Consider the linear regression model Yi=Xi+Ui, where Yi and Ui are scalar random variables, Xi is a vector of K random variables, and is a conformable vector of coefficients. Further, (Yi,Xi) i.i.d, E(YiXi)=Xi, and there is no perfect multicollinearity among the variables in Xi. Let the OLS estimator for using a random sample of N observations be ^=(i=1NXiXi)1i=1NXiYi. 1. (11 marks) Which of the following estimators are unbiased for ? Which are consistent for ? Explain. (Note: You don't need to show each of these properties from first principles; you may reference known results.) (i) ^ (ii) N1/2(^) (iii) ^+N/(N1) (iv) ^N/(N1) 2. (9 marks) For each of the following random variables, discuss if they have an asymptotic distribution or not. If yes, state the distribution. In each case, give an explanation why the random variable has or does not have an asymptotic distribution. (i) N1/4(^) (ii) N2/4(^) (iii) N3/4(^) Consider the linear regression model Yi=Xi+Ui, where Yi and Ui are scalar random variables, Xi is a vector of K random variables, and is a conformable vector of coefficients. Further, (Yi,Xi) i.i.d, E(YiXi)=Xi, and there is no perfect multicollinearity among the variables in Xi. Let the OLS estimator for using a random sample of N observations be ^=(i=1NXiXi)1i=1NXiYi. 1. (11 marks) Which of the following estimators are unbiased for ? Which are consistent for ? Explain. (Note: You don't need to show each of these properties from first principles; you may reference known results.) (i) ^ (ii) N1/2(^) (iii) ^+N/(N1) (iv) ^N/(N1) 2. (9 marks) For each of the following random variables, discuss if they have an asymptotic distribution or not. If yes, state the distribution. In each case, give an explanation why the random variable has or does not have an asymptotic distribution. (i) N1/4(^) (ii) N2/4(^) (iii) N3/4(^)