1. Consider the simple linear regression model Y 30+3 1X+U. Suppose that SLR.1-SLR.4 hold. Define Z, =g(Xi) for some known function g, and consider the following estimator for3 1: B1= il(Zi - Z)Yi Zi=(Zi - Z) Xi Assume that 1 (Z, - Z) X, is always nonzero so that 1 is well defined. (i) Show that 1 is a linear estimator, i.e., we can write3 1 as B 1= CY for some Ci's, where Ci is a function of X1, ..., Xn. (ii) Show that B1Bit CUi. (iii) Show that 1 is an unbiased estimator. (iv) Suppose that Var(U|X) = 02 (SLR.5). Show that Vare 1(X1, ..., Xn) = 02 BI(Zi - Z)2 $1(Z - Z)X; 2 You can use the fact that, under SLR.1-SLR.2, n Var CUiX1, ..., Xn \\ = C?Var(UilXi). (v) Lets 1 be the OLS slope estimator from a regression of Y on X. Show that, under SLR.1- SLR.5, Varg 1(X1, ..., Xn)