Forecasting Value-at-Risk using Functional Volatility Incorporating an Exogenous Effect

This paper proposes a novel extension of log and exponential GARCH models, where time-varying parameters are approximated by orthogonal polynomial systems. These expansions enable us to add and study the effects of market-wide and external international shocks on the volatility forecasts and provide a flexible mechanism to capture various dynamics of the parameters. We examine the performance of the new model in both theoretical and empirical analysis. We investigate the asymptotic properties of the quasi-maximum likelihood estimators under mild conditions. The small-sample behavior of the estimators is studied via Monte Carlo simulation. The performance of the proposed models, in terms of accuracy of both volatility estimation and Value-at-Risk forecasts, is assessed in an empirical study of a set of major stock market indices. The results support the proposed specifications with respect to the corresponding constant-parameters models and to other time-varying parameter models. © 2023 Elsevier Inc.