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.
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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• Spring '17
• Raffaelle Giacomini

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