Regression variables

Regression variables $X_1, \cdots,X_k$ can be added to any state space model by extending the state with the regression coefficients ( $\gamma = \gamma_1, \cdots,\gamma_k$ ). We consider that the initial model has a state space form (SSF) identified by the state $\tilde\alpha_t$ and the system matrices $\left[ \tilde Z_t,\tilde H_t,\tilde T_t,\tilde V_t, \tilde S_t, \tilde a_{-1}, \tilde P_*,\tilde B,\tilde P_\infty \right]$ (see Ssf model ).

The vectors and the matrices of the extended model are defined in a straightforward way. We consider below fixed and variable coefficients.

State vector

The state is $\alpha_t=\begin{pmatrix}\tilde\alpha_t \ \gamma_t \end{pmatrix}$

Initialization

$a_{0} = \begin{pmatrix} \tilde a_{0} \\ 0 \end{pmatrix}$

$B = \begin{pmatrix} \tilde B\ & 0 \\ 0 & I \end{pmatrix}$

$P_*= \begin{pmatrix} \tilde P_* & 0 \\ 0 & 0 \end{pmatrix}$

$P_\infty= \begin{pmatrix} \tilde P_\infty & 0 \\ 0 & I \end{pmatrix}$

Dynamics

$T_t =\begin{pmatrix} \tilde T_t & 0 \\ 0 & I\end{pmatrix}$

Fixed (case 1) or variable (case 2) coefficients are defined by means of the innovations of the transition equation

$V_t= \begin{cases}\begin{pmatrix} \tilde V_t & 0 \\ 0 & 0\end{pmatrix} \\ \begin{pmatrix} \tilde V_t & 0 \\ 0 & \Omega_t\end{pmatrix}\end{cases}$

$S_t= \begin{cases}\begin{pmatrix} \tilde S_t & 0 \\ 0 & 0\end{pmatrix} \\ \begin{pmatrix} \tilde S_t & 0 \\ 0 & \Gamma_t\end{pmatrix}\end{cases}$

Measurement

$Z_t = \begin{pmatrix} \tilde Z_t & X_{1t} & \cdots && X_{kt}\end{pmatrix}$

Example: time varying trading days

The key point in defining time varying coefficients is the specification of the covariance matrix in the transition equation.

Experience shows that parsimonious specifications should be preferred.

As with fixed coefficients, we impose that the sum of the coefficients of all days is 0 for each period.

This is achieve by defining the covariance matrix by

$\Omega_{i,j}$

We can model trading days in different ways. We will consider below two solutions: the variables may be defined by the number of days in each period, corrected for their seasonal mean effect or their can defined using “contrasts” (differences between the number of days in the period and the number of “contrasting” days - usually Sundays).