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580 Chapter 14 Introduction to Time Series Regression and Forecasting

How accurate is this AR(1) forecast? Table 14.1 shows that the actual growth
rate of GDP in 2013:Q1 was 1.1%, so the AR(1) forecast is high by 0.9 percentage
point; that is, the forecast error is −0.9. The R2 of the AR(1) model in Equation (14.7)
is only 0.11, so the lagged value of GDP growth explains a small fraction of the
variation in GDP growth in the sample used to fit the autoregression. This low R2
is consistent with the poor forecast of GDP growth in 2013:Q1 produced using
Equation (14.7). More generally, the low R2 suggests that this AR(1) model will
forecast only a small amount of the variation in the growth rate of GDP.

The standard error of the regression in Equation (14.7) is 3.16; ignoring
uncertainty arising from estimation of the coefficients, our estimate of the RMSFE
for forecasts based on Equation (14.7) is therefore 3.16 percentage points.

The pth-Order Autoregressive Model

The AR(1) model uses Yt - 1 to forecast Yt, but doing so ignores potentially useful
information in the more distant past. One way to incorporate this information is to
include additional lags in the AR(1) model; this yields the pth-order autoregressive,
or AR(p), model.

The pth-order autoregressive model [the AR(p) model] represents Yt as a
linear function of p of its lagged values; that is, in the AR(p) model, the regressors
are Yt - 1, Yt - 2, c, Yt - p, plus an intercept. The number of lags, p, included in an
AR(p) model is called the order, or lag length, of the autoregression.

For example, an AR(2) model of GDP growth uses two lags of GDP growth
as regressors. Estimated by OLS over the period 1962–2012, the AR(2) model is

GDPGRt = 1 .63 + 0 .28 GDPGRt - 1 + 0 .17 GDPGRt - 2 . (14.12)
(0.40) (0.08) (0 .08)

The coefficient on the additional lag in Equation (14.1312) is significantly differ-
ent from zero at the 5% significance level: The t-statistic is 2.27 (p-value = 0.02).
This is reflected in an improvement in the R 2 from 0.11 for the AR(1) model in
Equation (14.7) to 0.14 for the AR(2) model. Similarly, the SER of the AR(2)
model in Equation (14.12) is 3.11, an improvement over the SER of the AR(1)
model, which is 3.16.

The AR(p) model is summarized in Key Concept 14.3.

Properties of the forecast and error term in the AR (p) model. The assumption that

the conditional expectation of ut is zero, given past values of Yt [that is,

E(ut 0 Yt - 1, Yt - 2, c) = 0], has two important implications.