ECON312 Problem Set 4 (1) Empirical Application— Dummy Variable

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Egerton University *

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312

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Statistics

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Nov 24, 2024

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Question I: Empirical Application— Dummy Variable (30 points) Use the GPA2 data set in the R folder for this exercise. (a) Consider the following equation: = 0 + 1 + 2 2 + 3 + 𝑐??? 𝑐??? 𝑐??? 𝑐??? 𝑐??? ?𝑎 ?𝑎 ?𝑎 ?𝑎 ?𝑎 𝛽 𝛽 ℎ ?𝑖𝑧? ?𝑖𝑧? ?𝑖𝑧? ?𝑖𝑧? ?𝑖𝑧? 𝛽 ℎ ?𝑖𝑧? ?𝑖𝑧? ?𝑖𝑧? ?𝑖𝑧? ?𝑖𝑧? 𝛽 ℎ ????𝑐 ????𝑐 ????𝑐 ????𝑐 ????𝑐 𝛽 ?𝑎? ?𝑎? ?𝑎? ?𝑎? ?𝑎? 𝛽 ? ??𝑎?? ??𝑎?? ??𝑎?? ??𝑎?? ??𝑎?? 𝛽 𝑎? 𝑎? 𝑎? 𝑎? 𝑎? ???? ???? ???? ???? ???? ? 𝑐??? 𝑐??? 𝑐??? 𝑐??? 𝑐??? ?𝑎 ?𝑎 ?𝑎 ?𝑎 ?𝑎 4 + 5 + 6 ℎ + where is cumulative college grade point average; is size of high school graduating class, in hundreds; is ℎ ?𝑖𝑧? ?𝑖𝑧? ?𝑖𝑧? ?𝑖𝑧? ?𝑖𝑧? ℎ ????𝑐 ????𝑐 ????𝑐 ????𝑐 ????𝑐 academic percentile in graduating class; is combined SAT score; is a ?𝑎? ?𝑎? ?𝑎? ?𝑎? ?𝑎? ? ??𝑎?? ??𝑎?? ??𝑎?? ??𝑎?? ??𝑎?? binary gender variable; and is a binary, which is one for students athletes. 𝑎? 𝑎? 𝑎? 𝑎? 𝑎? ???? ???? ???? ???? ???? ℎ What are your expectations for the coefficients in this equation? Which ones are you unsure about? Answer Size of High School Graduating Class (hsiz): It's unclear how the size of the high school graduating class would affect college GPA. It could be positive if smaller classes allow for more individualized attention or negative if competition is higher in smaller classes. Academic Percentile in Graduating Class (hsperc): Higher academic percentile might be associated with a higher college GPA. SAT Score (sat): A positive relationship is expected; higher SAT scores may indicate better preparation for college. Gender (fem): The effect could be uncertain without additional information. It may depend on various factors like gender-based academic performance trends. Athlete Status (athlete): Athletes may have different time commitments, potentially affecting their GPA. (b) Estimate the equation in part (a). What is the estimated GPA differential between athletes and non-athletes? Is it statistically significant? The R Code are as follows: # Assuming df is your dataset model <- lm(colgpa ~ hsiz + I(hsiz^2) + hsperc + sat + fem + athlete, data = df) summary(model)

Check the coefficient for the athlete variable to find the estimated GPA differential between athletes and non-athletes. Assess its statistical significance based on the p-value. (c) Drop from the model and re-estimate the equation. Now, what is the estimated ?𝑎? ?𝑎? ?𝑎? ?𝑎? ?𝑎? effect of being an athlete? Discuss why the estimate is different than that obtained in part (b). R code model_no_sat <- lm(colgpa ~ hsiz + I(hsiz^2) + hsperc + fem + athlete, data = df) summary(model_no_sat) Compare the coefficient for athlete with the previous model. Discuss why it might differ (d) In the model from part (a), allow the effect of being an athlete to differ by gender and test the null hypothesis that there is no ceteris paribus differences between women athletes and women non-athletes. R code model_interaction <- lm(colgpa ~ hsiz + I(hsiz^2) + hsperc + sat + fem * athlete, data = df) summary(model_interaction) Look for the interaction term (fem * athlete) and assess its significance. (e) Does the effect of on differ by gender? Justify your answer. ?𝑎? ?𝑎? ?𝑎? ?𝑎? ?𝑎? 𝑐??? 𝑐??? 𝑐??? 𝑐??? 𝑐??? ?𝑎 ?𝑎 ?𝑎 ?𝑎 ?𝑎 R code model_gender_interaction <- lm(colgpa ~ hsiz + I(hsiz^2) + hsperc + sat * fem + athlete, data = df) summary(model_gender_interaction) Check if the interaction term (sat * fem) is significant. (f) ) Use the Chow Test to test whether there are differences in the coefficients of the model in part (a) (excluding the dummy variable) across genders. (Hint: look ? ??𝑎?? ??𝑎?? ??𝑎?? ??𝑎?? ??𝑎?? at the R-document to learn about how to run regressions on a subset of data

Chow Test for Gender Coefficient Differences: R code # Assuming you have a gender variable (gen) in your dataset model_male <- lm(colgpa ~ hsiz + I(hsiz^2) + hsperc + sat + athlete, data = subset(df, gen == "Male")) model_female <- lm(colgpa ~ hsiz + I(hsiz^2) + hsperc + sat + athlete, data = subset(df, gen == "Female")) chow.test(model_male, model_female) This tests whether there are differences in coefficients between genders.

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