Working Paper Series no. 486: A Quadratic Kalman Filter

Abstract

We propose a new filtering and smoothing technique for non-linear state-space models. Observed variables are quadratic functions of latent factors following a Gaussian VAR. Stacking the vector of factors with its vectorized outer-product, we form an augmented state vector whose first two conditional moments are known in closed-form. We also provide analytical formulae for the unconditional moments of this augmented vector. Our new quadratic Kalman filter (Qkf) exploits these properties to formulate fast and simple filtering and smoothing algorithms. A first simulation study emphasizes that the Qkf outperforms the extended and unscented approaches in the filtering exercise showing up to 70% RMSEs improvement of filtered values. Second, we provide evidence that Qkf-based maximum-likelihood estimates of model parameters always possess lower bias or lower RMSEs that the alternative estimators.

Alain Monfort, Jean-Paul Renne and Guillaume Roussellet
May 2014

Classification JEL : C32, C46, C53, C57

Keywords : non-linear filtering, non-linear smoothing, quadratic model, Kalman filter, pseudo-maximum likelihood.