Anon

220 The Basics of financial economeTrics

If we require that c > 0 and that 0 < a 1 < 1, then the return process Rt is
stationary with variance equal to σ^2 =−ca/() 1 1 and the variance process
is positive.
To illustrate the above, Figure 11.3 provides a plot of simulated returns
that follow an ARCH(1) process assuming that c = 0.1 and a 1 = 0.1. Fig-
ure 11.4 shows that the volatility of the series changes in time and oscillates
between low and high values.
To improve the ability of ARCH models to fit realistic time series, we
can use a larger number of lags. We use the notation ARCH(m) model to
denote a model with m lags and describe this form of the model as follows:

Rtt=σεt (11.3)

σtt^2 =+ca 11 Ra^22 −−++ mtRm (11.4)

In equation (11.3) a random error, εt, is multipled by the time-varying vola-
tility σt; equation (11.4) prescribes that current volatility is a weighted aver-
age of past squared returns plus a constant. In order to ensure that σt^2 is
nonnegative and the model stationary, we require that

1. The parameters a 1 ,... , am be nonnegative


2. a 1 +...+ am < 1

0 200 400 600 800 1,000

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time Steps

Returns

FIGure 11.3 Simulated ARCH(1) Return Process with Parameters c = 0.1 and a 1 = 0.1