 
Summary: Supplementary materials for this article are available at http://pubs.amstat.org/toc/jasa/104/487.
Nonparametric TransitionBased Tests
for Jump Diffusions
Yacine AÏTSAHALIA, Jianqing FAN, and Heng PENG
We develop a specification test for the transition density of a discretely sampled continuoustime jumpdiffusion process, based on a
comparison of a nonparametric estimate of the transition density or distribution function with their corresponding parametric counterparts
assumed by the null hypothesis. As a special case, our method applies to pure diffusions. We provide a direct comparison of the two densities
for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative
transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their
power functions. We investigate the finitesample properties through simulations and compare them with those of other tests. This article
has supplementary material online.
KEY WORDS: Generalized likelihood ratio test; Jump diffusion; Local linear fit; Markovian process; Null distribution; Specification test;
Transition density.
1. INTRODUCTION
Consider a given parameterization for a jump diffusion de
fined on a probability space ( , ,P)
dXt = (Xt,)dt + (Xt,)dWt + Jt dNt, (1.1)
where Wt is a Brownian motion, Nt is a Poisson process with
stochastic intensity (Xt;) and jump size 1, and Jt, the
jump size, is a random variable with density (·;Xt,). These
