tailieunhanh - ECONOMETRICS phần 6

nơi mà sự bình đẳng thứ ba sử dụng thực tế n xi ei = 0: Kể từ khi rẽ hạn . trên dòng cuối cùng không i = 1 không phụ thuộc vào nó sau đó ước tính CLS giảm thiểu bậc hai ở phía bên phải (7,12) Đây là một (bình phương) khoảng cách Euclide trọng giữa b và: Đây là một trường hợp đặc biệt của Ga khoảng cách chung cân nhắc | CHAPTER 7. RESTRICTED ESTIMATION 139 this equation into SSEn 3 to obtain SSEn 3 X yi - x 3 i 1 X xi3 e - xi3 2 i 1 X e 3 - 3 0 Ê. 0 3 - 3 i 1 i 1 nd2 n 3 - 3 Qxx 3 - 3 where the third equality uses the fact that 22n 1 xiêi 0 Since the first term on the last line does not depend on 3 it follows that the CLS estimator minimizes the quadratic on the right-side of This is a squared weighted Euclidean distance between 3 and 3 It is a special case of the general weighted distance Jn 3 Wn n 3 - 3 W 3 - 3 for Wn 0 a k X k positive definite weight matrix. In summary we have found that the CLS estimator can be written as 3 argmin Jn 3 Qxx R p c More generally a minimum distance estimator for 3 is 3md Wn argmin Jn 3 Wn R fi c where Wn 0. We have written the estimator as 3md Wn as it depends upon the weight matrix W An obvious question is which weight matrix Wn is appropriate. We will address this question after we derive the asymptotic distribution for a general weight matrix. Computation A general method to solve the algebraic problem is by the method of Lagrange multipliers. The Lagrangian is L 3 A 2 Jn 3 Wn a R 3 - c which is minimized over 3 A The solution is 3md Wn 3 - WnR R W R r 3 - c See Exercise . 1 If we set Wn Q. . . then specializes to the CLS estimator 3md Qxx 3cls In this sense the minimum distance estimator generalizes constrained least-squares. CHAPTER 7. RESTRICTED ESTIMATION 140 Asymptotic Distribution We first show that the class of minimum distance estimators are consistent for the population parameters when the constraints are valid. Assumption R p c where R is k X q with rank R q. Theorem Consistency Under Assumption Assumption Assumption and Wn - W 0 pmd Wn - p as n 1. Theorem shows that consistency holds for any weight matrix so the result includes the CLS estimator. Similarly the constrained estimators are asymptotically normally distributed. Theorem Asymptotic .