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An empirical analysis on the stability of Japan’s aggregate import demand function
Japan and the World Economy 13 (2001) 135±144
An empirical analysis on the stability of Japan's aggregate import demand function Shigeyuki Hamoria,*, Yoichi Matsubayashib a
Faculty of Economics, Kobe University, 2-1 Rokkodai, Nada-Ku, Kobe 657-8501, Japan b Faculty of Economics, Wakayama University, Wakayama, Japan
Received 24 March 2000; received in revised form 17 August 2000; accepted 28 September 2000
Abstract This paper empirically analyzes the stability of the Japanese import demand function based on the concept of cointegration. The standard cointegration tests and the test developed by Gregory and Hansen (1996) are performed. These empirical results do not support the presence of a stable cointegrating relation among real GDP, real imports and relative import prices. Thus, we cannot say that there is a stable import demand function in Japan during the period analyzed. As a result, stimulation of domestic business conditions in Japan will not necessarily link to the quantity of imports. This is an important ®nding from the viewpoint of Japanese economic policy. # 2001 Elsevier Science B.V. All rights reserved. JEL classification: F10; F40 Keywords: Cointegration; Japanese import demand function; Regime shifts
1. Introduction This paper aims at empirically analyzing the relationship of Japan's import demand function. Japan's trade surplus often attracts international public attention and has caused political friction between Japan and other countries, such as the United States.1 Japan is criticized for this trade surplus, and it has been recommended that Japan should reduce this trade surplus by stimulating domestic business conditions. Such a recommendation tacitly assumes stability of Japan's import demand function. In other words, the quantity of imports to Japan is assumed to be closely related to domestic business conditions and the * Corresponding author. Tel./fax: 81-6-6338-1255. E-mail address: [email protected] (S. Hamori). 1 Also see Matsubayashi (1999).
0922-1425/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 2 - 1 4 2 5 ( 0 0 ) 0 0 0 6 3 - 3
136
S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
relative prices of imports: if domestic business conditions are stimulated, the quantity of imports will increase and as a result, the trade surplus will fall. It is necessary to verify the stability of the Japanese import demand function to see whether the assumption underlying the recommendation is satis®ed empirically. A great deal of empirical research has been done regarding import and export behavior. (See Feenstra (1988, 1995) and Sawyer and Sprinkle (1999) for a survey.) Kemp (1962) is the seminal work on the analysis of structural change in the import demand function. He analyzed the structural change of the Canadian import demand function for the period preceding and following World War II. Joy and Stolen (1975), Stern et al. (1979) and Volker (1992) analyzed the possibility of structural change in the import demand function of the United States. Heien (1968) and Warner and Kreinin (1983) analyzed the structural change of the import demand function for several other industrialized countries.2 This paper analyses the long-run relationship between the import series and its explanatory variables, such as real income and relative import prices. Our analysis is based on the concept of cointegration, which was introduced by Engle and Granger (1987). The concept of cointegration implies the long-run equilibrium among variables. This paper examines whether a cointegrating relation exists among the quantity of imports, real GDP and relative import prices. When empirical analysis veri®es the presence of a cointegrating relation, it is suggested that there is a long-run equilibrium relationship among the variables, con®rming the relationship of the import demand function indirectly through data characteristics. Carone (1996) is an example of an analysis of an import demand function based on the concept of cointegration. This paper also pays attention to the stability of the cointegrating relation itself. When a cointegrating relation is rejected, there are two possibilities. One is the case where there is no stable long-run relationship among the variables. The other is the case where a structural change has occurred in the cointegrating relation itself. To examine this second case, Gregory and Hansen (1996) developed a test of cointegration under structural change. This paper performs empirical analysis by focusing on this latter case. 2. Data The empirical work in this study is based on a sample of quarterly observations covering the ®rst quarter of 1973 to the ®rst quarter of 1998. This corresponds to the period of the ¯exible exchange rate regime.3 The logarithmic values of real imports, real GDP and the relative import prices (import prices divided by GDP de¯ator) are used. They are seasonally adjusted data. These data were acquired from the CD-ROM of the Annual Report on National Accounts (Economic Planning Agency). The number of observations (T) is 101 for each variable. Figs. 1±3 show the movements of each variable. 2 Economic Planning Agency (1995, 1996) empirically analyzed the import demand function in Japan based on the technique of Shiller lag. 3 An analysis of the import demand function during the period of the fixed exchange rate regime was also performed. However, all variables were not found to be stationary during this period and consequently cointegration could not be directly applied.
S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
137
Fig. 1. Log of real GDP. First quarter of 1973 to the first quarter of 1998.
3. Empirical results 3.1. Unit root test A unit root test is performed for each variable, by applying the augmented Dickey±Fuller test (ADF test) (Dickey and Fuller, 1979, 1981). The auxiliary regression is run with an intercept and a time trend (Model A), and with an intercept (Model B) as follows: Model A : Dxt a bt gxt
1
k X di Dxt
i
ut ;
i1
Model B :
Dxt a gxt
1
k X
di Dxt
i
ut
i1
where xt corresponds to the log of real GDP at time t (yt), the log of real imports at time t (imt), and the log of relative import prices at time t (rpt), D the difference operator and ut the serially uncorrelated error term with mean zero and constant variance. The augmentation terms are added to convert the residuals into white noise without affecting the distribution of the test statistics under the null hypothesis of a unit root. The null hypothesis and the alternative hypothesis are shown as follows: H0 : g 0;
HA : g < 0
The test statistic for a unit root is the t-ratio of g. The critical values are tabulated in Fuller (1976, p. 373, Table 8.5.2). In the case of a unit root test, the result sometimes changes with the specification of the regression equation. This paper used part of the algorithm developed by Campbell and Perron (1991).4 We test the significance of the time trend 4
Also see Dolado et al. (1990).
138
S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
Fig. 2. Log of real import. First quarter of 1973 to the first quarter of 1998.
term (b) when a unit root is not rejected in Model A. This critical value is tabulated in Dickey and Fuller (1981). As is pointed out by Campbell and Perron (1991), the result of Model B becomes more important when a unit root is not rejected and the time trend term is insignificant in Model A. Table 1 shows the results of the unit root test. Since the results of the ADF test depend on the lag length of the augmentation term (k), we consider the following four cases to evaluate the robustness of our empirical results: 1. four periods; 2. eight periods;
Fig. 3. Log of relative import prices. First quarter of 1973 to the first quarter of 1998.
S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
139
Table 1 P First quarter of 1973 to the P first quarter of 1998 (Model A : Dxt a bt gxt 1 ki1 di Dxt i ut ; k Model B : Dxt a gxt 1 i1 di Dxt i ut ) (* and ** show that the null hypothesis is rejected at 1 and 5% significance levels, respectively) t-value of b
Q-statistica
P-valueb
1.087 0.177 0.511 1.126 1.197 1.837 1.197
0.967 0.356 0.364 1.006 NA NA NA
0.622 0.285 0.328 0.638 0.588 0.252 0.588
0.961 0.991 0.988 0.959 0.964 0.993 0.964
4 8 AIC(4) SBIC(2) 4 8 AIC(4) SBIC(2)
2.414 2.104 2.414 2.813 0.286 0.089 0.286 0.477
2.442 2.165 2.442 2.809 NA NA NA NA
0.952 0.516 0.952 8.393 1.259 0.493 1.259 7.520
0.917 0.972 0.917 0.078 0.868 0.974 0.868 0.111
4 8 AIC, SBIC(5) 4 8 AIC(5) SBIC(4)
2.825 2.509 2.520 1.276 0.986 0.977 1.276
2.617 2.469 2.484 NA NA NA NA
0.238 0.405 0.297 0.243 0.264 0.150 0.243
0.993 0.982 0.990 0.993 0.992 0.997 0.993
Variable
Model
Lag (k)
yt
Model A
4 8 AIC(5)c SBIC(3)d 4 8 AIC, SBIC(4)
Model B
Model A
imt
Model B
Model A
rpt
Model B
Unit root test statistics
a
Q-statistic is a Ljung±Box statistic for the null hypothesis that there is no autocorrelation up to order 4. P-value is the probability value associated with the Q-statistic. c AIC(k) shows that the kth order of lag length is chosen by AIC. d SBIC(k) shows that the kth order of lag length is chosen by SBIC. b
3. the period based on AIC (Akaike Information Criterion5); 4. the period based on SBIC (Schwarz±Bayesian Information Criterion6). We also report the Ljung±Box Q-statistics and their P-values for diagnostic purposes in Table 1.7 The Q-statistic at lag m (Q(m)) is a test statistic for the null hypothesis that there is no autocorrelation up to order m in residuals, and is asymptotically distributed as w2 with degrees of freedom equal to the number of autocorrelations. As an example, let us consider the results of GDP in Table 1 in the case of a four period lag in the augmentation term. In model A, the statistic to test for the unit root is 1.087 and thus the null hypothesis of a unit root is not rejected. For the time trend, we ®nd that the test 5
See Akaike (1974). See Schwarz (1978). 7 See Ljung and Box (1979). 6
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S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
statistic is 0.967 and thus the time trend term is found to be insigni®cant. Thus, we need to carry out a unit root test using Model B. Table 1 shows that the test statistic of a unit root is 1.197 in Model B and thus the null hypothesis of a unit root is again not rejected. Moreover, as is clear from the Q-statistic (0.588) and its P-value (0.964), the null hypothesis that the residual term does not contain serial correlation cannot be rejected. Therefore, under this speci®cation, it is possible to determine that real GDP contains a unit root. A similar argument is formed in the other cases and other variables. As is clear from Table 1, the null hypothesis of a unit root is not rejected for each variable in every case. Thus, we can say that each variable has a unit root. 3.2. Cointegration test The cointegration test is carried out in this paper using two types of test procedures. One is the residual-based cointegration test developed by Engle and Granger (1987) (Engle± Granger test), and the other is the cointegration test developed by Johansen (1988) and Johansen and Juselius (1990) (Johansen test). This paper applies these procedures to the trivariate system (imt, yt and rpt). If a linear combination of the three variables is stationary, there is a cointegrating relation among them. For the Engle±Granger test, we specify the cointegrating regression as follows: imt a b1 yt b2 rpt vt
(1)
where vt is the error term with mean zero. Let the residuals obtained from the cointegrating regression (1) be wt . The test of no-cointegration can then be performed by an auxiliary regression of Dwt on wt 1 augmented by the necessary lagged values of Dwt as follows: Dwt pwt
1
p X bi Dwt
i
et
(2)
i1
where et is a serially uncorrelated error term with mean zero and constant variance. The null hypothesis and the alternative hypothesis are as follows: H0 : p 0;
HA : p < 0
The null hypothesis implies that there is no cointegrating relation among variables, whereas the alternative hypothesis implies that there is a cointegrating relation among variables. Since the results of the Engle±Granger test depend on the lag length of augmentation (k), we consider the following four cases to evaluate the robustness of our empirical results: 1. 2. 3. 4.
four periods; eight periods; the period based on AIC; the period based on SBIC.
Critical values are tabulated in Engle and Yoo (1987). Table 2 shows the empirical results of the Engle±Granger test. For example, when the lag order is 4, the test statistic is 2.362 and the null hypothesis cannot be rejected at a 5% signi®cance level. Similar
S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
141
Table 2 Cointegration test: Engle±Granger test (first quarter of 1973 to first quarter of 1998) (* and ** show that the null hypothesis of no-cointegration is rejected at 1 and 5% significance levels, respectively)a Lag
Test statisic
4 8 AIC(6)b SBIC(4)c
2.362 2.018 1.758 2.362
a
Cointegration equation: imt a b1 yt b2 rpt vt . AIC(k) shows that the kth order of lag length is chosen by AIC. c SBIC(k) shows that the kth order of lag length is chosen by SBIC. b
results can be obtained for other choices of lag length. Thus, the null hypothesis of nocointegration is not rejected in this test. For the Johansen test, we employ the trace test. Since the results of the Johansen test also depend on the lag length of the vector error correction model (VECM), we also consider the four cases mentioned above to evaluate the robustness of our empirical results. Critical values are tabulated in Osterwald-Lenum (1992). Table 3 indicates that we obtained mixed results and that the obvious conclusion is not attained. Toda (1995) pointed out that the results of the Johansen test clearly depend on the lag length of VECM. Our results are consistent with Toda's ®nding and thus we place more importance on the empirical results of our Engle±Granger test. 3.3. Cointegration test with regime shift Finally, we carry out the cointegration test, which explicitly takes into consideration the structural change in cointegration vector. Gregory and Hansen (1996) develop this procedure. We consider the following two models: Table 3 Cointegration test: Johansen test (first quarter of 1973 to first quarter of 1998) Lag
Hypothesized number of cointegrating equations
Test statistic
4
None At most 1 At most 2
22.873 7.072 2.531
8
None At most 1 At most 2
36.677** 17.312* 4.291*
AIC, SBIC(1)a
None At most 1 At most 2
37.126** 11.666 4.697*
*
Shows that the null hypothesis of no-cointegration is rejected at 1% significance level. Shows that the null hypothesis of no-cointegration is rejected at 5% significance level. a AIC(k) shows that the kth order of lag length is chosen by AIC. SBIC(k) shows that the kth order of lag length is chosen by SBIC. **
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S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
1. level shift; 2. regime shift. Level shift implies that there is a shift in the constant term of the cointegrating equation. Regime shift implies that there is a change in the slope of the cointegrating equation. When cointegration is rejected, there are two possibilities. One is the case where the long-run equilibrium does not exist among variables. The other is the case where there is a structural change in cointegrating vector. To cope with this second problem, Gregory and Hansen (1996) develop the procedure to test for cointegration, which allow for the possibility of regime shifts. They developed three test statistics designed to test the null of nocointegration against the alternative of cointegration in the presence of a possible regime shift. In particular, they consider the cases where the intercepts and/or slope coefficients have a single break of unknown timing. Critical values are tabulated in Gregory and Hansen (1996). Following Gregory and Hansen (1996), we compute the cointegration test statistic for each possible regime shift t 2 T, and take the smallest value (the largest negative value) across all possible break points. For the actual computation, we consider t as a step function over (0.15, 0.85) that jumps every 1/T period. Therefore, the test statistic is computed for each break point in the interval ([0.15T], [0.85T]). The test statistics shown as Za , Zt and ADF correspond to the minimum values of Za, Zt and ADF test statistic, where Za and Zt are the unit root test statistics developed by Phillips and Perron (1988). Table 4 shows the empirical results. As is clear from this table, the null hypothesis of nocointegration is not rejected in every case. Thus, even if we take into consideration the possibilities of structural change, the cointegrating relation cannot be found among three variables.
Table 4 Cointegration test with regime shift (first quarter of 1973 to first quarter of 1998) (* and ** show that the null hypothesis is rejected at 1 and 5% significance levels, respectively) Test
Lag
Model A: level shift ADF
Zt Za Model B: regime shift ADF Zt Za a b
Test statistic
Break point
4 8 AIC(4)a SBIC(3)b
3.816 3.350 3.816 4.052 2.799 16.352
0.386 0.723 0.386 0.733 0.446 0.446
4 8 AIC, SBIC(3)
4.320 3.399 4.988 3.660 24.785
0.663 0.673 0.673 0.416 0.416
AIC(k) shows that the kth order of lag length is chosen by AIC. SBIC(k) shows that the kth order of lag length is chosen by SBIC.
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143
4. Conclusion This paper empirically analyzes the stability of the Japanese import demand function based on the concept of cointegration. The analysis uses quarterly data for the sample period covering the ®rst quarter of 1973 to the ®rst quarter of 1998. This corresponds to the period of the ¯exible exchange rate regime. The variables analyzed are real import, real GDP and relative import prices. The results of standard cointegration tests do not suggest that there is a cointegrating relation among the three variables. The test developed by Gregory and Hansen (1996) is also performed, which is the cointegration test under structural change. These empirical results also do not support the presence of a cointegrating relation among the variables. Thus, we cannot say that there is a stable import demand function in Japan during the period analyzed. As a result, stimulation of domestic business conditions in Japan will not necessarily link to the quantity of imports. This is an important ®nding from the viewpoint of Japanese economic policy.8 Acknowledgements The authors would like to thank the editor, Ryuzo Sato, and an anonymous referee for many helpful comments and suggestions. References Akaike, K., 1974. A new look at the statistical model identification. IEEE Transactions on Automatic Control AC-19, 716±723. Campbell, Y., Perron, P., 1991. Pitfalls and opportunities: what macroeconomists should know about unit roots. NBER Macroeconomic Annual, 141±201. Carone, G., 1996. Modeling of the US demand for imports through cointegration and error correction. Journal of Policy Modeling 18, 1±48. Dickey, D., Fuller, W.A., 1979. Distribution of the estimates for autoregressive time series with a unit root. Journal of the American Statistical Society 74, 427±431. Dickey, D., Fuller, W.A., 1981. Likelihood ratio of the estimates for autoregressive time series with a unit root. Econometrica 49, 1057±1072. Dolado, J., Jenkinson, T., Sosvilla-Rivero, S., 1990. Cointegration and unit roots. Journal of Economic Survey 4, 249±273. Economic Planning Agency, 1995. Economic White Paper 1995. Economic Planning Agency, 1996. Economic White Paper 1996. Engle, R.F., Granger, C.W.J., 1987. Cointegration and error correction: representation estimation and testing. Econometrica 55, 251±276. Engle, R.F., Yoo, B.S., 1987. Forecasting and testing in co-integrated systems. Journal of Econometrics 35, 143± 159. Feenstra R.C., 1988. Empirical Methods in International Trade. MIT Press. Feenstra, R.C., 1995. Estimating the effects of trade policy. In Grossman G.M., Rogoff K. (Eds.), Handbook of International Economics 3. Fuller W.A., 1976. Introduction to Statistical Time Series. John Wiley & Sons. 8
These results are consistent with Mah (1994).
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Gregory, A.W., Hansen, B.E., 1996. Residual-based tests for cointegration in models with regime shifts. Journal of Econometrics 70, 99±126. Heien, D.M., 1968. Structural stability and the estimation of international import price elasticity. Kyklos 21, 695±712. Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231±254. Johansen, S., Juselius, K., 1990. Maximum likelihood estimation and inference on cointegration with applications to the demand for money. Oxford Bulletin of Economics and Statistics 52, 169±209. Joy, J., Stolen, J.D., 1975. The change in the US import demand function from the 1950s to the 1960s. Review of Economics and Statistics 57, 109±111. Kemp, M.C., 1962. The Demands for Canadian Imports: 1926±55. Toronto University, Toronto, Ont. Ljung, G., Box, B., 1979. On a measure of lack of fit in time series models. Biometrika 66, 265±270. Mah, J.S., 1994. Japanese import demand behavior: the cointegration approach. Journal of Policy Modeling 16, 291±298. Matsubayashi, Y., 1999. An empirical analysis of the external imbalance and macroeconomy. Japan±US comparison. Doctorial Dissertation. Kobe University, Kobe, submitted for publication. Osterwald-Lenum, M., 1992. A note with quintiles of the asymptotic distribution of the maximum likelihood cointegration rank test statistics. Oxford Bulletin of Economics and Statistics 54, 461±472. Phillips, P.C.B., Perron, P., 1988. Testing for a unit root in time series regression. Biometrika 75, 335±346. Sawyer, W.C., Sprinkle, R.L., 1999. The Demand for Imports and Exports in the World Economy. Ashgate. Schwarz, G., 1978. Estimating the dimension of a model. Annals of Statistics 6, 461±464. Stern, R.M., Baum, C.F., Greene, M.N., 1979. Evidence on structural change in the demand for aggregate US imports and exports. Journal of Political Economy 87, 179±192. Toda, H.Y., 1995. Finite sample properties of likelihood ratio tests for cointegrating ranks in vector autoregressions. Econometric Theory 11, 1015±1032. Volker, P.A., 1992. On the import demand function: a comment. Journal of Political Economy 90, 1295±1299. Warner, D., Kreinin, M.E., 1983. Determinants of international trade flows. Review of Economics and Statistics 64, 368±379.
An empirical analysis on the stability of Japan's aggregate import demand function Shigeyuki Hamoria,*, Yoichi Matsubayashib a
Faculty of Economics, Kobe University, 2-1 Rokkodai, Nada-Ku, Kobe 657-8501, Japan b Faculty of Economics, Wakayama University, Wakayama, Japan
Received 24 March 2000; received in revised form 17 August 2000; accepted 28 September 2000
Abstract This paper empirically analyzes the stability of the Japanese import demand function based on the concept of cointegration. The standard cointegration tests and the test developed by Gregory and Hansen (1996) are performed. These empirical results do not support the presence of a stable cointegrating relation among real GDP, real imports and relative import prices. Thus, we cannot say that there is a stable import demand function in Japan during the period analyzed. As a result, stimulation of domestic business conditions in Japan will not necessarily link to the quantity of imports. This is an important ®nding from the viewpoint of Japanese economic policy. # 2001 Elsevier Science B.V. All rights reserved. JEL classification: F10; F40 Keywords: Cointegration; Japanese import demand function; Regime shifts
1. Introduction This paper aims at empirically analyzing the relationship of Japan's import demand function. Japan's trade surplus often attracts international public attention and has caused political friction between Japan and other countries, such as the United States.1 Japan is criticized for this trade surplus, and it has been recommended that Japan should reduce this trade surplus by stimulating domestic business conditions. Such a recommendation tacitly assumes stability of Japan's import demand function. In other words, the quantity of imports to Japan is assumed to be closely related to domestic business conditions and the * Corresponding author. Tel./fax: 81-6-6338-1255. E-mail address: [email protected] (S. Hamori). 1 Also see Matsubayashi (1999).
0922-1425/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 2 - 1 4 2 5 ( 0 0 ) 0 0 0 6 3 - 3
136
S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
relative prices of imports: if domestic business conditions are stimulated, the quantity of imports will increase and as a result, the trade surplus will fall. It is necessary to verify the stability of the Japanese import demand function to see whether the assumption underlying the recommendation is satis®ed empirically. A great deal of empirical research has been done regarding import and export behavior. (See Feenstra (1988, 1995) and Sawyer and Sprinkle (1999) for a survey.) Kemp (1962) is the seminal work on the analysis of structural change in the import demand function. He analyzed the structural change of the Canadian import demand function for the period preceding and following World War II. Joy and Stolen (1975), Stern et al. (1979) and Volker (1992) analyzed the possibility of structural change in the import demand function of the United States. Heien (1968) and Warner and Kreinin (1983) analyzed the structural change of the import demand function for several other industrialized countries.2 This paper analyses the long-run relationship between the import series and its explanatory variables, such as real income and relative import prices. Our analysis is based on the concept of cointegration, which was introduced by Engle and Granger (1987). The concept of cointegration implies the long-run equilibrium among variables. This paper examines whether a cointegrating relation exists among the quantity of imports, real GDP and relative import prices. When empirical analysis veri®es the presence of a cointegrating relation, it is suggested that there is a long-run equilibrium relationship among the variables, con®rming the relationship of the import demand function indirectly through data characteristics. Carone (1996) is an example of an analysis of an import demand function based on the concept of cointegration. This paper also pays attention to the stability of the cointegrating relation itself. When a cointegrating relation is rejected, there are two possibilities. One is the case where there is no stable long-run relationship among the variables. The other is the case where a structural change has occurred in the cointegrating relation itself. To examine this second case, Gregory and Hansen (1996) developed a test of cointegration under structural change. This paper performs empirical analysis by focusing on this latter case. 2. Data The empirical work in this study is based on a sample of quarterly observations covering the ®rst quarter of 1973 to the ®rst quarter of 1998. This corresponds to the period of the ¯exible exchange rate regime.3 The logarithmic values of real imports, real GDP and the relative import prices (import prices divided by GDP de¯ator) are used. They are seasonally adjusted data. These data were acquired from the CD-ROM of the Annual Report on National Accounts (Economic Planning Agency). The number of observations (T) is 101 for each variable. Figs. 1±3 show the movements of each variable. 2 Economic Planning Agency (1995, 1996) empirically analyzed the import demand function in Japan based on the technique of Shiller lag. 3 An analysis of the import demand function during the period of the fixed exchange rate regime was also performed. However, all variables were not found to be stationary during this period and consequently cointegration could not be directly applied.
S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
137
Fig. 1. Log of real GDP. First quarter of 1973 to the first quarter of 1998.
3. Empirical results 3.1. Unit root test A unit root test is performed for each variable, by applying the augmented Dickey±Fuller test (ADF test) (Dickey and Fuller, 1979, 1981). The auxiliary regression is run with an intercept and a time trend (Model A), and with an intercept (Model B) as follows: Model A : Dxt a bt gxt
1
k X di Dxt
i
ut ;
i1
Model B :
Dxt a gxt
1
k X
di Dxt
i
ut
i1
where xt corresponds to the log of real GDP at time t (yt), the log of real imports at time t (imt), and the log of relative import prices at time t (rpt), D the difference operator and ut the serially uncorrelated error term with mean zero and constant variance. The augmentation terms are added to convert the residuals into white noise without affecting the distribution of the test statistics under the null hypothesis of a unit root. The null hypothesis and the alternative hypothesis are shown as follows: H0 : g 0;
HA : g < 0
The test statistic for a unit root is the t-ratio of g. The critical values are tabulated in Fuller (1976, p. 373, Table 8.5.2). In the case of a unit root test, the result sometimes changes with the specification of the regression equation. This paper used part of the algorithm developed by Campbell and Perron (1991).4 We test the significance of the time trend 4
Also see Dolado et al. (1990).
138
S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
Fig. 2. Log of real import. First quarter of 1973 to the first quarter of 1998.
term (b) when a unit root is not rejected in Model A. This critical value is tabulated in Dickey and Fuller (1981). As is pointed out by Campbell and Perron (1991), the result of Model B becomes more important when a unit root is not rejected and the time trend term is insignificant in Model A. Table 1 shows the results of the unit root test. Since the results of the ADF test depend on the lag length of the augmentation term (k), we consider the following four cases to evaluate the robustness of our empirical results: 1. four periods; 2. eight periods;
Fig. 3. Log of relative import prices. First quarter of 1973 to the first quarter of 1998.
S. Hamori, Y. Matsubayashi / Japan and the World Economy 13 (2001) 135±144
139
Table 1 P First quarter of 1973 to the P first quarter of 1998 (Model A : Dxt a bt gxt 1 ki1 di Dxt i ut ; k Model B : Dxt a gxt 1 i1 di Dxt i ut ) (* and ** show that the null hypothesis is rejected at 1 and 5% significance levels, respectively) t-value of b
Q-statistica
P-valueb
1.087 0.177 0.511 1.126 1.197 1.837 1.197
0.967 0.356 0.364 1.006 NA NA NA
0.622 0.285 0.328 0.638 0.588 0.252 0.588
0.961 0.991 0.988 0.959 0.964 0.993 0.964
4 8 AIC(4) SBIC(2) 4 8 AIC(4) SBIC(2)
2.414 2.104 2.414 2.813 0.286 0.089 0.286 0.477
2.442 2.165 2.442 2.809 NA NA NA NA
0.952 0.516 0.952 8.393 1.259 0.493 1.259 7.520
0.917 0.972 0.917 0.078 0.868 0.974 0.868 0.111
4 8 AIC, SBIC(5) 4 8 AIC(5) SBIC(4)
2.825 2.509 2.520 1.276 0.986 0.977 1.276
2.617 2.469 2.484 NA NA NA NA
0.238 0.405 0.297 0.243 0.264 0.150 0.243
0.993 0.982 0.990 0.993 0.992 0.997 0.993
Variable
Model
Lag (k)
yt
Model A
4 8 AIC(5)c SBIC(3)d 4 8 AIC, SBIC(4)
Model B
Model A
imt
Model B
Model A
rpt
Model B
Unit root test statistics
a
Q-statistic is a Ljung±Box statistic for the null hypothesis that there is no autocorrelation up to order 4. P-value is the probability value associated with the Q-statistic. c AIC(k) shows that the kth order of lag length is chosen by AIC. d SBIC(k) shows that the kth order of lag length is chosen by SBIC. b
3. the period based on AIC (Akaike Information Criterion5); 4. the period based on SBIC (Schwarz±Bayesian Information Criterion6). We also report the Ljung±Box Q-statistics and their P-values for diagnostic purposes in Table 1.7 The Q-statistic at lag m (Q(m)) is a test statistic for the null hypothesis that there is no autocorrelation up to order m in residuals, and is asymptotically distributed as w2 with degrees of freedom equal to the number of autocorrelations. As an example, let us consider the results of GDP in Table 1 in the case of a four period lag in the augmentation term. In model A, the statistic to test for the unit root is 1.087 and thus the null hypothesis of a unit root is not rejected. For the time trend, we ®nd that the test 5
See Akaike (1974). See Schwarz (1978). 7 See Ljung and Box (1979). 6
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statistic is 0.967 and thus the time trend term is found to be insigni®cant. Thus, we need to carry out a unit root test using Model B. Table 1 shows that the test statistic of a unit root is 1.197 in Model B and thus the null hypothesis of a unit root is again not rejected. Moreover, as is clear from the Q-statistic (0.588) and its P-value (0.964), the null hypothesis that the residual term does not contain serial correlation cannot be rejected. Therefore, under this speci®cation, it is possible to determine that real GDP contains a unit root. A similar argument is formed in the other cases and other variables. As is clear from Table 1, the null hypothesis of a unit root is not rejected for each variable in every case. Thus, we can say that each variable has a unit root. 3.2. Cointegration test The cointegration test is carried out in this paper using two types of test procedures. One is the residual-based cointegration test developed by Engle and Granger (1987) (Engle± Granger test), and the other is the cointegration test developed by Johansen (1988) and Johansen and Juselius (1990) (Johansen test). This paper applies these procedures to the trivariate system (imt, yt and rpt). If a linear combination of the three variables is stationary, there is a cointegrating relation among them. For the Engle±Granger test, we specify the cointegrating regression as follows: imt a b1 yt b2 rpt vt
(1)
where vt is the error term with mean zero. Let the residuals obtained from the cointegrating regression (1) be wt . The test of no-cointegration can then be performed by an auxiliary regression of Dwt on wt 1 augmented by the necessary lagged values of Dwt as follows: Dwt pwt
1
p X bi Dwt
i
et
(2)
i1
where et is a serially uncorrelated error term with mean zero and constant variance. The null hypothesis and the alternative hypothesis are as follows: H0 : p 0;
HA : p < 0
The null hypothesis implies that there is no cointegrating relation among variables, whereas the alternative hypothesis implies that there is a cointegrating relation among variables. Since the results of the Engle±Granger test depend on the lag length of augmentation (k), we consider the following four cases to evaluate the robustness of our empirical results: 1. 2. 3. 4.
four periods; eight periods; the period based on AIC; the period based on SBIC.
Critical values are tabulated in Engle and Yoo (1987). Table 2 shows the empirical results of the Engle±Granger test. For example, when the lag order is 4, the test statistic is 2.362 and the null hypothesis cannot be rejected at a 5% signi®cance level. Similar
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Table 2 Cointegration test: Engle±Granger test (first quarter of 1973 to first quarter of 1998) (* and ** show that the null hypothesis of no-cointegration is rejected at 1 and 5% significance levels, respectively)a Lag
Test statisic
4 8 AIC(6)b SBIC(4)c
2.362 2.018 1.758 2.362
a
Cointegration equation: imt a b1 yt b2 rpt vt . AIC(k) shows that the kth order of lag length is chosen by AIC. c SBIC(k) shows that the kth order of lag length is chosen by SBIC. b
results can be obtained for other choices of lag length. Thus, the null hypothesis of nocointegration is not rejected in this test. For the Johansen test, we employ the trace test. Since the results of the Johansen test also depend on the lag length of the vector error correction model (VECM), we also consider the four cases mentioned above to evaluate the robustness of our empirical results. Critical values are tabulated in Osterwald-Lenum (1992). Table 3 indicates that we obtained mixed results and that the obvious conclusion is not attained. Toda (1995) pointed out that the results of the Johansen test clearly depend on the lag length of VECM. Our results are consistent with Toda's ®nding and thus we place more importance on the empirical results of our Engle±Granger test. 3.3. Cointegration test with regime shift Finally, we carry out the cointegration test, which explicitly takes into consideration the structural change in cointegration vector. Gregory and Hansen (1996) develop this procedure. We consider the following two models: Table 3 Cointegration test: Johansen test (first quarter of 1973 to first quarter of 1998) Lag
Hypothesized number of cointegrating equations
Test statistic
4
None At most 1 At most 2
22.873 7.072 2.531
8
None At most 1 At most 2
36.677** 17.312* 4.291*
AIC, SBIC(1)a
None At most 1 At most 2
37.126** 11.666 4.697*
*
Shows that the null hypothesis of no-cointegration is rejected at 1% significance level. Shows that the null hypothesis of no-cointegration is rejected at 5% significance level. a AIC(k) shows that the kth order of lag length is chosen by AIC. SBIC(k) shows that the kth order of lag length is chosen by SBIC. **
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1. level shift; 2. regime shift. Level shift implies that there is a shift in the constant term of the cointegrating equation. Regime shift implies that there is a change in the slope of the cointegrating equation. When cointegration is rejected, there are two possibilities. One is the case where the long-run equilibrium does not exist among variables. The other is the case where there is a structural change in cointegrating vector. To cope with this second problem, Gregory and Hansen (1996) develop the procedure to test for cointegration, which allow for the possibility of regime shifts. They developed three test statistics designed to test the null of nocointegration against the alternative of cointegration in the presence of a possible regime shift. In particular, they consider the cases where the intercepts and/or slope coefficients have a single break of unknown timing. Critical values are tabulated in Gregory and Hansen (1996). Following Gregory and Hansen (1996), we compute the cointegration test statistic for each possible regime shift t 2 T, and take the smallest value (the largest negative value) across all possible break points. For the actual computation, we consider t as a step function over (0.15, 0.85) that jumps every 1/T period. Therefore, the test statistic is computed for each break point in the interval ([0.15T], [0.85T]). The test statistics shown as Za , Zt and ADF correspond to the minimum values of Za, Zt and ADF test statistic, where Za and Zt are the unit root test statistics developed by Phillips and Perron (1988). Table 4 shows the empirical results. As is clear from this table, the null hypothesis of nocointegration is not rejected in every case. Thus, even if we take into consideration the possibilities of structural change, the cointegrating relation cannot be found among three variables.
Table 4 Cointegration test with regime shift (first quarter of 1973 to first quarter of 1998) (* and ** show that the null hypothesis is rejected at 1 and 5% significance levels, respectively) Test
Lag
Model A: level shift ADF
Zt Za Model B: regime shift ADF Zt Za a b
Test statistic
Break point
4 8 AIC(4)a SBIC(3)b
3.816 3.350 3.816 4.052 2.799 16.352
0.386 0.723 0.386 0.733 0.446 0.446
4 8 AIC, SBIC(3)
4.320 3.399 4.988 3.660 24.785
0.663 0.673 0.673 0.416 0.416
AIC(k) shows that the kth order of lag length is chosen by AIC. SBIC(k) shows that the kth order of lag length is chosen by SBIC.
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4. Conclusion This paper empirically analyzes the stability of the Japanese import demand function based on the concept of cointegration. The analysis uses quarterly data for the sample period covering the ®rst quarter of 1973 to the ®rst quarter of 1998. This corresponds to the period of the ¯exible exchange rate regime. The variables analyzed are real import, real GDP and relative import prices. The results of standard cointegration tests do not suggest that there is a cointegrating relation among the three variables. The test developed by Gregory and Hansen (1996) is also performed, which is the cointegration test under structural change. These empirical results also do not support the presence of a cointegrating relation among the variables. Thus, we cannot say that there is a stable import demand function in Japan during the period analyzed. As a result, stimulation of domestic business conditions in Japan will not necessarily link to the quantity of imports. This is an important ®nding from the viewpoint of Japanese economic policy.8 Acknowledgements The authors would like to thank the editor, Ryuzo Sato, and an anonymous referee for many helpful comments and suggestions. References Akaike, K., 1974. A new look at the statistical model identification. IEEE Transactions on Automatic Control AC-19, 716±723. Campbell, Y., Perron, P., 1991. Pitfalls and opportunities: what macroeconomists should know about unit roots. NBER Macroeconomic Annual, 141±201. Carone, G., 1996. Modeling of the US demand for imports through cointegration and error correction. Journal of Policy Modeling 18, 1±48. Dickey, D., Fuller, W.A., 1979. Distribution of the estimates for autoregressive time series with a unit root. Journal of the American Statistical Society 74, 427±431. Dickey, D., Fuller, W.A., 1981. Likelihood ratio of the estimates for autoregressive time series with a unit root. Econometrica 49, 1057±1072. Dolado, J., Jenkinson, T., Sosvilla-Rivero, S., 1990. Cointegration and unit roots. Journal of Economic Survey 4, 249±273. Economic Planning Agency, 1995. Economic White Paper 1995. Economic Planning Agency, 1996. Economic White Paper 1996. Engle, R.F., Granger, C.W.J., 1987. Cointegration and error correction: representation estimation and testing. Econometrica 55, 251±276. Engle, R.F., Yoo, B.S., 1987. Forecasting and testing in co-integrated systems. Journal of Econometrics 35, 143± 159. Feenstra R.C., 1988. Empirical Methods in International Trade. MIT Press. Feenstra, R.C., 1995. Estimating the effects of trade policy. In Grossman G.M., Rogoff K. (Eds.), Handbook of International Economics 3. Fuller W.A., 1976. Introduction to Statistical Time Series. John Wiley & Sons. 8
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