Editing Autoregressive model (section)

==''n''-step-ahead forecasting==

Once the parameters of the autoregression

:<math> X_t = \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t \,</math>

have been estimated, the autoregression can be used to forecast an arbitrary number of periods into the future. First use ''t'' to refer to the first period for which data is not yet available; substitute the known preceding values ''X''<sub>''t-i''</sub> for ''i=''1, ..., ''p'' into the autoregressive equation while setting the error term <math>\varepsilon_t</math> equal to zero (because we forecast ''X''<sub>''t''</sub> to equal its expected value, and the expected value of the unobserved error term is zero). The output of the autoregressive equation is the forecast for the first unobserved period. Next, use ''t'' to refer to the ''next'' period for which data is not yet available; again the autoregressive equation is used to make the forecast, with one difference: the value of ''X'' one period prior to the one now being forecast is not known, so its expected value—the predicted value arising from the previous forecasting step—is used instead. Then for future periods the same procedure is used, each time using one more forecast value on the right side of the predictive equation until, after  ''p'' predictions, all ''p'' right-side values are predicted values from preceding steps.

There are four sources of uncertainty regarding predictions obtained in this manner: (1) uncertainty as to whether the autoregressive model is the correct model; (2) uncertainty about the accuracy of the forecasted values that are used as lagged values in the right side of the autoregressive equation; (3) uncertainty about the true values of the autoregressive coefficients; and (4) uncertainty about the value of the error term <math>\varepsilon_t \,</math> for the period being predicted. Each of the last three can be quantified and combined to give a [[confidence interval]] for the ''n''-step-ahead predictions; the confidence interval will become wider as ''n'' increases because of the use of an increasing number of estimated values for the right-side variables.