage change in the base would lead to an equal response in the monetary aggregate. (7) Conventional monetary theory maintains the existence of a great ratio with respect to real money demand, that is, both [[delta].sub.1] and [[delta].sub.3] are both expected to equal 1. In addition, following McCallum (1989), the supply interest rate elasticity is expected to be greater than zero, that is, [[gamma].sub.2] > 0, and the demand elasticity is expected to be less than zero, [[delta].sub.2] < 0. The overall model may therefore be represented as follows:

(5) ([y.sub.t] [c.sub.t] [i.sub.t] [M.sub.t] [P.sub.t] [B.sub.t] [R.sub.t] 0 0 1 1

(-[[beta].sub.1] -[[gamma].sub.1] 0 [[delta].sub.1]
 1 0 0 0
 0 1 0 0
 0 0 1 1
 0 0 0 -[[delta].sub.3]
 0 0 -[[alpha] 0
 .sub.1]
(-[[beta].sub.2] -[[gamma].sub.2] -[[alpha] -[[delta].sub.2]
 .sub.2]

Finally, as recognized by Cutler et al. (1997), the system cannot accommodate the nominal interest rate, the real interest rate, and the price level because there exists perfect collinearity between the three variables. Because the central issue deals with the behavior the nominal interest rate within the money demand function, the real interest rate is removed. However, though the magnitude of the results change slightly, none of the main results are altered by the removal of the nominal interest rate and the inclusion of the real interest rate. (8)

III. EMPIRICAL RESULTS

Data

Given equation (5), the present study requires data for the seven endogenous variables. The data were obtained from the St. Louis Federal Reserve FRED database on a quarterly basis for the period 1959-2001. Formally, [y.sub.t] is real gross domestic product, [c.sub.t] is real personal consumption expenditures, and [i.sub.t] is real gross private domestic investment; all are in chained 1996 dollars. In addition, [M.sub.t] is the seasonally adjusted M2 money stock, [P.sub.t] is the Consumer Price Index, and [B.sub.t] represents the Board of Governor's seasonally adjusted monetary base measure. (9) Finally, [R.sub.t] is the three-month Treasury-bill rate. (10)

Empirical Methodology

The present article follows much of the recent literature by incorporating the Johansen maximum likelihood estimator approach. The popularity of the approach follows the results of Gonzolo (1994), where the author shows that the Johansen procedure yields more robust results when more than two variables are included, is less sensitive to the choice of lag, and is more robust to non-normal distributions. Furthermore, both Gonzolo (1994) and Haug (1996) show that the Johansen approach has the least size distortion and, therefore, has stronger small sample properties. (11)

The Johansen approach integrates both of the long- and short-run responses and maybe summarized by the following general k-order VAR model (see Johansen 1992 a, b):

(6) [DELTA][X.sub.t] = [mu] + [[i - 1].summation over (k - 1)] [[GAMMA].sub.i] [DELTA][X.sub.t-i] + [PI][X.sub.t-k] + [[member of].sub.t],

where [X.sub.t] is a vector of I(1) variables at time t, the [[GAMMA].sub.i] [DELTA][X.sub.t-I] terms account for the stationary variation related to the past history of the variables, and the [PI] matrix contains the cointegrating relationships. Furthermore the [PI] matrix may be separated into two components, such that [PI] = [alpha][beta]', where the cointegrating parameters are contained within the [beta] matrix and the [alpha] matrix describes the weights with which each variable enters the equation. Cointegration, then, requires that the [beta] matrix contains parameters such that [Z.sub.t], where [Z.sub.t] = [beta]'[X.sub.t], is stationary. Also, the [alpha] matrix is thought to represent the speed with which each variable changes to return the individual vectors to their respective long-run equilibrium and may be estimated from the error correction equations.

As was described in the introduction, the present analysis has two related lines of inquiry. The first involves whether the interest rate may be excluded from the [beta]' matrix. Johansen (1992b) suggests the use of log likelihood ratio statistics to test these zero restrictions. His method involves estimating the defined unrestricted vectors and comparing this result to a model which contains the additional [beta] matrix exclusion restrictions. (12) The ratio of the two models' log-likelihood statistics is distributed as [chi square] (n) where n represents the number of overidentifying restrictions. The second line of inquiry involves whether the nominal interest rate may be excluded from the [alpha] matrix. The Johansen approach also generalizes to restricting the short-run [alpha] matrix. More specifically, if [DELTA][X.sub.it] fails to respond to the defined long-run disequilibrium, that is, [[alpha].sub.i] =, 0, then [X.sub.it] is said to be weakly exogenous. (13)

Even if the cointegrated variable is found to be weakly exogenous, Granger causality may still exist. The feedback may come about from the associated VAR lags, that is, the [[GAMMA].sub.i']s. Strong exogeneity, therefore, imposes the additional requirement that [DELTA][X.sub.it] fail to respond to the incorporated (k) lags of [DELTA][X.sub.j], that is, all [[GAMMA].sub.i] = 0. Granger causality would then be found with the rejection of strong exogeniety. (14) Crowder (1998) offers straightforward tests for both weak exogeneity and strong exogeneity. Crowder maintains that weak exogeneity tests may be accomplished through a Student's t-test on the speed of adjustment parameter and strong exogeneity requires an additional F-test on the associated VAR lags. Both the overidentifying restriction tests and these are reported in this section.

Finally, as has been highlighted in the literature, cointegration and in particular money demand vector estimation can be sensitive to sample periods. Cutler et al. (2000) maintain that the systems approach yields efficiency gains due to the fact that the additional structure--of the additional equations--removes much of the noise surrounding the hypothesized M2 demand vector and therefore improves temporal estimation. To examine the temporal sensitivity of the present model and the corresponding results, the model was reestimated every four periods beginning with a sample of 1959Q1-1980Q4 and ending with the period 1959Q1-2001Q4.

In general, these results suggest that the main conclusions, in particular with respect to interest rates, are robust to changes in sample periods. In this case, the present results reenforce the results presented in Cutler et al. (2000) as an M2 demand function was estimated during the so-called missing M2 period of the early 1990s. However, the 1996-99 period was poorly estimated. These four periods, in general, reject the model, almost all of the great ratio predictions, as well as the weak and strong exogeneity tests. (15) However, if one imposes price homogeneity within the M2 demand equation during these periods, the results follow those reported next. In this case, it appears that the 1999-99 period was associated with a bit more noise and required more structure than other post-1980s periods.

Unit Root and Lag Length Tests

As is customary, the initial point of any analysis of cointegration is an examination of the number of unit roots contained within the individual variables. To determine the integrated level of each series, both augmented Dickey-Fuller (ADF) and Phillips-Perron tests were performed. Overall, both the ADF and the Phillips-Perron tests indicate that the seven variables fail to reject the presence of a single unit root. In addition, consistent with previous studies, they also suggest the absence of a second root. Therefore, the seven variables are estimated to be I(1) and may be cointegrated. (16)

The multivariate vector error collection (VEC) model also requires a lag structure to be selected. As was mentioned earlier, the choice is important because a lag structure that is too high may overparameterize and therefore reduce the power of the cointegration tests. However, a lag structure that is too low may not produce residuals that are white noise. To investigate the optimal lag structure, final prediction error and Akaike information criteria tests were performed for lags one through eight. Both tests suggested a lag length of four. However, it should be mentioned that other tests suggested alternative lag lengths. Therefore, the following empirical tests were examined with the alternative lag lengths. In general, the main conclusions were robust to the choice. Shorter lag options were unattractive because these yielded money demand income and price elasticities that were quite large. To examine the importance of longer lag options, I estimated the VEC model with eight lags and sequentially removed each lag and examined their relative importance. In this case, lags five through eight were insignificant. Finally, the choice of four lags is consistent with King et al. (1991) and Cutler et al. (1997).

The Estimated Number of Cointegrating Vectors

Estimation of similar levels of integration allows for the possibility of cointegration between the variables, but it does little to guarantee it or describe the precise manner in which the variables are related. Therefore, the next step is to determine the number of cointegrating relationships the seven variables are estimated to produce.

To determine the number of significant cointegrating vectors, it has become common to estimate the unrestricted matrix by Johansen's maximum likelihood method. The estimated stochastic matrix is then evaluated using log likelihood ratio statistics on the estimated eigenvalues to determine the number that are significant, that is, the trace test. However, an important issue in this process is whether to include a constant and trend term directly into the cointegrating vector. The present paper follows Johansen (1992b), who maintains that an examination of cointegration should begin with as general an approach as possible. Therefore the unrestricted matrix was estimated with both constant and trend terms to assess their relative importance. Because the trend term could easily be excluded from all cointegrating vectors and the constant was significant within the equations, the system was estimated without trend terms and with constant terms. However, the following results were robust to the exclusion of the constant.

Table 1 reports trace test results for the seven variables with four, six and eight lags. (17) Each of these suggests that the seven variables are well represented by four cointegrating vectors. In addition, the finding of four cointegrating vectors is consistent with King et al. (1991) and Cutler et al. (1997).

The Long-Run Estimates

As Johansen and Juselius (1992, 1994) describe, economic interpretation of the estimated cointegrating vectors becomes extremely difficult once the number of vectors exceeds one. Therefore, the next step in hypothesis testing involves determining whether the unconstrained vectors are consistent with the earlier proposed economic model.

To exactly identify the four equations, 16 restrictions are required. Two types of restrictions are incorporated. The initial set of four restrictions involves the choice of normalizing variables. These are motivated by the discussion in section II and the earlier matrix (equation [5]) where two of the vectors are expected to reflect the consumption and investment relationships, and the other two are hypothesized to reflect the behavior of the money market.

The second set of restrictions hypothesize that a particular variable(s) does not appear in a particular vector, that is, the Cowles Commission-type restrictions. These largely following King et al. (1991), where investment and money are omitted from the consumption vector, consumption and money from the investment vector, and consumption and investment from money demand. Analogously, consumption and investment are restricted to be zero within the hypothesized money supply vector, and the monetary base is removed from the consumption and investment vectors. In addition, the money demand and money supply vectors are differentiated by the zero restriction on the monetary base within the money demand vector and on price within the money supply vector. Finally, to complete the real and nominal-side distinction, equations (1) and (4)-(6) impose three overidentifying restrictions. Specifically, price is removed from both the consumption and investment functions, whereas real income is removed from the money supply relationship.

The estimated cointegrating vectors are reported in Table 2. Overall, the results suggest that the time-series data are generally consistent with the earlier hypothesized vectors. In addition, the data fail to reject the three overidentifying restrictions with a log-likelihood ratio statistic of 5.553 (p-value 0.136). Moreover, a cursory examination of the estimated coefficients suggests most of the earlier expectations. Specifically, although real income, price, and the monetary base all return coefficients that appear different than zero, they are only marginally different from one. These therefore suggest the existence of the great ratios. This is in contrast with the estimated interest rate behavior. These estimates are generally quite small and often of the wrong sign.

A more formal test for analyzing these would be to impose them on the particular vector and examine their likelihood ratio statistics. As was stated earlier, Johansen and Juselius's test allows the researcher the ability to make such specific restrictions on the hypothesized vectors. The last two rows of Table 2 report these likelihood ratio test statistics and their associated [chi square] p-values. The results generally confirm the existence of the great ratios and the limited long-run impact of the interest rate. For example, the likelihood ratio tests suggest that the interest rate variable may be excluded from all four hypothesized vectors. This position supports the estimation approach of Hendry and Mizon (1998) and Hendry and Ericsson (1991).

There are, however, two exceptions. The tests suggest the rejection of the great ratio in investment and the unity coefficient on price within the hypothesized money demand vector. The relatively high income elasticity within the investment vector may be suggested by the fact that investment is far more volatile than output and accounts for most business cycle episodes. In which case, reactions of investment to changes in income should be elastic, for examples, Culter et al. (1997, 2000).

The second rejection is the estimation of a money demand price elasticity less than unity. The finding is consistent with Cutler et al. (1997), Boughton and Tavlas (1990), and Boughton (1991). Each concludes that money demand price elasticity is less than one for many countries. Although the unrestricted values provide stronger log-likelihood ratio statistics, the following results do not change significantly if one imposes the unity restrictions.

The Short-Run Estimates

As was mentioned in the introduction, Granger causality within a cointegrated system contains an additional aspect. In addition to the possible response to the associated VAR lags, causality may come about from responses to the associated long-run disequilibrium. Specifically, the existence of cointegration requires that any deviation from long-run equilibrium produce an adjustment(s) to reattain the defined equilibrium. For example, if [[member of].sub.MDt] > 0, either M, P, y, R, or some combination of the four must respond to correct the disequilibrium. Equilibrium may be reattained by [M.sub.t] or [R.sub.t] decreasing or by [Y.sub.t] or [P.sub.t] increasing. Of course, any combination of movements would also reattain equilibrium. These responses are contained within the associated [alpha] matrix.

Table 3 reports the estimated [alpha] matrix. Specifically, the results represent the individual responses of the seven variables to past disturbances to the estimated cointegrating vectors within each of the VEC equations. The disequilibrium residuals ([[member of].sub.ct], [[member of].sub.it], [[member of].sub.MSt] and [[member of].sub.MDt]) were computed using the estimated [beta] matrix of Table 2. As described earlier, two tests are available to examine whether any of the included I(1) variables are weakly exogenous: (1) Student t-tests and/or (2) log likelihood ratio tests.

Overall, the results provide additional support for the hypothesized macroeconomic model with all four vectors producing an internal process that responds to vector disequilibrium correctly. Specifically, although the investment relationship has real income, real investment, and the interest rate available to correct disequilibrium, only the investment variable responds endogenously. This suggests that the vector may be correctly interpreted as representing an investment function and further suggests the weak exogeniety of real income and interest rate measures.

As for the consumption vector, it may be reequilibrated by real income, real consumption, and the interest rate. However, only real income is significant. Interestingly, although three of the four vectors have real income as an argument, by far the strongest and the only correct response exists within the consumption relationship. The linkage between real consumption and real income has recently been highlighted by Cochrane and Sbordone (1988), Harvey and Stock (1988), and Cochrane (1995). In particular, Cochrane (1995) maintains that consumption determines trend movements in income. In this case, one would expect that the relationship between consumption and income be more closely associated than real income and other variables. Also, the relatively strong response of the investment variable to disequilibrium within the consumption vector is further evidence of such a relationship. All of this may suggest the reinterpretation of the vector as representing permanent income.

As for the money market, both money supply and money demand produce endogenous movements that move the vector in the proper direction. However, only the demand equation produces the correct response in the money variable required for inversion to a demand equation. Therefore, although these results provide support for conventional money demand models, they fail to provide support for so-called buffer-stock models. (18) Finally, the supply equation produces only a response by the monetary base which suggests that the vector may reflect a monetary reaction function.

Consistent with the earlier [beta] matrix results, the interest rate failed to provide any significant responses. Specifically, the variable does not respond significantly to disequilibrium in any of the hypothesized vectors. Therefore, the interest rate was estimated to be weakly exogenous to all of the included equations. Interestingly, a similar result exists for the consumption variable. Both of these results may suggest that the two measures represent common trends. Formally, Harbo et al. (1998, 389) suggest that "the unexplained variation of a weakly exogenous variable is cumulated to a common trend." As was mentioned, the consumption result is consistent with Cochrane (1995, 242), where the author suggests that "consumption defines the trend [emphasis added] movements in (output)."

Finally, additional unexpected responses exit. These provide further justification for estimation under multiple markets. The exclusion of the cross-market behavior exhibited by consumption, investment, and the money equations would subject the estimation to the issues raised by Phillips (1991) and Johansen (1992a). For example, the lack of weak exogeneity of investment and money within the consumption vector suggests that these should be jointly estimated.

In addition to the [alpha] responses, the endogenous variables may be influenced by the VAR lags. To investigate these responses, Wald tests were performed on the four associated lags. These are reported in Table 4. The results further suggest the exogeniety of the interest rate to most of the remaining variables. Specifically, most of the Wald statistics reported in the final row of Table 4 are insignificant. Therefore, the interest rate may be viewed as being strongly exogenous to nearly all of the system's variables.

There is one exception, though. Specifically, the nominal interest rate variable does respond significantly to the associated price lags. This is not unexpected because the incorporated interest rate is a nominal value and one might expect inflation to impact nominal values. This does, however, provide an explanation for what drives the interest rate: Prices move nominal interest rates. In this case, the preceding results suggest that real interest rates are exogenous and stationary.

Table 5 highlights this relationship. Specifically, the table presents the estimated cointegrating relationship between the aggregate price level and the three-month Treasury bill rate. A trace test for the two-variable system with four lags and a constant rejects the [H.sub.0] of no cointegrating vector at the 1% critical level. Furthermore, log-likelihood ratio tests do not reject the Fisher model restriction of (1, - 1). In this case, the real interest rate, the difference between the two, would be stationary. Thereby providing corroborating evidence for Wu and Zhang (1997), Garcia and Perron (1994), and Mishkin (1992).

In the case of stationarity, the results of Schmidt (2000) and Ashenfelter and Card (1982) suggest that the short-run movements in the real rate are influenced by stickiness within the labor market. Specifically, extending the approach suggested by Mehra (1991) and Darrat (1994), Schmidt (2000) argues that deviations in the price-nominal wage relationship, those not associated with productivity gains produce significant real rate of return responses. In that case, real interest rates may be influenced by short-run deviations from the optimal unemployment rate. (19)

Finally, these tests provide additional information. For example, although consumption failed to respond to any of the associated long-run disequilibriums, the variable did respond to the lags of the price variable. This result in conjunction with the investment and output responses suggests that prices may Granger cause the real side of the economy. However, if one follows the previous consumption results, the significant money responses are consistent with permanent income Granger causing money and prices.

V. CONCLUSION

The article reexamined the M2 demand function within a macroeconomic system of equations. Specifically, the relevance of the interest rate was investigated. The results suggest that a stable and stationary M2 demand relationship exists. The results further suggest that the interest rate variable should be excluded from three of the four cointegrating vectors. One of the vectors where the interest rate could be excluded was the demand function. The results therefore suggest that the trivariate system of output, money, and prices are sufficient for reduction to stationarity and support the money demand models proposed by Hendry and Mizon (1998). The short-run results further suggest that the interest rate fails to respond to most of the system's remaining variables. In the end it does seem that real interest rates are determined largely outside the present model and suggests that they are stationary.

ABBREVIATIONS

ADF: Argumented Dicker-Fuller

GNP: Gross National Product

VAR: Vector Autoregressive

VEC: Vector Error Correction

TABLE 1
Cointegration Rank Tests: Unrestricted Trace Test (Variables: y, c, i,
M2, P, B, R)

[H.sub.0]: [H.sub.a]:

r = 0 r [greater than or equal to] 1
r [less than or equal to] 1 r [greater than or equal to] 2
r [less than or equal to] 2 r [greater than or equal to] 3
r [less than or equal to] 3 r [greater than or equal to] 4
r [less than or equal to] 4 r [greater than or equal to] 5
r [less than or equal to] 5 r [greater than or equal to] 6
r [less than or equal to] 6 r [greater than or equal to] 7

 Trace Trace Trace
 Statistic Statistic Statistic
[H.sub.0]: (4 Lags) (6 Lags) (8 Lags)

r = 0 176.371 190.846 243.197
r [less than or equal to] 1 105.08 130.379 141.590
r [less than or equal to] 2 71.093 83.392 82.298
r [less than or equal to] 3 46.567 49.911 45.679
r [less than or equal to] 4 24.775 25.415 24.681
r [less than or equal to] 5 8.9520 11.283 9.7601
r [less than or equal to] 6 0.1886 0.0601 0.0713

 90%
 Critical
[H.sub.0]: Values

r = 0 117.73
r [less than or equal to] 1 89.37
r [less than or equal to] 2 64.74
r [less than or equal to] 3 43.84
r [less than or equal to] 4 26.70
r [less than or equal to] 5 13.70
r [less than or equal to] 6 3.84

Notes: Sample: 1959Q4-2001Q4. The VEC system includes a constant term.
Critical values are from Johansen (1995). Bold represents rejection of
[H.sub.0] at the 10% level.

TABLE 2
Restricted Values of the VEC: Johansen & Juselius's [H.sub.6] Test
(Variables, four lags: y, c, i, M2, P, B, R)

Variable Consumption Investment

Income -1.070 -1.387
Consumption 1.000 1.000
Investment 0.000 0.000
M2 0.000 0.000
Price 0.000 0.000
Base 0.000 0.000
Interest rate 0.002 -0.007

Likelihood ratio tests:
Overidentifying [[beta].sub.1] [[gamma].sub.1]
restrictions: (X [3]) = 0.0 (X [4]) = 0.0 (X [4])
5.553 (0.136) 25.723 (0.000) 17.825 (0.001)

Great ratio [[beta].sub.1] [[gamma].sub.1]
restrictions = -1.0 (X [4]) = -1.0 (X [4])
 9.890 (0.042) 24.500 (0.000)
Interest rate
restrictions

[[beta].sub.2] = [[gamma].sub.2] [[beta].sub.2] [[beta].sub.2]
 = [[delta].sub.2] = 0.0 (x [4]) = 0.0 (X [4])
 = 0.0 (X [6]) 7.104 (0.131) 6.231 (0.182)
 10.478 (0.106)

[[beta].sub.2] = [[gamma].sub.2]
 = [[delta].sub.2]
 = [[alpha].sub.2]
 = 0.0 (X [7])
 16.039 (0.025)

Variable Money Supply

Income 0.000
Consumption 1.000
Investment 0.000
M2 1.000
Price 0.000
Base -0.933
Interest rate -0.034

Likelihood ratio tests:
Overidentifying [[alpha].sub.1]
restrictions: (X [3]) = 0.0 (X [4])
5.553 (0.136) 15.284 (0.009)

Great ratio [[alpha].sub.1]
restrictions = - 1.0 (X [4])
 9.865 (0.063)
Interest rate
restrictions

[[beta].sub.2] = [[gamma].sub.2] [[beta].sub.2]
 = [[delta].sub.2] = 0.0 (x [4])
 = 0.0 (X [6]) 8.402 (0.078)
 10.478 (0.106)

Variable Money Demand

Income -1.002
Consumption 0.000
Investment 1.000
M2 1.000
Price -0.740
Base 0.000
Interest rate -0.004

Likelihood ratio tests:
Overidentifying [[delta].sub.1] [[delta].sub.3]
restrictions: (X [3]) = 0.0 (X [4]) = 0.0 (X [4])
5.553 (0.136) 11.296 (0.023) 10.962 (0.027)

Great ratio [[delta].sub.1] [[delta].sub.3]
restrictions = - 1.0 (X [4]) = - 1.0 (X [4])
 5.554 (0.235) 10.885 (0.028)
Interest rate
restrictions

[[beta].sub.2] = [[gamma].sub.2] [[delta].sub.2]
 = [[delta].sub.2] = 0.0 (x [4])
 = 0.0 (X [6]) 5.715 (0.224)
 10.478 (0.106)

Notes: Sample: 1959Q4-2001Q4. The estimates have been normalized. The
(n) overidentifying restrictions are imposed on the estimated matrix
and the log-likelihood ratio tests are by the method suggested in
Johansen and Juselius (1992). The X (n) p-values are reported in
parentheses. Bold represents rejection of [H.sub.0] at the 5% level.

TABLE 3
Exogeneity Tests: Error Correction Terms (Variables, four lags: y, c,
i, M2, P, B, R)

 [[member of]. [[member of].
Variable sub.ct-1] sub.it-1]

Income 0.1703 -0.0114
 [2.005] [-0.878]
Likelihood ratio tests (X [4]): 11.472 6.4053
[[member of].sub.t-1] = 0 (0.0217) (0.1709)
Consumption -0.0336 0.016
 [-0.501] [1.564]
Likelihood ratio tests (X [4]): 8.2215 5.5894
[[member of].sub.t-1] = 0 (0.0838) (0.232)
Investment 1.2156 -0.293
 [3.182] [-5.028]
Likelihood ratio tests (X [4]): 12.623 27.221
[[member of].sub.t-1] = 0 (0.0133) (0.000)
Money 0.1845 0.0188
 [3.457] [2.312]
Likelihood ratio tests (X [4]): 15.652 10.846
[[member of].sub.t-1] = 0 (0.0035) (0.0284)
Price 0.0151 0.008
 [0.419] [1.453]
Likelihood ratio tests (X [4]): 5.7358 7.9799
[[member of].sub.t-1] = 0 (0.2198) (0.0923)
Base -0.0562 0.002
 [-0.927] [0.212]
Likelihood ratio tests (X [4]): 6.4345 5.6008
[[member of].sub.t-1] = 0 (0.169) (0.231)
Interest rate 5.6139 0.7098
 [0.719] [0.597]
Likelihood ratio tests (X [4]): 6.0749 5.9572
[[member of].sub.t-1] = 0 (0.1936) (0.2024)

 [[member of]. [[member of].
Variable sub.it-1] sub.it-1]

Income 0.0395 -0.0894
 [1.156] [-1.356]
Likelihood ratio tests (X [4]): 6.2419 6.4701
[[member of].sub.t-1] = 0 (0.1818) (0.1667)
Consumption 0.0061 -0.0013
 [0.227] [-0.026]
Likelihood ratio tests (X [4]): 5.5539 9.6962
[[member of].sub.t-1] = 0 (0.2351) (0.2065)
Investment 0.3106 -0.905
 [2.020] [-3.053]
Likelihood ratio tests (X [4]): 10.625 11.84
[[member of].sub.t-1] = 0 (0.0311) (0.0186)
Money 0.0503 -0.101
 [2.340] [-2.450]
Likelihood ratio tests (X [4]): 10.175 10.43
[[member of].sub.t-1] = 0 (0.0376) (0.0338)
Price 0.0062 0.0041
 [0.429] [0.145]
Likelihood ratio tests (X [4]): 5.686 5.5689
[[member of].sub.t-1] = 0 (0.2239) (0.2337)
Base 0.0684 -0.0806
 [2.802] [-1.713]
Likelihood ratio tests (X [4]): 11.383 7.9187
[[member of].sub.t-1] = 0 (0.0226) (0.0946)
Interest rate 4.1239 -8.766
 [1.313] [-1.448]
Likelihood ratio tests (X [4]): 6.6432 6.8824
[[member of].sub.t-1] = 0 (0.156) (0.1422)

Notes: Sample: 1959Q4-2001Q4. The estimates are reported with their
respective Student t-statistics (in brackets). The (n) overidentifying
restrictions are imposed on the estimated matrix and the log-likelihood
ratio tests are by the method suggested in Johansen and Juselius
(1992). The X (n) p-values are reported in parentheses. Bold represents
rejection of [H.sub.0] at the 5% level.

TABLE 4
Exogeneity Tests, Additional Lags (Variables, four lags): y, c, i, M2,
P, B, R)

Variable [SIGMA]d(y) [SIGMA]d(c) [SIGMA]d(i) [SIGMA]d(M)

Income 0.4397 0.0126 -0.0713 0.1863
 (0.788) (0.470) (0.686) (0.426)
Consumption -0.1723 0.1498 0.0469 0.1386
 (0.578) (0.341) (0.615) (0.461)
Investment 5.2902 -0.982 -0.753 1.4913
 (0.167) (0.009) (0.088) (0.104)
Money -0.2579 0.3271 -0.01 0.508
 (0.503) (0.161) (0.670) (0.000)
price -0.0511 0.0329 -0.0027 0.0491
 (0.126) (0.089) (0.235) (0.124)
Base -0.4707 0.2156 0.0573 0.0525
 (0.370) (0.251) (0.622) (0.472)
Interest rate 122.35 -64.424 -9.4564 -5.2527
 (0.092) (0.392) (0.365) (0.586)

Variable [SIGMA]d(P) [SIGMA]d(B) [SIGMA](R)

Income -0.5689 0.1321 0.0013
 (0.023) (0.402) (0.361)
Consumption -0.2936 0.0791 -0.003
 (0.004) (0.474) (0.354)
Investment -2.7874 1.4266 0.0156
 (0.004) (0.142) (0.024)
Money 0.1018 -0.095 -0.0004
 (0.060) (0.397) (0.000)
price 0.8242 0.0524 0.0044
 (0.000) (0.723) (0.000)
Base 0.1622 0.3114 -0.004
 (0.161) (0.082) (0.060)
Interest rate 8.2751 27.468 -0.113
 (0.042) (0.505) (0.003)

Notes: Sample: 1959Q4-2001Q4. The estimates are reported with their
respective Wald exclusionary tests' p-value in parentheses. Bold
indicates rejection of [H.sub.0] at the 5% level.

TABLE 5
Nominal Interest Rate and Inflation, four Lags

 [member of] [SIGMA][DELTA]
Variable CV [pi.sub.t-1] [(p).sub.t-j]

[Price.sub.t] 1.000 -0.0001 0.8812
 (-1.549)
[Interest.sub.t] -1.669 0.038 45.141
 (2.797)

 Wald test: Wald test:
 [DELTA] [DELTA]
 [(p).sub.t-j] [SIGMA][DELTA] [(R).sub.t-j]
Variable = 0 [(R).sub.t-j] = 0

[Price.sub.t] 65.226 0.0019 8.852
 (0.000) (0.002)
[Interest.sub.t] 3.295 0.1067 6.001
 (0.013) (0.000)

Notes: Sample: 1959Q4-2001Q4. Each equation contains a constant and
four lags variables. The [epsilon][pi.sub.t-1] was computed from the
normalized vector. Bold indicates rejection of [H.sub.0] at the 5%
level.

(1.) A search of leading economic journals suggests that the long run was integral to over 4,000 published articles. Furthermore, a search for cointegration produced over 2,000 entries in just over 15 years.

(2.) See, for example, Hafer and Jansen (1991), Friedman and Kuttner (1992), Baba et al. (1992), Stock and Watson (1993), Hoffman et al. (1995), Cutler et al. (1997), and Miyao (1997).

(3.) Separating out real money balances creates a difficulty because the integrated level of monetary aggregates and aggregate price measures is far from a settled issue. For example King et al. (1991) and Crowder (1998) argue that most monetary aggregates are I(2). However, Miller (1991), Konishi et al. (1993), Lastrapes and Selgin (1994), and Cutler et al. (1997) maintain that they are I(1). As is reported within the empirical section, the present study conforms with the studies which suggest that the variables are I(1). Following a similar theme, further efficiency gains may be possible if one separates out all real and nominal values. Therefore, real consumption and real investment were replaced with their nominal counterparts. The results from this exercise suggest that incorporating their real value is a valid restriction. Moreover once the (1 - 1) restrictions are imposed on the two vectors, the [chi square] (3) statistic is very close to the value reported in Table 2 of this text. This would suggest that the two models are quite similar. I thank an anonymous referee for raising the issue. The results are available on request.

(4.) More specifically, Phillips (1991) shows that the cointegrating coefficients can be optimally estimated without prior restrictions on the error correction's dynamic responses. However, the short-run estimates are potentially biased due to the omission. Therefore, although the interaction of the money demand and money supply responses continue to produce the long-run quantity theory result, the additional detail provide more accurate estimates of the individual short-run adjustments that occur, as the short-run properties are more susceptible to this error in specification Also, Artis and Lewis (1976) maintain that the specification bias is of greater concern during the highly volatile periods of the 1970s and 1980s than during other periods.

(5.) If the money supply relationship always returns its long-run relationship, these concerns are minimal. However, given the results of Baghestani and Mott (1997), such control seems unlikely. Their results indicate that there exist numerous periods when the money supply relationship, due primarily to movements in the money multiplier, is off its long-run level.

(6.) See, for example, King et al. (1995), (1991), and Cutler et al. (1997), who argue that the model underlies the real business cycle assumption that the economy is inherently driven by stochastic trends that are expected to influence consumption, investment and real output uniformly. In other words, while income and consumption may change, the ratio of the two should not. Canova et al. (1994) and Gali (1992) provide an alternative demand-side motivation for the model (minus the money supply representation) following the common IS-LM representation.

(7.) It is also consistent with the estimates provided by Baghestani and Mott (1997).

(8.) The stationary impact of inflation is removed in the first part of the estimation (Johansen) technique. Therefore, the inclusion of either nominal or real interest rates leads to similar results.

(9.) An alternative choice would be to use the St. Louis Fed's monetary base measure. Garfinkel and Thornton (1991) suggests that the two measures may differ significantly. Haslag and Hein (1990) argues that the St. Louis measure is superior in explaining GNP growth. However, in the end, Garfinkel and Thornton (1991, 32) argue that "there is little basis for choosing one measure over the other in empirical studies." Therefore, both measures were incorporated. The two measures produced very similar results. The St. Louis estimates are available on request.

(10.) There are a number of alternative rates, specification the federal funds rate, the Federal Reserve's M2 own rate, and the one-year Treasury bill rate. In addition, Baba et al. (1992) and Hetzel and Mehra (1989) suggest that the correct rate is more accurately represented by the monetary aggregate's relative own rate of return or the spread between the M2 rate and the alternative interest rates. Although the various rates generally produced similar results, there were a few outliers. Specifically, the Fed's M2 own rate did respond significantly (when the Board's base measure was used but not with the St. Louis measure) to disequlibirum in the money demand vector. Also, the Fed funds rate did respond to the lags of the St. Louis base measure, but not to lags of the Board's measure. The results from these other measures are available on request. Following Friedman (1956), it is possible that money demand is a function of along-term interest rate. Therefore, the system of equations was estimated with the 10-year Treasury rate. Although the interest rate did respond significantly to disequilibrium in both the money supply and demand vectors, the model was not well defined. In particular, the overidentifying [chi square] (4) statistics were quite large and easily rejected [H.sub.0]. Furthermore, imposing the zero restriction on the interest rate within the money demand function, that is, [[delta].sub.2] = 0, reduced the [chi square] statistics. This result suggests that if an interest rate is appropriate within the M2 demand function, it is most likely a short-term rate.

(11.) The results of Hargreaves (1994), however, suggest that the approach has stronger properties when the number of observations is above 100. The present study incorporates n = 165.

(12.) As will be highlighted, the approach may be generalized to encompass a variety of possible restrictions, that is, the great ratio restrictions.

(13.) The use of weak and strong exogeneity follows the definitions presented in Engle et al. (1983).

(14.) Following Ericsson et al. (1998), the [[GAMMA].sub.i] and [alpha] estimates provide additional information. Because these responses yield information on how the incorporated variables respond to associated disequilibrium, they provide information to policy makers as to the likely outcome of their policy. More specifically, most economic policy may be viewed as producing or responding to disequilibrium within an hypothesized economic relationship, specifically, alter [Z.sub.t] = [beta]'[X.sub.t]. Policy would therefore create a wedge between the cointegrated variables, which require some adjustment by the included variables to reattain the defined equilibrium. The ultimate question for policy makers is which variables and to what extent do the cointegrated variables move to clear the disequilibrium created by the new policy. The [[GAMMA].sub.i], and [alpha] estimates provide a sense of how the variables react.

(15.) These results are available on request

(16.) To save space, these results are included in a data appendix. The appendix is available on request.

(17.) Haug (1996) examines the small sample properties of numerous tests for the cointegrating rank and finds that the Johansen (1988) trace tests have the least amount of size distortions.

(18.) See Mizen (1994) or Laidler (1984) for a description of the theory(s) behind these models.

(19.) Theoretically, one could add a representation of the nominal wage-price vector into the present four-equation system. However, Johansen and Juselius (1992) and Baba et al. (1992) have pointed out that when cointegrated systems become too large, estimation and interpretation become increasingly difficult. In which case, the present model with its seven included variables would seem to push the envelope. A next step then, would be to create a smaller model that focuses solely on the dynamics of the real rate of return.

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MARTIN B. SCRMIDT, I have benefited from numerous comments from two anonymous referees. All remaining errors, of course, are nay own responsibility.

Schmidt: Associate Professor, Department of Economics, P.O. Box 8795, The College of William and Mary, Williamsburg, VA 23187-8795. Phone 1-757-221-2367, Fax 1-757-221-1175, E-mail schmidtm@wm.edu