The basics of financial econometrics

The Basics
of Financial
Econometrics


The Frank J. Fabozzi Series

Fixed Income Securities, Second Edition by Frank J. Fabozzi
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Handbook of Global Fixed Income Calculations by Dragomir Krgin
Managing a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. Fabozzi
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The Basics
of Financial
Econometrics
Tools, Concepts, and Asset
Management Applications

FRANK J. FABOZZI
SERGIO M. FOCARDI
SVETLOZAR T. RACHEV
BALA G. ARSHANAPALLI
WITH THE ASSISTANCE OF

MARKUS HÖCHSTÖTTER

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10╇ 9╇ 8╇ 7╇ 6╇ 5╇ 4╇ 3╇ 2╇ 1


FJF
To my son, Francesco, who I hope will read this book
SMF
To my family
STR
To my grandchildren Iliana, Zoya, and Svetlozar
BGA
To my wife Vidya and my children Priyanka and Ashish



Contents

Prefacexiii
Acknowledgmentsxvii
About the Authors

xix

Chapter 1
Introduction1

Financial Econometrics at Work
2
The Data Generating Process
5
Applications of Financial Econometrics to Investment Management 6
Key Points
10

Chapter 2
Simple Linear Regression

The Role of Correlation
Regression Model: Linear Functional Relationship
between Two Variables
Distributional Assumptions of the Regression Model
Estimating the Regression Model
Goodness-of-Fit of the Model
Two Applications in Finance
Linear Regression of a Nonlinear Relationship
Key Points

CHAPTER 3
Multiple Linear Regression

The Multiple Linear Regression Model 
Assumptions of the Multiple Linear Regression Model
Estimation of the Model Parameters

Designing the Model 
Diagnostic Check and Model Significance
Applications to Finance
Key Points

13

13
14
16
18
22
25
36
38

41

42
43
43
45
46
51
79

vii


viii

Contents

chapter 4
Building and Testing a Multiple Linear Regression Model

The Problem of Multicollinearity 
Model Building Techniques
Testing the Assumptions of the Multiple Linear Regression Model
Key Points

CHAPTER 5
Introduction to Time Series Analysis

What Is a Time Series?
Decomposition of Time Series
Representation of Time Series with Difference Equations
Application: The Price Process
Key Points 

chapter 6
Regression Models with Categorical Variables 
Independent Categorical Variables
Dependent Categorical Variables
Key Points

Chapter 7
Quantile Regressions

Limitations of Classical Regression Analysis 

Parameter Estimation
Quantile Regression Process
Applications of Quantile Regressions in Finance
Key Points

CHAPTER 8
Robust Regressions 

81

81
84
88
100

103

103
104
108
109
113

115

116
137
140

143

144
144
146
148
155

157

Robust Estimators of Regressions
158
Illustration: Robustness of the
Corporate Bond Yield Spread Model
161
Robust Estimation of Covariance and Correlation Matrices
166
Applications168
Key Points
170

Chapter 9
Autoregressive Moving Average Models

Autoregressive Models
Moving Average Models
Autoregressive Moving Average Models

171

172

176
178


Contents

ARMA Modeling to Forecast S&P 500 Weekly Index Returns
Vector Autoregressive Models
Key Points

ix
181
188
189

Chapter 10
Cointegration191
Stationary and Nonstationary Variables and Cointegration
Testing for Cointegration 
Key Points

chapter 11
Autoregressive Heteroscedasticity Model and Its Variants

Estimating and Forecasting Volatility 
ARCH Behavior
GARCH Model
What Do ARCH/GARCH Models Represent?
Univariate Extensions of GARCH Modeling
Estimates of ARCH/GARCH Models

Application of GARCH Models to Option Pricing 
Multivariate Extensions of ARCH/GARCH Modeling
Key Points

Chapter 12
Factor Analysis and Principal Components Analysis

Assumptions of Linear Regression
Basic Concepts of Factor Models
Assumptions and Categorization of Factor Models
Similarities and Differences between
Factor Models and Linear Regression
Properties of Factor Models
Estimation of Factor Models
Principal Components Analysis
Differences between Factor Analysis and PCA
Approximate (Large) Factor Models
Approximate Factor Models and PCA
Key Points

Chapter 13
Model Estimation

Statistical Estimation and Testing
Estimation Methods
Least-Squares Estimation Method
The Maximum Likelihood Estimation Method

192
196

211

213

214
215
223
226
226
229
230
231
233

235

236
237
240
241
242
244
251
259
261
263
264

265

265
267
268
278


x

Contents

Instrumental Variables
Method of Moments 
The M-Estimation Method and M-Estimators
Key Points

CHAPTER 14
Model Selection 

Physics and Economics: Two Ways of Making Science 
Model Complexity and Sample Size 
Data Snooping
Survivorship Biases and Other Sample Defects
Model Risk
Model Selection in a Nutshell
Key Points

Chapter 15
Formulating and Implementing Investment Strategies Using
Financial Econometrics
The Quantitative Research Process

Investment Strategy Process
Key Points

Appendix A
Descriptive Statistics

283
284
289
289

291

291
293
296
297
300
301
303

305

307
314
318

321

Basic Data Analysis

Measures of Location and Spread
Multivariate Variables and Distributions

321
328
332

Appendix B
Continuous Probability Distributions Commonly Used in
Financial Econometrics

343

Normal Distribution
344
Chi-Square Distribution
347
Student’s t-Distribution349
F -Distribution352
α-Stable Distribution
353

Appendix C
Inferential Statistics

Point Estimators
Confidence Intervals
Hypothesis Testing

359

359
369
372


Contents

Appendix D
Fundamentals of Matrix Algebra

xi

385

Vectors and Matrices Defined 
385
Square Matrices
387
Determinants388
Systems of Linear Equations
389
Linear Independence and Rank
391
Vector and Matrix Operations 
391
Eigenvalues and Eigenvectors
396

APPENDIX E

Model Selection Criterion: AIC and BIC
Akaike Information Criterion
Bayesian Information Criterion

Appendix F
Robust Statistics

399

400
402

405

Robust Statistics Defined
405
Qualitative and Quantitative Robustness
406
Resistant Estimators
406
M-Estimators408
The Least Median of Squares Estimator
408
The Least Trimmed of Squares Estimator
409
Robust Estimators of the Center
409
Robust Estimators of the Spread
410
Illustration of Robust Statistics

410

Index413



Preface

E

conometrics is the branch of economics that draws heavily on statistics for
testing and analyzing economic relationships. Within econometrics, there
are theoretical econometricians who analyze statistical properties of estima­
tors of models. Several recipients of the Nobel Prize in Economic Sciences
received the award as a result of their lifetime contribution to this branch of
economics. To appreciate the importance of econometrics to the discipline of
economics, when the first Nobel Prize in Economic Sciences was awarded in
1969, the co-recipients were two econometricians, Jan Tinbergen and Ragnar
Frisch (the latter credited for first using the term econometrics in the sense
that it is known today). The co-recipient of the 2013 Nobel Prize was Lars
Peter Hansen who had made major contributions to the field of econometrics.
Further specialization within econometrics, and the area that directly
relates to this book, is financial econometrics. As Jianqing Fan writes, the
field of financial econometrics
uses statistical techniques and economic theory to address a variety
of problems from finance. These include building financial models,
estimation and inferences of financial models, volatility estimation,
risk management, testing financial economics theory, capital asset
pricing, derivative pricing, portfolio allocation, risk-adjusted returns,
simulating financial systems, hedging strategies, among others.1

Robert Engle and Clive Granger, two econometricians who shared the 2003
Nobel Prize in Economics Sciences, have contributed greatly to the field of
financial econometrics.
Why this book? There is growing demand for learning and teaching
implementation issues related to the deployment of financial econometrics
in finance. The unique feature of this book is the focus on applications and
implementation issues of financial econometrics to the testing of theories
and development of investment strategies in asset management. The key
mes�sages expressed in this book come from our years of experience in
1â•›
“An Introduction to Financial Econometrics,” Unpublished paper, Department of
Operations Research and Financial Engineering, Princeton University, 2004.

xiii


xiv

Preface

designing, developing, testing, and operating financial econometric applica­
tions in asset management.
In this book we explain and illustrate the basic tools that are needed
to implement financial econometric models. While many books describe the
abstract mathematics of asset management, the unique feature of this book is
to address the question of how to construct asset management strategies using
financial econometric tools. We discuss all aspects of this process, including
model risk, limits to the applicability of models, and the economic intuition
behind models. We describe the critical issues using real life examples.
We start by discussing the process of applying financial econometrics

to asset management. The three basic steps of model selection, estimation,
and testing are discussed at length. We emphasize how in this phase eco­
nomic intuition plays an important role. Before designing models we have to
decide what phenomena we want to exploit in managing assets.
We then discuss the most fundamental financial econometric technique:
regression analysis. Despite its apparent simplicity, regression analysis is a
powerful tool the application of which requires careful consideration. We
describe different types of regression analysis, including quantile regressions
and regressions with categorical variables, their applicability, and the condi­
tions under which regression fails. We discuss the robustness of regression
analy�sis, introducing the concept and technique of robust regression. All
concepts are illustrated with real-life examples.
Next, we analyze the dynamic behavior of time series, introducing vec�tor
and scalar autoregressive models. We formalize mean-reversion, intro�ducing
the concept of cointegration, and describe the heteroscedastic behav�ior of
financial time series. We discuss the economic intuition behind each model,
their estimation, and methods for parameter testing. We also analyze the
limits of the applicability of autoregressive techniques, the advantage of
exploiting mean reversion when feasible, and the model risk associated with
autoregressive models. We again use real-life examples to illustrate.
Subsequently, we move to consider large portfolios and discuss the tech­
niques used to model large numbers of simultaneous time series, in particu­
lar factor models and principal components analysis. The issues associated
with the estimation and testing of large models and techniques to separate
information from noise in large sets of mutually interacting time series are
discussed.
Finally, we discuss the specific process of implementing a financial
econometric model for asset management. We describe the various steps of
this process and the techniques involved in making modeling decisions.
One important characteristic of model development today is the avail­

ability of good econometric software. Many building blocks of the pro­
cess of implementing a financial econometric application are available


Preface

xv

as off-the-shelf software. Most technical tasks, from optimization to the
estimaÂ�tion of regression and autoregressive models, are performed are per­
formed by econometric software. Using these software tools has become
common practice among those who develop financial applications. For this
reason we do spend much time discussing computational issues. These are
highly technical subjects that are handled by specialists. The general user
and/or developer of econometric applications do not spend time in rewriting
appli�cations that are commercially available. For this reason we focus on
the process of designing financial econometric models and we do not handle
the computational aspects behind basic techniques.





Frank J. Fabozzi
Sergio M. Focardi
Svetlozar T. Rachev
Bala G. Arshanapalli

Acknowledgments

W

e are grateful to Markus Höchstötter for his assistance in the prepara­
tion of several chapters and Appendices A, B, and C. His contribution
was of such significance that he is identified on the cover and title pages of
this book.
Chapter 15 is coauthored with Dr. K. C. Ma of KCM Asset Manage­
ment, and Professor of Finance and Director of the George Investments
Institute and Roland George Investments Program at Stetson University.
We also thank a number of doctoral students in the Department of
Applied Mathematics and Statistics at Stony Brook University, SUNY, for
taking time from their studies to review and provide feedback for various
chapters and appendices in this book. Below we have listed each student and
the chapters/appendices reviewed:
Fangfei (Sophia) Dong
Tetsuo Kurosaki
Tiantian Li
Barret Shao
Naoshi Tsuchida
Yuzhong (Marco) Zhang

Chapters 2, 3, and 12
Chapters 8, 9, 11, and 12
Chapter 13
Chapters 7 and 12
Chapters 8, 9, and 12
Chapters 2, 3, and 12

xvii



About the Authors

Frank J. Fabozzi is Professor of Finance at EDHEC Business School and a
member of the EDHEC Risk Institute. He has held various professorial posi­
tions at Yale and MIT. In the 2013–2014 academic year, he was appointed
the James Wei Visiting Professor in Entrepreneurship at Princeton University
and since 2011 has been a Research Fellow in the Department of Opera­
tions Research and Financial Engineering at the same institution. The edi­
tor of the Journal of Portfolio Management since 1986, Professor Fabozzi
has authored and edited many books in asset management and quantitative
finance. He serves on the advisory board of The Wharton School’s Jacobs
Levy Equity Management Center for Quantitative Financial Research, the Q
Group Selec�tion Committee, and from 2003 to 2011 on the Council for the
Department of Operations Research and Financial Engineering at Princeton
University. He is a Fellow of the International Center for Finance at Yale
University. He is a trustee for the BlackRock family of closed-end funds. He
is the CFA Institute’s 2007 recipient of the C. Stewart Sheppard Award and
an inductee into the Fixed Income Analysts Society Hall of Fame. Professor
Fabozzi earned a PhD in Economics in September 1972 from the City Uni­
versity of New York and holds the professional designations of Chartered
Financial Analyst (1977) and Certified Public Accountant (1982).
Sergio M. Focardi is a Visiting Professor at Stony Brook University, SUNY,
where he holds a joint appointment in the College of Business and the Depart­
ment of Applied Mathematics and Statistics. Prior to that, he was a Professor
of Finance at the EDHEC Business School in Nice. Professor Focardi is a
founding partner of the Paris-based consulting firm The Intertek Group. A

member of the editorial board of the Journal of Portfolio Management, he
has authored numerous articles and books on financial modeling and risk
management including the following Wiley books: Mathematical Methods in
Finance (2013), Probability and Statistics for Finance (2010), Quantitative
Equity Investing: Techniques and Strategies (2010), Robust Portfolio Opti­
mization and Management (2007), Financial Econometrics (2007), Financial
Modeling of the Equity Market (2006), The Mathematics of Financial Mod­
eling and Investment Management (2004), Risk Management: Framework,

xix


xx

About The Authors

Methods, and Practice (1998), and Modeling the Markets: New Theories
and Techniques (1997). He also coauthored three monographs published by
the Research Foundation of the CFA Institute: Challenges in Quantitative
Equity Management (2008), The Impact of the Financial Crisis on the Asset
ManÂ�agement Industry (2010), Trends in Quantitative Finance (2006). Pro­
fessor Focardi holds a degree in Electronic Engineering from the University
of Ge�noa and a PhD in Mathematical Finance and Financial Econometrics
from the University of Karlsruhe.
Svetlozar (Zari) T. Rachev is holds a joint appointment at Stony Brook Uni­
versity, SUNY, as a professor in the Department of Applied Mathematics
and Statistics and the College of Business. Previously he was Chair-Professor
in Statistics, Econometrics, and Mathematical Finance at the Karlsruhe
Institute of Technology (KIT) in the School of Economics and Business Engi­
neering, and is now Professor Emeritus at the University of California, Santa

Barbara in the Department of Statistics and Applied Probability. Professor
Rachev has published 14 monographs, 10 handbooks, and special edited
volumes, and over 300 research articles. His recently coauthored books pub­
lished by Wiley in mathematical finance and financial econometrics include
Financial Models with Lèvy Processes and Volatility Clustering (2011), A
Probability Metrics Approach to Financial Risk Measures (2011), Financial
Econometrics: From Basics to Advanced Modeling Techniques (2007), and
Bayesian Methods in Finance (2008). He is cofounder of Bravo Risk Man­
agement Group specializ�ing in financial risk-management software. Bravo
Group was acquired by Fi�nAnalytica for which he currently serves as Chief
Scientist. Professor Rachev completed his PhD in 1979 from Moscow State
(Lomonosov) Uni�versity, and his Doctor of Science degree in 1986 from
Steklov Mathematical Institute in Moscow.
Bala G. Arshanapalli is professor and Gallagher-Mills Chair of Business and
Economics at Indiana University Northwest. Before joining Indiana Univer­
sity, Professor Arshanapalli was the Virginia and Harvey Hubbell Professor
of Business and Finance at University of Bridgeport. He also held visiting
appointments at Concordia University and Purdue University. He served
on the board of Legacy Foundation. He serves on the editorial board of
International Journal of Bonds and Currency Derivatives and International
Journal of Economics and Finance. Professor Arshanapalli also served on
the editorial board of European Financial Management and International
Journal of Operations and Quantitative Management. His current research
interests include asset allocation, retirement planning, and empirical meth­
ods in financial time series modeling. Professor Arshanapalli has published


About The Authors

xxi

over 45 articles and his work has appeared in the Journal of Risk and Uncer­
tainty, Journal of Banking and Finance, International Journal of Money and
Finance, Journal of Portfolio Management, and Industrial and Labor Rela­
tions Review. He has consulted and taught executive development classes
for nationally recognized companies. Professor Arshanapalli earned a PhD
in Finance in 1988 from Northern Illinois University.



Chapter

1

Introduction

A

fter reading this chapter you will understand:

What the field of financial econometrics covers.
The three steps in applying financial econometrics: model selection,
model estimation, and model testing.
■⌀ What is meant by the data generating process.
■⌀ How financial econometrics is used in the various phases of investment
management.
■⌀
■⌀

Financial econometrics is the science of modeling and forecasting financial data such as asset prices, asset returns, interest rates, financial ratios,

defaults and recovery rates on debt obligations, and risk exposure. Some
have described financial econometrics as the econometrics of financial markets. The development of financial econometrics was made possible by three
fundamental enabling factors: (1) the availability of data at any desired
frequency, including at the transaction level; (2) the availability of powerful desktop computers at an affordable cost; and (3) the availability of
off-the-shelf econometric software. The combination of these three factors
put advanced econometrics within the reach of most financial firms such as
banks and asset management firms.
In this chapter, we describe the process and the application of financial
econometrics. Financial econometrics is applied to either time series data,
such as the returns of a stock, or cross-sectional data such as the market
capitalization1 of all stocks in a given universe at a given moment. With
the progressive diffusion of high-frequency financial data and ultra highfrequency financial data, financial econometrics can now be applied to
1â•›

A firm’s market capitalization, popularly referred to as “market cap,” is a measure
of the firm’s size in terms of the total market value of its common stock. This is found
by multiplying the number of common stock shares outstanding by the price per
share of common stock.

1