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Use the data in WAGEl for this exercise.(i) Use OLS to estimate the equation$\log (w a g e)=\beta_{0}+\beta_{1} e d u c+\beta_{2} e x p e r+\beta_{3} e x p e r^{2}+u$and report the results using the usual format.(ii) Is exper^{2} statistically significant at the 1$\%$ level?(iii) Using the approximation find the approximate return to the fifth year of experience. What is the approximate return to the twentieth year of experience?(iv) At what value of exper does additional experience actually lower predicted log(wage)? How many people have more experience in this sample?

i) .296ii) The $t$ statistic on exper' is about - $6.16,$ which has a p-value of essentially zero. Hence $\text {exper}^{2}$ is significant at 1$\%$ level (and much smaller significance levels).iii) $\approx 1.39 \%$iv) In the sample, there are 121 people with at least 29 years of experience. This is a fairly sizeablefraction of the sample.

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Chapter 6

Multiple Regression Analysis: Further Issues

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Sara S.

November 16, 2021

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Hi, everyone. Welcome back. Ah, we continue with comes your exercise number two in chapter six and we're going to use the data cell cold wage one for this exercise. So I have imported here and say that 1st 1 of these over less twist made the simple equation that relates that lug of wage thio interested plus education plus experience plus experience squared. So we have a quadratic turn for experience and went to report the results in usual format. Let's go first ahead and describe the data said to see what the variable star So the wage here, or the log of wage the wages outwardly average hourly earnings education is measured in here's as well as experiences measured in years. Potential experience. All right, We already have experienced squared here has been given or get it also generated. Of course. Let's let's go ahead and rung the regression. So regress lug of wage On what On education? Where's the constant? Jeremy's in play there. I say that Experience extremes. Let's see what? Okay. All right. Here it is on numbers of remember observations is the correct 1 526th recon. See right here. Ah, the Joint F tested very statistical significant, which means that the hypothesis but all the coefficient equal to zero is strongly rejected. And which means, of course, that a model does much, much better than a simple motor with Justin intercept. And this is also reflecting. The addresses are square almost 30% 29.623% of the variation in the luck of wage is explained by this two variables and democratic term. That's, you know, pretty good, given that the wages a function of many, many, many different observed observable and unavailable variables. And here the point is of interest. All our coefficients, local efficient, are statistically significant. Any level of significance are constant. Stream is not really statistically significant. And, uh, indeed the values theories bit more than one syndication away. By the way, we report the results here. These are estimated equation. All right? No, in part, two were being asked if the quadratic terms on experience experience where it is statistically significant. The 1% level Well, let's go back. Let's look at that. Whether we look first, we look at the T stat here. Maya's six. Well, we know that for a T stat. Um, it's a statistic. Thio Ah, show that ah coefficient is statistically given. The 1% level needs to be bid more than three right three point something here We have a very, very large value, so it must be very seriously significant. And it is. This reflected with P value of less than 0.1 which would be 1%. Indeed, this is significant any level of significance on 1% even less even the zero upon 1%. It's very, very, very significant. OK, in part three, we need to use the approximation, given that the difference in the expected wage is equal to 100 plus the estimate coefficient B that two plus two times would be the three times experience. Ah, no, This multiplied by the difference and experience that we're trying to, um, to find. So find the approximate return for the fifth year of experience and do the same to find the approximate return for the 20th year of experience. So using this equation, which is again an approximate approximation, Ah, one of the West made the return for the fifth year of experience. We started with experiencing all four and we increase experienced by one. So right, we need to look at the experience of someone has four years. Someone has five years and everything else is controlled for between these two individuals. And see what happens when we changed this year experienced by one. So we're gonna subsidy into that given equation are are estimated coefficients here. As you can see, B to B, I had to be the two head right, your 0.4 10 minus two times be the three. Remember, this is from the partial derivative that we, uh we learned in the textbook there's a partial derivative of the quadratic terms of the two comes, uh, in front times four, which is the experience, right? We're looking for and this whole thing is multiplied by adult experience were just able to one. So I didn't write it here, and this is approximately equal to 3.53%. Which means that the return that an extra year of experience when you have four years and you increase your experience by one here you get an estimate increasing your wage of abound. Uh, 3.53%. Remember, we need to multiply by 100 because this is a log level model. So we measure one unit increase of the independent variable. What effect? That's having person increase off the deep end. Grateful at least approximately because, you know, love difference is approximately equal that now we do the same for the 40th year of experience. So we start with experience it with 19. And of course, the experience it was 1 20 minus 19 was one. So if we do the same, we find that, uh um uh 20 the one extra year of experience when we're the 19th year the 20th gives at 1.39% increase smaller than before. And of course, this is expected because, as we saw in the estimated quadratic term here, this is negative. And the beata to have me that had to term is positive. Which means that the relation between experience and lug of ways it's nonlinear. But in what way? In an increasing in Kong cave way we talked about in the previous video, it's like this that we have a slug of wage on the Y axis experience of them. Exact is something like that, right? What's not well I don't do it now. Do it better. All right, I need it was like this. But I want to show you that there's a turnaround born here codestar where this affected maximal. Okay. And after that, the, um the ah, the partial derivative of the additional actually makes the predicted lug of ways lower. Which I don't know if it really makes sense. If you have I don't know, 40 29 years of experience. You get a lot of incredible. You have 35 years less, I don't know, but in a way, this is what it's in flight by a model. And now for the final part. At what value of experienced those additional experience, actually lower predicted lug waste. What what we just talked about. And how many people have more experience in the sample? Okay, to find this, remember, we learned it at some point in chapter six, almost in the beginning, the turnaround point, Or as it's called in mass, the inflection point of this Ah, increasing conch. A function is given by, uh, be the two hat divided by do times be the three hat here. This gives us around 28.7, let's say 29 years of experience, right? So this is the maximum points for the predict. Low wages is highest attn least the living wage. Based on the experience of this quadratic turn right, this relationship is at its highest point. And after that, it goes down. So the partial derivative becomes negative. Now, how many people have this experience in the sample that's easy? Just gonna count. Scores were going around the number of experience to 29 here. County of experience, greater or equal to 29 right, 101 121 people. And out of the 526 well, you know, just a showcase 1 26 divided by ah, 5 26 That's Ah, 24%. That's a sizable. You know, a fraction of the people have this, um, this level of experience. So this estimated equation is not really prone to out life. So a very few extreme observation or it, like of data set. So we should, you know, be fairly, um, this is a friendly, reasonable announces. We have sufficient data to conclude that indeed the maximum Ah, the maximum point of this relationship's on 29 years. And after that, the predicted lug of wage is going down

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