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).