Question 1;

Consider the regression V = XB+ E, where the regressors \\' are correlated with the error &, but this correlation is weak. Consider the decomposition of & to its projection on A and the orthogonal component : E = NT +U. Assume that (n A"'X, n 1/2xu) 4 (Q, (), where $ ~ N (0,02Q) and Q has full rank. Show that under the assumption of the drifting parameter DGP * = c/vn, where n is the sample size and c is fixed, the OLS estimator of 8 is consistent and asymptotically noncentral normal, and derive the asymptotic distribution of the Wald test statistic for testing the set of linear restriction RB = r, where R has full rank q.Consider a linear model with IID data y = Bate, where all variables are scalars. 1. Suppose that a and e are correlated, but there is an ( x 1 strong "instrument" = weakly correlated with e. Derive the asymptotic (as n - co) distributions of the 2SLS estimator of B, its t ratio, and the overidentification test statistic niZI-( zz)zin J =n wu where u = ) - BX are the vector of 2SLS residuals and Z is the matrix of instruments, under the drifting DGP w =co/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 / x 1 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 J, under the drifting DGP w = cw/vn and # = cx/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 z in the linear projection of a on 2. Also, specialize to the case / = 1.Consider estimation of a scalar parameter / on the basis of the moment function m(x, y, 0) = and IID data (r;, yi), i = 1. ..., n. Show that the second order asymptotic bias of the empirical likelihood estimator of 0 equals 0.Derive the second order bias of the Maximim Likelihood (ML) estimator ) of the parameter A > 0 of the exponential distribution f(y, A) = Xexp(-Ay), y 2 y