Consider the multiple regression model with three independent explanatory variables under the classical linear model assumptions MLR.1 through MLR.6: y=β0+β1x1+β2x2+β3x3+u You would like to test the null hypothesis: H0:β1−2β2=1 . [i] Let β^1and β^2 denote the OLS estimators of β1 and β2. Find Var(β^1−2β^2) in terms of the variances of β^1 and β^2 and the covariance between them. What is the standard error of β^1−2β^2? [ii] Write the t statistic for testing H0:β1−2β2=1. [iii] Define θ1=β1−2β2 and θ^1=β^1−2β^2. Write a regression equation involving β0, θ1, β2 and β3 that allows you to directly obtain θ^1 and its standard error. [iv] Explain how you could test the null hypothesis H0 by estimating the regression equation in part [iii] using STATA. What is the null hypothesis? What is the appropriate test statistic?
Consider the multiple regression model with three independent explanatory variables under the classical linear model assumptions MLR.1 through MLR.6:
y=β0+β1x1+β2x2+β3x3+u
You would like to test the null hypothesis: H0:β1−2β2=1 .
[i] Let β^1and β^2 denote the OLS estimators of β1 and β2. Find Var(β^1−2β^2) in terms of the variances of β^1 and β^2 and the
[ii] Write the t statistic for testing H0:β1−2β2=1.
[iii] Define θ1=β1−2β2 and θ^1=β^1−2β^2. Write a regression equation involving β0, θ1, β2 and β3 that allows you to directly obtain θ^1 and its standard error.
[iv] Explain how you could test the null hypothesis H0 by estimating the regression equation in part [iii] using STATA. What is the null hypothesis? What is the appropriate test statistic?
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