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52 ols estimation of an arp ordinary least squares

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5.2. OLS ESTIMATION OF AN AR(P) MODEL 97 5.2 Ordinary least-squares (OLS) estimation of an AR(p) model An alternative approach is to view the AR model as a regression model for X t with regressors X t - 1 , . . . , X t - p and error term Z t : X t = φ 1 X t - 1 + . . . + φ p X t - p + Z t , Z t WN(0 , σ 2 ) . Given observation for X 1 , . . . , X T , the regression model can be compactly written in matrix algebra as follows: X p +1 X p +2 . . . X T = X p X p - 1 . . . X 1 X p +1 X p . . . X 2 . . . . . . . . . . . . X T - 1 X T - 2 . . . X T - p φ 1 φ 2 . . . φ p + Z p +1 Z p +2 . . . Z T , Y = X Φ + Z. (5.2) Note that the first p observations are lost and that the effective sample size is thus reduced to T - p . The least-squares estimator (OLS estimator) is obtained as the minimizer of the sum of squares S (Φ): S (Φ) = Z 0 Z = ( Y - X Φ) 0 ( Y - X Φ) = T X t = p +1 ( X t - φ 1 X t - 1 - . . . - φ p X t - p ) 2 = T X t = p +1 ( X t - P t - 1 X t ) 2 -→ min Φ . (5.3) Note that the optimization problem involves no constraints, in particular causality is not imposed as a restriction. The solution of this minimization problem is given by usual formula: b Φ = ( X 0 X ) - 1 ( X 0 Y ) . Although equation (5.2) resembles very much an ordinary regression model, there are some important differences. First, the usual orthogonality as- sumption between regressors and error is violated. The regressors X t - j , j = 1 , . . . , p , are correlated with the error terms Z t - j , j = 1 , 2 , . . . . In addi- tion, there is a dependency on the starting values X p , ..., X 1 . The assumption of causality, however, insures that these features do not play a role asymp- totically. It can be shown that ( X 0 X ) /T converges in probability to b Γ p and
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98 CHAPTER 5. ESTIMATION OF ARMA MODELS ( X 0 Y ) /T to b γ p . In addition, under quite general conditions, T - 1 / 2 X 0 Z is asymptotically normally distributed with mean 0 and variance σ 2 Γ p . Then by Slutzky’s Lemma C.10, T ( b Φ - Φ) = ( X 0 X T ) - 1 X 0 Z T converges in distri- bution to N(0 , σ 2 Γ - 1 p ) Thus, the OLS estimator is asymptotically equivalent to the Yule-Walker estimator. Theorem 5.2 (Asymptotic property of the Least-Squares estimator) . Under the same conditions as in Theorem 5.1 the OLS estimator b Φ = ( X 0 X ) - 1 ( X 0 Y ) : T b Φ - Φ d ----→ N ( 0 , σ 2 Γ - 1 p ) , plim s 2 T = σ 2 where s 2 T = b Z 0 b Z/T and b Z t are the OLS residuals. Proof. See Chapter 13 and in particular section 13.3 for a proof in the mul- tivariate case. Additional details may be gathered from Brockwell and Davis (1991, chapter 8). Remark 5.1. In practice σ 2 Γ - 1 p is approximated by s 2 T ( X 0 X /T ) - 1 . Thus, for large T , b Φ can be viewed as being normally distributed as N(Φ , s 2 T ( X 0 X ) - 1 ) . This result allows the application of the usual t- and F-tests. Because the regressors X t - j , j = 1 , . . . , p are correlated with the errors terms Z t - j , j = 1 , 2 , . . . , the Gauss-Markov theorem cannot be applied. This implies that the least-squares estimator is no longer unbiased in finite sam- ples. It can be shown that the estimates of an AR(1) model are downward biased when the true value of φ is between zero and one. MacKinnon and
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