2. Consider the simple linear regression model with a continuous explanatory variable: Y = Bo + Bi* X + U (1) and assume that we have data from a randomized experiment. Given a random sample of size N > 2 from the population of interest, the OLS-estimator is Li= 22-1 (X; X) * (Y; Y) (2) = (X; X)" Under the stated assumptions this is an unbiased and consistent estimator of B1. In this question we explore the properties of an alternative estimator of B1. Let (Y1, X1) and (Y2, X2) be the first and second observations in the data. We define the "rise-over- -ROR run" estimator B1 of the slope of the regression line, B1, by ROR = Y2 Y1 X2 X1 (3) -ROR (a) Offer an interpretation of B1 Why is "rise-over-run" an appropriate name for this estimator? (b) Show that ~ROR U2 U Bi = Bit X2 X1 ROR ROR (c) Argue formally that B1 is an inconsistent estimator of B1. Indicate the results you are relying on to establish this result. (d) Show that Brom is an unbiased estimator of 31. To this end, first treat (X2, X1) as fixed and show that E B1 *|X2, X1 = B1. Then argue verbally why this implies that E [BROR] = B12 (e) Assume that var (U|X) = 02. Show that holding fixed the (X2, X) the following result is true: - 2*02 ROR. var (B1 * \X2, X) = x; - ,)?" ~ROR (f) Show that, holding fixed the values of X, B. has larger variance than the OLS- estimator By whenever N > 2. You will need to use the following result, which you can take as given: (X2 - X)?