A Non-Random Walk Down Wall Street

436 Pages·1999·19.03 MB·English

by  Andrew W. Lo

A Non-Random Walk Down Wall Street
Andrew W. Lo
A. Craig MacKinlay
Princeton University Press
Princeton and Oxford
Copyright © 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 0X20 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 0-691-05774-5 (alk. paper)
1. Stocks-Prices-Mathematical models. 2. Random walks
(Mathematics) I. MacKinlay, Archie Craig, 1955- . II. Title.
HG4915.L6 1999
332.63'222-dc21 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.
www.pup.princeton.edu
Printed in the United States of America
10 9 8 7 6 5
T~ my mother
ΑWL
To my parents
ACM
Contents
List of Fig~~res x~~~
List of Tables xv
Preface xxi
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 13
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 Size-Based 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
vii
v~~~ Contents
3 The Size and Power of the Variance Ratio Test in Finite Samples:
AMonte 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 Econometrίc 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 . 101
4.4.2 Nontrading and Index Autocorrelations 104
4.5 Extensions and Generalizations 105
Appendix A4: Proof of Propositions 108
5 When Are Cοηtrarian 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 124
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-Ask Spread . . 130
5.4 An Empirical Appraisal of Overreaction 132
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 147
6.1 Introduction 147
6.2 Long-Range Versus Short-Range Dependence 149
6.2.1 The Null Hypothesis 149
6.2.2 Long-Range Dependent Alternatives 152
6.3 The Rescaled Range Statistic . 155
6.3.1 The Modified R/S Statistic 158
6.3.2 The Asymptotic Distribution of Q„ . 160
6.3.3 The Relation Between Qn and Q,, 161
6.3.4 The Behavior of Qom, Under Long Memory
Alternatives . 163
6.4 R/S Analysis for Stock Market Returns 165
6.4.1 The Evidence for Weekly and Monthly Returns . 166
6.5 Size and Power 171
6.5.1 The Size of the R/STest 171
6.5.2 Power Against Fractionally-Differenced Alternatives 174
6.6 Conclusion 179
Appendix A6: Proof of Theorems 181
Part II 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 . 192
7.3 Squared Sharpe Measures 195
7.4 Implications for Risk-Based Versus Nonrisk-Based
Alternatives . 196
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 Financial Asset Pricing Models 213
8.1 Quantifying Data-Snooping Biases With Induced Order
Statistics . . 215
8.1.1 Asymptotic Properties of Induced Order Statistics 216
8.1.2 Biases of Tests Based on Individual Securities . 219
x Contents
8.1.3 Biases of Tests Based on Portfolios of Securities 224
8.1.4 Interpreting Data-Snooping Bias as Power 228
8.2 Monte Carlo Results 230
8.2.1 Simulation Results for ~~ 231
8.2.2 Effects of Induced Ordering on F-Tests 231
8.2.3 F-Tests With Cross-Sectional Dependence . 236
8.3 Two Empirical Examples 238
8.3.1 Sorting By Beta 238
8.3.2 Sorting By Size 240
8.4 How the Data Get Snooped 243
8.5 Conch~sion 246
9 Maximizing Predictability in the Stock and Bond Markets 249
9.1 Introduction 249
9.2 Motivation . 252
9.2.1 Predicting Factors vs. Predicting Returns 252
9.2.2 Numerical Illustration 254
9.2.3 Empirical Illustration 256
9.3 Maximizing Predictability 257
9.3.1 Maximally Predictable Portfolio 258
9.3.2 Example: One-Factor Model . 259
9.4 An Empirical Implementation 260
9.4.1 The Conditional Factors 261
9.4.2 Estimating the Conditional-Factor Model 262
9.4.3 Maximizing Predictability 269
9.4.4 The Maximally Predictable Portfolios 271
9.5 Statistical Inference for the Maximal R2 . 273
9.5.1 Monte Carlo Analysis 273
9.6 Three Out-of-Sample Measures of Predictability 276
9.6.1 Naive vs. Conditional Forecasts 276
9.6.2 Merton's Measure of Market Timing 279
9.6.3 The Profitability of Predictability 281
9.7 Conclusion 283
Part III 285
10 An Ordered Probit Analysis of Transaction Stock Prices 287
10.1 Introduction 287
10.2 The Ordered Probit Model 290
10.2.1 Other Models of Discreteness 294
10.2.2 The Likelihood Function . 294
10.3 The Data 295
10.3.1 Sample Statistics . 297
10.4 The Empirical Specification 307
Contents xi
10.5 The Maximum Likelihood Estimates 310
10.5.1 Diagnostics 316
10.5.2 Endogeneity of ~ tk and IBSk 318
10.6 Applications . 320
10.6.1 Order-Flow Dependence 321
10.6.2 Measuring Price Impact Per Unit Volume of Trade . 322
10.6.3 Does Discreteness Matter? 331
10.7 A Larger Sample 338
10.8 Conclusion 344
11 Index-Futures Arbitrage and the Behavior of Stock Index Futures
Prices 347
11.1 Arbitrage Strategies and the Behaυior of Stock Index
Futures Prices . . 348
11.1.1 Forward Contracts on Stock Indexes (No Transac-
tion Costs) . 349
11.1.2 The Impact of Transaction Costs . 350
11.2 Empirical Evidence 352
11.2.1 Data 353
11.2.2 Behavior of Futures and Index Series 354
11.2.3 The Behavior of the Mispricing Series . 360
11.2.4 Path Dependence of Mispricing 364
11.3 Conclusion 367
12 Order Imbalances and Stock Price Movements on October 19 and
20, 1987 369
12.1 Some Preliminaries 370
12.1.1 The Source of the Data . 371
12.1.2 The Published Standard and Poor's Index 372
12.2 The Constructed Indexes 373
12.3 Buying and Selling Pressure 378
12.3.1 A Measure of Order Imbalance 378
12.3.2 Time-Series Results 380
12.3.3 Cross-Sectional Results 381
12.3.4 Return Reversals 385
12.4 Conclusion 387
Appendix A12 389
A12.1 Index Levels . 389
A12.2 Fifteen-Minute Index Returns 393
References 395
Index 417
List of Figures
4.1 First-order autocorrelation of temporally aggregated observed
individual and portfolio returns as a function of the per pe-
riod nontrading probability p, where q is the aggregation
value and ~ _ /-~/~ 97
5.1 Loci of nontrading probability pairs (pa, pb) that imply a con-
stant cross-autocorrelation ~áb(k), for ~ab(k) _ .05, .10, .15,
.20, .25, k = 1, q = 5 138
5.2 Cross-autocorrelation pab(k) as a function of p~ and pb, for
q=5,k=1 139
6.1 Distribution and density function of the range V of a Brow-
nian bridge 162
6.2 Autocorrelograms of equally-weighted CRSP daily and
monthly stock returns indexes and fractionally-differenced
process with d = 1/4 170
7.1 Distributions for the CAPM zero-intercept test statistic for
four alternatives with monthly data 203
7.2 Distributions for the CAPΜ zero-intercept test statistic for
four alternatives with weekly data 206
10.1 Illustration of ordered probit probabilities p= of observing a
price change of si ticks, which are determined by where the
unobservable "virtual" price change Zk falls 293
10.2 Histograms of price changes, time-between-trades, and dol-
lar volume for the period from January 4, 1988, to December
30, 1988 301
10.3 Comparison of estimated ordered probit probabilities of price
change, conditioned on a sequence of increasing prices
(1/1/1) versus a sequence of constant prices (0/0/0) 327
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