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ECONOMETRICS phần 7

được xây dựng bằng cách sử dụng một e ước tính sơ bộ phù hợp, có thể thu được bằng cách . rst ^ thiết lập W n = I: Kể từ khi ước tính GMM phụ thuộc vào ước tính giai đoạn rẽ ., thường là trọng lượng ma trận W n được cập nhật, và sau đó b recomputed. Ước tính này có thể được lặp nếu cần thiết. | CHAPTER 9. ADDITIONAL REGRESSION TOPICS 168 One motivation for the choice of NLLS as the estimation method is that the parameter 0 is the solution to the population problem mine E yi m Xi 0 2 Since sum-of-squared-errors function Sn 0 is not quadratic 0 must be found by numerical methods. See Appendix E. When m X 0 is differentiable then the FOC for minimization are n 0 X me pi b êi. i 1 9.7 Theorem 9.4.1 Asymptotic Distribution of NLLS Estimator If the model is identified and m X 0 is differentiable with respect to 0 pn 0 - 0 - N 0 Ve Ve E mw m 1 E m máie2 E mw m i 1 where m i me Xi 0o . Based on Theorem 9.4.1 an estimate of the asymptotic variance Ve is V e n Xmeimej n X meim eiêì i 1 i 1 1 n 1 nx m eim ei i 1 where m ei me Xi 0 and êj yi m xi 0 . Identification is often tricky in nonlinear regression models. Suppose that m xi 0 Ppi P 2Xi 7 where Xi 7 is a function of Xi and the unknown parameter 7. Examples include Xi 7 xl Xi 7 exp 7Xi and Xi 7 Xi1 g xi 7 . The model is linear when P2 0 and this is often a useful hypothesis sub-model to consider. Thus we want to test Ho P2 0. However under Ho the model is yi P1zi ei and both P2 and 7 have dropped out. This means that under Ho 7 is not identified. This renders the distribution theory presented in the previous section invalid. Thus when the truth is that P2 0 the parameter estimates are not asymptotically normally distributed. Furthermore tests of Ho do not have asymptotic normal or chi-square distributions. The asymptotic theory of such tests have been worked out by Andrews and Ploberger 1994 and B. Hansen 1996 . In particular Hansen shows how to use simulation similar to the bootstrap to construct the asymptotic critical values or p-values in a given application. Proof of Theorem 9.4.1 Sketch . NLLS estimation falls in the class of optimization estimators. For this theory it is useful to denote the true value of the parameter 0 as 0o . The first step is to show that 0 - 00. Proving that nonlinear estimators are .