Andrew W. Lo - A Non-Random Walk Down Wall Street

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. StocksPricesMathematical models. 2. Random walks

(Mathematics) I. MacKinlay, Archie Craig, 1955 . II. Title.

HG4915.L6 1999

332.63'222dc21

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

ISBN-13: 978-0-691-09256-0 (pbk.)

To my mother

AWL

To my parents

ACM

List of Figures

First-order autocorrelation of temporally aggregated observed individual and portfolio returns as a function of the per period nontrading probability p, where q is the aggregation value and

Loci of nontrading probability pairs (pa, pb) that imply a constant cross-autocorrelation .10, .15, .20, .25, k = 1, q = 5

Cross-autocorrelation pqab(k) as a function of pa and pb, for q = 5, k=l

Distribution and density function of the range V of a Brownian bridge

Autocorrelograms of equally-weighted CRSP daily and monthly stock returns indexes and fractionally-differenced process with d = 1/4

Distributions for the CAPM zero-intercept test statistic for four alternatives with monthly data

Distributions for the CAPM zero-intercept test statistic for four alternatives with weekly data

Illustration of ordered probit probabilities pi of observing a price change of si ticks, which are determined by where the unobservable virtual price change Z*k falls

Histograms of price changes, time-between-trades, and dollar volume for the period from January 4, 1988, to December 30, 1988

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)

Percentage price impact as a function of dollar volume computed from ordered probit probabilities, conditional on the three most recent trades being buyer-initiated, and the three most recent price changes being +1 tick each for the period from January 4, 1988, to December 30, 1988

Discreteness matters

Mispricing (percent of index value) for (a) December 1984 and (b) March 1987 S&P 500 futures contracts

Boundary of mean absolute mispricing as a function of time to maturity

Comparison of price indexes for NYSE stocks included in the S&P 500 Index and not included for October 19 and 20, 1987

Plot of dollar volume in each fifteen-minute interval on October 19 and 20, 1987 as a percent of the market value of the stocks outstanding separately for S&P and non-S&P stocks

Plot of fifteen-minute returns on S&P stocks versus the order imbalance in S&P stocks in the same fifteen minutes for October 19, 1987

Plot of fifteen-minute returns on S&P stocks versus the order imbalance in the same fifteen minutes for October 20, 1987

Plot of fifteen-minute returns on non-S&P stocks versus the order imbalance in the same fifteen minutes for October 19, 1987

Plot of fifteen-minute returns on non-S&P stocks versus the order imbalance in the same fifteen minutes for October 20, 1987

Comparison of various constructed indexes measuring the S&P Composite Index with the published S&P Index on October 19, 1987

List of Tables

Variance-ratio test of the random walk hypothesis for CRSP equal- and value-weighted indexes

Market index results for a four-week base observation period

Variance-ratio test of the random walk hypothesis for sizesorted portfolios

Means of variance ratios over all individual securities with complete return histories from September 2, 1962, to December 26, 1985 (625 stocks)

Spuriously induced autocorrelations are reported for nontrading probabilities 1 - p of 10 to 50 percent

Empirical sizes of nominal 1, 5, and 10 percent two-sided variance ratio tests of the random walk null hypothesis with homoskedastic disturbances

Empirical quantiles of the (Dickey-Fuller) t-statistic

Empirical sizes of nominal 1, 5, and 10 percent two-sided variance ratio tests of the random walk null hypothesis with homoskedastic disturbances

Empirical quan tiles of the (asymptotically) N( 0, 1) variance ratio test statistic z1 (q) under simulated IID Gaussian random walk increments

Power of the two-sided variance ratio test

Sample first-order autocorrelation matrix for the 5 1 sub vector of observed returns to twenty equally-weighted size-sorted portfolios

Estimates of daily nontrading probabilities implicit in 20 weekly and monthly size-sorted portfolio return autocorrelations

Estimates of the first-order autocorrelation m of weekly returns of an equal-weighted portfolio of twenty size-sorted portfolios

Sample statistics

Averages of autocorrelation coefficients for weekly returns on individual securities, for the period July 6, 1962, to December 31, 1987

Analysis of the profitability of the return-reversal strategy applied to weekly returns, for the sample of 551 CRSP NYSEAMEX stocks with nonmissing weekly returns from July 6, 1962, to 31 December 1987 (1330 weeks)

Autocorrelation matrices

Comparison of autocorrelation functions

Fractiles of the distribution Fv(v)

R/S analysis of daily equal- and value-weighted CRSP stock returns indexes from July 3, 1962, to December 31, 1987

R/S analysis of monthly equal- and value-weighted CRSP stock returns indexes from January 30, 1926, to December 31, 1987

Finite sample distribution of the modified R/S statistic under an IID null hypothesis

Power of the modified R/S statistic under a Gaussian fractionally differenced alternative with differencing parameter d = 1/3

Historical Sharpe measures for selected stock indices, where the Sharpe measure is defined as the ratio of the mean excess return to the standard deviation of the excess return

A comparison of the maximum squared Sharpe measure for two economies denoted A and B

Theoretical sizes of nominal 5 percent -tests of (i = 1,, n) for individual securities