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 (2012, 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 bold y equals left-parenthesis y 1 comma y 2 comma ellipsis comma y Subscript q Baseline right-parenthesis, which denotes the q-variate response values, and the k-dimensional vector bold x, which denotes the predictors. Suppose that the observation instances are tau 1 less-than tau 2 less-than ellipsis less-than tau Subscript n. The possibility that multiple observations are taken at a particular instance tau Subscript i is not ruled out, and the successive observation instances do not need to be regularly spaced—that is, left-parenthesis tau 2 minus tau 1 right-parenthesis does not need to equal left-parenthesis tau 3 minus tau 2 right-parenthesis. For t equals 1 comma 2 comma ellipsis comma n, suppose p Subscript t (greater-than-or-equal-to 1) denotes the number of observations recorded at instance tau Subscript t. For notational simplicity, an integer-valued secondary index t is used to index the data so that t equals 1 corresponds to tau equals tau 1, t equals 2 corresponds to tau equals tau 2, and so on. Consider the following model:

StartLayout 1st Row 1st Column bold upper Y Subscript t 2nd Column equals bold upper Z Subscript t Baseline alpha alpha Subscript t Baseline plus bold upper X Subscript t Baseline beta beta plus epsilon epsilon Subscript t Baseline 3rd Column Observation equation 2nd Row 1st Column alpha alpha Subscript t plus 1 2nd Column equals bold upper T Subscript t Baseline alpha alpha Subscript t Baseline plus bold upper W Subscript t plus 1 Baseline gamma gamma plus bold c Subscript t plus 1 Baseline plus eta eta Subscript t plus 1 Baseline 3rd Column State transition equation 3rd Row 1st Column alpha alpha 1 2nd Column equals bold c 1 plus bold upper A bold 1 delta delta plus bold upper W 1 gamma gamma plus eta eta 1 3rd Column Initial condition EndLayout

The following list describes these equations:

  • The observation equation describes the relationship between the left-parenthesis p Subscript t Baseline asterisk q right-parenthesis-dimensional response vector bold upper Y Subscript t and the unobserved vectors alpha alpha Subscript t, beta beta, and epsilon epsilon Subscript t. The q-variate responses are vertically stacked in a column to form this left-parenthesis p Subscript t Baseline asterisk q right-parenthesis-dimensional response vector bold upper Y Subscript t. The m-dimensional vectors alpha alpha Subscript t are called states, the k-dimensional vector beta beta is the regression coefficient vector associated with predictors bold x, and the left-parenthesis p Subscript t Baseline asterisk q right-parenthesis-dimensional vectors epsilon epsilon Subscript t are called the observation disturbances. The matrices bold upper Z Subscript t (of dimension left-parenthesis q asterisk p Subscript t Baseline right-parenthesis times m) and bold upper X Subscript t (of dimension left-parenthesis q asterisk p Subscript t Baseline right-parenthesis times k) correspond to the state effect and the regression effect, respectively. The elements of bold upper X Subscript t are assumed to be fully known. The states alpha alpha Subscript t and the disturbances epsilon epsilon Subscript t are random sequences. It is assumed that epsilon epsilon Subscript t is a sequence of independent, zero-mean, Gaussian random vectors with diagonal covariances, with the diagonal elements denoted by sigma Subscript t comma i Superscript 2 Baseline comma i equals 1 comma 2 comma ellipsis comma q asterisk p Subscript t Baseline.

  • The state sequence alpha alpha Subscript 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, alpha alpha Subscript t plus 1, is obtained by multiplying its previous instance, alpha alpha Subscript t, by an m-dimensional square matrix bold upper T Subscript t (called the state transition matrix) and by adding three more terms: a known nonrandom vector bold c Subscript t plus 1 (called the state input); a regression term bold upper W Subscript t plus 1 Baseline gamma gamma, where bold upper W Subscript t plus 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 eta eta Subscript t plus 1. The m-dimensional state disturbance vectors eta eta Subscript t are assumed to be independent, zero-mean, Gaussian random vectors with covariances upper Q Subscript t (not necessarily diagonal).

  • The initial condition describes the starting condition of the state evolution equation. The starting state vector alpha alpha 1 is assumed to be partially diffuse: it is the sum of a known nonrandom vector bold c 1, a mean-zero Gaussian vector eta eta 1, and the terms bold upper A bold 1 delta delta and bold upper W 1 gamma gamma. bold upper A bold 1 delta delta represents the contribution from a d-dimensional diffuse vector delta delta (a diffuse vector is a Gaussian vector with infinite covariance). The observation and state regression vectors beta beta and gamma gamma are also assumed to be diffuse. The m times d matrix bold upper A bold 1 is assumed to be completely known.

  • The observation disturbances epsilon epsilon Subscript t and the state disturbances eta eta Subscript t (for t greater-than-or-equal-to 1) are assumed to be mutually independent. Either the elements of the matrices bold upper Z Subscript t, bold upper T Subscript t, and upper Q Subscript t and the diagonal elements of the observation disturbance covariances sigma Subscript t comma i Superscript 2 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 theta theta.

  • The d-dimensional diffuse vector delta delta from the state initial condition together with the observation and state regression vectors beta beta and gamma gamma constitute the overall left-parenthesis d plus k plus g right-parenthesis-dimensional diffuse initial condition of the model. For more information, see the section Filtering, Smoothing, Likelihood, and Structural Break Detection.

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 bold upper X Subscript t and bold upper W Subscript 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 epsilon epsilon Subscript 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 bold upper T Subscript t in the state equation relates the state alpha alpha Subscript t at time t with the state alpha alpha Subscript t plus 1 at time t plus 1. In many situations, such as when the observations are taken at irregular time intervals, bold upper T Subscript t depends on information at both t and t plus 1. Therefore, it is more appropriate to denote the transition matrix as bold upper T Subscript t Superscript t plus 1. However, for simplicity, the former notation is used throughout this chapter. The same comment applies to the covariance matrix bold upper Q Subscript t of the disturbance term eta eta Subscript t.

For easy reference, Table 4 summarizes the information contained in the SSM equations.

Table 4: State Space Model: Notation

Notation Description
tau 1 comma tau 2 comma ellipsis comma tau Subscript n Baseline Distinct index values at which the observations are recorded
n Number of distinct index instances
p Subscript t Number of observations recorded at index tau Subscript t, t equals 1 comma 2 comma ellipsis comma n
q Number of response variables in the model
bold upper Y Subscript t Baseline equals left-parenthesis y Subscript t comma 1 Baseline comma y Subscript t comma 2 Baseline comma ellipsis comma y Subscript t comma p Sub Subscript t Subscript asterisk q Baseline right-parenthesis Vertically stacked vector of response values recorded at tau Subscript t
upper N equals q asterisk sigma-summation Underscript t equals 1 Overscript n Endscripts p Subscript t Total number of response values in the data set
k Number of predictor (regressor) variables in the observation equation
bold upper X Subscript t left-parenthesis p Subscript t Baseline asterisk q right-parenthesis times k matrix of predictor values recorded at tau Subscript t
beta beta k-dimensional regression vector that is associated with the predictors
epsilon epsilon Subscript t Baseline tilde upper N left-parenthesis 0 comma left-parenthesis sigma Subscript t comma 1 Superscript 2 Baseline comma ellipsis right-parenthesis right-parenthesis left-parenthesis q asterisk p Subscript t Baseline right-parenthesis-dimensional observation disturbance vector with diagonal covariance
m Dimension of the state vectors alpha alpha Subscript t
alpha alpha Subscript t m-dimensional state vector
bold upper Z Subscript t left-parenthesis q asterisk p Subscript t Baseline right-parenthesis times m matrix that is associated with alpha alpha Subscript t in the observation equation
bold upper T Subscript t m times m state transition matrix
bold c Subscript t m-dimensional state input vector
bold upper W Subscript t m times g design matrix associated with gamma gamma, the state regression vector
gamma gamma g-dimensional state regression vector
eta eta Subscript t Baseline tilde upper N left-parenthesis 0 comma upper Q Subscript t Baseline right-parenthesis m-dimensional state disturbance vector
d Dimension of the diffuse vector delta delta in the state initial condition
delta delta tilde upper N left-parenthesis 0 comma kappa normal upper Sigma right-parenthesis, kappa right-arrow normal infinity Diffuse vector in the state initial condition
bold upper A bold 1 m times d constant matrix associated with delta delta
eta eta 1 tilde upper N left-parenthesis 0 comma upper Q 1 right-parenthesis m-dimensional state disturbance vector in the initial condition
theta theta Parameter vector


Last updated: June 19, 2025