Research Method

Chapter 3 Research Method and Data Information

This chapter is divided into research method, variable selection and T-REITs index presentation. The first part shows the models applied in this study for empirical tests. The second part shows the approaches of variable selection and clarifies the data source. The last part is establishing T-REITs index for the use of empirical study since there is no official REITs index in Taiwan.

3.1 Research Method

The purpose of this study is to explore the long-run and short-run relationship between T-REITs, macroeconomy and commercial real estate market. In the long-run equilibrium section, we employ Johansen cointegration method to analyze T-REITs and other variables, respectively. Variables with cointegration relationship are analyzed through VECM approach for further study; and variables without cointegration relation are interpreted based on the VAR approach, discussing the short-term influence of each variable to T-REITs. Finally, we conduct Granger causality test to explore the lead/lag relation between variables.

3.1.1 Cointegratoin Test

Cointegration test was proposed by Engle and Granger in 1987, which is specifically for the analysis of the relationship between a set of economic variables.

When a set of non-stationary variables become stationary through a linear combination, the non-stationary time series are said to be cointegrated. Cointegration is commonly used to explain the long-run equilibrium relationship between economic variables. When there is a cointegration relationship between variables, the characteristics of these variables tend to adjusted to the balanced direction, and the effects from the external factors are only a short-term deviation from equilibrium.

Therefore, the error correction function must be applied to decrease the deviation gradually, so that the variables will eventually return to the long-run equilibrium value.

The short-run relation between variables is described in the VECM approach.

Engle and Granger (1987) propose a two-step method approach to test the cointegration relation. The hypothesis is that when two non-stationary variables Y and X become stationary in a first-order differential, the Engle-Granger

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cointegration test can be applied. The first step of regression can be written as follow.

Y α βX ε （3.1）

In this equation, Y represents the dependent variable; X represents the independent variable and ε is the error term. The second-step of Engle-Granger cointegration is to decide the integrated order of residuals using unit-root test. The regression residuals can be expressed as:

∆ε ρε ε （3.2）

The null hypothesis of the equation is that there is no cointegration relationship between the two variables. If the result can’t reject the null hypothesis of ρ 0, then the residual series has a unit-root, implying there is no cointegration between series Y and X .

However, the Engle-Granger two-step method can’t point out one or more of the cointegration relationship and the relative dynamics of the adjustment process over two variables, thus Johansen (1991) proposes a new method based on VAR approach.

This method uses maximum likelihood estimation to clearly point out the existence of one or more cointegration relationship provides a more robust interpretation of the multiple long-run equilibrium relationship between variables. The variables generate a long-run impact matrix after differential, and use the two likelihood ratio statistics to confirm the rank of matrix, which determines the number of cointegration vector.

Assuming a VAR model of order p and n variables can be expressed as:

Y A Y A Y A Y BX ε （3.3）

where Y is a k-vector of non-stationary I(1) variables; X is a d-vector of deterministic variables; ε is a vector of innovations generated by the equation. Let Π ∑ A I ，Γ ∑ A , the equation (3.3) can be rewritten as:

∆Y ∏Y ∏ Y ∏ Y ∏ ∆Y （3.4）

And equation (3.4) can further be expressed as:

∆Y ∏Y Γ Y BX （3.5）

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where ∏ is the long run impact matrix, and the rank of ∏ decides the number of cointegration existed in Y . The Johansen test is to examine whether the rank generated by ∏ can be rejected through the estimation of non-restricted VAR model to the Π matrix. In order to carry out tests of the rank to determine the number of cointegration, Johansen proposed Trace test and Maximum Eigenvalue test.

λ r T ln 1 λ （3.6）

where T = number of sample size, λ = max-Eigen statistic of matrix ∏. The null hypothesis is H ：rank≦r.

λ r Tln 1 λ （3.7）

The null hypothesis is H ：rank=r, representing number of r cointegration vectors.

The alternative hypothesis is H ：rank=r+1, representing number of r+1 cointegration vectors existing between variables.

3.1.2 Vector Error Correction Model

According to Granger’s representation theorem, when cointegration exists between two variables, an error correction term must be added to correct the short-term imbalance between variables, making the time series back to long-run equilibrium. The VECM approach restricts the long-run behavior of the endogenous variables to converge to their cointegrated relationships while allowing for short-run adjustment dynamics. In addition, since the traditional time-series model complies with the requirements of stationary, the non-stationary series have to be differentiated before analyzing. But the difference will result in the loss of long-term information.

The adoption of error correction model can also save the problem. Error correction model is derived from the VAR approach, it specifies the correction term should be added into VAR when cointegration exists, making the variables move toward the direction of long-run equilibrium. Therefore, the movement of series is not only affected by the changes of current variables and itself, but also affected by the previous imbalance. Under the assumption of one cointegration and no lags, the cointegration model between series Y and X is Y βX , and the two corresponding to the VECM can be expressed as:

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Only the right variables are the error correction terms generated by integrating, and α is the speed of adjustment.

From (3.8a) and (3.8b), and in the case of long-run equilibrium, the value of the error correction term should be 0, and the error correction term will not be 0 when Y and X deviate from the equilibrium. Generally, the error correction model corrects the short-term imbalances through the error correction term, implying that when imbalance occurred in the previous period, it will be partially corrected at current period. Therefore, the error correction term may be regarded as the speed of adjustment between variables and the long-run equilibrium value. In addition, the error correction model contains a variable differential, error correction term, and the short-term changes between variables under long-run equilibrium relationship, which avoids the spurious regression error and the long-run messages that can be ignored by differential.

3.1.3 Vector Autoregression Model

The groups of variables without cointegration relationship apply the VAR approach. The VAR is commonly used for forecasting systems of interrelated time series and for analyzing the dynamic impact of random disturbances on the system of variables. The VAR model uses its own information and characteristics to do the analysis. In each equation, the dependent variable begins with the lags of their own period and together with the lags of other variables. In each regression, the interpreted variable uses its own lags as the explanatory variable, viewed as endogenous variable, to reflect the dynamic relationships between variables. Because the VAR model can indicate the lags’ short-term impact on the dependent variable by studying the correlation between the lags of the dependent variable and the lags of other variables.

In this study, the VAR model is to explore the short-term impact changes under the long-run fluctuations.

On the assumption that Y is influenced by the lag of itself and other variables and all the series are stationary, it includes the following two regressions:

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3.1.4 Granger Causality Test

Besides using cointegration and VAR to explore the relationship between variables, this study also uses Granger causality test to explore the lead/lag relations between T-REITs and other variables. The causality between two variables A and B is defined as whether placing the lag of A into the prediction equation of B would provide better forecasting results than only place the lag of A into the equation. This means that when there are two series Y and X , the inclusion of the lag Y items would enhance the prediction accuracy of X , and will also enhance the overall explanatory power of X . At this point, we say Y leads X , and X leads Y vice versa. The significance of the test results is that one variable contributes to the forecast of another variable, and provides leading information. If there exists an interaction between the two variables, then the result indicates the feedback relationship between variables. Suppose two variables are stationary, but does not have a cointegration relationship, the Granger causality equation is defined as:

∆Y α α ∆X β ∆Y ε （3.10）

where Y is dependent variable; X is independent variable, and P is lag terms. The null hypothesis is H ：α α α 0. If the results reject the null hypotheses, meaning it refuses that X does not lead Y, then the results indicating that adding X in the equation is useful in predicting Y.

If there is cointegration between the two variables, there would be bias by using equation (3.10) directly. In order to avoid the bias, the variables deviate from the

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long-run equilibrium level needs to be taking into consideration. Therefore, we should use VECM to do the estimation by adding error correction term λ into the above VAR model, becoming equation (3.11).

∆Y α α ∆X β ∆Y λ ε （3.11）

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