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Encyclopedia of Complexity and Systems Science pp 3504–3524Cite as

Financial Forecasting, Sensitive Dependence

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Empirical studies show that there are at least some components in future asset returns that are predictable using information that is currentlyavailable. When the linear time series models are employed in prediction, the accuracy of the forecast does not depend on the current return or theinitial condition. In contrast, with nonlinear time series models, properties of the forecast error depend on the initial value or the history. Theeffect of the difference in initial values in a stable nonlinear model, however, usually dies out quickly as the forecast horizon increases. For bothdeterministic and stochastic cases, the dynamic system is chaos if a small difference in the initial value is amplified at an exponential rate. Ina chaotic nonlinear model, the reliability of the forecast can decrease dramatically even for a moderate forecast horizon. Thus, the knowledgeof the sensitive dependence on initial conditions in a particular financial time series offers...

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Abbreviations

Global Lyapunov exponent:

A global stability measure of the nonlinear dynamic system. It is a long‐run average of the exponential growth rate of infinitesimally small initial deviation and is uniquely determined in the ergodic and stationary case. In this sense, this initial value sensitivity measure does not depend on the initial value. A system with positive Lyapunov exponents is considered chaotic for both deterministic and stochastic cases.

Local Lyapunov exponent:

A local stability measure based on a short‐run average of the exponential growth rate of infinitesimally small initial deviations. Unlike the global Lyapunov exponent, this initial value sensitivity measure depends both on the initial value and the horizon for the average calculation. A smaller local Lyapunov exponent implies a better performance at the point of forecast.

Noise amplification:

In a stochastic system with the additive noise, the effect of shocks can either grow, remain, or die out with the forecast horizon. If the system is nonlinear, this effect depends both on the initial value and size of the shock. For a chaotic system, the degree of noise amplification is so high that it makes the forecast almost identical to the iid forecast within the next few steps ahead.

Nonlinear impulse response function:

In a stochastic system with the additive noise, the effect of shocks on the variable in subsequent periods can be summarized in impulse response functions. If the system is linear, the impulse response does not depend on the initial value and its shape is proportional to the size of shocks. If the system is nonlinear, however, the impulse response depends on the initial value, or the history, and its shape is no longer proportional to the size of shocks.

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Authors and Affiliations

  1. Vanderbilt University, Nashville, USA

    Mototsugu Shintani

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  1. Mototsugu Shintani
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Shintani, M. (2009). Financial Forecasting, Sensitive Dependence. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_209

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