# Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling

Continuous time models are well known in sociology through the pioneering work of Simon (1952); Coleman (1968); Doreian and Hummon (1976, 1979) and others. Although they have the theoretical merit in modeling time as a owing phenomenon, the empirical application is more diffcult in comparison to time series models. This is in part due to the diffculty in computing likelihood functions for sampled, discrete time measurements (daily, weekly etc.), as they occur in empirical research. With large sampling intervals, one cannot simply replace differentials by differences, since then one obtains strongly biased estimates of structural parameters. Instead one has to consider the exact transition probabilities between the times of measurement. Even in the linear case, these probabilities are nonlinear functions of the structural parameter matrices with respective identiffication and embedding problems (Hamerle et al.; 1991). For nonlinear systems, additional problems occur due to the impossibility of computing analytical transition probabilities for most models. There are competing numerical methods based on nonlinear filtering, partial differential equations, integral representations, Monte Carlo and Bayesian approaches. We compute the likelihood function of a nonlinear continuous-discrete state space model (continuous time dynamics of latent variables, discrete time noisy measurements) by using a functional integral representation. The unobserved state vectors are integrated out in order to obtain the marginal distribution of the measurements. The infinite-dimensional integration is evaluated by Monte Carlo simulation with importance sampling. Using a Langevin equation with Metropolis mechanism, it is possible to draw random vectors from the exact importance distribution, although the normalization constant (the desired likelihood function) is unknown. We discuss several methods of estimating the importance distribution. Most importantly, we obtain smooth likelihood surfaces which facilitates the usage of quasi Newton algorithms for determinating the ML estimates. The proposed Monte Carlo method is compared with Kalman filtering and analytical approaches using the Fokker-Planck equation. More generally, one can compute functionals of diffusion processes such as option prices or the Feynman-Kac formula in quantum theory.

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Singer, Hermann: Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hagen 2016.

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