8 Consider the multiple regression model with three independent variables, under the classi- cal linear model assumptions MLR.1 through MLR.6:
y 5 b0 1 b1x1 1 b2x2 1 b3x3 1 u.
You would like to test the null hypothesis H0: b1 2 3b2 5 1.
(i) Let bˆ1 and bˆ2 denote the OLS estimators of b1 and b2. Find Var(bˆ1 2 3bˆ2) in terms of the variances of bˆ1 and bˆ2 and the covariance between them. What is the standard error of bˆ1 2 3bˆ2?
(ii) Write the t statistic for testing H0: b1 2 3b2 5 1.
(iii) Define u1 5 b1 2 3b2 and uˆ1 5 bˆ1 2 3bˆ2. Write a regression equation involving b0,
u1, b2, and b3 that allows you to directly obtain uˆ1 and its standard error.
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