Exercise Set 04
Technical exercise 1: practice with interaction terms
Let \(Y_t\) be the regressand and \(X_t^\prime=\left(1,X_{1t},X_{2t},X_{1t}*X_{2t}\right)\) be the list of regressors, so that the regression line is given by \[\widehat{Y}_t=\widehat{\beta}_0+\widehat{\beta}_1X_{1t}+\widehat{\beta}_2X_{2t}+\widehat{\beta}_3 X_{1t}*X_{2t}\] Suppose \(X_{1t}\) and \(X_{2t}\) are both continuous regressors.
- What would be the fitted value if \(X_{1t}\) is fixed at \(x_1\) and \(X_{2t}\) is fixed at \(x_2\)? Call this fitted value \(\widehat{Y}_A\).
- What would be the fitted value if \(X_{1t}\) is fixed at the same value \(x_1\) but \(X_{2t}\) is fixed at a different value \(x_2^\prime=x_2+1\)? Call this fitted value \(\widehat{Y}_B\).
- Obtain a simplified expression for \(\widehat{Y}_B-\widehat{Y}_A\). What happens to this expression when \(x_1=0\)? What happens when \(x_1\neq 0\)?
- In this situation, is it possible to provide an interpretation for the coefficient of the interaction term alone? Explain why or why not.
- Mapping \(Y\) to
stndfnl
, \(X_1\) toatndrte
, and \(X_2\) topriGPA
, along with what you have seen in the previous items, provide the best interpretation (for communication purposes) of the coefficient onpriGPA
alone. - If compare students who have the same attendance rate of 80 percent, how do those students who have prior college GPA higher by 1 point compare in terms of their standardized final exam scores?
Technical exercise 2: more on interaction terms
Let \(Y_t\) be the regressand and \(X_t^\prime=\left(1,X_{1t},X_{2t},X_{1t}*X_{2t}\right)\) be the list of regressors, so that the regression line is given by \[\widehat{Y}_t=\widehat{\beta}_0+\widehat{\beta}_1X_{1t}+\widehat{\beta}_2X_{2t}+\widehat{\beta}_3 X_{1t}*X_{2t}\] Suppose \(X_{1t}\) and \(X_{2t}\) are both dummy variables. Note that there would be four subgroups:
- Those \(t\) for which \(X_{1t} =X_{2t}=1\)
- Those \(t\) for which \(X_{1t} =X_{2t}=0\)
- Those \(t\) for which \(X_{1t}=1\), but \(X_{2t}=0\)
- Those \(t\) for which \(X_{1t}=0\), but \(X_{2t}=1\)
- Discuss how you are going to interpret \(\widehat{\beta}_0\), \(\widehat{\beta}_1\), \(\widehat{\beta}_2\), and \(\widehat{\beta}_3\). Repeat Item 4 of Technical exercise 1 for this case.
- Return to the article entitled, “Is Economics a Good Major for Future Lawyers? Evidence from Earnings Data”. Refer to Table 3 of that paper. Do you think it makes sense to create interaction terms for the majors (like the dummy variable electrical engineering multiplied by the dummy variable chemistry)? Explain why or why not.
What you will be expected to do
You will be submitting to my email a zip file (not rar, not 7z) with filename surname_exset04.zip
, replacing surname
with your actual surname, and making sure it contains
- Scanned PDF solutions to the technical exercises (do be mindful of the size of the file, keep under 15 MB if possible) with filename
surname_tech04.pdf
- Your qmd file with filename
surname_exset04.qmd
and - The HTML file associated with your qmd file.