The Basics of Financial Econometrics

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4 The Frank J. Fabozzi Series Fixed Income Securities, Second Edition by Frank J. Fabozzi Focus on Value: A Corporate and Investor Guide to Wealth Creation by James L. Grant and James A. Abate Handbook of Global Fixed Income Calculations by Dragomir Krgin Managing a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. Fabozzi Real Options and Option-Embedded Securities by William T. Moore Capital Budgeting: Theory and Practice by Pamela P. Peterson and Frank J. Fabozzi The Exchange-Traded Funds Manual by Gary L. Gastineau Professional Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. Fabozzi Investing in Emerging Fixed Income Markets edited by Frank J. Fabozzi and Efstathia Pilarinu Handbook of Alternative Assets by Mark J. P. Anson The Global Money Markets by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry The Handbook of Financial Instruments edited by Frank J. Fabozzi Interest Rate, Term Structure, and Valuation Modeling edited by Frank J. 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5 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

6 Cover image: hana / Datacraft / Getty Images Cover design: Wiley Copyright 2014 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmit ted in any form or by any means, electronic, mechanical, photocopying, recording, scan ning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) , outside the United States at (317) , or fax (317) Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at Library of Congress Cataloging-in-Publication Data: ISBN (Hardcover) ISBN (epdf) ISBN (epub) Typeset in 10/12 pt Sabon LT Std by Aptara Printed in the United States of America

7 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

9 Contents Preface Acknowledgments About the Authors xiii xvii xix Chapter 1 Introduction 1 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 13 The Role of Correlation 13 Regression Model: Linear Functional Relationship between Two Variables 14 Distributional Assumptions of the Regression Model 16 Estimating the Regression Model 18 Goodness-of-Fit of the Model 22 Two Applications in Finance 25 Linear Regression of a Nonlinear Relationship 36 Key Points 38 CHAPTER 3 Multiple Linear Regression 41 The Multiple Linear Regression Model 42 Assumptions of the Multiple Linear Regression Model 43 Estimation of the Model Parameters 43 Designing the Model 45 Diagnostic Check and Model Significance 46 Applications to Finance 51 Key Points 79 vii

10 viii Contents chapter 4 Building and Testing a Multiple Linear Regression Model 81 The Problem of Multicollinearity 81 Model Building Techniques 84 Testing the Assumptions of the Multiple Linear Regression Model 88 Key Points 100 CHAPTER 5 Introduction to Time Series Analysis 103 What Is a Time Series? 103 Decomposition of Time Series 104 Representation of Time Series with Difference Equations 108 Application: The Price Process 109 Key Points 113 chapter 6 Regression Models with Categorical Variables 115 Independent Categorical Variables 116 Dependent Categorical Variables 137 Key Points 140 Chapter 7 Quantile Regressions 143 Limitations of Classical Regression Analysis 144 Parameter Estimation 144 Quantile Regression Process 146 Applications of Quantile Regressions in Finance 148 Key Points 155 CHAPTER 8 Robust Regressions 157 Robust Estimators of Regressions 158 Illustration: Robustness of the Corporate Bond Yield Spread Model 161 Robust Estimation of Covariance and Correlation Matrices 166 Applications 168 Key Points 170 Chapter 9 Autoregressive Moving Average Models 171 Autoregressive Models 172 Moving Average Models 176 Autoregressive Moving Average Models 178

11 Contents ix ARMA Modeling to Forecast S&P 500 Weekly Index Returns 181 Vector Autoregressive Models 188 Key Points 189 Chapter 10 Cointegration 191 Stationary and Nonstationary Variables and Cointegration 192 Testing for Cointegration 196 Key Points 211 chapter 11 Autoregressive Heteroscedasticity Model and Its Variants 213 Estimating and Forecasting Volatility 214 ARCH Behavior 215 GARCH Model 223 What Do ARCH/GARCH Models Represent? 226 Univariate Extensions of GARCH Modeling 226 Estimates of ARCH/GARCH Models 229 Application of GARCH Models to Option Pricing 230 Multivariate Extensions of ARCH/GARCH Modeling 231 Key Points 233 Chapter 12 Factor Analysis and Principal Components Analysis 235 Assumptions of Linear Regression 236 Basic Concepts of Factor Models 237 Assumptions and Categorization of Factor Models 240 Similarities and Differences between Factor Models and Linear Regression 241 Properties of Factor Models 242 Estimation of Factor Models 244 Principal Components Analysis 251 Differences between Factor Analysis and PCA 259 Approximate (Large) Factor Models 261 Approximate Factor Models and PCA 263 Key Points 264 Chapter 13 Model Estimation 265 Statistical Estimation and Testing 265 Estimation Methods 267 Least-Squares Estimation Method 268 The Maximum Likelihood Estimation Method 278

12 x Contents Instrumental Variables 283 Method of Moments 284 The M-Estimation Method and M-Estimators 289 Key Points 289 CHAPTER 14 Model Selection 291 Physics and Economics: Two Ways of Making Science 291 Model Complexity and Sample Size 293 Data Snooping 296 Survivorship Biases and Other Sample Defects 297 Model Risk 300 Model Selection in a Nutshell 301 Key Points 303 Chapter 15 Formulating and Implementing Investment Strategies Using Financial Econometrics 305 The Quantitative Research Process 307 Investment Strategy Process 314 Key Points 318 Appendix A Descriptive Statistics 321 Basic Data Analysis 321 Measures of Location and Spread 328 Multivariate Variables and Distributions 332 Appendix B Continuous Probability Distributions Commonly Used in Financial Econometrics 343 Normal Distribution 344 Chi-Square Distribution 347 Student s t-distribution 349 F -Distribution 352 α-stable Distribution 353 Appendix C Inferential Statistics 359 Point Estimators 359 Confidence Intervals 369 Hypothesis Testing 372

13 Contents xi Appendix D Fundamentals of Matrix Algebra 385 Vectors and Matrices Defined 385 Square Matrices 387 Determinants 388 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 399 Akaike Information Criterion 400 Bayesian Information Criterion 402 Appendix F Robust Statistics 405 Robust Statistics Defined 405 Qualitative and Quantitative Robustness 406 Resistant Estimators 406 M-Estimators 408 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 Index 413

15 Preface 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 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 1 An Introduction to Financial Econometrics, Unpublished paper, Department of Operations Research and Financial Engineering, Princeton University, xiii

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Robust Equity Portfolio Management + Website The Frank J. Fabozzi Series Fixed Income Securities, Second Edition by Frank J. Fabozzi Focus on Value: A Corporate and Investor Guide to Wealth Creation by