SSM Procedure

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 that governs the underlying state. It can be described as , where and the (one-dimensional) state follows a random walk:

Here and . The initial condition is fully diffuse. Note that if , the resulting trend is a fixed constant.

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: controls the level variation, and controls the slope variation. If , the resulting trend is called an integrated random walk. If both and , then the resulting model is the deterministic linear time trend. Here , , and . The initial condition is fully diffuse.

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, , 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 , , and . The initial condition is partially diffuse with .

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

First the state space form for the stationary case—that is, when and , 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 denote the effective autoregressive order, and let denote the effective moving average order of the model. Similarly, let be the effective autoregressive polynomial, and let be the effective moving average polynomial in the backshift operator with coefficients and , obtained by multiplying the respective nonseasonal and seasonal factors. Then, a random sequence that follows an ARMA(p,q)(P,Q) model with a white noise sequence has a state space form with state vector of size . The system matrices are as follows: , and the transition matrix , in a blocked form, is given by

where if and is an dimensional identity matrix. The covariance of the state disturbance matrix , where is the variance of the white noise sequence and the vector contains the first m values of the impulse response function—that is, the first m coefficients in the expansion of the ratio . The covariance matrix of the initial state, , is computed as

where denotes the Kronecker product and the 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 follows an ARIMA(p,d,q)(P,D,Q) model with a white noise sequence . As in the notation for the stationary case, with slight abuse of notation, let denote the effective differencing order, and let be the effective differencing polynomial in the backshift operator with coefficients . It can be shown that has a state space form with state vector size . 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 () has been added to distinguish the entities from the nonstationary case. where the only nonzero value, 1, is at the index , and the transition matrix, , in a blocked form, is given by

The state disturbance matrix is given by

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 .