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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 7.12 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 7.12 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 7.13 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. 7.5 Computation A general method to solve the algebraic problem 7.13 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 7.14 See Exercise 7.5. 1 If we set Wn Q. . . then 7.14 specializes to the CLS estimator 3md Qxx 3cls In this sense the minimum distance estimator generalizes constrained least-squares. CHAPTER 7. RESTRICTED ESTIMATION 140 7.6 Asymptotic Distribution We first show that the class of minimum distance estimators are consistent for the population parameters when the constraints are valid. Assumption 7.6.1 R p c where R is k X q with rank R q. Theorem 7.6.1 Consistency Under Assumption 1.5.1 Assumption 3.16.1 Assumption 7.6.1 and Wn - W 0 pmd Wn - p as n 1. Theorem 7.6.1 shows that consistency holds for any weight matrix so the result includes the CLS estimator. Similarly the constrained estimators are asymptotically normally distributed. Theorem 7.6.2 Asymptotic .