Consider the multiple regression model with three independent variables, under the classical linear model assumptions MLR.1 through MLR.6:
y0 1x1 2x2 3x3 u.
You would like to test the null hypothesis H0: 1 32 1.
(i) Let ˆ1 and ˆ2 denote the OLS estimators of 1 and 2. Find Var(ˆ1 3ˆ2) in terms of the variances of ˆ1 and ˆ2 and the covariance between them. What is the standard error of ˆ1 3ˆ2?
(ii) Write the t statistic for testing H0: 1 32 1.
(iii) Define 1 1 32 and ˆ1 ˆ1 3ˆ2. Write a regression equation involving 0, 1, 2, and 3 that allows you to directly obtain ˆ1 and its standard error