Efficient generalized cross-validation for state space models

For a signal plus noise model with a state space representation, an efficient procedure is given for obtaining the trace of the influence matrix, where the influence matrix expresses the estimated signal vector as a linear combination of the observed data. This allows an O(n) evaluation of the generalized cross-validation criterion function. Our approach is very efficient, requiring the addition of only one equation to the ordinary Kalman filter, and extends to models with linear regressors in the observation equation. Important applications are to spline smoothing in regression and to variance components models for seasonal time series.