SSM Procedure

The filtering pass sequentially computes the quantities shown in Table 5 for and .

Table 5: KFS: Filtering Phase

Quantity	Description
	One-step-ahead prediction of the response values
	One-step-ahead prediction residuals
	Variance of the one-step-ahead prediction
	One-step-ahead prediction of the state vector
	Covariance of
	-dimensional vector
	-dimensional symmetric matrix
	Estimates of , , and by using the data up to
	Covariance of
	-dimensional symmetric matrix
	needed in the marginal likelihood computation

Here the notation denotes the conditional expectation of given the history up to the index : . Similarly denotes the corresponding conditional variance. The quantity is set to missing whenever is missing. Note that 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, are called one-step-ahead predictions (and are called one-step-ahead residuals) throughout this document. In the diffuse case, the conditional expectations must be appropriately interpreted. The vector and the matrix contain some accumulated quantities that are needed for the estimation of , , and . Of course, when (the nondiffuse case), these quantities are not needed. In the diffuse case, because the matrix is sequentially accumulated (starting at ), it might not be invertible until some . The filtering process is called initialized after . 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 for some other reason. In the diffuse case if the marginal likelihood is to be computed (see the section Likelihood Computation and Model-Fitting Phase), an additional matrix () is computed at each step by sequential accumulation.

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 are estimated by maximum likelihood. This requires repeated evaluation of the likelihood at different trial values of . After 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.