Semi-Nonparametric Estimation and Misspecification Testing of Diffusion Models

We propose novel misspeci\u85cation tests of semiparametric and fully parametric univariate di¤usion models based on the estimators developed in Kristensen (Journal of Econometrics, 2010). We \u85rst demonstrate that given a preliminary estimator of either the drift or the di¤usion term in a di¤usion model, nonparametric kernel estimators of the remaining term can be obtained. We then propose misspeci\u85cation tests of semparametric and fully para-metric di¤usion models that compare estimators of the transition density under the relevant null and alternative. The asymptotic distribution of the estimators and tests under the null are derived, and the power properties are analyzed by considering contiguous alternatives. Test directly comparing the drift and di¤usion estimators under the relevant null and alter-native are also analyzed. Markov Bootstrap versions of the test statistics are proposed to improve on the \u85nite-sample approximations. The \u85nite sample properties of the estimators are examined in a simulation study