Investigation of the long-run Quantity Theory of Money relationship

INVESTIGATION

OF THE

LONG-RUN

QUANTITY

OF MONEY

RELATIONSHIP

GEORGIOS

THEORY

KARRAS

ABSTRACT This paper uses mainly cointegration techniques to investigate the long-run Quantity Theory of Money in five countries: Canada, Germany, Japan, the United Kingdom, and the United States. The central hypothesis, several variants of which are being tested, is that money (Ml or M2), real income and the price level are cointegrated. A pattern consistent with the Quantity Theory emerges for Germany and Japan, whereas the other three data sets seem to reject the hypothesis.

1. INTRODUCTION This paper investigates the long-run Quantity Theory of Money for five countries: Canada, Germany, Japan, the United Kingdom, and the United States. In its simplest form, the Quantity Equation states that MV = PY, where M is the money supply, V is the velocity of money, P is the price level, and Y is real output. This equation however, is nothing more than an identity: is satisfied. What advances

there will always exist a Vsuch that the Quantity Equation the Quantity Equation to the Quantity Theory is the

Direct all correspondence to: Georgios Karras, The University of Illinois at Chicago, Department of Economics (M/C 144), College of Business Administration, Box 4348, Chicago, IL 60680. Copyright 0 1992 by JAI Press, Inc. International Review of Economics and Finance, 1(2):159-168 All rights of reproduction in any form reserved. ISSN: 1059-0560

GEORGIOS KARRAS

160

assumption that velocity is constant, which implies a close relationship between the growth rate of nominal GNP and the rate of growth of the money supply. More realistically, velocity must be stationary (or equivalently the logarithms of M, P, and Y must be cointegrated) for the Quantity Theory to constitute a useful approximation to the real world. If velocity can be shown to be stationary, a given increase (or decrease) in the rate of growth of the money supply will produce more or less the same increase (or decrease) in the rate of growth of nominal GNP (i.e., the rate of inflation plus the rate of real growth). If velocity is not stationary, the relationship between money growth and nominal GNP growth will not be as close, or even predictable, because some changes will simply be absorbed by chang& in velocity. This can be tested using recent econometric techniques, and mainly cointegration. * The rest of the paper is organized as follows: Section 2 establishes the link between cointegration and the Quantity Theory, and presents preliminary tests on the series used. Section 3 performs the cointegration tests. Section 4 tests velocity for stationarity. Finally, Section 5 presents the conclusions.

2. METHODOLOGY: COINTECRATION QUANTITY THEORY

AND THE

The idea behind cointegration is simple: two or more (nonstationary) time series are said to be cointegrated if they “move together”, in the sense that there exists a linear combination of them which is stationary. In fact, this linear combination can be interpreted as the long-run (or equilibrium) error? In its strongest form, the Quantity Theory (QT) implies that m, + v = pr + y,, where m is the money supply, v (a constant) is velocity, p is the price level, y is real income, and all variables are in logarithms. In a weaker (and more useful for our purposes) form we could write m, + V, = pf + y, and the QT now requires that vt = P, + y, - mt be stationary. In terms of the definition of cointegration, the elements of xt = [Pr y, md’ must be cointegrated with cointegration vector CX’= [l 1 -11, so that vt = o(‘xt is stationary. This is the hypothesis that will be tested in this paper. Section 3 tests whether y, p, and m are cointegrated at all (a necessary condition for QT), and Section 4 tests whether the specific vector [I 1 -11 cointegrates the variables (a necessary and sufficient condition for QT). Both tests are performed because the cointegrating vector is not necessarily unique. Engle and Granger (1987) show that a consistent estimate of the cointegrating vector can be obtained by regressing one of the variables on the rest, effectively imposing the normalization of a unit coefficient on the series used as the “dependent variable” in the regression. They, and Engle and Yoo (1987) develop a cointegration test based on the Fuller (1976) and Dickey-Fuller (198 1) testing procedures. Here, the following series are of interest: the nominal interest rate (r), the log of the Ml money supply (ml), the log of the M2 money supply (m2), the log of the price level

The Long-Run

Quantity

Theory of Money

Relationship

161

(p), the log of the real money supply (ml - p and m2 - p), the log of the real GNP @), and the log of the nominal GNP (y + P).~ The series are investigated for five countries: Canada, Germany, Japan, the United Kingdom, and the United States, with the exception of M2 which was included only for Canada, Japan, and the United Kingdom. Changes in the definitions of the M2 series for Germany and the United States within the periods used, made older observations not comparable with more recent ones. Quarterly observations are us3L4 The first step is to test the series for stationarity. Three tests are employed: the Fuller (1976) t-test, the Dickey-Fuller (1981) F-test, and the Phillips-Perron (1987, 1988) Z(t) test. Models with and without trend are estimated except for the interest rate which was tested only with models without trend. The results of the stationarity tests are not reported here because of space considerations (they are available upon request). The main findings are as follows: the nominal interest rate is non-stationary for all five countries but its first difference is stationary and with high significance levels. Of the other variables (ml, m2, ml - p, m2 -p, p, y + p, and y) only the first differences were tested for stationarity since the levels are obviously non-stationary. In practice, all three tests turned out to be insensitive to the model specification with respect to trend. Problematic series are the first differences of (i) p for Germany and the United States, (ii) ml -p for the U.S., and (iii) m2 for the U.K., for which the null of a unit root cannot be rejected by the Dickey-Fuller procedures even at the 10% level. (Note, however, that the Phillips-Perron tests reject the unit root hypothesis for these series.) All the other 27 variables are clearly stationary when differenced. Recognizing that sound application of cointegration techniques requires that all variables involved be integrated of the same order, we would have to either refrain from estimating cointegration equations that include the German or U.S. p, the U.S. ml -p, or the U.K. m2; or place a greater confidence on the Phillips-Perron statistics. In the following sections, it becomes evident that the second option has been chosen. A justification that may be offered is thelow power of the Dickey-Fuller tests, which means that the probability of accepting the null of nonstationarity when the null is in fact false, is high, which statistically justifies 4 acceptances out of 41 trials.5 The next sections examine the cointegration hypothesis implied by the Quantity Theory, and the behavior of velocity for the five countries. Both the Ml and M2 series were used for Canada, Japan, and the U.K., but due to the qualitative similarity of the results, only the Ml results are reported. The only exception is the direct tests for velocity (section 4) where both the Ml and M2 velocities are examined. All unreported results are available upon request. 3.

TESTING

FOR COINTEGRATION

A necessary condition for the Quantity Theory is that m,, p,, and y, be cointegrated. Table 1 presents the results of several cointegration tests. For robustness, two different normalizations were tried: one with m and one with p as the dependent variable. What

162

GEORGIOS KARRAS Table I.

Cointegration Tests

Unrestricted corrrmy

const.

Canada

,tll

.71

P Germany

,nl

-.9 1

P

-.96

IIll

1.82

P InI

-4.83

U.K.

P

-1.45

.91

In 1

.30

P

COIISt

Canada

ml -p

2.12 -0.47

Germany

Y + I’ ml -p

Japan

Y+P

1.10

ml -p

5.02 -1.92

ml -p

.63

.95

.80

-.31

.94

1.06

.99

1.15

-.44

.98

.97

-4.65*

-3.98*

.85

.98

1.67

-6.58**

-3.98*

-.55

.97

1.42

6.62**

-5.31**

1.25

.99

.57

-3.18

-1.25

1.10

-.77

.98

.20

-1.27

-2.03

1.18

-.32

.73 .85 1.01 .81

.63

P

.93

-1.34

ml -p

-1.97

DF

ADF

.35

-1.16

-1.74

.20

-1.28

-2.61

-5.331”

-4.32**

.98

.98

-4.49*

-1.25

.97

.68

-3.46

-1.80

DF

ADF -2.36

Models

,n

Y

R2

.I2

.05

.28

-1.27

.95

.33

-1.39

-1.60

.91

1.09*

-5.33**

-4.14**

.99

1.13*

-5.42**

-4.39**

.63

1.39*

-5.51**

-5.36**

.98

1.65**

+5.39**

-3.73*

1.34 1.33 .89 .52 1.16 .25 1.13

6.05

Y+P

DW

Restricted

1.67

Y+P U.S.

R2

-2.20

Y+P U.K.

Y

1.13

-4.38

cour1rr-y

,n

.74

-2.82

Japan

U.S.

p

Models

-.I8 1.40

DW

.10

.I2

4.47

-2.01

.99

.29

-1.89

-1.83

.08

.47

-2.70

-2.58

.98

.90

-4.23*

-1.27

4

Note:

The secondary

model is A:, = pz,_t ,+, 2 Qi A&, + u, where G are the residuals from the OLS regression i-l

of m 1 (orp) onp (or ,nl), y. and a vector of ones. DF is the r-statistic for p with q = 0; ADF is the r-statistic for p with (I = 4. Critial values from Engle and Yoo (1987). DW is the Durbin-Watson *Significant **Significant

statistic.

at 5%. at 1%.

is of interest here is the behavior of the error terms, and this is what Table 1 tests: if the residuals are stationary the variables are cointegrated, whereas if they are nonstationary, the variables are not cointegrated. Testing appears to be robust across the two different normalizations. Testing is also fairly robust with respect to the statistic chosen (DF or ADF). Only for the model for the U.S. do the two tests contradict each other; due to the quarterly data, however, higher credibility should probably be attached to the ADF statistic. In particular, non-cointegration (the null) cannot be rejected for Canada, the U.K., and

The Long-Run

Quantity

Theory of Money

Relationship

163

the U.S., regardless of the normalization chosen. However, the non-cointegration hypothesis can be rejected for Germany and Japan using any of the two models. To further check robustness over different normalizations we can try two restricted models for each country: one with real money (m - p) and one with nominal income 0, + p) as the dependent variables. Results are again reported in Table 1 and now, since N = 2, we can also use the DW statistic for our cointegration tests (critical values for this test are reported by Engle and Granger, 1987). Once more, Germany and Japan easily reject the null of non-cointegration, while Canada, and the U.K. fail to reject it. For the U.S. them - p model cannot reject non-cointegration but they +p model is inconclusive. However, given the small significance of the DW test, not rejecting the null is probably the right thing to do. Thus, again only Germany and Japan appear to support the Quantity Theory’s implications. A possible reason why the models for Canada, the U.K., and the U.S. fail to support the QT could be that these models are not well specified. In fact, the velocity of money is often assumed to be a function of the interest rate, of the form V~= CL,+ cry, where r1 is the nominal interest rate. Now the Quantity Theory implies that y,, pt, m,, and rf are cointegrated, and the cointegrating vector is (II= [l 1 -1 cl’. This was tested using restricted and unrestricted models similar to the ones of Table 1 but including the interest rate. (To preserve space the results are not reported but are also available on request.) The addition of the interest rate has no significant impact on the empirical findings. The coefficients have generally the right sign, including the coefficients of r (negative in the m and m - p specifications and positive in the p or y + p specifications). Noncointegration is again strongly rejected only for Germany and Japan.

4. A DIRECT TEST So far, we have only tested for the existence of a cointegrating vector (any cointegrating vector), which is only a necessary condition for the Quantity Theory. We have not yet examined whether it is the vector [ 1 1 -11 that cointegrates p, y, and m. The simplest way to test this necessary and sufficient condition is to test whether vlt = y, + pI - Al,, and ~2~= y, + pt - m2,, i.e., the logarithms of the Ml and M2 velocities, are stationary. This is the most restricted model in the sense that it forces the operating vector to be 01’ = [ 1 1 -11 and then it tests whether (Xis indeed a cointegrating vector. Another way to motivate this direct test would be the following: since the number of variables is three (N= 3), thecointegrating vector need not be unique.6 So, even if Section 3 (above) estimated a cointegrating vector different than 06 this should not exclude a from consideration. It follows, of course, that if the results of Section 3 are correct, then only for Germany and Japan could velocity as defined here, be stationary. Table 2 gives the results and they are very encouraging. The purpose of including the 1lagged terms is to whiten out E[ and satisfy the conditions of the Fuller and Dickey-Fuller tests. However, the “right” value for 1 (not necessarily the same for all series) cannot be determined a priori and, moreover, the computed

164

GEORGIOS KARRAS Table 2.

A Direct Test for Velocity

Models without Trend TF Canada

FDF

SQ

Models with Trend

,I

1

PP

TF

FDF

SQ

,z

1

PP

vl

-1.24

.82

.90

57

4

-1.68

-1.05

.98

.90

56

4

-0.59

v2

-1.92

2.34

.91

51

4

-1.12

-1.60

1.17

.98

60

2

-1.35

Germany

VI

-2.54

3.51

.29

58

4

-3.52*

-4.23**

8.94*

.63

5-I

4

-5.51**

Japan

~1

-1.40

2.80

.77

42

6

-3.12*

-4.45**

10.12**

.80

41

6

-6.82**

v2

-.I8

.28

.97

46

4

-1.52

-4.21*

11.43**

.95

45

4

-3.10

vl

-1.69

1.44

.59

51

4

-2.21

-1.31

1.53

.58

56

4

-1.65

v2

-2.24

2.61

.99

55

6

-1.35

-2.23

2.50

.99

48

6

-1.63

Vl

-1.61

2.77

.30

54

6

-2.28

1.64

.21

53

6

-3.94*

U.K.

U.S. Note:

.42

The model without trend is A_q=u+hr,_,

t1

+iAq_I-i+&,

r-l

and the model with trend is

1 AX,= a + Pt + A&_, + z

$i AX,_, + El.

TF is the r-statistic for h; critical values from Fuller (1976). FDF is the F-statistic for the hypothesis

(u, A)

= (0,O) or (p, h) = (0,O); critical values from Dickey and Fuller (1981). SQ is the significance level of the Box-Pierce statistic. PP is the Phillips-Perron Z(t) statistic. **: significant at 1%. *: significant at 5%

statistics are very sensitive to different choices of 1 (Schwert, 1987). The Box-Pierce statistic was chosen as an “objective” indicator of whether enough (but not too many) lags were included. SQ is the significance level of this statistic.’ The Phillips-Perron tests indicate that velocity is stationary for Germany (Ml) and Japan (Ml and M2) regardless of whether a trend is included or not. The Dickey-Fuller tests lead to the same conclusion only for the trend models. It is also interesting to note that all tests show velocity to be nonstationary for the other three countries (the only exception is the Phillips-Pen-on test for the U.S. model with trend, which however, cannot be safely interpreted as more than circumstantial evidence). That the [ 1 1 -11 vector appears to be a cointegrating vector for Germany and Japan but not for Canada, the U.K. and the U.S. reinforces the findings of the last section. In addition, the significance levels are very good, justifying our preference for the models with trend. Having shown that the vector [ 1 1 -11 indeed works for Germany and Japan when imposed on the data, we can now go one step further and ask whether the same vector could independently and freely be estimated from the data. For this purpose, we estimate “money demand” equations for these two countries on the levels of the series. Note that this would be impossible for the other three countries because of the Granger-Newbold (1974) objection: the error term would not be stationary there (since the series are not

The Long-Run

Quantity

Theory of Money Table 3.

Relationship

165

Generalized Least Squares

Process for tlze

Country

error term

Corut.

Germany

AR( 1)

ml

AR(l)

ml -p

r

P

-.85

1.10’

C.61)

C.09)

1.17!

AR(l)

Iill ml -p

-.04 (57) -1.75

-1.04 (58)

AR(4)

ml-p

l.lO!

-.I7

1.13!

C.05)

C.03) -.17

C.12) 1.34

C.03)

C.05)

1.13!

1.23’

C.08)

C.20)

-2.00

1.51

C.31) AR(4)

AR(4)

III1

ml-p

AR(I)

ml

ml-p

tnl

AR(4)

ml -p

ml

1.38

.84

C.09)

C.03) .78

.75

2.38

2.20

AR(4)

ml -p

In1

Note:

ml-p

-.Ol C.04)

2.44

-.02

~27)

CM)

1.51

.76

.74

.84

.79 C.04) .74

~24)

C.04)

1.61

.98

.83 C.07)

-.04 C.04)

1.91

-.03

C.28)

C.04)

DW is the Durbin-Watson statistic. **significant standard errors in parentheses.

.78

.93

.72

at 1%. !: insignificantly

.I4 C.13)

.98

.13 C.13)

.93

.13 C.13)

.95

.38** C.12)

.89

.36* C.12)

.95

.37** C.12)

C.04)

C.04)

.I3 C.13)

C.03)

C.07)

.44** C.12)

C.06)

1.80

C.29) AR(4)

.94’

.49** C.11)

C.03)

C.09)

.37** C.11)

-.18

C.26) AR(4)

.81

.97

C.06)

.42** C.11)

I .06! C.16)

C.48) AR(l)

.96

C.03)

.94!

.09 C.12)

-.18

c.24) AR(l)

.93

C.06)

C.09)

.03 C.12)

1.14

C.45) AR(l)

.99

C.41)

2.14

.43** C.11)

-.27

-1.17

;

.41** C.11)

.79

C.09)

c.27) Japan

.96

C.09)

c.24) AR(4)

R2

C.21) 1.38

-1.41 C.26)

AR(l)

Y

.89

.36** C.12)

different from one. Estimated

cointegrated) which would make the regressions “spurious” and standard statistical inference invalid, whereas for Germany and Japan the error terms are stationary and we only have to correct for serial correlation. This same point can also be illustrated by the Durbin-Watson statistics in Table 1: those for Canada, the U.K., and the U.S. are generally unacceptably low, whereas for Germany and Japan they are close to acceptable values.

166

GEORGIOS

KARRAS

Several alternative filtering procedures were tried but only two are reported: AR( 1) and AR(4). In the first step, the OLS residuals are used to obtain ; from s, = p&i + uI (the AR( 1) model), o;;, = p&_JA+ut (the AR(4) model). Then, all series are transformed using the filters 1 - pL or 1 - pL4. Note that filtered OLS is identical to Generalized Least Squares (GLS) with the first observations dropped. This means that the estimated standard errors are now consistent (and they are reported) and can be used to test whether the coefficients take the values that the Quantity Theory requires. The results are in Table 3. The first thing to note is that the AR(4) representations are more successful than AR(l), which is not surprising because of the quarterly data. The signs are all according to the theory, though the interest rate fails to be significant for Japan. German data are seen to support the [l 1 -11 vector for b p m] more decisively than data for Japan. However, the nonuniqueness problem should be kept in mind. The fact that for Japan a vector different than [ 1 1 -11 cointegrates the series does not mean that it is the only one. In the beginning of this section it was shown that this vector gives the desired results. Finally, the m models (unrestricted) perform much better than the m - p ones (restricted). Summing up, this section’s conclusion is that only the German and Japanese data sets satisfy the necessary (that y, p, and m are cointegrated) and sufficient (that the cointegrating vector is [ 1 1 -11) conditions for the long-run Quantity Theory.

5.

CONCLUSION

This paper tested some of the long-run implications of the Quantity Theory of Money for five countries, to find that they are clearly satisfied only by two of them (Germany and Japan). Several tests were tried and, interestingly, Germany and Japan passed all of them, while the rest (Canada, the U.K. and the U.S.) failed them all. I believe this works in favor of the results shown here. A practical implication of these results is that of the five countries we examined, only Germany and Japan can afford to ignore changes in velocity when they try to determine the long-run relationship between money growth and nominal GNP growth. For the other three countries (and especially for Canada and the U.K.) velocity fluctuations should not be ignored. What can account for the significantly different behavior of velocity in the countries examined? Although beyond the scope of the present study, two tentative explanations can be offered. The first, and most popular in the literature* focuses on changes in the degree of financial sophistication caused by deregulation and innovations. For this to be a valid argument, it must be that the financial environments of Germany and Japan have been subject to fewer changes that those of Canada, the U.K., and the U.S. A second possible explanation (and a more subtle one) may be the different behavior of the five countries’ central banks. It is interesting (and perhaps suggestive) that the Quantity Theory is supported for the two countries whose monetary policies are best described as “monetarist” (commitment to monetary targeting and controlling inflation) rather

The Long-Run

Quantity

Theory of Money Relationship

167

than “Keynesian “? It is clear, however, that further research is needed in order to shed light on the sources of these differences. In addition, there is a criticism against which this paper (or, for that matter, any other on the same subject) is defenseless: the Quantity Theory states MV = PY and not (Ml)(W) = (CPI)(GNP) [or even @42)(V2) = (CPI)(GNP)]. The paper tested the latter and not the former. Is Ml (or M2) a good measure of “the” money supply? Is CPI a good measure of “the” price level? The question is not raised in the sense of measurement errors (they do not matter if assumed to be Z(O)),but in the sense of good proxies. The results of the paper should be taken as conditional on the (Ml)(Vl) = (CPI)(GNP) or (M2)(V2) = (CPI)(GNP) formulations of the Quantity Theory. NOTES 1. For a recent empirical examination of the stability of Ml-velocity and the demand for money in the U.S., as well as an extensive bibliography, see Rasche (1987). Friedman and Schwartz (1982, especially chapter 6) take a more historical approach and focus both on the U.K. and the U.S. Both studies defend the velocity approach-Rasche by finding that “the hypothesis of unitary real income elasticity of the demand for real balances is not rejected’ and Friedman and Schwartz by stating that “analysis of the behavior of velocity is an analysis of the demand for money.” Cointegration tests on the U.S. data have been applied by Taylor (1989). 2. Formally, the elements of the vector xt = [xl,, .... x~r] are cointegrated if, (i) xi1 _ 1(l), for all i = 1,..., N and (ii) there exists a vector of constants (II, not identically equal to zero, such that zt = a’xtm I(O), i.e. zt is stationary. et is called the cointegrating vector (Engle and Granger, 1987). 3. The data are from the I.M.F. International Financial Statistics. Income was proxied by the GNP for all countries, Ml by “money” (line 34), M2 by Ml + QM where QM is “quasi-money” (line 35), the price level by the CPI (line 64), and the interest rate by the Treasury bill rate for Canada, the U.K., and the U.S., and by the Public Authorities yield for Germany and Japan. 4. Annual observations may be better suited for investigation of long-run relationships. Data availability, however, dictated the choice of frequency. 5. Indeed, how else can it be explained that for Germany Am and A(m - p) appear to be stationary with Ap non-stationary? Or that Ay and Ab + p) (again for Germany) are stationary with Ap non-stationary? For the U.S., one line of argument could be as follows: since Ay and A(y + p) contain no unit root Ap has to be stationary as well, and since Am is stationary, A(m -p) can not contain a unit root. A second reason for preferring the Phillips-Perron tests is that they are valid for a wider class of models than the Dickey-Fuller tests. I would like to thank an anonymous referee for bringing this to my attention. 6. If the number of variables is N, then the number of coin&rating vectors, r, satisfies r 5 N - 1, which means that the cointegrating vector a will be unique (up to a scalar multiple) only if N = 2 (Granger, 1986; Engle and Granger, 1987). 7. Alternatively, Akaike’s Final Prediction Error (FPE) or the closely related Akaike Information Criterion (AX) can be used in order to determine the “best” value for 1. For the series tried here they give virtually the same results. 8. See Tatom (1990). 9. OECD (1972) and OECD (1973) outline the Japanese and German monetary policy

GEORGIOS KARRAS backgrounds roughly at the beginning of the period examined in this paper. For Germany, see also Kahn and Jacobson (1989), and von Hagen (1989).

ACKNOWLEDGMENTS

I wish to thank Stephen Cosslett, Paul Evans, J. Huston McCulloch, an anonymous referee, and participants at the Econometrics seminar at the Ohio State University for valuable comments.

REFERENCES

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