14.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
14.1.4 Synopsis of this chapter . . . . . . . . . . . . . . . . . . . . . . . . 342
14.2 Content of chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
14.2.1 Properties of OLS with serially correlated errors; Serial correlationrobust inference after OLS
. . . . . . . . . . . . . . . . . . . . . . 342
14.2.2 Testing for autocorrelation . . . . . . . . . . . . . . . . . . . . . . . 347
14.2.3 R Application for stationary and weakly dependent time series
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
14.3 Answers to activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
14.4 Overview of chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
14.5 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 352
14.6 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 353
14.6.1 Additional exam based exercises . . . . . . . . . . . . . . . . . . . 353
14.6.2 Solutions to additional exam based exercises . . . . . . . . . . . . 355
15 Trends, seasonality and highly persistent time series in regression analysis357
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
15.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 357
15.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
15.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
15.1.4 Synopsis of this chapter . . . . . . . . . . . . . . . . . . . . . . . . 359
15.2 Content of chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
15.2.1 Trends and seasonality . . . . . . . . . . . . . . . . . . . . . . . . 360
15.2.2 Using highly persistent time series in regression analysis . . . . . 366
15.2.3 Testing for unit roots . . . . . . . . . . . . . . . . . . . . . . . . . . 370
15.2.4 Spurious Regression, Cointegration and error correction models . 375
15.2.5 R Application for non-stationary and persistent time series models 380
15.3 Answers to activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
15.4 Overview of chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
15.5 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 389
15.6 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 390
15.6.1 Additional exam based exercises . . . . . . . . . . . . . . . . . . . 390
15.6.2 Solutions to additional exam based exercises . . . . . . . . . . . . 393
A Syllabus
397
B Sample examination paper and examiner’s commentary
402
8
0
Preface
0.1 What is econometrics, and why study it?
Econometrics is the application of statistical methods to the quantification and critical assessment of hypothetical economic relationships using data. It is with the aid of econometrics that we discriminate between competing economic theories and put numerical
clothing onto the successful ones. Econometric analysis may be motivated by a simple
desire to improve our understanding of how the economy works, at either the microeconomic or macroeconomic level, but more often it is undertaken with a specific objective in
mind. In the private sector, the financial benefits that accrue from a sophisticated understanding of relevant markets and an ability to predict change may be the driving factor. In
the public sector, the impetus may come from an awareness that evidence-based policy
initiatives are likely to have the greatest impact.
It is now generally recognised that nearly all professional economists, not just those
actually working with data, should have a basic understanding of econometrics. There
are two major benefits. One is that it facilitates communication between econometricians
and the users of their work. The other is the development of the ability to obtain a
perspective on econometric work and to make a critical evaluation of it. Econometric
work is more robust in some contexts than in others. Experience with the practice of
econometrics and a knowledge of the potential problems that can arise are essential for
developing an instinct for judging how much confidence should be placed on the findings
of a particular study.
Such is the importance of econometrics that, in common with intermediate macroeconomics and microeconomics, an introductory course forms part of the core of any
serious undergraduate degree in economics and is a prerequisite for admission to a
serious Master’s level course in economics or finance.
0.2 Prerequisite for studying this subject
The prerequisite for studying this subject is a solid background in mathematics and elementary statistical theory. The mathematics requirement is a basic understanding of
multivariate differential calculus. With regard to statistics, you must have a clear understanding of what is meant by the sampling distribution of an estimator, and be familiar
with the principles of statistical inference and hypothesis testing. As many examples
and exercises relate to economics, it would be helpful if you have some knowledge of
economics, but this is not essential.
9
Prerequisites: If taken as part of a BSc degree, the following course(s) must be passed
before this course may be attempted.
• EC1002 Introduction to economics AND
• ST104A Statistics 1 (half course) AND
• MT105A Mathematics 1 (half course) OR MT1174 Calculus OR MT1186 Mathematical methods
Co-requisites: Students may only take EC2020 Elements of Econometrics at the same
time as, or after, the following courses, not before:
• ST104B Statistics 2 (half course)
• AND MT105B Mathematics 2 (half course)†
† If you have passed MT1174 Calculus OR MT1186 Mathematical Methods, you do not
need to take MT105B Mathematics 2. Please refer to your Regulations for further details.
0.3 Aims and objectives
The aims of this course are:
• To understand the nature and scope of econometrics as a social science.
• To develop an understanding of the use of regression analysis and related techniques for quantifying economic relationships and testing economic theories.
• To provide a good foundation for further study in econometrics and statistical techniques.
• To equip students to read and evaluate empirical papers in professional journals.
• To provide students with practical experience of using mainstream regression programmes to fit economic models.
0.4 Learning outcomes
At the end of this course, and having completed the Essential reading and activities, you
should be able to:
• describe and apply the classical regression model and its application to crosssection data
10
• describe and apply the:
– Gauss-Markov conditions and other assumptions required in the application
of the classical regression model
– reasons for expecting violations of these assumptions in certain circumstances
– tests for violations
– potential remedial measures, including, where appropriate, the use of instrumental variables
• recognise and apply the advantages of logit, probit and similar models over regression analysis when fitting binary choice models
• competently use regression, logit and probit analysis to quantify economic relationships using R
• describe and explain the principles underlying the use of maximum likelihood estimation
• apply regression analysis to time-series models using stationary time series, with
awareness of some of the econometric problems specific to time series applications (for example, autocorrelation) and remedial measures
• recognise the difficulties that arise in the application of regression analysis to nonstationary time series, know how to test for unit roots, and know what is meant by
cointegration.
0.5 Overview of the learning resources
0.5.1
Essential reading
The Essential reading for the course is:
Wooldridge, J. Introductory Econometrics: A Modern Approach. (Cengage, 2020)
seventh edition [ISBN 9781337558860]).
Detailed reading references in this subject guide refer to the edition of the set textbook
listed above. A new edition of this textbook may have been published by the time you
study this course. You can, however, use a more recent edition; use the detailed chapter
and section headings to identify relevant readings. Also check the virtual learning environment (VLE) regularly for updated guidance on readings. You must have a copy of
the textbook to be able to study this course.
11
0.5.2
Further reading
As long as you read the Essential reading you may want to do some supplementary
readings. You may be interested in consulting the following textbooks:
• Dougherty, C. Introduction to econometrics. (Oxford: Oxford University Press,
2016) 5th edition [ISBN 978-0-19-967682-8].
• Gujarati, D.N. and D.C. Porter Basic econometrics. (McGraw-Hill, 2009, International edition) 5th edition [ISBN 978-0-07-127625-2].
• Maddala, G.S. and K. Lahiri. Introduction to econometrics (Wiley, 2009) 4th edition
[ISBN 978-0-470-01512-4].
• Stock, J.H. and M.W. Watson. Introduction to econometrics (Pearson,2012) 3rd
edition [ISBN 13: 978-1-4082-6433-1].
0.5.3
The subject guide
The subject guide is intended for you to use in conjunction with the main textbook. It
will act to highlight important aspects of the content in the chapters, provide further
clarifications where deemed useful and also to extend it in certain places. For the final
examination, you will need to be familiar with the material in both the textbook and the
subject guide.
The subject guide provides a range of activities that will enable you to test your understanding of the basic ideas and concepts. We want to encourage you to try the exercises
you encounter throughout the material before working through the solutions provided on
the virtual learning environment (VLE). With econometrics the motto has to be “practice,
practice, practice. . . .” It is the best way to learn the material and prepare for examinations. The course is rigorous and demanding, but the skills you will be developing will be
rewarding and well recognized by future employers.
Each chapter in the subject guide begins with an overview of the topic covered and
a checklist of learning outcomes anticipated as a result of studying the associated, relevant, material in the textbook. The readings from the textbook (by section headings)
are supplemented by a discussion and various activities that should be attempted along
the way. This includes multiple choice questions (MCQs) that are available on the VLE,
exercises from the textbook, and any additional exercises. Worked out solutions for the
exercises can be found on the VLE (downloadable pdf files). At the end of each chapter
you will find an overview of the material covered, a list of key terms and concepts, and
12
a reminder of the learning outcomes. It concludes with additional exam based exercises
that will enable you to test your knowledge and understanding with worked out solutions.
To support students with the pre-requisites for EC2020 Elements of Econometrics,
the first chapter provides a maths and statistics refresher. Students are encouraged to
review this material and undertake the quizzes on the VLE to assess their preparation
before taking the course. Even though we will refer back to this material throughout the
course, students will benefit from reviewing this material beforehand. On the VLE you
will find accompanying MCQs to test the degree of familiarity as required for the course.
A suggested approach for students studying EC2020 Elements of Econometrics,
in particular independent learners, is to split the material in ten 2-week blocks.
1. Chapter 2 (Nature of econometrics and economic data) and Chapter 3 (Simple
regression model)
2. Chapter 4 (Multiple regression analysis: Estimation)
3. Chapter 5 (Multiple regression analysis: Inference)
4. Chapter 6 (Multiple regression analysis: Further issues and use of qualitative information)
5. Chapter 7 (Multiple regression analysis: Asymptotics), and Chapter 8 (Heteroskedasticity)
6. Chapter 9 (Instrumental variable estimation and two-stage least squares)
7. Chapter 10 (Measurement errors) and Chapter 11 (Simultaneous equation models)
8. Chapter 12 (Binary response models and Maximum Likelihood Estimation)
9. Chapter 13 (Regression analysis with time series data) and Chapter 14 (Autocorrelation in time series regression)
10. Chapter 15 (Trends, seasonality, and highly persistent time series in regression
analysis)
Throughout the course we discuss many empirical examples that help to motivate
each econometric method and should make the course more engaging. To provide you
with practical experience we have accompanied the material with an introduction to
R in RStudio and provide R applications that show how to implement the approaches
alongside. The subject guide will refer to this material where appropriate. Although not
examinable, it provides you, in our opinion, with valuable transferable skills.
13
0.5.4
Online study resources
In addition to the subject guide and the Essential reading, it is crucial that you take advantage of the study resources that are available online for this course, including the
VLE and the Online Library.
A free online supplement accompanying the textbook is available at
www.cengagebrain.com
It provides students access to a solutions manual which includes detailed steps and solutions to odd-numbered problems and computer exercises in the text.
You can access the VLE, the Online Library and your University of London email account
via the Student Portal at: http://my.londoninternational.ac.uk
You should have received your login details for the Student Portal with your official
offer, which was emailed to the address that you gave on your application form. You
have probably already logged into the Student Portal in order to register! As soon as
you registered, you will automatically have been granted access to the VLE, Online
Library and your fully functional University of London email account.
If you forget your login details at any point, please email uolia.support@london.ac.uk
quoting your student number.
0.5.5
The VLE
The VLE, which complements this subject guide, has been designed to enhance your
learning experience, providing additional support and a sense of community. It forms an
important part of your study experience with the University of London and you should
access it regularly.
The VLE provides a range of resources for EMFSS courses:
• Electronic study materials: All of the printed materials which you receive from
the University of London are available to download, to give you flexibility in how
and where you study.
• Discussion forums: An open space for you to discuss interests and seek support
from your peers, working collaboratively to solve problems and discuss subject
material. Some forums are moderated by an LSE academic.
• Videos: Recorded academic introductions to many subjects; interviews and debates with academics who have designed the courses and teach similar ones at
LSE.
14
• Recorded lectures: For a few subjects, where appropriate, various teaching sessions of the course have been recorded and made available online via the VLE.
• Audio-visual tutorials and solutions: For some of the first year and larger later
courses such as Introduction to Economics, Statistics, Mathematics and Principles
of Banking and Accounting, audio-visual tutorials are available to help you work
through key concepts and to show the standard expected in examinations.
• Self-testing activities: Allowing you to test your own understanding of subject
material.
• Study skills: Expert advice on getting started with your studies, preparing for
examinations and developing your digital literacy skills.
Note: Students registered for Laws courses also receive access to the dedicated
Laws VLE.
Some of these resources are available for certain courses only, but we are expanding
our provision all the time and you should check the VLE regularly for updates.
0.5.6
Making use of the Online Library
The Online Library (http://onlinelibrary.london.ac.uk) contains a huge array of journal
articles and other resources to help you read widely and extensively.
To access the majority of resources via the Online Library you will either need to use
your University of London Student Portal login details, or you will be required to register
and use an Athens login.
The easiest way to locate relevant content and journal articles in the Online Library
is to use the Summon search engine.
If you are having trouble finding an article listed in a reading list, try removing any
punctuation from the title, such as single quotation marks, question marks and colons.
For further advice, please use the online help pages (http://onlinelibrary.london.ac.uk/
resources/summon) or contact the Online Library team: onlinelibrary@shl.london.ac.uk
0.6 Examination advice
Important: the information and advice given here are based on the examination structure used at the time this subject guide was written. Please note that subject guides
may be used for several years. Because of this we strongly advise you to always check
both the current Programme regulations for relevant information about the examination,
and the VLE where you should be advised of any forthcoming changes. You should
15
also carefully check the rubric/instructions on the paper you actually sit and follow those
instructions.
Examination structure: The structure of the examination is as follows:
Candidates should answer eight out of 9 questions in three hours: all of the questions in
Section A (8 marks each) and three questions from Section B (20 marks each).
A calculator may be used when answering questions on this paper and it must comply
in all respects with the specification given with your Admission Notice.
Guidance on examination technique:
Remember, it is important to check the VLE for:
• up-to-date information on examination and assessment arrangements for this course
• where available, past examination papers and Examiners’ commentaries for the
course which give advice on how each question might best be answered.
16
1
Maths and statistics refresher
1.1 Introduction
The material in this chapter permits you to revise some basic mathematical tools crucial
for understanding regression analysis. It also provides a refresher of the basic statistical
theory that is used throughout the course and are central to econometric analysis.
It should be noted that the material does not replace the prerequisite courses in
mathematics, probability and statistics. By studying the essential readings listed, you
will revise the material using notation that will be used throughout the course. At the end
of each block you will be asked to complete associated quizzes on the VLE that will test
your understanding.
1.1.1
Essential reading
Readings: Wooldridge Appendices A,B and C
1.1.2
Synopsis of this chapter
The refresher is comprised of three modules and covers the key terms and concepts
indicated.
A Basic mathematical tools required for understanding regression analysis
Key terms and concepts:
• Summation Operator
• Average and Median
• Linear Function; Intercept and Slope
• Marginal effect
• Percentage Change, Percentage Point Change, and Proportionate Change
• Nonlinear Function: Quadratic, Exponential and Natural Logarithm
• Diminishing Marginal Effect
• Elasticity and Semi-Elasticity
• Derivative and Partial Derivative
B Fundamentals of probability required in econometrics
Key terms and concepts:
17
• Random Variable (discrete and continuous)
• Probability Density Function (pdf) and Cumulative Distribution Function (cdf)
• Joint Distributions and Independence
• Conditional Distribution
• Expected Value, Median, Variance and Standard Deviation
• Covariance and Correlation
• Conditional Expectation and Law of Iterated Expectation
• Variance of Sums of Random Variables
• Normal Distribution, Standard Normal Distribution, t Distribution, F Distribution and Chi-Squared distribution.
• Degrees of Freedom
C Fundamentals of mathematical statistics required in econometrics
Key terms and concepts:
• Population versus Sample
• Random Sampling
• Estimator and Estimate
• Unbiasedness of an Estimator and Bias
• Sampling Variance of an Estimator (Efficiency)
• Mean Squared Error (MSE)
• Consistency of an Estimator
• Sufficient conditions of Consistency
• Law of Large Numbers (LLN)
• Asymptotic Normality of an Estimator
• Central Limit Theorem (CLT)
• Estimators: Ordinary Least Squares, Maximum Likelihood Estimation
• Confidence Interval (Interval Estimation)
• Fundamentals of Hypothesis Testing
– Null and Alternative Hypothesis
– One-sided and Two-sided Alternative
– Type I and Type II Error
18