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A researcher would like to predict the dependentvariable YY from the two independentvariables X1X1 and X2X2for a sampleof N=13 subjects. Use multiple linear regres...

Question

A researcher would like to predict the dependentvariable YY from the two independentvariables X1X1 and X2X2for a sampleof N=13 subjects. Use multiple linear regression tocalculate the coefficient of multiple determination and teststatistics to assess the significance of the regression model andpartial slopes. Use a significance level α=0.02.X1X1X2X2YY40.666.75927.272.442.36556.269.117.371.225.43065.640.941.259.144.868.544.340.679.537.354.755.65866.359.446.127.262.942.430.967.244.539.522.664.3

A researcher would like to predict the dependent variable YY from the two independent variables X1X1 and X2X2for a sample of N=13 subjects. Use multiple linear regression to calculate the coefficient of multiple determination and test statistics to assess the significance of the regression model and partial slopes. Use a significance level α=0.02. X1X1 X2X2 YY 40.6 66.7 59 27.2 72.4 42.3 65 56.2 69.1 17.3 71.2 25.4 30 65.6 40.9 41.2 59.1 44.8 68.5 44.3 40.6 79.5 37.3 54.7 55.6 58 66.3 59.4 46.1 27.2 62.9 42.4 30.9 67.2 44.5 39.5 22.6 64.3 18.4 R2= F= P-value for overall model = t1= for b1, P-value = t2= for b2, P-value = What is your conclusion for the overall regression model (also called the omnibus test)? The overall regression model is statistically significant at α=0.02. The overall regression model is not statistically significant at α=0.02. Which of the regression coefficients are statistically different from zero? neither regression coefficient is statistically significant the slope for the first variable b1 is the only statistically significant coefficient the slope for the second variable b2 is the only statistically significant coefficient both regression coefficients are statistically significant



Answers

Use the data in ELEM94 95 to answer this question. See also Computer Exercise $\mathrm{C} 10$ in Chapter $4 .$
(i) Using all of the data, run the regression lavgsal on bs, lenrol, lstaff, and lunch. Report the coefficient on $b s$ along with its usual and heteroskedasticity-robust standard errors. What do you
conclude about the economic and statistical significance of $\hat{\beta}_{b s} ?$
(ii) Now drop the four observations with $b s>.5,$ that is, where average benefits are (supposedly
more than 50$\%$ of average salary. What is the coefficient on $b s ?$ Is it statistically significant
using the heteroskedasticity-robust standard error?
(iii) Verify that the four observations with $b s>.5$ are $68,1,127,1,508,$ and $1,670 .$ Define four
dummy variables for each of these observations. (You might call them $d 68, d 1127, d 1508,$
and $d 1670 .$ ) Add these to the regression from part (i) and verify that the OLS coefficients
and standard errors on the other variables are identical to those in part (ii). Which of the four
dummies has a $t$ statistic statistically different from zero at the 5$\%$ level?
(iv) Verify that, in this data set, the data point with the largest studentized residual (largest $t$ statistic on the dummy variable) in part (iii) has a large influence on the OLS estimates. (That is, run
OLS using all observations except the one with the large studentized residual.) Does dropping,
in turn, each of the other observations with $b s>.5$ have important effects?
(v) What do you conclude about the sensitivity of OLS to a single observation, even with a large
sample size?
(vi) Verify that the LAD estimator is not sensitive to the inclusion of the observation identified in
part (iii).

But one this is the estimated equation. The usual S standard errors are in green and in round brackets. The hetero skid elasticity, robust standard Iran's are in blue and in square bracket. As you can see the robot standard Iran's are somehow larger in all cases on variable lock of E. X. B pp. Yeah the robust T statistic is 1.5 not too high. So we are not very convinced if performance its link to spending hard to. The f statistic has a value of 132.7 and a p value almost zero. Therefore there is the strong evidence of heterocyclic elasticity for three and four. This is the same equation estimated by weighted least square usual. Standard herons are in round and green brackets and weighted least square standard Iran's are in blue square brackets. We don't care about it. Their standard barrel over there. Constant term the intercept. So I don't report the centered errol oh, weighted least square for it. Yeah. Oh sorry the round one hour the usual wait at least square and the square one is robust. Type all part three. Mhm. We have to compare the estimates from weighted least square approach to L. S. Approach. Yeah. And we see that the old L S n W L. S coefficients on lunch or the same. But for other variables they differ. The waiting list square estimate on lock of spending is much larger than the old L. S coefficient. Based on this estimate a 10% increase in spending. Let me write that down. That is equivalent to an increase of 0.1 in lot of spending. Let's do you a pulling 65 percentage point increase in the math past rate the weighted least square T statistic is much larger too. Before we get 1.5 now we get 3.8. More than double past four. We compare the robot standard Iran's with the usual W. L. S. Standard Iran's and we find that for the key variable lack of spending, the robust standard error is somewhat larger than the usual one robust scented errol, our lock of enrollment also somewhat higher and for lunch, the robust version is slightly lower than the usual one, part five, the weighted least square estimated is more precise and by precise, I mean, robust standard errol is smaller than the robust version of the old LS estimate. For the first one we got 1.82 and for the second one two point 35

Part one. This is an estimation result for the linear trend model. Using the first 119 Observations. And excluding the last 12 months, We have our square of .28 and the standard error of the regression Is 288.33 are too. If we estimate an A. R. One model excluding the last 12 months, this is what we get. Oh, our number of observations or degree of freedom. Now Number of observations we use in the model is 118 one less than the previous equation. The R Square is now lower .17 and the standard error of the regression is now over 300. So based on the R. Square and they're centered heroines of the re question, we can conclude that the linear trend model is a better fit to the data. A good model would have a high are square and a low send it errol of regression. When we compare the two models by measures of a rod's. We have R. M. S. E. Of the linear trend equation of 300 15 point right. And so that the route mean square of peril and for their mean absolute error O. M. E. We have 200,000 1.9 For the linear trend model. For the a. r. one model, The two values are 388.6 and 246.1 respectively. The A. R. one has higher measures of arrows compared to the linear trend. So again we conclude that the linear trend is a better fit to the data. In part four, we test this joint significance of their monthly dummies. The f statistic with 11 and 107 degrees of freedom is 1.15 with a p value of 0.3 28 Mhm. So we are unable to reject the null hypothesis, meaning we don't need to account for the seasonality in for casting C H. N. I am p.

So this is another problem with statistical inference. And this time we have an interesting regression equation that we're gonna look at that has a bunch of linear terms that were used to working with. But we also have the addition of a quadratic term or ah, squared term. So you'll read about this in the problem. But just a heads up brandy in a slight new element to this problem. So part one of this problem asked you to estimate the following a regression model. And as I typically do all just write it out in the function form and here that the dependent variable is the education of the individual in the sample. Uh, that would be the highest level of education completed by that person. Okay? And their education is a function of both of their parents highest completed education. So there mother's education and their fathers, education and as well as their own ability. And this is a measure that's hard to explain and not really in the data, but just think of it is innate ability. And also here's the quadratic term I was mentioning at the beginning is ability squared and while we have this quadratic term in here. You just think of this as well. Someone's ability might influence or increase their eventual educational attainment over their life. But you could also think that potentially there's decreasing returns to the ability you have as a person from birth. So at really high levels of ability, um, your educational attainment won't increase as much as it waas. If you had lower levels of ability compared to someone who had a little bit more than you, so does that decreasing returns to ability type. Um, concept is captured in this quadratic term here, so that's just some background not relevant for actually solving the problem. Uh huh. But para one asked you to report the results of this regression here and the usual form, and so I'll just write out the, um, the intercept First, I didn't include the intercept here, but the intercept. I'll just right here. So here's the coefficient for that intercept, and then it's standard error. Then I'll go along and start with the mother's education. Variable here, start 0.19 It's positive. Positive coefficient makes sense, So if your have a mother with more education, you're likely to also have a higher educational attainment If you're in the sample, how about the fathers Education? So also, you should get a positive coefficient here, which also makes sense both of these coefficients standard errors tell us that they're both these educational variables air statistically significant for sure. How about the ability? So we hope this will be positive. And it is in a 0.401 there with a again, a small standard error so definitely statistically significant here. And finally, the square of ability is positive there. So this is actually interesting. So I hypothesized just a minute or two ago that there might be decreasing returns to ability. Um, meaning that really high levels of ability. You don't really You're not predicted. Thio, uh, have, ah that much higher educational attainment, however, looks like I might have been a little wrong because this is a positive coefficient and it is statistically significant, so that would actually represent increasing returns to ability. So it hi, higher levels of ability. You actually are predicted to have even higher and higher increases in your educational attainment across individuals. That's interesting. Very interesting result there. So that's the first part of part one. The other part they want you to do in this first little part is test the null hypothesis, the following the hypothesis that education is linearly related thio ability. So the null hypothesis you can think of as that the coefficient for ability squared. Cool. Zero. Right? So the null hypothesis is that education is just linearly related to ability. So that's another way of saying that you're representing that Is that the coefficient on that quadratic term is zero, which gives the alternative is that the relationship is quadratic, so just changes a little bit to say. The alternative hypothesis is the This coefficient of the quadratic term does not equal zero. Right. So to test this hypothesis, you have to do one of our favorite tests. We use this a lot is the F test, right? So you have toe run your unrestricted regression, which includes the quadratic terms. So it includes the squared ability term and also run the restricted aggression without the quadratic term and just plug in the F statistic formula, those sums of squared residuals. So I'll just tell you what your statistics should be, actually, all right at the actual numbers that I got here. Um, and we'll look at the F statistic together, so new calculation should look somewhat like this. Um, this is the sum of squared residuals here for the restricted regression. So without the quadratic term, and then subtract off of that the sum of squared residuals Tibet, and write this. Okay, here we go. Sum of squared residuals for the unrestricted regression. So that's including the quadratic term. We only have one restriction here. So put one here and then in the dominator again, as we always do. You repeat that sum of squared residuals for the unrestricted regression and divided by our sample size. Minus are independent are independent variables plus one. So ah, that would be five. And so once you do all this calculation, she got the out of statistic of me to steal this really quick. All right, it does kind of look like this will be a rather large statistic at this point. So I'm expecting this to be relatively large and sure enough, So let's let's just say this is about equal to 37 should equal about 37 so I'll circle that. So if the statistic is that large. That's obviously going to reject the null hypothesis so we can reject the null and accept the alternative hypothesis, which is that the, uh, quadratic term should be in the regression. And another way of saying that is that education is quadratic lee related to ability, right? Which we could have guessed up here just from this coefficient. The fact that it's statistically significant, but that's a good check. All right, so part two, you have to dio a, um, kind of a similar a similar exercise. But this time we're testing whether, well, that's what color you dread. Whether beta one equals beta two, which is saying, does a year of Mother's education have the same impact on on individuals? Education? Is that the same impact as the Year of Fathers education? So, looking at whether these coefficients are are equal the coefficients on these these two variables. So what you should do for Part two is, uh, first of all, run the regression self. Just run the regression from part one. The way that's laid out, Make sure you include include the quadratic ability term, right, because we found out from part one that it, uh, that we rejected that null hypothesis. So just from the regression that's it's given and after that, Um, I'm not sure what software package you use, but there should be some sort of option for this and the taxes you use. I use data, something you can dio. The syntax is just test whether to coefficients are equal and the syntax is as following. So for our purposes, it would go test mother or mud M o T H education test, but a Duke equal father education because those are the those the variable names in the data set. So that's just be in the command line in stadia. And if you're in another software package, there should be another way to do that sort of test and trying to look that up if you don't know it. But run that test. What should pop out of that test on? This is what state he gives out. Hopefully other packages do a similar thing. It gives you that the F statistic equals 3.75 so that looks like it's gonna be significant, right? Um so, which means, if it's significant means it looks like we might reject the null hypothesis. So reject the concept that ah, year of Mother's Education has the same impact on on individuals education as another year of fathers education. But they asked for the P value. So I'll just write down the P value for this sober doing what the problem asks, and you should get a P value of 0.531 I'll just circle that because that will be our answer for part two. And so that's not quite at the 5% level of significance. So you could almost make a case either way. Um, depending what significance level you're you've planned on using or think is appropriate. Um, you could either say that at the 10% level. It looks like we reject the hypothesis that those coefficients on Mother education and Father Education are the same. But you could also make the argument that we would we are, except at the 5% or 1% level that they are the same. So depending on what what, you're comfortable with what you think is appropriate. That's the conclusion there. Great. That's part two. Part three. Ask you toe ad the two tuition variables to college tuition variables to the regression from Part one. So I'm just going to write right out how this would kind of go. So you just regress the regression for part one so regress education on the mother's education, father's education, ability and squared ability and the first tuition variable. So the college tuition at age 17 noted by two it 17 there and the other tuition variable, which is almost the same but tuition at age 18 and and her air term. So that should be the aggression you run for part three, and you have to determine whether these two new variables are jointly statistically significant. Is there a whole go back to our old friend, right? The statistic and do what we know how to do so. First number here would be the some of some of squared residuals for, um, some square residuals for the restricted regression, which means not including the tuition variables and this number would be the sum of squared residuals for the unrestricted aggression, which would include the tuition of variables here. Of course, we have to restrictions so there to there and finish out this statistic by filling in the denominators numbers, um, with our sample says here of of this. And now we have a total of six, right? We have. So we have six independent variables, So six plus one is seven here. So our F statistic, once you run these numbers should be 0.83 That's a low ebb. Statistics. So that looks like we're not going toe. Uh huh. It was like, we're going to say that the the two tuition variables here are not jointly significant, so Ah, it's right. Variables or not Big capital letters Not jointly. Significant. So that low that low ab statistic value is all we need. Toe tell us that that's part three. Two more parts. This problem. Do you have part four? Part four asked you to find just the simple correlation between those two tuition variables. We just added, So regards what program you use. They're gonna have a way for you. They should have a way for you to find out. Tests this correlation pretty easily. This is the status. Intacs, I guess. And you should get First of all, what's our hypothesis for this correlation? So tuition at age 17 vs tuition at age 18 mine would be that they're super, super, highly, positively correlated. Right, um, of your expectation. Tuition shouldn't waver too much from year to year. And when I ran ran this. I did get exactly that. So I got a 0.98 which is very, very, very high. Positive correlations. Almost one. So really not much difference here over a year between these two amounts. So that's good. That's good check. And once we have that, the problem asked you to explain why using instead of these individual variables, why, instead of using those new aggression while you want, why, we might want to instead just use an average of them. So just to be here, to note, that is average. So yeah, so why might we want to use an average of these two variables And the one? The main reason I think of is that it's think about it. It would cut down on unnecessary variation. Um, so I'm not how to say this. So variation cut down, and I want you to think of that is unnecessary variation. Right, because we're not getting really any more information from using tuition 18 variable in addition to the tuition 17. Right When we're thinking of, um, explaining the variation in in educational use of education. So cut down the necessary variation. That would be one reason because right there, almost perfectly correlated. So we just want to include we do want to make sure you include this tuition cost information, but an average would cut down this unnecessary variation. Me just right. Anus necessary. Unnecessary variation is cut down. Okay, so then I asked what happens when you use the average right? So you have to kind of go back and you have to create a new variable. What I did was I created a variable called average a PG Tuition equals are existing variables that were given in the data data set. So just some divided by two. So, however, which way you do that in your in your statistical software, create that average variable and run a new regression So the progression will run here in Part four is the same as three, except instead of using instead of using these blue for this, instead of using both of these variables will use the average. So what you want to do is regress everything in and part one. Plus, let's see that'll be beta beta, five beta, five times their newly created average tuition variable. What state are term? Right? So that's our our new regression. That's what you want to run here. And what you should find out have joined this regression. Is that Yeah, the P value for beta five. So a beta five pat let me switch colors again. So it's a little little clear here. You should get that beta five hat. So the coefficient on the average tuition variable. Um, first of all, it's it's it's positive, but the P value this is the most interesting part of the P value equals but point to. And if you look closely at the regression up in part three, they didn't ask you to look at this. But it might be interesting interesting to check this out. The P values for tuition 17 intuition 18 were between point A and 180.0.9. So this average here has brought the P value down toe 0.2. So, still not arguably not still statistically significant there, but at I have to say compare so compared that p value to when the tuition of variables were used separately. It was the P values for them are about 0.82 point nine approximately. So you could come away with FIS regression with a possible conclusion that creating this average variable is cut down on unnecessary variation. It also picks up three idea that it's closer to having some effect on eventual educational attainment, which lines up with what we'd expect. You know, the cost of cost of college does have an impact on whether we decide to go back for a year, or whether to differ. And things like that that should be part for should be good in after you do that. And then kind of we have Parts five, which asked if the if the findings for the average tuition variable we just found in Part four make sense when interpreted. Causeway. So I'll just write down what I got for the actual coefficient. Uh, I'll straight down for average tuition. So my estimated coefficient from Part four equaled 16 and 0.12 is the standard error. So again, uh, not quite statistically significance. But what we're interesting here is the coefficient right and especially the fact that it's positive. So when you think about this interpretation. This is saying that a, um if average tuition goes up by others to say one unit, so I'm not sure what the units are. But if average tuition goes up, then your education or individuals education on average is predicted to also go up. That's not what we expect, right? So we expect, as the cost of going to school goes up on average, uh, controlling for other factors that were controlling for we expect education thio, um, go down or this coefficient to be negative. So kind of brings up the question like What's going on here, But I might be going on also is something that OLS isn't very OLS estimation like this is not very good at picking up, which is, if you think about if people who complete more education. So to say, a person here has, uh, just say higher education eventually gets more education, but also went to also might have gone thio institutions or toe colleges with systematically higher tuition than our other are other person over here, um, you end up getting not quite high oven education and went to institutions with lower average tuition. So if you think about compared to these two different scenarios, let's say, and there's there's may be good reason to believe that, um, people who complete mawr education systematically tend institutions that charge higher average tuition. So that would be one reason of explaining and maybe the most possible reason for explaining this. It's, ah positive coefficient on average tuition, which is not what we would expect normally. But OLS is not good at picking up this sort of this interplay here. This kind of self selection. Um, yeah, that's what that's how you can end in the problem.


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