Exogenous Threshold Variable and MMI Stocks

In this section we first apply the threshold GARCH model to the data set that contains 20 stocks from MMI. We assume that the return series follows the threshold GARCH model where the trigger variable is either exogenous (VIX)

or endogenous (volume).

Since the return series in our threshold GARCH model is assumed to be a zero mean process, we first remove the mean from the returns. In addition to a constant we also filter the AR effect to order 5:

rt=Rt−µ−

5

P

j=1 δjRt−j

whereRtis the observed return,µis the mean, andδj is the coefficient of AR

variables.

The threshold variable (VIX or volume) is given in this threshold GARCH model, so we also need to determine the threshold value for this variable. Now the model that needs to be estimated has a set of parameters as a function of the threshold value.

rt=σtεt    σ2 t =ω0+α0r2t−1+β0σt2−1 if yt−1 ≤y∗ σ2 t =ω1+α1r2t−1+β1σt2−1 if yt−1 > y∗

To estimate the threshold value we divide the sample of threshold variable into 40 intervals and the 39 grid points correspond to 2.5th percentile point to 97.5th percentile point. We use only the first lag of VIX and volume as the threshold variable since we believe the most recent observation of them pro- vides the most up-to-date information on the condition of market and individ- ual stocks. The robust standard errors we compute for the volatility parame- ters are Bollerslev-Wooldridge standard errors. Since we use the grid search to obtain the threshold valuey∗, we are not able to compute its standard error.

There are other estimation methods available for the threshold model that pro- vide such statistics. We may explore alternative methods in the future studies. The estimation results in Table 3.7 are based on 4787 observations from Jan. 02, 1990 when VIX is available. The estimated threshold value and the probability that volatility is in regime 2 is given by y∗, for example the esti- mated threshold is y92.5 for IBM, it means that the threshold value is the 92.5

percentile point of VIX price, so the volatility is in a volatile regime with prob- ability of 7.5%. The parameters in the threshold GARCH model are significant for most of the stocks. For some stocks the sum of estimated parameters is greater than 1 in one regime, but consider the probabilityπ, we will still have the stationary process. The probability is given by the location of the thresh- old value in the sample space of the threshold variable, it is very clear in our estimation results since we use 2.5 percentile as the increment. For exam- ple the estimated parameters of PG in regime 2 have a sum of 1.0026, but the threshold value estimated isy80, that means there is only 20% chance that the

conditional variance shifts to the regime 2, therefore with a stationary regime 1 the stationarity condition for return process still holds. We observe that for some stocks the estimated parameters in 2 regimes are very similar, it is not surprising because we use VIX as a threshold variable for all 20 stocks in our sample, some stocks may follow closely with the market, while others may be less affected by the market conditions. Nonetheless when we use the mar- ket condition as a threshold, it indeed separates the returns in low volatility regime from that in the high volatility regime.

We plot the MMM return series in different regimes in Figure 3.1 and 3.2. Figure 3.1 contains a graph of return series that is divided into 2 regimes, while Figure 3.2 provides the threshold value of the VIX price to separate two

regimes.

From Figure 3.1 we see that given the threshold value, the high volatility regime identifies the 3 periods of clustering of extreme returns. The volatile pe- riods are separated from the less volatile periods, and the clustering of volatile periods confirm the presence of the disruptive events during the periods, such as the clustering of volatile periods at the starting point of the graph corre- sponds to the 1990-1991 Persian Gulf Crisis, the clustering of volatile periods at the middle of the graph corresponds to the periods from 1997 Asian crisis to 2000 dot.com bubble, and the clustering at the end of the graph corresponds to the 2008 subprime mortgage crisis. The use of the VIX as the threshold variable enable us to find the periods in which the VIX market and the stock are volatile since the markets tend to move together, meanwhile we will miss the stock specific information so that some volatile events are ignored simply because of the involatile VIX at that point in time. Nevertheless those stock specific events tend to be non-persistent, we observe that some large negative returns are not identified as in the high volatility regime simply because it is a rare event. Figure 3.2 shows the VIX price and regimes divided by the VIX price. The estimated threshold value for VIX price is 20.17, when the price of VIX is above the threshold value, people view the market as unstable, there- fore we observe large price movement in the VIX prices. The empirical results reveal the potential of the threshold variable to identify the regimes in the volatility process hence provide better forecast.