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Figure 2-2
The sample autocorrelation
function of the Lake Huron
residuals of Figure 1.10
2.5 Forecasting Stationary Time Series 63
showing the bounds
and the
model ACF ρ(i) (.791) i 0 10 20 30 40
. Lag
ˆρ(i) ± 1.96n −1/2 w 1/2
ii
ACF
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Sample ACF
*
95% Conf Bds
*
* Model ACF
*
*************************************
i 1, 2,.... In Figure 2.2 we have plotted the sample ACF of the Lake Huron
residuals y1,...,y98 from Figure 1.10 together with 95% confidence bounds for
ρ(i), i 1,...,40, assuming that data are generated from the AR(1) model
Yt .791Yt−1 + Zt
(2.4.13)
(see equation (1.4.3)). The confidence bounds are computed from ˆρ(i) ± 1.96n−1/2 w 1/2
ii , where wii is given in (2.4.12) with φ .791. The model ACF, ρ(i) (.791) i ,
is also plotted in Figure 2.2. Notice that the model ACF lies just outside the confidence
bounds at lags 2–6. This suggests some incompatibility of the data with the
model (2.4.13). A much better fit to the residuals is provided by the second-order
autoregression defined by (1.4.4).
2.5 Forecasting Stationary Time Series
We now consider the problem of predicting the values Xn+h,h > 0, of a stationary
time series with known mean µ and autocovariance function γ in terms of the
values {Xn,...,X1}, up to time n. Our goal is to find the linear combination of
1,Xn,Xn−1,...,X1, that forecasts Xn+h with minimum mean squared error. The best
linear predictor in terms of 1,Xn,...,X1 will be denoted by PnXn+h and clearly has
the form
PnXn+h a0 + a1Xn +···+anX1. (2.5.1)