COINTEGRATION TESTS OF THE MONETARY ...

INTERNATIONAL ECONOMIC JOURNAL Volume 10, Number 4, Winter 1996

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COINTEGRATION TESTS OF THE MONETARY EXCHANGE RATE MODEL: THE CANADIAN - U.S. DOLLAR, 1970 - 1994 PANAYIOTIS F. DIAMANDIS University of Crete DIMITRIS A. GEORGOUTSOS Athens University of Economics and Business GEORGIOS P. KOURETAS* University of Crete

Using data on the Canadian-U.S. dollar rate, we reexamine the monetary model of exchange-rate determination for the recent float in three ways. First, we test its longrun validity, using Johansen’s multivariate cointegration techniques. Second, we examine and test the model for the presence of speculative bubble, and finally we test for parameter stability of Johansen’s results using the Hansen-Johansen recursive tests. [F31]

1. INTRODUCTION The purpose of this paper is to overcome certain shortcomings in previous studies which have cast doubt on the empirical validity of the monetary model of the exchange rate determination as an appropriate framework for analyzing movements of the Canadian-U.S. dollar exchange rate. Thus, studies by Backus (1984) and Florentis et al. (1994) found little statistical evidence in support of the monetary model, and it is argued therefore that the Canadian-U.S. exchange rate can be better approximated by a random walk model. In contrast, Boothe (1983) offers the only statistical evidence in favor of the monetary model for describing the Canadian foreign exchange market for the 1971-1978 period. Several explanations have been offered for the apparent breakdown of the model, ranging from *Part of this work was completed while the third author was a Visiting Scholar at Michigan State University. The hospitality of the Department of Economics is gratefully acknowledged. The third author also gratefully acknowledges financial support by the Fulbright Foundation for a Senior Research Fellowship and NATO for a Postdoctoral Research Scholarship. We have benefited from comments and discussions by Richard Baillie, Rowena Pecchenino, Robert Rasche, Peter Schmidt and Jeff Wooldridge. We also wish to acknowledge the constructive comments provided by an anonymous referee and an editor of this journal. The usual disclaimer applies.

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inconsistent estimates of the coefficients to the instability in the underlying money demand functions. The above mentioned studies have employed econometric methodologies which do not take into account the presence of nonstationarities in the variables implied by the monetary model. As it has been shown, all standard testing procedures are inappropriate for making any valid statistical inference when nonstationary variables are involved in the estimation procedure. The recently developed theory of cointegration provides a suitable framework within which we can test for a long-run equilibrium relationship. In this paper we reexamine the monetary model for the Canadian-U.S. dollar exchange rate during the recent float by applying several recent developments in the econometrics of unit roots and cointegration. First, in addition to the standard Dickey-Fuller test for the presence of a unit root in the data, we implement the new KPSS test (Kwiatkowski, Phillips, Schmidt and Shin, 1992), which tests for the null of stationarity. Second, we further analyze the stochastic properties of the data, testing for the presence of possible stuctural breaks by employing the recursive tests developed by Banerjee et al. (1992) as well as the sequential tests due to Zivot and Andrews (1992). The above mentioned new testing procedures are necessary since the standard unit root tests have been shown to have low power against alternatives of trend stationarity or stationarity around a shifting trend. Third, we use Johansen’s multivariate cointegration technique to test for the existence of a longrun relationship underpinning the monetary equation, and we were able to identify one cointegration vector. In addition, we have tested a set of restrictions associated with the monetary model and we were unable to reject them. Therefore, it is shown that, in contrast to previous works, the monetary model is a valid framework for analyzing the long-run movements of the Canadian-U.S. dollar exchange rate. Given this finding, we further test for the possible existence of explosive bubble components in the behavior of the exchange rate, but no such finding was detected. A final novel feature of our analysis is the implementation of three tests, proposed by Hansen and Johansen (1993), for the evaluation of parameter constancy of the cointegration results. Although the dimension of the cointegration space is shown to be sample dependent, the estimated coefficients do not exhibit instabilities in recursive estimations. The plan of this paper is as follows: Section 2 presents the basic monetary model of exchange-rate determination. Data sources and time series properties of the individual variables are described in section 3. Johansen’s maximum likelihood technique for modelling and testing cointegration is presented in section 4. Section 5 reports the tests results, with the concluding remarks given in section 6. 2. THE MONETARY MODEL A typical monetary approach reduced form equation may be written as (see Baillie and MacMahon, 1989, for a further discussion):

MONETARY EXCHANGE RATE MODEL

et = β 0 + β 1 mt + β 2 mt∗ + β 3 y t + β 4 yt∗ + β 5 it + β 6 it∗

85

(1)

where, e is the price of the domestic currency in terms of the foreign currency, m is the money supply, y is the real income, and i is the interest rate. All the variables except the interest rate are in logarithms. An asterisk indicates a foreign country variable (the U.S. in our case). The early, flexible-price monetary model (Frenkel, 1976; Bilson, 1978) relies on the twin assumptions of continuous purchasing power parity (PPP) and the existence of stable money demand functions for the domestic and foreign economies. The first assumption implies that the real exchange rate, i.e., the exchange rate adjusted for differences in national price levels, cannot vary, by definition. However, a major characteristic of the recent experience with floating exchange rates has been the wide fluctuations in the real rates of exchange between many of the major currencies, which results to shifts in international competitiveness. The sticky-price monetary model (Dornbusch, 1976), allows for substantial overshooting of both the nominal and the real price-adjusted exchange rates beyond their long-run equilibrium (PPP) levels, since the exchange rates and the interest rates, which are considered the jump variables in the system, compensate for sluggishness in the goods prices. This, variant of the monetary model is considered to be more accurate in explaining the observable facts. According to equation (1), an increase in domestic (foreign) money supply will lead the domestic currency to depreciate (appreciate). An increase in domestic (foreign) real income will raise the money demand, causing the domestic currency to appreciate (depreciate). Finally, an increase in the home (foreign) interest rate will result in a depreciation (appreciation) of the exchange rate via a reduction of the demand for money. It is interesting to note that the last two effects are the opposites of the ones that are expected in the Mundell - Fleming fixed price model. This model, generally regards an increase in domestic real income as leading to a worsening trade balance and therefore to a depreciation of the exchange rate. In addition, a relative increase in the domestic interest rate leads to capital imports, and hence to an appreciation of the exchange rate. The Dornbusch’s (1976) sticky price model accords, in the long run, with the implications of the monetary approach but has Keynesian features in the short-run (Dornbusch’s model assumes β5 < 0 and β6 > 0). Different signs of the coefficients in equation (1) will also be produced under imperfect substitutability between the assets of the two countries (Girton and Roper, 1977). Associated with equation (1), which may be called the unrestricted version of the monetary model, is a set of coefficients restrictions that are regularly imposed and tested. The most important restriction is whether proportionality exists between the exchange rate and the relative monies (β1 = - β2 = 1). Moreover, equal and opposite coefficients on relative income (β3 = - β4) and interest rates (β5 = -β6) are

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quite often tested. A further assumption underlying the flexible-price monetary model is that uncovered interest parity (UIP) holds continuously-that is, the domestic-foreign interest differential is just equal to the expected rate of depreciation of the domestic currency. The UIP relationship is usually invoked when domestic and foreign assets are perfect substitutes, there is no uncertainty and there is absence of capital controls and transaction costs. Thus, denoting with E t [et+1] the expectation of the next period’s level of the exchange rate conditional on information available in period t, the UIP relationship is given as follows: it - t∗ i = Et [et+1 ] - et

(2)

An important extention of the monetary model is derived if we use the UIP condition (2) and assume that the interest rate elasticities are equal in the two countries. Then the forward-looking solution for the exchange rate is given by the equation:

et =

1 β5 β5 -t-1 ∑ Tj=0 ( ) j Et [ Lt + j ] + ( )T - t Et et + T 1 +β 5 1 +β 5 1 +β 5

(3)

where, Lt = β0 + β1mt + β2mt* + β3yt + β4yt* If the transversality condition holds, then the last term of the right side expression in (3) will tend to zero as the date T, for which expectations are being formed, becomes more distant. The remaining part of (3) provides the market fundamentals solution for the exchange rate, which we shall denote by Stf. However, if the discounted expected terminal value of the exchange rate does not vanish, then the general solution to equation (1), when combined with (2), is:

et = Stf +

B tS

(4)

where StB stands for the deviation of the exchange rate from its fundamental value, Stf, and it satisfies Et StB+ 1 = (1 +β 5 )StB / β 5 . If β5 > 0, we have a positive bubble, which implies a continual devaluation of the currency. This factor will finally dominate Stf in the explanation of the exchange rate’s behavior. A straightforward test for the existence of an explosive bubble component, StB, can

MONETARY EXCHANGE RATE MODEL

87

be constructed in the following way: If we subtract Lt from both sides of (3), we can derive, after some manipulation, that:

et - Lt = ∑ ∞j=1 [β 5 / ( 1 +β 5 )] j E t ∆Lt + j + StB

(5)

Now, if the variables entering the Lt expression are first-difference stationary, I(1), it suffices to show that et and Lt are cointegrated in order to reject the presence of an explosive bubble (MacDonald and Taylor, 1993).1 The empirical validity of the monetary model can only be tested in the long-run, given the short-run deviations of the exchange rate from its PPP. Therefore, we employ tests of cointegration which provide evidence for the existence of a long-run relationship among the exchange rate, money supplies, real incomes and interest rates, even if the individual variables are nonstationary. Therefore, under such a statistical specification, even Dornbusch’s sticky price model can be consistent with the presence of a long-run equilibrium even though a temporary overshooting of exchange rates is implied. 3. DATA DEFINITIONS AND STOCHASTIC PROPERTIES The data for this study are taken from the International Monetary Fund’s International Financial Statistics CD-ROM, and run from June 1970 (when the Canadian dollar was allowed to float) to May 1994. In particular, the exchange rate is end-of-period units of Canadian dollar per unit of U.S. dollar, the monetary aggregate is M1, the income measure is industrial production, and the interest rates are longterm rates proxied by ten-year government bond yields. A prerequisite for testing for cointegration is that all variables are nonstationary, and, therefore, we begin our analysis by examining for the order of integration of individual time series. We use two alternative tests. First, we implement the standard augmented Dickey-Fuller (ADF) test which tests for the null of a unit root in the series against the alternative of stationarity. Since this is a parametric test that requires the appropriate choice of the autoregressive lag structure to correct for autocorrelation, we applied it by choosing that number of lags for which no serial correlation was detected in the residuals of the regression, based on a Lagrange Multiplier test. Recently, DeJong et al. (1992) have shown that the Dickey-Fuller class of tests has low power against plausible trend stationary alternatives. Thus, we also applied the KPSS test developed by Kwiatkowski, Phillips, Schmidt and Shin Since m, m*, y, y * are the variables entering in L, the above-mentioned test requires that e is cointegrated with those four variables. 1

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(1992), which tests for the null of stationarity against the alternative of a unit root. Briefly, the KPSS test uses the components model:

yt = α + δt + t X + t ,v X t = Xt - 1 + u t

(6)

where yt is the sum of a deterministic trend t, a random walk Xt, and a stationary error v t. Also ut are i.i.d. (0, σu2). Since vt is assumed to be stationary, for the null hypothesis that yt is trend stationary we simply require that σu2 = 0. When σu2 equals zero, Xt is a constant and is added to the intercept of equation (6). We can also set δ = 0, in which case we test the null hypothesis that yt is stationary around a level rather than around a trend. The results of the ADF and KPSS tests on the individual time series are found in Table 1. For all the variables besides the Canadian and U.S. industrial production, we find the unambiguous result that they are realizations of an I(1) process. For the Canadian and U.S. output, there is an indication that may be trend stationary. We must note, though, that the KPSS results are sensitive to the lag length, and as is clear from Table 1, the null of stationarity is rejected with three lags, but not at twelve.2 We further pursue the investigation for the stochastic properties of the variables by testing for the presence of structural breaks in the data. Perron (1989) argues that standard unit root tests have low power against alternatives of stationarity around a structural break. We conduct this test by adopting the methodology developed by Banerjee et al. (1992), which, unlike Perron’s procedure, treats the break point as unknown. The proposed test consists of estimating a sequence of ADF t-statistics, where both a constant and a time trend are included in the model, by applying a recursive methodology. The minimum and maximum t-values of the obtained sequence are compared to critical values computed from the results of Monte Carlo replications on a random walk model (Banerjee et al., 1992, Table 1). We are unable to reject the null hypothesis of a random walk process with a constant drift when the estimated t-values are higher from the critical ones. Table 1 reports our results, and we unambiguously cannot reject the null hypothesis of a unit root without a structural break for all the individual series. We reached the same conclusion when we applied the Zivot and Andrews (1992) approach (see Table 1). Perron’s (1989) methodology is employed sequentially, implying that the break of the series occurs every time at a different date. The null hypothesis is the same in both the recursive and the sequential procedures although the alternative differs. In the sequential approach it is stationary around a shifting trend while in the recursive it is stationary around a constant trend. 2 Evidence from Monte Carlo simulation suggests that a smaller number of lags cause large distortions, while a larger number of lags tend to reduce the power of the test.

MONETARY EXCHANGE RATE MODEL

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Table 1. Tests for a Unit Root in the Data. Levels τµ e y m i y* m* i*

ττ

–1.19[12] (0.106) –1.88[5]

–2.24[12] (0.08) –3.12[5]

(0.118)

(0.08)

–0.43[12] (0.058) –1.74[5] (0.124) –1.08[5]

–2.06[12] (0.09) –1.66[5] (0.107) –1.20[5]

(0.094)

(0.15)

0.39[12] (0.269) –1.89[5] (0.180)

–2.60[12] (0.178) –1.82[5] (0.186)

First Differences ηµ

ητ (12) (ητ (3))

1.59

0.34

2.08

0.12 (0.35 )

2.29

0.22

0.54

0.42

2.21

0.07 (0.20 )

2.32

0.20

0.46

0.44

τµ

–3.52[12] (0.191) –5.47[5]

ηµ

tmax

tmin

t (a)

0.12

–0.07

–3.29

–3.23

0.11

–0.01

–2.74

–3.01

0.13

–0.03

–3.25

–5.01

0.11

–0.09

–3.15

–4.33

0.04

1.91

–2.11

–2.45

0.13

0.98

–3.23

–2.31

0.11

–0.01

–2.18

–2.18

(0.117) –3.93[12] (0.132) –6.68[5] (0.158) –5.75[5] (0.568) –3.79[12] (0.251) –6.85[5] (0.514)

Notes: The symbols e, y, m, i, denote, respectively, the spot exchange rate, industrial production, the narrow money supply, and the long-term interest rate. An asterisk denotes U.S. variables. indicates significance at the 5% level. τµ and ττ denote the standard Augmented Dickey-Fuller tests for the null of non-stationarity, when a constant and a constant and a time trend are included in the equation, respectively. Numbers in brackets after the ADF statistics indicate the lag length used in the autoregression to ensure residual whiteness while numbers in parenthesis indicate marginal significance level of the Lagrange Multiplier with 12 degrees of freedom. Critical values at the 5% significance level for the two tests and for a size T=283 and T=276 are equal to –2.8724 and –3.4282 and –2.8729 and –3.4291 respectively (MacKinnon, 1991). ηµ and ητ is the KPSS test for the null of stationarity when a constant and a constant and a time trend are included in the equation. Both tests are calculated with a lag length equal to 12. For the Canadian and U.S. ouputs, we also calculated the tests with 3 lags. The 5% critical values for the two tests are equal to 0.4627 and 0.148, respectively (Sephton, 1995, Table 2). Tests of the first differenced variables exclude the time trend. tmax (min) refers to the Banerjee et al. (1992) results from recursive estimations starting at 1972.11. The estimates correspond to the maximum and minimum values of the augmented with 12 lags sequence of Dickey-Fuller statistics, where both a time trend and a constant are included. Critical values at the 5% significance level are: tmax = –1.99, tmin = –4.33 (Banerjee, et al., Table 1). t(a): Zivot and Andrews (1992) sequential test. Critical value at 5% significance level, –5.08 (Zivot and Andrews, 1992, Table 4).

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Given the evidence for all the above mentioned tests, we treat all the variables involved in the monetary model as I(1) processes, and we proceed to examine whether a stable linear combination between the exchange rate and the respective monetary variables can be found. 4. JOHANSEN’S MAXIMUM LIKELIHOOD PROCEDURE For any set of I(1) variables, Johansen (1988) and Johansen and Juselius (1990) have developed a maximum-likelihood testing procedure for the number of cointegrating vectors, which also includes testing procedures for linear restrictions on the cointegration parameters, such as those implied by the monetary model. Any p-dimensional vector {Xt} that follows a Gaussian VAR process with lag order k+1 and drift µ, can always be written as:

k ∆Xt = ∑i=1 Γi ∆Xt-i + Γk+1 X t - k - 1 + φDt + µ +ε t , (t = 1 , . .) . , (7)T

where: εt is an independently and identically distributed p-dimensional vector with zero mean and variance-covariance matrix Λ, Dt is a vector of centered seasonal dummies that sum to zero over a full year by construction and this implies that they do not change the distribution of the rank tests, and T is the sample size. The rank of matrix Γk+1 gives the dimension of the cointegrating vector. If its rank is 0 < r < p, then Γk+1 can be decomposed into: - Γk+1 = α β'

(8)

where both α and β are p × r matrices. The rows of β' form the r d i s t i n c t cointegrating vectors, and, if we think of the elements of the r × 1 vector β'Xt-k-1 as “error correction” terms, then the elements of matrix (-α) express the speed of adjustment of the dependent variables towards the equilibrium state. Johansen (1988) suggested a procedure for deriving maximum likelihood estimates of α and β, as well as two likelihood ratio test statistics to determine the rank of the cointegration space. With the trace statistic, the null hypothesis is that there are, at most, r cointegrating vectors, while with the maximum eigenvalue statistic, we test for the presence of r versus r + 1 c o i n t e g r a t i n g vectors. Recently, Sephton and Larsen (1991) have shown that Johansen’s test may be characterized by sample dependency. Hansen and Johansen (1993) have suggested

MONETARY EXCHANGE RATE MODEL

91

methods for the evaluation of parameter constancy in cointegrated VAR models, that allow us to test the constancy of both the cointegrating space and vector(s) formally using estimates obtained from the Johansen FIML technique. Three tests have been cons tructed under the two VAR repres entations. In the “Zrepresentation,” all the parameters of model (7) are re-estimated during the recursions, while under the “R-representation,” the short-run parameters Γi, i = 1 . . . k, are fixed to their full-samples values, and only the long-run parameters in (8) are re-estimated.3 The first test is called the Rank test, and we examine the null hypothesis of sample independency of the cointegration rank of the system. This is accomplished by first estimating the model over the full sample, and the residuals corresponding to each recursive subsample are used to form the standard sample moments associated with Johansen’s reduced-rank approach. The eigenvalue problem is then solved directly from these subsample moment matrices. The obtained sequence of trace statistics is scaled by the corresponding critical values, and we accept the null hypothesis that the chosen rank is maintained regardless of the subperiod, it has been estimated. A second test deals with the null hypothesis of the constancy of the cointegration space for a given cointegration rank. Hansen and Johansen propose a likelihood ratio test that is constructed by comparing the likelihood function from each recursive subsample with the likelihood function computed under the restriction that the cointegrating vector estimated from the full sample falls within the space spanned from the estimated vectors of each individual sample. The test statistic is a χ 2 distributed with (p-r)r degrees of freedom, where p stands for the number of endogenous variables and r for the cointegraion rank. The final test examines the constancy of the individual e lements of the cointegrating vectors. A problem arises when the cointegration rank is greater than one, since the elements of the vectors cannot be identified unless certain restrictions are imposed. However, because there is a unique relationship between the eigenvalues and the cointegrating vectors, when these vectors have undergone a structural change, this will be reflected in the estimated eigenvalues. Hansen and Johansen (1993) have also derived the asymptotic distribution of the eigenvalues which allows one to test for the statistical significance of their deviations at different points in time. 5. COINTEGRATION RESULTS The results from both likelihood ratio tests for the complete model are shown in Table 2. In applying the maximum likelihood procedure, we have to establish the lag

3

The motivation for the “R-representation” is that, by fixing the estimates of the short-run parameters, we reduce the variance of the parameters that cointegration analysis is primarily interested in, i.e. the long-run ones (Hansen and Johansen, 1993).

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length for the VAR model (7). This was done by setting as an upper limit a model of thirteen periods and testing down against alternative models with shorter lag structure, using the likelihood ratio test corrected for the degrees of freedom proposed by Sims (1980). The chosen model was then tested for the presence of serial correlation in the residuals, using the portmantaeu Ljung-Box Q statistic. If the residuals in any equation were shown to be non-white, we sequentially chose a higher lag structure until they were whitened. For our case, we found that a seventh-order lag satisfied these criteria.4 We have also included in the estimation procedure eleven centered seasonal dummies, which are necessary to account for short-run effects that could otherwise violate the Gaussian assumption. Finally, we have also included a constant that appears only in the cointegrating vector.5 Table 2. Johansen - Juselius Maximum Likelihood Cointegration Tests r

(n-r)

Tr*

95%

Tr

95%

λmax*

95%

λmax

95%

r