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Econometrics Procedures
 

UCM Procedure

Models with Dependent Lags

The state space form of a UCM consisting of the lags of the dependent variable is quite different from the state space forms considered so far. Let us consider an example to illustrate this situation. Consider a model that has random walk trend, two simple time-invariant regressors, and that also includes a few—for example, k—lags of the dependent variable. That is,

StartLayout 1st Row 1st Column y Subscript t 2nd Column equals 3rd Column sigma summation Underscript i equals 1 Overscript k Endscripts phi Subscript i Baseline y Subscript t minus i plus mu Subscript t Baseline plus beta 1 x Subscript 1 t plus beta 2 x Subscript 2 t plus epsilon Subscript t 2nd Row 1st Column mu Subscript t 2nd Column equals 3rd Column mu Subscript t minus 1 Baseline plus eta Subscript t EndLayout

The state space form of this augmented model can be described in terms of the state space form of a model that has random walk trend with two simple time-invariant regressors. A superscript dagger (dagger) has been added to distinguish the augmented model state space entities from the corresponding entities of the state space form of the random walk with predictors model. With this notation, the state vector of the augmented model alpha Subscript t Superscript dagger Baseline equals left bracket alpha Subscript t Superscript Super Superscript prime Superscript Baseline y Subscript t Baseline y Subscript t minus 1 Baseline ellipsis y Subscript t minus k plus 1 Baseline right bracket Superscript prime and the new state noise vector zeta Subscript t Superscript dagger Baseline equals left bracket zeta Subscript t Superscript Super Superscript prime Superscript Baseline u Subscript t Baseline 0 ellipsis 0 right bracket Superscript prime, where u Subscript t is the matrix product upper Z Subscript t Baseline zeta Subscript t. Note that the length of the new state vector is k plus normal l normal e normal n normal g normal t normal h left parenthesis alpha Subscript t Baseline right parenthesis equals k plus 4. The new system matrices, in block form, are

upper Z Subscript t Superscript dagger Baseline equals left bracket 0 0 0 0 1 ellipsis 0 right bracket comma upper T Subscript t Superscript dagger Baseline equals Start 3 By 4 Matrix 1st Row 1st Column upper T Subscript t Baseline 2nd Column 0 3rd Column ellipsis 4th Column 0 2nd Row 1st Column upper Z Subscript t plus 1 Baseline upper T Subscript t Baseline 2nd Column phi 1 3rd Column ellipsis 4th Column phi Subscript k Baseline 3rd Row 1st Column 0 2nd Column upper I Subscript k minus 1 comma k minus 1 Baseline 3rd Column Blank 4th Column 0 EndMatrix

where upper I Subscript k minus 1 comma k minus 1 is the k minus 1 dimensional identity matrix and

upper Q Subscript t Superscript dagger Baseline equals Start 3 By 3 Matrix 1st Row 1st Column upper Q Subscript t Baseline 2nd Column upper Q Subscript t Baseline upper Z Subscript t Superscript prime Baseline 3rd Column 0 2nd Row 1st Column upper Z Subscript t Baseline upper Q Subscript t Baseline 2nd Column upper Z Subscript t Baseline upper Q Subscript t Baseline upper Z Subscript t Superscript prime Baseline 3rd Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column 0 EndMatrix

Note that the T and Q matrices of the random walk with predictors model are time invariant, and in the expressions above their time indices are kept because they illustrate the pattern for more general models. The initial state vector is diffuse, with

upper P Subscript asterisk Superscript dagger Baseline equals Start 2 By 2 Matrix 1st Row 1st Column upper P Subscript asterisk Baseline 2nd Column 0 2nd Row 1st Column 0 2nd Column 0 EndMatrix comma upper P Subscript normal infinity Superscript dagger Baseline equals Start 2 By 2 Matrix 1st Row 1st Column upper P Subscript normal infinity Baseline 2nd Column 0 2nd Row 1st Column 0 2nd Column upper I Subscript k comma k Baseline EndMatrix

The parameters of this model are the disturbance variances sigma Subscript epsilon Superscript 2 and sigma Subscript eta Superscript 2, the lag coefficients phi 1 comma phi 2 comma ellipsis comma phi Subscript k Baseline, and the regression coefficients beta 1 and beta 2. As before, the regression coefficients get estimated during the state smoothing, and the other parameters are estimated by maximizing the likelihood.

Last updated: July 09, 2026