The Time-Variance Relationship: Evidence on Autocorrelation in Common Stock Returns

Abstract

IT IS WELL ESTABLISHED THAT, under stationarity and intertemporal independence, the variance of returns has a precise functional relationship with,the differencing interval. However, in the voluminous literature on the properties of returns distributions, relatively scant attention has been paid to the time-variance relationship as a test of intertemporal returns independence, although, as far back as 1949, Holbrook Working [27] suggested such an application. The only exceptions appear to be Mandelbrot and Wallis [16] who mention the time-variance relationship as a tool for investigating long cycle dependence with variable periodicity, and Poole {21] and Young [28] who use it in empirical tests (using flexible exchange rates, and common stock returns, respectively). The time-variance relationship may be an especially robust tool for random walk analysis since, like spectral analysis, it does not require a priori specification of precise autocorrelation schemes. Furthermore the relationship can be used to gain insight into a variety of issues such as: the impact of specialist trading on stock price volatility (see Barnea [3] and Schwartz and Whitcomb [24]); the variance of net present value under dependent cash flow streams (where the time-variance relationship provides an alternative to the Markovian model of Bussey and Stevens [5]); and the finding by Altman, Jacquillat and Levasseur [2] and Pogue and Solnik [20] that the market model R2 falls as the differencing interval is shortened.In Section I, we allow for a general autoregressive structure, and thereby determine the effect of autocorrelation patterns on the time-variance relationship for both total returns and market model residuals. We also show that autocorrelation accounts for the observed lower market model R2 for shorter differencing intervals. Section II reports the results of our tests for autocorrelation on a sample of NYSE stocks. Section II contains our concluding remarks. The effects of nonstationarity and a special case multiple order autocorrelation scheme are considered in, respectively, Appendices A and B. I. THE TIME-VARIANCE RELATIONSHIPConsider a differencing interval of T years, broken into shorter intervals n years long. Then, the T year interval contains T/ n (integer) short periods, t = 1,..., T/n. Let the random variable r, be the (log) return, expressed as a rate per n over the tth short period, r, = log,[(P, + D,)/P, ,J. Return per annum over the T year differenc-

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