Analysis of panel data under continuous time-inhomogeneous Markov chains with stochastic covariates intensity

This paper revisits Kalbfleisch and Lawless (1985) and develops a novel approach for analysis of panel data under continuous time-inhomogeneous Markov chains whose intensity matrix depends on stochastic covariates. Unlike the mentioned paper, the new model assumes continuous observation of the Markov chains. Distributional properties are developed for derivation of likelihood function of sample paths and for numerical study. It is shown that the transition probability satisfies the Kolmogorov backward equation in the presence of covariates. The derived likelihood function generalizes the formula (2) on p.99 in Andersen and Keiding (2002) regarding the likelihood function of continuously observed time-inhomogeneous Markov chain with time-fixed covariates. Under piecewise constant transition intensity matrix, the score function and observed information matrix of the regression coefficients of covariates are presented explicitly in terms of tensor matrix representations. In particular, the information matrix is positive definite regardless the values of covariates and the statistics of sample paths. These appealing features allow maximum likelihood (ML) recursive estimation of the coefficients using fast convergent Newton-Raphson method. Importantly, for establishing the consistency and large sample properties of the ML estimates (MLE). Also for deriving empirical estimates of the standard errors. The theoretical results generalize that of presented in the aforementioned papers and Albert (1962). Numerical study confirms the accuracy of the MLE.