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Consider a model where the return to education depends upon the amount of work experience (and vice versa):$\log (w a g e)=\beta_{0}+\beta_{1} e d u c+\beta_{2} e x p e r+\beta_{3} e d u c \cdot e x p e r+u$(i) Show that the return to another year of education (in decimal form), holding exper fixed, is $\beta_{1}+\beta_{3}$ exper.(ii) State the null hypothesis that the return to education does not depend on the level of exper. What do you think is the appropriate alternative?(iii) Use the data in WAGE2 to test the null hypothesis in (ii) against your stated alternative.(iv) Let $\theta_{1}$ denote the return to education (in decimal form), when exper $=10 : \theta_{1}=\beta_{1}+10 \beta_{3}$ . Obtain $\hat{\theta}_{1}$ and a 95$\%$ confidence interval for $\theta_{1}$ . (Hint: Write $\beta_{1}=\theta_{1}-10 \beta_{3}$ and plug this into the equation; then rearrange. This gives the regression for obtaining the confidence interval for $\theta_{1}$ .

i) Holding Exper (and the elements in u) fixed, one will have:$\begin{aligned} \Delta \log (w a g e) &=\beta_{1} \Delta e d u c+\beta_{3}(\Delta e d u c) E x p e r \\ &=\left(\beta_{1}+\beta_{3} \text {Exper}\right) \Delta e d u c \end{aligned}$ $\frac{\Delta \log (\text {wage})}{\Delta e d u c}=\left(\beta_{1}+\beta_{3}$ Exper \right$)$This is the estimated change in proportion in wage variable given one more year of education.ii) $\beta_{3}>0 $ is the appropriate alternative.iii) $\log ($wage$)=\beta_{0}+\beta_{1} e d u c+\beta_{2}$ exper $+\beta_{3}($educ$\times$ exper $)+u$iv) $\hat{\theta}_{1} \approx .0761$ and $\operatorname{se}\left(\hat{\theta}_{1}\right) \approx .0066$

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Chapter 6

Multiple Regression Analysis: Further Issues

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Bedasa T.

May 27, 2021

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Merkineh G.

June 18, 2021

log(wage) =Bo+B1educ+B2exper+B3edu*exper+u. A. show that the return to another year of education(in decimal form)holding experience is fixed, is B1+B3exper.

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moving on with computer exercise number three, we're going to consider a model where the return to education depends upon the amount of work experience and vice versa. So the, um, regression model that we're gonna use is lug of wage, which is a dependent variable equals when intercept term plus b, the one times education plus B the two time experience plus B the three times education times experience. This is an interaction term, which is the most negative factor that says at supposedly the level the return to education will depend on the level of experience, right? This is the key part. Depends on the level of experience. Plus, of course, a disturbance term. It's standard I'm supposed to be where we don't really need it. But for the testing is supposed to be idea normal, distributed and homeless. Fantastic. In part A. We need to show that the return to another year of education in decimal form holding experience fixed is beata one plus B. The three times experience this is very easy. Is using the standard causal interpretation interpretation of Cherries parabolas uh, we're gonna we're gonna hold experience and the elements of you fixed so we can eso it out of the analysis once we apply the difference Operator, the Delta operator. And here, of course, difference Operator is not the same as in time. Siri's where we make the difference of, Ah, 10 trees, value minded. It's lag, but it's just simply the difference between, you know, change is like the It's like the partial derivative. Basically, in fact, is that this great analog of the partial derivative So what we're going to Dio is gonna take partial derivatives. But I'm just gonna show you here using a discreet for him, because this how the book does it. But of course, you can just take the partial derivatives, the the result in exactly the same, at least in in decimal foreign approximate terms. An approximate number. Who's here? We're just treating the equation indiscreet for a brother than continues, So we played a different separator. The difference in luck of ways equals to be the one times a difference Education. Let's be the three times a difference in education time experience, of course, the elements of experience and you have dropped out whatever he's not multiplied by equations dropped out exactly as we do with the partial derivatives One we treat This thing's in constant in this case. Ah, And if we rearrange this equation with a common factor we get be the one must be the three time experience Inferences, times, tilt education If we divided by little education, of course, we get what we want. Delta lug of waged by adult education equals B the one let's be the three time experience. And this translates to the approximate proportional change in wage given one more year of education. So the, uh, in terms of partial derivatives, it would be given tiny change in education here, we want to do you know, this is a discreet case measured in years s so we want to do the, you know, basically a partial derivative, but with a difference equals people with one, and we go to one. Uh, we can easily substitute everywhere. Everything cancels out. This is exactly what we get. Nice and easy, no important be wanted to stay. The no hypothesis that the return to education does not depend on the level of experience what we think is the appropriate alternative. So, as we said before, ah, the inclusion of the interaction term means that we want to test or we want to model the r. I want to create a model that education might depend on the level of experience. Now, if we want to taste monopolistic does not depend. Then the novel be That'd be the three equals zero. So this interaction term doesn't really matter doesn't make a difference, is not statistically significant, right? It was how if we think the educational experience interact positively so that people with more experience more productive when given another year of education than the alternative your property don't mind, it would be be the three great in the zero. Now, if we think that the reverse holds true than the alternative would be be the street less than zero. By another case, we really need to restrict a priori our T test, meaning that we don't have to explicitly specify one sided teeth test. But we can just say that the alternative is better. Three is not April 20 And how are we gonna know that, Thea, whether whether the test, um, I give us an answer for less than more than zero. Well, it will depend on the tea staff that we're gonna get. So if the estimated T statistic for a coefficient is minus 3.5, for example, they suggest that there's a negative relationships, a bit of threes. Lesson zero before decide is plus four, for example, that the alternate would be be the three greater than zero and it will be accepted. It would get such a high on A T stat. Now, in part three, winning to actually test what we're talking about and use the data set and waits to to test There's no hypothesis against alternative here and stay down. I have important our data set. As you can see, we've worked with this Data said before aware with these two, Which one there Very similar. So what we're gonna do, we're gonna regress, lug of wage education, times experience, Times in direction term. Now let me show you how you do it Here. Say you put See that education? Hashtag See that experience? Okay And let's see there, see what happens. All right, Here it is, right. Number of observation Ah, very significantly insignificant Join F test are squared just r squared off 13.13 point 2% of the variation in the lug of wages explained by this model. And here this is the interaction term right here. Coefficient of 0.0 32 a T tested 2.9 Well, you know, it's higher than the critical value of 1.6 with 5% plays in the critical value for 1%. Indeed, the P value is 0.0 36. So what does this mean? I, uh, written down here the estimated equation, right? And, um, given the t stat that we obtained What? Pea valued. Listen, 4%. What? Usual. We don't report for person. Listen, 5%. So if we, uh, if for no happens to be the three balls 20 r alternative, his biggest trees greater than zero, then we can reject the null hypothesis B 23 close to zero against the alternative at the 5% level of significance. Because, as you can see here, the P value obtain is less than five. Of course, we can also state less than four, but we don't usually, you know, report 4% level, But what we can do it like nothing is just a matter of Um um that's just how things are done in journalists. Usually usually the levels. Which means that 1% 2% and five usually But we can say that we can reject with 4% significance level. Okay, now in parts for Let's Eat a one d Know the return to education in dismal form when experience equals 2 10 So theater one, it was be the one place 10 beater three because, as we saw before, we found the return of education is be the one place either three times the level experience. So we want it now. See what happens when experienced equals 10. So be the one place can be the three we need to obtain. Obtain ah estimated value for theta one theta one had in 95% confidence interval for feta one. We're gonna follow the hint and rewrite Be the one as theater one minus 10 Be the three and plug this into the equation. Rearranged. Well, that's very easy. It's very straightforward. I don't want to go through the it's it really is. You can do it is just a simple manipulation. So if we like it, it would plug it in and rearrange. We get a lot of ways. It was to be the zero place theater. One education. Just be the two time extreme. Just be the three education times experience buying his 10. Right. So what we need to do here is we to going to run the aggression of lug wage with this variable salt right hand side. And we want the coefficient of education. Okay, so, first of all, we need to generate a new variable called Let's Call It, uh, you know, experience. Um, 10 you know, And it's gonna be cool too. Ah, Experience. Minus 10. This is a plight everywhere. Okay, USA takes from women's world That win. Great. That's great. Congrats. Now let's move on. And we're gonna run the regression here, but with, ah, education. And here instead of interaction term. See, experience Got VC experience. Uh, 10. I called it. All right. Here it is. Nice. Look at that justice Where? Okay, everything looks fine. Everything looks fine. Thea goodness of fit, of course, is the same as before. We just evaluate this, um, modeling at the level of experience. It was 10. And we're interesting days coefficient right here. Ah, look at that extremely high tea step. Which means that the coefficient significant at, you know, and the level of significance even a 0.1 Ah, again, interaction term here is the same as before. And we basically get, uh, almost the same things before, But this one here is more even more significant, Right? Let's write down the estimated equation. Well, we don't need it right down. We just obtain value. Feta had one equals two approximately equal to 0.761 And the standard ever, Which you can see right here. Right. This is the standard homo scholastics on the error. So this thing is not written in stone there, Manu many ways of dense than the air. They're just the standard, most tender homos. Fantastic one with no correction. If I just did the same thing and then re busts on the errors that we get larger center, which is supposed to more robust. So since we've been these value for theater one, which is the average right, is the average for the average estimated value. So he's supposed to meet the average effect, partial effect and the standard every 0.6 except than the 95 confidence interval with either one is gonna be the average partial effect, minus two times the same division and plus two times 10 deviation, which is here about 2 10 0.0 63 to 0.0 89. And as they say statistically, Slifkin, this means that is statistically significant differently. Zero. It is very fine. Bharati test And it's also very funny. The conference interval did. That does not include the value of zero, and this is a final answer.

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