2. Consider the linear model with two independent variables: y=x+z+u, where x,y, and z are n1 observation vectors and u is an error vector. Assume that x and z are non-random and that u has zero mean and covariance matrix 2In. You are asked to investigate the properties of three alternative estimators for . Let b denote the estimate resulting from a least-squares regression of y on x and z. Let b be the estimate obtained by regression y on z alone. Let b be the estimate resulting from the following step-wise procedure: First one regresses y on x alone, obtaining the estimate a; then one regresses the residual vector yax on z, obtaining b. (a) Derive expressions for the mean and for the variance of each of the three estimators. (b) Show that Var(b)Var(b)Var(b). 2. Consider the linear model with two independent variables: y=x+z+u, where x,y, and z are n1 observation vectors and u is an error vector. Assume that x and z are non-random and that u has zero mean and covariance matrix 2In. You are asked to investigate the properties of three alternative estimators for . Let b denote the estimate resulting from a least-squares regression of y on x and z. Let b be the estimate obtained by regression y on z alone. Let b be the estimate resulting from the following step-wise procedure: First one regresses y on x alone, obtaining the estimate a; then one regresses the residual vector yax on z, obtaining b. (a) Derive expressions for the mean and for the variance of each of the three estimators. (b) Show that Var(b)Var(b)Var(b)