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236 Chapter 7 Multivariate Time Series
Writing ˆγij (h) for the (i, j)-component of ˆƔ(h), i, j 1, 2,..., we estimate the
cross-correlations by
ˆρij (h) ˆγij (h)( ˆγii(0) ˆγjj(0)) −1/2 .
If i j, then ˆρij reduces to the sample autocorrelation function of the ith series.
Derivation of the large-sample properties of ˆγij and ˆρij is quite complicated in
general. Here we shall simply note one result that is of particular importance for
testing the independence of two component series. For details of the proof of this and
related results, see TSTM.
Theorem 7.3.1 Let {Xt} be the bivariate time series whose components are defined by
and
Xt1
Xt2
∞
k−∞
∞
k−∞
αkZt−k,1, {Zt1} ∼IID 0,σ 2
1 ,
βkZt−k,2, {Zt2} ∼IID 0,σ 2
2 ,
where the two sequences {Zt1} and {Zt2} are independent,
k |αk|
< ∞, and
k |βk| < ∞.
Then for all integers h and k with h k, the random variables n1/2 ˆρ12(h)
and n1/2
ˆρ12(k) are approximately bivariate normal with mean 0, variance
∞
j−∞ ρ11(j)ρ22(j), and covariance ∞ j−∞ ρ11(j)ρ22(j + k − h), for n large.
[For a related result that does not require the independence of the two series {Xt1}
and {Xt2} see Theorem 7.3.2 below.]
Theorem 7.3.1 is useful in testing for correlation between two time series. If one
of the two processes in the theorem is white noise, then it follows at once from the
theorem that ˆρ12(h) is approximately normally distributed with mean 0 and variance
1/n, in which case it is straightforward to test the hypothesis that ρ12(h) 0. However,
if neither process is white noise, then a value of ˆρ12(h) that is large relative to n −1/2 does
not necessarily indicate that ρ12(h) is different from zero. For example, suppose that
{Xt1} and {Xt2} are two independent AR(1) processes with ρ11(h) ρ22(h) .8 |h| .
Then the large-sample variance of ˆρ12(h) is n −1 1 + 2 ∞
k1 (.64)k 4.556n −1 .It
would therefore not be surprising to observe a value of ˆρ12(h) as large as 3n −1/2
even though {Xt1} and {Xt2} are independent. If on the other hand, ρ11(h) .8 |h|
and ρ22(h) (−.8) |h| , then the large-sample variance of ˆρ12(h) is .2195n −1 , and an
observed value of 3n −1/2 for ˆρ12(h) would be very unlikely.