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

This preview shows page 15 - 22 out of 34 pages.

15 / 32
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
Estimation of ARMA Models 17 / 32
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
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
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
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.

You've reached the end of your free preview.

Want to read all 34 pages?

• Fall '14