ABSTRACT
Just about the time that the nonlinear time series models were being developed time series analysis in econometrics took to another path of development. This development occurred because of the need to model data in economics and in particular, in finance where heteroscedasticity is the norm. Hence, the autoregressive moving average (ARMA) model with Gaussian noise and constant variance is inadequate in describing such data. Consider the classical regression model
yt =XTt β + t , (6.1)
where β is a p×1 vector of regression parameters, {t} is an independent noise sequence, and Xt is a p × 1 vector of explanatory variables. The classical solution to the heteroscedasticity problem is to assume that the variance of t is given by σ2Zt−1 where Zt−1 is an exogenous variable. As argued by Engle (1982) this solution is unsatisfactory in the time series context as it fails to recognize that the variance, like the mean, can also evolve over time. Let the time series be denoted {yt}. Denote by Ft−1 all the information available up to time t−1. In many situations we consider only the time series yt itself and hence Ft−1 = {yt−1, · · ·}. Engle (1982) proposed that the conditional variance of t can be modeled as
t = √ ht at (6.2)
where ht = h(yt−1, . . . , yt−q,α) . (6.3)
Here h( ) is a non-negative function of past yt’s, α a q× 1 vector of parameters, and at are independent identically distributed white noise with mean 0 and variance 1. In many applications, in particular in financial time series,
yt = t = √ ht at . (6.4)
This will be assumed from now on unless otherwise stated. In this case Ft−1 = {yt1 , · · ·} = {t−1, · · ·}. The unconditional mean of t is, from (6.2),
E(t) = E( √ ht at)
= E( √ ht)E(at)
= 0
because at is independent of √ ht and E(at) = 0. Furthermore, the con-
ditional variance of t given past t’s is just
E(2t |Ft−1) = E(ht a2t |Ft−1) = E(ht |Ft−1) · E(a2t ) = ht .
