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Abstract
Marginal likelihoods are central to Bayesian comparison of nonlinear state space models, but commonly used estimators require posterior simulation for each model, with computational costs that grow with the sample size $T$. We propose a deterministic estimator that requires no posterior simulation. A grid filter integrates out the latent states, and a Laplace approximation with a second-order correction integrates over the static parameters. Under derivative-growth and posterior-concentration conditions, the corrected Laplace approximation has error $O(T^{-2})$, and the size of the correction indicates the error that remains. In stochastic volatility experiments, the estimator is more accurate than Chib’s method, the Gelfand–Dey estimator and importance sampling with a fitted proposal at equal computing time for $T\geq 2{,}000$. Applications to two exchange rates and to $T=19{,}266$ five-minute stock returns provide the strongest support for heavy-tailed observations and a transitory volatility factor.