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The data set 401 $\mathrm{KSUBS}$ contains information on net financial wealth (nettfa), age of the survey respondent $(a g e),$ annual family income (inc), family size (fsize), and participation in certain pension plans for people in the United States. The wealth and income variables are both recorded in thousands of dollars. For this question, use only the data for single-person households (so fsize$=1 )$ .(i) How many single-person households are there in the data set?(ii) Use OLS to estimate the modelnettfa$=\beta_{0}+\beta_{1} i n c+\beta_{2} a g e+u$and report the results using the usual format. Be sure to use only the single-person households in the sample. Interpret the slope coefficients. Are there any surprises in the slope estimates?(iii) Does the intercept from the regression in part (ii) have an interesting meaning? Explain.(iv) Find the $p$ -value for the test $\mathrm{H}_{0} : \beta_{2}=1$ against $\mathrm{H}_{1} : \beta_{2}<1 .$ Do you reject $\mathrm{H}_{0}$ at the 1$\%$ significance level?(v) If you do a simple regression of netfa on inc, is the estimated coefficient on $i n c$ much different from the estimate in part (ii)? Why or why not?

(i) there are $2017$ single people in the sample of $9,275$(ii) SEE SOLUTION(iii) The intercept obtained above does not seems to be interesting because it provides the estimatednettfa for inc $=0$ and age $=0 .$ It can conclude that there is none of them even close to thesevalues in the appropriate population.(iv)From the output obtained in part (ii), it can conclude that the p-value for age $\left(\beta_{2}\right)$ is 0.000 that isless than the considered level of significance $0.01 .$ Therefore, the null hypothesis may berejected at 1$\%$ level of significance.

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Chapter 4

Multiple Regression Analysis: Inference

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Bear H.

April 18, 2020

Faiza N.

May 4, 2021

Is there anyway I can get the R studio code for this question?

Sonya R.

May 14, 2021

What is the function for part 2? I wrote reg nettfa inc age on stata and got different results with the ones discuss here

Jonas K.

May 31, 2021

Sonya R - you need to use this in stata to "regr nettfa inc age if fsize==1"

Dylan L.

October 26, 2021

Hello Are we able to see the video for the coding of this in STATA?It would be much appreciated!

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This is another statistical inference problem. And this time we're gonna look at a different equations. So we're interested in, um, net financial wealth of people in a survey, and it's a pretty simple model. We're just going to assume that the net financial wealth of people is just a function of their current family income, their annual family income and their age. So part one is pretty simple. It just ask you to find a number of single person households in the data set. And so, just in whatever program you like using, you can use the syntax to figure out the number of single person households. And I'll just give that number here. So scope black here. So if you write out the correct syntax and there's different ways to do that, depending on whatever program you use, you should get the number of single person households in the sample, and I just abbreviate household as HH. You should get that it is equal to 2017 and will be you will be using that sample in the rest of the problem. So there's the answer for part one. Pretty simple part two. Ask you thio, use classic OLS to estimate the following model. I wrote it out. In this function form, your problem probably has it in the standard econometric form, and it says, to be sure to use on Lee, you're single person households in the sample. So again, right your syntax to make that happen, just use the sample of 2017 single person households. So let's throughout the different numbers here, all right out there coefficients. And then I'll have the, um, standard airs underneath the coefficient. So for income, we have the positive coefficient that makes sense. So that's just saying, as your annual income goes up holding age, constant people are reporting that their net financial wealth is more so standing there era, for this is 0.6. That's gonna be very statistically significant and age. We would maybe also expect age to have a positive coefficient, right? So as you as you get older, holding holding income constant, you likely will have a higher net financial wealth than people who are younger. So let's put that there also with a small, small, standard error here. It's very going to be very statistically significant here, So after we've estimated this it says to, uh, interpret the slope coefficients, so let's just go to income first. So this 0.799 this just represents that one more dollar in a respondents income that's a holding age. Fixed is results in about 80 more sense in predicted net financial wealth. So I guess I could just write that out a little bit. So if you have one more dollar of income, you have about plus 80 cents more of you're wealth. So there's that interpretation on the income coefficient for age. Let's see what this one means. So 0.843 is a coefficient, and this means that holding income fixed if someone is one year older. So let's just say, plus one year, one year older, his order net financial income is British had increased by about $843. And just be careful with the units there. So the coefficient itself is 0.843 But since the net financial wealth variable is given in thousands of dollars, that's what that's what we we get for the impact. All right, And then I also asked you, are there any surprises in this these slope estimates, so we've kind of already answered this. But first of all, both of these coefficients are positive, you know, holding other things constant. If you have a higher annual income, you're protected toe have a higher net financial wealth and also holding your income constant as you get older. Your also predicted toe gain or have a higher net financial wealth. There's no surprises here. Part three moves along and just asks if the intercept from the regression that we estimated, which I haven't given. So let's just write that out. All spread out prepared? A not And hopefully you get this answer after you estimate in your own program. But your bid not should be negative 43 zero there and then if the standard error of 4.8 And so does this have any interesting meeting? So remember that the nurse steps the estimated intercept when we run these sorts of regressions. That is the value of our dependent variable, with all of the explanatory variables set to zero right, So we can think about this. This number negative 43.4 as the net financial wealth when income equals zero and when age equals zero. So what kind of interpretation we use here? So with someone's if if someone is predicted to have net financial wealth of around negative $43,000 when they have no annual income and ages zero, that doesn't give us any any kind of interpretation, it doesn't have any relevant value for our respondents that we have in our sample or in our population. So just using common sense, we can say there's no there's no interesting meeting here, so just write that in red and that's just given. Given what we know about these different variables, we know that that combination of having no income, having no age that doesn't give us anything to this intercept is is meaningless. And it's just the results of the calculations. All right, part four for Part four. It asked you to find the P value for the test of a null hypothesis that Beta two, which is the coefficient on age that beta two equals one against the null hypothesis which is baited to is less than one. So I'll write out first of all, the the null hypothesis. So now hypothesis debated. Two equals one. That's the coalition on age again have in the alternative is that it's less than one. So if you want to find the p value, um, for that test, what we need first is our t statistic and to get our t statistic. So our our first step here through a little one in green. So first step, we need t statistic. We need t equals br coefficient. So baited thio minus ones that are null hypothesis. Here the beta two equals one divided by the standard error. So, uh, I'll just write that is standard error of beat it too. And so plug these values in. Right. So Beta two, we had 0.843 and then again, minus once, that's the numerator. And then our standard error was put 0.92 And so our T statistic here for this hypothesis test is going to be about negative 1.71 That's our first step in determining the P value second step points we have. This t statistic is to just and you could do this. Go on, Google go on the internet and there are many ways you could find out p values from different T statistics. So the two things you need. So just say p here, So P value for a T statistic of negative 1.71 and the second parameter you need to put in here is the degrees of freedom involved in your regression here, which is basically given that we have almost no variables in our regression is basically a pivot to your sample size, so I just put in 2000. It's a little over 2000, but we'll get roughly the same results here. If you want to be really accurate instead of 2000, put in 2017 and once you input that into whatever never place you find on Google on the Internet, you'll get a P value of about 0.44 So that's decent statistical significance there. Let's circle that with red. It's not too bad, Onda question asked. Do you reject the null hypothesis that the 1% significance level, And so even though we have a P value of 04 that is greater than 0.1, So this means that we can reject then all hypothesis, so that'll be your answer for part four. And then finally, part five asked if you do a simple regression of the Net family income. Let's write this down. So net Family income, simple regression of the income. Sorry, Net financial wealth. A simple regression of the financial wealth on income. So this time you drop age from the regression again. I'm just saying that in the Function Forum, yeah, might be a little harder to look at than the econometric form, but you can write it out in the other way, if you if you feel it's easier. And it asked once you run that regression. Is your estimated coefficient on the income variable much different from the estimate, in part to when you had aged included in the model. And what you should get for this coefficient in the simple regression is 0.8 to 1 about, and so you just compare that coefficient. Let's say beta beta equals 0.8 to 1. You want to compare that coefficient to the coefficient you got in part two, right? Which is 0.843 I'm sorry, that's the wrong one. Go back. There's income. So 0.799 was the one we've gotten in Part two and 0.8 to 1 is the one we got for here in part five. Those really aren't too different. And why would that be so? The main reason why these might be different is that age. The correlation between income and age, given that we have just a sample of single person households isn't very high. Um, so we can write that down. So a reason for the not a huge difference between the simple regression estimate and the multiple regression estimate is just low correlation between income and age. Four are single person sample and you don't have toe. You can run that if you want. You can find out in your program run. You can run like a simple correlation between income and age and your sample. And, um, it is low about 0.4. So that explains why those parameter estimates on income aren't different between the simple on the multiple regression

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