Impacts of Multiple Period Lags in Dynamic Logic Models

This paper will provide an introduction to a new field of research, viz the sensitivity of the solution trajectory of a dynamic logit model (belonging to the class of discrete choice models) in the light of a multi-period lag structure. It is well known from recent advances in the area of chaos and turbulence theory that the stability of a dynamic system is critically dependent on various factors, such as threshold values of parameters, initial conditions and also the lag structure. This paper aims to identify the consequences of different lag structures in dynamic logit models (including also dynamic spatial interaction models). Various simulation experiments will be used to show that the onset of instability of the solution trajectory tends to decrease as the number of time lags increases (depending also on the growth rate of the system). Acknowledgement The second author would like to thank Andrea Gerali of the University of Bergamo for computer assistance. Also the CNR grant n.90.01247.CT11 is greatfully acknowledged. 1Theorv of Turhulence in Social Sciences Modelling tradition in the social sciences was usually based on linear static systerns models. Sometimes also dynamic linear models were used in order to describe the growth or decline of certain phenomena, but non-linear dynamic models were rather an exception. Although linear dynamic models are not necessarily very restrictive for well defined and regular movements of phenomena, they have severe shortcomings in case of highly irregular movements (e.g., in case of non-periodic evolution; see Broek 1986). It is in this context that the theory of chaos or turbulence has recently become an important analytical tooi. An important feature of chaos theory is that it is essentially concerned with deterministic, non-linear dynamic systerns which are able to produce complex motions of such a nature that they are sometimes seemingly random. In particular, they incorporate the feature that small uncertainties may grow exponentially (although all time paths are bound), leading to a broad spectrum of different trajectories in the long run, so that precise or plausible predictions are under certain conditions very unlikely. In this context, a very important characteristic of non-linear models which can generate chaotic evolutions is that such models exhibit strong sensitivity to initial conditions. Points which are initially close will diverge exponentially over time. Hence, even if we knew the underlying structure exactly, our evaluation of the current state of the system is subject to measurement error and, hence it is impossible to predict with confidence beyond the very short run. Similarly, if we knew the current state with perfect precision, but the underlying structure only approximately, the future evolution of the system would also be unpredictable. The equivalence of the two situations has been demonstrated by e.g. Crutchfield et al. (1982). After a series of interesting studies on chaotic features of complex systems in physics, chemistry, biology, meteorology and ecology, chaos theory has also been introduced and investigated in the social sciences. The main purpose of the use of this theory in the social sciences was to obtain better insight into the underlying causes of unforeseeable evolutions of complex dynamic social systems.