Co-integration between Two Stock Markets: Evidence from China

Testing for relationships between Shanghai and Shenzhen Stock Exchange markets: from co-integration perspective

1. INTRODUCTION

The connection between stock markets is of vital importance to international equity investments, both in terms of managing risk and maximizing returns. We analyze the connections that exist amongst the two Chinese mainland stock markets (Shanghai and Shenzhen). Specifically, we conducted a cointegration analysis through two methods (Engle-Granger and Johansen test) of two Chinese stock markets to test for the existence.

Cointegration can be viewed as the statistical expression of the nature of equilibrium relationships, with cointegrated variables sharing common stochastic trends.

The results of the study will aid us to gain insight into how Chinese stock markets’ cointegration contributes to portfolio diversification strategy, primarily in the context of the institutional investor.

2. LITERATURE REVIEW

According to Engle and Granger (1987), cointegration has emerged as a powerful technique for investigating common trends in multivariate time series, providing a sound methodology for modeling both long run and short run dynamics in a system. The interest in cointegration literature has increased significantly as a result of this work and has given rise to other important contributions to the subject.

Phylaktis, K., & Ravazzolo, F.(2005) also examined at the association between stock prices and exchange rate for Pacific Basin countries dur­ing 1980-1998 using Johansen’s cointegration test and multivariate Granger causality tests. The result showed that exchange and stock prices were related posi­tively and found that the financial crisis had an interim effect on the long-run relationship.

Lin and Tang (2013) investigate the relationships between Shanghai and Shenzhen stock market and reveal the evidence of cross-correlations between the two stock markets. The findings show that Shanghai and Shenzhen stock market are cointegrated.

3.1 BACKGROUND OF STUDY

Chinese stock markets attract foreign investors because of rapid development and potential.

Chinese stock markets are relatively new and has been established for less than 35 years but have been grown tremendously fast:

Shanghai market(SSE) -1990

Shenzhen market(SZSE) -1991

Shanghai and Shenzhen stock markets were established about 10 years after Chinese market-oriented reforms in 1978.

Two types of shares (A shares and B shares) traded in the Chinese stock markets. A shares are only for the domestic investors while B shares are restricted to international investors, including overseas Chinese residing in Hong Kong, Macau and Taiwan which provides an ideal testing ground of hypothesis of linkage between these markets.

3.2 PURPOSE OF THE RESEARCH

1. To capture the volatility dynamics in these two markets and study the relationship among them.
2. Identify how well mainland Chinese stock markets function relative to each other and the co-integration between them

•         Engle-Granger test (1987)

•         Johansen test (1991)

3.3 Hypothesis Testing

$H_0$

: There is no cointegration between the Shanghai stock market and Shenzhen market

4.0 RESEARCH METHODOLOGY

4.1 Cointegration

The regression theory, and the AR or VAR models are appropriate for modeling I(0) data. We know that if we have two non-stationary series, it is not good to regress them on one another. This is the problem of Spurious regression. However, sometimes, we need to look at non-stationary processes, how they are related to one another. This takes us to the concept of cointegration.

We introduce the statistical concept of cointegration that is required to make sense of regression models and VAR models with I(1) data.

For illustration, let consider two non-stationary process $X_t$

and $Y_t$

, both I(1), (as shown in fig. 1) that are not quite increasing at the same rate but they are both increasing. The distance between these two series is not relatively constant through time. Suppose we find a particular parameter  $\beta$

  and multiply $X_t$

by it, such that it causes $X_t$

  to rotate and makes their distance to be relatively constant through time (fig. 2).

                      Figure 1

             Figure 2

If there is truly some relationship between $X_t$

and $Y_t$

which is constant through time. Then the difference  $Y_t–\beta X_t$

 will turn out to be a stationary process. We then say that the two processes $X_t$

and $Y_t$

are cointegrated.

Definition

Two $I\left(1\right)$

nonstationary series are said to be cointegrated if there exists a stationary linear combination of the nonstationary random variables.

Let $Y_t={(y_{1t},\dots y_{\mathit{nt}})}^{‘}$

be an nx1 of or $I\left(1\right)$

time series. $y_{1t},\dots y_{\mathit{nt}}$

are said to be cointegrated if there exists an nx1 vector $\beta ={({\beta }_1,\dots ,{\beta }_n)}^{/}$

such that  

${\mathrm{ }\beta }^{/}{Y}_t={\beta }_1{y}_{1t}+\dots {+{\beta }_{n}y}_{\mathit{nt}}~\mathit{ I}(0$

)

The linear combination is referred to as long-run equilibrium relationship. The idea is that two $I\left(1\right)$

time series with a long-run equilibrium relationship cannot deviate far from the equilibrium since economic forces will act to restore the equilibrium relationship. 

There are two main cointegration methods that have consistently been used throughout past studies which are:

1. Engle-Grangers Two Step Estimation Method (only applied for two variable)

2. Johansen’s Maximum Likelihood Method using either the Trace Statistic and/or the Maximum Eigenvalue Statistic (can be applied to more than two variables).

We will be dealing majorly with Eagle-Grangers method in this work.

4.2 Engle-Granger Cointegration Test

This is a residual based test.

Engle and Granger consider the regression model for $y_{1t}$

$y_{1t}={\mu }_t+{{\beta }_{2}y}_{2t}+\dots {+{\beta }_{n}y}_{\mathit{nt}}+{\epsilon }_t$

Where ${\mu }_t$

is the deterministic term.

$H_0$

: The time series $y_{1t},\dots y_{\mathit{nt}}$

are not cointegrated

• We check whether the residual ${\epsilon }_t$

  is $I\left(0\right)$

  or $I\left(1\right)$

-         If ${\epsilon }_t$

is $I\left(0\right)$

, then $y_{1t},\dots y_{\mathit{nt}}$

are cointegrated

-         If ${\epsilon }_t$

is $I\left(1\right)$

, then $y_{1t},\dots y_{\mathit{nt}}$

are not cointegrated

Steps:

• Run ordinary least square regression to estimate ${\beta }_2,\dots ,{\beta }_n$
• Apply a unit root test to the estimated OLS residual using ADF or PP test without the deterministic terms.

4.3 Johansen’s Cointegration Test

The Johansen process is a maximum likelihood method that determines the number of cointegrating vectors in a non-stationary time series Vector Autoregression (VAR) with restrictions imposed, known as a vector error correction model (VEC). Johansen’s estimation model is as follows:

$∆X_t=\mathit{ \mu }+\sum _{i=1}^p{⎾}_i∆X_{t–i}+\alpha {\beta }^{‘}{X}_{t–i}+{\epsilon }_t$

where,

$X_t$

= (nx1) vector of all the non-stationary series

${⎾}_i$

= (nxn) matrix of coefficients

$\alpha$

= (nxr) matrix of error correction coefficients where r is the number of cointegrating relationships in the variables, so that 0 < r < n. This is a measure of the speed at which the variables adjust to their equilibrium.

$\beta$

= (n x r) matrix of r cointegrating vectors, so that 0 < r < n. This gives the long-run cointegrating relationship between the variables.

The Johansen test has two types of statistics:

1. Maximum eigenvalue statistics
2. Trace statistics

The setup is based on a vector error correction model (VECM).

$∆y_t=\mathit{ c}+\mathrm{\Pi }{y}_{t–i}+\sum _{i=1}^p{A}_i∆y_{t–i}+{\epsilon }_t$

Where Δ is the first difference operator, and $y_t$

is a vector of the time series at time t. C is a vector of constants.

Let r be the rank of  Π, which is the number of co-integrating vectors.

For both test statistics, it is a stepwise testing procedure, begins with the null hypothesis of no co-integration against the alternative of co-integration.

The Trace test is a joint test that tests the null hypothesis of no cointegration (H0: r = 0) against the alternative hypothesis of cointegration (H1: r > 0).

${\lambda }_{\mathit{trace}}\left(r\right)=–T\sum _{i=r+1}^g\ln (1–\widehat{{\lambda }_i}$

)

The Maximum Eigenvalue test examine each eigenvalue individually. It tests the null hypothesis that the number of cointegrating vectors equal r as against the alternative hypothesis of r+1 cointegrating vectors.

${\lambda }_{\mathit{max}}\left(r,r+1\right)=–T\ln (1–\ln (1–{\widehat{\lambda }}_{r+1}$

)

 r = number of cointegrated vectors under the null

${\widehat{\lambda }}_i$

= estimated ith ordered eigenvalue from the $\alpha {\beta }^{‘}$

matrices

5.0 DATA ANALYSIS

Data source:

we employ the 5 minutes frequency of Shanghai composite and Shenzhen component Index from April 1^st 2008 to Oct 13^th, 2019 from Yahoo Finance, covering T=2376 transaction days with 243 days yearly excluding the holidays and weekends).

Stock return was calculated as

$r_t=\log \left(P_{t+1}\right)–\log \left(P_t\right)$

Where $r_t$

and $P_t$

denote the time series return and price at the transaction time t respectively.

First glance of the two markets

Shanghai Vs Shenzhen

The two markets Shanghai and Shenzhen display similar trend. There seems to be a relationship between the two markets. We will investigate the relationship.

Table 1: Descriptive statistics of return series.

First glance of the two markets (in one graph)

Daily Rate of return for each market

The time series plot of the daily rate of returns of the two shows stationarity. In order to carry cointegration test between the two market variables, we need to ensure they are both non-stationary, hence we take the log of the rate of returns of the two series.

Table2: Descriptive Statistics of daily rate of return for each market

Daily Rate of return for both markets

Both markets rate of return shows similar trend and looks stationary

Next, we take the log transformation of the rate of returns of the two market to ensure they are both non-stationary. So, we test for non-stationarity of the transformed data using unit root test.

The ADF test shows that the log transformation of the two series are non-stationary. Hence, we can go further to carry out our cointegration test using Enger-Granger cointegration test.

Engle-Granger Test

1. Check the linear relationship between these two time series
2. Check the residual through ADF test 
3. If the residuals are stationary, then we can conclude that the two series are cointegrated.

R-squared is 0.7939, which indicates a linear relationship between the returns of the two markets. Approximately 80% of the relationship can be explained by the model due to the two stock markets.

Now, we check for the stationarity of the residual from the above regression.

We can see that the test statistics is -2.0954, which is smaller than 5% significance level -1.95, so we reject the null hypothesis of non-stationarity of the residual. Hence, the residuals are stationary. So, we can conclude that the two markets are cointegrated. So this means that there exist long-term equilibrium relationship between the two stock markets.

              Residual plot under Engle –Granger Test

6.0 CONCLUSION

We investigate the relationships between Shanghai and Shenzhen stock market during the 2008 to 2019 period and the result shows that, the two stock markets are cointegrated. That is, they have long time equilibrium relationship and they will tend to move together in the future.

References

• Alexander, C., 2001. Market Models: A guide to financial data analysis. John Wiley & Sons Ltd Chichester (UK)..
• Barunik, J., 2011. Lecture: Introduction to Cointegration.
• Engle, R. F. and Granger, C.W.J, 1987. Co-integration and Error Correction:Representation, Estimation and Testing. Econometrica, Vol. 55(Issue 2), pp. 251-276.
• Maggiora and Skerman, 2009. Johansen cointegration analysis of american and european stock market indices:An empirical study.
• Phylaktis, K., & Ravazzolo, F., 2005. Stock prices and exchange rate dynamics. Journal of International Money, 24(7).
• Tsay, R.S., , 2010. Analysis of Financial Time Series, Third Edition. John Wiley & Sons, New York..