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5.1 Preliminary Estimation 141
Large-Sample Distribution of Yule–Walker Estimators:
For a large sample from an AR(p) process,
ˆφ ≈ N φ,n −1 σ 2 Ɣ −1
p .
If we replace σ 2 and Ɣp by their estimates ˆσ 2 and ˆƔp, we can use this result to
find large-sample confidence regions for φ and each of its components as in (5.1.12)
and (5.1.13) below.
Order Selection
In practice we do not know the true order of the model generating the data. In fact,
it will usually be the case that there is no true AR model, in which case our goal
is simply to find one that represents the data optimally in some sense. Two useful
techniques for selecting an appropriate AR model are given below. The second is
more systematic and extends beyond the narrow class of pure autoregressive models.
• Some guidance in the choice of order is provided by a large-sample result (see
TSTM, Section 8.10), which states that if {Xt} is the causal AR(p) process defined
by (5.1.1) with {Zt} ∼iid 0,σ 2 and if we fit a model with order m>pusing
the Yule–Walker equations, i.e., if we fit a model with coefficient vector
ˆφm ˆR −1
m ˆρm, m>p,
then the last component, ˆφmm, of the vector ˆφm is approximately normally distributed
with mean 0 and variance 1/n. Notice that ˆφmm is exactly the sample
partial autocorrelation at lag m as defined in Section 3.2.3.
Now, we already know from Example 3.2.6 that for an AR(p), process the partial
autocorrelations φmm, m>p, are zero. By the result of the previous paragraph,
if an AR(p) model is appropriate for the data, then the values ˆφkk, k>p, should
be compatible with observations from the distribution N(0, 1/n). In particular,
for k>p, ˆφkk will fall between the bounds ±1.96n−1/2 with probability close to
0.95. This suggests using as a preliminary estimator of p the smallest value m
such that
ˆφkk
−1/2 < 1.96n for k>m.
The program ITSM plots the sample PACF ˆφmm,m 1, 2,... together with
the bounds ±1.96/ √ n. From this graph it is easy to read off the preliminary
estimator of p defined above.
• A more systematic approach to order selection is to find the values of p and φp
that minimize the AICC statistic (see Section 5.5.2 below)
AICC −2lnL(φp,S(φp)/n) + 2(p + 1)n/(n − p − 2),