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I am working through the ISLR book and wanted help with an exercise question since the solutions are not provided in my copy.

3.7 Exercises: 3 - Suppose we have a data set with five predictors, $X_1$=GPA, $X_2$=IQ, $X_3$=Gender (0 for Male, 1 for Female), $X_4$=Interaction between GPA and IQ, and $X_5$=Interaction between GPA and Gender. The response is starting salary afer graduation. Suppose we use least squares to fit the model, and get $\hat{\beta}_0$=50, $\hat{\beta}_1$=20, $\hat{\beta}_2$=0.07, $\hat{\beta}_3$=35, $\hat{\beta}_4$=0.01, $\hat{\beta}_5$=-10.

Which answer is correct, and why?

i. For a fixed value of IQ and GPA, males earn more on average than females.

ii. For a fixed value of IQ and GPA, females earn more on average than males.

iii. For a fixed value of IQ and GPA, males earn more on average than females provided that the GPA is high enough.

iv. For a fixed value of IQ and GPA, females earn more on average than males provided that the GPA is high enough.

My answer:

I first think of the full formula of the regression equation:

$Salary \approx 50 + 20(GPA) + 0.07(IQ) + 35(Gender) + 0.01(GPA)(IQ) - 10(GPA)(Gender)$

This means that the effect that GPA has on starting salary is complicated. For a male it seems simple, start with 50k and each GPA point adds 20k, and IQ points add a bit. But for females there's a negative takeaway of added GPA. Yes it adds 20k but it also takes away 10k. And an extra 35k is added to the female salary.

I don't know what to make of this result. If I had to guess, I would say the answer is 3. But the interactions are confounding. Any help is appreciated.

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These type of exercises are clearly understood when you have established very precisely the base category. Once you have done that, all goes smooth, as least squares is simply describing the mean of the dependent variable for each subgroup.

In your case, the base category (implicit in $\beta_{0}$) are males with zero GPA and zero IQ (unless the variables GPA and IQ are centered - a reasonably thing to do- in which case the levels of GPA and IQ would be the mean of the sample). You can find the base category by replacing all regressors with zero.

Having established that, now the marginal change in the base category should be clear (or, the ceteris paribus change)

  • GPA is the average change on wages of one more unit of GPA on males with originally zero GPA and zero IQ.

  • GPA*G is the average change on wages of one more unit of GPA on females with originally zero GPA and zero IQ.

  • IQ is the average change on wages of one more unit of IQ on males with originally zero GPA and zero IQ. Notice this does not depend on gender, so it also applies to female. This is because there is no IQ*G in the model.

  • G is the average change on wages of being a female with zero GPA and zero IQ (compared with males with zero GPA and zero IQ).

  • GPA*IQ is the average change on wages for an extra unit of IQ, given a certain level of GPA, and vice-versa. Notice this does not depend on gender.

To answer the question, it is clear from the sign of $\beta_{3}$ that on average women earn more than men if both have zero GPA and zero IQ. However, as GPA increases, average wages become relatively higher for men ($\beta_{5} < 0$). In consequence, if GPA is high enough, men will earn more than women, on average. Hence, the answer is (iii).

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  • $\begingroup$ Thank you for adding clarity to this question. The marginal change from the base category was a sticking point. $\endgroup$ – Pierre Lafortune Sep 5 '16 at 13:34

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