tailieunhanh - foundations of econometrics phần 7

có tính đến một thực tế rằng β cũng đã được ước tính. Nếu ma trận thông tin không chặn đường chéo, trong nhiều trường hợp khác, nó không phải là, nó sẽ cần thiết để đảo ngược toàn bộ ma trận để có được bất kỳ khối nghịch đảo. | The Covariance Matrix of the ML Estimator 413 taking account of the fact that p has also been estimated. If the information matrix were not block-diagonal which in most other cases it is not it would have been necessary to invert the entire matrix in order to obtain any block of the inverse. Asymptotic Efficiency of the ML Estimator A Type 2 ML estimator must be at least as asymptotically efficient as any other root-n consistent estimator that is asymptotically Therefore at least in large samples maximum likelihood estimation possesses an optimality property that is generally not shared by other estimation methods. We will not attempt to prove this result here see Davidson and MacKinnon 1993 Section . However we will discuss it briefly. Consider any other root-n consistent and asymptotically unbiased estimator say 0. It can be shown that plim n1 2 0 00 plim nr 2 0 00 v where v is a random k -vector that has mean zero and is uncorrelated with the vector plimn1 2 00 . This means that from we have Var plim n1 2 Ô 00 Var plim n1 2 0 00 Var v . nn nn Since Var v must be a positive semidefinite matrix we conclude that the asymptotic covariance matrix of the estimator must be larger than that of 0 in the usual sense. The asymptotic equality bears a strong and by no means coincidental resemblance to a result that we used in Section when proving the Gauss-Markov Theorem. This result says that in the context of the linear regression model any unbiased linear estimator can be written as the sum of the OLS estimator and a random component which has mean zero and is uncorrelated with the OLS estimator. Asymptotically equation says essentially the same thing in the context of a very much broader class of models. The key property of is that v is uncorrelated with plim n1 2 ớ 00 . Therefore v simply adds additional noise to the ML estimator. The asymptotic efficiency result is really an asymptotic version of the .