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14.3 Autoregressions 581

Autoregressions Key Concept

The pth-order autoregressive model [the AR(p) model] represents Yt as a linear 14.3
function of p of its lagged values:

Yt = b0 + b1Yt - 1 + b2Yt - 2 + g + bpYt - p + ut, (14.13)

where E(ut 0 Yt - 1, Yt - 2, c) = 0. The number of lags p is called the order, or the

lag length, of the autoregression.

The first implication is that the best forecast of YT + 1 based on its entire his-
tory depends on only the most recent p past values. Specifically, let YT + 1͉T =
E(YT + 1 ͉ YT,YT - 1, c) denote the conditional mean of YT + 1, given its entire his-
tory. Then YT + 1͉T has the smallest RMSFE of any forecast, based on the history
of Y (Exercise 14.5). If Yt follows an AR(p), then the best forecast of YT + 1 based
on YT,YT - 1, c is

YT + 10T = b0 + b1YT + b2YT - 1 + g + bpYT - p + 1, (14.14)

which follows from the AR(p) model in Equation (14.13) and the assumption that

E(ut 0 Yt - 1, Yt - 2, c) = 0. In practice, the coefficients b0, b1, c, bp are unknown,

so actual forecasts from an AR(p) use Equation (14.14) with estimated coeffi-
cients.

The second implication is that the errors ut are serially uncorrelated, a result
that follows from Equation (2.27) (Exercise 14.5).

Application to GDP growth. What is the forecast of the growth rate of GDP in
2013:Q1, using data through 2012:Q4, based on the AR(2) model of GDP growth
in Equation (14.12)? To compute this forecast, substitute the values of the GDP
growth in 2012:Q3 and 2012:Q4 into Equation (14.12): GDPGR2013:Q1͉2012:Q4 =
1 .63 + 0 .28 GDPGR2012:Q4 + 0 .17 GDPGR2012:Q3 = 1.63 + 0.28 * 0.15 + 0.17 *
2.75 ≅ 2.1%, where the 2012 values for GDPGR are taken from the fourth col-
umn of Table 14.1. The forecast error is the actual value, 1.1%, minus the forecast,
or 1.1% − 2.1% = −1.0%, slightly greater in absolute value than the AR(1) fore-
cast error of −0.9 percentage point.