The demand for nominal and real money balances in a large macroeconomic system.

I. Introduction Cointegration has become a popular empirical technique used to estimate the existence and stability of money demand relationships. A common approach has been to estimate a single equation system with real money balances (M1), real income and an interest rate, although Hafer and Jansen [13], Friedman and Kuttner [10] and Stock and Watson [29] find only minimal support for this specification. Baba, Hendry and Starr [6] estimate a more complicated Ml relation by including specific opportunity cost variables and find stronger support for the M1 relation using post WWII data. There have been four reasons put forth to explain the lack of a cointegrating relation between M1, real income and the interest rate. Stock and Watson [29] conclude that a stable money demand relation exists in data from 1900-89 but not for the post-war period alone. Their explanation is that sufficient variation exists in the longer sample to identify money demand, but excessive multicollinearity among variables during the post-war period prevents unique identification. Baba, Hendry and Starr [6] (BHS) maintain that misspecification of the M1 relation is the reason for poor results. They augment the money demand function with inflation, a measure of long term bond yield and risk, and learning curve weighted yields on newly introduced M1 and M2 assets. BHS claim their specification is able to account for the missing money period of 1975-76, the velocity decline of 1982-83 and the M1 explosion of 1985-86 while standard M1 models cannot. Friedman and Kuttner [10] offer a third reason why the simple money demand equation fails. They estimate a single equation model for several periods and conclude that a money demand vector exists over the 1960.2-79.4 period for M1. However, when data is included up to 1990.4, they are unable to estimate the money demand relationship. Friedman and Kuttner conclude that the money demand relationship may have broken down in the 1980s. A fourth explanation relates to the work of Phillips [27] and Johansen [18], who demonstrate that if important variables are omitted from a cointegration analysis, an incorrect number of vectors and biased estimates of the coefficients will result. In addition, the estimates will generally be inefficient. To demonstrate these points in a macroeconomic context, Cutler, Davies, Rhodd and Schwarm (CDRS) [7] consider money demand in conjunction with consumption, investment and import markets, and are able to identify a simple money demand representation better than using a single money market alone. All of the research cited above used real money balances which constrains the price elasticity to unity prior to estimation. In this paper, we use a model similar to CDRS, except that we separate money and prices to allow for a separate estimation of the price elasticity. In several time periods, the elasticity is estimated to be less than unity. Therefore, real money balances may be not always be consistent with the data and implied economic behavior. In addition, the results for our money market improve relative to the CDRS specification. We obtain more reasonable elasticities for consumption and investment, and we have greater success in identifying all markets. Our macroeconomic system, with its multiple markets, originates with King, Plosser, Stock and Watson [21].(1) They combine real GNP, consumption, investment, real money balances, nominal interest rates and the inflation rate into cointegrating vectors that represent money, consumption and investment markets. We augment their model by including imports and an import market as a fourth vector. We also use the ex-post real interest rate instead of the inflation rate and separate money and prices in order to test for a unitary price elasticity in the money market. To demonstrate gains from systems estimation, we compare single equation and systems estimates using a method developed by Johansen and Juselius [19]. Examining the period 1960.2-90.4, we use a more systematic rolling regression approach to deal with the sensitivity of sample periods. We find more support than other researchers for an Ml vector during the post-war period, which may be due to the efficiency gains of full system estimation as well as splitting Ml and the price level. We also conclude that the price elasticity of money demand is not unity for the late 1980s, which suggests that monetary decisions are based more on nominal than real money balances in this later period. Section II presents the four market macroeconomic system we use to reduce the biases and inefficiencies associated with estimating partial systems. Section III presents the empirical procedures, results and the speeds of adjustment. Then, three special issues are analyzed in section IV: the role of the price level in creating equilibrium in the system; the specific gains in efficiency from using a full system; and the implications of a price elasticity in the money demand function that is not unity. Section V contains a brief conclusion. II. Economic Model We are motivated to construct a more complete system than has been typically used to analyze money demand. Our approach is expressed in the following simple macroeconomic relations: [C.sub.t] = [a.sub.0] + [a.sub.1][Y.sub.t] + [a.sub.2][r.sub.t] + [e.sub.t] (1) [I.sub.t] = [b.sub.0] + [b.sub.1][Y.sub.t] + [b.sub.2][r.sub.t] + [u.sub.t] (2) [M.sub.t] = [d.sub.0] + [d.sub.1][Y.sub.t] + [d.sub.2][r.sub.t] + [d.sub.3][P.sub.t] + [v.sub.t] (3) I[M.sub.t] = [g.sub.0] + [g.sub.1][Y.sub.t] + [g.sub.2][r.sub.t] + [z.sub.t] (4) where Y is real GNP, C is real consumption, I is real investment, r is the expost real interest rate, M is money balances, P is the price level, IM is real imports and t is a time subscript.(2) The variables [e.sub.t], [u.sub.t], [v.sub.t] and [z.sub.t] are error terms. This system of equations implies four cointegrating vectors: equation (1) gives a simple consumption function; (2) is the investment function; (3) is money demand; and (4) is an import equation. The first two equations are standard and simple representations of their respective markets, but the import and money demand functions are somewhat different than traditional versions. We captured the effects of the traditional import demand model, which includes income and relative prices, through conditioning variables discussed below. The real interest rate is also included to make the import equation a variation of the consumption function, with the same variables driving import and domestic demand after adjusting for foreign relative price and interest rate effects. The main variation in the money demand equation is that the price level and nominal money balances are used instead of real money balances. These four equations represent logically distinct but clearly interrelated markets.(3) Each equation represents a long-run equilibrium relation with the error terms capturing disequilibria in each market; hence each is a separate cointegrating vector. As an example, the model predicts that C, Y and r must satisfy equation (1) over time such that [e.sub.t] is a stationary process. Thus, the seven variables in the system are combined into four equations whose residuals are unique stationary processes. The King et al. system may be viewed as a special case of the above system. They use consumption, investment, income, a nominal interest rate, real money balances (M2) and inflation to create a restricted version of equations (1), (2) and (3). Their consumption vector is represented as a "great ratio," which in effect imposes the restriction that [a.sub.1] = 1 in the system above. The investment vector is constructed in a similar fashion, with [b.sub.1] = 1, and their money vector is similar to ours except that the coefficient [d.sub.3] is restricted to unity. Their model differs from ours in two additional respects. King et al. reflect the real interest rate by using the inflation rate and nominal interest rate which, on the surface, is not of great consequence. Also, we introduce a foreign sector by examining the role of imports explicitly and exports implicitly through conditioning, whereas King et al. examine a closed system. III. Empirical Procedures and Results In this section, we first review the procedures used to analyze our system and then focus on an overview of the results. The overview presents results of single and joint tests done on the full system and also on single equation models. At the end of this section, issues concerned with the increased efficiency and reduced bias obtained from full system estimation are addressed and the speed of adjustment coefficients are evaluated. Empirical Procedure The seven variables included in our system are [X[prime].sub.t] = [[Y.sub.t][C.sub.t][I.sub.t][M.sub.t][P.sub.t][r.sub.t]I[M.sub.t]], which are expected to lead to four cointegrating vectors.(4) (Each variable is defined in the appendix). However, there are other associated variables, such as government expenditures and foreign sector variables, that influence movements in [X.sub.t]. Johansen and Juselius [19] and Baba, Hendry and Starr [6] have pointed out that when cointegrated systems become too large, estimation and interpretation can be difficult. Therefore, it has become common practice to introduce conditioning variables that do not affect the size of the system. Conditioning variables are used to eliminate unwanted influences that might affect the estimates of the cointegrating vectors. Since they are not in any hypothesized vectors, it would be inappropriate to include them in the system; conditioning variables are, however, influential and their effects need to be included. We have chosen real government expenditures and real exports because they help determine the level of real GNP and other variables in our system. Foreign variables are used to specify the import sector correctly. We elected to include a foreign interest rate and the Canadian dollar/dollar, deutschmark/dollar, and yen/dollar exchange rates to represent relative interest rates and rates of inflation among major trading partners. Additionally, due to the possible importance that government deficits have on economic decisions through the Ricardian effect, we include the real federal government deficit.(5) The conditioning vector ([Z[prime].sub.t] = [RG.sub.t][EX.sub.t][EURO.sub.t][CAN.sub.t][GER.sub.t][JAN.sub.t][DEF.sub.t] therefore includes real government expenditures, exports, a foreign interest rate, the C$/$, DM/$ and Yen/$ exchange rates, and the real federal deficit, respectively. The steps involved in the Johansen approach are, briefly, to difference [X.sub.t] and regress [Delta][X.sub.t] on [Delta][x.sub.t-1], [Delta][X.sub.t-2], ..., [Delta][X.sub.t-k-1] and [Delta][Z.sub.t] and save the residual vector.(6) Then [X.sub.t-k] is regressed on the future [Delta][X.sub.t-1], [Delta][X.sub.t-2], ..., [Delta][X.sub.t-k-1] and [Delta][Z.sub.t]. The resulting residual vector is also saved. This latter vector is nonstationary and contains the elements of the cointegrating vectors. Using reduced rank regression techniques, the covariance matrix of the two residual vectors is calculated and associated eigenvalues and cointegrating vectors are estimated [1]. Our system cannot accommodate both the real and nominal interest rates along with the price level because perfect correlation exists between the two interest rates and the difference in the price level. The latter variable enters the model as part of the [Delta][X.sub.t-i] in the initial regressions discussed above. Therefore, the stationary impact of inflation has been taken out in the first part of the Johansen technique, so that using nominal or real interest rates leads to similar results. We chose to use the real interest rate because it appears in three of the four markets. Testing for the Number of Cointegrating Vectors and Parameter Values The unrestricted matrix of cointegrating vectors was estimated using Johansen's method. Three levels of hypothesis testing were then performed on these results: (1), the trace test was used to determine the number of cointegrating vectors in the system; (2), Johansen and Juselius' [19] [H.sub.5] test was employed to determine if restrictions derived from equations (1)-(4) are valid individually; and, (3), joint [H.sub.5] tests based on parameter values from test results in (2) were performed to determine whether multiple markets existed simultaneously. The first tests are concerned with whether the estimated number of cointegrating vectors is consistent with the model. Our seven variable system is predicted to yield money, consumption, investment and import markets, so we expect four significant eigenvalues and associated eigenvectors. In most cases, the exact number of vectors predicted by our model is found with a significance level of one percent. We felt this critical level was usually appropriate because statistical tests on the number of vectors in large systems appear to over-estimate the number of cointegrating equations [9]. Johansen and Juselius [19] have pointed out that unconstrained vectors may be linear combinations of the true vectors and therefore might not have much economic interpretation. The second stage of hypothesis testing is then to determine whether estimates of the unconstrained vectors reduce to the following form: [Mathematical Expression Omitted]. (5) The first column vector contains the cointegrating coefficients for the consumption market, while the other columns include appropriate values for the investment, money and import markets respectively. Johansen and Juselius' [19] [H.sub.5] test restricts all coefficients on a specified number of vectors. Under the null hypothesis, the [H.sub.5] test statistic is [[Chi].sup.2] with (p - r)[r.sub.1] degrees of freedom. The letter p is the number of variables, r is the total number of vectors and [r.sub.1] is the number of restricted vectors. Tests can be made on single vectors, so that one vector is restricted while other vectors are freely estimated. It is also possible to restrict multiple vectors to see if more than one market can be identified simultaneously. Referring to (5), this would imply restricting the entire matrix of coefficients simultaneously. One remaining issue concerns the acceptance levels for the [H.sub.5] test. Johansen and Juselius [19] use probability values ranging from .15 and above to accept hypotheses concerning purchasing power parity and interest rate parity. We were similarly concerned with a Type II error in that we wanted to find structural equations, so we used larger probability values to accept a restriction. Estimation Results The Estimated Number of Cointegrating Vectors. Table I first presents the estimated number of cointegrating vectors for the full system of variables, conditioning on [Delta][Z.sub.t], over the following periods: 1960.2-76.4, 1960.2-77.4, 1960.2-78.4, ..., 1960.2-90.4. We use this series of rolling estimates to allow a more careful analysis of changes that occur in the money, consumption, investment and import markets over time. Table I. Estimated Number of Cointegrating Vectors for Various Subperiods for the Full System and Single Equation Models Full System Single Equation (1) (2) (3) (4) (5) M1 C I IM 1960.2-90.4 4 1 0 1 0 1960.2-89.4 4 1 0 1 0 1960.2-88.4 4 1 0 1 0 1960.2-87.4 4 1 0 1 0 1960.2-86.4 4 1 0 1 0 1960.2-85.4 4 1 0 1 0 1960.2-84.4 4 1 0 1 0 1960.2-83.4 4 1 0 1 0 1960.2-82.4 4 2 0 0 0 1960.2-81.4 4 1 0 1 0 1960.2-80.4 4(*) 1 0 1 0 1960.2-79.4 4(*) 1 0 2 0 1960.2-78.4 4 1 0 2 1(*) 1960.2-77.4 4 2 0 2 1(*) 1960.2-76.4 4 2 0 2 1(*) Notes: The Full System in column 1 includes Y, C, I, M1, P, r and IM and uses differences of real government expenditures, real government deficit, real exports, the Eurodollar rate and the C$/$, DM/$ and YEN/$ exchange rates as conditioning variables. The Single Equation model for the money market in column 2 uses Y, M1, P and r with no conditioning on any variables. The consumption, investment and import markets use respectively: C, Y, r; I, Y, r; and IM, Y, r. The number of cointegrating vectors was computed using Johansen's technique with four lags. The trace test was used at the one percent level to determine the number of vectors unless an * appears, for which a five percent level was used. Five vectors were estimated for 1982.(4) at the one percent level but we used four vectors. Initial support for the full system is fairly strong. Using the trace test at significance levels of one percent, we found that between 1960.2-76.4, ..., 1960.2-90.4, four vectors were predominately estimated, which is the expected number. The only exceptions were that five vectors were identified for 1960.2-82.4, and the five percent level was needed to identify four vectors for 1960.2-79.4 and 1960.2-80.4. We also estimated the standard single equation system for money demand using income, a real interest rate, nominal money balance, and the price level, but without the conditioning variables. Column 2 in Table I shows the results. Theory predicts one vector and, in most cases, we observed one vector. However, the estimated parameter values are unstable and at times have the wrong signs. We will explore this issue in detail below. The results for single equation analyses of the consumption, investment and import vectors, equations (1), (2) and (4) in our system, appear in columns 3, 4 and 5 in Table I. We could not identify vectors in any time periods for consumption and could only estimate the import relation in the late 1970s. This is quite contrary to our success in the full system estimation. We were, however, often able to estimate the investment vector in the single equation analysis. Estimated Parameter Values Using Johansen and Juselius' [H.sub.5] Test. Table II presents normalized parameter values and probability levels for the [H.sub.5] test on the full system. Each vector was tested individually for all subperiods. We found reasonable parameter values at acceptable probability levels for money and consumption during the entire sample. Except for two periods, we [TABULAR DATA FOR TABLE II OMITTED] were able to find acceptable values for the import market. The conclusion for the investment market is also strong as we are able to find this market over most periods. Besides accepting that a vector for a particular market exists, the stability of its parameters across time is of concern. Each of the four markets analyzed all have real GNP (our [Y.sub.t]) as an argument, with an elasticity of -1.0 in the cointegrating vectors for money, and between -0.7 and -0.8 for the consumption equations. It appears that these latter values incorporate a tax effect. The original specification of equation (1) was [a.sub.1] = b(1 - [Tau]) where b is the MPC and [Tau] is the tax rate. Thus, our estimates of [a.sub.1] at 0.7 and 0.8 would be consistent with reasonable MPCs of 0.9 to 0.95 and overall average tax rates of 0.2 to 0.3.(7) The elasticity of real GNP was between -1.5 and -2.0 in the import vector. While substantially higher than other responses, this latter elasticity is plausible given the growth of imports as a proportion of income over this era and is also consistent with other estimates [2; 30]. The interest rate semi-elasticities are consistent across time periods for the consumption and money relations. In the investment market, a structural shift appeared after 1986.4, when the income elasticity declined from -1.7 to -1.4. We hypothesize that high real interest rates in the 1980s might have decreased the influence of real output on investment decisions and made investment more sensitive to changes in the real interest rate, as the semi-elasticity of the interest rate increased to 0.08 at the same time that the income elasticity declined. Also, the large elasticity on income appears plausible because investment is far more volatile than output and accounts for most business cycle episodes, so reactions of investment to changes in income should be elastic. [TABULAR DATA FOR TABLE III OMITTED] In addition to tests for a single vector by itself, it is important to test for the joint existence of multiple vectors. Table III presents [H.sub.5] results when we restrict multiple vectors simultaneously using parameter values from the single tests. It also includes tests on multiple vectors imposing the restriction that the coefficient on price in the money equation is equal to one. Attempts to identify all four markets jointly failed regardless of the coefficient imposed on price; however, we were able to find the money, consumption and import markets simultaneously from 1984.4-90.4. In addition, we found the money and consumption equations jointly for most periods, but the choice of the restriction given to the price coefficient was important. For periods after 1987, these equations are found only by allowing the price coefficient to be less than unity. The earlier years generally permitted a price coefficient of unity. Statistical Issues Reflections on the Improved Performance Obtained from Full System Models. Our introductory discussion indicated that single equation approaches can result in inefficient and possibly biased results relative to a full system estimation. Our ability to identify vectors during many sample periods with the full system, but not using single equation models, may reflect the greater precision of full system estimation. Also, this inefficiency of single equation estimates may partly explain why Hafer and Jansen, Friedman and Kuttner, and Stock and Watson do not find support for M1 with post WWII data. In addition, they imposed the restriction of a unitary price elasticity without testing by using real money balances, a restriction that we found was not valid in all time periods. Similar efficiency gains were seen in the full system results for consumption and imports. As Table I indicates, we never found a consumption vector in the single equation system and were only able to estimate the import vector in the late 1970s. In contrast, our results in Table II demonstrate success in estimating relatively stable markets for both consumption and imports. Phillips [27] showed that leaving out variables that are cointegrated with included variables results in biased estimates of the cointegrating vectors. Because income is also in the cointegrating vectors for investment and consumption, single equation estimates of money demand are unreliable. This suggests that all of our single equation estimates are biased and points to another advantage of using a full system. Stock and Watson [29] analyze the robustness of the Johansen technique for the money (real M1) market alone and conclude that small changes in data result in large changes in the estimated parameters. Our single money market results support Stock and Watson's findings, as the unrestricted income elasticities in that analysis ranged from 1.05 to -10. However, we find a much more stable money market using a full system. For the periods 1988-90, the acceptable ranges were narrow, between -0.9 and -1.1. Unfortunately, continued exploration with single market [H.sub.5] tests ([Alpha] = .15) in the earlier 1980s revealed income elasticities greater than -2.0. We then explored the range of income elasticities in the context of a joint [H.sub.5] test using the money, consumption and import markets together for periods in the earlier 1980s. The ranges in these analyses were usually about four tenths, so that, for example, the subperiod 1987 had an acceptable range of -0.7 to -1.0. (The 1985 range was the only one that was slightly higher, at -0.6 to -1.5). We conclude that, for some systems, efficiency gains are only realized when joint tests in the full system are used.(8) We identified the money market in every period, a substantial improvement over CDRS and most other money demand estimates, but the precision of our income elasticity declined in the early periods.(9) For the late 1970s, our single vector [H.sub.5] tests in the full system resulted in very large ranges for the income elasticity and we could not identify any markets jointly. This problem was noted by Hoffman, Rasche and Tieslau [16], who maintain that the small degrees of freedom available in estimations ending in the 1970s substantially reduces the efficiency of cointegration results. Whereas CDRS did not have this problem, we suspect that explicitly introducing the price level as a system variable, with its substantial volatility in the 1970s, reduced the precision of the results in this paper. Speed of Adjustment Coefficients. Our previous analysis emphasized long-run equilibrium relationships among system variables. However, cointegrated systems also have an error correction representation which relates each variable's growth rate to previous departures from equilibrium in all markets. The behavior of the money market illustrates the usefulness of this part of cointegration analysis. The money vector is represented by the restricted coefficients of the cointegrating vector, shown in the following equation: - 1.0[Y.sub.t] + 1.0M[1.sub.t] - .7[P.sub.t] + 0.02[r.sub.t] = [v.sub.t]. (6) If [v.sub.t] is positive, then the money market is above equilibrium. The adjustment back to equilibrium requires that either income rise, money demand fall, the price level rise or interest rates fall. [TABULAR DATA FOR TABLE IV OMITTED] Theoretically, any combination of the above movements will clear the market. Therefore, income and price variables should respond positively to [v.sub.t], and money and interest rates should move negatively to eliminate positive values of [v.sub.t]. We present error correction results for 1960.2-78.4 and 1960.2-90.4 for several reasons. The 1978 estimation results included the successful identification of all four vectors individually. The coefficients on these vectors were also typical of those found throughout the late 1970s. The last period, 1960.2-90.4, is a time when all four vectors are again identified individually and provides an overview of the entire sample. Table IV reports the relationships between the seven variables in the system and the disequilibrium errors in the associated markets. The disequilibrium residuals were calculated by multiplying the estimated vectors from Table II for each market by the relevant variables. Our analysis of the money market reveals that M1 moves to clear that market in both periods because the estimated speed of adjustment is -0.04 in 1978 and significant, and, in 1990, it is -0.08 and significant. M1 has a positive coefficient in the cointegrating vector for money, as shown in equation (6) above; thus, a positive disequilibrium requires that M1 fall to re-equilibrate the money market, just as the negative sign on the speed of adjustment coefficient suggests. Additionally, the price level and income have correct signs and are significant for the 1978 analysis. However, the 1990 price coefficient is negative, and therefore causes greater disequilibrium, while the income variable is insignificant. These issues will be discussed below. In the investment market, movements in investment itself are equilibrating in both periods. The investment variable has a positive sign in its cointegrating vector, which implies that the speed of adjustment should be negative. For the 1978 and 1990 periods, there are significant speeds of adjustment of -0.12 and -0.21, respectively. The real interest rate helps to clear the investment market in both periods as well because the estimates are negative and significant at the ten percent level. Movements in the real interest rate are insignificant in the consumption and import markets for both periods, which indicates that factors from the investment market mainly dictate changes in the real interest rate. The income variable is insignificant in both periods. The consumption variable is negative and significant in the 1978 analysis and therefore clears its own market, but it becomes insignificant in 1990.(10) The speed of adjustment parameter for income in 1990, however, is positive and significant, which suggests that income has become an equilibrating variable for the consumption market. The import market equilibrates through movements in the import variable in both periods, with a negative speeds of adjustment that are significant at a minimum of ten percent. The other potential market clearing variables, the real interest rate and income, are insignificant in both periods. The role of the income variable demonstrates the extended insights that can be gained from a full system analysis. Income appears in all four markets and probably cannot move to maintain equilibrium simultaneously in each market. For 1990, output moves to re-equilibrate the consumption market since the speed of adjustment is estimated to be 0.22 and significant at the five percent level. We also observe that output causes greater disequilibrium in the import market as the estimated speed of adjustment is -0.013. This is expected since an increase in imports reduces the demand for domestic goods and thus causes a decrease in domestic output. Therefore, greater pressure is put on imports to clear its own market. These results offer insight into previous single equation models, such as that of Hoffman and Rasche [15]. Using monthly data, they found negative speeds of adjustment for money and the interest rate as we did, but they also obtained a significantly negative speed of adjustment coefficient on the income variable, which indicates that income is causing disequilbria in the money market. We suspect that their estimate is biased by ignoring the role of income in other markets, as we found that income was not significant in 1990 and was correct in the 1978 period. IV. Reflections on Economic Issues Three special issues are reported in this section. The first has to do with the role of price in creating equilibrium in the money market. Secondly, we discuss specific improvements from the increased efficiency of using a full systems estimation. The section concludes with a discussion of the implications of a changing price elasticity in the money market. The Role of Price in Equilibrating the Money Market The speed of adjustment analysis provided several perspectives on the role of the price variable in equilibrating the money market. For the 1978 analysis, the estimated speed of adjustment was 0.02 and significant at the ten percent level. However, for 1990, the estimate is -0.025 and significant at the five percent level. A positive speed of adjustment is needed to re-equilibrate the money market, so movements in the price level caused disequilibria in the money market in 1990 but not in 1978. We hypothesize that the relative size of shocks to aggregate supply and demand account for the change in the sign of the speed of adjustment coefficient across time. During the 1970s, relatively large supply shocks were caused by changes in the price of oil. Blanchard and Quah [3], for instance, estimate the relative size of aggregate supply and demand shocks and find that supply shocks were large in the mid to late 1970s. The reversal of the speed of adjustment coefficient in the 1980s leads us to conclude that supply shocks were relatively large compared to demand shocks in the 1970s but not in the 1980s. To see the linkage between aggregate supply and demand shifts and the speed of adjustment directions, consider an outward shift in aggregate demand. This causes output to rise and [v.sub.t] in equation (6) to be negative. The error correction model suggests that APt should fall to help re-equilibrate the money market, but the outward shift in aggregate demand will cause increased pressure on prices. As output rises and [v.sub.t] becomes negative, inflation also rises, causing the money market to go further out of equilibrium. Thus, for the 1980s, when the speed of adjustment for the price variable was negative and significant, aggregate demand shocks appear to be more important. In contrast to the later years examined, the supply shocks of the 1970s had a tendency to offset demand shocks. As output rises in response to a positive supply shock, [v.sub.t] becomes negative. But, for changes in the price variable to re-equilibrate the money market, inflation must fall. It does with a supply shock, as the outward shift in aggregate supply causes a reduction in inflation and helps clear the money market. As shown above, demand shocks cause the opposite response and lead to disequilibrating behavior. Thus, we conclude that the larger supply shocks in this earlier era led to a correct speed of adjustment coefficient. However, in the later period when demand shocks were relatively larger, the inflation variable caused disequilibrium in the money market. The Stability of the Money Demand Functions Given the general failure of single equation estimates of M1 demand, researchers have emphasized better specification of this market. Baba, Hendry and Starr, in particular, maintain that poor single equation results are due to misspecification of the money market. BHS therefore construct a more sophisticated model for M1 using a volatility measure of bond holdings, a learning-adjusted maximum yield on instruments in M2 and averages of short and long-term bond yields.(11) They concluded that this specification successfully yields reasonable and stable parameter values in three periods where simpler specifications fail: the missing money phenomenon of 1974-76; the great velocity decline of 1982-83; and the M1 explosion of 1985-86. In their view, the rejection of parameter constancy was seen as evidence of misspecification rather than a shift in the money demand function. Our systems approach offers another avenue to deal with misspecification problems in the three periods analyzed by BHS. The full system results suggest that the M1 vector is easily identified for data through 1976 and also for the 1982-83 and 1985-86 periods. The income elasticities and interest rate semi-elasticities are stable for the entire 1980s, thereby giving support for a simple money demand representation. Table III indicates that the joint tests for the money, consumption and import markets also provide strong support for the stability of the money market. The full system estimates thus do not appear to be adversely affected by these three periods and successfully address the BHS concern of parameter constancy. Implications of the Price Elasticity in the Money Demand Equation The last issue of interest is the estimate of the price elasticity in our unrestricted money demand equation. We find that, by and large, the price elasticity can be restricted to unity for the years 1960.2-76.4, . . ., 87.4, indicating that economic agents made decisions based on real money balances during these years. However, from 1988 to 1990, we cannot accept this restriction. The first implication of a price elasticity less than one is that the level of real money balances does not remain the same in the face of one percent increases in the price level and nominal money balances. We hypothesize that, as prices increase, agents do not increase their M1 holdings as much because they can easily hold wealth in alternative assets and use telephone transfers or ATM machines. Thus, the fall in the price elasticity may be due to technological change which allows agents to more easily switch money holdings from M2 to M1; also, the timing of this change in our results, which occurs only in the late 1980s, seems consistent with the increased use of automatic banking. A second implication of a price elasticity less than unity is that the price level is not the correct deflator of nominal money balances. Therefore, dividing M1 by P before estimation is inappropriate. The proper specification is to model the demand for nominal money balances and include the price level to determine if agents react to real money balances, as we have done. Laidler [23, 161] states: The macroeconomic proposition that the demand for money is a demand for real balances is closely related in logic to the microeconomic proposition that equilibrium demands and supplies of goods and services depend only on their relative prices and not on the level of money prices. If the latter hypothesis were falsified, then so would the former, . . . Laidler suggests that testing by Meltzer [25], Laidler [22] and Goldfeld [11] provide support for a unitary price elasticity. However, he does acknowledge that using cointegration results, Boughton and Tavlas [5] and Boughton [4] conclude that the price elasticity is less than one in many countries. Laidler initially doubts the validity of these results but does concede that dynamic modeling may offer insights that standard regression analysis cannot. In contrast to Laidler, who taps misperception by economic agents as the source of a low price elasticity, we propose that technology and its consequent impact on the data series as the reason. V. Conclusion This paper presents the results of a Johansen cointegration analysis of a large macroeconomic system using rolling regressions from 1976 to 1990. The results of extensive testing of hypotheses indicate the presence of four markets, including money, consumption, investment and import markets. We found increased efficiency and reduced bias in estimates using the full system relative to single equation models. Also, we were able to find a simple M1 demand relation for years during which most simple models fail according to Baba, Hendry and Starr. Thus, a full system approach provides an alternative modeling strategy for stable money demand models besides creating more complex single equation models. The speed of adjustment coefficients were generally consistent with on hypothesized markets. However, our full system estimation provided a richer set of relations than possible with single equation models. In particular, income moved to re-equilibrate the consumption market, which is perhaps the dominant market for that variable, but it did not have a significant influence on the money market. M1 acted to clear the latter market. We additionally suggested that shifts in aggregate demand cause the price level to move in directions opposed to clearing the money market. Finally, we found that the price elasticity was not unitary in the samples including 198890, which we felt may have had to do with the increased use of automatic banking facilities. We concluded that it is often inappropriate to assume that the price elasticity in a money demand model is unity and use real money balances. Appendix The data was acquired from the Citibase data tape and transformed as follows (the original Citibase labels are included):(12) M1 = log (ml) - FMI P = price level - log (GNP implicit price deflator) - GD R = U.S. Treasury 10 year rate (% per anum) - FYGL NG = log (federal government expenditures) - GGFEX RG = log (federal government expenditures) - log (GNP deflator) Y = log (real GNP) - GNP82 I = log (real gross private investment) - GPI82 C = log (real consumption) - GC82 r = real interest rate (R - inflation rate) IM = log (real imports) - G1M82 EX = log (exports) - GEX GER = log (Deutsche Mark per U.S. $) - EXRGER JAN = log (Yen per U.S. $) - EXRJAN EURO = log (Eurodollar deposit rate, London 3-month) - FYUR90 CAN = log (Canadian dollar per U.S. $) - EXRCAN DEF = log (real deficit) - GGFNET - P 1. Other examples of estimating multiple vector systems are Johansen and Juselius [19] and Dibooglu and Enders [8]. Both papers were interested in observing interest rate and purchasing power parity simultaneously. 2. We used the real interest rate in the money market instead of the nominal interest rate for reasons related to the Johansen estimation technique. This issue is explored in section III below. 3. This system passes standard rank condition tests for identification as well as the Johansen and Juselius [20] procedure designed to identify cointegration models. In addition, none of the seven variables could be excluded from the system using Johansen and Juselius' [19] weak exogeneity test and significance levels of .06 except the real interest rate. We suspect that this occurred because of the high degree of collinearity with the Eurodollar deposit rate, which was used as a conditioning variable. We chose to retain the real interest rate as a system variable for theoretical purposes and also because it performed well in the speed of adjustment analysis. 4. The seven variables are all I(1) based on univariate ADF tests. See King et al. [21] for a thorough discussion. Miller [26] analyzes the price level and finds that the null is rejected by the DF test but not by the ADF test. Further analysis reveals that second-differencing is suggestive of over-differencing since the coefficients of the lagged levels exceed negative one in absolute value. We concur with Miller that the price level is better represented as a stationary series in first-differences. 5. We thank an anonymous referee for suggesting inclusion of the Canadian exchange rate and the real deficit as conditioning variables. The former represents one of our largest trading partners and the latter reflects possible Ricardian effects. In the error correction model reported below, the growth rate of consumption is inversely related to changes in the real deficit, suggesting possible Ricardian impacts. See Seater [28] for a review of this literature. 6. We used Microfit 3.0 to estimate and test the cointegrating relations. 7. Our estimates of [a.sub.1] did not yield these plausible results prior to including the Canadian exchange rate and the deficit as conditioning variables. 8. CDRS are able to identify the money market for the years 1960.2-75.4, . . ., 78.4 and for 1960.2-84.4, . . ., 90.4. They find acceptable income elasticities for money demand that are consistently between -0.8 and -1.1 for the money market alone. 9. This paper also provides other improvements over the CDRS specification. CDRS encountered severe multicollinearity problems between disequilibrium errors in the late 1980s, which reduced the number of significant vectors to two. Our model does not have this problem. 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