Estimation in Financial Models - RiskLab

3.2 Discrete models
In section 1.2 we dealt with three kinds of stochastic volatility models, following
an AR(p)-process, an ARCH or even a GARCH process. Now we
discuss Maximum Likelihood (ML) estimation for these models and derive
the properties of the estimators. Finally, we treat the Bayesian analysis.
3.2.1 AR models
In our representation of estimation theory for the AR(1)-process we follow
[67] x5. The parameter estimation problem for linear time series (i.e. ARMA
processes) is to be found in many textbooks, see for instance [16], [34], [64].
Below we discuss a slightly dierent approach which will turn out to be
useful in more general models. The AR(1) case should therefore be seen as a
pedagogic example with respect to this more general methodology.
Consider the autoregressive AR(1) model
X n = X n,1 + " n ; n 1; (3.62)
where 2 IR is the parameter to be estimated, X 0 is a random variable
with E(X 0 ) = 0 and f" n ;n 1g is `noise', where " n N (0; 2 ) i.i.d. with
known 2 .
From equation (3.62) we conclude
X n = " n + " n,1 + :::+ n,1 " 1 + n X 0 ;
and thus the probabilistic properties of fX n g essentially depend on the joint
distribution of X 0 ;" 1 ;" 2 ;:::.We have
and
Var X n = 2 (1 + 2 + :::+ 2(n,1) )+ 2n Var X 0 (3.63)
E X n X n,k = 2 k (1 + 2 + :::+ 2(n,k,1) )+ k 2(n,k) Var X 0 : (3.64)
In the case jj < 1, we obtain from (3.63) and (3.64) by choosing X 0
N (0; 2
)
1, 2
E X n =0; Var X n = 2
1 , 2 ; E X n X n,k = 2 k
1 , 2 ;
and fX n g has a stationary distribution.
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