thứ tự của Taylor g chức năng phi tuyến (θ) là một ước lượng là giá trị thực sự của xung quanh tham số θ. Giải thích lý do tại sao thời hạn cuối cùng là tiệm không đáng kể so với nhiệm kỳ thứ hai. 5,14 Sử dụng Taylor đa biến để mở rộng đầu tiên, cho thấy rằng, nếu ma trận hiệp phương sai tiệm cận của | 348 Instrumental Variables Estimation Sketch a proof of the result that expression 2 U PPwX - pPwXi U a0 is asymptotically distributed as  k 2 when the vector u is IID 0 CTqI and is asymptotically uncorrelated with the instruments W. Here k 2 k ki where X has k columns and Xi has ki columns. The IV variant of the HRGNR evaluated at 3 3 can be written as c PUPwX U 1PWXb residuals where c is an n-vector of which every component equals 1 and ci is an n n diagonal matrix with tth diagonal element equal to the tth element of the vector y X . Verify that this artificial regression possesses all the requisite properties for hypothesis testing namely that The regressand in is orthogonal to the regressors when 3 3iv The estimated OLS covariance matrix from evaluated at 3 3iv is equal to n n k times the HCCME Varh 3lV given by The HRGNR allows one-step estimation The OLS parameter estimates b from are such that 3iv 3 b. Show that nR2 from the modified IVGNR is equal to the Sargan test statistic that is the minimized IV criterion function for model divided by the IV estimate of the error variance for that model. Consider the following OLS regression where the variables have the same interpretation as in Section on DWH tests y X3 MwYC u. Show that an F test of the restrictions 0 in is numerically identical to the F test for Ỏ 0 in . Show further that the OLS estimator of 3 from is identical to the estimator 3iv obtained by estimating by instrumental variables. Show that the difference between the generalized IV estimator 3iv and the OLS estimator 3ols for which an explicit expresion is given in equation has zero covariance with 3ols itself. For simplicity you may treat the matrix X as fixed. Using the same methods as those in Sections and show that the nonlinear version of the IVGNR satisfies the three conditions analogous to those set out in Exercise
