Duration Time Series

In this section we focus our attention on duration time series, i. e. sequences of random durations, indexed by their successive numbers in the sequence and possibly featuring temporal dependence. In practice these data are generated, for example, by randomly occurring transactions on credit cards, by claims ran­domly submitted to insurance agencies at unequal intervals, or by assets traded at a time varying rate on stock markets. According to the traditional time series analysis the ultimate purpose of our study is to model and estimate the dynamics of these stochastic duration processes.

There are two major characteristics which account for the distinct character of duration time series. Unlike the familiar time series data, duration sequences are not indexed by time, but as mentioned earlier, by numbers indicating their position in the sequence. Such indices are necessarily integer valued, and in this respect dynamic durations belong to traditional discrete time series observed at a fixed frequency. Since the duration indices correspond to arrivals of some random events, researchers often employ the notion of an operational time scale with unitary increments set by the event arrivals.

Unlike the duration data discussed earlier in the text, duration time series do not represent patterns exhibited by a sample of individuals over a fixed span of calendar time. For example, in a sample of unemployed people, we may encounter individuals who, during the sampling period, experienced not one, but several unemployment spells. Yet, the study aims at finding the probabilistic structure of durations common to all individuals. At the individual level too few consecutive durations are available for inference on the dynamics, anyway. In contrast, the time series of durations represent times between many outcomes of the same repeated experiment (for example trading a stock), and always concern the same statistical individual (for example the IBM stock).

While the traditional duration analysis is essentially applied to cross section data, the analysis of stock trading dates, claim arrivals, or transactions on a bank account require a time series approach adapted to the specific features of durations. This field of research is quite recent. It originated from a growing interest in quote-by-quote data provided by electronic systems implemented on financial markets and has been developed in parallel to progressing computer capacities allowing for treatment of large data sets. The number of transactions on a particular stock in a stock market concluded during one day may be very large indeed and easily exceed several hundred thousands.

We begin this section with insights into the dynamics of durations in the simple case of the Poisson process. Next we cover some recent developments in this field including the Autoregressive Conditional Duration (ACD) model and the Stochastic Volatility Duration (SVD) model.

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