The SSM Procedure

Trend Models for Regular Data

These models are applicable when the data type is either regular or regular with replication. A good reference for these models is Harvey (1989).

Random Walk Trend

This model provides a trend pattern in which the level of the curve changes with time. The rapidity of this change is inversely proportional to the disturbance variance $\sigma ^2$ that governs the underlying state. It can be described as $\mb {Z} \alpha _{t}$ , where $Z= (1)$ and the (one-dimensional) state $\alpha _{t}$ follows a random walk:

$\alpha _{t+1} = \alpha _{t} + \eta _{t+1}, \; \; \eta _{t} \sim N(0, \sigma ^2)$

Here $\mb {T} = 1$ and $\mb {Q} = \sigma ^2$ . The initial condition is fully diffuse. Note that if $\sigma ^{2} = 0$ , the resulting trend is a fixed constant.

Local Linear Trend

This model provides a trend pattern in which both the level and the slope of the curve change with time. This variation in the level and the slope is controlled by two parameters: $\sigma ^{2}_{1}$ controls the level variation, and $\sigma ^{2}_{2}$ controls the slope variation. If $\sigma ^{2}_{1} = 0$ , the resulting trend is called an integrated random walk. If both $\sigma ^{2}_{1} = 0$ and $\sigma ^{2}_{2} = 0$ , then the resulting model is the deterministic linear time trend. Here $\mb {Z}= (1 \; 0)$ , $\mb {T}= (1 \; 1,\; 0\; 1)$ , and $\mb {Q} = \mr {Diag} (\sigma ^{2}_{1}, \; \sigma ^{2}_{2})$ . The initial condition is fully diffuse.

Damped Local Linear Trend

This trend pattern is similar to the local linear trend pattern. However, in the DLL trend the slope follows a first-order autoregressive model, whereas in the LL trend the slope follows a random walk. The autoregressive parameter or the damping factor, $\phi$ , must lie between 0.0 and 1.0, which implies that the long-run forecast according to this pattern has a slope that tends to 0. Here $\mb {Z}= (1 \; 0)$ , $\mb {T}= (1 \; 1,\; 0\; \phi )$ , and $\mb {Q} = \mr {Diag}(\sigma ^{2}_{1}, \; \sigma ^{2}_{2})$ . The initial condition is partially diffuse with $\mb {Q_{1}} = \mr {Diag}(0, \; \sigma ^{2}_{2}/(1-\phi *\phi ))$ .

ARIMA Trend

This section describes the state space form for a component that follows an ARIMA(p,d,q) ${\times }$ (P,D,Q) $_{\mi {s}}$ model. The notation for ARIMA models is explained in the TREND statement.

First the state space form for the stationary case—that is, when $d=0$ and $D=0$ , is explained. A number of alternate state space forms are possible in this case; the one described here is based on Jones (1980). With slight abuse of notation, let $p = p + s *P$ denote the effective autoregressive order, and let $q = q + s Q$ denote the effective moving average order of the model. Similarly, let $\phi$ be the effective autoregressive polynomial, and let $\theta$ be the effective moving average polynomial in the backshift operator with coefficients $\phi _{1}, \; \ldots , \; \phi _{p}$ and $\theta _{1}, \; \ldots , \; \theta _{q}$ , obtained by multiplying the respective nonseasonal and seasonal factors. Then, a random sequence $\xi _ t$ that follows an ARMA(p,q) ${\times }$ (P,Q) $_{\mi {s}}$ model with a white noise sequence $a_ t$ has a state space form with state vector of size $m = \max (p, q+1)$ . The system matrices are as follows: $\mb {Z} = \left[1 \; 0 \; \ldots \; 0 \right]$ , and the transition matrix $\mb {T}$ , in a blocked form, is given by

$\mb {T} = \left[ \begin{tabular}{cc} $0$ & $I_{m-1}$ \\ $\phi _{m}$ \; \ldots & $ \phi _1$ \end{tabular} \right]$

where $\phi _{i} = 0$ if $i > p$ and $I_{m-1}$ is an $(m-1)$ dimensional identity matrix. The covariance of the state disturbance matrix $\mb {Q} = \sigma ^{2} \psi \psi ^{}$ , where $\sigma ^{2}$ is the variance of the white noise sequence $a_ t$ and the vector $\psi = \left[\psi _{0} \ldots \psi _{m-1} \right]^{}$ contains the first $m$ values of the impulse response function—that is, the first $m$ coefficients in the expansion of the ratio ${\theta } / {\phi }$ . The covariance matrix of the initial state, $\mb {Q}_1$ , is computed as

$\mi {vec}(\mb {Q}_{1}) = (\mb {I} - \mb {T}\bigotimes \mb {T})^{-1} \mi {vec}(\mb {Q})$

where $\bigotimes$ denotes the Kronecker product and the $\mi {vec}$ operation on a matrix creates a vector formed by vertically stacking the rows of that matrix.

A number of alternate state space forms are possible in the nonstationary case also. The form used by the SSM procedure utilizes the state space form for the stationary case as a building block. Suppose that a random sequence $\xi _ t$ follows an ARIMA(p,d,q) ${\times }$ (P,D,Q) $_{\mi {s}}$ model with a white noise sequence $a_ t$ . As in the notation for the stationary case, with slight abuse of notation, let $d = d + s*D$ denote the effective differencing order, and let $\Delta$ be the effective differencing polynomial in the backshift operator with coefficients $\Delta _{1}, \; \ldots , \; \Delta _{d}$ . It can be shown that $\xi _ t$ has a state space form with state vector size $m^{\dagger } = m + d$ . In what follows, the system matrices and related quantities in the nonstationary case are described in terms of similar entities in the stationary case. A superscript dagger ( $\dagger$ ) has been added to distinguish the entities from the nonstationary case. $\mb {Z}^{\dagger } = \left[0 \; 0 \; \ldots \; 1 \; \ldots \; 0 \right]$ where the only nonzero value, 1, is at the index $m + 1$ , and the transition matrix, $\mb {T}^{\dagger }$ , in a blocked form, is given by

$\mb {T}^{\dagger } = \left[ \begin{tabular}{ccc} $\mb {T}$ & 0 & 0 \\ $\mb {Z} \mb {T}$ & $\Delta _{1} \ldots $ & $\Delta _{d}$ \\ 0 & $I_{d-1}$ & 0 \end{tabular} \right]$

The state disturbance matrix $\mb {Q}^{\dagger }$ is given by

$\mb {Q}^{\dagger } = \left[ \begin{tabular}{ccc} $\mb {Q}$ & $ \mb {Q} \mb {Z}^{}$ & 0 \\ $\mb {Z} \mb {Q}$ & $\mb {Z} \mb {Q}\mb {Z}^{} $ & 0 \\ 0 & 0 & 0 \end{tabular} \right]$

Finally, the initial state is partially diffuse: the first $m$ elements are nondiffuse and the last $d$ elements are diffuse. The covariance matrix of the first $m$ elements is $\mb {Q}_1$ .