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Use the data in LAWSCH85 for this exercise.(i) Using the same model as in Problem 4 in Chapter $3,$ state and test the null hypothesis that the rank of law schools has no ceteris paribus effect on median starting salary.(ii) Are features of the incoming class of students- namely, $L S A T$ and $G P A-$ individually or jointly significant for explaining salary? (Be sure to account for missing data on $L S A T$ and $G P A . )$(iii) Test whether the size of the entering class (clsize) or the size of the faculty (faculty) needs to be added to this equation; carry out a single test. (Be careful to account for missing data on clsize and faculty.)(iv) What factors might influence the rank of the law school that are not included in the salary regression?
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(a) see explanation(b) LSAT: No, GPA: Yes, both: YES(c) 0.95$($ approx $)$(d) See explanation
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Multiple Regression Analysis: Inference
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October 26, 2021
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So for this four part problem, we start off by stating and testing the null hypothesis that the rank of law schools has no calories pair of US effect on medium starting salary. So just really quickly stating the hypothesis would just be that beta five equals zero. So that's the null hypothesis. And how do we test us? No hypothesis. So we have to test it by, of course, estimating the equation. So I'll just write out what that looks like with the associate numbers. And hopefully you get the same numbers after running this in your statistical software program. So our dependent variable is the log of salary and the numbers we get the intercept is 8.34 and then the coefficient on L sat is 0.47 Here's the all set variable. Then on G p A. We have 0.248 so positive that makes sense. Both l sat and GPA have positive coefficients, which makes sense. Next, we have, um, a 0.95 on here and just a couple more but below here, Um 038 coefficient on cost on the log of cost specifically also makes sense. That's a positive coefficient. The cost of the law school goes up. It would stand to reason that your salary after law school might be a little bit higher, and then our last variable here is rank of the law school, and that is estimated to have a negative coefficient. And just, um, note that this negative coefficient here it makes sense because the what we mean by rank of law school means if the number is lower. So if it's let's say one or two, that's a better rank. But it's a lower number, so this negative coefficient means that as high as Ah law school is ranked more highly, then it is estimated that the students will end up having a bigger salary after law school here, the coefficients. It's also add in the standard errors here, so let's go with that. So here we go 0.53 here, then standard error of below L sat looks like it might not be a significant when looked at with the coefficient. The standard air for GPA looks like Deepa is a significant variable. Also significant here, probably with this standard air cost, maybe not too significant and then rank. Check out that small standard errors that's going to tell us that rank is very statistically significance, this estimation process. And finally, it's at in the last little little bits. Info here. Sample sizes 1 36 and R squared equals 0.842 Thank you. So to end up this part one, we are testing the following coefficient. We're testing rank coefficient. So we're testing If it is, uh, statistically significance and the standard air is much, much smaller than the coefficient. So we can conclude that rank indeed does have we can conclude that it does have an effect on the median starting salary of law school grad students. Right Part two. You have to look at whether the features of incoming class of students, namely the L SAT and GPA, if they're individually or jointly significant for explaining salary. So let's look at those two Elsad looking here else at these are looking at the how the standard error stacks up against the coefficient. There more or less equivalent. They're very close, so doesn't look like else. It is individually significant for explaining salary. Whoever G P a. Here, the standard air is much smaller than the coefficient, so we couldn't include the g p. A is individually significant for explaining salary. And so that's right this out here. So l SATs we're gonna say no, not individual, Individually significant. However, G p a, we can say is individually significant for explaining salary and since g p A as then we can also conclude that l sat and G p A would also be jointly significant. So since G p. A is individually significant, it also means that the combination of L sat and GPA are jointly significant in explaining log of salary the outcome. So for part three, the we, uh, sorry. The problem asked you to test whether the size of the entering class or the size of the faculty needs to be added to this equation. So that's just another test of joint significance. So adding class size here and faculty if adding them to the original equation. Uh, if that should be done, if they should be added so the single test we should dio is the test of joint significance, and this is different from part two because we need to actually know this formula for the test of joint significance. So to do that, we need to do a nap test on me. F test. I'll just write it out here for you. Ah, long, long, kind of fractional expression here. But I'll just ride it out, and then you can go through it step by step and figure out what exactly this is trying to say while we're going to use this right, almost there, got a couple more letters and numbers and then we can go ahead and talk about what this actually means. So this is the F statistic, and this is our test for joint significance. So this F statistic is referring to two different regressions that you run, which involves either, including these two new variables class, size and faculty were excluding them. This first term in the F statistic, This s s r with a sub script are that is the sum of squared residuals for the restricted model. So this is this which means this is the sum of squared residuals you get after you run the restricted regression, which means you do not include class, size and faculty. So you're not including class, size and faculty in that progression So you have to file that away. Second one here. Similar sum of squared residuals. But this time, this subscript you are beneath tells you this is the unrestricted model. And this is the model that includes little check. Mark includes class, size and faculty in it. So you have toe also file that away. Note that it also pops up down here. So you use that number twice. Oh, I missed a little. Should be a division sign here. Sorry. Finally, up here we have Q. That's the number of restrictions which is just to equals two in our case, because we are restrictions are two variables class, size and faculty. Then we just have two more things to know. This is and this is just our number of observations in our sample, which in our case, is 131 and then this K here is our independent variables in the unrestricted model that you run. So it's too little quickly. Write that out and again, just to remind the unrestricted model is the model where you include class, size and faculty. No restrictions. You include those variables. Okay, so you have to create this statistic in order to pull off the test of joint significance and what you should get if you run both the restricted and unrestricted regressions, save those sum of squared residuals for each. You should get the following a statistic. You should get 0.95 and just a disclaimer. This is a pretty low at statistic, and it does result in a P value that is not very significant at typical significance levels. The P value here illegal about 0.39 so you can safely say just going back to our variables here. I can safely say that class, size and faculty are not join me significant. That's what we can end Part three on after you write it out. Artist. And they're so not joining Significant. Finally, Part four doesn't include much math. More just have to think in economic terms about what factors might influence the rank of the law school that aren't included in the salary regression. So think about law school rank and what we might be missing from our original regression. So what might be missing? And when you're thinking about this sort of question, you want to think about other things that might affect both the ranking of the law school and also salaries. And, um, example that we already have in the regression is, let's say L sat and GPA scores. So we know that l sat and GPA scores way up top here, draw some green arrows. Those air, you know, correlated or connected associate ID with law school rank. Because if students have higher scores, the ranking of a law school likely be higher. And they might also associate very well with salary after law school, since higher scores might me and you do better in law school and have a better knowledge and be ableto land a higher salary job after law school. So those are examples of what we have included that are correlated, are associated with both the ranking and salary after law school. Something else You might want to think that they're missing, and this is what part for it's getting at. You might think about maybe things like gender. So these, your demographic things, gender of students or maybe race of students sorts of things that maybe we don't have a good idea of. If they would affect the ranking of the law school or if they would affect the salary you might get after out of law school. But nonetheless, we might think that those could be useful things to control for in the very least, even if there's not greats economic theory behind including them, um, so that's part four, and that will conclude the problem.
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