1. Given a random sample, {(yi,xi),i=1,,n}, from the bivariate random variable (Y,X), the linear conditional mean: E(YX=x)=0+0x is to be estimated by OLS, ^n and ^n. The following classical assumptions are usually assumed to be hold: A1 yi=0+0xi+ei,i=1,,n. A2 xi are nonstochastic and nonconstant. A3 E(ei)=0. A4 E(ei2)=02, and E(eiej)=0 for all i=j. A5 ei are i.i.d. N(0,02). Prove the following results: (a) The sampling distribution of yi is N(0+0xi,02). (b) ^n and ^n are both linear estimators. (c) ^n is a BLUE (Best Linear Unbiased Estimator). (d) The sampling distributions of ^n and ^n are N(0,^n2) and N(0,^n2), respectively. (e) i=1nu^i2/02=(n2)^n2/02n22. Also, ^n and ^n are independent of ^n2. Note that you have to specify which the classical assumption(s) is (are) necessary in you proves. 1. Given a random sample, {(yi,xi),i=1,,n}, from the bivariate random variable (Y,X), the linear conditional mean: E(YX=x)=0+0x is to be estimated by OLS, ^n and ^n. The following classical assumptions are usually assumed to be hold: A1 yi=0+0xi+ei,i=1,,n. A2 xi are nonstochastic and nonconstant. A3 E(ei)=0. A4 E(ei2)=02, and E(eiej)=0 for all i=j. A5 ei are i.i.d. N(0,02). Prove the following results: (a) The sampling distribution of yi is N(0+0xi,02). (b) ^n and ^n are both linear estimators. (c) ^n is a BLUE (Best Linear Unbiased Estimator). (d) The sampling distributions of ^n and ^n are N(0,^n2) and N(0,^n2), respectively. (e) i=1nu^i2/02=(n2)^n2/02n22. Also, ^n and ^n are independent of ^n2. Note that you have to specify which the classical assumption(s) is (are) necessary in you proves