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### Summary

### Summary

For over half a century, financial experts have regarded the movements of markets as a random walk--unpredictable meanderings akin to a drunkard's unsteady gait--and this hypothesis has become a cornerstone of modern financial economics and many investment strategies. Here Andrew W. Lo and A. Craig MacKinlay put the Random Walk Hypothesis to the test. In this volume, which elegantly integrates their most important articles, Lo and MacKinlay find that markets are not completely random after all, and that predictable components do exist in recent stock and bond returns. Their book provides a state-of-the-art account of the techniques for detecting predictabilities and evaluating their statistical and economic significance, and offers a tantalizing glimpse into the financial technologies of the future.

The articles track the exciting course of Lo and MacKinlay's research on the predictability of stock prices from their early work on rejecting random walks in short-horizon returns to their analysis of long-term memory in stock market prices. A particular highlight is their now-famous inquiry into the pitfalls of "data-snooping biases" that have arisen from the widespread use of the same historical databases for discovering anomalies and developing seemingly profitable investment strategies. This book invites scholars to reconsider the Random Walk Hypothesis, and, by carefully documenting the presence of predictable components in the stock market, also directs investment professionals toward superior long-term investment returns through disciplined active investment management.

### Author Notes

Andrew W. Lo is the Harris & Harris Group Professor of Finance at the Sloan School of Management, Massachusetts Institute of Technology

A. Craig MacKinlay is Joseph P. Wargrove Professor of Finance at the Wharton School, University of Pennsylvania

### Reviews 1

### Choice Review

The phrase "random walk" was initially applied to security prices by Paul Samuelson in 1965 but was popularized by Burton Malkiel, whose A Random Walk Down Wall Street is a popular press classic currently in its sixth edition (CH, Apr'96). Malkiel explained how successive stock prices are independent of each other, and because stock prices patterns are virtually nonexistent, studying past price behavior would not lead to superior investment results. The obvious investment strategy becomes buy-and-hold, which minimizes transaction costs and taxes, or the acquisition of index funds that track the market. This book mimics Malkiel's title, but any similarity in volumes immediately ends after the introduction. Instead, in this collection of their research papers Lo (Sloan School, MIT) and MacKinlay (Wharton School, Univ. of Pennsylvania) shows security prices do not necessarily follow a random walk (i.e., they are predictable). While the authors readily admit achieving superior investment returns remains exceedingly difficult, their position is that such performance is possible. Since this book brings together important research on efficient financial markets, it should be in every library serving graduate programs in investments. General readers and most undergraduates will find most of the material virtually unfathomable. H. Mayo; The College of New Jersey

### Table of Contents

List of Figures | p. xiii |

List of Tables | p. xv |

Preface | p. xxi |

1 Introduction | p. 3 |

1.1 The Random Walk and Efficient Markets | p. 4 |

1.2 The Current State of Efficient Markets | p. 6 |

1.3 Practical Implications | p. 8 |

Part I p. 13 | |

2 Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test | p. 17 |

2.1 The Specification Test | p. 19 |

2.1.1 Homoskedastic Increments | p. 20 |

2.1.2 Heteroskedastic Increments | p. 24 |

2.2 The Random Walk Hypothesis for Weekly Returns | p. 26 |

2.2.1 Results for Market Indexes | p. 27 |

2.2.2 Results for Size-Based Portfolios | p. 30 |

2.2.3 Results for Individual Securities | p. 32 |

2.3 Spurious Autocorrelation Induced by Nontrading | p. 34 |

2.4 The Mean-Reverting Alternative to the Random Walk | p. 38 |

2.5 Conclusion | p. 39 |

Appendix A2 Proof of Theorems | p. 41 |

3 The Size and Power of the Variance Ratio Test in Finite Samples: A Monte Carlo Investigation | p. 47 |

3.1 Introduction | p. 47 |

3.2 The Variance Ratio Test | p. 49 |

3.2.1 The IID Gaussian Null Hypothesis | p. 49 |

3.2.2 The Heteroskedastic Null Hypothesis | p. 52 |

3.2.3 Variance Ratios and Autocorrelations | p. 54 |

3.3 Properties of the Test Statistic under the Null Hypotheses | p. 55 |

3.3.1 The Gaussian IID Null Hypothesis | p. 55 |

3.3.2 A Heteroskedastic Null Hypothesis | p. 61 |

3.4 Power | p. 68 |

3.4.1 The Variance Ratio Test for Large q | p. 69 |

3.4.2 Power against a Stationary AR(1) Alternative | p. 70 |

3.4.3 Two Unit Root Alternatives to the Random Walk | p. 73 |

3.5 Conclusion | p. 81 |

4 An Econometric Analysis of Nonsynchronous Trading | p. 85 |

4.1 Introduction | p. 85 |

4.2 A Model of Nonsynchronous Trading | p. 88 |

4.2.1 Implications for Individual Returns | p. 90 |

4.2.2 Implications for Portfolio Returns | p. 93 |

4.3 Time Aggregation | p. 95 |

4.4 An Empirical Analysis of Nontrading | p. 99 |

4.4.1 Daily Nontrading Probabilities Implicit in Autocorrelations | p. 101 |

4.4.2 Nontrading and Index Autocorrelations | p. 104 |

4.5 Extensions and Generalizations | p. 105 |

Appendix A4 Proof of Propositions | p. 108 |

5 When Are Contrarian Profits Due to Stock Market Overreaction? | p. 115 |

5.1 Introduction | p. 115 |

5.2 A Summary of Recent Findings | p. 118 |

5.3 Analysis of Contrarian Profitability | p. 121 |

5.3.1 The Independently and Identically Distributed Benchmark | p. 124 |

5.3.2 Stock Market Overreaction and Fads | p. 124 |

5.3.3 Trading on White Noise and Lead-Lag Relations | p. 126 |

5.3.4 Lead-Lag Effects and Nonsynchronous Trading | p. 127 |

5.3.5 A Positively Dependent Common Factor and the Bid-Ask Spread | p. 130 |

5.4 An Empirical Appraisal of Overreaction | p. 132 |

5.5 Long Horizons Versus Short Horizons | p. 140 |

5.6 Conclusion | p. 142 |

Appendix A5 p. 143 | |

6 Long-Term Memory in Stock Market Prices | p. 147 |

6.1 Introduction | p. 147 |

6.2 Long-Range Versus Short-Range Dependence | p. 149 |

6.2.1 The Null Hypothesis | p. 149 |

6.2.2 Long-Range Dependent Alternatives | p. 152 |

6.3 The Rescaled Range Statistic | p. 155 |

6.3.1 The Modified R/S Statistic | p. 158 |

6.3.2 The Asymptotic Distribution of Q[subscript n] | p. 160 |

6.3.3 The Relation Between Q[subscript n] and Q[subscript n] | p. 161 |

6.3.4 The Behavior of Q[subscript n] Under Long Memory Alternatives | p. 163 |

6.4 R/S Analysis for Stock Market Returns | p. 165 |

6.4.1 The Evidence for Weekly and Monthly Returns | p. 166 |

6.5 Size and Power | p. 171 |

6.5.1 The Size of the R/S Test | p. 171 |

6.5.2 Power Against Fractionally-Differenced Alternatives | p. 174 |

6.6 Conclusion | p. 179 |

Appendix A6 Proof of Theorems | p. 181 |

Part II p. 185 | |

7 Multifactor Models Do Not Explain Deviations from the CAPM | p. 189 |

7.1 Introduction | p. 189 |

7.2 Linear Pricing Models, Mean-Variance Analysis, and the Optimal Orthogonal Portfolio | p. 192 |

7.3 Squared Sharpe Measures | p. 195 |

7.4 Implications for Risk-Based Versus Nonrisk-Based Alternatives | p. 196 |

7.4.1 Zero Intercept F-Test | p. 197 |

7.4.2 Testing Approach | p. 198 |

7.4.3 Estimation Approach | p. 206 |

7.5 Asymptotic Arbitrage in Finite Economies | p. 208 |

7.6 Conclusion | p. 212 |

8 Data-Snooping Biases in Tests of Financial Asset Pricing Models | p. 213 |

8.1 Quantifying Data-Snooping Biases With Induced Order Statistics | p. 215 |

8.1.1 Asymptotic Properties of Induced Order Statistics | p. 216 |

8.1.2 Biases of Tests Based on Individual Securities | p. 219 |

8.1.3 Biases of Tests Based on Portfolios of Securities | p. 224 |

8.1.4 Interpreting Data-Snooping Bias as Power | p. 228 |

8.2 Monte Carlo Results | p. 230 |

8.2.1 Simulation Results for [theta subscript p] | p. 231 |

8.2.2 Effects of Induced Ordering on F-Tests | p. 231 |

8.2.3 F-Tests With Cross-Sectional Dependence | p. 236 |

8.3 Two Empirical Examples | p. 238 |

8.3.1 Sorting By Beta | p. 238 |

8.3.2 Sorting By Size | p. 240 |

8.4 How the Data Get Snooped | p. 243 |

8.5 Conclusion | p. 246 |

9 Maximizing Predictability in the Stock and Bond Markets | p. 249 |

9.1 Introduction | p. 249 |

9.2 Motivation | p. 252 |

9.2.1 Predicting Factors vs. Predicting Returns | p. 252 |

9.2.2 Numerical Illustration | p. 254 |

9.2.3 Empirical Illustration | p. 256 |

9.3 Maximizing Predictability | p. 257 |

9.3.1 Maximally Predictable Portfolio | p. 258 |

9.3.2 Example: One-Factor Model | p. 259 |

9.4 An Empirical Implementation | p. 260 |

9.4.1 The Conditional Factors | p. 261 |

9.4.2 Estimating the Conditional-Factor Model | p. 262 |

9.4.3 Maximizing Predictability | p. 269 |

9.4.4 The Maximally Predictable Portfolios | p. 271 |

9.5 Statistical Inference for the Maximal R[subscript 2] | p. 273 |

9.5.1 Monte Carlo Analysis | p. 273 |

9.6 Three Out-of-Sample Measures of Predictability | p. 276 |

9.6.1 Naive vs. Conditional Forecasts | p. 276 |

9.6.2 Merton's Measure of Market Timing | p. 279 |

9.6.3 The Profitability of Predictability | p. 281 |

9.7 Conclusion | p. 283 |

Part III p. 285 | |

10 An Ordered Probit Analysis of Transaction Stock Prices | p. 287 |

10.1 Introduction | p. 287 |

10.2 The Ordered Probit Model | p. 290 |

10.2.1 Other Models of Discreteness | p. 294 |

10.2.2 The Likelihood Function | p. 294 |

10.3 The Data | p. 295 |

10.3.1 Sample Statistics | p. 297 |

10.4 The Empirical Specification | p. 307 |

10.5 The Maximum Likelihood Estimates | p. 310 |

10.5.1 Diagnostics | p. 316 |

10.5.2 Endogeneity of [Delta]t[subscript k] and IBS[subscript k] | p. 318 |

10.6 Applications | p. 320 |

10.6.1 Order-Flow Dependence | p. 321 |

10.6.2 Measuring Price Impact Per Unit Volume of Trade | p. 322 |

10.6.3 Does Discreteness Matter? | p. 331 |

10.7 A Larger Sample | p. 338 |

10.8 Conclusion | p. 344 |

11 Index-Futures Arbitrage and the Behavior of Stock Index Futures Prices | p. 347 |

11.1 Arbitrage Strategies and the Behavior of Stock Index Futures Prices | p. 348 |

11.1.1 Forward Contracts on Stock Indexes (No Transaction Costs) | p. 349 |

11.1.2 The Impact of Transaction Costs | p. 350 |

11.2 Empirical Evidence | p. 352 |

11.2.1 Data | p. 353 |

11.2.2 Behavior of Futures and Index Series | p. 354 |

11.2.3 The Behavior of the Mispricing Series | p. 360 |

11.2.4 Path Dependence of Mispricing | p. 364 |

11.3 Conclusion | p. 367 |

12 Order Imbalances and Stock Price Movements on October 19 and 20, 1987 | p. 369 |

12.1 Some Preliminaries | p. 370 |

12.1.1 The Source of the Data | p. 371 |

12.1.2 The Published Standard and Poor's Index | p. 372 |

12.2 The Constructed Indexes | p. 373 |

12.3 Buying and Selling Pressure | p. 378 |

12.3.1 A Measure of Order Imbalance | p. 378 |

12.3.2 Time-Series Results | p. 380 |

12.3.3 Cross-Sectional Results | p. 381 |

12.3.4 Return Reversals | p. 385 |

12.4 Conclusion | p. 387 |

Appendix A12 | p. 389 |

A12.1 Index Levels | p. 389 |

A12.2 Fifteen-Minute Index Returns | p. 393 |

References | p. 395 |

Index | p. 417 |