Question 171 175

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Consider a linear model with IID data where all variables are scalars. 1. Suppose that r and e are correlated, but there is an / x 1 strong "instrument" z weakly correlated with e. Derive the asymptotic (as n - co) distributions of the 2SLS estimator of 8, its t ratio, and the overidentification test statistic J =n u'z(Z'Z)-12'U where U = V - BX' are the vector of 2SLS residuals and Z is the matrix of instruments, under the drifting DGP w = cw/vn, where w is the vector of coefficients on z in the linear projection of e on z. Also, specialize to the case { = 1. 2. Suppose that r and e are correlated, but there is an f x I weak "instrument" z weakly correlated with e. Derive the asymptotic (as n - co) distributions of the 2SLS estimator of 8, its t ratio, and the overidentification test statistic 7, under the drifting DGP w = cw/vn and * = c/vn, where w is the vector of coefficients on z in the linear projection of e on z, and * is the vector of coefficients on a in the linear projection of a on 2. Also, specialize to the case { = 1.Derive the second order bias of the Maximim Likelihood (ML) estimator & of the parameter > > () of the exponential distribution f (v. A) = 1 0, Xexp(-Ay). # 20 #