SSM Procedure

Smoothing Phase

After the filtering phase of KFS produces the one-step-ahead predictions of the response variables and the underlying state vectors, the smoothing phase of KFS produces the full-sample versions of these quantities—that is, rather than using the history up to left-parenthesis t comma i minus 1 right-parenthesis, the entire sample bold upper Y is used. The smoothing phase of KFS is a backward algorithm, which begins at t equals n and i equals q asterisk p Subscript n and goes back toward t equals 1 and i equals 1. It produces the following quantities:

Table 8: KFS: Smoothing Phase

Quantity Description
y overTilde Subscript t comma i Baseline equals normal upper E left-parenthesis y Subscript t comma i Baseline vertical-bar bold upper Y right-parenthesis Interpolated response value
upper F overTilde Subscript t comma i Baseline equals normal upper V normal a normal r left-parenthesis y Subscript t comma i Baseline vertical-bar bold upper Y right-parenthesis Variance of the interpolated response value
ModifyingAbove alpha alpha With tilde Subscript t Baseline equals normal upper E left-parenthesis alpha alpha Subscript t Baseline vertical-bar bold upper Y right-parenthesis Full-sample estimate of the state vector
bold upper P overTilde Subscript t Baseline equals normal upper C normal o normal v left-parenthesis alpha alpha Subscript t Baseline vertical-bar bold upper Y right-parenthesis Covariance of ModifyingAbove alpha alpha With tilde Subscript t
left-parenthesis ModifyingAbove delta delta With caret ModifyingAbove beta beta With caret ModifyingAbove gamma gamma With caret right-parenthesis Superscript prime Baseline equals bold upper S Subscript n comma p Sub Subscript n Subscript Superscript negative 1 Baseline bold b Subscript n comma p Sub Subscript n Subscript Full-sample estimates of delta delta, beta beta, and gamma gamma
bold upper S Subscript n comma p Sub Subscript n Subscript Superscript negative 1 Covariance of left-parenthesis ModifyingAbove delta delta With caret ModifyingAbove beta beta With caret ModifyingAbove gamma gamma With caret right-parenthesis Superscript prime


Note that if y Subscript t comma i is not missing, then y overTilde Subscript t comma i Baseline equals normal upper E left-parenthesis y Subscript t comma i Baseline vertical-bar bold upper Y right-parenthesis equals y Subscript t comma i and upper F overTilde Subscript t comma i Baseline equals normal upper V normal a normal r left-parenthesis y Subscript t comma i Baseline vertical-bar bold upper Y right-parenthesis equals 0 because y Subscript t comma i is completely known, given bold upper Y. Therefore, y overTilde Subscript t comma i provides nontrivial information only when y Subscript t comma i is missing—in which case y overTilde Subscript t comma i represents the best estimate of y Subscript t comma i based on the available data. The full-sample estimates of components that are specified in the model equations are based on the corresponding linear combinations of ModifyingAbove alpha alpha With tilde Subscript t. Similarly, their standard errors are computed by using appropriate functions of bold upper P overTilde Subscript t.

If the filtering process remains uninitialized until the end of the sample (that is, if bold upper S Subscript n comma p Sub Subscript n Subscript is not invertible), some linear combinations of delta delta, beta beta, and gamma gamma are not estimable. This, in turn, implies that some linear combinations of alpha alpha Subscript t are also inestimable. These inestimable quantities are reported as missing. For more information about the estimability of the state effects, see Selukar (2010).

Last updated: June 19, 2025