1. Consider the linear regression model y = X0 (a) Obtain the formula for B that minimises Q ($) = u'Wu, where W is a full rank matrix. [Hint: You can use the chain rule fo matrix differentiation]...

1. Consider the linear regression model y = X0 (a) Obtain the formula for B that minimises Q ($) = u'Wu, where W is a full rank matrix. [Hint: You can use the chain rule fo matrix differentiation] Under which conditions is this the GIV1M estimator? (b) Show that this simplifies to the OLS estimator if W = I (idenity matrix). (c) Show it simplifies if W = SF'. Under which conditions is this the GLS estimator? (d) Show that this simplifies to the 2SLS estimator if W = Z where Z correspond to some valid instruments. Would this estimator still be unbiased if Var(u) # erq? 1. What essential assumptions must Z satisfy for the IV esimator to be consistent for 0? Explain. 2. Show that given just identification the 2SLS estimator reduces to the IV estimator.
3. Give a real-world example of a situation where IV estimation is needed because of inconsticency of OLS and specify suitable instruments. 4. Consider a probit model where, for a given vector of independent variables, = 110i] = (1) Consider a GMM estimator based on the result that, (2) What assumption do we need to get the following moment condition? : -ERN = (3) Show how to construct a GMM estimator based on these results. (let say a 2-step, with / as the weighting matrix in the first step) What would be the moment condition for a nonlinear least squares esti-mation of this model?