## Abstract

The paper evaluates whether a monetary aggregate can serve as a useful predictor of inflation, using recent developments in the principle of cointegrated variables. M2 but not M1 is cointegrated with relevant price, transactions, and rate of return variables. However, deviations of M2 from its long-run equilibrium value do not significantly enhance inflation forecasts based on conventional output-gap models, a result that stands in contrast to the Federal Reserve’s P* relationship. Nevertheless, changes in M2 do contain information about future inflation, consistent with the view that the demand for money reflects the forward-looking behavior of private agents.

## I. Introduction

Through much of the 1980s the U.S. Federal Reserve System relied less on monetary aggregates and more on a broad range of economic and financial variables to guide the implementation of monetary policy. This trend culminated in 1987 when the Federal Reserve decided to cease specifying an annual growth range for M1, 1/ the variable that had been viewed historically as the most reliable monetary aggregate in indicating changes in nominal income growth. More recently, however, the apparent long-run relationship between M2 and the price level has raised interest within the Federal Reserve in this broader monetary aggregate, the so-called P^{*} relationship. 2/

The analysis of Hallman __et al.__ builds upon the traditional quantity theory of money and specifies the long-run equilibrium price level as _{t} V2*/^{*} is the average value of M2 velocity for the period 1955:1 to 1988:1 and _{t} ≡ M2_{t} V2_{t}/Y_{t}, where V2_{t} is actual M2 velocity and Y_{t} is real GNP. Their empirical analysis finds that deviations of P_{t} from _{t} from V2^{*} and Y_{t} from

This paper recasts the analysis of Hallman __et al.__ in a more general framework based on a modern theory of money demand and the principle of cointegrated variables. Specifically, this paper undertakes an examination of whether a monetary aggregate can serve as a reliable predictor of movements in the price level. Most modern theories of money demand predict the existence of a relationship among a measure of money and such variables as the demand for goods and services, the corresponding price level, and the competing and own rates of return on money balances. More importantly, money demand theory can illustrate the forward-looking behavior of private agents that can reveal information about their expectations of future prices and expenditure plans. This aspect of the choice of how much money to hold also bears relevance to the forecasting of inflation.

The econometric methodology followed in this paper draws on recent developments in the theory of cointegrated variables and involves two basic steps. 3/ The first is to search for a long-run relationship among a monetary aggregate and appropriately defined transactions, price Level, and interest rate variables in which a linear combination of these variables shares a common trend with the monetary aggregate. If such a relationship is found, the set of variables is said to be cointegrated. The second step relies on the existence of cointegration and formulates an error-correction model that represents the process of adjustment: among the variables to their long-run relationship. 4/

The empirical analysis presented here finds that M2 but not M1 is cointegrated with relevant transactions, price, and rate of return variables. On that basis, the appendix proceeds to consider the short-run dynamics of M2 and price level variables. The analysis confirms the existence of a well-behaved demand equation for this broader monetary aggregate. However, the inflation equations collapse to simple auto-regressive models. This result suggests that M2 rather than the price level adjusts in the short run to restore the long-run relationship among these variables. Thus, relatively little information about the inflation process is gained by analyzing changes in M2 alone.

While a long-run relationship exists among M2, the price level and other determinants of money demand, the inflation process may be better represented by considering developments in both the goods and money-markets rather than by those in the money market alone. Empirically, this is achieved by substituting for the transactions variable in the long-run relationship a measure of potential output. This substitution is made because the real demand for goods and services is not expected to deviate persistently from its potential supply in view of the economy’s tendency to close any output gap over time and its intertemporal budget constraint. Deviations from the long-run relationship between money, potential output, price level, and interest rate variables thus reflect the process of adjustment in both the goods and money markets. Dynamic price equations based on this long-run relationship indicate that, when the deviations from this long-run relationship are decomposed into a term related to the velocity gap in the Federal Reserve’s analysis and an output gap, the velocity gap term is insignificant and the traditional Phillips curve approach to inflation based on an output gap remains useful—that is, contrary to the Federal Reserve’s analysis, deviations of M2 from its long-run relationship to the determinants of money demand do not enhance inflation forecasts. However, changes in M2 do contain some information about future inflation, consistent with the view that the demand for M2 reflects forward-Looking behavior by households and businesses.

The remainder of the paper is organized as follows: Section 2 outlines the basic principles of money demand theory and highlights their relevance to forecasting inflation. The third section searches for long-run relationships among monetary aggregates and the measurable determinants of money demand. Section 4 examines the short-run adjustment in the money market by estimating dynamic equations for M2 and the level price variables. The fifth section extends the analysis of inflation to include developments in the goods and money markets which substantially improves the dynamic price equations. Section 6 offers some conclusions.

## II. Theoretical Considerations

While the objectives of this paper are primarily empirical, economic theory provides an important foundation for the econometric methodology based on cointegration used below by providing guidance concerning the variables to use in the search for a long-run relationship between, __inter alia__, a monetary aggregate and price level variable. More importantly, money demand theory can illustrate the forward-looking behavior of private agents that can reveal information about their expectations of future prices and expenditure plans. This aspect of the choice of how much money to hold is particularly relevant to the forecasting of inflation.

Most modern theories of money demand predict a relationship among a measure of money and such variables as the demand for goods and services, price level, and certain interest rates. Moreover, money demand theory serves to identify other factors that influence money demand and to clarify which assets yield monetary services. In general, monetary assets are distinguished by the yield of a nonpecuniary return arising from their role as a medium of exchange or the relatively low cost at which they can be converted to a medium of exchange. These characteristics explain why monetary assets are held even though they are clearly dominated in their rates of return by some financial assets. Thus, changes in factors such as regulations and payments technology, by altering the monetary services of assets, influence both the quantity of money demanded and potentially the appropriate measure of money.

The basic principles of money demand theory can be derived from a dynamic optimization model of a representative household. The demand for money fundamentally reflects choice at the margin between holding an additional unit of money and the alternative uses of wealth (e.g., spending on goods and services or holding a financial asset). At an optimal intertemporal plan for consumption expenditures, portfolio allocations and money balances, the household is indifferent to small changes in its plan. This condition yields predictions about money demand that can be tested empirically.

The specific formulation of the representative household’s dynamic optimization problem used here emphasizes the forward-looking aspect of its choice of how much money to hold. Consider a hypothetical household that seeks to maximize

where c_{t} and l_{t} are the household’s consumption and Leisure during t, δ is the rate of time preference, and Ω_{0} is the set of available information when the intertemporal plan is formed. 5/ The utility function, U(.,.), is assumed to be quasiconcave and thus to yield unique, positive values for c_{t} and l_{t}. For simplicity, labor is taken to be supplied inelastically.

Money indirectly enters the optimization problem. Real money balances enable the household to increase its leisure by reducing the amount of “shopping” time, s_{t}, required to purchase c_{t}. 6/ The units of time are chosen so that there is one unit of time per period available for shopping and leisure together, l_{t} = 1 - s_{t}. Given prevailing regulations and payments technology the time required for undertaking transactions increases with the amount of consumption and decreases with the quantity of real balances held by the household (up to a satiation level). Specifically,

where Ψ(.,.) has partial derivatives Ψ_{1} > 0 and Ψ_{2} ≤ 0. M_{t-1} is the nominal stock of money held at the beginning of period t, and m_{t-1} = M_{t-1}/P_{t} where P_{t} is the money price of the consumption good. 7/ In addition to services from its role as the medium of exchange, money held at the beginning of period t can yield a pecuniary return of (1 + RM_{t}) at the beginning of period t+1.

As an alternative to consumption or holding real money balances, the household can assemble a portfolio of bonds. These securities can be purchased at par in period t and be redeemed for (1 + RB_{t}) units of money in period t+1. B_{t-1} denotes the number (possibly negative) of bonds held by the household at the beginning of period t, while b_{t-1} = B_{t-1}/P_{t}.

In the above setting, the household’s budget constraint is

Here y_{t} is real labor income and π_{t} is the inflation rate, (P_{t + 1} - P_{t})/P_{t}.

At the optimal intertemporal plan for c_{t}, m_{t}, and b_{t}, the household is indifferent to small changes in this plan. 8/ Two such perturbations are considered. First, consumption is reduced by one unit and real money balances are raised by one unit during period t. As a result, utility changes by

During period t+1, more leisure results from the (1 + π_{t})^{−1} increase in real money balances at the beginning of the period. In addition, consumption is raised during period t+1 by (1 + π_{t})^{−1} (1 + RM_{t}) to restore real money balances to their original paths. These changes in utility are given by

At the optimal plan, (4) and (5) must be equal.

The second perturbation to the plan is similar to the first except that bonds are used to substitute consumption intertemporally rather than money balances. The change in utility in period t is again given by (4), while that in period t+1 is

At the optimal plan, (4) and (6) must be equal, and thus so must be (5) and (6).

The two perturbations when combined determine a portfolio-balance relationship among E(m_{t} | Ω_{t}), E(c_{t + 1} | Ω_{t}), E(Π_{t} | Ω_{t}), RB_{t}, and RM_{t}. 9/ The equality between (5) and (6) can be expressed as the implicit function

The second term is simply the ratio of the expected marginal utility of the transactions services of money to the expected marginal utility of consumption. If (7) is solvable for m_{t}, the demand for money balances satisfies the relationship.

since l_{t+1} = 1 - Ψ(c_{t+1}, m_{t}). Strictly speaking, (8) is not a money demand function, but rather a first order condition that the demand for money must satisfy at an optimal holding of money balances.

Several difficulties obviously exist in generalizing from a money demand theory for a representative household to that for the economy as a whole, including the problem of aggregating over households and the existence of other types of economic agents. While the former problem generally is not addressed in macroeconomics, a model of a representative firm’s demand for money analogous to that above could be developed. In that case, an expression similar to (7) would relate the difference between the competing and own rates of return on holding money balances to the monetary services of the assets. This consideration implies that the relevant transactions variable for the economy as a whole is much broader than consumption.

Two basic conclusions about money demand are derived from the above model and its qualifications. First, the demand for a stock of money at a point in time reflects real expenditure plans, expectations about the price level, and the competing and own rates of return on money balances. The appropriate transactions variable and corresponding price level go well beyond household consumption. Therefore, the amount of money private agents choose to hold may contain information useful in predicting movements in price level variables such as the GNP deflator or that for total domestic demand. Second, changes in such factors as regulations and payments technology (represented by shifts in Ψ(.,.) in the above model) alter the monetary services of assets, influencing both the quantity of money demanded and potentially the set of assets that yield a monetary return.

## III. Testing for Cointegration

The potentially measurable determinants of money demand include real transactions and price level variables, as well as competing and own rates of return. However, the precise empirical quantities that correspond to the theoretical concepts are not clearcut, as is indeed the case for money. The search for a long-run relationship that includes a monetary aggregate and price level variable thus covers, in addition to two monetary aggregates (M1 and M2), several measures of real transactions and corresponding price levels. These latter variables are real GNP (GNP82), real domestic demand excluding federal government expenditures (DDXGF82), real total domestic demand (DDT82), and the implicit deflators associated with these variables (PGNP, PDDXGF, PDDT). Two competing rates of return are used, the three-month Treasury bill rate (RTB) and six-month commercial paper rate (RCP). The own rate of return on M1 (RM1) is a moving weighted average of the nominal yields of the components of M1, where the weights reflect component shares in the aggregate. Two own rates of return on M2 are considered, one corresponds to that for M1 (RM2) while the other is a moving weighted average of the yields on the non-M1 components of M2 (RM2NM1). Finally, a time trend serves as a rough proxy for technological progress in the payments system. 10/

The theory of cointegrated variables provides tests for the existence of a long-run relationship among a monetary aggregate and the determinants of money demand. To implement these tests, long-run equations (i.e., equations in level terms only) are estimated for the various sets of variables. The residuals of these equations are then examined to determine whether deviations from the long-run relationship tend to persist or to diminish over time. Specifically, the equations’ residuals are tested for their order of integration. 11/ If the null hypothesis that the residuals are integrated or order zero cannot be rejected, the set of variables is said to be cointegrated. 12/

### 1. Time series properties of data

An important preliminary to the tests for cointegration is to examine the order of integration of each time series to be included in the long-run relationship, and in particular to test for the presence of a unit root. 13/ The tests for cointegration used in this paper are appropriate only if each time series (or linear combination of two series) is integrated of order one.

The tests to detect the presence of unit roots in the individual time series presented forthwith include the Augmented Dickey-fuller (ADF) test and two associated with Phillips and Perron, z(α) and z(t_{α}). 14/ The former is well known. The latter two are the autoregressive coefficient and corresponding t-statistic both corrected for serial correlation in the regression’s error term. Moreover the Phillips-Perron tests are based on several alternative hypotheses about the time series properties of the variables, which permit testing for __inter alia__ the presence of a non-zero drift and non-zero mean in the time series under consideration. 15/ The testing methodology is amplified in Perron (1986).

The strategy begins with the regression equation

which permits testing the null hypothesis of a unit root with drift against the alternative hypothesis of a stationary series with a deterministic trend. 16/ The results for selected variables are presented in Table 1, which includes in addition to the ADF, z(α) and z(tα) tests, a modified F-test, z(Φ_{3}), of the null hypothesis H_{0}: (μ ≠ 0, β = 0, α = 1). 17/ The tests statistics indicate that the null hypothesis of a unit root cannot be rejected at the 5 percent significance level for any series. However, the inability to reject the null hypothesis could reflect the low power of these tests.

A second regression equation,

yields more powerful tests than those presented in Table 1. Before considering the test statistics based on this regression, however, it must be determined that μ^{*} is zero, since the test statistics are not invariant with respect to this parameter. The modified F-test, z(Φ_{2}), of the null hypothesis H_{0}: (μ = β = 0, α = 1) that is reported in Table 1 can be used to make this determination. With the exception of the three interest rate variables, the null hypothesis can be rejected leading to the conclusion that the other time series tested appear to have a unit root with drift.

Since the null hypothesis of μ equal to zero cannot be rejected for the three rate of return variables, these time series were tested for the presence of a unit root using the second regression equation. The results are presented in Table 2 and again the null hypothesis of a unit root cannot be rejected in each case.

Finally, a more powerful test of the null hypothesis of a unit root is possible for the own rate of return on M1, since the null hypothesis of a zero mean cannot be rejected for this series. A modified F-test, z(Φ_{1}), of the null hypothesis H_{0}: (μ^{*} = 0, α^{*} = 1) is reported in Table 2. A third regression equation

affords more powerful tests for the presence of a unit root in RM1 than those presented in Table 2. However, as seen in Table 3, the null hypothesis of a unit root still cannot be rejected.

To summarize, the Phillips-Perron tests presented in this subsection indicate one cannot reject the hypothesis that all of the basic time series variables considered in this appendix have unit roots. In contrast to the other variables, there is no evidence of drift in the rate of return variables—and in the case of the own rate of return on M1 the mean is zero (Chart 1). A further conclusion is that since price level measures are integrated of order one, the corresponding inflation measures are integrated of order zero.

### 2. Cointegration tests

The tests for cointegration used in this paper are the ADF, z(α) and z(t_{α}) tests applied to the residuals from the estimated long-run relationships. These relationships are derived from vector autoregressions (VARs) with lags over three quarters. 18/ The choice of variables to include in the VARs was guided by the theoretical framework above and the statistical properties of the individual time series. The residuals from the long-run relationships were then calculated for use in the cointegration tests.

The results in Table 4 indicate that, for a selection of transactions and price level variables, M1 is not cointegrated with these variables and the own and competing rates of return. The absence of cointegration in the case of M1 may stem from the regulatory reforms implemented in the late 1970s and 1980s. In light of this evidence, no attempt was made to estimate a dynamic money demand equation for this aggregate.

Tables 5 and 6 present test statistics for two types of long-run relationships that include M2. In the first, the cointegrating regressions relate M2 to a selection of transactions and price level variables. These long-run relationships are consistent with the velocity approach to the demand for M2 that underlies the P^{*} relationship of Hallman __et al.__. In this case, it is not possible to reject the null hypothesis of no cointegration. The second relationship includes in addition to those variables in the first the own and competing rates of return. 19/ The addition of these variables to the long-run relationships leads to the rejection of the null hypothesis of no cointegration.

The above results, which suggest that it would be inappropriate to include residuals from a velocity-based long-run relationship in a dynamic equation that represents the demand for M2, appears to reflect two factors. First, as is well known, empirical estimates of the speed with which money balances adjust to gaps between the desired or “long-run” demand and actual holdings is typically very slow. Second, in the early years of the sample period, the own rate of return on M2 was subject to regulation while in more recent years it has exhibited inertia with the result that throughout the period there has been significant movements in the opportunity cost of holding M2. 20/ These opportunity cost movements, some of which persisted, resulted in offsetting movements in velocity. When these two factors are taken together, the power of the statistics presented here is such that the data is interpreted as being consistent with the absence of a tendency for money demand to revert to a long-run equilibrium value. The inclusion of the own and competing rates of return in the cointegrating regression incorporates the second factor and the statistical tests then point to the rejection of the hypothesis of no cointegration.

Two representative long-run relationships derived from the VARs for M2 are presented below, one for the case in which the transactions variable is real GNP and the other in which it is real total domestic demand. These relationships are:

respectively. To allow for the endogeneity of the transactions and price level variables, their lagged values are used as predictors of their current values. 21/

There are a couple of noteworthy features in these equations. First, in both cases the income and price elasticities fall within two standard deviations of unity. This implies that, abstracting from shifts in relative rates of return, velocity measures will tend to be stable over the long run. Moreover, the equations imply that homogeneity with respect to the price level is satisfied for M2. Second, the coefficients on the own and competing rate of return variables are approximately of equal and opposite signs, indicating that the demand for nominal M2 is in this respect primarily affected by the opportunity cost of holding money. 22/ Converting the reported semi-elasticity values to elasticity values, at 10 percent nominal interest rates the coefficients imply own and competing elasticities of 0.4 when DDT82 is the scale variable and 0.5 to 0.6 when GNP82 is the scale variable.

## IV. Short-run Dynamics of Money Demand and Prices 23/

This section presents error-correction representations of the demand for nominal M2 for the cases where the scale variable is real GNP and total domestic demand (with corresponding price variables). These representations elucidate the short-run dynamics of money demand and result in equations for nominal M2 which can be evaluated with a view to determining whether a stable money demand function exists. 24/ The regressions are run on quarterly data from 1959:4 to 1989:4.

LM test for autocorrelation 25/

ARCH: 26/ F(4,107) = 0.93

Heteroskedasticity: 27/ F(10,104) = 1.239

ARCH: F(4,106) = 0.69

Heteroskedasticity: F(10,103) = 1.672

The t-ratios are in parentheses and ECM refers to the residuals derived from the corresponding long-run equation estimated above. Although differing in detail, both equations are broadly similar in terms of overall performance. Specifically, they both satisfy a broad range of statistical tests.

Concerning the specifics of the equations, with particular attention on equation (11) which includes the GNP scale variable, the coefficient on the error-correction term (ECM), which can be interpreted as a speed of adjustment parameter, indicates that in common with much of the empirical literature in this area that the speed with which money demand adjusts to shocks is very slow. 28/

Given the concern with the ability of monetary aggregates to track macroeconomic variables, an important question is how the equation_performs over time. In Chart 2, the one-step residuals (Y_{t} - X_{t}’β_{t} = u_{t} where β_{t} is the estimated β using data up to and including t) are graphed together with their current standard errors (±2α_{t}). This graph indicates that the residuals have tended to become larger in recent decades—the standard-error bounds have widened indicating that the variance has increased—and that on occasions the residuals have exceeded the bounds. This suggests caution in using M2 as an indicator.

Further insight into the behavior of equation (11) can be found by considering the one-step ahead forecasts for the period 1987:3 to 1989:4 (Chart 3). The forecasting bounds are never exceeded. Moreover, a X^{2} test comparing within and post-sample residual variances for parameter constancy cannot reject at the 5 percent confidence level the hypothesis of parameter constancy. However, the forecasts are static. The chart indicates a systematic tendency in recent quarters to overpredict money demand, implying the possibility of cumulative errors, which potentially could be serious.

Charts 4 and 5 track the recursive least squares coefficients for the ECM term and the lagged difference in the own rate of return. Chart 4 shows that the coefficient on the ECM term shifted in the early 1980s, possibly reflecting the impact of financial deregulation. The coefficient on the own rate of return variable 29/ clearly shows the impact of financial deregulation in the early 1980s when the coefficient became significant—prior to that, the variable would have exhibited insufficient variation. 30/

To this point the focus has been on estimating the demand function for nominal M2 balances. As noted earlier, this paper is primarily concerned with the possibility that M2 might be a useful predictor of inflation. As a first step to evaluating this possibility, the residuals from the long-run relationship were included in an error-correction formulation of inflation behavior to see if information obtained from an empirical analysis of the money market could enhance inflation forecasts. The outcome of the resulting testing down procedure was an autoregressive process in inflation—in particular, the ECM term derived from the relevant long-run money demand equation had no significant impact. 31/

## V. Adjustment in the Money and Goods Markets and Inflation

While a stable long-run relationship exists between M2 and the price level, the inflation process may be better represented by developments in both the goods and money markets rather than by those in the money market alone. This can be achieved by substituting a measure of potential output for actual demand in the long-run relationship. Specifically, consider the form of long-run equations (9) and (10)

Substituting potential output (YP) results in,

Setting equation (12) equal to equation (13),

That is, the residuals in equation (14) can be decomposed into those from the estimated equation for money-market equilibrium and an output gap measure. The residual ê is the counterpart to the difference between P and P^{*} in the framework of Hallman __et al.__. Since e has already been shown to be I(0) and tests indicate that the output gap measures are also I(0), it follows that ê is also I(0), 32/ indicating that M2 is cointegrated with measures of the price level, own and competing rates or return, and potential output.

The following error-correction representations of the inflation process were estimated for the cases of the GNP and total domestic demand deflators:

ARCH: F(4,108) = 0.66

Heteroskedasticity: F(6,109) = 0.536

ARCH: F(4,105) = 1.32

Heteroskedasticity: F(12,100) = 0.799.

The period of both regressions is 1960:1 to 1989:4. ECMGAP refers to the residuals derived from the corresponding long-run equations.

Concerning their statistical properties, there is evidence of significant first-order autocorrelation in equation (15). Since there is no such evidence in equation (16), the problem may arise from the differing definitions of the alternative scale and inflation variables. 33/

As regards the content of the equations, in contrast to the earlier result where the inflation equation degenerated into an autoregressive process with no role for developments in the money market, the residuals from the long-run relationship that also includes goods market behavior are significant. Moreover, in both cases, knowledge of the behavior of changes in the stock of M2 contain information about future inflation, suggesting that the decision of how much money to hold reflects forward-looking behavior by private agents. Changes in the competing rate of return to holding M2 are also helpful when the domestic demand deflator is under consideration.

To elaborate on these results, equations (15) and (16) were re-estimated allowing for the decomposition of the residuals in equation (14). This resulted in the following two equations.

ARCH: F(4,108) = 0.36

Heteroskedasticity: F(8,107) = 0.225

ARCH: F(4,106) = 1.12

Heteroskedasticity: F(10,103) = 0.371

GNPGAP and DDTGAP refer to the relevant output gaps. 34/

Concerning their statistical properties, there is again evidence of significant first-order autocorrelation in the equation focusing on the GNP deflator (equation (17)).

As regards the form of these equations, the most notable feature is that the ECM term is not included in the final equation. This implies that deviations of money from its long-run relationship—the analogue of Hallman __et al.’s__ velocity gap—are not useful in the present context in predicting inflation, a result that is consistent with the earlier empirical analysis. The close parallels between the equations above and those reported by Hallman __et al.__ naturally suggest the question of where the discrepancy arises. In this connection it is worth noting that the velocity gap term reported in Hallman __et al.’s__ preferred equation is significant only at about the 7 percent level. Moreover, their regressions terminated in 1988:1 and inspection using recursive least squares indicated that the additional seven quarters of data in the regressions presented here if added to the time series used in their regressions would further weaken their result.

That the primary influence of money on inflation is captured by the output gap has been noted before. Specifically, Gordon (1985) showed how to decompose “excess” nominal GNP growth (his analogue of the output gap) into the sum of “excess” money growth and actual velocity growth. If money’s influence on inflation arises solely ‘hrough this decomposition, then the money and velocity terms, when entered in place of his output gap measure, should have the same coefficient values. When estimated in unrestricted form, he finds the two estimated coefficient values to be statistically indistinguishable and concludes that money has no influence in predicting inflation apart from its effect on the output gap.

It should be emphasized that this is a statistical rather than a theoretical result. It reflects the endogeneity of M2, a broad monetary aggregate, and indicates that the price level leads money (M2) rather than vice versa. This conclusion was confirmed by some Granger causality tests. 35/ Finally, note that there is a long-run relationship between money and prices as evidenced by equations (9) and (10).

While M2 may not be a useful anchor for the price level, it is worth noting that __changes__ in the stock of M2 do contain information about future inflation as evidenced from equations (17) and (18). One explanation for this result is that the demand for money reflects forward-looking behavior by private agents.

Finally, Charts 6 and 7 present the one-step residuals and one-step ahead forecasts for the period 1987:3 to 1989:4 for equation (17). By both criteria, except for an outlier around the time of the first oil-price shock (Chart 6), the equation is well-behaved. In particular, the one-step ahead forecasts indicate that thee is no persistent tendency to over- or underpredict inflation suggesting that dynamic forecasts on the basis of this equation would perform in a relatively satisfactory manner.

## VI. Concluding Observations

The principal results of this paper may be briefly summarized as follows. First, a dynamic optimization model indicates that, although sensitive to changes in regulation and payments technology, the demand for money reflects real expenditure plans, expectations about the price level, and own- and competing-rates of return to holding money.

Second, empirical tests indicate that all of the basic time series considered in this appendix have unit roots (I(1)). The tests also indicate that for the case of M1, money, price, income, and the relevant rate of return variables are not cointegrated. For the case of M2, these variables are cointegrated, although the evidence suggests that if a velocity framework is adopted, and thus the rate of return variables are excluded, the relevant variables are not cointegrated. The velocity approach underlies the Federal Reserve’s P^{*} relationship.

Third, focusing on M2, estimates of long-run relationships that include this aggregate are not inconsistent with a unitary income elasticity of the demand for money and homogeneity with respect to prices. The empirical analysis also indicates that there is a stable well-defined money-demand function for M2. However, the estimated inflation equations based on information obtained from money market behavior collapse to autoregressions indicating that M2 adjusts in the short run to the price level rather than vice versa.

Fourth, when its scope is expanded to allow for developments in both money and goods markets, the empirical analysis points to the conclusion that deviations of money from its long-run relationship do not significantly enhance inflation forecasts based on conventional output gap models. However, changes in M2 do contain information about future inflation, suggesting that the demand for M2 reflects forward-looking behavior by private agents.