Forecasting Details

The ARIMA Procedure

Forecasting Details

If the model has input variables, a forecast beyond the end of the data for the input variables is possible only if univariate ARIMA models have previously been fit to the input variables or future values for the input variables are included in the DATA= data set.

If input variables are used, the forecast standard errors and confidence limits of the response depend on the estimated forecast error variance of the predicted inputs. If several input series are used, the forecast errors for the inputs should be independent; otherwise, the standard errors and confidence limits for the response series will not be accurate. If future values for the input variables are included in the DATA= data set, the standard errors of the forecasts will be underestimated since these values are assumed to be known with certainty.

The forecasts are generated using forecasting equations consistent with the method used to estimate the model parameters. Thus, the estimation method specified on the ESTIMATE statement also controls the way forecasts are produced by the FORECAST statement.

Infinite Memory Forecasts

If METHOD=CLS is used, the forecasts are infinite memory forecasts, also called conditional forecasts. The term conditional is used because the forecasts are computed by assuming that the unknown values of the response series before the start of the data are equal to the mean of the series. Thus, the forecasts are conditional on this assumption.

The series x_t can be represented as

$x_{t} = a_{t} + \sum_{i=1}^{{\infty}}{{\pi}_{i}x_{t-i}}$

where ${{\phi}(B)/{\theta}(B)=1-\sum_{i=1}^{{\infty}}{{\pi}_{i}B^i}}$ .

The k-step forecast of x_t+k is computed as

$\hat{x}_{t+k} =\sum_{i=1}^{k-1}{\hat{{\pi}}_{i}\hat{x}_{t+k-i}} +\sum_{i=k}^{{\infty}}{\hat{{\pi}}_{i}x_{t+k-i}}$

where unobserved past values of x_t are set to zero, and ${\hat{{\pi}}_{i}}$ is obtained from the estimated parameters ${\hat{{\phi}}}$ and ${\hat{{\theta}}}$ .

Finite Memory Forecasts

For METHOD=ULS or METHOD=ML, the forecasts are finite memory forecasts, also called unconditional forecasts. For finite memory forecasts, the covariance function of the ARMA model is used to derive the best linear prediction equation.

That is, the k-step forecast of x_t+k, given (x₁, ... ,x_t-1), is

$\tilde{x}_{t+k} =C_{k,t} V^{-1}_{t} (x_{1},{ ... },x_{t-1})'$

where C_k,t is the covariance of x_t+k and (x₁, ... ,x_t-1), and V_t is the covariance matrix of the vector (x₁, ... ,x_t-1). C_k,t and V_t are derived from the estimated parameters.

Finite memory forecasts minimize the mean-squared error of prediction if the parameters of the ARMA model are known exactly. (In most cases, the parameters of the ARMA model are estimated, so the predictors are not true best linear forecasts.)

If the response series is differenced, the final forecast is produced by summing the forecast of the differenced series. This summation, and thus the forecast, is conditional on the initial values of the series. Thus, when the response series is differenced, the final forecasts are not true finite memory forecasts because they are derived assuming that the differenced series begins in a steady-state condition. Thus, they fall somewhere between finite memory and infinite memory forecasts. In practice, there is seldom any practical difference between these forecasts and true finite memory forecasts.

Chapter Contents
Previous
Next
Top