15 32 Properties of b \u03b3 and b \u03c1 b \u03b3 n and b \u03c1 are biased but consistent as n

15 32 properties of b γ and b ρ b γ n and b ρ are

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Properties of b γ and b ρ b γ n and b ρ are biased but consistent as n → ∞ under certain conditions. Theorem 3 (CLT of b ρ ) If { X t } is the stationary process X t - μ = X j = -∞ ψ j Z t - j , { Z t } ∼ IID (0 , σ 2 ) , where j = -∞ | ψ j | < and E Z 4 < , then for each h , we have n ( b ρ ( h ) - ρ ( h )) = ⇒ N (0 , W h ) , where b ρ ( h ) = [ b ρ (1) , · · · , b ρ ( h )] T and ρ ( h ) = [ ρ (1) , · · · , ρ ( h )] T . W h is a non-negative definition matirx (see Theorem 7.2.1 of B & D for the expression of W ). 16 / 32
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Estimation of ARMA Models 17 / 32
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ARMA Modeling The determination of an appropriate ARMA(p,q) model to represent an observed stationary time series involves a number of inter-related problems. The choice of p and q (Order Selection); Estimation of ARMA coefficients and the white noise variance, given p and q ; Check goodness of fit. Throughout we assume the data has been adjusted by subtraction of the mean. Therefore, we only consider zero-mean ARMA models. The modeling procedure often starts with plotting the estimated ACF and PACF of the data. If ACF decreases to 0 quickly, the process is suspected to be a MA process. If PACF decreases to 0 quickly, the process is suspected to be an AR process. First obtain preliminary estimators of φ ’s and/or θ ’s. More efficient esimation: maximum likelihood estimators . Need preliminary estimators as initial values for optimization. 18 / 32
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Preliminary Estimation of AR(p) Preliminary estimation of AR coefficients and PACF estimators can be obtained by the Yule-Walker Equation and Durbin-Levinson Algorithm. Let { X t } be the zero-mean causal autoregressive process X t = φ 1 X t - 1 + · · · + φ p X t - p + Z t , { Z t } ∼ WN (0 , σ 2 ) . Our aim is to find estimators of the coefficient vector φ = ( φ 1 , . . . , φ p ) T and the white noise variance σ 2 based on the observations X 1 , . . . , X n . Multiplying both sides by X t - j , j = 0 , . . . , p and taking expectations, we obtain the Yule-Walker equations, Γ p φ = γ p , and σ 2 = γ (0) - φ T γ p . We replace the covariance γ ( j ) by the corresponding sample covariances b γ ( j ) , we obtain a set of equations for the so-called Yule-Walker estimators b φ and b σ 2 of φ and σ 2 , namely b Γ p b φ = b γ p and b σ 2 = b γ (0) - b φ T b γ p . 19 / 32
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Distribution of b φ Theorem 4 If { X t } is the causal AR(p) process with { Z t } ∼ IID (0 , σ 2 ) , and b φ is the Yule-Walker estimator of φ , then n 1 / 2 ( b φ - φ ) = ⇒ N (0 , σ 2 Γ - 1 p ) , Moreover, b σ 2 P -→ σ 2 . 20 / 32
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Order Misspecification If the true order is p and we attempt to fit a process of order m for m n - 1 , we should expect the estimated coefficient vector φ m = ( b φ m 1 , · · · , b φ mm ) T to have a small value of b φ mm for each m > p , since b φ mm is the estimator of the PACF at lag m . The following theorem is extremely useful in helping us to identify the appropriate order of the process to be fitted.
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