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Consider two separate linear regression models and For concreteness, assume that the vector yi contains observations on the wealth ofn randomly selected individuals in Australia and y2 contains observations on the wealth of n randomly selected individuals in New Zealand. The matrix Xi contains n observations on ki explanatory variables which are believed to affect individual wealth in Australia, and he matrix X2 contains n observations on k2 explanatory variables which are believed to affect individual wealth in New Zealand. Notice that we are not assuming that the same regressors appear in both X and X2. For simplicity, we assume that Var(u1,2. In econometrics we often wish to combine two or more linear regression equations into a single equation. (a) Combine the linear regression models given by (1) and (2) into a single model of the form where y, u and B are partitioned vectors and Xis a partitioned matrix. Carefully specify the dimensions of each sub-matrix (sub-vector) of y.u, B and Xin (3). Note: The partitioning must satisfy the conformability conditions for matrix addition and matrix multiplication. (b) The formula for the OLS estimator of B in (3), which we denote by b, is i) Derive the partitioned matrix X X.Specify the dimensions of each element of the partitioned matrix. Hint: Let the partitioned matrix Ai A2 Then ii) Use the properties of partitioned matrices stated in lectures to derive the partitioned matrix (Xx)-1.Specify the dimensions of each element of the matrix iii) Derive the partitioned vector X'y. Specify the dimensions of each element of the vector. iv) Show that the partitioned vector b in (4) may be written as (kixl) b2 (k2x) where (c) What is the practical implication of the result in (b) iv) above? (d) Assume that the matrix X in (3) is an nxk matrix. Prove that if the column rank of Xis less than k, then X'X is a singular matrix and the OLS estimator of B in (3) is not defined. Consider two separate linear regression models and For concreteness, assume that the vector yi contains observations on the wealth of n randomly selected individuals in Australia and y2 contains observations on the wealth of n randomly selected individuals in New Zealand. The matrix Xi contains n observations on ki explanatory variables which are believed to affect individual wealth in Australia, and he matrix X2 contains n observations on k2 explanatory variables which are believed to affect individual wealth in New Zealand. Notice that we are not assuming that the same regressors appear in both Xi and X2. For simplicity, we assume that Var(ui) = ơiln, i = 1,2 In econometrics we often wish to combine two or more linear regression equations into a single equation (a) Combine the linear regression models given by (1) and (2) into a single model of the form where y, u and β are partitioned vectors and X is a partitioned matrix. Carefully specify the dimensions of each sub-matrix (sub-vector) of y,u, B and X in (3). Note: The partitioning must satisfy the conformability conditions for matrix addition and matrix multiplication. (b) The formula for the OLS estimator of B in (3), which we denote by b, is i) Derive the partitioned matrix XX Specify the dimensions of each element of the partitioned matrix. Hint: Let the partitioned matrix Ai A2 As As Then [42] ii) Use the properties of partitioned matrices stated in lectures to derive the partitioned matrix (Xx)- Specify the dimensions of each element of the matrix 4 mar iii) Derive the partitioned vector ry. Specify the dimensions of each element of the vector iv) Show that the partitioned vector b in (4) may be written as where (4 mar (c) What is the practical implication of the result in (b) iv) above? (d) Assume that the matrix Xin (3) is an nxk matrix. Prove that if the column rank of Xis less than k, then XX is a singular matrix and the OLS estimator of B in (3) is not defined.. (6 mat

Answer 1

a)

Consider y 0 We know that, Дх be the estimated regression line. 2 2 i-1 1697.80-572x43 43 14 157.42 B2.33 2 2 572 2.33 Thus the linear regression can be given bv. 48.01-2.33x or x3.7 y 48.01-2.33x3.7 y-39.389 Thus the point estimate of mean permeability when compressive strength is 3.7 is 39.389. Residual 46.1- 39.389 Residual - 6.711 Hence the corresponding residual is 6.711

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