Autoregressive Heteroscedasticity Model and Its Variants 229

Estimates of ARCH/GARCH Models

ARCH/GARCH models can be estimated with the maximum likelihood
estimation (MLE) method described in Chapter 13. To illustrate, consider
the MLE method applied to the simplest ARCH(1) model described in
equations (11.1) and (11.2). The MLE method seeks the values of the
model’s parameters that maximize the probability computed on the data
(likelihood).
It is convenient to condition on the first observation and compute the
conditional likelihood given the first observation. The assumptions of the
ARCH(1) model imply that each return has the following normal condi-
tional distribution:

fRRR

R

tt

t

t

t

(,− ...,)=−exp







(^10) 
2
2

1

2 πσ^2 σ

If the ARCH model is correctly specified, all these conditional probabilities
are mutually independent and therefore the likelihood is simply the product
of the likelihoods:

L

R

t

t

t t

T

=−







= 

∏

1

2 2

2

2

1 πσ σ

exp

And the log-likelihood is

(^) loglL Tog() log t R
t
T
t
t t
T
=− − ()−

∑∑
2

2

1

1 2

2

2

1

πσ

σ

(11.14)

This expression contains the variable σt, which can be recovered for every t
from equation (11.2):

σtt=+ca 11 R^2 −

We have therefore the expression of the log-likelihood in function of the
parameters c, a 1. These parameters can be determined as those values that
maximize the log-likelihood. Maximization is performed with numerical
methods.
In practice, estimation is performed by most standard econometric
packages such as E-Views and by software such as MATLAB. The user needs
only to input the time series of residuals, the specification of the model (i.e.,
ARCH, GARCH, E-GARCH, or the other models described in this chapter),
and the number of lags and the software performs the estimation and the
volatility forecasts.