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

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

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

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

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

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

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 $\mb{Q}_1$.