VARDLM Procedure

@VARDLM generates the state-space system matrices for an N-variable vector autoregressive process whose lag coefficients are provided in a VECTOR[RECT]. You can also use it to set up the framework (proper dimensions, all the fixed values put in place) based upon an input number of lags (LAGS option) to fill out the non-fixed elements later.

@VARDLM( options ) lagMats

The simplest state-space (dynamic linear model) representation of

\begin{equation} {\bf{Y}}_t = \Phi _1 {\bf{Y}}_{t - 1} + \Phi _2 {\bf{Y}}_{t - 2} + \ldots + {\rm{ }}\Phi _p {\bf{Y}}_{t - p} + {\bf{U}}_t \end{equation}

has state-vector

\begin{equation} {\bf{X}}_{\rm{t}} {\rm{ = }}\left[ {\begin{array}{*{20}c} {{\bf{Y}}_t } \\ {{\bf{Y}}_{t - 1} } \\ \vdots \\ {{\bf{Y}}_{t - p + 1} } \\ \end{array}} \right] \end{equation}

which will be a vector of length \(Np\) blocked by lag and

\begin{equation} {\bf{A}} = \left[ {\begin{array}{*{20}c} {\Phi _1 } & {\Phi _2 } & \ldots & {\Phi _p } \\ {I_N } & {0_N } & \ldots & {0_N } \\ {0_N } & \ddots & {0_N } & {0_N } \\ {0_N } & {0_N } & {I_N } & {0_N } \\ \end{array}} \right] \end{equation}

\begin{equation} {\bf{F}} = \left[ {\begin{array}{*{20}c} {I_N } \\ {0_N } \\ \vdots \\ {0_N } \\ \end{array}} \right] \end{equation}

with \({\bf{C}} = {\bf{F}}\). All the submatrices in \(\bf{A}\) and \(\bf{F}\) are dimension \(N \times N\).

Note that the only thing that depends upon the free coefficients is the first block of rows in \({\bf{A}}\). In this form, there is no "measurement" error as the only shocks (\(\bf{u}\)) are in the state equation.

Parameters

LagMats (optional, input)

VECTOR[RECT] of \(N \times N\) VAR lag coefficient matrices on consecutive lags

Options

LAGS=number of lags in model (input)

Can be used as an alternative to the LagMats parameter to set up the framework of the matrices if you don't yet know the coefficients.

A=DLM system "A" matrix (output; input/output if used with POKE)

F=DLM system "F" matrix (output)

C=DLM system "C" matrix (output)

POKE/[NOPOKE]

If POKE, assumes that the A option matrix has the framework set up and only changes the first block of rows to insert the coefficients out of LagMats This avoids repeating the set up of the fixed elements.