224 The Basics of financial economeTrics
is noisy and unreliable. To avoid too many lags, Tim Bollerslev introduced
a variant of ARCH models called the generalized autoregressive conditional
heteroscedasticity model or GARCH model.^7
In a GARCH model, volatility depends not only on the past values of
the process as in ARCH models but also on the past values of volatility. The
GARCH(p, q) model, where p is the number of lags or the past value of Rt^2
and q the number of lags for the variance in the model, is described by the
following pair of equations:
Rdtt=+σεt (11.8)
σσ^2 tt=+ca 11 Ra^22 −−++ ptRbpt++ 1122 −−+bqtσ q (11.9)
Conditions to ensure that σ is positive and that the process is stationary
are the same as for ARCH models; that is
- c > 0
- All parameters a 1 ,... , ap, b 1 ,... , bq must be nonnegative
- The sum of all parameters must be less than 1:
aa 11 ++ pq++bb+< 1If these conditions are met, then the squared returns can be represented as
an autoregressive process as follows:
Rctt^2 =+()ab 11 + Ra^22 −− 1 ++ ()mm+bRtm+wt (11.10)
To illustrate, let’s simulate a GARCH(1,1) process assuming that
c = 0.1, a 1 = 0.4, and b 1 = 0.4. Figure 11.7 shows 1,000 simulated returns
obtained for this GARCH(1,1) process. Figure 11.8 represents the cor-
responding volatility. As can be seen, the conditional heteroscedasticity
effect is quite strong. In fact, we can see from this figure that volatility
periodically goes to a minimum of about 0.4 and then rises again to higher
values. This periodic oscillation of the value of volatility is the essence of
ARCH/GARCH models.
(^7) Tim Bollerslev, “Generalized Autoregressive Conditional Heteroscedasticity,” Jour-
nal of Econometrics 31 (1986): 307–327.