The SSM Procedure

Filtering Pass

The filtering pass sequentially computes the quantities shown in Table 27.5 for $t = 1, 2, \ldots , n$ and $i = 1, 2, \ldots , q*p_{t}$ .

Table 27.5: KFS: Filtering Phase

Quantity	Description
$\hat{y}_{t, i} = \mr {E}( y_{t, i} \| y_{t, i-1}, \ldots y_{t, 1}, \mb {Y}_{t-1}, \ldots , \mb {Y}_{1} )$	One-step-ahead prediction of the response values
$\nu _{t,i} = y_{t, i} - \hat{y}_{t, i}$	One-step-ahead prediction residuals
$F_{t, i} = \mr {Var}( y_{t, i} \| y_{t, i-1}, \ldots y_{t, 1}, \mb {Y}_{t-1}, \ldots , \mb {Y}_{1} )$	Variance of the one-step-ahead prediction
$\hat{\pmb {\alpha }}_{t, i} = \mr {E}( \pmb {\alpha }_{t} \| y_{t, i-1}, \ldots y_{t, 1}, \mb {Y}_{t-1}, \ldots , \mb {Y}_{1} )$	One-step-ahead prediction of the state vector
$\mb {P}_{t, i} = \mr {Cov}( \pmb {\alpha }_{t} \| y_{t, i-1}, \ldots y_{t, 1}, \mb {Y}_{t-1}, \ldots , \mb {Y}_{1} )$	Covariance of $\hat{\pmb {\alpha }}_{t, i}$
$\mb {b}_{t, i}$	$(d + k+ g)$ -dimensional vector
$\mb {S}_{t, i}$	$(d + k + g)$ -dimensional symmetric matrix
$\left( \hat{\pmb {\delta }} \; \; \hat{\pmb {\beta }} \; \; \hat{\pmb {\gamma }} \right)_{t,i}^{} = \mb {S}_{t, i}^{-1}\mb {b}_{t, i}$	Estimates of $\pmb {\delta }$ , $\pmb {\beta }$ , and $\pmb {\gamma }$ by using the data up to $(t,i)$
$\mb {S}_{t, i}^{-1}$	Covariance of $\left( \hat{\pmb {\delta }} \; \; \hat{\pmb {\beta }} \; \; \hat{\pmb {\gamma }} \right)_{t,i}^{}$

Here the notation $\mr {E}( y_{t, i} | y_{t, i-1}, \ldots y_{t, 1}, \mb {Y}_{t-1}, \ldots , \mb {Y}_{1} )$ denotes the conditional expectation of $y_{t, i}$ given the history up to the index $(t, i-1)$ : $(y_{t, i-1}, \ldots y_{t, 1}, \mb {Y}_{t-1}, \ldots , \mb {Y}_{1})$ . Similarly $\mr {Var}( y_{t, i} | y_{t, i-1}, \ldots y_{t, 1}, \mb {Y}_{t-1}, \ldots , \mb {Y}_{1} )$ denotes the corresponding conditional variance. The quantity $\nu _{t,i} = y_{t, i} - \hat{y}_{t, i}$ is set to missing whenever $y_{t,i}$ is missing. Note that $\hat{y}_{t, i}$ are one-step-ahead forecasts only when the model has only one response variable and the data are a time series; in all other cases it is more appropriate to call them one-measurement-ahead forecasts (since the next measurement might be at the same time point). Despite this, $\hat{y}_{t, i}$ are called one-step-ahead predictions (and $\nu _{t,i}$ are called one-step-ahead residuals) throughout this document. In the diffuse case, the conditional expectations must be appropriately interpreted. The vector $\mb {b}_{t, i}$ and the matrix $\mb {S}_{t, i}$ contain some accumulated quantities that are needed for the estimation of $\pmb {\delta }$ , $\pmb {\beta }$ , and $\pmb {\gamma }$ . Of course, when $(d+k+g) = 0$ (the nondiffuse case), these quantities are not needed. In the diffuse case, because the matrix $\mb {S}_{t, i}$ is sequentially accumulated (starting at $t=1, i=1$ ), it might not be invertible until some $t= t_{*}, i=i_{*}$ . The filtering process is called initialized after $t= t_{*}, i=i_{*}$ . In some situations, this initialization might not happen even after the entire sample is processed—that is, the filtering process remains uninitialized. This can happen if the regression variables are collinear or if the data are not sufficient to estimate the initial condition $\pmb {\delta }$ for some other reason.

The filtering process is used for a variety of purposes. One important use of filtering is to compute the likelihood of the data. In the model-fitting phase, the unknown model parameters $\pmb {\theta }$ are estimated by maximum likelihood. This requires repeated evaluation of the likelihood at different trial values of $\pmb {\theta }$ . After $\pmb {\theta }$ is estimated, it is treated as a known vector. The filtering process is used again with the fitted model in the forecasting phase, when the one-step-ahead forecasts and residuals based on the fitted model are provided. In addition, this filtering output is needed by the smoothing phase to produce the full-sample component estimates and for the structural break analysis.