Over the past few years heavy-tailed phenomena have attracted the interest of various researchers in time series analysis, extreme value theory, econometrics, telecommunications, and various other fields. The need to consider time series with heavy-tailed distributions arises from the observation that traditional models of applied probability theory fail to describe jumps, bursts, rapid changes and other erratic behavior of various real-life time series. In joint work with Thomas Mikosch, we have studied tail behavior of the marginal distributions for a host of nonlinear time series models such as GARCH and stochastic volatility models. These processes, which are commonly used for modeling financial time series consisting of log-returns, tend to have Pareto-like tails. The relationship between tail-heaviness and serial dependence in the data has been particularly intriguing. In order to detect nonlinearities in financial time series, econometricians often recommend examining the autocorrrelation function (ACF) of not only the time series itself, but also powers of the absolute values. On the other hand, it is believed that many financial time series have infinite third, fourth or fifth moments. If the process has an infinite fourth moment, then the variance of the squares of the process does not exist, in which case the ACF may be of little diagnostic value. The theoretical development that Mikosch and I have provided for the sample ACF confirms that one must view the graphs of the ACF with extreme caution.

Allpass time series are special cases of ARMA processes that exhibit interesting features. First, the allpass process is uncorrelated. Second, as long as the process is nonGaussian, the process is not independent. This can often be seen from inspection of the ACF of the squares of the process. So in many respects, allpass processes mimic properties-lack of serial correlatation and bursty behavior-that are often associated with the nonlinear time series models in finance. In work with Jay Breidt and graduate student Beth Andrews, we are exploring identification and efficient estimation procedures for fitting allpass models. Allpass models are widely used in the engineering literature for fitting noncausal and noninvertible models. Two-stage procedures for detecting noncausality or noninvertiblity are currently under investigation.

The modeling of time series of count data often requires models and techniques that go beyond classical time series analysis. The need for a flexible class of models for count data that can be easily fitted is clearly demonstrated in applications such as modeling of disease incidence, as for example in the modelling of polio counts in the U.S., and the number of asthma presentations at an emergency room in a hospital. Other emerging areas of application include finance, where the response variable is the number of transactions that occur in a small time interval; and spatial-temporal modeling in ecology, where the response might correspond to the number of rare species in a given region at a fixed time. In work with William Dunsmuir, Ying Wang, and Sarah Streett, we have considered two types of models, parameter-driven and observation-driven, for time series of counts. The fitting of parameter-driven models is often computationally intensive especially if a large number of explanatory variables is included in the model. While observation-driven models tend to be easier to fit, their theoretical properties can be difficult to derive. Addressing these issues has been a constant theme in our research.

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