Time Varying Volatility Models

7.2 ARCH and GARCH Models
69
Suppose Zj has a normal distribution. Since the kurtosis of a normal distribution is equal to 3, we have K(Zt) = E[z^^]/(E[zf])^ =3 . For s^, it holds that
68
7 Time Varying Volatility Models
ized ARCH) model; Glosten, Jagannathan, and Runkle (1993) and Zakoian (1994) extended it to the TGARCH (threshold GARCH) model; and Nelson (1991) extended it to the EGARCH (exponential GARCH) model. This chapter focuses on the ARCH-type modeling approach and the causality technique developed by Cheung and Ng (1996).
7.2 ARCH and GARCH Models
We begin with a brief review of the ARCH family of Statistical models. The ARCH model was originally designed by Engle (1982) to model and forecast the conditional variance. The process allows the conditional variance to change over time as a function of past errors while the unconditional variance remains constant. Let variable y^ have the following AR(k) process: yt=^o+Z!li^iyt.i+^t^ (7.1)
^ Let X be a random variable with mean EpC], and let g(») be a convex function. Here, Jensen's inequality implies E[g(X)] > g(E[X]). For example: g(X)=X^ is convex, hence EP(^]>(E[X])\
7 Time Varying Volatility Models
7.1 Background
Since the seminal work by Engle (1982), the ARCH model has reached a remarkable level of sophistication.^ The ARCH model has become one of the most prevalent tools for characterizing changing variance. Consider the basic nature of the forecasting problem. When the volatility of stock retums is constant, the confidence interval for the stock retum is a function of the sample variance or sample Standard deviation. Here, the volatility implies the conditional variance of asset retums. Yet the shocks that affect the stock retums are also likely to affect the volatility of the stock retums, hence the sample variance or Standard deviation will not be constant. For this reason, the development of a reasonably accurate confidence interval for forecasting requires an understanding of the characteristics of volatility in relationship to the stock retums. The ARCH process explicitly recognizes the difference between unconditional and conditional variance and allows the latter to change over time as a function of past errors. Data has also shown that the percentage changes in stock prices have fatter tails than the percentage changes predicted by stationary normal distributions (Kon 1984). The ARCH model recognizes the temporal dependence in the second moment of stock retums and exhibits a leptokurtic distribution for the unconditional errors from the stock-retum-generating process. Under the recognized phenomenon known as volatility clustering, a period of increased (decreased) volatility isfrequentlyfoUowed by a period of high (low) volatility that persists for some time. As the ARCH model takes the high persistence of volatility into consideration, it has often been extended to more complex models used to characterize changing variance as a function of time. Bollerslev (1986) extended it into the GARCH (general^ A survey article by Bollerslev, Chou, and Kroner (1992) cited more than 300 papers applying ARCH, GARCH, and other related models. ARCH and GARCH models were shown to successfully model time-varying volatility in financial time-series data.
Thus, the conditional variance of y^ is equal to the conditional variance of The kurtosis (K) of s^ is defined as foUows: K[Ej = E[E',]/iE[s',]f. (7.4)
70
ห้องสมุดไป่ตู้
7 Time Varying Volatility Models
The GARCH model developed by Bollerslev (1986) is an extension of the ARCH model. The ARCH(p) process specifies the conditional variance solely as a linear function of past sample variances, whereas the GARCH(p,q) process allows lagged conditional variances to enter as well. This corresponds to some sort of adaptive learning mechanism. The variance dynamics is thus specified as follows:
The conditional variance at time t depends on two factors: a constant ( o ) and past news about volatility taken as the squared error from the past (the ARCH term, i.e.,^^ a-s^ ). The p of the ARCH(p) refers to the number of ARCH terms in equation (7.4). The condition o > 0, a^ >0 guarantees the non-negativity of variance. As equation (7.6) clearly shows, the conditional variance is the weighted average of the squared values of past errors. For the ARCH model, it holds that Var^_JyJ = Et_i[s^] = afE^_Jzf] = af, where a^ is the conditional variance of y^ and is called volatility.
£t=cj,Zt'
(7.2)
where z^ is identical and independent distribution (i.i.d.) with E[zj = 0 and E[z^] = l , and z^ and a^ are statistically independent. It thus holds that \.,[y'] = Ki[(yt -E,_,(y,))^] = E , j 8 f ] . (7.3)
K(s.) = - H # ^
E[aJ]E[z^] (E[af])^(E[zn)3E[a?] (7.5)
(£[<])
2i\2
>3.
where the second equality follows from the independence of a^ and z^, and the inequality in the fourth line is implied by Jensen's inequality.^ Equation (7.5) shows that the distribution of s^ has a fatter tail than normal as long as G^ is not constant (Campbell, Lo and MacKinlay, 1997). This is consistent with the idea that the percentage changes in stock prices have fatter tails than the percentage changes predicted by a stationary normal distribution (Kon 1984). The ARCH(p) model is specified as follows: G^,=(iy-{-Y,l=i^i^ii^ ^ - 0 ' a i > 0 . (7.6)