Estimation and Inference based on Summary Statistics for State Space Models

This paper introduces a theoretically robust framework for conditional mean estimation associated with maximum likelihood estimation (MLE) in state space models with nonstationary time series, integrating the classical Kalman filter with Bayesian inference. We propose a dierence-based approach using optimally designed summary statistics from observable data, overcoming the intractability of traditional Kalman filter statistics reliant on latent states. Our formulation creates a direct mapping between feasible statistics and structural parameters, enabling clean separation between state dynamics and measurement noise. Under mild regularity conditions, we prove the consistency and asymptotic normality of the estimators, achieving the Cramrr-Rao lower bound for eciency. The methodology extends to non-Gaussian innovations with finite moments, ensuring robustness. Monte Carlo approximations preserve asymptotic eciency under controlled sampling rates, while finite sample bounds and robustness to model misspecification and data contamination confirm reliability. These theoretical advances are complemented by a comprehensive simulation study demonstrating superior performance compared to conventional approaches. These results advance the theoretical foundations of state space modelling, providing a statistically ecient and computationally feasible alternative to conventional approaches.