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The values of the determinants and tests of stability of the money demand function of the United States.

Executive Summary

Using multivariate cointegration methodology, this paper examines the long-run stability of the U.S. money demand function using both nominal and real M1 and M2 monetary aggregates. The estimated results, based on the Johansen cointegration technique, found evidence that the U.S. M1 and M2 demand functions, nominal or real, have long-run stability. The magnitudes of the estimated parameters are different for different specifications of the models. The estimated real income elasticity and interest rate elasticity, over the specifications of the money demand functions, are more sensitive to the choice of the interest rate and less sensitive to the scale variable. The implication of this stability is that there is at least one common factor which influences money supply, real income and interest rates. This finding comes as no surprise, given the coordinated monetary policy of the U.S. Federal Reserve to control output and avoid inflation.

The empirical results of this study, with respect to M1 money demand, are in contrast with the empirical evidence presented in some of the earlier studies.


Tests on stability of the demand for money function conducted in the studies prior to the mid-1980s were controversial. Recent developments in cointegration technique provided a more precise way to reexamine the stability issue. The objective of this paper is to apply the cointegration technique, proposed by Soren Johansen (1988), to test the stability of the U.S. money demand function.

There are several advantages in applying the cointegration technique to study the long-run behavior of money demand. It allows us to investigate independently the long-run demand for money without the need for deciding which of the many postulates of its short-run dynamics should be explicitly explored. The error correction technique that is complementary to cointegration methodology gives us this full flexibility to model short-run dynamics in several ways. For instance, in cointegration methodology, the debate whether to include current income or permanent income (or both) as arguments in the money demand function does not arise, because permanent income is the steady state value of the current income.


For analytical purposes we start with Fisher's equation of exchange, and use the transformed version of this identity as our money demand function for cointegration tests. Following this discussion, empirical results of the cointegration test are presented.

Fisher's equation of exchange (MV = Py), in natural logs, (see Dickey et al, 1991), is:

In M + In V - In P - In y = 0 (1)

where M is nominal money, V is velocity, P is the price level, and y is real income (Q/P, Q=Output and P=Price). The theory of money demand transformed this identity into an equation by making V a function of some economic variables. However, the form and the arguments of the money demand function are different for different model specifications. Since V is not observable, it is proxied by V*.

In V * = In V + e (2)

where e is the random error associated with V. It is postulated that V * is a function of some observable variables other than y, M, or P.

In M + In V * - In P - In y = e (3)

If V * is a perfect proxy for V, the expected value of e should be zero, i.e. e is stationary. However, the proxy, V*, may deviate from its true value in the short-run, but should converge to its true value in the long run. If the relationship among the variables in (3) is not stationary it implies that either V * is not a good proxy for V or that there is no long-run relationship among these variables, i.e. the long-run money demand function does not exist.

The Fisher equation, in essence, implies a long-run relationship between money, real income, opportunity cost of holding real money balances, and velocity. This theoretical assertion has been tested since the late 1960s with limited success. As a matter of fact, the empirical literature on money demand function is voluminous--and also contentious.

A linear combination of the above variables,

[b.sub.1] In M + [b.sub.2] In V * + [b.sub.3] In P + [b.sub.4], In y (4)

is hypothesized to be stationary. There may or may not be a prior knowledge of the cointegrating vector, [b.sub.1], [b.sub.2], [b.sub.3], [b.sub.4]. If not known, it has to be estimated.

Next, the Fisher equation hypothesizes that the cointegrating vector [1,1,-1,-1], exists. This vector combines the four series into a single series e. Given this assumption, we can test for cointegration by applying the unit root test to e. If e is stationary, a long-term relationship between the concerned variables is valid. However, detecting cointegrating relationships among variables without any theoretical basis is relatively hard. If theory can be used to assign values a priori to some of the coefficients, finding cointegrating relationships is not difficult. In the present case, Fisher's equation fully specifies the cointegrating vector, and applying the usual unit root tests for cointegration is appropriate and easy to estimate the cointegrating vector.

Engle and Granger (1987), Stock and Watson (1988) and Johansen (1988) have suggested alternative tests for cointegration, and techniques for estimating the cointegrating vectors. The three procedures are quite different, but their focus is on identifying the most stationary linear combination of the vector of the time series concerned. If the numbers of time series involved are more than two, more than one linear combination can exist.

Unit Root Tests

Since the early 1980s the income velocity of M1 (M1/GDP) has been drifting upward, and hence, it is thought to be non-stationary. As such, the macroeconomic relationship between money (M1 measure) and income is not stable over time. The income velocity of M2 (the broader money measure) appears to be around a stationary mean, and formal tests have shown that M2 and income are cointegrated (Engle and Granger, 1987). These results are presented as evidence for the non-existence of a stable long-run relationship between M1 and income, and the existence of a stable money demand (Md) for M2 (Hallman, Porter, and Small, 1989).

According to the Fisher equation M1 is the relevant measure of money, and it also suggests that there should be at least one cointegrating relationship between Ml, V, y, and R The specification of the [M.sup.d] is critical for conducting cointegrating tests.

The long-run nominal Md function, following Dickey et al, is:

[M.sup.d] = f(P, Q, Z) (5)

where M and Q represent money and output in nominal terms, and Z stands for all other variables, such as the expected rate of inflation and expected real rates of interest, etc.

Assuming the absence of money illusion, real money demand can be expressed as:

[m.sup.d] = f(y, z) (6)

where [m.sup.d] = [M.sup.d]/P and y=Q/P

It is also assured that the real demand for money is homogeneous of degree one in real income, so that the reciprocal of the income velocity of money is:

md/y = g(Z) (7)

The equilibrium condition is:

[m.sup.d] [m.sup.s] (8)

and g(Z) is observed as the ratio of real money stock to real income.

The reciprocals of the velocities of M1 and M2 can be written as:

m1/y = g1(Z) and m2/y = g2(W),

where Z is a subset of W (signifying all other variables), as M1 is part of M2. Since velocity is not observable, its proxy must be specified to conduct cointegration tests. Further, if M1 and income are not found to be cointegrated, but M2 and income are, then a stable relationship must exist between g1(Z) and g2(W).

The cointegration tests are conducted on U.S. time series quarterly data (M1, M2, GDP, GDP deflator, 3-month T-bill rate, rates on 10-year T-note) covering the period 1961.1 to 1996.1. The data were collected from various issues of the International Financial Statistics (IFS). Ml, M2, and GDP data are seasonally adjusted.

Results and Discussion

The first step is the unit root test on the individual time series to determine the order of integration of each series. Unit root tests, employing the augmented Dickey-Fuller (D-F) procedure, are performed on the levels of the time series, Ml, Q, 3-month T-bill rate, yields on 10-year T-note. AM the variables are in nominal terms. The results of the unit root tests are in

Table 1. The null hypothesis is that the series under investigation has a unit root. The critical values by Mackinnon are used for the tests. The computed augmented D-F statistic below the critical value in absolute terms requires rejective of the null hypothesis.
Table 1:
Augmented Dickey-fuller (ADF) Unit Root Test

Series in     ADF Test    Series in their   ADF Test
levels       Statistic  first differences  Statistic

Ml             -2115                 Ml     -3.475
M2            -2.009                 M2     -4.263
Q             -0.510                  Q     -5.100
T-bill        -2.196             T-bill     -4.328
10-Yr. Note    1.994        10-Yr. Note     -5.543

Mackin non Critical Values for rejection of hypothesis
of a unit root

1%    -3.4770
5%    -3.4415
10%   -3.1415

The reported test statistics in Table 1 (column 2) indicate that the null hypothesis cannot be rejected for any of the series at the 5 percent significance level. Hence, it is concluded that each of the time series, in levels, has a unit root. The third column of Table 1 has the unit root test results on the first differences of each of the same series. The null hypothesis of a unit root can be rejected in the case of all the series 5 percent significance level. As the time series appear to be stationary in their first differences, no further tests are conducted. Hence, it is concluded that the order of integration of each of the time series is 1(1).

Since each of the time series concerned is integrated in the same order, there is a possibility for cointegration between them. Cointegration tests, employing the Johansen procedure, are conducted on real M1 and also on real M2 (in their natural logs), as the dependent variable, real GDP (in natural log form), and one of the nominal interest rate variables (3-month T-bill rate / yield on 10-year T-note) as the right-hand side variables. The lag lengths for the variables, after trying a wide range, are set at four.

The test results for M1 and M2 are presented in Tables 2 and 3, respectively. The results obtained when the yield on the 10-year T-note was used are included (in the Tables), but not when the 3-month T-bill yield Wd5 was used, as they are almost the same.

The first line in Tables 2 and 3 tests the hypothesis of no cointegration, i.e. there is no long-run equilibrium relationship among the variables. As the likelihood ratio, 29.3017, exceeds the critical value of 24.31 at the 5 percent significance level, the null hypothesis of no co-integration is strongly rejected. The series in question are cointegrated.
Table 2:

Johansen Cointegration Test

Series: In m1, in y, R1OYr. Note
Lags interval: 1 to 4

Hypothesizes # of

Eigenvalue   Likelihood   5% Critical   Cointegrating
             Ratio (LR)         Value   equations

0144             29,302         24.31   None
0.045             8,150         12.53   At most 1
0.014             1.943          3.84   At most 2

Likelihood ratio test indicated one cointegrating equation
at 5 percent significance level.

Normalized Cointegrating Coefficients:

In m1        1n y            R10Yr. Note

1.000        0.938           0.037

Log Likelihood: 988.214

Table 3:
Johansen Cointegration Test

Series: In m2, In y, R10Yr. Note

Lags interval: 1 to 4

Eigenvalue  Log likelihood  5 percent  Hypothesized #
                Ratio (LR)   Critical  of
                                Value  cointegrating

0.139               25.841      24.31  None
0.037                5.455      12.53  At most 1
0.002                0.283       3.84  At most 2

LR test indicates one cointegrating equation at
5 percent significance level

Normalizing Cointegrating Coefficients:

In m2   in y  R10Yr. Note

1.000  0.958       -0.036

Log likelihood ratio: 1052.209

The second line tests the hypothesis of one cointegrating vector. The maximum eigenvalue test failed to reject the null hypothesis of one cointegrating vector. So there are two common trends and one cointegrating vector. Results indicate that real M1 is cointegrated with real income and either of the two nominal interest rates.

Next, the normalized cointegrating vector (the first element corresponds to the dependent variable, in ml, the second and third to In y and nominal yield on Two-year T-notes) is close to the theoretical expectation in Fisher's equation: that the coefficient of the real income in the long run should be equal to one in magnitude. The coefficient of In y is 0.94, close to one. The hypothesis that the normalized coefficient on output is unity is not rejected using either interest rate, and the hypothesis of a zero coefficient for the nominal interest rate is not rejected for both interest rates.

Conclusion and Implications

The U.S. data support the asserted long-run equilibrium relationship between real M1 (and also real M2), real income, and nominal interest rates. The results allow us to conclude that the real M1 and real M2 demand functions are stable. The implication of this stability is that there is at least one common factor which influences money supply, real income and interest rates. This finding comes as no surprise, given the coordinated monetary policy of the U.S. Federal Reserve to control output and avoid inflation.


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Augustine C. Arize, Texas A&M University-Commerce

Peter Harris, School of Management, New York Institute of Technology

Krishna M. Kasibhatla, North Carolina Agricultural and Technical State University

loannis N. Malindretos, Cotsakos College of Business, William Paterson University

Moschos Scoullis, Montclair State University
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Author:Arize, Augustine C.; Harris, Peter; Kasibhatla, Krishna M.; Malindretos, Ioannis N.; Scoullis, Mosch
Publication:Review of Business
Article Type:Report
Geographic Code:1USA
Date:Jun 22, 2012
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