Solution Manual for Introductory
Econometrics A Modern Approach 7th
by Wooldridge
3
, CHAPTER 1
TEACHING NOTES
You have substantial latitude about what to emphasize in Chapter 1. I find it useful to talk about
the economics of crime example (Example 1.1) and the wage example (Example 1.2) so that
students see, at the outset, that econometrics is linked to economic reasoning, if not economic
theory.
I like to familiarize students with the important data structures that empirical economists use,
focusing primarily on cross-sectional and time series data sets, as these are what I cover in a
first-semester course. It is probably a good idea to mention the growing importance of data sets
that have both a cross-sectional and time dimension.
I spend almost an entire lecture talking about the problems inherent in drawing causal inferences
in the social sciences. I do this mostly through the agricultural yield, return to education, and
crime examples. These examples also contrast experimental and nonexperimental data. Students
studying business and finance tend to find the term structure of interest rates example more
relevant, although the issue there is testing the implication of a simple theory, as opposed to
inferring causality. I have found that spending time talking about these examples, in place of a
formal review of probability and statistics, is more successful (and more enjoyable for the
students and me).
CHAPTER 1
SOLUTIONS TO PROBLEMS1.1
(i) Ideally, we could randomly assign students to classes of different sizes. That is, eachstudent is assigned a
different class size without regard to any student characteristics such asability and family background. For
reasons we will see in Chapter 2, we would like substantialvariation 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
withsmaller 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 parentstend to be more involved in their
children’s education.(iii) Given the potential for confounding factors – some of which are listed in (ii) –
finding anegative 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 thesubject of
multiple regression analysis.
1.3
It does not make sense to pose the question in terms of causality. Economists would assumethat 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 theconstraint that there are only 168 hours in
a week. We can then use statistical methods tomeasure the association between studying and working,
including regression analysis that wecover 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 EXERCISESC1.1
(i) The average of
educ
is about 12.6 years. There are two people reporting zero years ofeducation, and 19 people reporting 18 years
4
,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
dent
, the CPI was 56.9 in1976 and 184.0 in 2003.(iv) To convert 1976 dollars into 2003 dollars, we use the ratio of
the CPIs, which is. Therefore, the average hourly wage in 2003 dollars is roughly, which is a reasonable
figure.184/56.93.23
≈
3.23($5.90)
≈
$19.06 1
2 (v) The sample contains 252 women (the number of observations with
female
= 1) and 274men.
C1.3
(i) The largest is 100, the smallest is 0.(ii) 38 out of 1,823, or about 2.1 percent of the sample.(iii) 17(iv) The
average of
math4
is about 71.9 and the average of
read4
is about 60.1. So, at leastin 2001, the reading test was harder to pass.(v) The sample correlation between
math4
and
read4
is about .843, which is a very highdegree of (linear) association. Not surprisingly, schools that have high pass
rates on one testhave a strong tendency to have high pass rates on the other test.(vi) The average of
exppp
is about $5,194.87. The standard deviation is $1,091.89, whichshows rather wide variation in spending per
pupil. [The minimum is $1,206.88 and themaximum is $11,957.64.
5
, CHAPTER 2
TEACHING NOTES
This is the chapter where I expect students to follow most, if not all, of the algebraic derivations.
In class I like to derive at least the unbiasedness of the OLS slope coefficient, and usually I
derive the variance. At a minimum, I talk about the factors affecting the variance. To simplify
the notation, after I emphasize the assumptions in the population model, and assume random
sampling, I just condition on the values of the explanatory variables in the sample. Technically,
this is justified by random sampling because, for example, E(ui|x1,x2,…,xn) = E(ui|xi) by
independent sampling. I find that students are able to focus on the key assumption SLR.3 and
subsequently take my word about how conditioning on the independent variables in the sample is
harmless. (If you prefer, the appendix to Chapter 3 does the conditioning argument carefully.)
Because statistical inference is no more difficult in multiple regression than in simple regression,
I postpone inference until Chapter 4. (This reduces redundancy and allows you to focus on the
interpretive differences between simple and multiple regression.)
You might notice how, compared with most other texts, I use relatively few assumptions to
derive the unbiasedness of the OLS slope estimator, followed by the formula for its variance.
This is because I do not introduce redundant or unnecessary assumptions. For example, once
SLR.3 is assumed, nothing further about the relationship between u and x is needed to obtain the
unbiasedness of OLS under random sampling.
6
Econometrics A Modern Approach 7th
by Wooldridge
3
, CHAPTER 1
TEACHING NOTES
You have substantial latitude about what to emphasize in Chapter 1. I find it useful to talk about
the economics of crime example (Example 1.1) and the wage example (Example 1.2) so that
students see, at the outset, that econometrics is linked to economic reasoning, if not economic
theory.
I like to familiarize students with the important data structures that empirical economists use,
focusing primarily on cross-sectional and time series data sets, as these are what I cover in a
first-semester course. It is probably a good idea to mention the growing importance of data sets
that have both a cross-sectional and time dimension.
I spend almost an entire lecture talking about the problems inherent in drawing causal inferences
in the social sciences. I do this mostly through the agricultural yield, return to education, and
crime examples. These examples also contrast experimental and nonexperimental data. Students
studying business and finance tend to find the term structure of interest rates example more
relevant, although the issue there is testing the implication of a simple theory, as opposed to
inferring causality. I have found that spending time talking about these examples, in place of a
formal review of probability and statistics, is more successful (and more enjoyable for the
students and me).
CHAPTER 1
SOLUTIONS TO PROBLEMS1.1
(i) Ideally, we could randomly assign students to classes of different sizes. That is, eachstudent is assigned a
different class size without regard to any student characteristics such asability and family background. For
reasons we will see in Chapter 2, we would like substantialvariation 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
withsmaller 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 parentstend to be more involved in their
children’s education.(iii) Given the potential for confounding factors – some of which are listed in (ii) –
finding anegative 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 thesubject of
multiple regression analysis.
1.3
It does not make sense to pose the question in terms of causality. Economists would assumethat 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 theconstraint that there are only 168 hours in
a week. We can then use statistical methods tomeasure the association between studying and working,
including regression analysis that wecover 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 EXERCISESC1.1
(i) The average of
educ
is about 12.6 years. There are two people reporting zero years ofeducation, and 19 people reporting 18 years
4
,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
dent
, the CPI was 56.9 in1976 and 184.0 in 2003.(iv) To convert 1976 dollars into 2003 dollars, we use the ratio of
the CPIs, which is. Therefore, the average hourly wage in 2003 dollars is roughly, which is a reasonable
figure.184/56.93.23
≈
3.23($5.90)
≈
$19.06 1
2 (v) The sample contains 252 women (the number of observations with
female
= 1) and 274men.
C1.3
(i) The largest is 100, the smallest is 0.(ii) 38 out of 1,823, or about 2.1 percent of the sample.(iii) 17(iv) The
average of
math4
is about 71.9 and the average of
read4
is about 60.1. So, at leastin 2001, the reading test was harder to pass.(v) The sample correlation between
math4
and
read4
is about .843, which is a very highdegree of (linear) association. Not surprisingly, schools that have high pass
rates on one testhave a strong tendency to have high pass rates on the other test.(vi) The average of
exppp
is about $5,194.87. The standard deviation is $1,091.89, whichshows rather wide variation in spending per
pupil. [The minimum is $1,206.88 and themaximum is $11,957.64.
5
, CHAPTER 2
TEACHING NOTES
This is the chapter where I expect students to follow most, if not all, of the algebraic derivations.
In class I like to derive at least the unbiasedness of the OLS slope coefficient, and usually I
derive the variance. At a minimum, I talk about the factors affecting the variance. To simplify
the notation, after I emphasize the assumptions in the population model, and assume random
sampling, I just condition on the values of the explanatory variables in the sample. Technically,
this is justified by random sampling because, for example, E(ui|x1,x2,…,xn) = E(ui|xi) by
independent sampling. I find that students are able to focus on the key assumption SLR.3 and
subsequently take my word about how conditioning on the independent variables in the sample is
harmless. (If you prefer, the appendix to Chapter 3 does the conditioning argument carefully.)
Because statistical inference is no more difficult in multiple regression than in simple regression,
I postpone inference until Chapter 4. (This reduces redundancy and allows you to focus on the
interpretive differences between simple and multiple regression.)
You might notice how, compared with most other texts, I use relatively few assumptions to
derive the unbiasedness of the OLS slope estimator, followed by the formula for its variance.
This is because I do not introduce redundant or unnecessary assumptions. For example, once
SLR.3 is assumed, nothing further about the relationship between u and x is needed to obtain the
unbiasedness of OLS under random sampling.
6
