"Frontmatter". In: Analysis of Financial Time Series

290 VALUEATRISKare available, we assume thatk t = γ 0 + γ 1 x 1t +···+γ v x vt ≡ γ 0 + γ ′ x tln(α t ) = δ 0 + δ 1 x 1t +···+δ v x vt ≡ δ 0 + δ ′ x t (7.35)β t = θ 0 + θ 1 x 1t +···+θ v x vt ≡ θ 0 + θ ′ x t .If γ = 0, then the shape parameter k t = γ 0 , which is time-invariant. Thus, testing thesignificance of γ can provide information about the contribution of the explanatoryvariables to the shape parameter. Similar methods apply to the scale and locationparameters. In Eq. (7.35), we use the same explanatory variables for all the threeparameters k t , ln(α t ),andβ t . In an application, different explanatory variables maybe used for different parameters.When the three parameters of the extreme value distribution are time-varying, wehave an inhomogeneous Poisson process. The intensity measure becomes[(T 1 , T 2 ) × (r, ∞)] = T 2 − T 1T(1 − k t(r − β t )α t) 1/kt+, r >η. (7.36)The likelihood function of the exceeding times and returns {(t i , r ti )} becomes⎛⎞∏N η[L = ⎝1T g(r t i; k ti ,α ti ,β ti ) ⎠ × exp − 1 ∫ N]S(η; k t ,α t ,β t )dt ,T 0i=1which reduces to⎛⎞ [∏N ηL = ⎝1T g(r t i; k ti ,α ti ,β ti ) ⎠ × exp − 1 Ti=1]N∑S(η; k t ,α t ,β t )t=1(7.37)if one assumes that the parameters k t , α t ,andβ t are constant within each trading day,where g(z; k t ,α t ,β t ) and S(η; k t ,α t ,β t ) are given in Eqs. (7.32) and (7.31), respectively.For given observations {r t , x t | t = 1,...,N}, the baseline time interval T ,and the threshold η, the parameters in Eq. (7.35) can be estimated by maximizing thelogarithm of the likelihood function in Eq. (7.37). Again we use ln(α t ) to satisfy thepositive constraint of α t .Remark: The parameterization in Eq. (7.35) is similar to that of the volatilitymodels of Chapter 3 in the sense that the three parameters are exact functions of theavailable information at time t. Other functions can be used if necessary.7.7.5 Model CheckingChecking an entertained two-dimensional Poisson process model for exceedancetimes and excesses involves examining three key features of the model. The first