TimeSeriesBook.pdf

Converges to σ 2 j m ψ j 2 as t this term converges

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converges to σ 2 | j | >m ψ j 2 as T → ∞ . This term converges to zero as m → ∞ . The approximation error T ( X T - μ ) - T X ( m ) T - μ therefore
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4.1. ESTIMATION OF THE MEAN 75 converges in mean square to zero and thus, using Chebyschev’s inequality (see Theorem C.3 or C.7), also in probability. We have therefore established the third condition of Theorem C.14 as well. Thus, we can conclude that T ( X T - μ ) d ----→ X . Under a more restrictive summability condition which holds, however, within the context of causal ARMA processes, we can provide a less technical proof. This proof follows an idea of Phillips and Solo (1992) and is based on the Beveridge-Nelson decomposition (see Appendix D). 2 Theorem 4.3. For any stationary process X t = μ + X j =0 ψ j Z t - j with the properties Z t IID(0 , σ 2 ) and j =0 j 2 | ψ j | 2 < , the arithmetic average X T is asymptotically normal: T ( X T - μ ) d ----→ N 0 , X h = -∞ γ ( h ) ! = N 0 , σ 2 X j =0 ψ j ! 2 = N(0 , σ 2 Ψ(1) 2 ) . Proof. The application of the Beveridge-Nelson decomposition (see Theo- rem D.1 in Appendix D) leads to X T - μ = 1 T T X t =1 Ψ(L) Z t = 1 T T X t =1 (Ψ(1) - (L - 1)) e Ψ(L)) Z t = Ψ(1) 1 T T X t =1 Z t ! + 1 T e Ψ(L)( Z 0 - Z T ) T ( X T - μ ) = Ψ(1) T T t =1 Z t T ! + 1 T e Ψ(L) Z 0 - 1 T e Ψ(L) Z T . The assumption Z t IID(0 , σ 2 ) allows to apply the Central Limit Theo- rem C.12 of Appendix C to the first term. Thus, T T t =1 Z t T is asymp- totical normal with mean zero and variance σ 2 . Theorem D.1 also implies 2 The Beveridge-Nelson decomposition is an indispensable tool for the understanding of integrated and cointegrated processes analyzed in Chapters 7 and 16.
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76 CHAPTER 4. ESTIMATION OF MEAN AND ACF | Ψ(1) | < . Therefore, the term Ψ(1) T T t =1 Z t T is asymptotically normal with mean zero and variance σ 2 Ψ(1) 2 . The variances of the second and third term are equal to σ 2 T T j =0 e ψ 2 j . The summability condition then implies according to Theorem D.1 that T j =0 e ψ 2 j converges for T → ∞ . Thus, the variances of the last two terms converge to zero implying that these terms converge also to zero in probability (see The- orem C.7) and thus also in distribution. We can then invoke Theorem C.10 to establish the Theorem. Finally, the equality of h = -∞ γ ( h ) and σ 2 Ψ(1) 2 can be obtained from direct computations or by the application of Theo- rem 6.4. Remark 4.1. Theorem 4.2 holds for any causal ARMA process as the ψ j ’s converge exponentially fast to zero (see the discussion following equation (2.3) ). Remark 4.2. If { X t } is a Gaussian process, then for any given fixed T , X T is distributed as T ( X T - μ ) N 0 , X | h | <T 1 - | h | T γ ( h ) . According to Theorem 4.2, the asymptotic variance of the average de- pends on the sum of all covariances γ ( h ). This entity, denoted by J , is called the long-run variance of { X t } : J = X h = -∞ γ ( h ) = γ (0) 1 + 2 X h =1 ρ ( h ) !
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