Estimation in Financial Models - RiskLab

of time varying variances and covariances. Engle [22] introduced such a new
class of stochastic processes, called the class of AutoRegressive Conditional
Heteroscedastic (ARCH) processes where the conditional variances and covariances
depend on the past. These are discrete time stochastic processes
f" t g of the form
" t = z t t (3.67)
with z t i. i. d. , E(z t )=0, Var(z t )=1, and with time-varying positive t which
is time (t , 1) measurable. Assume in the following z t N(0; 1) and " t is a
scalar process. The extension to the multivariate case is straightforward, see
e.g. [13].
First we focus directly on the process f" t g (3.67) and assume that " t is itself
observable. However, note that in many applications " t corresponds to the
innovations for some other stochastic process y t , where y t = f(b; x t )+" t with
" t conditionally distributed N (0; 2 ), f a function of x t which is time (t , 1)
measurable and of a parameter vector b. In this context we will consider
later the ARCH regression model, see (3.75), where f(b; x t ) x 0 tb, that is
the conditional mean of y t is given as x 0 tb.
The conditional variance of " t equals t 2 . We will concentrate in the following
on some frequently used models for t 2 . Engle [22] suggests one possible
parametrization for t
2 as a linear function of the past q squared values of
the process
2 t = 0 +
qX
i=1
i " 2 t,i; (3.68)
where for the model to be well-dened 0 > 0 and i 0 for i = 1;:::;q.
This model is known as the linear ARCH(q) model.
Nowwe discuss Maximum Likelihood (ML) estimation in the linear ARCH(q)
model. For a discussion of the ML estimation see section 3.1.1. Apart from
some constants, the log-likelihood of the tth observation is
l t = , 1 2 log 2 t , 1 2 "2 t = 2 t ; (3.69)
where the term , 1 log 2 2 t arises from the transformation from z t to " t . The
log-likelihood function for the full sample " T ;" T ,1 ;:::;" 1 is
L = 1 T
TX
t=1
l t : (3.70)
To estimate the unknown parameters 0 ( 0 ; 1 ;:::; q ) we maximize the
log-likelihood function, that is the rst order derivatives are set equal to zero.
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