A Quadratic Kalman Filter

Journal of Econometrics

Alain Monfort, Jean-Paul Renne, Guillaume Roussellet


July 1, 2015

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 simulation study first emphasizes that the Qkf outperforms the extended and unscented approaches in the filtering exercise showing up to 70% RMSEs improvement of filtered values. Second, it provides evidence that Qkf-based maximum-likelihood estimates of model parameters always possess lower bias or lower RMSEs than the alternative estimators.

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