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This homework is to practice on multiple linear regression and matrix manipulation. Attach the complete R codes for Problem 2 at the end of the homework. Total: 90 points.

1. (40 points) Matrix calculation. Consider the multiple linear regression model in the matrix

 2 ^ form Y = X + with E( ) = 0 and Var( ) = In. Let be the least squares estimator of ^ ^ ^ , Y = X be the tted values, and e = Y Y be the residuals. (a) ^ ^ 2 and X. Detail the steps in (30 pts) Express Var( ), Var(Y), and Var(e) in terms of

obtaining the expressions.

(b) (10 pts) Show that eT 1n = 0:

2. (50 points) Data analysis. A commercial real estate company evaluates age (X1), operating expenses (X2, in thousand dollar), vacancy rate (X3, in percentage), total square footage (X4, in ten thousand square feet) and rental rates (Y , in thousand dollar) for commercial properties in a large metropolitan area in order to provide clients with quantitative information upon which to make rental decisions. The data are taken from 81 suburban commercial properties. Consider the multiple linear regression model

iid 2

Yi = 0 + 1X1i + 2X2i + 3X3i + 4X4i + i; i = 1; : : : ; n; with i N(0; ):

We use matrix manipulation to t the model. (For numerical results, keep three digits after the decimal point.)

1. In R, create the design matrix X and the response vector Y. What are their dimensions? Compute XT X, XT Y, XT X 1, and least squares estimator of . Copy your results here. Compute the tted values and residuals. Copy those for the rst 6 cases here. Based on the residuals, give an estimate of 2.

1. Test whether 2 = 0 or not at 0.01 signi cance level.

1. Consider a property with the following characteristics: X1 = 4; X2 = 10; X3 = 10; X4 = 8. Give a 95% con dence interval for the expected rental rate of this property.

1. Give a 95% prediction interval for the rental rate of the property in (c).

1. Continue from part (c), suppose you are also interested in another property with X1 = 5; X2 = 11; X3 = 8; X4 = 12, provide prediction intervals for the rental rate of the two properties with at 0.95 family con dence level.

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