Question

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 model nettfa$=\beta_{0}+\beta_{1} i n c+\beta_{2} a g e+u$ 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?

Name: Problem 8, Chapter 4: Introductory Econometrics
Uploaded: 2020-09-24T22:22:42-08:00
Duration: 15 min 53 s
Channel: Lyle Anderson

   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 model
nettfa$=\beta_{0}+\beta_{1} i n c+\beta_{2} a g e+u$
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?

Introductory Econometrics

Jeffrey M.… 6th Edition

Chapter 4, Problem 8

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Solved by Expert Lyle Anderson

09/24/2020

Step 1

This can be done by filtering the data set for instances where the family size (fsize) equals 1. The number of single-person households in the data set is 2017. Show more…

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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 model nettfa$=\beta_{0}+\beta_{1} i n c+\beta_{2} a g e+u$ 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?

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Key Concepts

Sample Selection and Data Subsetting

Sample selection involves restricting the dataset to a specific group of observations based on certain criteria. This process is crucial in ensuring that the analysis is relevant to the research question and can affect both the estimates and their interpretation, as it might alter the underlying distribution of the variables.

Interpretation of the Regression Intercept

The intercept in a regression equation represents the estimated value of the dependent variable when all independent variables are equal to zero. Its meaningfulness depends on whether this scenario is within a plausible range of the data; if not, the intercept may have limited substantive interpretation.

Statistical Significance and p-values

The p-value quantifies the probability of observing an estimated coefficient as extreme as the one obtained if the null hypothesis were true. A low p-value (often below a threshold such as 0.01) indicates that the observed effect is statistically significant, suggesting that the relationship captured by the coefficient is unlikely to be due to random variation.

Simple vs. Multiple Regression

Simple regression involves one independent variable, while multiple regression involves two or more. The distinction is important because omitting relevant variables in simple regression can lead to omitted variable bias, whereas multiple regression controls for other factors, potentially leading to different and more reliable estimates of the relationship between the key independent variable and the dependent outcome.

Interpretation of Regression Coefficients

Each coefficient in a regression model represents the expected change in the dependent variable for a one-unit change in the corresponding independent variable, holding other variables constant. This enables researchers to assess the impact of each predictor on the outcome and provides insights into the relationships among variables.

Ordinary Least Squares (OLS) Regression

OLS regression is a statistical method used to estimate the relationship between a dependent variable and one or more independent variables by minimizing the sum of squared differences between the observed outcomes and the predictions. It is a fundamental tool in econometrics and statistics for understanding and quantifying linear relationships in data.

Hypothesis Testing in Regression

Hypothesis testing in the context of regression involves assessing whether a regression coefficient is statistically significantly different from a specified value, typically using t-tests. This process helps determine whether the effects identified by the model are likely to be real or if they could have occurred by chance.

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Transcript

00:01 This is another statistical inference problem.

00:04 And this time we're going to look at a different equation.

00:07 So we're interested in net financial wealth of people in a survey.

00:14 And it's a pretty simple model.

00:16 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.

00:30 So part one.

00:32 Is pretty simple.

00:33 It just asks you to find the number of single person households in the dataset.

00:38 And so just in whatever program you like using, you can use the syntax to figure out the number of single person households.

00:53 And i'll just give that number here.

00:56 So let's go black here.

00:59 So if you write out the correct syntax, then there's different ways to do that depending on whatever program you use.

01:08 You should get the number of single person.

01:10 And households in the sample and i'll just abbreviate households as hh.

01:17 You should get that it is equal to 2017.

01:27 And we'll be using that sample in the rest of the problem.

01:33 So there's the answer for part one, pretty simple.

01:38 Part two ask you to use classic ols to estimate the following model.

01:46 I wrote it out in this function form.

01:50 Your problem probably has it in the standard econometric form.

01:56 And it says to be sure to use only your single person households in the sample.

02:00 So again, write your syntax to make that happen.

02:05 Just use the sample of 2017 single person households.

02:11 So let's write out the different numbers here, or write out the coefficients, and then i'll have the standard areas underneath the coefficient.

02:23 So for income, we have the positive coefficient.

02:27 That makes sense.

02:29 So that's just saying as your annual income goes up, holding age constant, people are reporting that their net financial wealth is more.

02:42 So standard error for this is 0 .06.

02:45 So that's going to be very statistically significant.

02:51 And age, we would maybe also expect age to have a positive coefficient.

02:55 Right.

02:56 So as you.

02:57 As you get older, holding income constant, you likely will have a higher net financial wealth than people who are younger.

03:11 So let's put that there.

03:14 Also, with a small standard error here, it's going to be very statistically significant here.

03:24 So after we've estimated this, it says to interpret the slope coefficients.

03:30 So let's just go to income first.

03:36 So this point 799.

03:37 And this just represents that one more dollar in a respondent's income, and that's holding age fixed, results in about 80 more cents in predicted net financial wealth.

03:56 So i guess i could just write that out a little bit.

04:00 So if you have one more dollar of income, you have about plus 80 cents more of your wealth.

04:22 So there's that interpretation on the income coefficient.

04:29 For age, let's see what this one means.

04:33 So 0 .843 is a coefficient.

04:35 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 or her net financial income is predicted to increase by about $843.

05:00 And just be careful with the units there.

05:03 So the coefficient itself is 0 .843.

05:08 But since the net financial wealth variable is given in thousands of dollars, that's what we get for the impact.

05:21 All right.

05:23 And then i also asked you, are there any surprise? in these slope estimates.

05:28 So we've kind of already answered this, but first of all, both of these coefficients are positive, you know, holding other things constant.

05:39 If you have a higher annual income, you're predicted to have a higher net financial wealth, and also holding your income constant.

05:48 As you get older, you're also predicted to gain or have a higher net financial wealth.

05:57 So no surprises here.

06:01 3 moves along and just asks if the intercept from the regression that we estimated, which i haven't given.

06:09 So let's just write that out.

06:11 I'll just write out for beta not.

06:14 And hopefully you get this answer after you estimate in your own program.

06:19 But your beta not should be negative 43.

06:26 So zero there...

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