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Course manual Econometrics: Week 1
1 About the course manual
At the beginning of every week we will upload a course manual
for the correspondingweek. We decided to create a course manual for
this course to have one source whereall information concerning both
the content and the organization of the course is madeavailable for
the student. Important information that is contained in the lecture
slideswill be repeated in this manual. It is therefore obligatory
for all student taking this courseto read the manual for every week
of the semester. E-mails containing questions thatcould be answered
by reading the manual will not be replied. The most important
partof the manual is this weeks, which contains background
information about the course,some tips at how to study for the
course, as well as important dates.
Besides some organizational information the course manual will
contain detailed readinginstructions for every week, including both
obligatory and voluntary literature. Further-more, obligatory and
additional exercises for each week will be provided. Finally,
startingfrom week 2 the manual will contain the solutions to the
obligatory exercises.
2 People
The course is taught by Jun.-Prof. Dr. Hans Manner, who gives
both the lectures and theexercise meetings. He has studied
econometrics at the University of Maastricht from 2001-2005,
followed by a PhD at the same university from 2006-2010. Since
April 2010 he worksat the Department for Social- and Economic
Statistics, being attached to the chair of Prof.Dr. Karl Mosler. He
can be reached by e-mail ([email protected]), byphone
(0221-4704130) or by coming to his office located at
Meister-Ekkehart-Str. 9,second floor during his office hours to be
found on his webpage.
Walter Orth assists in designing and preparing the course, and
may occasionally take overa class in case of emergency. In
particular, he is responsible for preparing the exercisesand their
solutions. Nevertheless, he can also be contacted in case students
have questionsregarding the content of the course. His e-mail
address [email protected] he can be sought during
his office hours to be found on his webpage. CurrentlyWalter Orth
is a PhD Student of Prof. Mosler. Before that he studied
mathematics andeconomics in Duisburg-Essen.
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3 General information about the course
Econometrics is an extremely important topic for all economics
students and for manybusiness students as well. Sufficient
knowledge of econometrics often is a prerequisite, orat least
extremely useful, for taking many economics and finance courses.
Additionally,writing a Masters thesis often requires performing an
empirical analysis using economet-ric techniques. The aim of this
course is to prepare the students for such tasks and toteach the
most relevant econometric techniques needed to analyze economic
data.
While being very useful, most students find econometrics to be a
rather difficult subject.First of all, it requires knowledge and
understanding of many topics from mathematicsand statistics.
Although all students have learned these things, only few actually
remem-ber the relevant things when needed. Econometrics itself
requires understanding oftencomplex theory and knowing when certain
things need to be applied. There are manysubtle issues of when and
how different methods may or may not be applied. Therefore,
mastering econometrics requires hard work, time, exercise and
reflecting on what hasbeen learned. We tried to design the course
in a way that finds a good balance betweenlectures/exercise
sessions and self study. We hope that following our
recommendationson how to study for this course, students will be
able to acquire sufficient knowledgeof econometrics not only to
pass the exam without difficulty, but also to apply
basiceconometric techniques in practice.
Students should nevertheless be aware that, for a masters course
in econometrics, thematerial covered will not be very advanced.
Students who have good prior knowledge, forexample those who have
taken the Profilgruppe Quantitative Methoden der Wirtschafts-und
Sozialwissenschaften in Cologne, are recommended to skip this
course and move
right away into an advanced econometrics course like Time series
analysis or AdvancedEconometrics. Those who do take the course,
find it interesting, and want to learnmore about econometrics are
recommended to take one of the follow up courses offeredby our
department. There are many interesting and important topics in
econometricsthat cannot be covered in a one-semester course and it
definitely pays off to have thisknowledge.
4 Prior knowledge
As mentioned above, prior knowledge in mathematics and
statistics is absolutely crucial.All students who have done their
bachelor studies in Cologne should be fine. In general,having heard
two statistics courses, which include an introduction to the linear
regressionmodel is expected. We recommend that you review your old
textbook or notes wheneveryou find it necessary. The most important
concepts will again be covered in this course,but this will be very
brief and without much explanation. Having taken an
additionalcourse in econometrics at the bachelor level would be
very helpful, but not necessary.Students who have studied
econometrics before should decide for themselves if they wantto
take this course, implying they will hear many things they have
learned before, or ifright away they want to take a more advanced
course. This depends on how intensivetheir former course was and
how well they performed. If you have any doubt on this issue
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feel free to contact the professor.
5 The textbook and reading
The main textbook for the course is A guide to modern
econometrics by Marno Verbeek.The course is based on the third
edition of the book, but if you have a different editionthere
should be no problem. Each week we will give you detailed reading
and we willfollow the structure of this book quite closely. This
means that the lecture slides and thecorresponding book chapters
are close to each other in terms of structure, content andnotation.
This is intended, as we believe that this will facilitate combining
self study andlectures. Reading the book is obligatory and you are
required to know everything fromthe weekly reading. We highly
recommend you to buy this book, as it will not only beextremely
useful for studying this course, but will also serve as a reference
in the future.If you are not willing or able to buy this book there
are currently 16 copies available atthe Lehrbuchsammlung, as well
as further copies at other libraries of the faculty.
There are other textbooks that you may want to consider when
studying for this course.If you have studied econometrics in the
past the book you have used back then shouldbe useful, also given
that it is familiar to you. We recommend three additional books
forwhich we will give some broad reading indication for each week.
The books Introduc-tory Econometrics by Jeffrey Wooldridge and
Introduction to Econometrics by JamesStock and Mark Watson are a
little bit easier than the Verbeek. Econometric Analysisby William
Greene is much more advanced, but also contains more details and
manyadditional topics.
Those who prefer to consult a German textbook we suggest
Einfhrung in die konome-trie by Walter Assenmacher or konometrie.
Eine Einfhrung by Ludwig von Auer.
6 gretl
Econometrics requires using a computer program to analyze data.
Many such programsexist and each programm has its pros and cons. We
have decided to use gretl, becauseit allows to use a wide range of
econometric techniques while at the same time beingfreeware. Many
other programs such as EVIEWS, STATA or Matlab may be more
suitable for certain specialized task, but they are expensive
and students cannot usethem at home. Given that this course is too
large for going to the computer lab thiswould be very
problematic.
gretlcan be downloaded for free from
http://gretl.sourceforge.net/. There you alsofind a lot of
additional information about the program and a lot of add ons. In
particular,you can directly include the complete data used in the
Verbeek book, but also data fromother books, which makes doing the
empirical exercises much easier. An introduction togretlwill be
given during the second exercise meeting.
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7 Studying for the course
From the above it should be rather clear how the ideal student
should study for thiscourse. You should read the obligatory
literature either before or after the correspondinglecture. If,
after both lecture and reading, a topic is not understood you may
want to lookup the topic in one of the other books. If you find the
course difficult and have problemsunderstanding everything you may
decide to read one of the other books continuously.
The exercises should at least be read and thought about before
the second class of theweek, but ideally student should have tried
to solve them by themselves. Just looking atthe professor
presenting the solution may seem fine to you, but experience has
shown thatfor most students this does not suffice. After the
exercise meeting it is recommended toreview the solutions of those
exercises that appeared difficult. Note that solutions to
allobligatory exercises will be provided after the meetings. Again,
those who have problemswith the material should try the additional
exercise and try to answer the review questions
(for which we will in general not provide the answers).
8 The exam
There will be two exams offered this semester. The first will
take place on 31.01.2012 at14.00h in Aula 1. This is during the
last week of the semester and may be at the sametime as other exams
or classes. Therefore the second exam date will be on 20.03.2012
at10.00h in HS C. The exams will last 60 minutes and you are
allowed to bring a pocketcalculator and one A4 piece of paper with
any notes you desire. However, you will not be
given a formula sheet, so all formulas you think you will need
must be included in thosenotes. There are hardly any past exams
available to help you study, since this course isrelatively new and
is taught by Jun.-Prof. Manner for the first time. However,
duringthe lectures you can expect to get some hints at what may be
useful for the exam. Thelast lecture of the semester will be spent
reviewing the most relevant material and shouldagain serve as a
preparation for the exam. For now it can be said that the main
focuswill be on understanding the material of the course rather
than exclusively being able toperform mechanical calculations
(although that may also be a part). The exam questionswill
certainly require students to interpret computer output and relate
it to theoreticalconcepts.
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9 This weeks reading
Obligatory reading: Appendix A and B, as well as Chapter 1 of
Verbeek
Additional reading: The appendices of Wooldridge are very
detailed about the mathemat-
ical and statistical basics. They also contain many examples.
Reading it all is probablya bit too much, but you may want to read
the sections corresponding to topics that youfind difficult.
Chapter 1 of Wooldridge gives you a nice introduction on what
econometrics is about.
Appendices A and B of Greene are very detailed and contain much
more information thanwe need. You may want to consult them if you
are looking for more concise secondaryreading.
10 Exercises (obligatory)
1)Calculate the following sums and products (as far as they are
defined):
A=
3 12 0
6 7 32 1 4
B= 6 +
3 71 2
C=
1 01 2
1 2
D=
37
1 9
E=4 2
3 59 7
39
F=
6 3 34 1 2
1 8 47 5 2
2)Let A and B be n nmatrices with full rank. Calculate
a) (AB)(B1A1)
b) (A(A1 + B1)B)(B+ A)1
3)Consider the matrix
X=
x1 x2 xn
,
where x1, x2, . . . , xn have Kentries each. Show that
ni=1
xix
i= XX .
4)
a) Let X1, X2, Y1, Y2 be random variables. Show that
cov{X1+ X2, Y1+ Y2}= cov{X1, Y1}+ cov{X1, Y2}+ cov{X2, Y1}+
cov{X2, Y2}
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b) Now, X is a column vector of random variables with m entries.
Note that thecovariance matrix of X is defined as V{X} =
E{(XE{X})(XE{X})} =E{XX} E{X}E{X}. Further, let A be an mmatrix of
constants. Show that
V{AX}= AV{X}A
11 Additional exercises
1)Show that(AB)1 =B1A1
for any two n nmatrices Aand B that have full rank (i.e. rank(A)
= rank(B) = n).
2)Show thatcov{aX1, bX2}= ab cov{X1, X2}
where aand bare constant (non-random) scalars and X1 and X2 are
random variables.
3)Show that E{(X E{X})(X E{X})}= E{XX} E{X}E{X}.
If you still feel unfamiliar with matrix algebra you may want to
work through exercises2, 3 and 4 in chapter 2 of the textbook
Mosler/Dyckerhoff/Scheicher: MathematischeMethoden fuer
Oekonomen.
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Course manual Econometrics: Week 2
1 General information
This week is concerned with introducing the linear regression
model along with modelassumptions, its estimation by ordinary least
squares and the properties of the OLSestimator both in finite
samples and asymptotically.
2 Reading
Obligatory reading: Verbeek, Sections 2.1, 2.2, 2.3 and 2.6.
Additional reading: Wooldridge, Chapters 2,3 and 5. The material
that we cover isdistributed over these Chapters in Wooldridge and
many topics we cover in the comingweeks are treated in between.
Therefore you have to look for the relevant sections beforeor read
these chapters completely after we have treated all the
subjects.
Greene, Chapter 2, Chapter 3.1 and 3.2, Chapter 4.1-4.6 and
4.9.
3 Exercises
During this weeks exercise meeting we will provide an
introduction to gretl. You mayprint and read the document A short
introduction to gretl, the content of which will beexplained in the
meeting. If you cant make it to the meeting make sure to go
throughall the steps in the manual and explore some of the features
of gretl by clicking throughsome of the menus.
Additionally, if time permits we will give you a short
introduction to alternative programsthat can be used for applying
econometric techniques. Examples of such programs areEVIEWS, STATA,
Matlab and R.
4 Additional exercises
1)
Look at the True/False questions available at
http://www.econ.kuleuven.be/gme/. Try
to answer 2.1, 2.3, 2.5 and 2.7.
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2)
Consider the simple linear regression model
yi=1+ 2xi+ i , i= 1, . . . , N ,
Assume that it holds that i N(0, 2) for i = 1, . . . , N , where
1, . . . , n are indepen-dent.
You observe the following values for xi and yi:
xi 5 10 0 15 5yi 5 5 5 12.5 17.5
a) Calculate estimates for 1 and 2 using the Ordinary Least
Squares method.
b) Estimate 2
. (Note: The estimated square root of2
is usually called the standarderror of the regression.)
c) Write down the estimated regression along with the estimated
standard errors of1 and 2.
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5 Solution to last weeks exercises
1)
A =
3 12 0
6 7 32 1 4
=
16 20 512 14 6
D =
37
1 9
=
3 277 63
E =
4 2 3 5
9 7
39
=
30 6
39
= 144
F =
6 3 34 1 2
1 8 47 5 2
=
6 43 1
3 2
1 8 47 5 2
=
34 28 1610 19 10
11 34 16
The matrices B and Care not defined.
2)
a) (AB)(B1A1) =B A(A1)(B1) =B AA1
= IB1
= BB1 = I
=I
b) (A(A1 + B1)B)(B + A)1 = (( AA1 = I
+ AB1)B)(B + A)1 = (B + A B1B = I
)(B +
A)1 = (B+ A)(B+ A)1 =I
3)
X= x1 x2 xn
xi=
x1ix2i
xKi
, i= 1, . . . , n
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XX=
x1 x2 xn
x1 x2 xn
=
x1 x2 xn
x1
x2
xn
= x1x
1KKmatrix
+x2x
2+ . . .+ xnx
n
=n
i=1
xix
i
4)
a)
cov{X1+ X2, Y1+ Y2}=E{((X1+ X2)E{X1+X2})((Y1+Y2)E{Y1+ Y2})}
=E{(X1E{X1}+X2E{X2})(Y1E{Y1}+ Y2E{Y2})}
=E{(X1E{X1})(Y1 E{Y1}) + (X1E{X1})(Y2E{Y2})+
(X2E{X2})(Y1 E{Y1}) + (X2E{X2})(Y2E{Y2})}
=E{(X1E{X1})(Y1 E{Y1})}+E{(X1E{X1})(Y2E{Y2})}+
E{(X2E{X2})(Y1 E{Y1})}+E{(X2E{X2})(Y2E{Y2})}
=cov{X1, Y1}+cov{X1, Y2}+cov{X2, Y1}+cov{X2, Y2}
b)
V{AX}= E{(AX E{AX})(AX E{AX})}
=E{(AX AE{X})(AX AE{X})}
=E{(A(X E{X}))(A(X E{X}))}
=E{A(X E{X})(X E{X})A}
=AE{(X E{X})(X E{X})
}A
=AV{X}A
6 Solution to last weeks additional exercises
1)
We only have to show that(AB)(B1A1) = I .
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Indeed, it holds that
(AB)(B1A1) =A BB1 = I
A1 =AA1 =I .
2)
cov{aX1, bX2}=E{aX1bX2} E{aX1}E{bX2}
=abE{X1X2} abE{X1}E{X2}
=ab(E{X1X2} E{X1}E{X2})
=ab cov{X1, X2}
3)
E{(X E{X})(X E{X})}
=E{XX XE{X} E{X}X + E{X}E{X}}
=E{XX} E{XE{X}} E{E{X}X}+E{E{X}E{X}}
=E{XX} E{X}E{X} E{X}E{X}+E{X}E{X}
=E{XX} E{X}E{X}
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Course manual Econometrics: Week 3
1 General information
In this week we mainly treat the problem of hypothesis testing
in the linear regressionmodel. Problems related to
multicollinearity and how to detect it are treated as well.Finally,
we look at how to make predictions with the linear regression
model.
2 Reading
Obligatory reading: Verbeek, Sections 2.4, 2.5, 2.7, 2.8 and
2.9
Additional reading: Wooldridge, Chapter 4, Section 6.4. The R2
treated on pages 80-81and multicollinearity on page 95.
Greene, Section 3.5 and Chapter 5. Section 5.5 can be
skipped.
3 Exercises
1)
Suppose you estimate a parameter vector by some estimator b and
that your estimatorhas the following property:
N(b ) N(0, A) ,whereAis some matrix. Assume further that there
is an estimator Awhich consistentlyestimates A. Now, consider the
following empirical results:
N= 10000 , b= 2
3
, A=
400 100100 900
Calculate asymptotic standard errors and t-statistics for b1 and
b2.
2)
Consider the following multiple linear regression model:
yi=1+2xi2+3xi3+i i= 1, . . . , N .
a) Explain how one can test the hypothesis that3= 1 .
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b) Explain how one can test the hypothesis that 2 +3 = 0 . As
one alternative,consider to rewrite the model in a way that allows
applying the standard t-test.
c) Explain how one can test the hypothesis that2=3= 0 .
d) Explain how one can test the hypothesis that2= 0 and 3 =
1.
3)
Load the data set dataweek3ex3.gdt into GRETL.
a) Assume you believe that there exists a linear relationship
between y and x2, x3,x4, and x5. Estimate a linear regression and
interpret the output. What are themost striking findings? What is
the most likely explanation for your findings?
b) Use the appropriate tools from the lecture to look for
evidence of multicollinearity
in your data.
4)
Load the data set hprice1.gdt from the GRETL introduction and
consider again OLSestimation of the linear regression model
log(price)i=1+2log(sqrft)i+3log(lotsize)i+4bedroomsi+i.
a) Test the hypothesis that 4= 0.
b) Test the hypothesis that 2
= 1.
c) Test the hypothesis that 2+4= 0.
d) Test the joint hypothesis that 4= 0 and 2 = 1.
4 Additional exercises
1)
Show that in the linear model
y=X + , N(0, 2I)
the Wald test for the general linear hypothesis H0: R=q is
asymptotically equivalentwith the F test.
2)
Look again at the True/False questions available at
http://www.econ.kuleuven.be/gme/.Try to answer 2.2, 2.4, 2.6 and
2.8.
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5 Solution to last weeks additional exercises
2)
a) The matrix containing the regressors is
X=
1 51 101 01 151 5
.
Calculating the OLS estimator (XX)1Xy yieldsb1 = 5 and b2= 1.
(Note thatthe parameter vector bis also sometimes denoted by
.)
b) For the standard error of regression we have to calculate the
residuals
ei = yi yi=yi (b1+b2xi) , i= 1, . . . , N .
In our case we have
yi 0 5 5 10 10ei 5 10 10 2.5 7.5
Thus, the estimator for 2 is
s2 = 1
N 2Ni=1
e2i = 95.8333 ,
so that the standard error of regression is given by
s=
95.8333 = 9.7895 .
c) The standard errors of b1 and b2 are the square roots of the
values on the main
diagonal of
V{b} =s2(XX)1. Here we have se(b1) = 5.3619 and se(b2) =
0.6191,
so that the estimated regression can be written down as
y= 5(5.3619)
1(0.6191)
x .
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Course manual Econometrics: Week 4
1 General information
This week is concerned with various rather practical, but
extremely important aspectsconcerning the use of the linear
regression model. We discuss how to interpret the es-timated
parameters of the model in different situations. Furthermore, we
study how toselect the set of regressors and how to test the
functional form.
2 Reading
Obligatory reading: Verbeek, read the entire Chapter 3 with the
exception of Section3.2.3
Additional reading: Wooldridge, the relevant material cannot be
found in one single placein this book, but is dispersed over
Chapters 2, 3 and 6. As mentioned before, it makes
sense to read the first 6 Chapters of Woolridge to cover the
material of about the first 4weeks of this course.
Greene, The relevant parts of Chapters 6 and 7.
3 Exercises
1)
Consider the simple regression
log(yi) =1+2log(xi) +i, i= 1, . . . , N . (1)
a) Show that 2 can be interpreted as elasticityofyi with respect
to xi.
b) Calculate the elasticity ofyi with respect to xi for the
alternative model
yi = 1+2xi+i, i= 1, . . . , N . (2)
Explain the essential difference between the elasticity of model
(1) and model (2).
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c) Consider now a third model:
log(yi) =1+2xi+i, i= 1, . . . , N . (3)
Interpret the coefficient 2.
2)
a) Suppose you want to investigate the question if a beer tax
will reduce traffic fatali-ties. Further assume that you have data
on the number of traffic fatalities and thebeer tax rate for
different regions of a country. Would it be sensible to include
i)the amount of beer consumption or ii) the number of miles driven
as explanatoryvariables in your regression?
b) Suppose you want to investigate the question if pesticide use
of farmers has an
effect on the health expenditures of farmers. When regressing
health expenditureson pesticide usage amounts, does it make sense
to include the number of doctorvisits as a control variable?
3)(Adapted from Stock/Watson)
Consider the results from a study comparing total compensation
among top executivesin a large set of U.S. public corporations in
the 1990s. Let Femalebe a dummy variablethat is equal to 1 for
females and 0 for males.
a) A simple regression of the logarithm of earnings on
Femaleyields
log(Earnings) = 6.48(0.01)
0.44(0.05)
Female
Interpret the coefficient ofFemale.
b) Two new variables, the market value of the firm (a measure of
firm size, in millionsof dollars) and stock return (a measure of
firm performance, in percentage points),are added to the
regression:
log(Earnings) = 3.86(0.03)
0.28(0.04)
Female+ 0.37(0.004)
log(MarketV alue) + 0.004(0.003)
Return
Interpret the coefficient of log(MarketValue). Further, explain
why the coefficientofFemalehas changed from the regression in
a).
c) What would happen to your regression if the market value of
firms is measured inbillions?
4)(Adapted from Stock/Watson)
Load the dataset CPS04.gdt into GRETL. The data are from the
Current PopulationSurvey of the U.S. Department of Labor.
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a) Run a regression of the logarithm of average hourly earnings
(AHE) on age (Age),gender (Female) and education (Bachelor).
IfAgeincreases from 33 to 34, how areearnings expected to
change?
b) Run a regression of log(AHE) on log(Age), Femaleand Bachelor.
IfAge increases
from 33 to 34, how are earnings expected to change?
c) Run a regression of log(AHE)onAge,Age2,Femaleand Bachelor.
IfAge increasesfrom 33 to 34, how are earnings expected to
change?
d) Do you prefer the regression in b) to the regression in a)?
Explain.
e) Do you prefer the regression in c) to the regression in a)?
Explain.
f) Do you prefer the regression in c) to the regression in b)?
Explain.
g) Plot the regression relation between Ageand log(AHE) from c)
for females with a
Bachelor degree.
4 Additional exercises
1)(Adapted from Verbeek)
Explain why it is inappropriate to drop two explanatory
variables from the model at thesame time on the basis of their
t-statistics only.
2)(Adapted from Stock/Watson)
Suppose you want to analyze the relationship of class size and
student performance asmeasured by some test score. Talking with a
teacher you get the following comment:
In my experience, students do well when the class size is less
than 20 students and dopoorly when the class size is greater than
25. There are no gains from reducing class sizebelow 20 students,
the relationship is constant in the intermediate region between 20
and25 students, and there is no loss to increasing class size when
it is already greater than25.
If the teacher is right, how should you choose the functional
form of your model?
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5 Solutions to last weeks exercises
1)
Given that V{N(b )} = V{Nb} A and using that A is a consistent
estimatorofA, in large samples we have
V{
Nb} A V{b} 1
NA
V{b} = 110000
400 100100 900
V{b1} = 40010000
= 0.04 , V{2} = 90010000
= 0.09
The standard errors are equal to the square roots of the
estimated variances so that we
have se(b1) = 0.2 and se(b2) = 0.3. For the t statistics we have
t1 =
b1
se(b1) = 10 andt2=
b2se(b2)
= 10.
2)
a) The hypothesis that 3 = 1 can be tested by means of a
t-test.The test statistic is
t= b3 1se(b3)
.
which - under the null hypothesis - has an approximate standard
normal distributionin large samples and atdistribution with (N3)
degrees of freedom in small samplesunder the assumption of
normality of the error term. At the 95 % confidence level,we reject
the null in large samples if|t| >1.96 (two-tailed test).
b) The hypothesis that 2+3 = 0 can also be tested by means of a
t-test.The test statistic is
t= b2+b3se(b2+b3)
.
To calculate se(b2+b3) we use the estimated covariance
matrixV{b} and the factthat V{b2+b3} =V{b2} +V{b3} + 2 cov{b2,
b3}.Alternatively, you can rewrite the model as
yi=1+ (2+3)xi2+3(xi3 xi2) +iyi=1+
2xi2+3(xi3 xi2) +iand apply the usual t-test for 2 .
c) The joint hypothesis that2=3= 0 can be tested by means of
theoverall F-testwhich is a special case of the general F-test. The
test statistic is
F = R2/2
(1 R2)/(N 3).
We compare the test statistic with the critical values from an F
distribution with
2 (the number of restrictions) and N-3 degrees of freedom.
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d) The joint hypothesis that2= 0 and3= 1 can be tested by means
of the generalF-test. For the general joint linear null hypothesis
H0 : R = qwe have in ourcase
R=
0 1 00 0 1
and q= (0, 1). The test statistic is
F =(Rb q)(R(XX)1R)1(Rb q)
Js2 ,
where one has to insert R and q, the estimated parameter vector
b, the regressormatrix X, the degrees of freedom J(here J= 2) and
the estimated error variances2 = 1
NK
Ni=1e
2i . Note that the matrixR has nothing to do with the
goodness-
of-fit measure R2.
3)
a) Model 1: OLS, using observations 1100Dependent variable:
Y
Coefficient Std. Error t-ratio p-value
const 0.0541118 0.0922596 0.5865 0.5589X2 0.621068 0.680645
0.9125 0.3638X3 0.402881 0.595978 0.6760 0.5007X4 0.497742 0.295766
1.6829 0.0957
X5 0.601589 0.0897158 6.7055 0.0000
Mean dependent var 0.000323 S.D. dependent var 1.554198Sum
squared resid 79.71127 S.E. of regression 0.916005R2 0.666672
AdjustedR2 0.652637F(4, 95) 47.50110 P-value(F)
7.09e22Log-likelihood 130.5559 Akaike criterion 271.1118Schwarz
criterion 284.1376 HannanQuinn 276.3836
The first three variables are not significant, but the R2 is
quite large indicating agood overall fit. The large standard errors
may partly be caused by multicollinearity.
b) First of all you compute the correlation matrix between the
regressors:
Correlation coefficients, using the observations 11005% critical
value (two-tailed) = 0.1966 for n = 100
X2 X3 X4 X51.0000 0.8816 0.4565 0.0328 X2
1.0000 0.0123 0.0816 X31.0000 0.1114 X4
1.0000 X5
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The correlation betweenx2 andx3 is quite large (0.88),
indicating multicollinearity.Also the correlation between x2 and x4
is notable.
Next, run the auxiliary regression of the each of the regressors
on the remainingones and look at the resulting R2j . You get R
22 = 0.97, R
23 = 0.96, R
24 = 0.89,
and R2
5 = 0.024. This indicates strong multicollinearity between the
first threevariables, but excludes x5.
4)
a) This test is automatically performed by GRETL. The
t-statistic is 1.342 and theassociated p-value is 0.1831. Thus, we
cannot rejectH0 at conventional significancelevels.
b) The t-statistic for this test is
t= 0.700232 10.0928652
= 3.228
Using the p-value finder of GRETL (see the GRETL introduction)
gives a p-valueof 0.00177876 for the two-tailed t-test, so that H0
is clearly rejected. Note that wehave to choose 88 4 = 84 degrees
of freedom.
c) Similar to exercise 2b) you have two options. First, in the
model window, you maygo to Tests Linear restrictions and typeb[2]+
b[4] = 0. Second, rearrange themodel by defining a new variable
defined as the difference ofbedroomsilog(sqrft)i(this can done in
GRETL via
Add
Define new variable
varname = lsqrft -
bdrms) and reestimate the model with the regressors
log(sqrft)i,log(lotsize)i andvarname. The standard t-test for the
coefficient oflog(sqrft)iin the revised modelcan then be applied.
The result is a t-statistic of 8.918 which corresponds to ap-value
of 8.68e-014 leading to a very clear rejection of the null
hypothesis.
d) Since we have a joint hypothesis we have to use the F-test
now. Go again to Tests Linear restrictions, and type b[2] = 1 and
b[4] = 0 (see Help forexplanations). The test statistic is F=
5.25722 with a p-value of 0.00706007. Thus- not surprisingly after
the result from part b) - we reject our joint hypothesis.
6 Solution to last weeks additional exercises
1)
The statistic for the F-test is given by (see lecture 3)
F =(Rb q)(R(XX)1R)1(Rb q)
Js2
=(Rb q)(Rs2(XX)1R)1(Rb q)
J
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The corresponding Wald statistic is (see again lecture 3)
W = (Rb q)(RV{b}R)1(Rb q)
Since the standard estimate for the covariance matrix V
{b
}under the assumption
N(0, 2I) is s2(X
X)
1, we see that F and Wonly differ by the factor 1/J.
Moreover,under H0 F FJNK and W
asympt.2J. Using the definition of the F distribution (see
lecture 1) we can write F as
F = J/J
NK/N K,
thus as a ratio of2 distributed random variables. For large N,
we can use the result thatdue to the definition of the 2
distribution (see lecture 1) and the Law of Large Numbers
NKN
K
=
NKi=1 U
2i
N
KN E{U2i } =V{Ui} = 1
so thatJF =
asympt.J
We reject the null hypothesis ifF > FJNK;1 or, equivalently,
ifJF > JFJNK;1. As
we have shown, the null distribution ofJF converges to a 2J
distribution as N so that asymptotically JFJNK;1 =
2J;1. Since the latter is the critical value of the
Wald test and J F =W, the F-test and the Wald test are
asymptotically equivalent andconsequently do not differ much in
large samples. In small samples, the F-test can beshown to be more
conservative as there can be cases where the Wald test rejects H0
and
the F-test does not but not vice versa.
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Course manual Econometrics: Week 5
1 General information
This week we treat the problem of heteroscedasticity, so a
violation of the assumptionof constant error variances. We study
the consequences for estimation and inferenceusing OLS, more
efficient estimation by generalized least squares and how to test
for
heteroscedasticity.
2 Reading
Obligatory reading: Verbeek, Sections 4.1-4.5
Additional reading: Wooldridge, Chapter 8.
Greene, Chapter 8.
3 Exercises
1)
Consider the modelyi=1+2xi2+i
where the error terms are uncorrelated but V{i|X}= 2x2
i2.
a) Explain how an appropriate Generalized Least Squares
estimator can be constructed.
Use the notation of the lecture and be specific about your
choice of ,P ,hi and
.
b) What is the interpretation of2?
c) How can you test the assumed relation of the error term and
xi2?
2)
Consider the modelyi = 1+2xi2+3di+i
where di is a dummy variable taking values 0 and 1. We assume
that the error terms are
uncorrelated but V{i|X}= 2
1 ifdi= 1 and V{i|X}= 2
2 ifdi= 0.
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a) Explain how an appropriate Generalized Least Squares
estimator can be constructed.Use the notation of lecture and be
specific about your choice of ,P ,hi and .
b) How can you get standard errors for your GLS estimator?
c) How can you graphically inspect your assumption for the error
term?
3)
Use the dataset 401ksubs.gdt which contains the variable net
total financial assets(nettfa, in $1000s) and several other
variables which may explain a personss financialwealth. Consider
the regression
nettfai=1+2inci+3agei+4age2
i+5marri+i
a) Estimate the model by OLS.
b) According to your model, at what age is financial wealth
supposed to be lowest?
c) Test for heteroskedasticity using the White test and the
Breusch-Pagan test.
d) Given your results from part c), does it make sense to
perform additionally theGoldfeld-Quandt test based on subsamples
for married and unmarried persons?
e) Use your insights from part c) to construct an appropriate
Weighted Least Squaresestimator of the model. Compare the parameter
estimates and standard errors withthe OLS approach.
f) Apply heteroskedasticity-robust standard errors to your OLS
estimator and to theWLS estimator as well. Why may the latter be
sensible?
4 Additional exercises
1)
Which of the following are consequences of
heteroscedasticity?
a) OLS is inconsistent.
b) OLS is biased.
c) OLS is inefficient.
d) s2(XX)1 is an inconsistent estimate of the covariance matrix
of the OLS estima-tor.
e) The t and Ftests as presented in lecture 3 are no longer
valid.
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5 Solution to last weeks exercises
1)
a) Generally, the elasticity of a variabley with respect to x is
defined as
el:= y/y
x/x=
y
xx
y
Hence the elasticity can be interpreted as the approximate
percentage change ofyfor each 1% change ofx.
For model (1),
el1= yixi
xiyi
= yixi
log yi/yilog xi/xi
= log yilog xi
so that we obtain aconstantelasticity ofel1= 2. Note that for
this interpretation
we must use the natural logarithm.
b) In case of model (2) the elasticity corresponds to
el2= yixi
xiyi
=2xiyi
That means the elasticity is not constant and instead depends on
the current valuesofxand y.
c) For model (3),
2= log(yi)
xi=log(yi)
yi
yi
xi=yi/yi
xiwhich is called the semi-elasticity of y with respect to x. In
words, the semi-elasticity is the approximate percentage change ofy
given a one unit increase ofx.(For further practice you might
calculate the elasticity for model (3) as well.)
2)
a) It would not be sensible to include the amount of beer
consumption as an explana-tory variable although it may be
significant and increase the goodness-of-fit of yourregression. If
you include beer consumption the coefficient of beer tax gives you
theeffect of the beer tax on traffic fatalities,holding beer
consumption constant. Since abeer tax is only supposed to work if
it reduces beer consumption, this does not makesense. In contrast,
including the miles driven in a region is sensible. It is likely
tobe an important explanatory variable and it still allows the
desired interpretationof the beer tax coefficient.
b) Including the number of doctor visits as a control variable
is not sensible althoughdoctor visits are likely to be highly
significant. This is because you would estimatethe effect of
pesticide usage on health expenditures, holding the number of
doctorvisits constant. Thus, you would only estimate the effect of
pesticide usage onhealth expenditures that did not arise together
with doctor visits. This is probably
not what you are interested in.
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3)
a) The earnings of females are estimated to be on average 44%
lower for females thanfor males. More formally, 0.44 is the
semi-elasticity of earnings with respect to
gender (see exercise 1).b) A 10% increase of the market value of
a firm is estimated to increase the earnings
of top executives by 3.7%, ceteris paribus. The coefficient
ofFemaleis now lower inabsolute terms implying 28% less earnings
for females holding the other variables,especially market value,
constant. Note that the regression from part a) suffers fromomitted
variable bias if i) gender is correlated with the omitted variables
like, forinstance,log(MarketValue), and ii) the omitted variables
have non-zero coefficients.Since log(MarketValue) is highly
significant, the omitted variable bias is likely tooccur due to a
negative correlation ofFemaleand log(MarketValue), i.e. larger
firmshaving less female top executives.
c) Since the coefficient oflog(MarketValue)is an elasticity it
has no dimension, so thatthe interpretation would not change.
Further, it is possible to show that running theregression with
market value measured in billions would only change the
intercept(here it would change by 0.37 log(1000)) and leave all
other coefficients unchanged.
4)
a) The coefficient ofAgeis 0.0244429. Since it represents a
semi-elasticity, this meansthat increasing Ageby one unit, for
instance from 33 to 34, is expected to lead to
a 2.4% increase in hourly earnings.
b) The coefficient of log(Age) is 0.724697 giving the estimated
elasticity of earningswith respect to age. From 33 to 34, age
increases by about 3%, so that the expectedincrease in earnings is
0.030.724697 = 0.02196, i.e. about 2.2%.
c) When Age increases from 33 to 34, the predicted change in
log(AHE) is
(0.147045340.00207056342)(0.147045330.00207056332) = 0.0083
,
meaning a 0.83% increase in earnings.
d) The regressions from a) and b) contain the same number of
parameters and canthus be compared by their goodness-of-fit as
measured by the R2. Thus, we wouldprefer b) which has a slightly
higher R2 (0.192685 vs. 0.192372).
e) The regression from c) is the same as a) but augmented with
Age2. Thus, theR2
mustbe higher for c) but this does not necessarily mean that we
prefer c). However,since Age2 is highly significant (p-value of
0.0031) and the Akaike InformationCriterion decreases from 10163.48
to 10156.74, we indeed prefer model c).
f) Noting that now no model is nested in the other one, we stick
again to the AIC.Since 10156.74 < 10160.39, we prefer again
model c).
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g) For females with a Bachelor degree, the regression relation
is
log(AHE) = 0.05873320.1797871 + 0.4050771 +
0.147045Age0.00207056Age2
log(AHE) = 0.2840232 + 0.147045Age0.00207056Age2
AHE=exp(0.2840232 + 0.147045Age0.00207056Age2 +
0.50.4568762)
=exp(0.388391039 + 0.147045Age0.00207056Age2)
Note that we added half the estimated error variance for
predicting AHE(see slide9, lecture 4). In GRETL, go to Tools Plot a
curve, and use the formulagiven above, i.e. exp(0.388391039 +
0.147045x0.00207056x2) and specify areasonable range for x(age),
for instance 20-60.
6 Solution to last weeks additional exercises
1)
Two explanatory variables may have low t-statistics if they are
highly correlated, even ifat least one of their true coeffcients is
nonzero. A joint test (F-test) takes this correlationinto account.
Put differently, if you drop one of the two variables from the
model, theremaining one may become (highly) significant and you
will not observe this if you removetwo variables at once. As an
example, assume you have both a short-term and a long-term interest
rate in a model explaining investments. Given the high correlation
betweenthe two interest rates, both may have fairly low
t-statistics. However, if you drop one ofthem, the remaining
interest rate will pick up much of the explanatory power of the
two,and (probably) will be statistically significant.
2)
The teacher has the hypothesis that there is no linear effect of
class size on studentperformance. Rather, there are three
categories (25), which have differenteffects but without any
effects of changes within a category. Such a relationship can
bemodelled using dummy variables. In this case, define
d1= 1 if 20 class size 25 , and d1= 0 otherwise
d2= 1 if class size >25 , and d2= 0 otherwise
Making a linear regression with these dummy variables gives
coefficients with the followinginterpretation. The coefficient ofd1
is the ceteris paribus effect of increasing class sizefrom below 20
(the base category) to 20-25 on test scores. The coefficient ofd2
is theeffect of increasing class size from below 20 to above 25 on
test scores. In practice, youshould compare the goodness-of-fit of
such a dummy variable specification with otheralternatives like the
simple linear specification.
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Course manual Econometrics: Week 6
1 General information
This week we treat a second common violation of the standard
assumptions, namelyautocorrelation. As last week, we look at
consequences for OLS, alternative estimation,tests for
autocorrelation and how to compute robust standard errors.
2 Reading
Obligatory reading: Verbeek, Sections 4.6-4.11
Additional reading: Wooldridge, Chapter 12.
Greene, Some parts of Chapter 19.
3 Exercises1)
The plots on the following page were generated from a
first-order autoregressive processesof the following form:
t = t1+t, t= 1, . . . , 100 , tN(0, 1)
The choices for are -0.9, 0 and 0.9. Which plot refers to which
value of?
2)
Verify the approximation of the Durbin-Watson statistic given in
the lecture,dw 2 2
where is an estimate of the first-order autocorrelation of the
error terms.3)
a) Explain why autocorrelation may arise because of an omitted
variable. Give exam-ples.
b) Imagine you have a linear regression model with the dependent
variable being an-nual stock returns observed at a monthly
frequency. What are the consequences?
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0 10 25 40 55 70 85 100
3
2
1
0
1
2
0 10 25 40 55 70 85 100
4
2
0
2
4
0 10 25 40 55 70 85 100
6
2
0
2
4
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4)
Load the dataset gasoline.gdt. Consider the following two
regressions which both try toexplain per capita gasoline
consumption in the US for the years 1960-1995:
log(Gt/Popt) =1+2log(P gt) +3log(Yt) +t
log(Gt/Popt) =1+2log(P gt) +3log(Yt) +4log(P nct) +5log(P
uct)+
6log(P ptt) +7log(P dt) +8log(P nt) +9log(P st) +10t+t
a) Estimate both models by OLS and plot the residuals against
time. What do thesegraphs tell about possible autocorrelation in
each of the two models?
b) Test for autocorrelation by i) regressing the residuals on
their first lag and applyinga t-test, ii) using the Durbin-Watson
test and iii) using the Breusch-Godfrey-testwith up to 3 lags. Do
the results fit to your interpretation in part a)?
c) Apply Newey-West (HAC) standard errors to the augmented
model.
5)
Load the dataset ukrates.gdtwhich is a time-series dataset
consisting of monthly short-term (variable rs) and long-term
interest rates (variable r20) on U.K. government secu-rities.
Consider the regression
rst = 1+2 r20t1+t
where rst = rst rst1 and r20t1=r20t1 r20t2. The model can be
interpreted
as a simple monetary policy reaction function.
a) Estimate the model by Ordinary Least Squares and make tests
for first-order au-tocorrelation using i) a regression of the
residuals upon their first lag and ii) theDurbin-Watson test.
b) Re-estimate your model using Newey-West (HAC) standard
errors.
c) Re-estimate your model by applying the Feasible Generalized
Least Squares esti-mator by Cochrane and Orcutt and interpret the
differences.
d) Re-estimate your model by applying the Prais-Winsten
estimator which is identicalto the Cochrane-Orcutt procedure but
additionaly uses the first observation.
4 Additional exercises
1)
Which of the following are consequences of autocorrelation?
a) OLS is inconsistent.
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b) OLS is biased.
c) OLS is inefficient.
d) s2(XX)1 is an inconsistent estimate of the covariance matrix
of the OLS estima-
tor.e) The t and Ftests as presented in lecture 3 are no longer
valid.
f) The model is misspecified.
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5 Solution to last weeks exercises
1)
a) Our assumption is that
V{|X}= 2Diag{x2i2}= 2
Thus,
=
x212 0
. . .
0 x2N2
.Therefore, hi=xi2.Since 1 =PP,
P =Diag{x1i2 }.
Applying the transformation matrix P to y and Xgives the
transformed model
P y= P X+P
yixi2
= 1xi2
+2+ ixi2
, i= 1, . . . , N
yi =1x
i1+2x
i2+
i , i= 1, . . . , N
The GLS estimator can be written as
=
Ni=1
1
x2i2xix
i
1 Ni=1
1
x2i2xiyi
or
=
Ni=1
xi x
i
1 Ni=1
xi y
i
wherex = (xi1, x
i2).
b) Since
V{i |X}= V{ixi2
|X}= 1
x2i2V{i|X}=
2x2i2x2i2
=2,
2 is the variance of the error in the transformed model and not
equal to the variance
in the original model.
c) Since the assumption is that the variance of the errors is
proportional toxi2,
V{i|X}= E{2i |X}=
2x2i2
we can simply use estimates of2i , for instance squared
residuals taken from OLS,to perform the auxiliary regression
2i =1+2x2i2 , i= 1, . . . , N
if our assumption is correct, 1 = 0 and 2 = 0 which can be
tested seperately by
t-tests.
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2)
a) Denote the number of observations where di = 1 as N1 and let
N2 be the numberof observations with di= 0. Then, our assumption is
that
V{|X}= 2 =2
21
2 0. . .21
2
22
2
. . .
0 2
2
2
N1colums
N2colums
Thus hi = 1
for i= 1, . . . , N 1 and hi= 2
for i = N1+ 1, . . . , N .The transformation matrix P is
P =
1
0. . .
1
2
. . .
0 2
The GLS estimator is
=N1
i=1
2
21xix
i+N
j=N1+1
2
22xix
i
1N1i=1
2
21xiyi+
Nj=N1+1
2
22xiyi
=
N1i=1
1
21xix
i+N
j=N1+1
1
22xix
i
1N1i=1
1
21xiyi+
Nj=N1+1
1
22xiyi
b) The covariance matrix ofcan be estimated as
V{}= 2(X1X)1,
where
2 = 1N 3
(y X)1(y X)
and
1 =
2
21
0. . .
2
21
2
22
. . .
0 2
22
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c) Before one would apply the GLS approach shown above, one
should inspect theunderlying assumption. This can be done by
estimating the model by OLS andplotting the residuals separately
for the subsamples with di= 1 and di = 0 respec-tively. If you
there are too much data for such a plot, a boxplot for the
residuals ofboth subsamples is preferable.
3)
a) Model 1: OLS, using observations 19275Dependent variable:
nettfa
Coefficient Std. Error t-ratio p-value
const 9.18133 10.0688 0.9119 0.3619inc 1.05132 0.0272112 38.6356
0.0000age 2.29337 0.488324 4.6964 0.0000
sq age 0.0385523 0.00560576 6.8773 0.0000marr 10.0488 1.33773
7.5118 0.0000
b) We are looking for a local minimum of our regression function
with respect to age.Taking the partial derivative with respect to
age yields the first-order condition:
2.29337 + 2 0.0385523age != 0
Solving this equation leads toage = 29.74. It is easily seen
that the second (partial)derivative is positive so that we have
indeed a minimum at age 30.
c) The null hypothesis of homoskedasticity is clearly rejected
by both the White testand the Breusch-Pagan test. Age and income
especially seem to be sources ofheteroskedasticity.
d) No. One the one hand, we have already seen in the
Breusch-Pagan test that mar-riage is likely to be another source of
heteroskedasticity. More importantly, theGoldfeld-Quandt test
considers only the heteroskedasticity induced by marriageand
neglects any other sources like age and income which are apparently
existentin our case.
e) The regression from the Breusch-Pagan test seems to be a
sensible starting point for
a Weighted Least Squares approach. However, if we would directly
use it we wouldhave no guarantee to get positive variance estimates
and thus could get negativeweights. Instead, we might consider the
same variables but using the multiplicativemodel presented in the
lecture. The corresponding auxiliary regression is
log(e2i ) =1+2inci+3agei+4age2i +5marri+vi
To do so in GRETL, go to Save Squared residuals in the model
window whichgenerates the corresponding new variable. Taking the
log of the squared residualsand performing the auxiliary regression
we get the interesting result that marriageis not a significant
source of heteroskedasticity anymore. Dropping marri gives the
following results:
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Model 3: OLS, using observations 19275Dependent variable: l
usq2
Coefficient Std. Error t-ratio p-value
const 5.24314 0.390843 13.4149 0.0000inc 0.0426766 0.000988221
43.1853 0.0000age 0.153820 0.0189863 8.1016 0.0000sq age 0.00243750
0.000217944 11.1841 0.0000
Using the predictions from this auxiliary regression (Save
Fitted values) andreversing the logarithmic transformation we
have
h2i =exp(5.24314 + 0.0426766inci 0.153820agei+ 0.00243750age2i
)Note that it does not matter if we keep or drop
1 since including it means multi-
plying all observations with a constant which does not change
the result. For WLS
estimation, go in GRETL to ModelsOther linear modelsWeighted
LeastSquares and choose 1/h2i as the weight variable (see Help for
details). Doing sogives
Model 4: WLS, using observations 19275Dependent variable:
nettfa
Variable used as weight: h sq inv
Coefficient Std. Error t-ratio p-value
const 0.337574 5.19623 0.0650 0.9482inc 0.550563 0.0225816
24.3810 0.0000age 0.840889 0.271119 3.1016 0.0019sq age 0.0167648
0.00339816 4.9335 0.0000marr 3.93802 0.586034 6.7198 0.0000
The difference in the parameter estimates is remarkably high. As
expected fromtheory, the standard errors are lower under the WLS
approach.
f) Go to Edit Modify Model and choose robust standard errors.
Under Con-figure you get several options for
heteroskedasticity-robust standard errors. Option
HC0 refers to the original White standard errors which you have
seen in the lec-ture. Applying this option to our models increases
the standard errors in the OLSmodel considerably while the WLS
standard errors remain on a similar magnitude.Nevertheless, using
robust standard errors within WLS may be sensible since themodel
for the variance of the error term will only be an approximation to
reality sothat heteroskedasticity may well still be present.
6 Solution to additional exercises
1)
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a) False. Homoskedasticity is not required for the consistency
of OLS.
b) False. Homoskedasticity is not required for unbiasedness of
OLS.
c) True. If the error terms are heteroskedastic, the WLS
estimator is more efficient in
theory. This, however, only holds if the variances of the error
terms are known. Inpractice, we have to specify a model for the
variance of the error term and estimateit. Then, there is no
guarantee that WLS is more efficient. However, WLS will beoften be
more efficient especially if the degree of heteroskedasticity is
high.
d) True. In the derivation of this formula we used the
assumption of homoskedasticity.
e) True. Although there are certain departures from
homoskedasticity where the usualt- and F-tests are still
asymptotically valid, they will be generally invalid. How-ever, the
t-statistic with heteroskedasticity-robust standard errors is
asymptoticallynormal distributed and the Wald test with a
heteroskedasticity-robust covariance
matrix can also be used for inference (see the exercises from
week 3 for the relationof F-tests and Wald tests). Note that exact
small sample distributions (i.e. t- andF-distributions) are no
longer available under autocorrelation even if we assumenormality
of the error term.
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Course manual Econometrics: Week 7
1 General information
This week we review the properties of OLS and the relevant
assumptions. We shall seeexamples when OLS cannot be saved anymore.
The instrumental variables estimator willbe introduced as a
solution for these cases.
2 Reading
Obligatory reading: Verbeek, Sections 5.1-5.3.1
Additional reading: Wooldridge, Chapter 15.
Greene, Chapter 12.
3 Exercises
1) (adapted from Stock/Watson)
The demand for a commodity is given byQ= 1+2P+u, whereQdenotes
the quantity,Pdenotes the the price, and udenotes factors other
than price that determine demand.Supply for the same commodity is
given by Q= 1+ 2P+v, where v denotes factorsother than price that
determine supply. Assume that u and v both have a mean of
zero,variances2
uand 2
v, and are uncorrelated, i.e. cov{u, v}= 0.
a) Solve the two equations to show how Qand P depend on uand
v.b) Calculate cov{P, u} and cov{P, v} and interpret the
results.
c) Derive cov{P,Q} and V{P}.
d) A random sample of observations (Qi, Pi), i = 1, . . . , N ,
is collected, and Qi isregressed on Pi. Use the answer from c) to
derive the asymptotic Ordinary LeastSquares regression coefficient
ofPi.
e) Suppose the OLS estimate from d) is used to estimate the
slope of the demandfunction, 2. Is the estimated slope
(asymptotically) correct, too large or too small?
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(Hint: Use the fact that demand curves usually slope down and
supply curves slopeup.)
2)
Consider the instrumental variable regression
modelyi=1+2x1i+3x2i+i
where x1i is correlated with i and z1i is a potential
instrument. Which assumption ofthe instrumental variables estimator
is not satisfied if
a) z1i is independent of (yi, x1i, x2i)?
b) z1i=x2i?
c) z1i=cx1i (where c is a constant) ?
3) (adapted from Stock/Watson)
During the 1880s, a cartel known as the Joint Executive
Committee (JEC) controlledthe rail transport of grain from the
midwest to eastern cities in the United States. Thecartel preceded
the Sherman Antitrust Act of 1890, and it legally operated to
increasethe price of grain above what would have been the
competitive price. From time to time,cheating by members of the
cartel brought about a temporary collapse of the
collusiveprice-setting agreement.
The data filerailway.gdtcontains weekly observations on the rail
shipping price and otherfactors from 1880 to 1886. Suppose that the
demand curve for rail transport of grain
is specified as log(Qt) = 1+ 2log(Pt) + 3Icet+ t , where Qt is
the total tonnage ofgrain shipped in week t, Pt is the price of
shipping a ton of grain by rail and Icet is abinary variable that
is equal to 1 if the Great Lakes are not navigable because of ice.
Iceis included because grain could also be transported by ship when
the Great Lakes werenavigable. Further, the variable cartel is a
dummy variable for the activity of the cartel.
a) Estimate the demand equation by OLS. What is the estimated
value of the demandelasticity and its standard error?
b) In exercise 1 we have analyzed that the interaction of supply
and demand is likely to
make the OLS estimator of the elasticity biased. Consider now
using the variablecartel as an instrumental variable for log(P).
Use economic reasoning to arguewhether cartelplausibly satisfies
the two conditions for a valid instrument.
c) Regress log(Pt) on cartelt and I cet. What do the results
tell you about the qualityofcartelas an instrument?
d) Estimate the demand equation by instrumental variable
regression. What is theestimated demand elasticity and its standard
error? Compare the results to theOLS estimates.
e) Perform the Durbin-Wu-Hausman test and interpret the
results.
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4 Additional exercises
1)
Consider the example from slide 3 of lecture 7,
yt= 1+2yt1+3yt2+t.
Show the result from the lecture, namely that the assumption
E{|X} can not hold forthis model. (Xdenotes as always the matrix of
regressors which are here lagged valuesofyt.)
5 Solution to last weeks exercises
1)
First plot: = 0; second plot: = 0.9; third plot: = 0.9.
If the autocorrelation is positive and high ( = 0.9), the
process tends to stay above(below) its mean (zero) in the next
period if it is above (below) its mean in the currentperiod. For =
0.9 the process tends to reverse its sign from one period to
another.
2)
dw=
T
t=2
(et et1)2
T
t=1 e
2t
=
T
t=2
(e2t2etet1+e
2t1)
T
t=1 e
2t
=
T
t=2
e2t
T
t=1
e2t
+
T
t=2
e2t1
T
t=1
e2t
2
T
t=2
etet1
T
t=1
e2t
22
T
t=2
etet1
T
t=1
e2t
as the sample size becomes large because both
T
t=2e2
tT
t=1e2t
and
T
t=2e2
t1T
t=1e2t
tend to 1.
Since
T
t=2etet1
T
t=1e2t
is an estimator of, dw tends to 22.
It can be shown that this estimate of
is very close to the estimate which results fromregressing et on
et1 by OLS.
3)
a) See exercise 4. Another example would be the omission of a
variable that describesseasonal patterns in monthly or quarterly
data. Consider for instance the case thatyou want to explain the
activity in the construction industry based on monthlydata. It
could then be that the residuals in your model would have a
tendencyto be negative in the winter months and positive in the
summer months. A win-ter/summer dummy variable could possibly solve
this problem. Another typical
example is an omitted lagged dependent variable.
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b) LetRtbe the annual return of a stock, i.e. the return over
the period [t12, t], wheretime is measured in months. AsRt+1refers
to the period [t11, t+1],RtandRt+1have 11 months in common and are
therefore not stochastically independent andprobably heavily
autocorrelated. The autocorrelation ofRt will probably
translateinto autocorrelation of the error term in a regression
explainingRtsince unexpected
(stock market) events in one month have a direct influence on Rt
in the followingtwelve month periods. Under autocorrelation,
routinely computed standard errorsand tests will be incorrect and
misleading, so that robust standard errors (HAC)are
recommended.
4)
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
1960 1965 1970 1975 1980 1985 1990 1995
residual
Regression residuals (= observed - fitted lGpc)
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1960 1965 1970 1975 1980 1985 1990 1995
residual
Regression residuals (= observed - fitted lGpc)
a) The graph on the left hand side refers to the first model
whereas the graph on theright hand side refers to the augmented
model. The model with more regressors
seems to suffer less from autocorrelation since the residuals
cross the zero linemore often. This is an example where the
omission of relevant variables leads toautocorrelation.
b) i) Regressing the residuals of our model on their first lag
(use thelags. . .optionin the model specification window to add the
lagged residual), we get estimatedfirst-order autocorrelations of
0.948469 (p value 2.49e-014) and 0.268612 (p value0.1147). Thus, we
clearly reject the null hypothesis of no autocorrelation for
thefirst model whereas we do not reject the null hypothesis for the
second model. Ofcourse, not rejecting the null hypothesis
(especially in small samples) does not meanthat the null hypothesis
is true.
ii) The Durbin-Watson statistics are 0.172878 and 1.373491,
respectively. Lookingat the bounds for the critical values given in
the lecture we see that we reject thenull hypothesis clearly for
the first model and that we are in the inconclusive regionfor the
augmented model (the bounds for K=10 are not too far away from
thebounds for K=9).
iii) Running auxiliary regressions from the residuals on their
first three lags yieldsR2 values of 0.847827 and 0.107939. Thus,
the Breusch-Godfrey test statistics are32 0.847827 = 27.13 and 32
0.107939 = 3.454. Using the p value finder of GRETLand applying 3
degrees of freedom, we get p values of 5.52923e-006 and
0.326771,
respectively.
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Thus, for each test, the results fit to the graphical analysis
from part a).
c) Applying Newey-West standard errors yields in our case
standard errors which arelower than the default standard errors.
More often, it is the other way around.
5)
a) A regression of the residuals on their first lag gives a
coefficient of 0.148841 whichis statistically significant from zero
with a p value of 0.0006. The Durbin-Watsonstatistic is 1.702273.
Going toTestsDurbin-Watsonpvalue gives 0.000292831,so that we also
reject H0.
b) The standard errors increase as expected since standard
errors that ignore autocor-relation are usually (asymptotically)
downward biased. Note that HAC standarderrors are also not unbiased
but they are at least asymptotically unbiased in contrastto
non-robust standard errors.
c) Going to Model Time series Cochrane-Orcutt we can perform the
FeasibleGeneralized Least Squares approach from the lecture. The
iterations lead to aslightly increased estimated (first-order)
autocorrelation. The coefficient ofr20t1is now somewhat lower than
with OLS estimation. The standard errors tend to besmaller than in
part b) pointing to a more efficient estimation than by doing
OLS.Note that a comparison with unadjusted standard errors (part
a)) is not meaningfulsince these standard errors are invalid under
autocorrelation.
d) Go to Model Time series Prais-Winsten. The results are very
similar.
This is no surprise since the information from one additional
observation shouldnot make a big difference especially if the
sample size is relatively large as it is thecase here.
6 Solution to last weeks additional exercises
1)
a) False.
b) False.
c) True. If the error terms are correlated, an appropriate GLS
estimator is moreefficient in theory. This, however, only holds if
the correlations (and the variances)of the error terms are known.
In practice, we have to specify a model for theautocorrelations of
the error term and estimate it. Then, there is no guarantee thatWLS
is more efficient. However, FGLS will be often be more efficient
especially ifthe degree of autocorrelation is high.
d) True. In the derivation of this formula we used the
assumption of uncorrelatederror terms.
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e) True. However, the t-statistic with HAC standard errors is
asymptotically normaldistributed and the Wald test with a HAC
covariance matrix can also be used forinference. Exact small sample
distributions (i.e. t- and F-distributions) are nolonger available
under heteroskedasticity even if we assume normality of the
errorterm.
f) Depends. A high degree of autocorrelation may point to
misspecification but thereis no general rule that there must be
misspecification. In practice, we should justcheck the functional
form and test additional regressors if these are available.
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Course manual Econometrics: Week 8
1 General information
This week we continue studying instrumental variables estimators
in the various situationwhen they are needed. We then generalize
the estimator and look at specification tests.
2 Reading
Obligatory reading: Verbeek, Sections 5.3.2-5.5
Additional reading: Wooldridge, Chapter 15.
Greene, Chapter 12.
3 Exercises
1)
Consider the following two equations:
yi=1+2xi+ui
xi=1+2yi+vi
ui andviare the error terms of the models and are assumed to be
uncorrelated with eachother having variances 2u >0 and
2v >0.
a) Show that xi is correlated with ui if2= 0.
b) What are the consequences of your finding?
2)
Consider the equations
yi=1+2x1i+3z2i+4z3i+ui
xi=1+2yi+3z2i+vi
and assume that cov{z2i, ui}= cov{z3i, ui}= cov{z2i, vi}=
cov{z3i, vi}= 0.
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a) What do the assumptions mean?
b) Can we consistently estimate 2?
c) Can we consistently estimate 2?
3)
a) Briefly describe the Two-stage Least Squares (2SLS)
approach.
b) Show that the 2SLS estimator derived in the lecture is
identical to the (generalized)Instrumental Variables estimator.
Consider the case of overidentification as well asthe case of exact
identification.
4)
Consider again the dataset from last week (railway.gdt) and the
corresponding regression
log(Qt) =1+2log(Pt) +3Icet+t
a) Use cartelt, cartelt1 and cartelt2 as instruments for log(Pt)
and estimate themodel with the instrumental variables
estimator.
b) Why may it be sensible to use cartelt1 and cartelt2 as
additional instruments?
c) Given your reasoning from part b), how do you interpret the
results from part a)?
d) What is the risk of using additional instruments in general?
What about the specificcase in this exercise?
e) Check your reasoning from part d) by performing the
specification test from theend of lecture 8.
f) Re-estimate the model by applying Newey-West (HAC) standard
errors. (Theseare easily generalized from the OLS case given in the
lecture to IV regressions.)
g) Test that the demand elasticity is equal to -1.
4 Additional exercises
1)
Why does the Instrumental Variable (IV) estimator lead to a
smaller R2 than the OLSone? What does this say of the R2 as a
measure for the adequacy of the model?
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5 Solution to last weeks exercises
a)
(1) Q = 1+2P+u
(2) Q = 1+2P+v P =1
2+
1
2Q
1
2v
Substituting P in (1):
Q = 1+2(1
2+
1
2Q
1
2v) +u| 2
2Q = 21 21+2Q 2v+2u
Q = 21 21
2 2+
2u 2v
2 2Substituting Q in (2):
P = 12+ 12
(1+2P+u) 12v| 2
2P = 1+1+2P+u v
P = 1 1
2 2+
u v
2 2
b)
cov(P, u) =cov
1 12 2
+ u v
2 2, u
= 12 2cov(u v, u)
= 1
2 2
V(u) cov(u, v) =0
=
2u2 2
= 0
cov(P, v) =cov
1 12 2
+ u v
2 2, v
=
1
2 2 cov(u v, v)
= 1
2 2
cov(u, v) =0
V(v)
=
2v2 2
= 0
Interpretation: Since cov(P, u) = 0 and cov(P, v) = 0, the
regressors are correlatedwith the error terms so that the
conditions for the consistency of OLS are not met.
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c)
cov(P, Q) =cov
1 12 2
+ u v
2 2,21 12
2 2+
2u 2v
2 2
=
1
(2 2)2 cov(u v, 2u 2v)
= 1
(2 2)2
cov(u, 2u) cov(u, 2v) =0
cov(v, 2u) =0
+cov(v, 2v)
=
22u+2
2v
(2 2)2
V(P) =V
1 12 2
+ u v
2 2
= 1
(2 2)2V(u) +V(v) 2 cov(u, v)
=0
=
2u+2v
(2 2)2
d)
2,OLS=
ni=1(Qi Q)(Pi P)n
i=1(Pi P)2
=
1n
ni=1(Qi Q)(Pi P)1nn
i=1(Pi P)2
=cov(Q, P)V(P) n cov(Q, P)V(P) c)= 2
2u+2
2v
2u+2v
=2
e)
2,OLS 2n
22u+2
2v
2u+2v
22u+
2v
2u+2v
=(2 2)
2u
2u+2v
>0 , if2 >0 and 2
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3)
a) The estimated demand elasticity is -0.63433 with a standard
error of 0.0819697.
b) The variable cartel can be reasonably assumed to be relevant
for the supply side
only. That means, cartel is likely to be uncorrelated with any
demand shocks andthus uncorrelated with the error term of the
demand equation. Further, cartelshould be correlated with prices
since usually cartels use their power to raise prices.Thus, both
conditions, exogeneity and relevance, for a valid instrument are
prettylikely to be met in this case.
c) With a t-ratio of 14.7 cartel is highly significant and seems
to have importantexplanatory power with respect to prices. This
result supports our assumptionthat cartel is a relevant
instrument.
d) Go to Model Instrumental variables Two-stage least squares
(which is
another name for the IV estimator given in the lecture) and
specify log(Qt) asthe dependent variable, log(Pt) and Icet as
independent variables and cartelt andIcet as instruments. The
demand elasticity is now estimated to be -0.872271 witha standard
error of 0.131355. In line with the results from exercise 1, the
resultpoints to an overestimation of the demand elasticity if OLS
is applied (In exercise1 we did not use logarithmic transformations
and no additional covariate but theresults would be similar).
Further, the IV estimate is closer to -1 which would bethe demand
elasticity expected from economic theory in monopolies. Finally,
thestandard error of the IV estimator is larger than that of OLS.
This is also expectedfrom theory and does not tell us that OLS
should be preferred.
e) Start with the regression from part c) and save the
residuals. Then, estimate theoriginal model by OLS as in a) but add
the residuals from the auxiliary regression.Under the null
hypothesis, that both OLS and IV are consistent, the coeffient
ofthis new variable should be zero. However, we get a coefficient
of 0.396247 witha t-ratio of 2.385 so that we reject the null
hypothesis at the 5% level. Note thatin the Durbin-Wu-Hausman test
the alternative hypothesis is inconsistency of OLSbut consistency
of IV. Thus, consistency of IV is assumed and not tested.
6 Solution to last weeks additional exercises
1)
Note that E{|X}= 0 means
E
1 x11 . . . x1K...
...N xN1 . . . xNK
= 0orE{i|xjk}= 0 for everyi, j = 1, . . . , N andk = 1, . . . ,
K . Further note that conditionalmean independence implies
uncorrelatedness, i.e.
E{i|xik}= 0 cov{i, xjk}= 0.
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In our model we have
cov{yt1, t1}= cov{1+2yt2+3yt3+t1, t1}
=V{t1} = 0
if we assume contemporaneous uncorrelatedness of the regressors
with the error term, i.e.
cov{yt1, t}= cov{yt2, t}= 0.
Consequently, the assumption E{|X} = 0 (which is necessary for
unbiasedness) is notmet in this case whereas E{ixi}= 0 (necessary
for consistency) might still hold.
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Course manual Econometrics: Week 9
1 General information
This week the method of maximum likelihood estimation is
introduced. This estimationtechnique is used for a wide range of
econometric models and is therefore extremely useful.Binary choice
models are also treated. Further models that required maximum
likelihoodestimation are covered in the following week.
2 Reading
Obligatory reading: Verbeek, 6.1, 7.1.1-7.1.4
Additional reading:
Wooldridge, Relevant parts of Chapter 17
Greene, Relevant parts of Chapter 16 and 23
3 Exercises
1)
Consider a random variable X with an exponential distribution.
The correspondingdensity function is given by ( >0):
f(x) =
ex x 0
0 x
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2)
Using the expressions from the score contributions on slide 16
of lecture 9, to derive theinformation matrix, the asymptotic
covariance matrix and an estimator for the varianceofin large
samples for the case thatis the maximum likelihood estimator in the
linearregression model with normally, independently and identically
distributed errors. (Hint:Use the fact that for a normally
distributed random variable Xwith zero mean it holdsthat E{X4} =
34, where 2 is the variance ofX.)
3) (adapted from Stock/Watson)
The dataset insurance.gdt provides data about the insurance
status of U.S. citizens rep-resented by the binary variable
insuredand further covariates.
a) Regressinsuredon the variableshealthy (a binary variable
regarding self reportedhealth status), age, male, married and
selfemp (binary variable; 1 = self em-ployed) using (i) the Logit
and (ii) the Probit specification. Compare the results.
Which specification would you prefer?
b) Compute the marginal effects of your regression evaluated at
the mean values ofthe regressors. Compare again the Logit and the
Probit model.
c) Compute the estimated probabilities that a healthy and
married man, aged 40, isinsured if he is (i) self-employed and (ii)
not self-employed. Interpret the differencebetween (i) and
(ii).
d) Imagine that people who are uninsured tend to start
self-employment (maybe be-cause they are jobless). Would it then be
correct to interpret the marginal effect
from part c) as a causal effect?
e) Augment your model by using age2 as an additional regressor.
Does this improveyour model? Compute the marginal effect of
age.
f) Include the additional regressor malemarried. For which
persons is the variableequal to one? Is the effect of marriage on
the probability to be insured higher formen or for women?
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4 Solution to last weeks exercises
1)
a)
cov(ui, xi) =cov(ui, 1+2yi+vi)
=cov(ui, 2yi)
=2cov(ui, 1+2xi+ui)
=2(2cov(ui, xi) +V ar(ui))
=22cov(ui, xi) +22u
cov(ui, xi) 22cov(ui, xi) =22u
cov(ui, xi) = 2
2u
1 22
b) The result from part a) shows that xi is endogenous so that
the fundamental as-sumption for the consistency of Ordinary Least
Squares (OLS) is not met. Thus,OLS is not an appropriate estimator
in this case.
2)
a) The assumptions mean that the variables z2 and z3 are
exogenous for both equa-tions, i.e. we can think ofz2 and z3 as
being generated from outside the model.
b) No. OLS would be inconsistent because of simultaneous
equations bias which canbe derived similarly to exercise 1. We
could estimate1 consistently if we wouldhave a valid instrument for
x. However, the variables which are relevant for xaccording to the
seccond equation, y and z2, can not serve as instruments since yis,
of course, endogenous and z2 is already included in the first
equation so that itdoes not provide an additional moment
condition.
c) Yes. Although OLS in inconsistent for the same reasons as in
b), we could use z3as an instrument for y and then consistently
estimate 2.
3)
a) The first stage of the 2SLS approach is an OLS regression of
all endogenous regres-sors on all instruments. For instance, if we
have a single endogenous regressor, sayxk, there is one first-stage
regression wherexk is regressed on the other explanatoryvariables
of the original equation (1, x2, . . . , xk1, xk+1, . . . , xK) and
the excluded in-struments. The vector of predicted values from the
first-stage regression, xk =Z(where Zis the matrix containing the
exogenous explanatory variables and the ex-cluded instruments and
are the OLS estimates from the first-stage regression), isthen
substituted for xk in the original equation and the modified
original equation
is then estimated by OLS.
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b) From slide 30 in lecture 8:
IV = (XX)1Xy
= ((Z(ZZ)1ZX)Z(ZZ)1ZX)1(Z(ZZ)1ZX)y
= (XZ(ZZ)1
ZZ(ZZ)1ZX)1XZ(ZZ)1
Zy
= (XZ(ZZ)1ZX)1XZ(ZZ)1Zy
If R = K, ZX (and XZ) is a square matrix and if we assume the
regulatorycondition that ZX is invertible (similar to the required
existence of (XX)1 forOLS) we have
IV = (ZX)1ZZ(XZ)1XZ(ZZ)1Zy
= (ZX)1ZZ(ZZ)1Zy
= (ZX)1Zy
4)
a) The coefficient for the demand elasticity is now -0.985085.
(Last week we haveestimated demand elasticites of -0.872271 by IV
without cartelt1 and cartelt2and -0.63433 by OLS.)
b) In general, additional instruments can help to improve the
efficiency of the IVestimator, i.e. they may reduce the variance of
the estimator.
c) As expected from part b), the standard error of the demand
elasticity is now0.119144 compared to 0.131355 when cartelt1 and
cartelt2 are not included.
d) The risk of using additional instruments is that the
exogeneity assumption is notmet for the additional instruments.
Then, the IV estimator is not consistent. In ourcase, it is hard to
argue that cartelt1 and cartelt2 are not exogenous if we
arguethatcarteltis exogenous. Since such reasoning is often
sensible, lagged instrumentsare quite often applied in
practice.
e) Save the residuals from your model and regress them by OLS on
the full set ofinstruments, i.e. a constant, Icet, cartelt,
cartelt1 and cartelt2. The R
2 from
this regression is 0.016274, so that our test statistic is 326
0.016274 = 5.305. Thetest statistic is 2 distributed with R K= 2
degrees of freedom under the nullhypothesis. Using GRETLs pvalue
finder we get a value 0.0704748 so that we donot reject the null
hypothesis at a significance level of 5%. Thus, there is no
strongevidence against the exogeneity of our additional
instruments.
f) Applying Newey-West standard errors increases the standard
error of the demandelasticity substantially to 0.212288. This is
not surprising given that the Durbin-Watson statistic is 0.461790
so that the first-order autocorrelation is about 1 0.461790/2 =
0.769 (see exercise 2 from week 6 and lecture 6, slide 15) pointing
to
considerable autocorrelation.
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g) Since we have evidence for autocorrelation we should not
usetorFdistributions but
we can use the asymptotic result that under the null hypothesis
t =2+1se(2) N(0, 1)
or, equivalently,t2 21(the latter is a Wald test). Doing this in
GRETL ( Tests Linear restrictions b[2] = 1) gives a p value of
0.943987 so that the null
hypothesis is not rejected.
5 Solution to last weeks additional exercises
1)
OLS minimizes the residual sum of squares and therefore
maximizes the R2. Any otherestimator, including instrumental
variables, results in a lowerR2. Note that we are oftennot
interested in obtaining an R2 that is as high a possible, but in
obtaining consistentestimates for the coefficients of interest that
are as accurate as possible. The R2 does not
tell us which estimator is the preferred one. The R2 tells us
how well the model fits thedata (in a given sample) and typically
is only interpreted in this way when the model isestimated by
OLS.
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Course manual Econometrics: Week 10
1 General information
This week a number of topics will be treated in somewhat less
detail. First, binary choicemodels will be studied in more detail
with emphasis on the measurement of the goodnessof fit and the
interpretation of an underlying latent variable. This
interpretation becomesuseful for other models introduced this week.
Next, the model is extended to multiplepossible outcomes, but we
restrict attention to ordered probit and logit models, where
oneneeds to assume a natural ordering of the variables. The more
general case of multinomialmodels is not treated, but the
interested student may read the relevant section in thebook.
Furthermore, so called count data models are treated, that allow
modeling thenumber of certain events. Finally, the important topic
of censored regression with thetobit model will be treated
briefly.
The treatment of these topics is necessarily somewhat
superficial. However, it is impor-tant to know they exist and what
the underlying principles are. If you are interestedin more detail
you may consider reading additional sections in Verbeek or look at
thetreatment of these models in other textbooks (which may be much
better and moredetailed).
2 Reading
Obligatory reading: Verbeek, 7.1.5-7.1.6, 7.2.1-7.2.4, 7.3 and
7.4.1-7.4.3. The part on thetruncated regression on page 234 may be
skipped.
Furthermore, you also have to read page 433 in Appendix B.
Additional reading: As usually, the treatment in the other books
is different from Verbeek,so if you are interested read:
in Wooldridge, the relevant parts of Chapter 17
and
in Greene, the relevant parts of Chapter 16 and 23
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3 Exercises
1) (adapted from Greene)
Consider the number of tickets demanded for events in a certain
sports arena. Denote
this variable by Y
. What is actually observed is the number of tickets sold (Y)
whichis equal to Y if the arena is not sold out and equal to 20000,
the maximum capacity,otherwise. Suppose that the mean number of
tickets sold is 18000 and that the arena issold out for 25% of all
events. Calculate the mean number of tickets demanded under
theassumption that Y N(, 2). (Hints: Use that E[Y|Y < a] =+
((a)/)
((a)/)
and that (0.675)/(0.675) = 0.424.)
2)
The dataset credit.gdt (see Verbeek, 7.2.3) is a sample of US
firms in 2005 containingtheir Standard & Poors credit ratings
and several explanatory variables. (There seemsto be an error in
the formulas in Verbeek, p. 217, where the expressions on the
righthand side lack a 1 in front of them.)
a) Use an Ordered Logit model to regress rating on booklev (book
leverage) andebit (earnings before interest and taxes/total
assets). Interpret the coefficients.
b) What is the probability that a firm with 50% book leverage
and 10% ebit has arating of 4?
c) What is the probability that a firm with 50% book leverage
and 10% ebit has arating of 4 or more?
d) Estimate a Logit model using the binary dependent variable
invgrade (Investmentgrade rating) and the same regressors as
before. Compare the results. What is youranswer for c) under this
model? (A rating of 4 or more is defined to be
investmentgrade).
3)
Use the dataset patents.gdt (see Verbeek, 7.3.2) which is a
sample of 181 internationalmanufacturing firms containing data on
the number of patent applications (patents),the expenditures for
research and development (R&D) and several other variables.
a) Regress patents on the logarithm of R&D expenditures, the
dummy variables forthe different industries (the reference category
is food, fuel, metal and others) andthe dummy variables for the
country (the reference category is Europe) by using(i) the Poisson
model and (ii) the negative binomial model (NegBin II). Use
theoption for robust standard errors.
b) Test for overdispersion in the Poisson model.
c) Interpret the coefficient of log(R&D) in the negative
binomial model.
d) Interpret the coefficient of the USA dummy in the negative
binomial model.
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e) Estimate a linear model instead (by OLS and with patents and
not log(patents) asthe dependent variable) and use
heteroscedasticity-robust standard errors. (i) Showthat
homoscedasticity can be excluded if the Poisson model is correctly
specified.(ii) What is the elasticity of the number of patents with
respect to R&D expendi-tures according to the linear model for
a US firm in the computers industry with
R&D expenditures o