The SSM Procedure

State Space Model and Notation

The (linear) state space model is described in the literature in a few different ways and with varying degree of generality. The description given in this section loosely follows the description given in Durbin and Koopman (2001, chap. 6, sec. 4). This formulation of SSM is quite general; in particular, it includes nonstationary SSMs with time-varying system matrices and state equations with a diffuse initial condition (these terms are defined later in this subsection).

Suppose that observations are collected in a sequential fashion (indexed by a numeric variable $\tau$ ) on some variables: the vector $\mb {y} = (y_{1}, y_{2}, \ldots , y_{q})$ , which denotes the $q$ -variate response values, and the $k$ -dimensional vector $\mb {x}$ , which denotes the predictors. Suppose that the observation instances are $\tau _{1} < \tau _{2} < \ldots < \tau _{n}$ . The possibility that multiple observations are taken at a particular instance $\tau _{i}$ is not ruled out, and the successive observation instances do not need to be regularly spaced—that is, $(\tau _{2} - \tau _{1})$ does not need to equal $(\tau _{3} - \tau _{2})$ . For $t = 1, 2, \ldots , n$ , suppose $p_{t}$ ( $\geq 1$ ) denotes the number of observations recorded at instance $\tau _{t}$ . For notational simplicity, an integer-valued secondary index $t$ is used to index the data so that $t = 1$ corresponds to $\tau = \tau _{1}$ , $t = 2$ corresponds to $\tau = \tau _{2}$ , and so on. Consider the following model:

$\begin{equation*} \begin{aligned} \mb {Y}_{t} & = \mb {Z}_{t} \pmb {\alpha }_{t} + \mb {X}_{t} \pmb {\beta } + \pmb {\epsilon }_{t} & \qquad \text {Observation equation} \\ \pmb {\alpha }_{t+1} & = \mb {T}_{t} \pmb {\alpha }_{t} + \mb {W}_{t+1} \pmb {\gamma } + \mb {c}_{t+1} + \pmb {\eta }_{t+1} & \qquad \text {State transition equation} \\ \pmb {\alpha }_{1} & = \mb {c}_{1} + \mb {A_{1}} \pmb {\delta } + \mb {W}_{1} \pmb {\gamma } + \pmb {\eta }_{1} & \qquad \text {Initial condition} \end{aligned}\end{equation*}$

The following list describes these equations:

The observation equation describes the relationship between the $(p_{t}*q)$ -dimensional response vector $\mb {Y}_{t}$ and the unobserved vectors $\pmb {\alpha }_{t}$ , $\pmb {\beta }$ , and $\pmb {\epsilon }_{t}$ . The $q$ -variate responses are vertically stacked in a column to form this $(p_{t}*q)$ -dimensional response vector $\mb {Y}_{t}$ . The $m$ -dimensional vectors $\pmb {\alpha }_{t}$ are called states, the $k$ -dimensional vector $\pmb {\beta }$ is the regression coefficient vector associated with predictors $\mb {x}$ , and the $(p_{t}*q)$ -dimensional vectors $\pmb {\epsilon }_{t}$ are called the observation disturbances. The matrices $\mb {Z}_{t}$ (of dimension $(q*p_{t}) \times m$ ) and $\mb {X}_{t}$ (of dimension $(q*p_{t}) \times k$ ) correspond to the state effect and the regression effect, respectively. The elements of $\mb {X}_{t}$ are assumed to be fully known. The states $\pmb {\alpha }_{t}$ and the disturbances $\pmb {\epsilon }_{t}$ are random sequences. It is assumed that $\pmb {\epsilon }_{t}$ is a sequence of independent, zero-mean, Gaussian random vectors with diagonal covariances, with the diagonal elements denoted by $\sigma ^{2}_{t,i}, i = 1, 2, \ldots , q*p_{t}$ .
The state sequence $\pmb {\alpha }_{t}$ is assumed to follow a Markovian structure described by the state transition equation and the associated initial condition.
The state transition equation postulates that a new instance of the state, $\pmb {\alpha }_{t+1}$ , is obtained by multiplying its previous instance, $\pmb {\alpha }_{t}$ , by an $m$ -dimensional square matrix $\mb {T}_{t}$ (called the state transition matrix) and by adding three more terms: a known nonrandom vector $\mb {c}_{t+1}$ (called the state input); a regression term $\mb {W}_{t+1} \pmb {\gamma }$ , where $\mb {W}_{t+1}$ is an $m \times g$ -dimensional design matrix with fully known elements and $\gamma$ is the $g$ -dimensional regression vector; and a random disturbance vector $\pmb {\eta }_{t+1}$ . The $m$ -dimensional state disturbance vectors $\pmb {\eta }_{t}$ are assumed to be independent, zero-mean, Gaussian random vectors with covariances $Q_{t}$ (not necessarily diagonal).
The initial condition describes the starting condition of the state evolution equation. The starting state vector $\pmb {\alpha }_{1}$ is assumed to be partially diffuse: it is the sum of a known nonrandom vector $\mb {c}_{1}$ , a mean-zero Gaussian vector $\pmb {\eta }_{1}$ , and the terms $\mb {A_{1}} \pmb {\delta }$ and $\mb {W}_{1} \pmb {\gamma }$ . $\mb {A_{1}} \pmb {\delta }$ represents the contribution from a $d$ -dimensional diffuse vector $\pmb {\delta }$ (a diffuse vector is a Gaussian vector with infinite covariance). The observation and state regression vectors $\pmb {\beta }$ and $\pmb {\gamma }$ are also assumed to be diffuse. The $m \times d$ matrix $\mb {A_{1}}$ is assumed to be completely known.
The observation disturbances $\pmb {\epsilon }_{t}$ and the state disturbances $\pmb {\eta }_{t}$ (for $t \geq 1$ ) are assumed to be mutually independent. Either the elements of the matrices $\mb {Z}_{t}$ , $\mb {T}_{t}$ , and $Q_{t}$ and the diagonal elements of the observation disturbance covariances $\sigma ^{2}_{t,i}$ are assumed to be completely known, or some of them can be functions of a small set of unknown parameters (to be estimated from the data). Suppose that this unknown set of parameters is denoted by $\pmb {\theta }$ .
The $d$ -dimensional diffuse vector $\pmb {\delta }$ from the state initial condition together with the observation and state regression vectors $\pmb {\beta }$ and $\pmb {\gamma }$ constitute the overall $(d+k+g)$ -dimensional diffuse initial condition of the model. See the section Filtering, Smoothing, Likelihood, and Structural Break Detection for more information.

Although this description of the state space model might appear involved, it conveniently covers many variants of the SSMs that are encountered in practice and precisely describes the most general case that can be handled by the SSM procedure. An important restriction about the preceding description of the model formulation is that it assumes that the matrices $\mb {X}_{t}$ and $\mb {W}_{t}$ that appear in the observation equation and the state equation respectively are free of unknown parameters and that the covariances of the observation disturbances $\pmb {\epsilon }_{t}$ are diagonal. In most practical situations, the model under consideration can be easily reformulated to a statistically equivalent form that conforms to this restriction.

Note: The transition matrix $\mb {T}_{t}$ in the state equation relates the state $\pmb {\alpha }_{t}$ at time $t$ with the state $\pmb {\alpha }_{t+1}$ at time $t+1$ . In many situations, such as when the observations are taken at irregular time intervals, $\mb {T}_{t}$ depends on information at both $t$ and $t+1$ . Therefore, it is more appropriate to denote the transition matrix as $\mb {T}_{t}^{t+1}$ . However, for simplicity, the former notation is used throughout this chapter. The same comment applies to the covariance matrix $\mb {Q}_{t}$ of the disturbance term $\pmb {\eta }_{t}$ .

For easy reference, Table 27.4 summarizes the information contained in the SSM equations.

Table 27.4: State Space Model: Notation

Notation	Description
$\tau _{1}, {\tau _2}, \ldots , \tau _{n}$	Distinct index values at which the observations are recorded
$n$	Number of distinct index instances
$p_{t}$	Number of observations recorded at index $\tau _{t}$ , $t = 1, 2, \ldots , n$
$q$	Number of response variables in the model
$\mb {Y}_{t} = ( y_{t,1}, y_{t,2}, \ldots , y_{t, p_{t}*q} )$	Vertically stacked vector of response values recorded at $\tau _{t}$
$N = q * \sum _{t = 1}^{n} p_{t}$	Total number of response values in the data set
$k$	Number of predictor (regressor) variables in the observation equation
$\mb {X}_{t}$	$(p_{t}*q) \times k$ matrix of predictor values recorded at $\tau _{t}$
$\pmb {\beta }$	$k$ -dimensional regression vector that is associated with the predictors
$\pmb {\epsilon }_{t} \sim N(0, (\sigma ^{2}_{t,1}, \ldots ) )$	$(q*p_{t})$ -dimensional observation disturbance vector with diagonal covariance
$m$	Dimension of the state vectors $\pmb {\alpha }_{t}$
$\pmb {\alpha }_{t}$	$m$ -dimensional state vector
$\mb {Z}_{t}$	$(q*p_{t}) \times m$ matrix that is associated with $\pmb {\alpha }_{t}$ in the observation equation
$\mb {T}_{t}$	$m \times m$ state transition matrix
$\mb {c}_{t}$	$m$ -dimensional state input vector
$\mb {W}_{t}$	$m \times g$ design matrix associated with $\pmb {\gamma }$ , the state regression vector
$\pmb {\gamma }$	$g$ -dimensional state regression vector
$\pmb {\eta }_{t} \sim N(0, Q_{t} )$	$m$ -dimensional state disturbance vector
$d$	Dimension of the diffuse vector $\pmb {\delta }$ in the state initial condition
$\pmb {\delta } \sim N(0, \kappa \Sigma )$ , $\kappa \rightarrow \infty$	Diffuse vector in the state initial condition
$\mb {A_{1}}$	$m \times d$ constant matrix associated with $\pmb {\delta }$
$\pmb {\eta }_{1} \sim N(0, Q_{1} )$	$m$ -dimensional state disturbance vector in the initial condition
$\pmb {\theta }$	Parameter vector