Chapter 5 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.