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692 Chapter 16 Additional Topics in Time Series Regression
Key Concept Iterated Multiperiod Forecasts
16.2 The iterated multiperiod AR forecast is computed in steps: First compute the
one-period-ahead forecast, then use that to compute the two-period-ahead fore-
cast, and so forth. The two- and three-period-ahead iterated forecasts based on
an AR(p) are
Yn T + 2∙T = bn0 + bn1YnT + 1∙T + bn2YT + bn3YT - 1 + g + bnpYT - p + 2 (16.10)
Yn T + 3∙T = bn0 + bn1Yn T + 2∙T + bn2Yn T + 1∙T + bn3YT + g + bnpYT - p + 3, (16.11)
where the bn’s are the OLS estimates of the AR(p) coefficients. Continuing this
process (“iterating”) produces forecasts further into the future.
The iterated multiperiod VAR forecast is also computed in steps: First com-
pute the one-period-ahead forecast of all the variables in the VAR, then use those
forecasts to compute the two-period-ahead forecasts, and continue this process
iteratively to the desired forecast horizon. The two-period-ahead iterated forecast
of YT + 2, based on the two-variable VAR(p) in Key Concept 16.1, is
Yn T + 2∙T = bn10 + bn11Yn T + 1∙T + bn12YT + bn13YT - 1 + g + bn1pYT - p + 2
+ gn11Xn T + 1∙T + gn12XT + gn13XT - 1 + g + gn1pXT - p + 2, (16.12)
where the coefficients in Equation (16.12) are the OLS estimates of the VAR
coefficients. Iterating produces forecasts further into the future.
dependent variable is Yt and the regressors are Yt - 2, Yt - 3, Xt−2, and Xt−3. The
coefficients from this regression can be used directly to compute the forecast of
YT + 2 using data on YT, YT - 1, XT, and XT−1, without the need for any iteration.
More generally, in a direct h-period-ahead forecasting regression, all predictors
are lagged h periods to produce the h-period-ahead forecast.
For example, the forecast of GDPGRt two quarters ahead using two lags each
of GDPGRt−2 and TSpreadt−2 is computed by first estimating the regression:
GDPGRt∙t - 2 = 0.57 + 0.34GDPGRt - 2 + 0.03GDPGRt - 3
(0.67) (0.07) (0.10)
+ 0.62TSpreadt - 2 - 0.01TSpreadt - 3. (16.13)
(0.47) (0.46)
