2. Consider the linear regression model y=X0+u, where y is the T1 vector of observations on the dependent variable, X is a Tk matrix containing the observations on the explanatory variables, 0 is the k1 vector of regression coefficients, and u denotes the error term. Assume that (2) is the true model for y,X is fixed in repeated samples with rank(X)=k and uN(0,02IT). The OLS estimator of 0 is ^T=(XX)1Xy. Suppose that a researcher believes that the regression coefficients satisfy R0=r where R is a nrk matrix of specified constants with rank(R)=nr and r is a nr1 vector of specified constants. As a result, she estimates (2) subject to R0=r via Restricted Least Squares (RLS). Define ^T=(R(XX)1R)1(R^Tr). (a) What is the interpretation of ^T within the context of the RLS estimation described above? [3 marks] (b) Assume that R0=r. Show that ^TN(0,02{R(XX)1R}1). [6 marks] (c) Suppose now that the restrictions do not hold that is, R0r=0. What is the sampling distribution of ^T ? Justify your answer. [5 marks] Now suppose that the model for y is given by (2) but assume: X is stochastic with rank(X)=k with probability one, and the conditional distribution of u given X is N(0,02IT). (d) A researcher claims that if the restrictions hold - that is, R0=r - then the sampling distribution of ^T is given by (4). Evaluate whether this claim is correct. Briefly justify your conclusion. [6 marks] 2. Consider the linear regression model y=X0+u, where y is the T1 vector of observations on the dependent variable, X is a Tk matrix containing the observations on the explanatory variables, 0 is the k1 vector of regression coefficients, and u denotes the error term. Assume that (2) is the true model for y,X is fixed in repeated samples with rank(X)=k and uN(0,02IT). The OLS estimator of 0 is ^T=(XX)1Xy. Suppose that a researcher believes that the regression coefficients satisfy R0=r where R is a nrk matrix of specified constants with rank(R)=nr and r is a nr1 vector of specified constants. As a result, she estimates (2) subject to R0=r via Restricted Least Squares (RLS). Define ^T=(R(XX)1R)1(R^Tr). (a) What is the interpretation of ^T within the context of the RLS estimation described above? [3 marks] (b) Assume that R0=r. Show that ^TN(0,02{R(XX)1R}1). [6 marks] (c) Suppose now that the restrictions do not hold that is, R0r=0. What is the sampling distribution of ^T ? Justify your answer. [5 marks] Now suppose that the model for y is given by (2) but assume: X is stochastic with rank(X)=k with probability one, and the conditional distribution of u given X is N(0,02IT). (d) A researcher claims that if the restrictions hold - that is, R0=r - then the sampling distribution of ^T is given by (4). Evaluate whether this claim is correct. Briefly justify your conclusion. [6 marks]
