Model Selection Testing for Diffusion Processes
with Applications to Interest Rate and Exchange Rate Models

Abstract

A model selection test for nonnested misspecified diffusion models is developed based on the Kullback-Leibler information criterion. A new asymptotic framework accounts for the high significance of diffusion functions relative to drift functions for high frequency data. The test examines the hypothesis that two competing models are equivalent. Our approach distinguishes the roles of diffusion and drift functions and shows the equivalence of models must be understood differently depending on the sampling frequencies. When the sampling frequency is high, it is of primary importance for a model to have a diffusion function close to the true diffusion function, and we compare drift functions when the models can not be distinguished by the diffusion functions. As the sampling frequencies become higher, the diffusion functions are more important, and the information for ranking the drift functions is weaker. The drift functions are useful only when we sample data for long enough. Our new asymptotics deals with the different rates of information in the diffusion and drift functions by considering both the sampling interval Δ and the sampling span T, and we show the sampling span must increase at a relative speed faster than 1/Δ² (or Δ²T→∞)  to ensure sufficient information to be collected for distinguishing two models by their drift functions. The limiting distribution of the test statistic is normal, and we compare different asymptotic approximations to the sampling distribution of the test statistic using subsampling, and nonparametric block bootstrap methods, as well as the standard normal approximation for the test statistics standardized by the heteroskedasticity autocorrelation consistent variance estimators. We apply our test to spot interest rate models and exchange rate models. We find that many popular models are observationally equivalent.