Testing the Random Walk Hypothesis

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Statistical Methods & Capital Markets
Testing Random Walk Hypothesis
Nicolas Mancini

* Table of Content

Abstract
Theoretical background
Methodology
Data & Results
Comparison
Conclusion
References

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I. Abstract

The aim of this paper is to test the random walk hypothesis by applying the runs test on time series of several selected stocks. The random walk theory is the theory that stock prices changes have the same distribution and are independent of each other, so the past movement or trend of a stock price or market cannot be used to predict its future movement. Shortly said it is the idea that stocks take a random and unpredictable path. The motivation behind this work is to analyze whether the random walk hypothesis is valid in different periods of time. Therefore we will use recent data and more distant data in order to draw a conclusion. Furthermore it could be interesting to compare results for companies that are not specifically active the same industry. The runs test will be carried out with the R software, a free software, though really powerful, for environmental and statistical computings and graphics. II. Theoretical background

Introduced for the first time by Burton Malkiel in his top-seller « A Random Walk Down Wall Street » in 1973, the model of random walk might be a reasonable approximation to the true dynamics of stock log prices. Random walk time series are time series {xt : t = 0,1,2,…} starting with x0, where any value over time can be calculated as follows : xt = xt-1 + at (for t >= 0) and where {at : t = 0,1,2,…} are sequences of independent and identically distributed random variables with zero mean.

The independence of increments implies that the probability of the increment at is not influenced by past values. A mathematical formulation for this would be : P(at at-1, at-2,…) = P(at). The identical distribution of increments means that the probability distribution of the random variables at (for t=>0) remains unchanged regardless of t. The increments are often assumed to be Gaussian, in other words to be normally distributed. The random walk hypothesis is expressed unofficially as: ‘’In efficient markets stock prices are random walks’’. Under efficient markets we understand markets where all participants (traders, investors, banks…) are thinking rationally and where they will only act in order to maximize profit. We also assume that there is no assymetric information, i.e. every participant has acces to all the important information at the same time. Furthermore the stock prices reflects all the available information, so that means that every security traded on the market is correctly valued given the available information.

Note that in case of modelling the random walk, we will use the logs of the stock prices rather than the stock prices itself due to suitability reasons. The log is more convenient because it is time additive, which means that a two-period log return is identical to the sum of the two individual log returns. To get the n-period log return, you can simply add the consecutive single period returns. Simple returns, conversely, are not time additive. Moreover, the logarithms and exponents are mathematically more convenient and easier to work with. Finally the logs are a good approximation of the reality (discrete returns), especially in short periods of time (i.e. daily). Claiming that log returns are idependent and identically distributed is just a rough assumption but making profit out of the knowledge of this dependence might not be possible due to transaction costs and similar issues. Technical analysis is a method diverging from the random walk hypothesis by assuming that ‘’past patterns in the price tend to recure and thus price movement is predictable to some extent.’’ – Mgr. Milan Basta, Ph.D. The random walk theory matters because it claims that traders or investors cannot...
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