STUDENT SOLUTIONS MANUAL
Jeffrey M. Wooldridge
Introductory Econometrics: A Modern Approach, 4e
CONTENTS
Preface iv
Chapter 1 Introduction 1
Chapter 2 The Simple Regression Model 3
Chapter 3 Multiple Regression Analysis: Estimation 9
Chapter 4 Multiple Regression Analysis: Inference 17
Chapter 5 Multiple Regression Analysis: OLS Asymptotics 24
Chapter 6 Multiple Regression Analysis: Further Issues 27
Chapter 7 Multiple Regression Analysis With Qualitative 34
Information: Binary (or Dummy) Variables
Chapter 8 Heteroskedasticity 42
Chapter 9 More on Specification and Data Problems 47
Chapter 10 Basic Regression Analysis With Time Series Data 52
Chapter 11 Further Issues in Using OLS With Time Series Data 58
Chapter 12 Serial Correlation and Heteroskedasticity in 65
Time Series Regressions
Chapter 13 Pooling Cross Sections Across Time. Simple 71
Panel Data Methods
Chapter 14 Advanced Panel Data Methods 78
Chapter 15 Instrumental Variables Estimation and Two Stage 85
Least Squares
Chapter 16 Simultaneous Equations Models 92
,Chapter 17 Limited Dependent Variable Models and Sample 99
Selection Corrections
Chapter 18 Advanced Time Series Topics 110
Appendix A Basic Mathematical Tools 117
Appendix B Fundamentals of Probability 119
Appendix C Fundamentals of Mathematical Statistics 120
Appendix D Summary of Matrix Algebra 122
Appendix E The Linear Regression Model in Matrix Form 123
, PREFACE
This manual contains solutions to the odd-numbered problems and computer exercises in
Introductory Econometrics: A Modern Approach, 4e. Hopefully, you will find that the
solutions are detailed enough to act as a study supplement to the text. Rather than just
presenting the final answer, I usually provide detailed steps, emphasizing where the
chapter material is used in solving the problems.
Some of the answers given here are subjective, and you or your instructor may have
perfectly acceptable alternative answers or opinions.
I obtained the solutions to the computer exercises using Stata, starting with version 4.0
and ending with version 9.0. Nevertheless, almost all of the estimation methods covered
in the text have been standardized, and different econometrics or statistical packages
should give the same answers to the reported degree of accuracy. There can be
differences when applying more advanced techniques, as conventions sometimes differ
on how to choose or estimate auxiliary parameters. (Examples include
heteroskedasticity-robust standard errors, estimates of a random effects model, and
corrections for sample selection bias.) Any differences in estimates or test statistics
should be practically unimportant, provided you are using a reasonably large sample size.
While I have endeavored to make the solutions free of mistakes, some errors may have
crept in. I would appreciate hearing from students who find mistakes. I will keep a list
of any notable errors on the Web site for the book,
academic.cengage.com/economics/wooldridge. I would also like to hear from students
who have suggestions for improving either the solutions or the problems themselves. I
can be reached via e-mail at .
I hope that you find this solutions manual helpful when used in conjunction with the text.
I look forward to hearing from you.
, CHAPTER 1
SOLUTIONS TO PROBLEMS
1.1 (i) Ideally, we could randomly assign students to classes of different sizes. That is, each
student is assigned a different class size without regard to any student characteristics such as
ability and family background. For reasons we will see in Chapter 2, we would like substantial
variation in class sizes (subject, of course, to ethical considerations and resource constraints).
(ii) A negative correlation means that larger class size is associated with lower performance.
We might find a negative correlation because larger class size actually hurts performance.
However, with observational data, there are other reasons we might find a negative relationship.
For example, children from more affluent families might be more likely to attend schools with
smaller class sizes, and affluent children generally score better on standardized tests. Another
possibility is that, within a school, a principal might assign the better students to smaller classes.
Or, some parents might insist their children are in the smaller classes, and these same parents
tend to be more involved in their children’s education.
(iii) Given the potential for confounding factors – some of which are listed in (ii) – finding a
negative correlation would not be strong evidence that smaller class sizes actually lead to better
performance. Some way of controlling for the confounding factors is needed, and this is the
subject of multiple regression analysis.
1.3 It does not make sense to pose the question in terms of causality. Economists would assume
that students choose a mix of studying and working (and other activities, such as attending class,
leisure, and sleeping) based on rational behavior, such as maximizing utility subject to the
constraint that there are only 168 hours in a week. We can then use statistical methods to
measure the association between studying and working, including regression analysis that we
cover starting in Chapter 2. But we would not be claiming that one variable ―causes‖ the other.
They are both choice variables of the student.
SOLUTIONS TO COMPUTER EXERCISES
C1.1 (i) The average of educ is about 12.6 years. There are two people reporting zero years of
education, and 19 people reporting 18 years of education.
(ii) The average of wage is about $5.90, which seems low in the year 2008.
(iii) Using Table B-60 in the 2004 Economic Report of the President, the CPI was 56.9 in
1976 and 184.0 in 2003.
(iv) To convert 1976 dollars into 2003 dollars, we use the ratio of the CPIs, which is
.9 3.23 . Therefore, the average hourly wage in 2003 dollars is roughly
3.23($5.90) $19.06 , which is a reasonable figure.
Jeffrey M. Wooldridge
Introductory Econometrics: A Modern Approach, 4e
CONTENTS
Preface iv
Chapter 1 Introduction 1
Chapter 2 The Simple Regression Model 3
Chapter 3 Multiple Regression Analysis: Estimation 9
Chapter 4 Multiple Regression Analysis: Inference 17
Chapter 5 Multiple Regression Analysis: OLS Asymptotics 24
Chapter 6 Multiple Regression Analysis: Further Issues 27
Chapter 7 Multiple Regression Analysis With Qualitative 34
Information: Binary (or Dummy) Variables
Chapter 8 Heteroskedasticity 42
Chapter 9 More on Specification and Data Problems 47
Chapter 10 Basic Regression Analysis With Time Series Data 52
Chapter 11 Further Issues in Using OLS With Time Series Data 58
Chapter 12 Serial Correlation and Heteroskedasticity in 65
Time Series Regressions
Chapter 13 Pooling Cross Sections Across Time. Simple 71
Panel Data Methods
Chapter 14 Advanced Panel Data Methods 78
Chapter 15 Instrumental Variables Estimation and Two Stage 85
Least Squares
Chapter 16 Simultaneous Equations Models 92
,Chapter 17 Limited Dependent Variable Models and Sample 99
Selection Corrections
Chapter 18 Advanced Time Series Topics 110
Appendix A Basic Mathematical Tools 117
Appendix B Fundamentals of Probability 119
Appendix C Fundamentals of Mathematical Statistics 120
Appendix D Summary of Matrix Algebra 122
Appendix E The Linear Regression Model in Matrix Form 123
, PREFACE
This manual contains solutions to the odd-numbered problems and computer exercises in
Introductory Econometrics: A Modern Approach, 4e. Hopefully, you will find that the
solutions are detailed enough to act as a study supplement to the text. Rather than just
presenting the final answer, I usually provide detailed steps, emphasizing where the
chapter material is used in solving the problems.
Some of the answers given here are subjective, and you or your instructor may have
perfectly acceptable alternative answers or opinions.
I obtained the solutions to the computer exercises using Stata, starting with version 4.0
and ending with version 9.0. Nevertheless, almost all of the estimation methods covered
in the text have been standardized, and different econometrics or statistical packages
should give the same answers to the reported degree of accuracy. There can be
differences when applying more advanced techniques, as conventions sometimes differ
on how to choose or estimate auxiliary parameters. (Examples include
heteroskedasticity-robust standard errors, estimates of a random effects model, and
corrections for sample selection bias.) Any differences in estimates or test statistics
should be practically unimportant, provided you are using a reasonably large sample size.
While I have endeavored to make the solutions free of mistakes, some errors may have
crept in. I would appreciate hearing from students who find mistakes. I will keep a list
of any notable errors on the Web site for the book,
academic.cengage.com/economics/wooldridge. I would also like to hear from students
who have suggestions for improving either the solutions or the problems themselves. I
can be reached via e-mail at .
I hope that you find this solutions manual helpful when used in conjunction with the text.
I look forward to hearing from you.
, CHAPTER 1
SOLUTIONS TO PROBLEMS
1.1 (i) Ideally, we could randomly assign students to classes of different sizes. That is, each
student is assigned a different class size without regard to any student characteristics such as
ability and family background. For reasons we will see in Chapter 2, we would like substantial
variation in class sizes (subject, of course, to ethical considerations and resource constraints).
(ii) A negative correlation means that larger class size is associated with lower performance.
We might find a negative correlation because larger class size actually hurts performance.
However, with observational data, there are other reasons we might find a negative relationship.
For example, children from more affluent families might be more likely to attend schools with
smaller class sizes, and affluent children generally score better on standardized tests. Another
possibility is that, within a school, a principal might assign the better students to smaller classes.
Or, some parents might insist their children are in the smaller classes, and these same parents
tend to be more involved in their children’s education.
(iii) Given the potential for confounding factors – some of which are listed in (ii) – finding a
negative correlation would not be strong evidence that smaller class sizes actually lead to better
performance. Some way of controlling for the confounding factors is needed, and this is the
subject of multiple regression analysis.
1.3 It does not make sense to pose the question in terms of causality. Economists would assume
that students choose a mix of studying and working (and other activities, such as attending class,
leisure, and sleeping) based on rational behavior, such as maximizing utility subject to the
constraint that there are only 168 hours in a week. We can then use statistical methods to
measure the association between studying and working, including regression analysis that we
cover starting in Chapter 2. But we would not be claiming that one variable ―causes‖ the other.
They are both choice variables of the student.
SOLUTIONS TO COMPUTER EXERCISES
C1.1 (i) The average of educ is about 12.6 years. There are two people reporting zero years of
education, and 19 people reporting 18 years of education.
(ii) The average of wage is about $5.90, which seems low in the year 2008.
(iii) Using Table B-60 in the 2004 Economic Report of the President, the CPI was 56.9 in
1976 and 184.0 in 2003.
(iv) To convert 1976 dollars into 2003 dollars, we use the ratio of the CPIs, which is
.9 3.23 . Therefore, the average hourly wage in 2003 dollars is roughly
3.23($5.90) $19.06 , which is a reasonable figure.
