(1) (4 points) Show that the sample variance 3: is biased for the true variance 02. (2) (3 points) How would you correct the bias? (3) (3 points) What is the bias of the infeasible variance estimator 3:,in = % 231:1(It \")2. Why am I calling this estimator infeasible? Now, let us turn to the largeT (or innitesample or asymptotic) properties of .33. Write the following: 1 T _ f 2% XV i=1 Zaa~t ,u) (Ymiz 8?; 423s: m22(7m%;m wwfW (I) - (a) (a) (5) Now, subtract 02 from the lefthand side and from the righthand side of Eq. (1) and standardize by x/T to obtain: ms: a?) = W 2(Y m% 2m is) We a? ,/ 13:1 (6*) (a*) (b*) (4) (4 points) Show that s: is consistent for 02 by applying the WLLN to (a), (b) and (c) in Eq (1). (5) (4 points) Show that MTG: U2) is asymptotically normal by ap plying the WLLN, the CLT and Slutsky's theorem to (cf), (5\") and (c*) in Eq. (2) Problem 2 (Asymptotic methods). (30 points) Assume an iid sample {x1, X2, ..., x7} from some distribution with expected value u and variance o. A natural estimator for the true variance (i.e., o2) of the random variable which generates the data is the sample variance, namely s, = X)2, where X defines the sample mean, i.e., X = T Lt-1: First, let us focus on the finite-T (or finite-sample) properties of s2