Detecting Linear and Nonlinear Dependence in Stock Returns: New Methods Derived from Chaos Theory

Approaches to describing the behavior ofstock prices were long dominated by a simple model, a geometric random walk with uncorrelated innovations (Fama, 1970). An implication of this model is that stock returns are independent and identically distributed (iid) random variables. Early tests found little evidence of economically significant short-horizon autocorrelations and predictability, thereby supporting the weak form of the efficient markets hypothesis, which specifies that the price histories of stocks contain no useful information on future prices (for a review of the early literature, see Fama, 1970). Later research, however, produced a substantial body ofliterature challenging the adequacy ofthis model. Lo and MacKinlay (1988) and Conrad and Kaul (1989) calculated positive autocorrelations for short-horizon returns, while evidence of the predictability of long-horizon returns was developed by Shiller (1984), DeBondt and Thaler (1985), Summers (1986), Fama and French (1988), Poterba and Summers (1988) and Jegadeesh (1991), among others (see Fama, 1991, for a survey of this research). Other deviations from the random walk model have been explained by calendar anomalies, such as day-of-the-week effects (French, 1980; and Ariel, 1987 and 1990) and the January effect, especially for small stocks (Keim, 1983; Roll, 1983; and Lakonishok and Smidt, 1988). Recent research into an explanation ofstock return behavior has drawn on the field of nonlinear dynamics, including chaos theory. Nonlinearity has been of interest in part because it can generate a richer range of behavior than can linear models. If, as argued by Ramsey (1989), simple linear models are inadequate to describe economic and financial time series, then exploration of more complex, nonlinear types of models may reveal additional useful information about the behavior of stock returns. Research undertaken on nonlinear structure in stock returns includes the use of a bilinear model by Granger and Andersen (1978) and the bispectrum of daily returns for several common stock series by Hinich and Patterson (1985). Models of conditional heteroskedasticity in stock returns have been developed by French et al.