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Stationarity of Random Data: Some Implications for the Distribution of Stock Price Changes

Published online by Cambridge University Press:  19 October 2009

Bruce D. Fielitz
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Extract

This paper has discussed the importance of stationary data in statistical applications and has at the same time suggested one method for testing for stationarity. An application of the testing procedure is made to common stock prices. The results indicate that these data could be nonstationary in the usual sense of stability of the mean and mean-square values despite efforts to transform the data into a stationary form using first differences.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 6 , Issue 3 , June 1971 , pp. 1025 - 1034
Copyright
Copyright © School of Business Administration, University of Washington 1971

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References

1 Bendat, Julius S. and Piersol, Allan G., Measurement and Analysis of Random Data (New York: John Wiley and Sons, Inc., 1966), pp. 219–223Google Scholar.

2 Kendall, M.G. and Stuart, Allan, The Advanced Theory of Statistics (London: Charles Griffen and Company, Ltd., 1966), Vol. II, pp. 477–480, Vol.III, pp. 357–358.Google Scholar

3 If the mean value and the autocorrelation of a series do not vary over time, the data are said to be weakly stationary. For the case where all possible moments and joint moments are invariant over time, the data are said to be strongly stationary. For many practical applications, vertification of weak stationarity will justify an assumption of strong stationarity.

4 Bendat and Piersol, Measurement and Analysis, p. 219.

5 To insure that the R observations of the sequence, say , are independent observations of the same random variable, Xstr, one should allow a space between each interval or skip intervals so that the data are effectively independent of adjacent intervals.

6 For any set of time ordered values, say , the number of reversals, A, in magnitude of the set may range from 0 to 1/2 R(R-1) if the order of magnitude of Xstr is of 1, 2,…, R or R, (R−1),…, 1, respectively. Also the possible orderings in magnitude of Xstr, is a permutation of 1, 2,…, R, or in other words, R!

7 Bendat and Piersol, Measurement and Analysis, pp. 221–222.

8 Ibid., p. 171.

9 Kendall, and Stuart, , Advanced Theory, Vol. II, pp. 477–480, Vol III, pp. 357–358.Google Scholar

10 Moore, Arnold, “A Statistical Analysis of Common Stock Prices,” unpublished Ph.D. dissertation, University of Chicago, 1962Google Scholar.

11 The assumptions discussed earlier indicate that if the mean and mean-square values are stable, the mean and autocorrelation function are stationary, or in other words, the weak condition of stationarity holds.

12 Since Var , it follows directly that even if the mean values are stable, when the mean-square values are not stable, the variance is not stable.

13 Kendall, M. G., “The Analysis of Economic Time – Series — Part I: Prices,” The Random Character of Stock Market Prices, ed. Cootner, Paul H., (Cambridge, Mass.: The M.I.T. Press, 1964), pp. 85–99Google Scholar.

14 Osborne, M.F.M., “Brownian Motion in the Stock Market,” Operations Research, VII (March–April 1959), pp. 145–173CrossRefGoogle Scholar.

15 Alexander, Sidney S., “Price Movements in Speculative Markets: Trends or Random Walks,” Industrial Management Review, II (May 1961), pp. 7–26Google Scholar.

16 Mandelbrot, Benoit, “The Variation of Certain Speculative Prices,” The Random Character of Stock Market Prices, ed. Cootner, Paul H., (Cambridge, Mass.: The M.I.T. Press, 1964), pp. 307–332Google Scholar.

17 Fama, Eugene, “The Behavior of Stock Market Prices,” Journal of Business, XXXVIII (January 1965), pp. 34–105CrossRefGoogle Scholar.

18 Hagin, Robert L., “An Empirical Evaluation of Selected Hypotheses Related to Price Changes in the Stock Market,” unpublished Ph.D. dissertation, U.C.L.A., 1966Google Scholar.

19 Mandelbrot, “Variation,” pp. 307–332.

20 Fama, “Behavior.” pp. 34–105.

21 Hagin, “Empirical Evaluation.”

22 Fama, “Behavior,” pp. 34–105.

23 One should note that the above presented empirical results of instability in the variance of stock price changes do not necessarily establish an infinite variance in the price changes. A referree has pointed out that a random walk that has two or more step sizes, and selected step size by other than random means, would produce an instable, though not an infinite, variance. However, if the. Distribution characterizing stock price changes does indeed belong to the non-Gaussian family of stable Paretian distributions, the above described test will enable the researcher to detect the empirical instability of the variance which is characteristic of the non-Gaussian stable Paretian process.

24 Fama, “Behavior,” pp. 34–105.

25 Cootner, Paul H., The Random Character of Stock Market Prices (Cambridge, Mass.: The M.I.T. Press, 1964)Google Scholar.

26 Fama, “Behavior,” pp. 34–105.

27 Bruce D. Fielitz, “On the Behavior of Stock Price Relatives as a Random Process with an Application to New York Stock Exchange Prices,” unpublished D.B.A. dissertation, Kent State University, 1969.