Refer to the dataset wagefinal.dta. Use Stata to compute the required test statistic.Consider the simple linear regression model: $ \text{wage} = \beta_0 + \beta_1 \text{educ} + u $ (5), where $ \text{educ} $ is years of education. Assume $ E(u|\text{educ}) = 0 $. Use Stata to perform the F-statistic version of the Breusch-Pagan test for heteroskedasticity, testing the null hypothesis that the error variance is constant against the alternative that it depends linearly on $ \text{educ} $. The F-statistic value obtained is
.(NOTE: Write your answer in number format, with 2 decimal places of precision level; do not write your answer as a fraction. Use a period for the decimal separator.)

Question

Refer to the dataset wagefinal.dta. Use Stata to compute the required test statistic.Consider the simple linear regression model: $ \text{wage} = \beta_0 + \beta_1 \text{educ} + u $ (5), where $ \text{educ} $ is years of education. Assume $ E(u|\text{educ}) = 0 $. Use Stata to perform the F-statistic version of the Breusch-Pagan test for heteroskedasticity, testing the null hypothesis that the error variance is constant against the alternative that it depends linearly on $ \text{educ} $. The F-statistic value obtained is 
.(NOTE: Write your answer in number format, with 2 decimal places of precision level; do not write your answer as a fraction. Use a period for the decimal separator.)

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To tackle this task, start by recognizing that the Breusch-Pagan test assesses whether the variance of the regression residuals from wage ~ education depends on education itself, via a linear specification of the variance.

First, you would run the simple linear regression of wage on educ to obtain the OLS residuals. In Stata, this is done with: regress wage educ. The output will provide the residuals (e). You can save these residuals to a new variable, or let Stata handle them internally for the subsequent steps.

Next, you form the auxiliary regression used in the Breusch-Pagan approach. Regress the squared residuals on a constant and the regressor(s) that you want to test for heteroskedasticity. For the F-statistic version testing variance as a linear function of educ, you would run: regress e^2 c.educ, where e^2 denotes the squared residuals from the previous regression and c.educ is the centered or uncentered education term depending on the exact specification you are following. In Stata, this can be implemented by: gen e2 = e^2; regress e2 educ. The key idea is to capture whether the variance of e is related to educ.

Then, retrieve the F-statistic that tests the null hypothesis that the coefficients on the explanatory variables in the auxiliary regression (excluding the constant) are jointly zero. In the classic Breusch-Pagan formulation, this F-statistic corresponds to the R-squared value from the auxiliary regression and the number of regressors in that auxiliary model, typically using the formula F = [(R^2)/(k)] / [((1 - R^2)/(n - k - 1))], but Stata provides the exact F-statistic in the output when you use the test command or simply interpret the regression results for e^2 on educ.

In Stata, you can implement the F-statistic version directly by leveraging the estat hettest command after your regress, but with explicit linear specification you might prefer a manual approach: after running regress wage educ, run gen e2 = e^2; regress e2 educ; display _f; or use: test educ if you used the exact auxiliary regression. The key is to extract the F-statistic value associated with the hypothesis that the coefficient on educ in the auxiliary regression is zero (and, if you included a constant, the joint test on the regressors that appear in the variance model).

Important practical notes:
- Ensure that you are using the correct dataset wagefinal.dta in Stata and that educ is measured in years of schooling as assumed.
- If you have a sample size n and you include only educ as the regressor in the auxiliary regression, then the number of restrictions k is 1 (the coefficient on educ).
- The F-statistic is reported in Stata’s output for the auxiliary regression; you should read off the value and format it to two decimal places, as requested. Do not present the result as a fraction.

Because the problem provides no numerical dataset here, the actual numerical value of the F-statistic cannot be computed without running these steps on wagefinal.dta. If you perform the steps outlined above in Stata and locate the F-statistic in the auxiliary regression (or via the appropriate test command), you would then report that value with two decimal digits, e.g., 3.45, 1.00, etc., depending on your sample.

In summary, the procedure is:
1) Regress wage on educ to obtain residuals.
2) Regress the squared residuals on educ (and a constant) to obtain an R-squared for the auxiliary regression.
3) Convert that R-squared into the F-statistic for the test of homoskedasticity against the alternative that variance changes linearly with educ.
4) Report the F-statistic value rounded to two decimals, using a period as the decimal separator.

AS.440.606.50.SP25 Final Exam 1B

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