Predictive Density Evaluation

This chapter discusses estimation, specification testing, and model selection of predictive density models. In particular, predictive density estimation is briefly discussed, and a variety of different specification and model evaluation tests due to various authors including Christoffersen and Diebold [Christoffersen, P., Diebold, F.X. (2000). "How relevant is volatility forecasting for financial risk management?". Review of Economics and Statistics 82, 12-22], Diebold, Gunther and Tay [Diebold, F.X., Gunther, T., Tay, A.S. (1998). "Evaluating density forecasts with applications to finance and management". International Economic Review 39, 863-883], Diebold, Hahn and Tay [Diebold, F.X., Hahn, J., Tay, A.S. (1999). "Multivariate density forecast evaluation and calibration in financial risk management: High frequency returns on foreign exchange". Review of Economics and Statistics 81, 661-673], White [White, H. (2000). "A reality check for data snooping". Econometrica 68, 1097-1126], Bai [Bai, J. (2003). "Testing parametric conditional distributions of dynamic models". Review of Economics and Statistics 85, 531-549], Corradi and Swanson [Corradi, V., Swanson, N.R. (2005a). "A test for comparing multiple misspecified conditional distributions". Econometric Theory 21, 991-1016; Corradi, V., Swanson, N.R. (2005b). "Nonparametric bootstrap procedures for predictive inference based on recursive estimation schemes". Working Paper, Rutgers University; Corradi, V., Swanson, N.R. (2006a). "Bootstrap conditional distribution tests in the presence of dynamic misspecification". Journal of Econometrics, in press; Corradi, V., Swanson, N.R. (2006b). "Predictive density and conditional confidence interval accuracy tests". Journal of Econometrics, in press], Hong and Li [Hong, Y.M., Li, H.F. (2003). "Nonparametric specification testing for continuous time models with applications to term structure of interest rates". Review of Financial Studies, 18, 37-84], and others are reviewed. Extensions of some existing techniques to the case of out-of-sample evaluation are also provided, and asymptotic results associated with these extensions are outlined.