The Basicsof Financial

Econometrics

The Frank J. Fabozzi SeriesFixed Income Securities, Second Edition by Frank J. FabozziFocus on Value: A Corporate and Investor Guide to Wealth Creation by James L. Grant and James A. AbateHandbook of Global Fixed Income Calculations by Dragomir KrginManaging a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. FabozziReal Options and Option-Embedded Securities by William T. MooreCapital Budgeting: Theory and Practice by Pamela P. Peterson and Frank J. FabozziThe Exchange-Traded Funds Manual by Gary L. GastineauProfessional Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. FabozziInvesting in Emerging Fixed Income Markets edited by Frank J. Fabozzi and Efstathia PilarinuHandbook of Alternative Assets by Mark J. P. AnsonThe Global Money Markets by Frank J. Fabozzi, Steven V. Mann, and Moorad ChoudhryThe Handbook of Financial Instruments edited by Frank J. FabozziInterest Rate, Term Structure, and Valuation Modeling edited by Frank J. FabozziInvestment Performance Measurement by Bruce J. FeibelThe Handbook of Equity Style Management edited by T. Daniel Coggin and Frank J. FabozziThe Theory and Practice of Investment Management edited by Frank J. Fabozzi and Harry M. MarkowitzFoundations of Economic Value Added, Second Edition by James L. GrantFinancial Management and Analysis, Second Edition by Frank J. Fabozzi and Pamela P. PetersonMeasuring and Controlling Interest Rate and Credit Risk, Second Edition by Frank J. Fabozzi,

Steven V. Mann, and Moorad ChoudhryProfessional Perspectives on Fixed Income Portfolio Management, Volume 4 edited by Frank J. FabozziThe Handbook of European Fixed Income Securities edited by Frank J. Fabozzi and Moorad ChoudhryThe Handbook of European Structured Financial Products edited by Frank J. Fabozzi and Moorad ChoudhryThe Mathematics of Financial Modeling and Investment Management by Sergio M. Focardi and Frank J. FabozziShort Selling: Strategies, Risks, and Rewards edited by Frank J. FabozziThe Real Estate Investment Handbook by G. Timothy Haight and Daniel SingerMarket Neutral Strategies edited by Bruce I. Jacobs and Kenneth N. LevySecurities Finance: Securities Lending and Repurchase Agreements edited by Frank J. Fabozzi and Steven V. MannFat-Tailed and Skewed Asset Return Distributions by Svetlozar T. Rachev, Christian Menn, and Frank J. FabozziFinancial Modeling of the Equity Market: From CAPM to Cointegration by Frank J. Fabozzi,

Sergio M. Focardi, and Petter N. KolmAdvanced Bond Portfolio Management: Best Practices in Modeling and Strategies edited by Frank J. Fabozzi,

Lionel Martellini, and Philippe PriauletAnalysis of Financial Statements, Second Edition by Pamela P. Peterson and Frank J. FabozziCollateralized Debt Obligations: Structures and Analysis, Second Edition by Douglas J. Lucas,

Laurie S. Goodman, and Frank J. FabozziHandbook of Alternative Assets, Second Edition by Mark J. P. AnsonIntroduction to Structured Finance by Frank J. Fabozzi, Henry A. Davis, and Moorad ChoudhryFinancial Econometrics by Svetlozar T. Rachev, Stefan Mittnik, Frank J. Fabozzi, Sergio M. Focardi, and Teo Jasic Developments in Collateralized Debt Obligations: New Products and Insights by Douglas J. Lucas,

Laurie S. Goodman, Frank J. Fabozzi, and Rebecca J. ManningRobust Portfolio Optimization and Management by Frank J. Fabozzi, Peter N. Kolm,

Dessislava A. Pachamanova, and Sergio M. FocardiAdvanced Stochastic Models, Risk Assessment, and Portfolio Optimizations by Svetlozar T. Rachev,

Stogan V. Stoyanov, and Frank J. FabozziHow to Select Investment Managers and Evaluate Performance by G. Timothy Haight, Stephen O. Morrell,

and Glenn E. RossBayesian Methods in Finance by Svetlozar T. Rachev, John S. J. Hsu, Biliana S. Bagasheva, and Frank J. FabozziThe Handbook of Municipal Bonds edited by Sylvan G. Feldstein and Frank J. FabozziSubprime Mortgage Credit Derivatives by Laurie S. Goodman, Shumin Li, Douglas J. Lucas,

Thomas A. Zimmerman, and Frank J. FabozziIntroduction to Securitization by Frank J. Fabozzi and Vinod KothariStructured Products and Related Credit Derivatives edited by Brian P. Lancaster, Glenn M. Schultz, and

Frank J. FabozziHandbook of Finance: Volume I: Financial Markets and Instruments edited by Frank J. FabozziHandbook of Finance: Volume II: Financial Management and Asset Management edited by Frank J. FabozziHandbook of Finance: Volume III: Valuation, Financial Modeling, and Quantitative Tools edited by Frank J. Fabozzi Finance: Capital Markets, Financial Management, and Investment Management by Frank J. Fabozzi and

Pamela Peterson-DrakeActive Private Equity Real Estate Strategy edited by David J. LynnFoundations and Applications of the Time Value of Money by Pamela Peterson-Drake and Frank J. FabozziLeveraged Finance: Concepts, Methods, and Trading of High-Yield Bonds, Loans, and Derivatives

by Stephen Antczak, Douglas Lucas, and Frank J. FabozziModern Financial Systems: Theory and Applications by Edwin NeaveInstitutional Investment Management: Equity and Bond Portfolio Strategies and Applications by Frank J. Fabozzi

The Basicsof Financial

EconometricsTools, 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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Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:ISBN 978-1-118-57320-4 (Hardcover)ISBN 978-1-118-72743-0 (ePDF)ISBN 978-1-118-72723-2 (ePub)

Typeset in 10/12 pt Sabon LT Std by Aptara

Printed in the United States of America.10 9 8 7 6 5 4 3 2 1

FJFTo my son, Francesco, who I hope will read this book

SMFTo my family

STRTo my grandchildren Iliana, Zoya, and Svetlozar

BGATo my wife Vidya and my children Priyanka and Ashish

vii

Preface xiii

Acknowledgments xvii

About the Authors xix

ChAPter 1Introduction 1

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

ChAPter 2Simple Linear regression 13

The Role of Correlation 13Regression Model: Linear Functional Relationship

between Two Variables 14Distributional Assumptions of the Regression Model 16Estimating the Regression Model 18Goodness-of-Fit of the Model 22Two Applications in Finance 25Linear Regression of a Nonlinear Relationship 36Key Points 38

ChAPter 3Multiple Linear regression 41

The Multiple Linear Regression Model 42Assumptions of the Multiple Linear Regression Model 43Estimation of the Model Parameters 43Designing the Model 45Diagnostic Check and Model Significance 46Applications to Finance 51Key Points 79

Contents

viii Contents

ChAPter 4Building and testing a Multiple Linear regression Model 81

The Problem of Multicollinearity 81Model Building Techniques 84Testing the Assumptions of the Multiple Linear Regression Model 88Key Points 100

ChAPter 5Introduction to time Series Analysis 103

What Is a Time Series? 103Decomposition of Time Series 104Representation of Time Series with Difference Equations 108Application: The Price Process 109Key Points 113

ChAPter 6regression Models with Categorical Variables 115

Independent Categorical Variables 116Dependent Categorical Variables 137Key Points 140

ChAPter 7Quantile regressions 143

Limitations of Classical Regression Analysis 144Parameter Estimation 144Quantile Regression Process 146Applications of Quantile Regressions in Finance 148Key Points 155

ChAPter 8robust regressions 157

Robust Estimators of Regressions 158Illustration: Robustness of the

Corporate Bond Yield Spread Model 161Robust Estimation of Covariance and Correlation Matrices 166Applications 168Key Points 170

ChAPter 9Autoregressive Moving Average Models 171

Autoregressive Models 172Moving Average Models 176Autoregressive Moving Average Models 178

Contents ix

ARMA Modeling to Forecast S&P 500 Weekly Index Returns 181Vector Autoregressive Models 188Key Points 189

ChAPter 10Cointegration 191

Stationary and Nonstationary Variables and Cointegration 192Testing for Cointegration 196Key Points 211

ChAPter 11Autoregressive heteroscedasticity Model and Its Variants 213

Estimating and Forecasting Volatility 214ARCH Behavior 215GARCH Model 223What Do ARCH/GARCH Models Represent? 226Univariate Extensions of GARCH Modeling 226Estimates of ARCH/GARCH Models 229Application of GARCH Models to Option Pricing 230Multivariate Extensions of ARCH/GARCH Modeling 231Key Points 233

ChAPter 12Factor Analysis and Principal Components Analysis 235

Assumptions of Linear Regression 236Basic Concepts of Factor Models 237Assumptions and Categorization of Factor Models 240Similarities and Differences between

Factor Models and Linear Regression 241Properties of Factor Models 242Estimation of Factor Models 244Principal Components Analysis 251Differences between Factor Analysis and PCA 259Approximate (Large) Factor Models 261Approximate Factor Models and PCA 263Key Points 264

ChAPter 13Model estimation 265

Statistical Estimation and Testing 265Estimation Methods 267Least-Squares Estimation Method 268The Maximum Likelihood Estimation Method 278

x Contents

Instrumental Variables 283Method of Moments 284The M-Estimation Method and M-Estimators 289Key Points 289

ChAPter 14Model Selection 291

Physics and Economics: Two Ways of Making Science 291Model Complexity and Sample Size 293Data Snooping 296Survivorship Biases and Other Sample Defects 297Model Risk 300Model Selection in a Nutshell 301Key Points 303

ChAPter 15Formulating and Implementing Investment Strategies Using Financial econometrics 305

The Quantitative Research Process 307Investment Strategy Process 314Key Points 318

APPendIx Adescriptive Statistics 321

Basic Data Analysis 321Measures of Location and Spread 328Multivariate Variables and Distributions 332

APPendIx BContinuous Probability distributions Commonly Used in Financial econometrics 343

Normal Distribution 344Chi-Square Distribution 347Student’s t-Distribution 349F -Distribution 352α-Stable Distribution 353

APPendIx CInferential Statistics 359

Point Estimators 359Confidence Intervals 369Hypothesis Testing 372

Contents xi

APPendIx dFundamentals of Matrix Algebra 385

Vectors and Matrices Defined 385Square Matrices 387Determinants 388Systems of Linear Equations 389Linear Independence and Rank 391Vector and Matrix Operations 391Eigenvalues and Eigenvectors 396

APPendIx eModel Selection Criterion: AIC and BIC 399

Akaike Information Criterion 400Bayesian Information Criterion 402

APPendIx Frobust Statistics 405

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

Index 413

xiii

Econometrics 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 estima­tors 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 co­recipients 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 co­recipient 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

Preface

1 “An Introduction to Financial Econometrics,” Unpublished paper, Department of Operations Research and Financial Engineering, Princeton University, 2004.