TimeSeriesBook.pdf

0 the last equality follows from the observation that

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= 0 . The last equality follows from the observation that proposition 13.1 implies plim X 0 X T = Γ p nonsingular and that proposition 13.2 implies plim Z X T = 0. Thus, we have established that the Least-Squares estimator is consistent. Equation (13.7) further implies: T (vec b Φ - vec Φ) = T ( (( X 0 X ) - 1 X 0 ) I n ) vec Z = X 0 X T - 1 I n 1 T ( X 0 I n ) vec Z As plim X 0 X T = Γ p nonsingular, the above expression converges in distribution according to Theorem C.10 and Proposition 13.2 to a normally distributed random variable with mean zero and covariance matrix ( Γ - 1 p I n )( Γ p Σ)( Γ - 1 p I n ) = Γ - 1 p Σ
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254 CHAPTER 13. ESTIMATION OF VAR MODELS Proof of Theorem 13.2 Proof. b Σ = ( Y - b Φ X 0 )( Y - b Φ X 0 ) 0 T = ( Y - Φ X 0 + (Φ - b Φ) X 0 )( Y - Φ X 0 + (Φ - b Φ) X 0 ) 0 T = 1 T ( Z + (Φ - b Φ) X 0 )( Z + (Φ - b Φ) X 0 ) 0 = ZZ 0 T + Z X T - b Φ) 0 + (Φ - b Φ) X 0 Z 0 T + (Φ - b Φ) X 0 X T - b Φ) 0 Applying Theorem C.7 and the results of propositions 13.1 and 13.2 shows that Z X - b Φ) 0 T p ----→ 0 and - b Φ) X 0 X T T - b Φ) 0 p ----→ 0 . Hence, T ( Y - b Φ X 0 )( Y - b Φ X 0 ) 0 T - ZZ 0 T ! = T b Σ - ZZ 0 T p ----→ 0 13.4 The Yule-Walker Estimator An alternative estimation method can be derived from the Yule-Walker equa- tions. Consider first a VAR(1) model. The Yule-Walker equation in this case simply is: Γ(0) = ΦΓ( - 1) + Σ Γ(1) = ΦΓ(0) or Γ(0) = ΦΓ(0)Φ 0 + Σ Γ(1) = ΦΓ(0) .
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13.4. THE YULE-WALKER ESTIMATOR 255 The solution of this system of equations is: Φ = Γ(1)Γ(0) - 1 Σ = Γ(0) - ΦΓ(0)Φ 0 = Γ(0) - Γ(1)Γ(0) - 1 Γ(0)Γ(0) - 1 Γ(1) 0 = Γ(0) - Γ(1)Γ(0) - 1 Γ(1) 0 . Replacing the theoretical moments by their empirical counterparts, we get the Yule-Walker estimator for Φ and Σ: b Φ = b Γ(1) b Γ(0) - 1 , b Σ = b Γ(0) - b Φ b Γ(0) b Φ 0 . In the general case of a VAR(p) model the Yule-Walker estimator is given as the solution of the equation system b Γ( h ) = p X j =1 b Φ j b Γ( h - j ) , k = 1 , . . . , p, b Σ = b Γ(0) - b Φ 1 b Γ( - 1) - . . . - b Φ p b Γ( - p ) As the the least-squares and the Yule-Walker estimator differ only in the treatment of the starting values, they are asymptotically equivalent. In fact, they yield very similar estimates even for finite samples (see e.g. Reinsel (1993)). However, as in the univariate case, the Yule-Walker estimator always delivers, in contrast to the least-square estimator, coefficient estimates with the property det( I n - ˆ Φ 1 z - . . . - ˆ Φ p z p ) 6 = 0 for all z C with | z | ≤ 1. Thus, the Yule-Walker estimator guarantees that the estimated VAR possesses a causal representation. This, however, comes at the price that the Yule-Walker estimator has a larger small-sample bias than the least-squares estimator, especially when the roots of Φ( z ) get close to the unit circle (Tjøstheim and Paulsen, 1983; Shaman and Stine, 1988; Reinsel, 1993). Thus, it is generally preferable to use the least-squares estimator in practice.
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256 CHAPTER 13. ESTIMATION OF VAR MODELS
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Chapter 14 Forecasting with VAR Models 14.1 Forecasting with Known Parameters The discussion of forecasting with VAR models proceeds in two steps. First, we assume that the parameters of the model are known. Although this as- sumption is unrealistic, it will nevertheless allow us to introduce and analyze important concepts and ideas. In a second step, we then investigate how the
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