A Test for Strong Hysteresis

Laura Piscitelli - University of Strathclyde

The mathematical definition of systems with hysteresis, that is nonlinear input-output systems with memory, is different from the kind of system typically found in economics. Economic theory (and modelling practice) has almost always specified simple dynamic systems with regular leads and lags in their responses, corresponding to input-output systems with unit or zero (or at least stable) roots. These models cannot capture the "selective memory" feature of hysteretic behaviour, that is the influence of certain past events only (typically, non-dominated sequences of previous peaks and troughs). There is therefore a difficulty in testing for and validating economic models which contain hysteretic behaviour. The appropriate empirical tests simply have not been developed. In particular, the usual unit vs. zero (or stable) root tests used in econometric analysis are unable to detect hysteretic behaviour, or to distinguish it from more conventional economic behaviour. The purpose of this paper is to propose a new way of testing for hysteresis, by drawing on some ideas in the mathematical/control theory literature and adapting them to fit into the typical economic framework with elements of hysteresis.

The paper proposes the following two-step procedure: a) to compute from x, the relevant input variable, the outcomes of the potentially hysteretic behaviour h(x) given x; b) we then test the significance of the hysteresis in the output variable of interest y by testing the significance of h(x) in the regression of y on h(x) and possibly other variables. To give concrete form to this test, the paper shows how it works in the case of the Preisach model, that is a model of strong hysteresis in which the hysteretic behaviour emerges from a system of microeconomic decisions, each of them optimising but influenced in different ways by current and past behaviour to give the "selective memory" feature which is characteristic of hysteresis. A fundamental role is therefore played by the distribution of the micro-foundations elements within the macroeconomic outcomes. The hysteresis variable in this case can be computed assuming it contains uniform, normal, Poisson or exponentially distributed stochastic terms, and the sensitivity of the test to different distributional assumptions is compared. An important development here would be to extend this sensitivity analysis by some Monte Carlo studies, and the use of order statistics.

An application of the previously specified procedure is realised by using a set of european exchange rates against the US dollar. Results suggest a wider importance of hysteresis in the dynamics of economic variables than previously thought.

Scheduled for Session 1.1 Computation And Econometrics - I