Random Walk Hypothesis

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Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test Andrew W. Lo A. Craig MacKinlay University of Pennsylvania In this article we test the random walk hypothesis for weekly stock market returns by comparing variance estimators derived from data sampled at different frequencies. The random walk model is strongly rejected for the entire sample period (19621985) and for all subperiod for a variety of aggregate returns indexes and size-sorted portofolios. Although the rejections are due largely to the behavior of small stocks, they cannot be attributed completely to the effects of infrequent trading or timevarying volatilities. Moreover, the rejection of the random walk for weekly returns does not support a mean-reverting model of asset prices. Since Keynes’s (1936) now famous pronouncement that most investors’ decisions “can only be taken as a result of animal spirits-of a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of benefits multiplied by quantitative probabilities,” a great deal of research has been devoted to examining the efficiency of stock market price formation. In Fama’s (1970) survey, the vast majority of those studies were unable to reject the ‘“efficient markets” This paper has benefited considerably from the suggestions of the editor Michael Gibbons and the referee. We thank Cliff Ball, Don Keim, Whitney K. Newey, Peter Phillips, Jim Poterba, Krishna Ramaswamy. Bill Schwert, and seminar participants at MIT, the NBER-FMME Program Meeting (November 1986). Northwestern University, Ohio State University, Princeton University, Stanford University, UCLA, University of Chicago, University of Michigan, University of Pennsylvania, University of Western Ontario, and Yale University for helpful comments. We are grateful to Stephanie Hogue, Elizabeth Schmidt, and Madhavi Vinjamuri for preparing the manuscript. Research support from the Geewax-Terker Research Program in Investments, the National Science Foundation (Grant No. SES-8520054), and the University of Pennsylvania Research Fund is gratefully acknowledged. Any errors are of course our own. Address reprint requests to Andrew Lo, Department of Finance, Wharton School, University of Pennsylvania, Philadelphia, PA 19104. The Review of Financial Studies 1988, Volume 1, number 1, pp. 41-66. © 1988 The Review of Financial Studies 0021-9398/88/5904-013 $1.50

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hypothesis for common stocks. Although several seemingly anomalous departures from market efficiency have been well documented,1 many financia1 economists would agree with Jensen’s (1978a) belief that “there is no other proposition in economics which has more solid empirical evidence supporting it than the Efficient Markets Hypothesis.” Although a precise formulation of an empirically refutable efficient markets hypothesis must obviously be model-specific, historically the majority of such tests have focused on the forecastability of common stock returns. Within this paradigm, which has been broadly categorized as the “random walk” theory of stock prices, few studies have been able to reject the random walk model statistically. However, several recent papers have uncovered empirical evidence which suggests that stock returns contain predictable components. For example, Keim and Stambaugh (1986) find statistically significant predictability in stock prices by using forecasts based on certain predetermined variables. In addition, Fama and French (1987) show that long holding-period returns are significantly negatively serially correlated, implying that 25 to 40 percent of the variation of longer-horizon returns is predictable from past returns. In this article we provide further evidence that stock prices do not follow random walks by using a simple specification test based on variance estimators. Our empirical results indicate that the random walk model is generally not consistent with the stochastic behavior of weekly...
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