A Quadratic Kalman Filter
39 Pages Posted: 21 Dec 2013 Last revised: 25 Nov 2014
Date Written: November 1, 2014
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.
Keywords: Non-linear filtering, non-linear smoothing, quadratic model, Kalman filter, Pseudo-maximum likelihood
JEL Classification: C32, C46, C53, C57
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