Question

a) In lecture we derived the estimate of B in WLS as Derive A.wls when p-1. (It should have a form similar to simple linear regression.) (Hints: Notice that we can write a weighted average as analogues of the sums of squares identities we've used; you should derive these if you need to use them.) . You may need to use weighted b) Assume we have the following data: T1 T2 y2 That is, we have a total of n observations, but the values of X and Y are equivalent for observations n - 1 and n. Consider the following weights: Wi This is a weighting variation called replication weights if multiple observations can have identical values (that is, for some i j, yi -yj and X, - Xj), instead of storing both rows, we store a single row with those values along with a count of the number of replicate observations Show that the following is true under these conditions: LD Tn Note that this is more generally true with any replication weights, but you only need show it with the specific weights defined above.) c) Consider fitting the following two models based upon the data described in part b). where W is wi on the diagonal as describe in part b). In (1), you fit OLS on the whole data set. In (2), you fit WLS excluding the nth row, and weighting the n 1th observation double. Show that A,ols β1,wls. (Hints: Be careful about summation indices and be careful about sample size. You'll want to use your results from both previous parts.)