1. Suppose we have a standard linear model equation, in matrix form, on a sample with n observations. The model has k predictive variables and a constant term. So: y = XB+ where y is n x 1, X is n x (k+1) and of full rank, is (k+1) 1, e is n x 1, E[e|X] = 0, and Var[e|X] = o In. Further, 3 = (X'X)X'y, and e=y - X. (a) Show that is an unbiased estimator of 3. [11 marks] (b) Suppose k=2 in the linear model above, and a researcher is working on an empirical application where economic theory says E[y x] = f(x) = = x; # x. (i) Although the function f(.) is non-linear, we can still test this theory using an F-test. Explain why this is possible. [5 marks] (ii) State the null and alternative hypotheses for testing this theory, and find the formula for the restricted sum of squares for this particular specification. [10 marks] (iii) Derive the form of the elements of the X'X matrix needed for this regression model. [7 marks]