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Refer to the example used in Section $4-4 .$ You will use the data set TWOYEAR.(i) The variable $p h s r a n k$ is the person's high school percentile. (A higher number is better. For example, 90 means you are ranked better than 90 percent of your graduating class.) Find the smallest, largest, and average phsrank in the sample.(ii) Add phsrank to equation $(4.26)$ and report the OLS estimates in the usual form. Is phsrank statistically significant? How much is 10 percentage points of high school rank worth in terms of wage?(iii) Does adding phsrank to $(4.26)$ substantively change the conclusions on the returns to two- and four-year colleges? Explain.(iv) The data set contains a variable called $i d .$ Explain why if you add $i d$ to equation $(4.17)$ or $(4.26)$ you expect it to be statistically insignificant. What is the two-sided p-value?

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Chapter 4

Multiple Regression Analysis: Inference

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there's a problem. Also works with inference in econometrics and again, as we go through numbers one through four we're gonna at a variable drop variable and see just how inference changes given what you decided to include in your in your estimation regression. So part one here is relatively simple. All you have to do is find the smallest, largest and average high school percentile rank in the data set in the sample. And so I'm just going to write out the The answer is that you should get for these. So for the smallest or the minimum percentile, rank should be zero. That makes sense. The maximum is 99 again. Makes sense. This can all be checked in your statistical software program using certain commands, descriptive statistics, that sort of approach, Um, and the average What we don't really know. But again, a simple command in your program should give you an average of 56.16 as the average percentile. That's part one Part two have already kind of preloaded on here. In part two is asking you Thio, Add the high school percentile rank variable to equation 4.26 in the textbook and to estimate the equation, then to report the OLS estimates. So this is equation 4.26 here, just adding in high school percent tell rank high school percentile. So once you estimate this equation in your whatever program you use, you should get the following numbers and I'll write out the coefficients for you. Oh, I won't worry about the intercept. The beta, not we won't worry about that, but we'll start with the that That J C variable here. Yeah, which is that Uh huh. Some of two and four year colleges, Um, the number of years and 22 and four year colleges. So here's the starting with J. C. We have a negative coefficient here of negative 0.93 says a Maybe a little surprising, Um, but we'll keep going with that and put the standard error of 0.70 so that will probably give us non statistical significance. So even though there's a negative coefficient here, it doesn't look like it's going to be statistically significant for the total college variable. You have a positive coefficients here, and standard air is pretty small, so that's going to be significant what we have for experience experience. We have again a positive 0.49 coefficient. That's good that it's positive and definitely statistically significant, with a small standard error here. Then finally, r percent of high school rank so positive it's a good sign. We'd expect that as the as your percentile in your high school increases thin, your predicted wage would increase also. So there's our coefficient, and there's our standard error. The standard error. I'll put a zero. Here it's 00.30 It's like our standard error isn't that much smaller than the coefficient. So that answers our first question that they want answered. Which is is PHS rank statistically significant in this new regression? And so we can say sneaky little bracket here, Not statistically significant, just kind of abbreviate that for you on the T statistic on that is about is over one, maybe one of the quarter, 1.25 So not statistically significant for PHS rank. That's regression. The next question you have to ask for Part two. Here is how much is 10% points of the high school rank worth in terms of wage. So what you have to do for. To answer that question is take the coefficient estimate here, so take 0.3 I'm bringing out here, then multiply it by 10. So, times 10 right, So 10 percentage point increase in rank equals 0.3 So you just almost move the decimal over one spot. So that's equals a 0.3% increase in wages is how you interpret that estimate. Right, Because our outcome, our dependent variable, is logged. And so this 10% point increase has a percentage change interpretation here. So that's the final answer for part two, part three Slightly switches of what? You're what you're looking at. It asked you if adding PHS rank into that equation so into our, uh, into this equation here, so does adding PHS rank substantively change the conclusions on the returns to and for your colleges. So where you're looking now at J. C, right? That's the variable that represents the two and four year colleges. And so what we find is, and you can compare to this answer Thio the estimates given in Equation 4.26 So I'll just write that here for you. So compare. Compare a pair of these estimates. Thio Equation 4.26 specifically the J C. Estimate. That's what we're really looking at. So again, this J. C. This T statistic is about 1.3 in our estimates, and this is smaller than equation 4.26 Um, yeah, in equation 4.26 the T statistic is definitely little bit smaller. Sorry is larger. And equation 4.26 So we have a drop in the statistical significance test here for Casey. Um, but the coefficient magnitude here is very similar to 4.26 The coefficient again is point. 00 right. Er sorry. Negative 0.93 and in in the equation 4.26 It is negative 0.102 So we have Alright, this in in red. So here J c coefficient is about course I should say it's about equals negative 0.93 and then in 4.26 So, in equation 4.26 the coefficient equals yeah, uh, the coefficient equals negative point. 0102 So very similar coefficients. And so adding in PHS rank what we can say is that adding in PHS rank here to the regression doesn't really change are coefficient, so it doesn't really substantively change. Change the conclusions on the returns to two and four year colleges. Those are very similar, Um, and again, the T statistic here, the statistical thing against also ah doesn't change too much. There's no significance there. Yeah, that's part three. Part four asks about a variable in the data set called ID. And this question this part of the question just asks Why? If you add, I d to either equation 4.17 or 4.26 if you add it, why you would expect it to be statistically insignificant. So when you look at what I'd stands for, the key is that it is a worker i d number. And why would that be? While they give us a hint is toe Well, I would expect it to be statistically insignificant. And just because these types of IEDs that are assigned, we would expect it to be relatively randomly assigned, so I'll just give that a big underline. So the key is the random part idea number is randomly assigned, then including it in the regression in equation. She mean it's basically un correlated with any other variables in the model. So, as another way of thinking about that is, it would be uncorrelated with our explanatory variables and to really be uncorrelated with the outcome is well with log of wage. So if we were added Thio equation 4.17 4.26 we would expect it to be statistically insignificant circle that just to emphasize that conclusion there and the last little part for part for it is asking you what is a two sided p value and what you should find if you add Ah, if you add I d to the regression, you should just get the T statistic already in green here, T is about 0.54 So give that a check mark because that is not statistically insignificant. And so we're right about our conclusion that including this ID number really won't be statistically significant again. You go from the fact that it's likely randomly assigned to your sample population, which means that it's likely uncorrelated with both your explanatory variables but also your outcome and so you can make that conclusion to be pretty certain about that

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