Time Series Analysis of a Contagious Process

Abstract

Between 1968 and 1972 there were 326 attempts to hijack aircraft worldwide, including 137 attempts in the United States. Those attempts were often attributed to imitation of previous hijacking incidents, and aircraft hijacking, like many other types of behavior, was described as a ''contagious'' phenomenon. In this article a general mathematical model is developed for serial dependence of rates at which events occur and is applied to data on the hijacking attempts in the U.S. during the 1968-1972 period.The general model leads to a class of autoregressive moving average (ARMA) time series models for Poisson-distributed variates. The ARMA parameters are constrained in such a way that clear interpretation of a process as either contagious or self-inhibiting is possible. The time series models can be generalized to distinguish effects of various subsets of prior incidents (e.g., previous successful and unsuccessful hijacking attempts) and to allow for effects of exogenous time series on the rate of occurrences. Procedures for maximum likelihood estimation and hypothesis testing are described in the Appendix.Unlike many other contagion models, ours is a model for a stationary stochastic process rather than for an epidemic. The underlying probability model is appropriate for aircraft hijacking data since it takes into account the rarity of hijackers in the U.S. population and the fact that the entire U S . population was in contact with those "infected" individuals as a result of the nationwide publicity that hijacking attempts received. As a model for a stationary process, it takes into account the fact that aircraft hijacking neither began in 1968 nor ended in 1972.Parameter estimates for the models suggest that successful hijackings in the United States did indeed increase the subsequent rate of hijacking attempts during the 1968-1972 period; each successful hijacking generated about .5 new attempts, with a median delay of about 33 days. Unsuccessful attempts had neither a stimulating nor an inhibiting effect.

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