Autoregressive Heteroscedasticity Model and Its Variants 221
Figures 11.5 and 11.6 provide the plot of a simulated series of returns
that follow an ARCH(3) process; that is, a process that follows an ARCH
model with three lags. The model’s coefficients assumed in the plots are
c = 0.1, a 1 = 0.6, a 2 = 0.2, and a 3 = 0.1. Notice that the assumed values for
the parameters satisfy the two conditions for the model to be positive sta-
tionary: (1) the assumed value for the three parameters for three lags are all
positive and (2) their sum is 0.9 which is less than one. It can be shown that
Rt^2 is an autoregressive process.^5 By establishing that Rt^2 is an autoregressive
process, if returns follow an ARCH process, we prove that squared returns
(^5) Let’s substitute (^) σtt^2 =+ca 11 Ra^22 −−++ mtR m into Rtt=σεt and take conditional
expectations. As lagged values of Rt are known and εt has zero mean and unit vari-
ance, we obtain:
ER()tt^2 RR^22 −− 1 tm=+()ca 11 Ra^22 tm−−++REtm ()(ε^2 t =+ca 11 Ratm^22 −−++ Rtm
(^) This relationship shows that Rt^2 is an autoregressive process. The process Rt^2 repre-
sents squared returns and therefore must be positive. To ensure that Rt^2 be positive,
we require that ca>≥ 00 ,,i im= 1 ,,.... It can be demonstrated that this condition
guarantees stationarity. Taking expectations, we see that the unconditional variance is:
ca/1()−∑ i
0 200 400 600 800 1,000
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time Steps
Volatility
FIGure 11.4 Plot of Volatility Relative to Figure 11.3
Note: Periods of high and low volatility alternate but the frequency of changes is high.