high inflation rates and the long-run money demand function: evidence from cointegration tests

TAUFIQ CHOUDHRY University of Wales

Swansea, United Kingdom

High Inflation Rates and the Long-Run Money Demand Function: Evidence from Cointegration Tests"

This paper attempts to determine whether there exists a stationary long-run money demand function in Argentina, Israel, and Mexico. Tests based on the Johansen method of cointegration reveal strong support for a stationary money demand function in the long run in all three countries. This result only holds when the annualized rate of change of the exchange rate (currency depreciation) is included in the money demand function.

1. Introduction One of the more heavily investigated subjects in macroeconomics is the

money demand function. Without spending a large amount of space and time, one could say that a simple real money demand function may be represented as follows:

(M/P) a = f (y, i ) .

This function states that demand for real money balances (M/P) a where M is the nominal money, P is the price level, depends upon a variable that reflects the level of transactions in the economy such as real income or real wealth (y) and a variable that represents the opportuni ty cost of holding money such as the rate of interest or the rate of inflation (i). According to the Cambr idge and the Keynesian approach the relationship be tween real money demand and the level of real income or real wealth is direct, and the relationship be tween real money demand and the rate of interest or rate of inflation is inverse.

Thus, research on the money demand function assumes that there exists a stationary long-run equilibrium relationship be tween real money balances, real income or real wealth, and the opportunity cost of holding real money balances (Fr iedman 1956). The purpose of this paper is to de termine whether there exists a stationary long-run real money demand function in

*I thank Gerald P. Dwyer, Donald F. Gordon, Myles S. Wallace, John T. Warner, and the seminar participants at Temple University for helpful comments and suggestions on an earlier draft. I also thank two anonymous referees for numerous useful comments. All remaining errors and omissions are my responsibility alone.

Journal of Macroeconomics, Winter 1995, Vol. 17, No. 1, pp. 77-91 Copyright © 1995 by Louisiana State University Press 0164-0704/95/$1.50

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three high inflation countries. The countries used in the empirical work are Argentina, Israel and Mexico. 1 The time periods selected are the following: Argentina from August 1975 to March 1987, Israel from April 1974 to December 1988, and Mexico from January 1976 to December 1987. 2 During the stated periods these countries experienced high and variable inflation rates, large balance of payments deficits, and recurrent currency devaluations (Dornbusch 1989; Livitian and Piterman 1986; and Looney 1985).

Cointegration tests (Johansen method) are used to test the hypothesis of a stationary long-run money demand function. A detailed analysis of the concept of cointegration is provided in Engle and Granger (1987), Johansen and Juselius (1990), and Dickey, Jansen and Thornton (1991). Briefly, cointegration implies that in a long-run relationship between different nonstationary variables, it is required that these variables should not move too far away from each other. Individually these variables might drift apart in the short run, but in the long run they are constrained. In simple words, two or more nonstationary time series are cointegrated if a linear combination of these variables is stationary (converges to an equilibrium over time). The main idea behind cointegration is a specification of models that include beliefs about the movements of variables relative to each other in the long run, such as the long-run money demand function. Thus, if the money demand function describes a stationary long-run relationship among real money balances, real income and the opportunity cost of holding money it can be interpreted to mean that the stochastic trend in real money balances is related to the stochastic trends in real income and the opportunity cost of holding money. In other words, if these variables are cointegrated they will be constrained to an equilibrium relationship in the long run. While it is possible for deviations from the equilibrium to exist, they are mean reverting. Applica- tions of the cointegration test in the estimation of the money demand function are explicitly analyzed in Johansen and Juselius (1990) and Dickey, Jansen and Thornton (1991). As noted in footnote 1, Melnick (1990) applies cointegration tests in his empirical work on Argentina's money demand. But Melnick uses the Engle-Granger (1987) procedure which may provide weak

1Some of the earlier empirical work on the estimation of the money demand function in the stated countries are the following: Diz (1970), Khan (1977), Melnick (1990, 1988), Ungar and Zilberfarb (1980), and Zilberfarb (1983). However, with the exception of Melnick (1990) these studies used standard normal regression procedures, the Chow test, the Brown-Durbin-Evans test, etc., to investigate the money demand function. These procedures are inappropriate if the data applied are nonstationary; tests of cointegration constitute the proper testing technique. Melnick applies the cointegration procedure in the estimation of money demand function for Argentina.

2Lack of the price level and the exchange rate data before the 1970s made it impossible to go back any further in time.

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results when more than two variables are involved (Gonzalo 1994). In this paper, the empirical work is carried out by means of the Johansen procedure of conducting cointegration tests because this procedure provides more robust results when there are more than two variables (Gonzalo 1994).

Results presented in this paper indicate strong support for a stationary long-run money demand function for all three countries. However, a stationary long-run money demand function in all three countries is only ensured if the annualized rate of change of the exchange rate (currency depreciation) is included in the relationship. This result may imply currency substitution in these countries (Amber and McKinnon 1985; Abel et al. (1979); and Frenkel 1977, 1980). According to Frenkel (1977) in economies which are characterized by high inflation and flexible exchange rate, foreign currency becomes an important substitute for domestic currency. Assuming there is only choice between two assets, namely, domestic and foreign currency, the proportion in which the two currencies are held will depend very much on the rate of currency depreciation. The risk averse economic agent will respond to a higher rate of depreciation by raising the proportion of foreign money in his or her portfolio. Thus, during currency substitution the relevant cost of holding domestic money is the expected rate of change in the exchange rate.

Both the rate of inflation and the rate of change of the exchange rate may be required in the money demand function of high inflation countries. Ramirez-Rojas (1985) indicates that, during the stated time periods in Ar- gentina, Israel and Mexico, the demand for foreign currency, especially United States dollars by domestic residents, increased beyond the require- ments of international trade and tourism. According to Ramirez-Rojas, in order to estimate a money demand function in these countries it may be necessary to include a measure of currency substitution in the money demand function. Domowitz and Elbadawi (1987) state that if foreign money is a substitute for domestic money, the omission of a variable that measures the return to foreign currency in the demand function for real money may bias the model towards overstating the influence of inflation. Abel et al. (1979) point out that the presence of rate of change of the exchange rate (currency depreciation) in the money demand function in high inflation economies can be interpreted in more than one manner. First, the rate of change of the exchange rate may act as a proxy for the domestic rate of inflation. 3 Second, it may measure the expected rate of currency depreciation, and thus of the opportunity cost of holding domestic currency as opposed to foreign cur-

3In this paper domestic inflation rate is included as a separate variable, so currency depreciation may not be used as a proxy for inflation.

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rency. 4 Abel et al. (1979) also show that since both goods and foreign assets can be substi tuted for domestic currency, both the rate of inflation and the rate of change of the exchange rate need to be in the money demand function, s Frenkel (1977, 1980) also notes that if commodit ies and domestic money are substituted, then the opportuni ty cost of holding money is the expected rate of inflation. Similarly, as stated earlier, if foreign money and domestic money are substituted, then the opportunity cost of holding money is the expected rate of change in the exchange rate. In this study, based on the conclusions drawn by Abel et al. (1979) and Frenkel (1977, 1980) both the inflation rate and the rate of change of the exchange rate (currency depreciation) are included in the estimation of the money demand function.

Domest ic monetary autonomy that is ensured under a flexible exchange rate be tween two countries is reduced if currency substitution exists be tween the two countries (Bergstrand and Bundt 1990 and McKinnon 1982). I f individuals hold both domestic currency and foreign currency, then an in- crease in the opportunity cost of holding one currency can lead to higher demand for the other currency in both countries. Higher demand will cause a flow of that currency across borders, resulting in the loss of monetary control. Further , if the domestic resident substitutes foreign money for domestic money, then financing of the deficit through the revenue from seignorage will decrease, and the base of the inflation tax will be reduced. According to Ramirez-Rojas (1985) in economies where inflation is high and variable, policies oriented towards reducing currency substitution may have to be applied. One such policy is to p romote foreign currency deposits in the domestic financial system. This policy may reduce the outflow of foreign currency by reducing transaction cost and increasing the liquidity to the depositor. Another policy would be to maintain higher domestic interest rate.

4As advocated by Taylor (1991) and Frenkel (1977), the actual method used to form expectations is not the main concern. Following Taylor, the only assumption made is that the forecasting error is stationary, forecasting error being the difference between the expected rate of the variable and the actual rate that transpires. Thus, according to Taylor, assumption of stationary forecasting errors implies that expectations of a time series are not hopelessly different from the actual outcome, even when the series has accelerated growth rates. Taylor (p. 342), under several different assumptions concerning expectation formation, provides the plausibility of the assumption of a stationary forecasting error, provided the process being forecast contains a single unit root. Taylor (p. 339) is further able to show that a money demand function that includes both the inflation rate and currency depreciation must be a stationary function if the errors in forecasting the inflation rate and currency depreciation, and the random disturbance of the function, are stationary. Frenkel (1977) provides support for the notion that, during periods of high inflation, expectations may have been rational regardless of the method used to create the expectation.

~l'aylor (1991) confirms Abel et al.'s (1979) suggestion of using both the inflation rate and currency depreciation in the money demand function of a high inflation economy.

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This policy is to make domestic currency deposits more attractive compared to holding foreign currency. McKinnon (1982) also suggests policies oriented towards curbing currency substitution.

2. The Detn nnd the Unit Root T e ~ $ Results Since cointegration tests require a certain stochastic structure of the

time series involved, the first step in the estimation procedure is to determine if the variables are stationary or nonstationary in levels. For our purposes the variables should be nonstationary in levels (should contain a unit root). Dickey and Fuller (1979, 1981) provide one of the methods of detecting whether a time series contains a unit root or not. The following regression is used to check for unit roots:

k

AXt = a + lit + (p - 1)Xt-1 + Y~ ~ i ~ t - i + e t . (1) i=1

Equation (1) is known as the "augmented Dickey-Fuller test" (ADF) (Said and Dickey 1984). 6 The null hypothesis in the ADF test is a unit root (p=l). For X t to be stationary, ( p -1 ) should be negative and significantly different from zero. The necessary critical values required for the unit root hypothesis testing are provided in Fuller (1976).

Five variables are tested for unit roots: seasonally unadjusted monthly series of real M1, real M2, real industrial production (proxy for real income), the annualized rate of inflation, and the annualized rate of change of the exchange rate. 7 Real money balances are created by dividing the nominal balances by the CPI. The annualized inflation rate is created using the CPI, and the monthly average exchange rate between the relevant currency and the United States dollar is used to create the annualized rate of change of the exchange rate. 8 Thus, the rate of change of the exchange rate represents

6DeJong et al. (1992) and Schwert (1987, 1989) study the operating characteristics of the different unit root tests. They conclude that the augmented version of the test provided by Dickey and Fuller is the most useful in practice.

7In general, it may be preferable to use seasonally unadjusted data. Ghysels (1990) claims that seasonal-adjustment filters have at least three adverse effects on the power of the unit root test. First, the power of unit root tests may be reduced due to the smoothing effects of the filters. Second, the long leads and lags used in the filters may produce distant autocorrelation in the adjusted series. The third and final problem is induced by the nonlinear properties of seasonal- adjustment filters.

SThe annualized inflation rate is created by using the following formula, (1 + ((Pt - P,-1)/P,-a)) lz - 1 ,

where P is the Consumer Price Index. The same formula is used to create the annualized rate of change of the exchange rate.

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TABLE 1. Unit Root Test

Argentina Variables Two Unit Roots One Unit Root Real M1 -4.22a/(12) -2.67/(12) Real M2 -5.42a/(6) -2.30/(9) Inflation Rate -5.82a/(9) -2.60/(12) Currency Depreciation -5.91~/(9) -2.51/(6) Real Income -2.67c/(9) - 1.33/(12)

Israel Variables Two Units Roots Single Unit Root Real M 1 - 4.59a/(6) - 1.47/(6) Real M2 - 16.94~/(0) - 1.49/(2) Inflation Rate --2.97b/(9) --2.56/(12) Currency Depreciat ion --3.00b/(2) --2.40/(2) Real Income - 4 . 4 P / ( 2 ) -1.33/(12)

Mexico Variables Two Unit Roots Single Unit Root Real M 1 - 12.19a/(0) - 1.75/(6) Real M2 -9.83~/(0) - 1.71/(12) Inflation Rate -3.05b/(9) -0.97/(12) Currency Depreciat ion -3.79a/(6) - 1.22/(6) Real Income -3.67a/(6) -2.56/(12)

NOTE: a, b, & c imply rejection of the null at 1%, 5%, & 10% respectively. Data in the parentheses represent the number of significant lags.

the rate of depreciation of the relevant currency against the dollar. The data used are obtained from the IMF tapes and statistical books.

Table 1 presents the unit root tests results. All variables are in logarithms except the rate of inflation and the rate of change of the exchange rate. 9 Based on the evidence presented by Schwert (1987, 1989), up to fourteen lags (k) are included in ADF tests involving Israel, and twelve lags are used for Argentina and Mexico. Insignificant lags (standard F-test) were dropped from the regressions. I f the elimination of lags produced serial correlation, then the lags were added back on. Following the suggestion by Dickey and Pantula (1987) the unit root tests are first conducted for two roots and, if two

9Application of the inflation rate and the rate of change of the exchange rate rather than their logarithms assumes that the absolute rather than the percentage change in these variables is what matters for money demand. Friedman and Schwartz (1982) describe the advantages of using the opportunity cost of holding money (interest rates, inflation rate, etc.) in absolute rather than in logarithms in the estimation of money demand function.

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roots are rejected, then a single unit root is tested for. When the null hypothesis is two roots the time trend is excluded from the ADF test. Both the constant and the time trend are included when the null hypothesis is one unit root. 1° In this manner, the two unit root test allows for an alternative hypothesis of stationarity with a nonzero intercept on the differenced series, while the single unit root test allows for the alternative hypothesis of trend stationarity and a nonzero intercept on the series in levels. Trend stationarity implies that the deviation of the series from a linear function of time follows a stationary process. All tests include monthly seasonal dummies in order to eliminate seasonality. These seasonal dummies enter exogenously in the relationship. As Table 1 shows, variables for all three countries are able to reject the null hypothesis of two unit roots but are unable to reject the null hypothesis of a single unit root. Thus, all the variables are stationary after first differencing, but are nonstationary in levels. In other words, all the variables contain a single unit root.

3. Cointegration Tests Results There exists more than one method of conducting cointegration tests.

Cointegration tests in this paper are conducted by means of the method developed by Johansen (1988), and Johansen and Juselius (1990). The Jo- hansen method applies the maximum likelihood principle to determine the presence of cointegrating vectors in nonstationary time series. This method detects the number of cointegrating vectors and allows for tests of hypotheses regarding elements of the cointegrating vectors. Johansen (1988) provides two different tests, the trace test and the maximum eigenvalue test to determine the number ofcointegrating vectors. If a nonzero vector or vectors are indicated by these tests, a stationary long-run relationship is implied, n According to Dickey, Jansen and Thornton (1991) cointegrating vectors are obtained from the reduced form of a system where all of the variables are assumed to be jointly endogenous. Thus, cointegrating vectors cannot be interpreted as representing structural equations. However, cointegrating vectors may be due to constraints that an economic structure imposes on the long-run relationship between the jointly endogenous variables. Osterwald- Lenum (1992) provides the appropriate critical values required for these cointegration tests.

A likelihood ratio test is used to select the number of lags required in the cointegration test. As in the unit root tests, lags are not omitted if their

l°The ADF tests are conducted in this manner because ,all the series seem to contain a linear time trend in levels, while the differenced series does not.

llMore detailed analyses of the Johansen method are provided in Dickey, Jansen-and Thornton (1991).

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TABLE 2. Cointegration Tests

Monetary Variable: Narrow Definition of Money (M1). Argentina

Vectors Trace Test Eigenvalue Test r = 0 55.15 b 29.94 b

r <= 1 25.12 12.41 r <= 2 12.80 8.75 r <= 3 4.05 4.05

Israel Vectors Trace Test Eigenvalue Test r = 0 65.28 a 43.36 a r <= 1 21.93 11.26 r <= 2 10.67 6.57 r <= 3 4.10 4.10

Mexico Vectors Trace Test Eigenvalue Test r = 0 75.97 a 44.62 ~ r <= 1 13.35 12.09 r <= 2 9.25 7.18 r <= 3 2.07 2.07

NOTE: a, b, & c imply significance at 1%, 5%, & 10% respectively.

exclusion introduces serial correlation. In all cases twelve lags were indicated for the cointegration tests. Since there seems to be a linear trend in all the nonstationary series, cointegration tests are conducted with the inclusion of a deterministic linear trend. 12 Seasonality is eliminated by exogenously in- cluding monthly seasonal dummies in all regressions. I f a cointegrating vector or vectors are indicated by the trace and the maximum eigenvalue tests as statistically significant, this implies a stationary long-run equilibrium relationship between the variables. I f more than one vector is found, then the test may indicate more than one relationship (Johansen and Juselius 1990; and Dickey, Jansen and Thornton 1991.)

Tables 2 and 3 present results from the cointegration tests (the trace

12We also tested the hypothesis that a linear trend is not required in the tests by means of the chi-square tests. The hypothesis was rejected, implying the presence of a linear trend. Johansen and Juselius (1990) show the application of the chi-square test in the stated hypothesis testing.

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TABLE 3. Cointegration Tests


Monetary Variable: Broad Definition of Money (M2). Argentina

Vectors Trace Test Eigenvalue Test r = 0 87.10 ~ 38.68 a r <= 1 18.42 10.15 r <= 2 6.27 5.62 r <-- 3 3.65 3.65

Israel Vectors Trace Test Eigenvalue Test r = 0 56.13 ~ 36.33 a r <= 1 19.98 9.99 r <= 2 8.99 5.67 r <= 3 4.32 4.32

Mexico Vectors Trace Test Eigenvalue Test r = 0 54.28 b 27.35 b r <= 1 26.93 15.13 r <= 2 11.80 7.85 r <= 3 3.95 3.95

NOTE: See note at the end of Table 2.

and the maximum eigenvalue test) using real M1 and real M2, respectively. Both definitions of money provide similar results. For Argentina, using real M1, both the trace test and the maximum eigenvalue tests indicate one cointegrating vector as significant at the 5% level. For Israel and Mexico, one vector is indicated as significant at 1% by both tests. For all three countries we have strong evidence of a stationary relationship between real M 1, real income, the inflation rate and the rate of change of the exchange rate. Results obtained using real M2 are provided in Table 3. Both tests confirm one vector as significant for all the countries. Thus, results from real M2 also indicate a stationary relationship in all three countries. Cointegration tests with only the inflation rate or the rate of change of the exchange rate in the relationship fail to produce a stationary long-run money demand function. The cointegration tests produce a significant and stationary money demand function only when the inflation rate and the rate of change of the exchange rate are included. Thus, cointegration tests fail to find any significant vector or vectors when the function lacks either the inflation rate or the rate of change in the

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exchange rate. This is true of demand functions for both the real M 1 and the real M2 in all three countries, la

4. Long-Run Elasticities The estimated cointegrating vectors are given economic meaning by

means of normalizing on the real money balances. The normalized equations are obtained by dividing each cointegrating vector by the negative of the cointegrating vector on real money balances. These normalized equations are obtained from reduced forms, and may represent money demand, money supply, or some more complicated interaction (Johansen and Juselius 1990 and Dickey, Jansen and Thornton 1991). However, these normalized equations appear to be money demand and they show signs on the variables that are consistent with money demand. Part I of Table 4 presents the implied long-run elasticities obtained from these normalized equations. Using the chi-square test, all the variables are tested for significanceA 4 In such a test the null hypothesis is that the coefficient on the relevant variable is equal to zero. With the exception of real income in the demand function for real M1 in Israel, all the variables are significantly different from zero in every relationship tested. It is clear from the normalized equations and the significance tests that both the rate of inflation and the rate of change of the exchange rate are required in the demand function for both definitions of money in all three countries.

A significant presence of the rate of change of the exchange rate in the demand function for real money balances may provide evidence of currency substitution in these countries. Currency substitution reduces domestic monetary control that is ensured under a flexible exchange rate and, in countries where the government engages in inflationary finance, currency substitution reduces both the financing of deficit by means of seignorage and the base of

13For instance, in the case of Israel, the demand function for real M1 without the rate of change exchange rate yields the following results:

Vectors: r = 0 r < = 1 r < = 2 Trace test: 18.81 8.59 2.79

Eigenvalue test: 10.22 5.80 2.79 Similarly, the Israeli real M1 demand function without the inflation rate provides the following result:

Vectors: r = 0 r < = 1 r < = 2 Trace test: 20,33 9.30 3.21

Eigenvalue test: 10,95 6.17 3.21 The remaining ten results are similar. They are not provided simply because of the lack of space and time, but are available on request.

14Johansen and Juselius (1990) show the application of the standard chi-square test to determine the significance of individual variables.

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TABLE 4. Part I: Implied Long-Run Elasticities

Null Hypothesis: Elasticites are equal to zero. Real M1

Real Income Inflation Rate Currency Deprec Argentina 0.95 a - 0 . 0 6 b - 0 . 0 3 b Israel 0.43 - 0 . 6 7 a - 0 . 0 3 b Mexico 0.47 b - 1.81 ~ - 0 . 0 1 '~

Real M2 Real Income Inflation Rate Currency Deprec

Argentina 0.78 b - 0 . 3 5 b - 0 . 0 6 b Israel 0.37 ¢' - 0 . 1 8 b - 0 . 0 5 b Mexico 1.38 b - 1.28 a - 0 . 0 2 a

NOTE: a, b, & c imply significantly different from zero at 1%, 5%, & 10% respectively.

Part II: Real Income Elasticity Test

Null Hypothesis: Income Elasticity is equal to unity. Real M1 Real M2

Argentina 2.92 c' 8.57 a Israel 7.53 a 6.67 a Mexico 7.90 a 5.78 b

NOTE: a, b, & c imply reject of the null at 1%, 5%, & 10% respectively.

the inflation tax. Means other than monetary expansion may have to be applied in order to finance the deficit (Ramirez-Rojas 1985).

In all of the six relationships tested, in absolute value the long-run inflation rate semi-elasticity of money (real M 1 and real M2) demand is larger than the long-run currency depreciation semi-elasticity of money demand in all three countries. Converting these semi-elasticities to elasticities, once again the inflation elasticity is larger in value (absolute) than the currency depreciation elasticity. 1~ Thus, according to results obtained, currency substitution may exist in these countries, but the inflation rate plays a larger role in the determination of demand for both the real M1 and the real M2 in all three countries. The size of the currency depreciation coefficients (long-run semi-elasticities) are small, ranging from - 0 . 0 1 to -0 .06 , but all the coefficients are significantly different from zero at 1% or 5% level. The overall currency depreciation elasticity ranges from - 0 . 1 2 to -0 .34 . Inflation semi-

l'SFull elasticities are not provided here for two reasons. First, both the semi-elasticities and full elasticities provide the same story, and second, they are not provided in order to save time and space, but are available on request.

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elasticities ranged from -0.06 to - 1.81, while the range of the full elasticities are from -0.33 to -4.74.

The chi-square test is further used to check if the equilibrium real income elasticity of real money balance is equal to unity. In this test the null hypothesis is that the coefficient on real money balance is equal to the coefficient on the real income. Part II of Table 4 presents the results of these tests. The results are presented in the form of the estimated chi-square. In all six cases the null hypothesis is rejected. Thus, equilibrium real income elasticity is different from unity in all six money demand functions. This result also implies that the cointegrating vectors reject any stationary linear combination of income velocity of money (real M1 and real M2), the inflation rate, and the rate of change of the exchange rate in all three countries.

5. Conclusion This paper attempts to determine whether there exists a stationary

long-run money demand function in Argentina, Israel and Mexico. The time period involved ranges from the mid 1970s to the late 1980s. During the stated periods, these countries experienced high and variable inflation. Cointegration tests (Johansen procedure) are applied to test the hypothesis of a stationary long-run money demand function. The function tested includes real money balances, real income, the rate of inflation, and the rate of change of the exchange rate (currency depreciation). Results provide a strong support for a stationary function in all three countries. This result only holds with the inclusion of currency depreciation in the money demand function. The results show that both the rate of inflation and the rate of change of the exchange rate belong in the money demand function. Nor- malized equations obtained show the right sign on all the variables and they are individually significant. Exchange rate sensitivity of file demand for money may indicate currency substitution. Currency substitution reduces the domestic monetary control that is ensured by a flexible exchange rate and also reduces the base of the inflation tax and the financing of deficit by means of seignorage. Under such conditions policies oriented towards reducing currency substitution may have to be applied. Two such policies may be the promotion of foreign currency deposits in the domestic financial system and the maintance of higher domestic interest rate.

Received: March 1993 Final version: April 1994

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high inflation rates and the long-run money demand function: evidence from cointegration tests

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