Density and conditional distribution-based specification analysis

The technique of using densities and conditional distributions to carry out consistent specification testing and model selection amongst multiple diffusion processes has received considerable attention from both financial theoreticians and empirical econometricians over the last two decades. In this chapter, we discuss advances to this literature introduced by Corradi and Swanson (J Econom 124:117-148, 2005), who compare the cumulative distribution (marginal or joint) implied by a hypothesized null model with corresponding empirical distributions of observed data. We also outline and expand upon​ further testing results from Bhardwaj et al. (J Bus Econ Stat 26:176-193, 2008) and Corradi and Swanson (J Econom 161:304-324, 2011). In particular, parametric specification tests in the spirit of the conditional Kolmogorov test of Andrews (Econometrica 65:1097–1128, 1997) that rely on block bootstrap resampling methods in order to construct test critical values are first discussed. Thereafter, extensions due to Bhardwaj et al. (J Bus Econ Stat 26:176-193, 2008) for cases where the functional form of the conditional density is unknown are introduced, and related continuous time simulation methods are introduced. Finally, we broaden our discussion from single process specification testing to multiple process model selection by discussing how to construct predictive densities and how to compare the accuracy of predictive densities derived from alternative (possibly misspecified) diffusion models. In particular, we generalize simulation steps outlined in Cai and Swanson (J Empir Financ 18:743-764, 2011) to multifactor models where the number of latent variables is larger than three. We finish the chapter with an empirical illustration of model selection amongst alternative short-term interest rate models.