State space framework

State space model

The general linear gaussian state-space model can be written in many different ways. The measurement equation and the state equation considered in JD+ 3.0 are presented below.

$y_t = Z_t \alpha_t + \epsilon_t,\quad \epsilon_t \sim N\left(0, \sigma^2 H_t\right), t \gt 0$ $\alpha_{t+1} = T_t \alpha_t + \mu_t, \quad \mu_t \sim N \left(0, \sigma^2 V_t \right), t \ge 0$

$y_t$ is the observation at period t, $\alpha_t$ is the state vector. $\epsilon_t, \mu_t$ are assumed to be serially independent at all time points and independent between them at all time points.

The residuals of the state equation will be modelled as

$\mu_t = S_t \xi_t, \quad \xi_t \sim N\left( 0, \sigma^2 I\right)$

In other words, $V_t=S_t S_t’$

The initial ($\equiv t=0$) conditions of the filter are defined as follows:

$\alpha_{0} = a_{0} + B\delta + \mu_{0}, \quad \delta \sim N\left(0, \kappa \sigma^2 I \right),\: \mu_{0} \sim N\left(0,\sigma^2 P_*\right)$

where $\kappa$ is arbitrary large. $P_*$ is the variance of the stationary part of the initial state vector and $BB’= P_\infty$ models the diffuse part.

The definition used in JD+ is quasi-identical to that of Durbin and Koopman[1].

In summary, up to the scaling factor $\sigma^2$, the model is completely defined by the following quantities (possible default values are indicated in brackets):

$\mathbf{Z_t}, \mathbf{H_t} [=0]$ $\mathbf{T_t}, \mathbf{V_t} [=S_t S_t'], \mathbf{S_t} [=Cholesky(V)]$ $\mathbf{a_{-1}}[=0], \mathbf{P_*} [=0], \mathbf{B} [=0], \mathbf{P_\infty} [=BB']$

Bibliography

[1] DURBIN J. AND KOOPMAN S.J. (2012): “Time Series Analysis by State Space Methods”, second edition. Oxford University Press.