Title of Dissertation Time-varying systematic risk in Thailand's stock market: Evidence from multivariate GARCH and Kalman filter estimates. The purpose of this study was to use multivariate GARCH and the Kalman filter to estimate time-varying systematic risk or beta.
Motivation and Research Question
Therefore, the systematic risk of the market can be estimated more precisely by using more proper models. Therefore, the study will provide an explanation of the volatile beta of each stock sector.
Literature Review
Regarding the GARCH model, this article compares the time-varying systematic risk results of three types of GARCH models: GARCH, BEKK GARCH and GARCH-GJR. The study evaluates the performance of the model by calculating the mean squared error (MSE) and the mean absolute error (MAE).
Scope of the Study
The first is the Kalman filter Random Walk model (RW model), the time update in this model depends on the function of the previous value of itself. The last one is the Kalman filter autoregressive (1) model (AR(1) model) in which the time update function is formed by the AR(1) model.
Capital Asset Pricing Model (CAPM)
This means that the variance of the OLS estimator is minimal and the value from the estimate is not different from the true value between the dependent variable and the independent variable (two values of the estimate).
Systematic Risk Estimation Using OLS
The daily return of each stock sector as well as the daily return of the market are regressed using the least squares method. It was confirmed on CAPM that the beta of each stock sector was the share of the stock sector's sensitivity to non-diversifying risk or systematic risk.
Systematic Risk Estimation Using the Rolling OLS
The following is a beta comparison between OLS and current OLS methods of the Property and Construction Proprietary Sector (PROPCON). The systematic risk of this stock sector was quite similar to the market risk, hovering around 0.9450.
Theoretical Background of Multivariate GARCH
According to the multivariate GARCH (1,1) process, the conditional variance of each variable comes from the lagged squared error (ε21t-1 and ε22t-1), and the product of lagged errors (ε1t-1ε2t-1. The biggest disadvantage. of them is the number of parameters needed for the estimation can be quite large. When the number of variables becomes larger, complication in the estimation can become a problem of the model.
In order to solve the problem of a large number of parameters, many researchers have tried to minimize the size of the model by finding a suitable constraint on the general form, and one of the most popular constraints is to diagonalize the system . In this way, the conditional variance (hijt) contains only lags of itself and the cross products of εitεjt. With this specification, the model is easy to estimate even if there are large numbers of variables.
However, the model assumes no interactions between the variances, which is the problem with this specification.
Methodology
Multivariate GARCH VECH model
Multivariate GARCH BEKK model
However, there was a difference from the normal form of the VECH model multivariate GARCH with respect to the term Si,t-1 designed to capture potential asymmetry in the conditional variance. From the raw data which was the daily closing price on the SET index and SET industry index (equity sector index), the daily returns of both market portfolio (SET index) and industry indices were calculated into continuous compound returns before adding them into the estimation is set. . Therefore, the variable Ri was the return of each industry index and Rm the return of the SET index.
The research step for the multivariate GARCH can be divided into two main steps. First, the time-varying beta was estimated using the multivariate GARCH in two models, namely the VECH model and the BEKK model. Their shapes were mentioned earlier and the study estimated systematic risk using these models.
Next, the study plotted the beta estimated by these two methods to see the pattern of systematic risk in each stock sector and trend and to see how each model captured the change in the market.
Results and Contribution
Next, the study compared the beta of each stock sector by plotting the beta using the two models of the multivariate GARCH. Moreover, the time-varying betas using these two models had similar patterns for this stock sector. The results of the plotting also confirmed that this stock sector had the lowest beta in the market.
The beta calculated using the OLS of the resource equity sector was greater than the market beta. The beta was around 1.1862, meaning that this stock sector had higher systematic risk than the market. Using the OLS model, the beta of the service stock sector was 0.66, which was lower than the market beta.
This implies that the technology capital sector had lower systematic risk than market risk.
Kalman Filter: Theoretical Background
Methodology
Random Walk Model
We assume that Y depends on a single unobservable datum called the state of nature. This natural state also changes over time, and the previous value of the natural state also affects the next natural state. Phadke (1981) discussed a good example of the Kalman model in the case of statistical quality control.
In this study, the number of defects observed in a sample obtained at time t represented by Yt, while Ɵ1,t. This is an example where both the time equation and the measurement equation can be the matrix. Yt stands for the share sector's return at time t, Ft is the market return at time t,.
And Ɵit is the time equation or the time-varying beta is the function that depends on the delay of itself as shown in equation (4.4).
Random Coefficient Model
Autoregressive Model (AR(1))
In this paper, the time-varying beta of the stock sector index will be estimated using these three Kalman filter models. In each model, the beta will be estimated eight times, as there are eight equity sector indices in the Thai stock market. There are 1,842 closing price observations in each stock sector and also 1,842 closing price observations of the market index or SET index.
The state space model in EViews will be used to estimate the time-varying beta version.
Results and Contributions
Results from the Kalman Filter Random Walk (RW) Model
According to equations (4.3) and (4.4), it can be seen that the time update or betas in this model varied over time in a random walk pattern. The results in the table also show that the return on the market index has affected the equity sector index. But in this model, the beta of the consumer product was significant, which is different from the following two models.
Results from the Kalman Filter Random Coefficient (RC) Model In this model, the time update or beta depends on the random coefficient
Since we allow the coefficient C1,t to vary over time, it can be deduced that beta is also time-varying. According to the table, the results for most of the industries showed significant betas, which may explain that the market return also affected the return of the stock sector index except for the consumer products stock sector.
Results from the Kalman Filter Autoregressive (AR(1)) Model The time update and measurement update were estimated using the set of
From the C2 coefficient, the results showed that the time update or in this study are the time-varying systematic risks (beta) in all industries. Regarding the significant effect of market returns on the time update (Ɵt), it is found that the time update betas are mostly significant, except for the consumption product. After explaining the results of the table, the time-varying beta plot was developed from these three models and they were divided by equity sector as shown.
AGRO
CONSUMP
INDUS
PROPCON
RESOURCE
SERVICE
TECH
Measure of Forecasting Accuracy
- In-sample Forecasting
In most cases, the multivariate GARCH VECH model produced smaller or equal values of both RMSE and MAE than the BEKK model, except in the financial equity sector (FIN). Most of all, the RMSE and MAE of the Kalman filter RC model and the RW model were quite similar and larger than the values of the Kalman filter AR(1) model. Therefore, the Kalman filter AR(1) model was inferred to be the best forecasting performance model among these three models of the Kalman filter in terms of in-sample forecasting.
Next, this study compared the performance of the AR(1) Kalman filter model, which is the best forecasting model among the three Kalman filter models, with running OLS. From the overall forecasting results in the sample, we can conclude that the multivariate GARCH VECH model is better than the running OLS for most industries. First, we compared the forecast accuracy between the VECH model and the BEKK model of the multivariate GARCH method.
The results also suggested, as with the in-sample forecast, that the VECH model was better than the BEKK model for the multivariate GARCH.
Comparison of Time-varying Beta Plot
The pattern of the beta plot of this stock sector was similar to the others. The most volatile beta was the multivariate GARCH VECH model, while the Kalman filter AR model had a smaller volatility range and the rolling OLS provided the trend line of the beta. This stock sector provided good evidence of the time-varying beta of the multivariate GARCH and Kalman filter.
According to the time-varying beta plot in Figure 5.5, we can observe the obvious downward trend of the beta in the real estate and construction stock sector since the end of 2011 using the multivariate GARCH VECH model and the Kalman filter AR(1) model. However, when considering the beta from the rolling OLS, the time-varying beta of this model response was too slow for the impact of the market. As seen in the plot, there was an increasing trend in the time-varying beta of the technology stock sector.
Next, this study also plotted the out-of-sample time-varying beta of the three models.
