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60 Chapter 2 Stationary Processes
where T is the k × 2k matrix
⎡
⎢
T ⎢
⎣ .
0 ··· 0 0 Y1 Y2 ··· Yk
0 ··· 0 Y1 Y2 ··· Yk 0
0 Y1 Y2 ··· Yk 0 ··· 0
Yi Xi − ¯Xn, i 1,...,n, and Yi 0 for i n + 1,...,k. Then for any real k × 1
vector a we have
a ′ ˆƔka n −1 (a ′ T )(T ′ a) ≥ 0, (2.4.7)
and consequently the sample autocovariance matrix ˆƔk and sample autocorrelation
matrix
ˆRk ˆƔk/γ (0) (2.4.8)
are nonnegative definite. Sometimes the factor n −1 is replaced by (n − h) −1 in the
definition of ˆγ (h), but the resulting covariance and correlation matrices ˆƔn and ˆRn
may not then be nonnegative definite. We shall therefore use the definitions (2.4.4)
and (2.4.5) of ˆγ (h) and ˆρ(h).
Remark 1. The matrices ˆƔk and ˆRk are in fact nonsingular if there is at least one
nonzero Yi, or equivalently if ˆγ(0) >0. To establish this result, suppose that ˆγ(0) >0
and ˆƔk is singular. Then there is equality in (2.4.7) for some nonzero vector a, implying
that a ′ T 0 and hence that the rank of T is less than k. Let Yi be the first nonzero
value of Y1,Y2,...,Yk, and consider the k × k submatrix of T consisting of columns
(i + 1) through (i + k). Since this matrix is lower right triangular with each diagonal
element equal to Yi, its determinant has absolute value |Yi| k 0. Consequently, the
submatrix is nonsingular, and T must have rank k, a contradiction.
Without further information beyond the observed data X1,...,Xn, it is impossible
to give reasonable estimates of γ (h) and ρ(h) for h ≥ n. Evenforh slightly
smaller than n, the estimates ˆγ (h) and ˆρ(h) are unreliable, since there are so few pairs
(Xt+h,Xt) available (only one if h n − 1). A useful guide is provided by Box and
Jenkins (1976), p. 33, who suggest that n should be at least about 50 and h ≤ n/4.
The sample ACF plays an important role in the selection of suitable models for
the data. We have already seen in Example 1.4.6 and Section 1.6 how the sample
ACF can be used to test for iid noise. For systematic inference concerning ρ(h),
we need the sampling distribution of the estimator ˆρ(h). Although the distribution
of ˆρ(h) is intractable for samples from even the simplest time series models, it can
usually be well approximated by a normal distribution for large sample sizes. For
linear models and in particular for ARMA models (see Theorem 7.2.2 of TSTM for
exact conditions) ˆρk ( ˆρ(1),..., ˆρ(k)) ′ is approximately distributed for large n as
.
⎤
⎥
⎦ ,