Properties of a simple bilinear stochastic model: Estimation and predictability

Abstract

We analyze the properties of arguably the simplest bilinear stochastic multiplicative process, proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise. By construction, it has no linear predictability (zero two-point correlation) but a certain nonlinear predictability (non-zero three-point correlation). It can thus be considered as a paradigm for testing the existence of a possible nonlinear predictability in a given time series. We present a rather exhaustive study of the process, including its ability to produce fat-tailed distribution from Gaussian innovations, the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super-exponential sensitivity of the inversion of the innovations in the presence of a large impulse. Our study emphasizes the conditions under which a degree of predictability can be achieved and describe a number of different attempts, which overall illuminates the properties of the process. In conclusion, notwithstanding its remarkable simplicity, the bilinear stochastic process exhibits remarkably rich and complex behavior, which makes it a serious candidate for the modeling of financial time series among others. IntroductionDaily asset returns in liquid markets exhibit two key statistical properties: (i) price changes are not auto-correlated beyond a few minutes, but (ii) the absolute values of changes are autocorrelated over long time scales leading to long-term persistence of the volatility. Hsieh (1995) noted that nonlinear processes can generate this type of behavior while linear process cannot. Volatility clustering, also called ARCH effect (Engle, 1982) is a clear manifestation of the existence of non-linear dependences between returns observed at different lags. (FI)-GARCH (Baillie et al

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