UCM Procedure

This section details the options for specifying an ARMA model for the irregular component. The specification of ARMA models requires some notation, which is explained first.

Let B denote the backshift operator—that is, for any sequence , . The higher powers of B represent larger shifts (for example, ). A random sequence follows a zero-mean ARMA(p,q)(P,Q) model with nonseasonal autoregressive order p, seasonal autoregressive order P, nonseasonal moving average order q, and seasonal moving average order Q, if it satisfies the following difference equation specified in terms of the polynomials in the backshift operator, where is a white-noise sequence and s is the season length:

The polynomials and are of orders p, P, q, and Q, respectively, which can be any nonnegative integers. The season length s must be a positive integer. For example, satisfies an ARMA(1,1) model (that is, and ) if

for some coefficients and and a white-noise sequence . Similarly, satisfies an ARMA(1,1)(1,1) model if

for some coefficients and and a white-noise sequence . The ARMA process is stationary and invertible if the defining polynomials and have all their roots outside the unit circle—that is, their absolute values are strictly larger than 1.0. It is assumed that the ARMA model specified for the irregular component is stationary and invertible—that is, the coefficients of the polynomials and are constrained so that the stationarity and invertibility conditions are satisfied. The unknown coefficients of these polynomials become part of the model parameter vector that is estimated using the data.

The notation for a closely related class of models, autoregressive integrated moving average (ARIMA) models, is also given here. A random sequence is said to follow an ARIMA(p,d,q)(P,D,Q) model if, for some nonnegative integers d and D, the differenced series follows an ARMA(p,q)(P,Q) model. The integers d and D are called nonseasonal and seasonal differencing orders, respectively. You can specify ARIMA models by using the DEPLAG statement for specifying the differencing orders and by using the IRREGULAR statement for the ARMA specification. For an example of ARIMA(0,1,1)(0,1,1) model specification, see Example 30.8. Brockwell and Davis (1991) can be consulted for additional information about ARIMA models.

You can use options of the IRREGULAR statement to specify the desired ARMA model and to request printed and graphical output. A few examples of the IRREGULAR statement are given next.

The following statement specifies an irregular component that is modeled as an ARMA(1,1) process. It also requests plotting its smoothed estimate.

irregular p=1 q=1 plot=smooth;

The following statement specifies an ARMA(1,1)(1,1) model. It also fixes the coefficient of the first-order seasonal moving average polynomial to 0.1. The other coefficients and the white-noise variance are estimated using the data.

irregular p=1 sp=1 q=1 sq=1 s=12 sma=0.1 noest=(sma);

You can specify the following ARMA model options: