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694 Chapter 16 Additional Topics in Time Series Regression

Key Concept Direct Multiperiod Forecasts

16.3 The direct multiperiod forecast h periods into the future based on p lags each of
Yt and an additional predictor Xt is computed by first estimating the regression

Yt = d0 + d1Yt - h + g + dpYt - p - h + 1 + dp + 1Xt - h
+ g + d2pXt - p - h + 1 + ut, (16.15)

and then using the estimated coefficients directly to make the forecast of YT + h
using data through period T.

Which Method Should You Use?

In most applications, the iterated method is the recommended procedure for
multiperiod forecasting, for two reasons. First, from a theoretical perspective,
if the underlying one-period-ahead model (the AR or VAR that is used to
compute the iterated forecast) is specified correctly, then the coefficients are
estimated more efficiently if they are estimated by a one-period-ahead regres-
sion (and then iterated) than by a multiperiod-ahead regression. Second, from
a practical perspective, forecasters are usually interested in forecasts not just
at a single horizon but at multiple horizons. Because they are produced using
the same model, iterated forecasts tend to have time paths that are less erratic
across horizons than do direct forecasts. Because a different model is used at
every horizon for direct forecasts, sampling error in the estimated coefficients
can add random fluctuations to the time paths of a sequence of direct multi-
period forecasts.

Under some circumstances, however, direct forecasts are preferable to iter-
ated forecasts. One such circumstance is when you have reason to believe that the
one-period-ahead model (the AR or VAR) is not specified correctly. For exam-
ple, you might believe that the equation for the variable you are trying to forecast
in a VAR is specified correctly, but that one or more of the other equations in the
VAR is specified incorrectly, perhaps because of neglected nonlinear terms. If the
one-step-ahead model is specified incorrectly, then in general the iterated multi-
period forecast will be biased, and the MSFE of the iterated forecast can exceed
the MSFE of the direct forecast, even though the direct forecast has a larger vari-
ance. A second circumstance in which a direct forecast might be desirable arises