(c)In a regression model +E, switching the independent and dependent varia bles and running a least pr ralid est imator of 1 d)r tends te 5. Inst ead of regressor matrix X, its rotation definded Z= XA with a K x K matrix A is used as a regressor. Show that the residuals from this regression are the same as those from the regression using regressor X 2 Finite sample properties of the OLS estimator 1. Suppose that the regression model is gi=a+BT i +Ei every E: has densit f(a)=exp(x Note that E(Ei)=A for all i. Show that the least squares estimat or of B is unbiased but that the lse of a is not Suppose that you uncessarily included a constant term in a bivariate linear regression model. In other words, the true model is i=B. C;+ Ei, whereas you est imated the model a+Bxz+∈z (a) Is the Ols estimator of B unbiased (b)Is the Ols estimator of B more efficient than that from the true model? 3.(Extension of 2). The true model is But the model 2z2+Bx1+ was estimated2 (c) In a regression model yi = αxi + εi , switching the independent and dependent variables and running a least squares provide a valid estimator of 1 α . (d) R¯2 tends to favor larger models. 5. Instead of regressor matrix X, its rotation definded Z = XA with a K × K matrix A is used as a regressor. Show that the residuals from this regression are the same as those from the regression using regressor X. 2 Finite sample properties of the OLS estimator 1. Suppose that the regression model is yi = α + βxi + εi , every εi has density f (x) = exp − x λ  /λ, x ≥ 0. Note that E (εi) = λ for all i. Show that the least squares estimator of β is unbiased but that the LSE of α is not. 2. Suppose that you uncessarily included a constant term in a bivariate linear regression model. In other words, the true model is yi = βxi + εi , whereas you estimated the model yi = α + βxi + εi . (a) Is the OLS estimator of β unbiased? (b) Is the OLS estimator of β more efficient than that from the true model? 3. (Extension of 2). The true model is yi = β ′ xi + εi . But the model yi = α ′ zi + β ′ xi + εi was estimated.