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 have received considerable attention from both financial theoreticians and empirical econometricians over the last two decades. One reason for this interest is that correct specification of diffusion models describing dynamics of financial assets is crucial for many areas in finance including equity and option pricing, term structure modeling, and risk management, for example. In this paper, we discuss advances to this literature introduced by Corradi and Swanson (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, Corradi and Swanson (BCS: 2008) and Corradi and Swanson (2011). In particular, parametric specification tests in the spirit of the conditional Kolmogorov test of Andrews (1997) that rely on block bootstrap resampling methods in order to construct test critical values are first discussed. Thereafter, extensions due to BCS (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 (2011) to multifactor models where the number of latent variables is larger than three. These final tests can be thought of as continuous time generalizations of the discrete time "reality check" test statistics of White (2000), which are widely used in empirical finance (see e.g. Sullivan, Timmermann and White (1999, 2001)). We finish the chapter with an empirical illustration of model selection amongst alternative short term interest rate models.