Supplementary Appendix for “ Generalized Autoregressive Score Models with Applications ”

In this Supplementary Appendix we present additional new material related to the main paper “Generalized Autoregressive Score Models with Applications”. We refer to the model as the GAS model. For reference purposes, we first give a short review of the relevant equations for the general GAS model. Appendix A presents more existing models that can be represented as special cases of GAS models. Appendix B formulates new models including unobserved components models, models with time-varying higher order moments, time-varying multinomial model and dynamic mixture models. In Appendix C we present the simulation results for the two illustration models of the main paper : the Gaussian copula model with time-varying correlations and the marked point process model. Basic GAS model specification Let N × 1 vector yt denote the dependent variable of interest, ft the time-varying parameter vector, xt a vector of exogenous variables (covariates), all at time t, and θ a vector of static parameters. Define Y t = {y1, . . . , yt}, F t = {f0, f1, . . . , ft}, and X t = {x1, . . . , xt}. The available information set at time t consists of {ft , Ft} where Ft = {Y t−1 , F t−1 , X }, for t = 1, . . . , n. We assume that yt is generated by the observation density yt ∼ p(yt | ft , Ft ; θ). (1) Furthermore, we assume that the mechanism for updating the time-varying parameter ft is given by the familiar autoregressive updating equation