Stock Market Prices Do Not Follow Random Walks: Evidence From A Simple Specification Test

Andrew W. Lo

A. Craig MacKinlay

NBER Working Paper Series

National Bureau Of Economic Research

1050 Massachusetts Avenue

Cambridge, MA 02138

February 1987


In this paper, 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 (1962-1985) and for all sub-periods for a variety of aggregate returns indexes and size—sorted portfolios. Although the rejections are largely due to the behavior of small stocks, they cannot be ascribed to either the effects of infrequent trading or time-varying volatilities. Moreover, the rejection of the random walk cannot be interpreted as supporting a mean—reverting stationary model of asset prices, but is more consistent with a specific nonstationary alternative hypothesis.

Stock Market Prices Do Not Follow Random Walks: Evidence From A Simple Specification Test – Introduction

Since Keynes’ now famous pronouncement in his General Theory 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,”1 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” hypothesis for common stocks. Although several seemingly anomalous departures from market efficiency have been well-documented,3 many financial economists would agree with Jensen’s (1978) belief that “there is no other proposition in economics which has more solid empirical evidence supporting it than the Efficient Markets Hypothesis.”

Although the precise formulation of an empirically refutable efficient markets hypothesis is obviously 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 statistically reject the random walk model. However, several recent papers have uncovered new empirical evidence which suggests that stock returns contain stationary or mean-reverting components. For example, Keim and Stambaugh (1986) find statistically significant predictable components in stock prices using forecasts based upon certain predetermined variables. In addition, Fama and French (1986) show that long holding-period returns are significantly negatively serially correlated, implying that 25 to 15 percent of’ the variation of longer-horizon returns is predictable from past returns.

In this paper, we provide further evidence that stock prices do not follow random walks by using a simple specification test based upon variance estimators. Our empirical results indicate that the random walk model is generally not consistent with the stochastic behavior of weekly returns, especially for the smaller capitalization stocks. However, in contrast to the negative serial correlation which Fama and French (1986) find for longer horizon returns, we find significant positive serial correlation for weekly and monthly holding-period returns. For example, using 1216 weekly observations from September 6, 1962 to December 26, 1985 we compute the weekly first-order autocorrelation coefficient of the equal-weighted CRSP index to be 30 percent! This empirical puzzle becomes even more striking when we show that it cannot possibly be attributed to either the effects of infrequent trading or heteroscedasticity.

Of course, these results do not necessarily imply that the stock market is inefficient or that prices are not rational assessments of ‘fundamental’ values. As Leroy (1973) and Lucas (1978) have shown, rational expectations equilibrium prices need not even form a martingale sequence, of which the random walk is a special case. Therefore, without a more explicit economic model of the price—generating mechanism, a rejection of the random walk hypothesis has few implications for the efficiency of market price
formation. Although our test results may be interpreted as a rejection of some economic model of efficient price formation, there may exist other plausible models which are consistent with the empirical findings. Our more modest goal in this study is to employ a test which is capable of distinguishing among an interesting set of alternative stochastic price processes.

Stock market prices

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