Fast filtering and smoothing for multivariate state space models (Q62653)

A multivariate Gaussian linear state space model is considered defined by NEWLINE\[NEWLINEy_t=z_t\alpha_t+\varepsilon_t,\;\alpha_{t+1}=T_t\alpha_t+R_t\eta_t,NEWLINE\]NEWLINE where \(Y_n=(y_1,\dots,y_n)\) are observations, \(\alpha_t\) are unobservable, \(\varepsilon_t\sim N(0,H_t)\) and \(\eta_t\sim N(0,Q_t)\) are independent, and \(z_t\), \(T_t\) and \(R_t\) are fixed matrices. The initial distribution of \(\alpha_1\) is \(N(a,P)\), where \(P=\kappa P_\infty+P_*\) and \(\kappa\to\infty\) (diffuse prior). A general construction of Kalman filter smoother estimators for NEWLINE\[NEWLINE\hat\varepsilon_t=E(\varepsilon_t |Y_n),\;\hat\eta_t=E(\eta_t |Y_n),\;\hat\alpha_t=E(\alpha_t |Y_n)NEWLINE\]NEWLINE is described. The authors propose a new algorithm for the computation of the estimators based on a univariate representation of the model in the case when \(H_t\) is a diagonal matrix. The main motivation of this approach is computational efficiency. Maximum likelihood estimation of the model parameters is discussed. Applications to vector spline smoothing and bid-ask spreads modelling are considered.