"Frontmatter". In: Analysis of Financial Time Series

88 CONDITIONAL HETEROSCEDASTIC MODELSEstimationTwo likelihood functions are commonly used in ARCH estimation. Under the normalityassumption, the likelihood function of an ARCH(m) model isf (a 1 ,...,a T | α)= f (a T | F T −1 ) f (a T −1 | F T −2 ) ··· f (a m+1 | F m ) f (a 1 ,...,a m | α)[ ]T∏ 1= √ exp − a2 tt=m+1 2πσt2 2σt2 × f (a 1 ,...,a m | α),where α = (α 0 ,α 1 ,...,α m ) ′ and f (a 1 ,...,a m | α) is the joint probability densityfunction of a 1 ,...,a m . Since the exact form of f (a 1 ,...,a m | α) is complicated, itis commonly dropped from the prior likelihood function, especially when the samplesize is sufficiently large. This results in using the conditional likelihood function[ ]f (a m+1 ,...,a T | α, a 1 ,...,a m ) =T∏t=m+11√2πσt2exp− a2 t2σ 2 twhere σt2 can be evaluated recursively. We refer to estimates obtained by maximizingthe prior likelihood function as the conditional maximum likelihood estimates(MLE) under normality.Maximizing the conditional likelihood function is equivalent to maximizing itslogarithm, which is easier to handle. The conditional log likelihood function isl(a m+1 ,...,a T | α, a 1 ,...,a m ) =T∑t=m+1− 1 2 ln(2π)− 1 2 ln(σ t 2 ) − 1 2,at2σt2.Since the first term ln(2π) does not involve any parameters, the log likelihood functionbecomes[]T∑ 1l(a m+1 ,...,a T | α, a 1 ,...,a m ) =−2 ln(σ t 2 ) + 1 at22 σt2 ,t=m+1where σt2 = α 0 + α 1 at−1 2 +···+α mat−m 2 can be evaluated recursively.In some applications, it is more appropriate to assume that ɛ t follows a heavytaileddistribution such as a standardized Student-t distribution. Let x v be a Student-tdistribution with v degrees of freedom. Then Var(x v ) = v/(v − 2) for v>2, and weuse ɛ t = x v / √ v/(v − 2). The probability density function of ɛ t is( ) −(v+1)/2Ɣ((v + 1)/2)f (ɛ t | v) =Ɣ(v/2) √ 1 + ɛ2 t, v > 2, (3.7)(v − 2)π v − 2