Question: Consider the multiple regression model with three independent variables, under the classical linear model assumptions MLR.1 through MLR.6:

You would like to test the null hypothesis \(\mathrm{H}_{0}: \beta_{1}-3 \beta_{2}=1\)

1. Let \(\hat{\beta}_{1}\) and \(\hat{\beta}_{2}\) denote the OLS estimators of \(\beta_{1}\) and \(\beta_{2} .\) Find \(\operatorname{Var}\left(\hat{\beta}_{1}-3 \hat{\beta}_{2}\right)\) in terms of the variances of \(\hat{\beta}_{1}\) and \(\hat{\beta}_{2}\) and the covariance between them. What is the standard error of \(\hat{\beta}_{1}-3 \hat{\beta}_{2} ?\)
2. Write the \(t\) statistic for testing \(\mathrm{H}_{0}: \beta_{1}-3 \beta_{2}=1\).
3. Define \(\theta_{1}=\beta_{1}-3 \beta_{2}\) and \(\hat{\theta}_{1}=\hat{\beta}_{1}-3 \hat{\beta}_{2} .\) Write a regression equation involving \(\beta_{0}, \theta_{1}, \beta_{2}\), and \(\beta_{3}\) that allows you to directly obtain \(\hat{\theta}_{1}\) and its standard error.

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