1. Consider the linear regression model y=x11+x22 + , where y, x and x2 are N x 1 column vectors, and e is an N x 1 vector of regression errors; B1 and B2 are scalar parameters. Assume that x1 is endogenous, while X2 is exogenous, i.e. E[xe] 0, while E[x2] = 0. (a) [25%] Show that the OLS estimator B2 of 32 cam be expressed as =2+ (x2(IN Px)x2)2 (IN - Px), - where I is the identity matrix of dimension N and the matrix Px =x1(xx)1. (b) [25%] Use the result in (a) to show that B2 is unbiased if the re- gressors X and x2 are orthogonal, irrespective of the endogeneity of X1. (c) [25%] Suppose an N x 1 vector z is available that is considered as an instrument for X. State the conditions that z needs to satisfy to be a valid instrument for X, and explain how it can be used to obtain an unbiased estimator for 3' = (B1, B2). (d) [25%] Show that the IV estimator for B2 is equivalent to the OLS estimator for B2 when X and x2 are orthogonal.