Likelihood, Filtering, and Smoothing

The Kalman filter and smoother (KFS) algorithm is the main computational tool for using SSM for data analysis. This subsection briefly describes the basic quantities generated by this algorithm and their relationship to the output generated by the SSM procedure. For proper treatment of SSMs with a diffuse initial condition or when regression variables are present, a modified version of the traditional KFS, called diffuse Kalman filter and smoother (DKFS), is needed. A good discussion of the different variants of the traditional and diffuse KFS can be found in Durbin and Koopman (2001). The DKFS implemented in the SSM procedure closely follows the treatment in de Jong and Chu-Chun-Lin (2003). Additional details can be found in these references.

The state space model equations (see the section State Space Model and Notation) imply that the combined response data vector has a Gaussian probability distribution. This probability distribution is proper if , the dimension of the diffuse vector in the initial condition, is 0 and if , the number of regression variables in the observation equation, is also 0 (the regression parameter is also treated as a diffuse vector). Otherwise, this probability distribution is improper. The KFS algorithm is a combination of two iterative phases: a forward pass through the data, called filtering, and a backward pass through the data, called smoothing, that uses the quantities generated during filtering. One of the advantages of using the SSM formulation to analyze the time series data is its ability to handle the missing values in the response variables. The KFS algorithm appropriately handles the missing values in . For additional information about how PROC SSM handles missing values, see the section Missing Values.

Filtering Pass

The filtering pass sequentially computes the quantities shown in Table 27.5 for and .

Table 27.5: KFS: Filtering Phase

Quantity

Description

One-step-ahead prediction of the response values

One-step-ahead prediction of the state vector

Covariance of

-dimensional vector

-dimensional symmetric matrix

Estimate of and by using the data up to

Covariance of

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.

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.

Likelihood Computation and Model Fitting Phase

In view of the Gaussian nature of the response vector, the likelihood of , , can be computed by using the prediction-error decomposition, which leads to the formula

where , denotes the determinant of , and denotes the transpose of the column vector . In the preceding formula, the terms that are associated with the missing response values are excluded and denotes the total number of nonmissing response values in the sample. If is not invertible, then a generalized inverse is used in place of , and is computed based on the nonzero eigenvalues of . Moreover, in this case . When has a proper distribution (that is, when ), the terms that involve and are absent and the preceding likelihood is proper. Otherwise, it is called the diffuse likelihood or the restricted likelihood.

When the model specification contains any unknown parameters , they are estimated by maximizing the preceding likelihood function. This is done by using a nonlinear optimization process that involves repeated evaluations of at different values of . The maximum likelihood (ML) estimate of is denoted by . When the restricted likelihood is used for computing , the estimate is called the restricted maximum likelihood (REML) estimate. Approximate standard errors of are computed by taking the square root of the diagonal elements of its (approximate) covariance matrix. This covariance is computed as , where is the Hessian (the matrix of the second-order partials) of evaluated at the optimum .

Let denote the dimension of the parameter vector . After the parameter estimation is completed, a table, called Likelihood Computation Summary is printed. It summarizes the likelihood calculations at as shown in Table 27.6.

Table 27.6: Likelihood Computation Summary

Quantity

Formula

Nonmissing response values used

Estimated parameters

Initialized diffuse state elements

Normalized residual sum of squares

Full log likelihood

In addition, the Likelihood Based Information Criteria table reports a variety of information-based criteria, which are functions of , , and . Table 27.7 summarizes the reported information criteria in smaller-is-better form:

Table 27.7: Information Criteria

Criterion

Formula

Reference

AIC

Akaike (1974)

AICC

Hurvich and Tsai (1989)

Burnham and Anderson (1998)

HQIC

Hannan and Quinn (1979)

BIC

Schwarz (1978)

CAIC

Bozdogan (1987)

Forecasting Phase

After the model-fitting phase, the filtering process is repeated again to produce the model-based one-step-ahead response variable forecasts (), residuals (), and their standard errors (). In addition, one-step-ahead forecasts of the components that are specified in the MODEL statements, and any other user-defined linear combinations of , are also produced. These forecasts are set to missing as long as the index (that is, until the filtering process is initialized). If the filtering process remains uninitialized, then all the quantities that are related to the one-step-ahead forecast (such as and ) are reported as missing. When the fitted model is appropriate, the one-step-ahead residuals form a sequence of uncorrelated normal variates. This fact can be used during model diagnostic process.

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 , the entire sample is used. The smoothing phase of KFS is a backward algorithm, which begins at and and goes back toward and . It produces the following quantities:

Table 27.8: KFS: Smoothing Phase

Quantity

Description

Interpolated response value

Variance of the interpolated response value

Full-sample estimate of the state vector

Covariance of

Full-sample estimate of and

Covariance of