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Data used in this paper, relating to the U.S. or Canadian dollar exchange rate and U.S. and Canada macroeconomic variables, are taken from International Financial Statistics and run from January 1980 to December 2000. The chosen monetary aggregate is seasonally unadjusted M2. The income measure is seasonally adjusted industrial production. The three-month treasury bill rate is used for the short-term interest rate, and the logarithmic change of the consumer price index over the preceding 12 months is used for the unobservable expected inflation rate. S&P 500 stock index and Toronto Stock Exchange 300 index are used for the share price. To get the real share prices, they are divided by the CPI with the base year of 1990. All data are expressed in logarithm except the interest rate.
Preliminary Test: Unit Roots
Prior to testing for cointegration, one needs to examine the time series properties of the variables. They should be integrated of the same order to be cointegrated. In other words, variables should be stationary after differencing each time series the same number of times. Most macroeconomic variables have been found to be non-stationary in their levels and stationary in first differences, which means that they are I(1).
In testing for stationarity, the augmented Dickey-Fuller [1979] (ADF) test and the Kwiatkowski, Phillips, Schmidt, and Shin [1992] (KPSS) test are implemented. The Dickey-Fuller type unit root tests are criticized because their failure to reject the null hypothesis may be attributed to their low power against weakly stationary alternatives. However, KPSS tests the null hypothesis of stationary against the alternative of a unit root. Thus the KPSS test is a complement procedure to the ADF test. To implement the ADF test, (2) one estimates the regression:
[DELTA][X.sub.t] = [alpha] + [beta][X.sub.t-1] + [summation over (k)(j=1)] [[gamma].sub.j][DELTA][X.sub.t-j] + [[epsilon].sub.t], (5)
where [DELTA] is the difference operator, X is the series being tested, k is the number of lagged differences, and e is an error term. If the t-statistics is less than the critical values, then the null hypothesis of a unit root ([beta] = 0) cannot be rejected. However, if the t-ratio is larger than the critical value, the null hypothesis of non-stationarity can be rejected. KPSS test statistics is:
[[zeta].sub.u] = [T.sub.2] [SIGMA] ([S.sup.2.sub.t]/[s.sub.2](L)), (6)
where
[S.sub.t] = [summation over (t)(i=1)] et,
and
[S.sub.2] = [T.sup.-1] [summation over (T)(t=1)] [e.sup.2.sub.t] + [2T.sup.-1] [summation over (L)(s = 1)] (1 - s/(L + 1)) [summation over (T)(t = s+1)] [e.sub.t][e.sub.t-s]
[S.sub.t] is the partial sum process of the residual e, T is the number of observations, and L is the lag length. If the test statistic is greater than the critical values, the null hypothesis of stationarity is rejected in favor of the unit root alternative.
Table 1 and 2 present that all series are first-difference stationary. When these series are also tested with a trend term for non-stationary test, none of the variables are trend stationary. Hence, all variables are non-stationary in levels.
Model Specification Test
It is necessary to determine the appropriate lag length (k) before the cointegration tests are conducted. Rather than the information-based rules such as Akaike information criteria and Schwartz criteria, this paper uses the general-to-specific modeling strategy that chooses between a model with k lags and a model with k + 1 lags. For instance, the procedure to choose the optimal lag length is to test down from an k lags system until k - 1 can be rejected at the 5 percent level, using a likelihood ratio statistics. Then, check the residuals for whiteness. If the residuals at this stage are non-white, choose a higher lag structure until the residuals are whitened.
In this paper, two models are initially estimated with k, arbitrarily set equal to 13. Then this unrestricted model is tested against a restricted model where k is reduced to 12. Although the results are not reported, both models are specified with k = 12.
Multivariate Cointegration
In order to model the short-run relationship, one first needs to examine if there is a long-run relationship. This paper uses the multivariate cointegration technique proposed by Johansen [1988] and Johansen and Juselius [1990] in order to test whether there is a long-run relationship between the exchange rate and the fundamental variables. A number of researchers [Boothe and Glassman, 1987; Baillie and Selover, 1987; McNown and Wallace, 1989], using the Engle-Granger [1987] two-step procedure, tested for cointegration and were not able to reject the null of no cointegration. However, recent studies [MacDonaId and Taylor, 1993, 1994; McNown and Wallace, 1994; Moosa, 1994], using the Johansen [1988] maximum likelihood method, showed strong evidence of cointegration for the monetary model.
According to MacDonald and Taylor [1993, 1994] and Moosa [1994], the Johansen and Juselius [1990] method is preferred to the simpler regression-based Engle and Granger [1987] method because it fully captures the underlying time series properties of the data, provides a test statistics for the total number of cointegrating vectors, and permits direct hypothesis testing on the coefficients of the cointegrating vectors. In addition, its results are invariant with respect to the direction of normalization, because it makes all of the variables explicitly endogenous. Since Johansen [1988] and Johansen and Juselius [1990] provide detailed description of the test procedure, this paper does not present the test procedure.
Table 3 shows that the results of Johansen maximum likelihood estimation. As the Dornbusch-Frankel model in Table 3 shows, the trace test shows that the null hypothesis of r = 0 is rejected at the 95 percent level. The maximum eigenvalue test shows that the null of r = 0 against the alternative r = 1 is rejected at the 95 percent level. Therefore, there is one cointegrating relationship in the Dornbusch-Franked model.
Dornbusch-Frankel model with share price shows that the null of r = 2 can be rejected at the 95 percent level, while the maximum eigenvalue test shows that the null of r = 1 cannot be rejected at the 95 percent significance level. In this case, there is the contradiction between two tests for cointegration rank. According to Cheung and Lai [1993], the trace statistic shows more robustness to both skewness and excess kurtosis in innovations than the maximum eigenvalue statistic. Therefore, based on the trace test, Dornbusch-Franekl model has three cointegrating vectors between exchange rate and fundamental variables. So these test results are supportive of the long-run properties of the monetary models. As we know, a cointegrating vector implies a long-run relationship among jointly endogenous variables arising from constraints implied by the economic structure on the long-run relationship. It means that the more the number of cointegrating vectors, the more stable will be the system of non-stationary cointegr ated variables.
The values of the coefficients in these cointegrating vectors are reported in Table 4. The estimated cointegrating vectors are given economic meaning by means of normalizing on the exchange rate. The vector that makes economic sense is that the estimated coefficients are close to and have the same signs as those predicted by economic theory. However, according to Dickey, Jansen, and Thornton [1991], cointegration analysis does not give estimates with structural interpretation regarding the magnitude of the parameters of the cointegrating vectors. Because cointegrating vectors merely imply long-run, stable relationship among jointly endogenous variables, they cannot be interpreted as structural equations. Cointegration relationships that do not make any economic sense need not to be discarded.
Error Correction and Dynamic Forecasting
After obtaining the long-run cointegration relationships using the Johansen approach, the short-run VAR in error-correction form (VECM) can be estimated with the cointegration relationships explicitly included and then construct out-of-sample forecasts by using this short-run dynamic equation. Following the general-to-specific approach to modeling, one can estimate a 12th order autoregressive distributed lag of the nominal exchange rate on economic fundamental variables and one lag of the error-correction term for the period 1980:1 to 1998:12 as:
[DELTA][S.sub.t] = [[GAMMA].sub.1][DELTA][X.sub.t-1] + ... + [[GAMMA].sub.k][DELTA][X.sub.t-11] + [ECM.sub.t-1] + [PHI][D.sub.t] + [[epsilon].sub.t], (7)
where [ECM.sub.t-1] is one lag of an error-correction term, and [D.sub.t] includes seasonal dummies and an intercept. The error-correction terms are obtained from the co-integrating relationship which are normalized against the exchange rate in each of the models.
Theory implies that the error-correction term is negative and significantly different from zero. The coefficient is an estimate of the speed of adjustment back to the long-run equilibrium relationship. A negative coefficient on the error-correction term implies that in the event of a deviation between the actual and long-run equilibrium level, there would be an adjustment back to the long-run relationship in subsequent periods to eliminate this discrepancy. Since all the variables in the above model are I(0), statistical inference using standard t and F tests is valid. The paper can achieve the final parsimonious specification by removing the insignificant regressors and testing whether this reduction in the model is supported by F-test. Finally, the resultant model can be checked by performing diagnostic tests on the residuals.
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