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144 Chapter 5 Modeling and Forecasting with ARMA Processes
computed ˆφ11 and ˆv1 above using the Durbin–Levinson algorithm. The Yule–Walker
AR(1) model obtained by ITSM for {Xt} is therefore (not surprisingly)
Xt − 0.4219Xt−1 Zt, {Zt} ∼WN(0, 0.1479), (5.1.14)
and the corresponding model for {Yt} is
Yt − 0.1336 − 0.4219(Yt−1 − 0.1336) Zt, {Zt} ∼WN(0, 0.1479). (5.1.15)
Assuming that our observed data really are generated by an AR process with
p 1, (5.1.13) gives us approximate 95% confidence bounds for the autoregressive
coefficient φ,
0.4219 ± (1.96)(.1479)
(.17992) √ (0.2194, 0.6244).
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Besides estimating the autoregressive coefficients, ITSM computes and prints out
the ratio of each coefficient to 1.96 times its estimated standard deviation. From these
numbers large-sample 95% confidence intervals for each of the coefficients are easily
obtained. In this particular example there is just one coefficient estimate, ˆφ1 0.4219,
with ratio of coefficient to 1.96×standard error equal to 2.0832. Hence the required
95% confidence bounds are 0.4219 ± 0.4219/2.0832 (0.2194, 0.6244), as found
above.
A useful technique for preliminary autoregressive estimation that incorporates
automatic model selection (i.e., choice of p) is to minimize the AICC (see equation
(5.5.4)) over all fitted autoregressions of orders 0 through 27. This is achieved by
selecting both Yule-Walker and Find AR model with min AICC in the Prelim-
ACF
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
Figure 5-1
The sample ACF of
the differenced series
0 5 10 15 20 25 30
{Yt } in Example 5.1.1. Lag