Computing the observed Fisher information matrix for linear Gaussian state space models and its application to specification tests

This article develops a recursive formula for computing the exact score vector and the observed Fisher information matrix in linear Gaussian state space models. The method provides closed-form analytical expressions for both first- and second-order derivatives, avoiding reliance on numerical differentiation or asymptotic approximations. These results facilitate the construction of Wald, Lagrange multiplier (LM), and information matrix (IM) tests within the state-space framework. Monte Carlo experiments demonstrate that these analytical derivatives ensure numerical stability and improved finite-sample properties of the resulting test statistics. An empirical application to expected inflation illustrates the practical utility of the proposed methodology.