Identifying nonlinear covariate effects in semimartingale regression models

Summary LetX _t be a semimartingale which is either continuous or of counting process type and which satisfies the stochastic differential equationdX _t=Y_t(t, Z_t) dt+dM_t, whereY andZ are predictable covariate processes,M is a martingale and is an unknown, nonrandom function. We study inference for by introducing an estimator for and deriving a functional central limit theorem for the estimator. The asymptotic distribution turns out to be given by a Gaussian random field that admits a representation as a stochastic integral with respect to a multiparameter Wiener process. This result is used to develop a test for independence ofX from the covariateZ, a test for time-homogeneity of , and a goodness-of-fit test for the proportional hazards model (t,z)=₁(t)a ₂(z) used in survival analysis.Research supported by the Army Research Office under Grant DAAL03-86-K-0094Research supported by the Air Force Office of Scientific Research under Contract F49620-85-C-0007