TimeSeriesBook.pdf

H b γ h converges in probability as t to γ h b γ h

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h , b Γ( h ) converges in probability as T → ∞ to Γ( h ) : b Γ( h ) p ----→ Γ( h ) Proof. A proof can be given along the lines given in Proposition 13.1. As for the univariate case, we can define the long-run covariance matrix J as J = X h = -∞ Γ( h ) . (11.1) As a non-parametric estimator we can again consider the following class of estimators: ˆ J T = T - 1 X h = - T +1 k h T b Γ( h ) where k ( x ) is a kernel function and where b Γ( h ) is the corresponding estimate of the covariance matrix at lag h . For the choice of the kernel function and the lag truncation parameter the same principles apply as in the univariate case (see Section 4.4 and Haan and Levin (1997)).
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224 CHAPTER 11. ESTIMATION OF COVARIANCE FUNCTION 11.2 Testing the Cross-Correlation of two Time Series The determination of the asymptotic distribution of b Γ( h ) is complicated. We therefore restrict ourselves to the case of two time series. Theorem 11.4. Let { X t } be a bivariate stochastic process whose components can be described by X 1 t = X j = -∞ α j Z 1 ,t - j with Z 1 t IID(0 , σ 2 1 ) X 2 t = X j = -∞ β j Z 2 ,t - j with Z 2 t IID(0 , σ 2 2 ) where { Z 1 t } and { Z 2 t } are independent from each other at all leads and lags and where j | α j | < and j | β j | < . Under these conditions the asymp- totic distribution of the estimator of the cross-correlation function ρ 12 ( h ) be- tween { X 1 t } and { X 2 t } is T ˆ ρ 12 ( h ) d -→ N 0 , X j = -∞ ρ 11 ( j ) ρ 22 ( j ) ! , h 0 . (11.2) For all h and k with h 6 = k , ( T ˆ ρ 12 ( h ) , T ˆ ρ 12 ( k )) 0 converges in distribution to a bivariate normal distribution with mean zero, variances j = -∞ ρ 11 ( j ) ρ 22 ( j ) and covariances j = -∞ ρ 11 ( j ) ρ 22 ( j + k - h ) . This result can be used to construct a test of independence, respectively uncorrelatedness, between two time series. The above theorem, however, shows that the asymptotic distribution of T ˆ ρ 12 ( h ) depends on ρ 11 ( h ) and ρ 22 ( h ) and is therefore unknown. Thus, the test cannot be based on the cross-correlation alone. 1 This problem can, however, be overcome by implementing the following two-step procedure suggested by Haugh (1976). First step: Estimate for each times series separately a univariate invertible ARMA model and compute the resulting residuals ˆ Z it = j =0 ˆ π ( i ) j X i,t - j , i = 1 , 2. If the ARMA models correspond to the true ones, these resid- uals should approximately be white noise. This first step is called pre-whitening. 1 The theorem may also be used to conduct a causality test between two times series (see Section 15.1).
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11.3. SOME EXAMPLES FOR INDEPENDENCE TESTS 225 Second step: Under the null hypothesis the two time series { X 1 t } and { X 2 t } are uncorrelated with each other. This implies that the residu- als { Z 1 t } and { Z 2 t } should also be uncorrelated with each other. The variance of the cross-correlations between { Z 1 t } and { Z 2 t } are there- fore asymptotically equal to 1 /T under the null hypothesis. Thus, one can apply the result of Theorem 11.4 to construct confidence intervals based on formula (11.2). A 95-percent confidence interval is therefore given by ± 1 . 96 T - 1 / 2 . The Theorem may also be used to construct a
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