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Statistical Analysis of Price Series Obscured by Averaging Measures

Published online by Cambridge University Press:  19 October 2009

Barr Rosenberg
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Extract

When measures such as the average or the midrange are used to report a typical value for the price series in each interval, the stochastic character of the underlying price process is subtly transformed. Fortunately, the spurious serial dependence introduced by averaging measures is sufficiently well understood to allow direct tests of many hypotheses to be made from averaged data. Moreover, a simple autoregressive transformation of the averaged data can be used to unscramble the effects of averaging on the lower-frequency components of the spectrum of the underlying process. These statistical devices are presented and are then illustrated by applications to the Cowles Commission Common- Stock Indexes, a massive collection of New York Stock Exchange price indexes tabulated in the form of monthly midranges.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 6 , Issue 4 , September 1971 , pp. 1083 - 1094
Copyright
Copyright © School of Business Administration, University of Washington 1971

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References

1 Alfred Cowles and Associates, Common-Stock Indexes (Bloomington, Indiana: Principia Press, 1939).Google Scholar

2 Cowles, Alfred and Jones, Herbert, “Some A Posteriori Probabilities in Stock Market Action,” Econometrica, 5 (July 1937), pp. 280–294.CrossRefGoogle Scholar

3 Cowles, Alfred, “A Revision of Previous Conclusions Regarding Stock Price Behavior,” Econometrica, 28 (October 1960), pp. 909–915CrossRefGoogle Scholar; and Holbrook Working, “Note on the Correlation of First Differences of Averages in a Random Chain,” in the same issue, pp. 916–918.

4 Daniels, H. E., “Autocorrelation between First Differences of Mid-Ranges,” Econometrica, 34 (January 1966), pp. 215–219.CrossRefGoogle Scholar

5 Rosenberg, Barr, “The Distribution of the Midrange — A Comment,” Econometrica (January 1970).Google Scholar

6 Ibid.

7 Kendall, Maurice and Stuart, Alan, The Advanced Theory of Statistics, Vol. III (New York: Hafner Publishing Co., 1966), p. 432.Google Scholar

8 This figure is the standard error of the estimated kurtosis for a sample of 815 independent, normally distributed observations, Kendall and Stuart, Vol. I, 2nd Edition, p. 243. The actual null hypothesis is that the first differences are distributed with kurtosis equal to 2.798, and with a first-order serial correlation of .315 and higher-order serial correlations equal to zero. The lower null kurtosis will tend to reduce the standard error slightly, while the first-order serial correlation may increase it somewhat. Neither effect is important here, for the estimated kurtosis differs from the null value by more than 75 times the standard error.

9 Excepting end effects, the transfer function is

The expression for the amplitude factor, , and the expression for the phase shift follows directly from this.

10 The amplitude factor and phase lead for the filter (with the factor γ included) were calculated from the formulas in the text. The amplitude factor for the midrange recording procedure was calculated by applying the midrange procedure to an actual sinusoid, taking first differences, and comparing the amplitude of these to the amplitude of first differences in the sinusoid. The odd periodicity of N + .2 recording intervals was chosen so that the ends of recording intervals would not always coincide with the extreme values of the sinusoids, since in this case the midrange reduces to the average of the beginning and ending prices in all recording intervals.