Cox’s factoring of regression model likelihoods for continuous-time processes. (English) Zbl 0899.62114

Summary: D. R. Cox [Biometrika 62, 269-276 (1975; Zbl 0312.62002)] showed that the likelihood of regression models for discrete-time processes factors into a partial likelihood and a product of conditional laws for the covariates, given the history. J. Jacod [Stochastic Processes Appl. 26, 47-71 (1987; Zbl 0632.62088)] constructed a partial likelihood for continuous-time regression models in terms of the predictable characteristics of the response process. Here we prove a factorization of the likelihood, analogous to Cox’s, assuming both the response and the covariates to be semimartingales. The result is useful for counting process regression modelling and inference, and also for regression involving continuous processes and diffusions with jumps.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G48 Generalizations of martingales
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