TimeSeriesBook.pdf

The theorem below establishes that these estimators

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The theorem below establishes that these estimators lead under rather mild conditions to consistent and asymptotically normally distributed esti- mators. Theorem 4.4. Let { X t } be the stationary process X t = μ + X j = -∞ ψ j Z t - j with Z t IID(0 , σ 2 ) , j = -∞ | ψ j | < and j = -∞ j | ψ j | 2 < . Then we have for h = 1 , 2 , . . . b ρ (1) . . . b ρ ( h ) d ----→ N ρ (1) . . . ρ ( h ) , W T where the elements of W = ( w ij ) i,j ∈{ 1 ,...,h } are given Bartlett’s formula w ij = X k =1 [ ρ ( k + i ) + ρ ( k - i ) - 2 ρ ( i ) ρ ( k )][ ρ ( k + j ) + ρ ( k - j ) - 2 ρ ( j ) ρ ( k )] . Proof. Brockwell and Davis (1991, section 7.3) Brockwell and Davis (1991) offer a second version of the above theorem where j = -∞ j | ψ j | 2 < is replaced by the assumption of finite fourth moments, i.e. by assuming E Z 4 t < . As we rely mainly on ARMA pro- cesses, we do not pursue this distinction further because this class of process automatically fulfills the above assumptions as soon as { Z t } is identically and independently distributed (IID). A proof which relies on the Beveridge- Nelson polynomial decomposition (see Theorem D.1 in Appendix D) can be gathered from Phillips and Solo (1992).
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4.2. ESTIMATION OF ACF 79 0 5 10 15 20 -1 -0.5 0 0.5 1 order correlation coefficient estimated ACF lower bound for confidence interval upper bound for confidence interval Figure 4.1: Estimated autocorrelation function of a WN(0,1) process with 95 percent confidence interval for sample size T = 100 Example: { X t } ∼ IID(0 , σ 2 ) The most important application of the above theorem, is related to the case of a white noise process for which ρ ( h ) is equal to zero for | h | > 0. The above theorem then implies that w ij = 1 , for i = j ; 0 , otherwise. The estimated autocorrelation coefficients converge to the true autocorrela- tion coefficient, in this case zero. The asymptotic distribution of T b ρ ( h ) converges to the standard normal distribution. This implies that for large T we can approximate the distribution of b ρ ( h ) by a normal distribution with mean zero and variance 1 /T . This allows the construction of a 95 percent confidence interval for the case that the true process is white noise. This confidence interval is therefore given by ± 1 . 96 T - 1 2 . It can be used to verify if the observed process is indeed white noise. Figure 4.1 plots the empirical autocorrelation function of a WN(0,1) pro- cess with a sample size of T = 100. The implied 95 percent confidence is therefore equal to ± 0 . 196. As each estimated autocorrelation coefficient falls within the confidence interval so that we can conclude that the observed times series may indeed represent a white noise process.
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80 CHAPTER 4. ESTIMATION OF MEAN AND ACF Instead of examining each correlation coefficient separately, we can test the joint hypothesis that all correlation coefficients up to order N are equal to zero, i.e. ρ (1) = ρ (2) = . . . = ρ ( N ) = 0, N = 1 , 2 , . . . . As each T ˆ ρ ( h ) has an asymptotic standard normal distribution and uncorrelated with T ˆ ρ ( k ), h 6 = k , the sum of the squared estimated autocorrelation coefficients is χ 2 distributed with N degrees of freedom. This test statistic is called
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