Department of Economics S3412
Columbia University Summer 2022
SOLUTIONS to Problem Set 2
Introduction to Econometrics
1. [graded] For many years, housing economists believed that households spend a constant
fraction of income on housing, as in
housing expenditure = ï¢ (income) + u
The file housing.dta contains housing expenditures (housing) and total expenditures
(total) for a sample of 19th century Belgian workers collected by Edouard Ducpetiaux1
The differences in housing expenditures from one observation to the next are in the
variables dhousing; the differences in total expenditures are in variable dtotal.
(a) Compute the means of total expenditure and housing expenditure in this sample
(b) Estimate ï¢ using total expenditure for total income.
(c) If income rises by 100 (it averages around 900 in this sample) what change in
estimated expected housing expenditure results according to your estimate in (b)?
(d) Interpret the R
(e) What economic argument would you make against housing absorbing a constant
share of income?
(f) What are some determinants of housing captured by u?
1 Edouard Ducpetiaux, Budgets Economiques de Classes de Ouvrieres en Belgique (Brussels, Hayaz 1855)
total 162 902.8239 411.6408 377.06 2822.54
housing 162 72.54259 57.26064 7.25 450.52
Variable Obs Mean Std. Dev. Min Max
. sum housing total
c) Housing expenditure is expected to increase by 7.49
d) Since this regression does not contain a constant, we cannot necessarily interpret
in the usual way (i.e. 64.6% of the variations in housing expenditures can be
explained by the variations in income). To see this, run the regression including a
constant; the R2
is now 12.3%!
e) The relationship more likely to be non-linear
f) Price, mortgage interest rates, location, etc. (answers will vary here)
2. [graded] Use Table 2 to answer the following questions. Table 2 presents the results of four
regressions, one in each column. Estimate the indicated regressions and fill in the values
(you may either handwrite or type the entries in; if you choose to type up the table, an
electronic copy of Table 2 in .doc format is available on the course Web site). For example,
to fill in column (1), estimate the regression with colGPA as the dependent variable and
hsGPA and skipped as the independent variables, using the â€œrobustâ€ option, and fill in the
(a) Fill out the table with necessary numbers, some will be on STATA output some you
will need to calculate yourself.
(b) Common sense predicts that your high school GPA (hsGPA) and the number of
classes you skipped (skipped) are determinants of your college GPA (colGPA). Use
regression (2) to test the hypothesis (at the 5% significance level) that the coefficients
on these two economic variables are all zero, against the alternative that at least one
coefficient is nonzero.
at least one coef. is nonzero
The p-value for the F-statistic =.00<.05, thus we reject at the 5% significance level. We tend
to conclude at least one coefficient is nonzero.
(c) Find the F-statistic for regression (3) and explain what is it testing?
The F-statistic for regression (3) is 12.07; it is jointly testing whether all of the coefficients are
total .0749545 .0043495 17.23 0.000 .0663651 .0835439
housing Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 1380396.94 162 8520.96874 Root MSE = 54.901
Adj R-squared = 0.6463
Residual 485275.167 161 3014.13147 R-squared = 0.6485
Model 895121.769 1 895121.769 Prob > F = 0.0000
F( 1, 161) = 296.98
Source SS df MS Number of obs = 162
. reg housing total, noconstant
> 12\Problem Set 2 Fall 2012\housing.dta”, clear
equal to 0, that is – if all the regressors jointly have no explanatory power.
(d) Find the F-statistic for regression (4) and explain what is it testing?
The F-statistic for regression (4) is 11.14 (this is not from the table); it is jointly testing whether
all of the coefficients are equal to 0, that is – if all the regressors jointly have no explanatory
(e) Are bgfriend (whether you have a boy/girlfriend) and campus (whether you live on
campus) jointly significant determinants of college GPA? Use regression (2) and (4)
to test your hypothesis. (i.e. use homoskedasticity-only F stat formula, eq.7.14 in the
book, instead of directly testing with STATA)
at least one coef. is nonzero
where and are obtained from regression (4) and (2) respectively; the number of
restrictionsis q=2; is the number of regressors in the unrestricted regression; q and
are degrees of freedom for the F distriction(here we performnon-robust regressions).
F(2, 135, Î±=0.05) =3.06>2.62, thus we cannot reject at the 5% significance level. We tend to
conclude that bgfriend and campus jointly have no explanatory power. Alternatively, we can use
the Stata command: di fprob(2, 135, 2.62) to find the associated p-value = .077>.05, again we
cannot reject at the 5% significance level
Definitions of Variables in GPA4.dta (data is from Wooldridge textbook)
colGPA Cumulative College Grade Point Average of a sample of 141 students
at Michigan State University in 1994.
hsGPA High School GPA of students.
skipped Average number of classes skipped per week.
PC = 1 if the students owns a personal computer
= 0 otherwise.
bgfriend = 1 if the student answered â€œyesâ€ to having a boy/girl friend
= 0 otherwise.
campus = 1 if the student lives on campus.
= 0 otherwise.
College GPA Results Dependent variable: colGPA
Regressor (1) (2) (3) (4)
Skipped -.077 -.065 -.065 -.071
(.025) (.025) (.025) (.026)
PC __ .128
bgfriend __ __ .084
campus __ __ __ -.124
F-statistics testing the hypothesis that the population coefficients on the indicated
regressors are all zero:
hsGPA, skipped 20.90
hsGPA, skipped, PC __ 15.47
hsGPA, skipped, PC, bgfriend, __ __ 12.07
bgfriend, campus __ __ __ 2.55
Regression summary statistics
.211 .234 .241 .252
.223 .250 .263 .278
Regression RMSE .331 .326 .324 .322
n 141 141 141 141
Notes: Heteroskedasticity-robust standard errors are given in parentheses under estimated
coefficients, and p-values are given in parentheses under F- statistics. The F-statistics are
Following questions will not be graded, they are for you to practice and will be discussed at
the recitation by your teaching assistant:
1. SW Empirical Exercise 6.1
6.1. Regressions used in (a) and (b)
Beauty 0.133 0.166
Intercept 4.00 4.07
SER 0.545 0.513
2 0.036 0.155
(a) The estimated slope is 0.133
(b) The estimated slope is 0.166. The coefficient does not change by a large amount. Thus, there
does not appear to be large omitted variable bias.
(c) The first step and second step are summarized in the table
Intro 0.12 0.03
OneCredit â€“0.37 0.57
Female 0.19 â€“0.14
Minority 0.08 â€“0.15
NNEnglish 0.02 â€“0.24
Intercept â€“0.11 4.05
Regressing the residual from step 2 onto the residual from step 1 yield a coefficient on
Beauty that is equal to 0.166 (as in (b)).
(d) Professor Smithâ€™s predicted course evaluation = (0.166 ï‚´ 0) + (0.011 ï‚´ 0) + (0.634 ï‚´ 0) âˆ’
(0.173 ï‚´ 0) âˆ’ (0.167 ï‚´ 1) âˆ’ (0.244 ï‚´ 0) + 4.068 = 3.901
2. SW Empirical Exercises 7.1
SER 9.99 9.07
2 0.029 0.200
(a) The estimated slope is 0.60. The estimated intercept is 1.08.
(b) The estimated marginal effect of Age on AHE is 0.59 dollars per year. The 95%
confidence interval is 0.59 ï‚± 1.96 ï‚´ 0.04 or 0.51 to 0.66.
(c) The results are quite similar. Evidently the regression in (a) does not suffer from
important omitted variable bias.
(d) Bobâ€™s predicted average hourly earnings = (0.59 ï‚´ 26) + (âˆ’ 3.66 ï‚´ 0) + (8.08 ï‚´ 0)
âˆ’ ï€°ï€®ï€¶ï€³ = $14.17. Alexisâ€™s predicted average hourly earnings = (0.59 ï‚´ 30) + (âˆ’ 3.66 ï‚´ 1)
+ (8.08 ï‚´ 1) âˆ’ ï€°ï€®ï€¶ï€³ = $21.49.
(e) The regression in (b) fits the data much better. Gender and education are important
predictors of earnings. The R
are similar because the sample size is large (n =
(f) Gender and education are important. The F-statistic is 781, which is (much) larger than
the 1% critical value of 4.61.
(g) The omitted variables must have non-zero coefficients and must correlated with the
included regressor. From (f) Female and Bachelor have non-zero coefficients; yet there
does not seem to be important omitted variable bias, suggesting that the correlation of
Age and Female and Age and Bachelor is small. (The sample correlations are
Female) = âˆ’0.03 and
(Age,Bachelor) = 0.00).
3. SW Exercises 7.1
SER 9.99 9.07
2 0.029 0.200
(a) The estimated slope is 0.60. The estimated intercept is 1.08.
(b) The estimated marginal effect of Age on AHE is 0.59 dollars per year. The 95% confidence
interval is 0.59 ï‚± 1.96 ï‚´ 0.04 or 0.51 to 0.66.
(c) The results are quite similar. Evidently the regression in (a) does not suffer from important
omitted variable bias.
(d) Bobâ€™s predicted average hourly earnings = (0.59 ï‚´ 26) + (âˆ’ 3.66 ï‚´ 0) + (8.08 ï‚´ 0) âˆ’ ï€°ï€®ï€¶ï€³ =
$14.17. Alexisâ€™s predicted average hourly earnings = (0.59 ï‚´ 30) + (âˆ’ 3.66 ï‚´ 1) + (8.08 ï‚´ 1) â€“
ï€°ï€®ï€¶ï€³ = $21.49.
(e) The regression in (b) fits the data much better. Gender and education are important predictors
of earnings. The R
and are similar because the sample size is large (n = 7711).
(f) Gender and education are important. The F-statistic is 781, which is (much) larger than the
1% critical value of 4.61.
(g) The omitted variables must have non-zero coefficients and must correlated with the included
regressor. From (f) Female and Bachelor have non-zero coefficients; yet there does not seem
to be important omitted variable bias, suggesting that the correlation of Age and Female and
Age and Bachelor is small. (The sample correlations are (Age, Female) = âˆ’0.03 and
(Age,Bachelor) = 0.00).
2 R Cor Cor
Get Professional Assignment Help Cheaply
Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?
Whichever your reason is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.
Why Choose Our Academic Writing Service?
- Plagiarism free papers
- Timely delivery
- Any deadline
- Skilled, Experienced Native English Writers
- Subject-relevant academic writer
- Adherence to paper instructions
- Ability to tackle bulk assignments
- Reasonable prices
- 24/7 Customer Support
- Get superb grades consistently
Online Academic Help With Different Subjects
Students barely have time to read. We got you! Have your literature essay or book review written without having the hassle of reading the book. You can get your literature paper custom-written for you by our literature specialists.
Do you struggle with finance? No need to torture yourself if finance is not your cup of tea. You can order your finance paper from our academic writing service and get 100% original work from competent finance experts.
While psychology may be an interesting subject, you may lack sufficient time to handle your assignments. Don’t despair; by using our academic writing service, you can be assured of perfect grades. Moreover, your grades will be consistent.
Engineering is quite a demanding subject. Students face a lot of pressure and barely have enough time to do what they love to do. Our academic writing service got you covered! Our engineering specialists follow the paper instructions and ensure timely delivery of the paper.
In the nursing course, you may have difficulties with literature reviews, annotated bibliographies, critical essays, and other assignments. Our nursing assignment writers will offer you professional nursing paper help at low prices.
Truth be told, sociology papers can be quite exhausting. Our academic writing service relieves you of fatigue, pressure, and stress. You can relax and have peace of mind as our academic writers handle your sociology assignment.
We take pride in having some of the best business writers in the industry. Our business writers have a lot of experience in the field. They are reliable, and you can be assured of a high-grade paper. They are able to handle business papers of any subject, length, deadline, and difficulty!
We boast of having some of the most experienced statistics experts in the industry. Our statistics experts have diverse skills, expertise, and knowledge to handle any kind of assignment. They have access to all kinds of software to get your assignment done.
Writing a law essay may prove to be an insurmountable obstacle, especially when you need to know the peculiarities of the legislative framework. Take advantage of our top-notch law specialists and get superb grades and 100% satisfaction.
What discipline/subjects do you deal in?
We have highlighted some of the most popular subjects we handle above. Those are just a tip of the iceberg. We deal in all academic disciplines since our writers are as diverse. They have been drawn from across all disciplines, and orders are assigned to those writers believed to be the best in the field. In a nutshell, there is no task we cannot handle; all you need to do is place your order with us. As long as your instructions are clear, just trust we shall deliver irrespective of the discipline.
Are your writers competent enough to handle my paper?
Our essay writers are graduates with bachelor's, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college degree. All our academic writers have a minimum of two years of academic writing. We have a stringent recruitment process to ensure that we get only the most competent essay writers in the industry. We also ensure that the writers are handsomely compensated for their value. The majority of our writers are native English speakers. As such, the fluency of language and grammar is impeccable.
What if I don’t like the paper?
There is a very low likelihood that you won’t like the paper.
- When assigning your order, we match the paper’s discipline with the writer’s field/specialization. Since all our writers are graduates, we match the paper’s subject with the field the writer studied. For instance, if it’s a nursing paper, only a nursing graduate and writer will handle it. Furthermore, all our writers have academic writing experience and top-notch research skills.
- We have a quality assurance that reviews the paper before it gets to you. As such, we ensure that you get a paper that meets the required standard and will most definitely make the grade.
In the event that you don’t like your paper:
- The writer will revise the paper up to your pleasing. You have unlimited revisions. You simply need to highlight what specifically you don’t like about the paper, and the writer will make the amendments. The paper will be revised until you are satisfied. Revisions are free of charge
- We will have a different writer write the paper from scratch.
- Last resort, if the above does not work, we will refund your money.
Will the professor find out I didn’t write the paper myself?
Not at all. All papers are written from scratch. There is no way your tutor or instructor will realize that you did not write the paper yourself. In fact, we recommend using our assignment help services for consistent results.
What if the paper is plagiarized?
We check all papers for plagiarism before we submit them. We use powerful plagiarism checking software such as SafeAssign, LopesWrite, and Turnitin. We also upload the plagiarism report so that you can review it. We understand that plagiarism is academic suicide. We would not take the risk of submitting plagiarized work and jeopardize your academic journey. Furthermore, we do not sell or use prewritten papers, and each paper is written from scratch.
When will I get my paper?
You determine when you get the paper by setting the deadline when placing the order. All papers are delivered within the deadline. We are well aware that we operate in a time-sensitive industry. As such, we have laid out strategies to ensure that the client receives the paper on time and they never miss the deadline. We understand that papers that are submitted late have some points deducted. We do not want you to miss any points due to late submission. We work on beating deadlines by huge margins in order to ensure that you have ample time to review the paper before you submit it.
Will anyone find out that I used your services?
We have a privacy and confidentiality policy that guides our work. We NEVER share any customer information with third parties. Noone will ever know that you used our assignment help services. It’s only between you and us. We are bound by our policies to protect the customer’s identity and information. All your information, such as your names, phone number, email, order information, and so on, are protected. We have robust security systems that ensure that your data is protected. Hacking our systems is close to impossible, and it has never happened.
How our Assignment Help Service Works
1. Place an order
You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.
2. Pay for the order
Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.
3. Track the progress
You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.
4. Download the paper
The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.