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.
4
,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
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
den
t
, 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
math
4 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
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.
4
,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
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
den
t
, 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
math
4 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
