Estimation of the parameters and volatilities in a GARCH(1,1) model could be based on data from the entire period. However, if we do not consider some extraordinary events occurring during the period while estimating the parameters, the estimates so obtained may not be meaningful. Hence, we want to detect some change points and find several such events according to historical incidents. Consequently, we will divide 163 days into several subperiods. We then estimate the parameters of the GARCH(1,1) model for each subperiod. At the same time, we hope that volatilities estimated with change points can have better performance than those ignoring such changes. That is, volatilities with change points will yield smaller values of accuracy measures.
5.1 Empirical Volatility Measurement
By using methods in the Western Electric Handbook (1956) (see methods 1 to 5 in section 3.3), we find some candidates of change points.
Since there is no point exceeding the 3-sigma limit from Figure 5.1, we say that the whole pattern is random by method 1 of Western Electric rules. In Table 5.1, there are two candidates of change points by using method 2 but the LRT values are less than 9.488. That means the candidates of change points can not change data structure.
By method 3, as indicated in Table 5.2, there is only one candidate with the LRT value less than the critical value. Absence of change points detected by method 5 (Five consecutive points are in a rising or falling trend) produces a natural pattern. It is much similar to the result of method 1 above.
We can observe from Table 5.3 that when period 1 (Jan. 2, 2003 to May 14, 2003)
and period 2 (May 15, 2003 to Aug. 29, 2003) had been decided, the value of the likelihood ratio test is bigger than 9.488.
2
According the LRT, the only change point occurred on May 14, 2003. The LRT statistic yields a value bigger than the critical value of 9.488. However, other candidate points are not change points according to the LRT. Our finding coincides with the news announcement on May 14, 2003 that the epidemic SARS had reached the countryside of Mainland China.4 Hence, we can separate 163 days into two subperiods.
From Figure 5.2, it is clear and observable that the 87 th point is a possible change point of events. Before the 87 th day, the returns had more substantial variations than the remainder. The epidemic SARS has since subsided and consequently the stock market became more stable.
After making sure of the change point, we can rewrite separate formulae for separate subperiods as follows:
0.014 1.79319 1 ,
for the GARCH(1,1) model of period 1, and
0.014 1.50269 1 ,
for the GARCH(1,1) model of period 2. These formulae indicate that the parameters change significantly. Meanwhile, if we ignore the change and fit the GARCH(1,1) __________________________
4 Data source: http://sars.health.gov.tw/article.asp?channelid=H&serial=189&click=
model for the entire period, we have
which is quite different from either (5.1) or (5.2). The estimates of parameters for different models are shown in Table 5.4.
For an GARCH model, the standardized shocks
are iid and N(0,1). And the Ljung-Box (1978) Q-statistic is expressed as below
2
where is the lag-l sample autocorrelation of or a .Table 5.7 shows the value of the Ljung-Box Q-statistic for the squares of the standardized shocks. It is also noted that the Q-statistic has also been computed for the standardized shocks themselves and the adequacy of the models has been established as well. We can check this table and find the fitted models are all adequate.
ˆl
ρ at t2
Table 5.5 reports the in-sample forecasting accuracy criteria RMSE, MAE, and LE, respectively. It compares the performance of the fitted models. The RMSE criterion is not very robust, and in practice easily influenced by a few large values.
The MAE criterion is less susceptible to these values.
Under the accuracy measures for the GARCH(1,1) model, we find higher values
for RMSE, MAE and LE when no change point was detected. For example, RMSE for the GARCH(1,1) model with change point is 0.0190417 versus 0.0234972 without change point. Thus, detecting structural change is meaningful in fitting the GARCH(1,1) models to financial data.
5.2 Some Issues of Interest
There are some interesting issues derived from the empirical results. We will illustrate them as follows.
Besides estimating volatility, GARCH types of models have some interesting and useful properties. If we standardized the return,
ˆ then will be distributed as N(0,1) when the normality assumption is reasonable for . We can verify this by calculating some statistics, such as mean, standard deviation, skewness and excess kurtosis, as given in Table 5.6. Skewness characterizes the degree of asymmetry of a distribution around its mean. Positive (Negative) skewness indicates a distribution with an asymmetric tail extending towards more positive (negative) values. The skewness of normal distribution is zero.
A significant degree of asymmetry (skewness regardless of sign) is bigger than 2 standard errors of skewness (ses). The ses can be estimated by using the following formula (Tabachnick & Fidell, 1996):
zt
rt
ses 6
= N , where N is the number of observations.
The excess kurtosis of a standard normal random variable is zero. A distribution with positive excess kurtosis is said to have heavy tails. It implies that the distribution has more mass on the tails than a normal distribution. A significant degree of mass tail (kurtosis regardless of sign) is bigger than 2 standard errors of kurtosis (sek). The sek can be estimated roughly using the following formula (Tabachnick & Fidell, 1996):
sek 24
= N , where N is the number of observations.
From Table 5.6, it is clear that the GARCH(1,1) models have heavy tails due to positive excess kurtosis. Hence, the GARCH(1,1) models are not reasonable for modeling volatilities. Since no proper detection of change point is done, the GARCH(1,1) model without change points can be affected by the oscillation of prices coming from some unusual events. In order to get more insight, we combine Table 5.6, Figure 5.3.1 and Figure 5.3.2. Figure 5.3.1 and Figure 5.3.2 are the quantile-quantile (q-q) plots. The q-q plot is a graphical technique for determining if the data have come from a certain distribution. Realized volatility may be fitted with standard normal distribution as shown in Figure 5.3.2. However, it will provide only daily reference and will have no capability of prediction. Meanwhile, it seems that the GARCH(1,1) models are fitted somewhat poorly with the standard normal distribution as shown in Figure 5.3.1. From Table 5.6, we see that the GARCH(1,1) models with or without change points have positive skewness. That means that standardized returns under the GARCH(1,1) models are slightly skewed to the right.
The values of Jaque-Bera (JB) test5 in Table 5.6 show that the GARCH(1,1) model with change points is fitted better with standard normal distribution than the GARCH(1,1) model without change points. Furthermore, the volatilities under the GARCH(1,1) models are overvalued because their variances of standardized returns are less than 1, as shown in Table 5.6. In short, the GARCH(1,1) models with change points will be a useful model for our purpose, even though it is not perfect.
__________________________
5 Jarque and Bera (1980,1987) showed the Jaque-Bera (JB) test of normality:
2
, where S = skewness coefficient and K =excess kurtosis coefficient. For a normally
distributed variable, S = 0 and K = 0. Therefore, the JB test of normality is a test of the joint hypothesis that S and K are 0 and 0, respectively.