The 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 $(t, i-1)$ , the entire sample $\mb {Y}$ is used. The smoothing phase of KFS is a backward algorithm, which begins at $t = n$ and $i = q * p_{n}$ and goes back toward $t=1$ and $i=1$ . It produces the following quantities:

Table 27.8: KFS: Smoothing Phase

Quantity	Description
$\tilde{y}_{t, i} = \mr {E}( y_{t, i} \| \mb {Y} )$	Interpolated response value
$\tilde{F}_{t, i} = \mr {Var}( y_{t, i} \| \mb {Y} )$	Variance of the interpolated response value
$\tilde{\pmb {\alpha }}_{t} = \mr {E}( \pmb {\alpha }_{t} \| \mb {Y} )$	Full-sample estimate of the state vector
$\tilde{\mb {P}}_{t} = \mr {Cov}( \pmb {\alpha }_{t} \| \mb {Y} )$	Covariance of $\tilde{\pmb {\alpha }}_{t}$
$\left( \hat{\pmb {\delta }} \; \; \hat{\pmb {\beta }} \; \; \hat{\pmb {\gamma }} \right)^{} = \mb {S}_{n, p_{n}}^{-1}\mb {b}_{n, p_{n}}$	Full-sample estimates of $\pmb {\delta }$ , $\pmb {\beta }$ , and $\pmb {\gamma }$
$\mb {S}_{n, p_{n}}^{-1}$	Covariance of $\left( \hat{\pmb {\delta }} \; \; \hat{\pmb {\beta }} \; \; \hat{\pmb {\gamma }} \right)^{}$

Note that if ${y}_{t, i}$ is not missing, then $\tilde{y}_{t, i} = \mr {E}( y_{t, i} | \mb {Y} ) = {y}_{t, i}$ and $\tilde{F}_{t, i} = \mr {Var}( y_{t, i} | \mb {Y} ) = 0$ because ${y}_{t, i}$ is completely known, given $\mb {Y}$ . Therefore, $\tilde{y}_{t, i}$ provides nontrivial information only when ${y}_{t, i}$ is missing—in which case $\tilde{y}_{t, i}$ represents the best estimate of ${y}_{t, i}$ 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 $\tilde{\pmb {\alpha }}_{t}$ . Similarly, their standard errors are computed by using appropriate functions of $\tilde{\mb {P}}_{t}$ .

If the filtering process remains uninitialized until the end of the sample (that is, if $\mb {S}_{n, p_{n}}$ is not invertible), some linear combinations of $\pmb {\delta }$ , $\pmb {\beta }$ , and $\pmb {\gamma }$ are not estimable. This, in turn, implies that some linear combinations of $\pmb {\alpha }_{t}$ are also inestimable. These inestimable quantities are reported as missing. For more information about the estimability of the state effects, see Selukar (2010).