Higher-Order Marginal Likelihood Approximation for Nonlinear State Space Models

Bayesian comparison of nonlinear state space models requires integrating over a growing latent state path and a low-dimensional parameter vector. We propose a marginal likelihood approximation that integrates out the states with a numerical or simulation-based evaluator and applies a second-order corrected Laplace approximation to the integral over the static parameters. The correction is expressed through smoothing cumulants of the complete-data log density and, under derivative-growth and posterior-concentration conditions, reduces the log marginal likelihood error from $O(T^{-1})$ to $O(T^{-2})$. In stochastic volatility experiments, it removes 64\% to 99\% of the Laplace error at far lower cost than simulation-based estimators. Applications to bivariate exchange-rate volatility models and to a sample of $T = 19{,}266$ five-minute stock returns find the strongest support for heavy-tailed observations and for a transitory volatility factor.