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$$\begin{array}{l}{\text { Use the data in GPA2 for this exercise. }} \\ {\text { (i) Consider the equation }}\end{array}$$$$\begin{aligned} \text {colgpa}=& \beta_{0}+\beta_{1} h s i z e+\beta_{2} h s i z e^{2}+\beta_{3} h s p e r c+\beta_{4} s a t \\ &+\beta_{5} \text {female}+\beta_{6} \text {athlete}+u \end{aligned}$$$$\begin{array}{l}{\text { where colgpa is cumulative college grade point average; hsize is size of high school graduating }} \\ {\text { class, in hundreds; hsperc is academic percentile in graduating class; sat is combined SAT }} \\ {\text { score; female is a binary gender variable; and athlete is a binary variable, which is one for }}\end{array}$$ $$\begin{array}{l}{\text { student-athletes. What are your expectations for the coefficients in this equation? Which ones }} \\ {\text { are you unsure about? }}\end{array}$$$$\begin{array}{l}{\text { (ii) Estimate the equation in part (i) and report the results in the usual form. What is the estimated }} \\ {\text { GPA differential between athletes and nonathletes? Is it statistically significant? }}\end{array}$$$$\begin{array}{l}{\text { (iii) Drop sat from the model and reestimate the equation. Now, what is the estimated effect of being }} \\ {\text { an athlete? Discuss why the estimate is different than that obtained in part (ii). }}\end{array}$$$$\begin{array}{l}{\text { (iv) In the model from part (i), allow the effect of being an athlete to differ by gender and test the }} \\ {\text { null hypothesis that there is no ceteris paribus difference between women athletes and women }} \\ {\text { nonathletes. }} \\ {\text { (v) Does the effect of sat on colgpa differ by gender? Justify your answer. }}\end{array}$$

The p-value of the coefficient of female $\bullet$ sat is 0.6796 which is greater than critical p-value of0.05 at 5$\%$ level of significance indicating that there is no statistically significant difference ofeffect of sat on colgpa by gender.

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Chapter 7

Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables

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So the first part of this model this problem? Sorry. We want to talk about what? Our expectation for the coefficients of each of the variables in our model are. Okay, so I kind of just want to do this very simply and just talk about the direction of each variable, a possible direction. So I personally went to a really big high school, and we had a a lot of, um, students that had a very high success rates because our high school had access to so many resource is because it was so large. So I'm going to say that, um from my experience, there is a positive relationship between the size of the high school and a student's college G p A. Because they may be more prepared. But on the other hand, you could argue that it will have a negative relationship, because if the high school is large and there aren't many teachers and classes are gonna be crowded and there's not really access to resource is so, um, the impact or the relationship between high school sizes. High schools get bigger college G p ace drop because students are less prepared. So maybe that's what's going on here? So you can argue either way, as long as you have economic intuition, you have you convey this answer, Correct. Now, when we have this high school size squared but we're talking about is basically imposing diminishing marginal returns on the size of the high school. So, um, when we do this, the directionality of, uh, the relationship between college G p A in high school size should remain the same or pretty similar. So, um, I'm going to say this is positive as well. And, um, the high school percentile there. Ah, the academic percentile in the graduating class. I'm going to say that it's has a negative relationship because as your percentile increases, if you're in the 99th percentile, um, then that means that you are sorry, I was confusing. Okay, so if you're in the 99th percentile, that means you're in the top 1% of students. Um, I think that's how it works. So because of this, I'm assuming that as your percentile increases, then your college G p A will also increase because students that do well in high school do better in college is my assumption now s a t scores, I'm going to say a pretty similar thing. S e T scores are what's used by a lot of colleges to predict high school or sorry, college performance. So I'm gonna say that's a positive relationship being female or being male. Um, I really don't think there will be any relationship. Or at least it's because men and women are equal. Um, so I'm going to say there's a zero relationship, right? There is no relationship here and then whether there is a relationship between being an athlete and their college d p g p A. A lot of research has shown that, um has shown that athletes are high performing people. Um, and they do very well in high pressure situations that some non athletes may not. So I'm going to say being an athlete is going to have a positive relationship now, noticed that what I'm talking about these I am just giving some reference for where my ideas air coming from these air, just sitting hypotheses out thatyou contest. It doesn't mean that they have to be right. They have to be wrong. Um, like, in our data, maybe s a T scores. We find that there is a negative relationship between S A T scores and college G p A. That's perfectly all right, as long as you have some reasoning to back up your thought process and ideas. Okay. And now, um, we want to actually figure out what's going on. So we're going to, um we ran this regression. I you did this in our I included all the variables we had in our model. And what we see is that, um, the impact of high school size is actually negative. All right? So as high school size increases, um, their college GPA drops now. This may be because, um, resource is our limited for the number of students that high school has, and these students may not be performing as well. Now, high school squared Notice that this is now positive, right? But the difference between high school and high school squared is really, really, really small. Um, it goes from negative 0.0 five. Yeah. 2.5 That's not a very large change, but it is a change. So just keep that in mind. Also, the significance of this drop, which just means that our predictor is not as strong as just high school size now, the high school percentile. Um, this is interesting as, um, high school percentile increases college GPS drops. So that makes me believe that my understanding of percentiles was actually wrong. Um, and that if you're in the 99th percentile, um, that means that you are in the bottom 1% of your class, but we'll see that you have to go to the code book, and we don't have that here. We just have that hatreds. Perk is academic percentile on graduating class s a T score. This is positive. That's as we expected. Ah, Female is positive. Um, and then athlete is also positive. So, again, your hypotheses don't need to be right as long as they have reasoning behind them. And now we're asked to find the estimated G p a differential between athletes and non athletes. Okay, so the athletes versus non athletes. So athlete is a binary variable zero and 10 means not an athlete. One means an athlete. So when we go from 0 to 1, what we're saying is when we go from being a non athlete to being an athlete, there is about. There's a 0.17 g p a point change change in your g p A. So athletes in this situation, um, earn a g p a 0.17 points higher than nonathletes. Now, that isn't to say that we have a causal interpretation. It doesn't mean that athletes that being an athlete makes you have a 0.17 point higher G p a. It just means that, um, we see a positive relationship between being an athlete and having, uh, Andrew G p. A. Okay. And now, um, the next thing we want to do is drop s a t from our variable. So and now we want to see if there's a change in athlete. All right, so we have a coefficient. Beta for athlete is equal to 0.0 five. So I'm gonna call you this athlete to and then the beta for athlete one is point 169 So 0.5 point 16 Stein, they are different. Let's just say they're different without doing this t test. And now we want to see, um whether why is it's so first of all, is it statistically significant. It's no longer statistically significant. Whereas before it was statistically significant at the zoo. Zero level. It's a really, really small p value, but it is no longer significant. So why is this happening? What is going on? Um, it's because, um, it's because of something called admitted variable bias. So animated variable bias. What we're trying to do in any regression. Sorry. In any regression, what we want to do is estimate the coefficient of some variable asbestos possible. So if we have some model Okay, we have a constant. We have beta one plus the beta one X beta, one x one plus beta, two x. You all right? This is our model, as we have it. And we have beta three x three plus our air. This is our composite air. And this is the model that we have right now. So are composited. Error basically means that we aren't able to capture X three in our model. We don't have it in our data. So we want to do is estimate What is the impact of not having X three in our data? This is called the emitted variable bias formula. And what we see is the expected value of beta hat. One is equal to the true value of beta one plus beta, three times the co variance between X one and X three over the variance of X one. Now, this is basically just to say that there is a change in the expected value of beta one if we don't have X three in our model because if X three is not our model, that means we don't have the true beta. Okay, so, um, as long as the co variance as long as this and this are not equal to zero, that means we have omitted variable bias of some sort. Um, in this situation, the because these two values are different and makes me believe that we have omitted variable bias. If we consider this to be the true bait about you, um, or one of these to be considered the true beta value. Even if they're not considered the true beta value, there's still some difference in them which is attributable to dropping s a t from our model. Um, so that would be ah. And now to backtrack a little bit. What is the estimated effect of being an athlete. Um, the estimated effect. Even though this isn't coddle this going from being a non athlete to being an athlete is associate ID with a plus 0.5 increase in G P A. And you guys can put that in nice English. But this is just for a short hand, just for now. All right. And now what we want to do is, um, see whether there is a difference between being a woman athlete and being a woman non athlete. So are no hypothesis for this or no hypothesis would be that the coefficient of woman athletes is equal to the coefficient of woman, non athletes and what we want to do. You see if there is a difference between these two, So we run this regression, and in this model, I now have interaction terms that I calculated before this is athlete times, female athlete times, not female and not athlete times, not female. The reason we don't include um, no, actually end, uh, female is because we want to avoid this. Perfect. Um, no perfect Kalinin Garrity assumption is because we do not want perfect colony area. Our model and notice That is also why we dropped the athlete variable and the female variable independently. Um, and what we want to do is test whether there's a difference between the two. So, um, there is there a difference between women athletes, women athletes right here and woman non athletes. So because we don't include women non athletes in our model, that means that, um, we have a that woman non athletes is our is our reference category. Alright, women, non athletes is our reference. So whenever we talk about athlete and female, athlete and male, non athlete and mail any of these variables we're talking the coefficients from these variables is talking about going from being women and non athlete to athlete and female. So what we see is we have, um so women non athlete to woman an athlete goes from zero to 0.175 All right, that's great. But how do we test to make sure that these air these two are statistically different? Um, we would look at the standard errors. Okay? The standard error, which would give us our T statistic, which would give us our p value. So because this p value if we assume that we're at a significant level of 0.5 because 0.37 is less than 0.5 It means that 1.751 to the negative 1st 0.175 is statistically different than zero. So, um, we tested this. No hypothesis. We can reject the know. And, um, we show that there is a difference between women athletes and women. Non athletes. Now, what we want to do finally is tests or check the effect of S a T on college and see if it differs by gender. So in other words, we want to just run in the same thing with an interaction turn. All right, this essay, tee times, female. Um, I put back into our model everything that we the athlete in the female variables because we don't need we need that for our, um, for our model, we no longer have to avoid perfect linearity. If we were doing female an s A. T and not female in S a t, then we would, um we would basically be able to create this female variable again. But because we don't have that, we can include female times as Etienne, our model and female and athlete independently. So, um, does the effect of or does the effect of s a t on college g p a. Deferred by gender. And what we see is because, um, this value over here 5.12 So the beta of s 80 times female is equal to zero point are, uh, 5.12 to the negative fifth, Um, and this is our P value is 0.69 which is pretty high. We can't really reject it at any point. So this is this basically shows that, um, s A T does not differ by gender when we're referring to its impact on college G p A. So no.

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