F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym A Non-Random Walk Down Wall Street F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym A Non-Random Walk Down Wall Street Andrew W. Lo A. Craig MacKinlay Princeton University Press Princeton and Oxford F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym Copyright O 1999 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock. Oxfordshire OX20 1SY All Rights Reserved Fifth printing, and first paperback printing, 2002 Paperback ISBN 0-691-09256-7 The Library of Congress has cataloged the cloth edition of this book as follows Lo, Andrew W. (Andrew Wen-Chuan) A non-random walk down Wall Street / Andrew W. Lo and A. Craig MacKinlay. p. cm. Includes bibliographical references and index. ISBN 0-69 1-05774-5 (alk. paper) 1. stocks-prices-~athescal models. 2. Random walks (Mathematics) I. MacKinlay, Archie Craig, 1955- . 11. Title. HG4915.L6 1999 332.63'2224~21 98-31390 British Library Cataloging-in-Publication Data is available This book was composed in ITC New Baskerville with LATEX by Archetype Publishing Inc., 15 Turtle Pointe Road, Monticello, IL 61856 Printed on acid-free paper. - Printed in the United States of America ISBN-13: 978-0-691-09256-0 (pbk.) F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym To my mother AWL To my parents ACM F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym Contents List of Figures xiii List of Tables xv Preface xzci 1 Introduction 3 1.1 The Random Walk and Efficient Markets . . . . . . . . . . 4 1.2 The Current State of Efficient Markets . . . . . . . . . . . 6 1.3 Practical Implications . . . . . . . . . . . . . . . . . . . . . 8 Part I 2 Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test 17 2.1 The Specification Test . . . . . . . . . . . . . . . . . . . . 19 2.1.1 Homoskedastic Increments . . . . . . . . . . . . . 20 2.1.2 Heteroskedastic Increments . . . . . . . . . . . . . 24 2.2 The Random Walk Hypothesis for Weekly Returns . . . . . 26 2.2.1 Results for Market Indexes . . . . . . . . . . . . . . 27 2.2.2 Results for SizeBased Portfolios . . . . . . . . . . . 30 2.2.3 Results for Individual Securities . . . . . . . . . . . 32 2.3 Spurious Autocorrelation Induced by Nontrading . . . . . 34 2.4 The Mean-Reverting Alternative to the Random Walk . . . 38 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Appendix A2: Proof of Theorems 41 F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym ... Vll l contents 3 The Size and Power of the Variance Ratio Test in Finite Samples: A Monte Carlo Investigation 47 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 The Variance Ratio Test . . . . . . . . . . . . . . . . . . . 49 3.2.1 The IID Gaussian Null Hypothesis . . . . . . . . . . 49 3.2.2 The Heteroskedastic Null Hypothesis . . . . . . . . 52 3.2.3 Variance Ratios and Autocorrelations . . . . . . . . 54 3.3 Properties of the Test Statistic under the Null Hypotheses . 55 3.3.1 The Gaussian IID Null Hypothesis . . . . . . . . . . 55 3.3.2 A Heteroskedastic Null Hypothesis . . . . . . . . . 61 3.4 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.1 The Variance Ratio Test for Large q . . . . . . . . . 69 3.4.2 Power against a Stationary AR(1) Alternative . . . . 70 3.4.3 Two Unit Root Alternatives to the Random Walk . . 73 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4 An Econometric Analysis of Nonsynchronous Trading 85 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 A Model of Nonsynchronous Trading . . . . . . . . . . . . 88 4.2.1 Implications for Individual Returns . . . . . . . . . 90 4.2.2 Implications for Portfolio Returns . . . . . . . . . . 93 4.3 Time Aggregation . . . . . . . . . . . . . . . . . . . . . . 95 4.4 An Empirical Analysis of Nontrading . . . . . . . . . . . . 99 4.4.1 Daily Nontrading Probabilities Implicit in Auto- correlations . . . . . . . . . . . . . . . . . . . . . . . 1 01 4.4.2 Nontrading and Index Autocorrelations . . . . . . 104 4.5 Extensions and Generalizations . . . . . . . . . . . . . . . 105 Appendix A4: Proof of Propositions 108 5 When Are Contrarian Profits Due to Stock Market Overreaction? 115 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 A Summary of Recent Findings . . . . . . . . . . . . . . . 118 5.3 Analysis of Contrarian Profitability . . . . . . . . . . . . . 121 5.3.1 The Independently and Identically Distributed Bench- mark . . . . . . . . . . . . . . . . . . . . . . . . . . .12 4 5.3.2 Stock Market Overreaction and Fads . . . . . . . . 124 5.3.3 Trading on White Noise and Lead-Lag Relations . 126 5.3.4 Lead-Lag Effects and Nonsynchronous Trading . . 127 5.3.5 A Positively Dependent Common Factor and the Bid-Askspread . . . . . . . . . . . . . . . . . . . . .1 30 5.4 An Empirical Appraisal of Overreaction . . . . . . . . . . 132 F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym Contents ix 5.5 Long Horizons Versus Short Horizons . . . . . . . . . . . 140 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Appendix A5 143 6 Long-Term Memory in Stock Market Prices 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Long-Rangeversus Short-Range Dependence . . . . . . . 6.2.1 The Null Hypothesis . . . . . . . . . . . . . . . . . 6.2.2 Long-Range Dependent Alternatives . . . . . . . . 6.3 The Rescaled Range Statistic . . . . . . . . . . . . . . . . . 6.3.1 The Modified R/S Statistic . . . . . . . . . . . . . . 6.3.2 The Asymptotic Distribution of Q, . . . . . . . . . . 6.3.3 The Relation Between Q, and . . . . . . . . . . 6.3.4 The Behavior of 4, Under Long Memory Alternatives . . . . . . . . . . . . . . . . . . . . . . 6.4 R/S Analysis for Stock Market Returns . . . . . . . . . . . 6.4.1 The Evidence for Weekly and Monthly Returns . . . 6.5 Size and Power . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 TheSizeoftheRISTest . . . . . . . . . . . . . . . 6.5.2 Power Against Fractionally-Differenced Alternatives 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A6: Proof of Theorems Part I1 185 7 Multifactor Models Do Not Explain Deviations from the CAPM 189 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.2 Linear Pricing Models. Mean-Variance Analysis. and the Optimal Orthogonal Portfolio . . . . . . . . . . . .1 92 7.3 Squared Sharpe Measures . . . . . . . . . . . . . . . . . . 195 7.4 Implications for Risk-Based Versus Nonrisk-Based Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . .1 96 7.4.1 Zero Intercept F-Test . . . . . . . . . . . . . . . . . 197 7.4.2 Testing Approach . . . . . . . . . . . . . . . . . . . 198 7.4.3 Estimation Approach . . . . . . . . . . . . . . . . . 206 7.5 Asymptotic Arbitrage in Finite Economies . . . . . . . . . 208 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 212 8 Data-Snooping Biases in Tests of F i c i a l A sset Pricing Models 213 8.1 Quantifying Data-Snooping Biases With Induced Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 15 8.1.1 Asymptotic Properties of Induced Order Statistics . 216 8.1.2 Biases of Tests Based on Individual Securities . . . . 219 F©o Cr ogpeynreigrahlt ,q Puerirnieces,t ocno nUtnaicvte wrseitbym