In equation $(4.42)$ of Chapter $4,$ using the data set BWGHT, compute the $L M$ statistic for testing
whether motheduc and fatheduc are jointly significant. In obtaining the residuals for the restricted
model, be sure that the restricted model is estimated using only those observations for which all vari-
ables in the unrestricted model are available (see Example 4.9).

Answer

In order to find the joing effect, use the method of LM restricted and unrestricted F-test.
First run the simple regression taking colgpa as a dependent variable on cigs, parity, and faminc,
using only the $1,191$ observations with non-missing observations on motheduc and fatheduc.
After obtaining the residuals, $\widetilde{\mu}_{i}$ regress these residual on $\mathrm{cigs}_{\mathrm{i}}$ $\mathrm{parity}_{\mathrm{i}}$ $\mathrm{faminc}_{\mathrm{i}}$ $\mathrm{motheduc}_{\mathrm{i}}$ $\mathrm{fatheduc}_{\mathrm{i}}$ using $1,197$ observations with nonmissing values for both motheduc and fatheduc.
The R-squared from this regression is unrestricted, $R_{u^{\prime}}^{2}$ which is about.0024.
With $1,191$ observations, the chi-square statistic is calculated as the following
$(1,191)(.0024) R_{u}^{2} \approx 2.86$
The p-value from the $\mathrm{X}_{2}^{2}$ distribution is about. $239,$ which is very close to $0.242,$ the $p$ -value for the comparable $F$ test.

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Introductory Econometrics

Chapter 5

Multiple Regression Analysis: OLS Asymptotics

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Video Transcript

Hi, everyone. So this problem for Chapter five is just a more practice in, um, doing an f test. So, looking at whether more than one explanatory variables in your model are jointly significant and I've promoted our equation, I don't usually do this, but I preloaded the equation will be estimating, um, onto the screen here. And this problem is just very closely tied to, um, equation 4.42 So this goes back to Ah, a problem in chapter four. Um, and so this is what it is. Just a looking at body weight here. And this is, um, sorry. Birth weight of a bit of a child of an infant and looking at that birth weight as a function of, um, how much the mother smoked during pregnancy. That's this variable, the birth order of the child. So where the child fits into the the family in general. So how many other kids are surrounding it? Um and then the family income here is the third variable. And then that the two variables that arm or more more of interests and we'll be doing our test on is the mother's education Father's education. So all five of these explanatory variables are, um uh, thought to have a potential effect on the birth weight of a child. So, um, the hypothesis. Um Oh, sorry. First, I'll say that the problem asks you to with this equation using the daddy's the data set B w g h t. So the data set you'll be using, um, to do this problem is that's its name. It's the he bought birth weight data set. The problem asked you to complete compute the statistic for testing whether mother's education here and father's education, whether they're jointly significant. And so when you hear joint significance, you should immediately think all right, we will probably have to do an f test. Um, Thio find a solution to that problem and estimate estimate that joint significance. So the first thing you want Thio just make sure you you stay up front is that you're going tohave. We're gonna have to restrictions in the model to restrictions. Um, the restrictions again are going to be the the mothers and the fathers. Education. Okay. And you're not hypothesis. Also, you can stay and, you know, hypothesis could basically just be stated as, um, Beta four equals zero comma so and beta five equals zero. Right? So once you have that all set up, um, that's where Oh, and you can Also, something else you can also do is, um, specify and just state the degrees of freedom. And you're gonna wanna look at the degrees of freedom from the for the unrestricted model, which is this model that I've already written up here. But the degrees of freedom are gonna have end. So your sample size n minus minus six. So let's put that in there. And has that six number come about? Um, artist rates equals n minus K minus one. Where K is the number of explanatory variables in the unrestricted models. So end minus five. Minus one. Right? Okay. We stayed all those things up front. That will be useful as we go through the problem and computer F statistic toe. Look at whether those two education variables air jointly. Significant. The problem also want you to really think hard about, um, making sure that when you estimate the restricted model for the F statistic. So when you dio um, let's just say when you think about the number of observations which I'll just I'll just say upfront here. You can figure this out pretty easily. Hopefully, um, the number of observations in this sample is 1181. So you have that. And just remember that that sample sizes in the unrestricted model. So we haven't taken the education variables out of the model yet. They're all still included in the bottle. The problem is just asking you to make sure that you use that same number of observations. When you estimate the restricted model at this way, it has toe work. Um, so when I guess I can write this by saying, um, you just have also have to make sure you use that also in the restricted model, right. That's because some observations for which, um, some observations for the mothers and fathers education here. Some of them might be missing data for one or both of these. So that's why you would have, um, potentially a greater number of observations in the restricted model. So just make sure you use those two, um, in both of those unrestricted interviews and models used the same number of people. All right, so once we have all this set up. What we What we need to dio is just writer Absolutely right part are happy his So I'll just write it out explicitly for us. And we want to use the r squared form of of the statistic here. I'm not sure I've done to my videos before. I typically use the sum of squared residuals form, but, um, here we will use the R squared form. So the R squared form of the F statistic is as follows. So you have the r squared from your unrestricted model? Is that first term here minus the R squared from the restricted model? So that's taking out those two education variables divided by Q. That's the number of restrictions. And so that will equal to, in our case, divided by one minus again, the R squared from the unrestricted model, the full model, then that that is divided by the degrees of freedom in the under shifted models. Again, we've already set ourselves up. We've found out some information such as this degrees of freedom. Um, let me just read out really quickly what these will be, So Q is going to be too degrees of freedom. We found out appear and minus six. Right, So 1191 minus six. So you should have to be defeated. Will be 1185 here and now we just have to run those aggressions, right? So the first regression you'll run, um, have to write this out. So we are clear on what we have to do here. So there were, uh let's just say regression sub you are. So the unrestricted aggression is the full model, right? That's the first one you want to run, and then the second it's a little one here. First regression, you want to run second regression Want to run? Is the regression the restricted regression? So the regression dropping the to as education variables. So I'll just say that as drop Yeah, education variables and just a reminder for this part. Use again. Use the, um, the observations in the in the unrestricted model. So use sample and use that sample from the unrestricted model. Um, in your estimation, right. All right. So you don't have to run those regressions. Andi, just find the are squares, what you should get or the r squared for the full model or the under strict regression. You should get 0.3 87 And then when you run the restricted model, you should get, um, no pause here and think, Just make sure you think so. Restricting model. We're dropping a couple of our explanatory variables. So you would expect that are square would go Which way? Up or down. Um, if you're dropping potential explanatory power, the answer is I'll just I'll just write the R Squared three. Answer is down. Right, Because we're losing some explanatory power were explaining less of the variation and in our dependent variable birth weight, right, So that's what you should get for There are squares and now we have all the numbers we need to compete are f statistic. So just explicitly all louds. So horror Unrestricted model R squared is 0.387 This out front here minus 0.364 divided by two our restrictions and then bottom. Here we just have one minus point over 387 divided by, um 1185 1000 and 85 degrees of freedom in the under 16 model and that will pop out thio drum roll 1.42 So we're pretty much at the end of the problem. So again, remember, we want toe find out whether, um, these two variables Mother's education and fathers education are jointly significant, um, in determining the birth weight of their child. Right. So that's why we don't all this all this statistic work here. So this F statistic, if you if you look it up, it will be, um, well below the, um, the 5% critical value. So just say, using this F value that it is below the 5% critical value. So, basically, this EP statistic is not large enough to tell us that to tell us that we should reject the null hypothesis which waas That, uh, that mother, that the mother education and father education that those parameters were both zero. That was our null hypothesis up here. Right? So our f statistic, the bottom here is telling us that we can't with any certainty reject the null so can't reject the null at at any at any decent statistical significance level. So 5% but say that's a decent level. Can't reject no hypothesis. And so our final thing we can right here. Is that e the education of the mother of the child? So Mother's mother's education and the fathers education are jointly, insignificant. Okay, are jointly insignificant and then determining the birth weight of their child. It's going to shorten that up here, says a review that insignificant, generally insignificant, significant in determining, um, the birth weight of their child. There was a previous birth ways, um, be done w done of their child. And we could take a step back, I guess. And just think about that conclusion. Um, there might be some reason that you think that education of the mother and father might be significant in determining the birthday their child. And that would be maybe, um, the families diet. So how their how their dieting and how they're their lifestyle choices when they're raising a child or during the pregnancy period on. Do you think that if there, um, the couple is less educated, they might make some more lifestyle choices? Diet choices, whatever you I can think of that would lead to a lower birth weight of their child. But again, remember, we are controlling for Cem other relevant variables here, particularly I would argue the family income variable. So we're already controlling for their level of income here. Um, and of course, the other two variables here would be but important to control for. So, um, that's the final conclusion here. Right on band. That's the and the problem here.

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