| Sahiplik | Değer |
| İsim | II.II.1 OLS for Multiple Regression |
| Açıklama | II.II.1 OLS for Multiple Regression The general linear statistical model can be described in matrix notation as (II.II.1-1) where y is a stochastic T*1 vector, X is a deterministic (exogenous) T*K matrix, b is a K*1 vector of invariant parameters to be estimated by OLS, e is a T*1 disturbance vector, T is the number of observations in the sample, and K is the number of exogenous variables used in the right hand side of the econometric equation. It is furthermore assumed that (II.II.1-2) which is the equivalent matrix expression of the weak set of assumptions under section II.I.3. The least squares estimator minimizes ee (the sum of squared residuals). Solving the normal equations XXb = Xy with respect to b yields (II.II.1-3) where XX must be a non singular symmetric K*K matrix! Obviously, the OLS estimator is unbiased (II.II.1-4) since E(Xe) = 0 by assumption (X is exogenous). This result can be proved quite easily. Note that if X is not exogenously given (thus stochastic) the small sample property of unbiasedness only holds if E(Xe) = 0. Under the assumption of OLS it can be proved that the covariance matrix of the parameters is (II.II.1-5) The Gauss-Markov theorem states that if (II.II.1-6) then any other estimator (II.II.1-7) has a parameter covariance matrix which is at least as large as the covariance matrix of the OLS parameters (II.II.1-8) This important theorem therefore proves that the OLS estimator is a best linear unbiased estimator (BLUE). If D* is a K by T matrix which is independent from y and if (II.II.1-9) the parameter vector is by definition a linear estimator, and if (II.II.1-10) then it follows that (II.II.1-11) Evidently, it follows from (II.II.1-11) that the parameter vector can only be unbiased if DX = 0 and if E(D*e) = 0. Now what happens to the covariance matrix of this estimator? Obviously, we find (II.II.1-12) which proves the theorem (on comparing (II.II.1-12) with ... |
| Dosya Adı | Bağlantı www.edubilim.com_ekonometri1-22-6.doc |
| Dosya Boyutu | Link |
| Dosya Tipi | doc (Mime Tipi: link) |
| Ekleyen | admin |
| Ekleme tarihi: | 15.10.2008 16:13 |
| İnceleyenler | Herkes |
| Kontrol Eden | Editor |
| İndirme sayısı | 2 İndirme sayısı |
| En son güncelleme | 14.10.2008 16:14 |
| Anasayfa | http://www.edubilim.com |
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