# Common stochastic trends in Pacific rim stock markets.

In an economic environment where international considerations are prominent, knowledge of the structure of the international equity market is important. Individual investors are interested in knowing the relationship among different national stock prices because it facilitates international diversification. Economists also pay attention to the relationship because it affects capital flows, and investment and consumption decisions.One idiosyncrasy of national stock prices is that over long horizons they tend to move together and follow a common upward trend. For example, the existence of price comovements was found in Hilliard's (1979) spectral analysis of international equity market indices. Eun and Shim (I 989) also found a significant interdependent relationship among several national stock prices, using a vector autoregression (VAR) model. More recently, Kasa (1992) estimated an error-correction VAR model and computed a common stochastic trend in the equity markets of the U.S., Japan, England, Germany, and Canada.

This study complements Kasa's work by investigating the common stochastic trend among national stock prices of the U.S., Japan, Taiwan, Hong Kong, Singapore, and South Korea. Study of the above five East Asian stock markets is interesting because, given the right conditions--moves toward opening, deregulation, and having shown high economic growth--the stock markets in those countries have become the focus of international investment during the past decade. The U.S. market is included in the analysis so as to examine the inter-continental stock price relationship between North America and the Pacific-Rim of East Asia.

As noted by Engle and Granger (1987), the specifications for VAR models depend on nonstationarity and cointegration properties of the vector time series involved in the analysis. If the nonstationary time series are cointegrated and converge toward long-run equilibrium relationships, a VAR model in levels is inefficient and may lead to spurious regression results, while that in first difference is misspecified. Based on the Johansen maximum likelihood procedure (1988; 1991) and Johansen and Juselius (1990), an error-correction VAR model is estimated and long-run common stochastic trends computed. Further, maximum likelihood ratio tests are conducted to compare among countries their cointegration coefficients and adjustment speeds toward the long-run equilibrium relationship.

This study is organized as follows. It starts with a brief discussion on Johansen's maximum likelihood procedure used in the study, then describes the data and some preliminary analyses, followed by results of model estimation and selection. The study reports the long-run common trend, and the results of cointegration coefficient tests, along with the results of short-run adjustment speed tests. It then simulates short-run dynamics of the stock prices.

JOHANSEN'S PROCEDURE

Consider a conventional VAR model in levels

(1) [X.sub.t] = [mu] + [[pi].sub.1] [X.sub.t-1] + ... + [[pi].sub.k] [X.sub.t-k] + [[epsilon.sub.[upsilon]]

where: [X.sub.t] and [epsilon.sub.t] are of dimension p x 1, [epsilon.sub.t] ~ N(O,[conjunction]), and [[pi].sub.i]'s and p x p and [mu] p x 1 regression coefficients. Following Johansen (1988, 1991), and Johansen and Juselius (1990), Equation 1 can be reparametrized as:

(2) [delta][X.sub.t] = [mu] + [[gamma].sub.1] [delta][X.sub.t-1] + ... + [[gamma].sub.k-1] [delta][X.sub.t-k+1] + [pi][X.sub.t-k] + [[epsilon.sub.[upsilon]]

where [[gamma].sub.i] [equivalent] - (I - [[pi].sub.1] - ... [[pi].sub.i],

i = 1 , ..., k - 1 ,

[pi] [equivalent] - (I - [[pi].sub.1] - ... - [[pi].sub.k], and

[delta] is the difference operator.

The short-term dynamics of the model are captured by matrices [[gamma].sub.1] through [[gamma].sub.k-1], while matrix [pi]provides information about the long-run relationships among the series. If the series are nonstationary but cointegrated, the rank of [pi] denoted by [tau], is less than the dimension of [pi] (i.e., 0 < [tau] < [rho]), and equals the number of cointegration vectors in the system. In this case, there exist [tau] stationary cointegrating relationships among the series and [rho] - [tau] common unit roots dictating the long-run stochastic trend of the variables. The error-correction model, Equation 2, can be alternatively written as:

(3) [delta][X.sub.t] = [mu] + [[gamma].sub.1] [delta][X.sub.t-1] + ... + [[gamma].sub.k-1] [delta][X.sub.t-k+1] + [alpha][beta]' [X.sub.t-k] + [[epsilon.sub.[upsilon]]

where [pi] is now expressed as [alpha][beta]' with [alpha] and [beta] being [rho] x [tau] matrices. The matrix [beta] contains the r cointegration vectors and [beta][X.sub.t-k] represents r error correction terms capturing the deviations from the r stationary relationships. Each row of matrix [alpha] represents the adjustment speed of an individual series toward the r cointegrating relationships. Linear hypotheses regarding the adjustment speeds of different series can be formed by testing:

(4) [alpha] = A [psi],

where: A is [rho] x m,

[psi] is m x [tau], and

m is [rho] minus the number of restrictions.

For example, to test a single restriction that the adjustment speeds of the first two stock variables are the same in a system of six stock prices, the null hypothesis is:

[H.sub.O]: [[alpha].sub.1,i] = [[alpha].sub.2,i], ([inverted A] i = 1, 2, ..., [tau]).

That is, the first two rows of [alpha] are the same, element by element. In the matrix notation of Equation 4, the hypothesis can be written as:

[Mathematical Expression Omitted]

where [alpha] involves:

[Mathematical Expression Omitted]

and [psi] involves:

[Mathematical Expression Omitted]

Similarly, linear hypotheses regarding the cointegration coefficients can be formed by testing the restriction: (5) [beta] = H [phi]

where H is [rho] x m and [phi] is m x [tau]. For example, to test a single restriction that the cointegration coefficients of the first stock variable are all zero in a system c f six stock prices, the null hypothesis is:

[H.sub.O]: [[beta].sub.1,i] = O, ([inverted A] i = 1, 2, ..., [tau]).

In the matrix notation of Equation 5, the hypothesis can be written as

[Mathematical Expression Omitted]

where [beta] involves:

[Mathematical Expression Omitted]

and [phi] involves:

[Mathematical Expression Omitted]

Maximum likelihood procedures for the estimation of the error-correction VAR model in Equation 3, determination of the number of cointegration vectors r, and testing of the restriction in Equations 4 and 5 are discussed in Johansen (1988; 1991), and Johansen and Juselius (1990). For a textbook treatment of Johansen's procedure see Davidson and MacKinnon (1993). For an excellent informal presentation on the procedure, see Kasa (1992).

DATA AND PRELIMINARY ANALYSIS

Weekly data on national stock price indices are obtained for the United States and the five Asian countries, as extracted from The Far Eastern Economic Review. All the indices are based on local currencies. The period of the analysis extends from January 7, 1985 through May 18,1992, with a total of 2310 observations. In case of missing data (arising from holidays and special events), the weekly price is assumed to be the average of the recorded previous price and the next price. The six countries and their representative indices used in the analysis are shown in Table 1.

Table 1. THE REPRESENTATIVE STOCK PRICE INDICES Country Index USA Dow Jones Industrial Japan Nikkei Stock Average Taiwan Weighted Index Hong Kong Heng Seng Index Singapore(a) Fraser's Industrial Index (1985:1:7 - 1987:12:28) All Shares Index (1987:12:29 - 1991:9:2) Straits Times Index (1991:9:3 - 1992:5:18) South Korea Seoul Composite Index Note: (a)The index used in Singapore has been changed twice during the observation period; however, these data have been transformed to Fraser's Industrial Index.

Figure 1 contains plots of the observed stock price indices. [The figure also presents the computed common trends to be discussed in a later section. Al. series are measured in natural logs. The figure suggests that the stock indices may be nonstationary. Table 2 contains Dickey-Fuller (1979; 1981) tests of the null hypothesis that a unit root exists in each series. Tests are conducted against two alternatives: one consistent with fluctuations around a constant mean ([[tau].sub.[gamma]] test), the other with stationary fluctuations around a deterministic linear trend ([[tau].sub.[tau]] test). Both tests entertain four-lag differences and twenty-four-lag differences of the variable in question to purge possible serial correlations in the error term. In all cases the null hypothesis of a unit root cannot be rejected at the 5 percent significance level, suggesting the existence of a unit root in each country's stock price.(1)

[TABULAR DATA 2 OMITTED]

Model Estimation and Selection

The estimation of the error-correction VAR model in Equation 3 is undertaken with twelve lags, twenty-four lags, and thirty-six lags (k = 12, 24, 36). For each lag length specification, the model is estimated using the Johansen maximum likelihood procedure and the number of cointegration vectors r determined through Johansen's [[lambda].sub.max] and trace tests. The final selection of the model among alternative lag specifications is based on normality, and serial correlation tests on the estimated residuals, as well as on the out-of-sample forecast performance evaluations. Initially, the estimation involves only data from January 7, 1985 through May 27, 1991, with the remaining 51 (one-year) observations reserved for forecast evaluation purposes.

The result of the cointegration test is in Table 3. Both the trace test and [[lambda].sub.max] test indicate that there are one, two, and four cointegration vectors for the twelve-lag, twenty-four-lag, and thirty-six-lag models, respectively. Apparently, the number of cointegrating relationships is sensitive to the choice of lag length. Results of the diagnostic tests on the estimated residuals are in Table 4. Based on Ljung-Box Q-statistics (1978), all the estimated equations are free from serial correlation problems, regardless of the lag length specification. Yet, the results of the Jarque-Bera univariate normality test (1980) suggest that the model might leave something to be desired; none of the estimated equations pass the univariate normality test. However, as pointed out by a reviewer, a multivariate normality test is more appropriate because Johansen's procedure is based on this assumption. The results of Bera-John multivariate normality tests (1983) are presented in the lower half of Table 4. From there, it is clear that the model passes the skewness test, kurtosis test, and the overall normality test, regardless of the lag specification.

[TABULAR DATA 3 OMITTED]

[TABULAR DATA 4 OMITTED] With the last year sample not included in the estimation, one-step-ahead through fifty-one-step-ahead forecasts are obtained for each of the three lag specifications and compared with the observed data. The average absolute percentage forecast errors over the 51 weeks are reported in Table 5. The performances of the twelve-lag and twenty-four-lag models are impressive, with the latter having a slightly smaller overall error: averaged over the six countries, the forecast error is 1.57 percent for the twelve-lag model and 1.42 percent for the twenty-four-lag model. On the other hand, the performance of the thirty-six-lag model is relatively unimpressive, with the error ranging from 3.12 percent for Korea to 11.44 percent for Singapore. The twenty-fourlag model is chosen as the final model for the subsequent analysis of common trends, cointegration coefficient tests, and adjustment speed tests.

[TABULAR DATA 5 OMITTED]

COMMON TREND

The twenty-four-lag error-correction VAR model is used to compute the common trend for each of the six stock price indices. The procedure is to decompose each element of vector [X.sub.t] into: (1) a stochastic trend component arising from the unit root properties of [X.sub.t]; and (2) a stationary component due to the cointegrating relationship of the series. For the twenty-four-lag model, two cointegration vectors (i.e., r = 2) were identified.(2) Hence, there are four unit roots (i.e., [rho] - r = 4) commonly shared by all six stock price variables. By filtering out the stationary part and focusing on the trend component, the importance of the common unit roots to the long-run movement of each stock price series is examined. Several methods have been proposed to obtain a trend/stationary decomposition. For example, one can derive from the estimated error-correction VAR equations the "Common Trends Representation" of Stock and Watson (1988). Alternatively, the problem can be cast in a state space framework, as advocated by Aoki (1988) and employed by Cerchi and Havenner (1988).

This study adopts a more straightforward decomposition procedure developed in Kasa (1992). Based on the projection theorem, Kasa decomposes [X.sub.t] into direct sum of its orthogonal projections onto two subspaces (of [R.[rho]]) which are orthogonal complements to each other:

(6) [X.sub.t] = {[Beta][perpendicular to]([Beta][perpendicular to]'[Beta][perpendicular to] [sup.-1][Beta][perpendicular to]'} [X.sub.t] + {Beta([Beta]'[Beta]) [sup.-1][Beta]' [X.sub.t], where: [Beta] is a 6 x 2 (i.e., p X r) matrix containing the two cointegration vectors, and [Beta][perpendicular to] is a 6 x 4 (i.e., p x (p - r)) orthogonal complement of [Beta]. The first term on the right-hand-side of Equation 6 defines the common stochastic trend of the series, while the second term captures the stationary component of the variables. Following Kasa, the first projection operator {[Beta][perpendicular to]([Beta][perpendicular to]' [Beta][perpendicular to]) [sup.-1][Beta][perpendicular to]'} is said to consist of a 6 x 4 factor loading matrix, [Beta][perpendicular to], and a 4 x 6 market weight matrix, ([Beta][perpendicular to]' [Beta][perpendicular to][sup.-1][Beta][perpendicular to]'. Each row of the 4 x 6 market weight matrix pertains to a specific common unit root and [Beta][perpendicular to] is normalized such that the weights in each row sum up to unity. Given this construction, the ith column of the 6 x 4 factor loading matrix [Beta][perpendicular to] can be interpreted as the relative importance of the six stock prices to the part of the stochastic trend contributed by the ith unit root. Alternatively, it can be viewed as the relative importance to the six stock prices of that part of the common trend contributed by the ith unit root.

Table 6 reports the factor loading matrix [Beta][perpendicular to! computed from the decomposition. There are four unit roots shared by the six stock variables. For the first unit root, the common trend is most important to the U.S. variable; the loading factor is 0.95. This component of the common trend is also important to other price variables. For example, the loading factor is -0.40 for the Korea variable and 0.36 for the Hong Kong variable. The loading factors pertaining to the second common unit root are, in general, small. As to the part of the common trend contributed by the third unit root, Singapore, Japan, and Taiwan have relatively large loading factors (0.58, 0.44, and -0.41, respectively). Finally, the last column of the factor loading matrix indicates that the fourth unit impact trend is most significant for the Korea variable and Taiwan variable (1.38 and 1.16, respectively).

[TABULAR DATA 6 OMITTED]

Since there are four common unit roots in the system, the overall stochastic trend for each variable is determined jointly by the four unit roots. The computed common trend for each of the six stock variables is presented in Figure 1. The figure also contains plots for the corresponding observed time series (in log). The difference between the computed common trend and the observed series represents the stationary component of the variable. Strikingly, the computed common trend for each variable mimics the shape of the observed data series very well. Further, deviations of the computed common trend from the observed data are very small in magnitude for the U.S. and Taiwan variables; the average deviations are only 2.19 percent for the former and 2.15 percent for the latter. This result suggests that the stochastic trends dictated by the four common unit roots are very important to the long-run movement of the U.S. and Taiwan stock prices, whereas the stationary component plays only a minor role. Though not as dramatic as the case for the U.S. and Taiwan, the four common unit roots also have important effects on the formation of other stock price variables; the average deviations of the computed common trend from the observed data are 20.47, 20.89, 20.46, and 12.74 percent for Japan, Hong Kong, Singapore, and Korea, respectively.

Before proceeding, it is useful to examine whether the above results on common trends are consistent with standard theories of international asset pricing. Using a general present value model of stock prices, Kasa shows that the cointegration properties of stock prices should mirror those of their dividend payments.(3) More specifically, the author argues that each country's loading factor in the stock price trend, as reflected in [Beta][perpendicular to], should be similar to its loading factor in the dividend trend. To avoid problems associated with dividend data in the empirical analysis, Kasa also adopts a broader interpretation of national equity payments by using GNP as a proxy of dividends.

GNP variables are collected for the U.S., Japan, South Korea, and Taiwan. Due to data availability, total manufacturing production is used as a proxy of GNP for Singapore, and the value of exports for Hong Kong. All the variables (in logs) are quarterly data ranging from the first quarter of 1970 to the last quarter of 1991. The above data are used to estimate an error-correction VAR model for the GNP variables. Notice that this estimated GNP model is actually not directly comparable with the stock price model, because the time period used in the estimation and the frequency of the data are different.(4)

The estimation is undertaken with two-lag, four-lag, and eight-lag specifications. Based on the trace test and [gamma.sub.max] test, the number of cointegration vectors is found to be two, three, and five for the two-lag, four-lag, and eight-lag models, respectively. The overall forecasting error is 0.46 percent for the two-lag model, 0.35 percent for the four-lag model, and 1.09 percent for the eight-lag model.(5) Though the four-lag quarterly specification slightly outperforms the two-lag specification, the latter is chosen as the final GNP model because it is consistent with the twenty-four-lag weekly model adopted previously for the stock prices. In the twenty-four-lag weekly model the number of cointegration vectors was also found to be two.

The corresponding factor loading matrix [Beta][perpendicular to] is reported in Table 7. There are four unit roots shared by the six GNP variables. For the first unit root, the common trend is most important to the U.S. variable; the loading factor is 0.40. This component of the common trend is also important to other GNP variables. For example, the loading factors are 0.39 and -0.33 for the Korea and Japan variables, respectively. This result is consistent with the first column of Table 6 in that the common trend associated with the first unit root is important to all the stock price variables. The second column of Table 7 indicates that the loading factors pertaining to the second common unit root are, in general, larger than those associated with the stock price variables appearing in the second column of Table 6. However, the relative importance of the second unit root is the same between the GNP and stock price models: Hong Kong, Taiwan, Japan, Korea, Singapore, and USA. The third column of Table 7 shows that the loading factors associated with the third common unit root are large for the Singapore, Taiwan, and Japan GNP variables; a result consistent with that of the stock price model (third column of Table 6). A similar conclusion can be drawn from comparing the fourth columns of Tables 6 and 7. Thus, though not directly comparable between the weekly and quarterly models, the above results, in general, suggest that the computed stock price common trends are consistent with the theory in that they mirror those of the GNP (or dividend payments).

[TABULAR DATA 7 OMITTED]

"COMMON" STOCK REGION

The above common trend results indicate that the stationary component plays only a minor role in the long-run movement of the U.S. and Taiwan stock prices, while it exerts modest impacts on the remaining four prices. This leads one to wonder if the U.S. and Taiwan stock prices belong to an equilibrium regime different from the one dictating the other four variables. In this part of the study, tests are conducted on the individual country's cointegration coefficients to ascertain whether they are statistically different from zero. If a country's cointegration coefficients are zero, the long-run movement of its stock price is not governed by the equilibrium condition underlying the cointegration vectors. Hence, this country can be thought of as not belonging to the "common" stock region comprising the remaining countries.

The estimated cointegration vectors are reported in the upper half of Table 8 as Linear hypothesis regarding zero [Beta] coefficients for each individual country is tested using restriction defined in Equation 5. The resulting maximum likelihood ratio test statistics are reported in the lower half of Table 8. It is found that all the cointegration coefficients are significantly different from zero, except those pertaining to the U.S. and Taiwan variables. To exclude the U.S. and Taiwan markets from the "common" stock region of the remaining four countries, however, is too drastic an approach, because the joint hypothesis of zero A coefficients for the two markets is rejected.

[TABULAR DATA 8 OMITTED]

Adjustment Speed

The result on the common trend shows that, in the long-run, movements in the six stock prices have been dominated by their common unit roots. Since a great number of economic activities associated with international portfolio diversification are short-term in nature, it is also important to investigate the short-run dynamics of the stock prices. Before doing so, the speed of adjustment from short-term disequilibria toward the long-run common trend is examined. The adjustment speed toward the stationary component and, hence, its orthogonal complement, the common trend, is depicted by the [alpha] matrix reported in the upper half of Table 9. The focus here is to test whether the adjustment speeds of the four little dragons in Asia (Taiwan, Hong Kong, Singapore, and South Korea) are the same as those of the two economic superpowers (the U.S. and Japan), and whether the two superpowers have the same adjustment speed. Linear hypotheses regarding the adjustment speeds of different countries are tested using restriction defined in Equation 4.

[TABULAR DATA OMITTED]

Maximum likelihood ratio test statistics are reported in the lower half of Table 9. At the five percent critical level, the null hypothesis of equal adjustment speed is rejected for the U.S.-Hong Kong pair, U.S.-Korea pair, and Japan-Korea pair. However, at the one percent critical level, it is rejected only for the Japan-Korea pair. Since most countries have the same adjustment speed, the result suggests that the long-run comovement arising from the common unit roots also sheds light toward the short-run relationship among the six stock prices.

Short-Run Dynamics

To investigate the short-run interaction among the six stock prices, impulse analyses are conducted through shocking a specific price variable. Figures 2 and 3 present the impulse response functions over a one-year period of the individual prices to a standard one-period shock in the U.S. variable and in the Japan variable, respectively.(6)

Figure 2 shows that the short-run price adjustments to a shock in the U.S. market are similar in shape and magnitude for Taiwan, Hong Kong, and Singapore prices. This may be due to the commonalities in the ethnic background of the three countries. Figure 3 shows that the adjustment paths to a shock in the Japan market are similar in shape and magnitude for Taiwan and South Korea. This is supposedly explainable because both countries were under Japanese military domination for more than a half century before World War II, and have been under its immense economic influence since then. In general, both figures show that the responses of all countries to both shocks are very small (less than one percent). Also, the result appears to suggest that the U.S. and Japan markets are efficient in that most short-run responses in those markets have occurred within the first few weeks. However, it takes approximately twenty weeks for the Taiwan and Korea markets to reach new equilibrium states after the shock.

CONCLUSIONS

This study examines the common stochastic trends among national stock prices of the U.S. and five East Asian countries, including Japan, Taiwan, Hong Kong, Singapore, and South Korea. The stock price series are found to be nonstationary and yet cointegrated. Two cointegrating relationships are identified and the six stock price variables are found to share four common unit roots. The result shows that the stochastic trends dictated by the four common unit roots are important to the long-run movement of the stock prices, especially those of the U.S. and Taiwan. Though not conclusive, the result suggests that U.S. and Taiwan markets may not belong to a "common" stock region containing the remaining four countries. The result also shows that most variables have the same adjustment speed in moving from short-run disequilibria toward the common trend. Finally, impulse response analyses suggest that short-run adjustments to temporary shocks occur rather quickly for some countries, and that for others the pattern and magnitude of the adjustments may be influenced by commonalities in ethnic and other backgrounds of the countries involved.

In analogy to cointegration in means, a direction for future research is to consider the common trend in conditional variances of the stock prices. As pointed out in Bollerslev and Engle (1993), numerous papers have established the suitability of the so-called integrated generalized autoregressive conditional heteroskedastic (IGARCH) models for the stock prices. In the IGARCH models shocks to the conditional variance are persistent in the sense that they remain important for forecasts of all horizons. The multivariate IGARCH method developed in Bollerslev and Engle can be used as a framework to identify common persistence in conditional variances of the stock prices. Further, it may be possible to investigate simultaneously cointegration means and co-persistence in conditional variances within a given model. The point for this effect. work by Glosten, Jagannathan, and Runkle (1993) provides an interesting starting

FIGURE I The Observed Stock Prices and Computer Common Trends.

[ILLUSTRATION OMITTED]

FIGURE II Impulse Responses: Shocking the USA Market.

[ILLUSTRATION OMITTED]

FIGURE III Impulse Responses: Shocking the Japane Market.

[ILLUSTRATION OMITTED]

NOTES

(*) Direct all correspondence to: PinJ. Chung, Iowa State University, Department of Economics, Heady Hall, Ames, IA 50011-1070. Senior authorship equally shared. (**) We are indebted to Walter Enders, and Harvey Lapan for useful comments, and to the referees for constructive suggestions. The usual caveat applies. (1.) Other lag specifications (0, 8, 12, 16, and 20 lags) are also entertained. Conclusions on the unit root test are, in general, the same, regardless of the number of lags chosen. (2.) As pointed out by a reviewer, the number of cointegration vectors tends to be very sensitive to the observational interval. To investigate this issue, the weekly stock price data are transformed into monthly data by taking the last week of a month as the monthly observation. To be consistent with the twenty-four-lag weekly specification, a six-lag monthly model is estimated. The number of cointegration vectors for the monthly model is found to be the same as for the weekly model; namely two. Though data frequency does not appear to be a problem for the model chosen, this robust result is by no means general. For example, it is found that the number of cointegration vectors for the nine-lag monthly model is five, and yet the corresponding number for the thirty-six-lag weekly model was four. (3.) This result was derived under the assumptions that discount rates follow stationary stochastic processes, and that a transversality condition holds which rules out bubbles. (4.) The variables are quarterly because weekly data are not available. Also, a longer estimation period is used for the GNP model because of the degree of freedom considerations. The GNP data (measured in local currencies) for the U.S., Japan, and South Korea are from various issues of International Financial Statistics, published by the International Monetary Fund (IMF). Taiwan is not a member of the IMF and its GNP data are from the Directorate- General of Budget, Accounting & Statistics, Executive Yuan, Taiwan. Singapore reports quarterly GNP data to the IMF only after 1984, while Hong Kong does not report GNP to the IMF at all. The manufacturing production data for Singapore and the value of exports for Hong Kong are from the IMF. (5.) Detailed results on the trace test, [[gamma].sub.max] test, and forecasting evaluations are not reported here, but can be obtained from the authors. (6.) Results from shocking the remaining four variables can be obtained from the authors.

REFERENCES

[1] Aoki, M. 1988. "On Alternative State Space Representations of Time Series Models." Journal of Economic Dynamics and Control 12. 595-607. [2] Bera, A. and S. John. 1983. "Tests for Multivariate Normality with Pearson Alternatives." Communication in Statistics 12: 103-117. [3] Bollerslev, T and R.F. Engle. 1993, "Common Persistence in Conditional Variances." Econometrica 61: 167-186. [4] Cerchi, M. and A. Havenner. 1988. "Cointegration and Stock Prices: The Random Walk on Wall Street Revisited." Journal of Economic Dynamics and Control 12: 333-346, [5] Davidson, R. and J.G. MacKinnon. 1993. Estimation and Inference in Econometrics. New York: Oxford University Press. [6] Dickey, D.A. and W.A. Fuller. 1979. "Distribution of the Estimators for Autoregressive Time Series with a Unit Root." Journal of the American Statistical Association 74: 427-431. [7] ---. 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root." Econometrica 49. 1057-1072. [8] Engle, R.F. and C.W.J. Granger. 1987. "Co-integration and Error Correction: Representation, Estimation, and Testing." Econometrica 55: 251-276. [9] Eun, C. and S. Shim. 1989. "International Transmission of Stock Market Movements." Journal of Financial and Quantitative Analysis 24: 241-256. [10] Glosten, L.R., R. Jagannathan, and D. Runkle. 1993. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks." Journal of Finance 48: 1779-1801. [11] Hilliard, J. 1979. "The Relationship Between Equity Indices on World Exchanges." Journal of Finance 34: 103-114. [12] Jarque, C.M. and A.K. Bera. 1980. "Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals." Economics Letters 6: 255-259. [13] Johansen, S. 1988. "Statistical Analysis of Cointegration Vectors." Journal of Economic Dynamics and Control 12: 231-254. [14] ---. 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models." Econometrica 59. 1551-1580. [15] Johansen, S. and K. Juselius. 1990. "Maximum Likelihood Estimation and Inference on Cointegration-With Applications to the Demand for Money." Oxford Bulletin of Economics and Statistics 52: 169-2 1 0. [16] Kasa, K. 1992. "Common Stochastic Trends in International Stock Markets." Journal of Monetary Economics 29. 95-124. [17] Ljung, G.M. and G.E.P Box. 1978. "On a Measure of Lack of Fit in Time Series Models." Biometrika 66: 297-303. [181 Osterwald-Lenum, M. 1992. "A Note with Quantiles of the Asymptotic Distribution of the Maximum Likelihood Cointegration Rank Test Statistics." Oxford Bulletin of Economics and Statistics 54: 461-472. [19] Stock, J.H. and M.W. Watson. 1988. "Testing for Common Trends." Journal of the American Statistical Association 83- 1097-1107.

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Author: | Chung, Pin J.; Liu, Donald J. |
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Publication: | Quarterly Review of Economics and Finance |

Date: | Sep 22, 1994 |

Words: | 5205 |

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