Durations

Christian Gourieroux and Joann Jasiak

1 Introduction

Duration data represent times elapsed between random arrivals of events. They play an important role in many areas of science such as engineering, manage­ment, physics, economics, and operational research. In economics, duration data frequently appear in labor and health studies, insurance analysis, and finance. For example, a commonly used duration-based statistic is the average individual lifetime, called the life expectancy, provided yearly by national surveys. It serves a variety of purposes. The macroeconomists quote it as an indicator of the level of development and welfare of the society, while applied microeconomists con­sider it implicitly in designing and pricing contracts such as life insurances. For specific projects, duration data are collected from a number of different sources. Various longitudinal studies are conducted on the national level to record durations of individual unemployment spells for job search studies. Data on durations of hospital treatments provide information on the anticipated expenses of the health care system. In academic research, the business cycle analysis and macroeconomic forecasting require studies of durations of recessions and expan­sions measuring times elapsed between subsequent turning points of the economy. Finally, in the private sector, businesses collect their own duration data. For example, insurance agencies record the times between reported car accidents to determine individual insurance premia or bonus-malus schemes, and learn about the attitude of their customers with respect to risk.

The probability theory often defines the distributional properties of durations with respect to the distribution of delimiting random events. In particular, the arrival frequency of these events has some major implications for research. For illustration, let us compare the durations of job searches to durations between consecutive transactions on a computerized stock market, like the New York Stock Exchange (NYSE). While a job search may last from a few days up to several months, a duration between trades may only amount to a fraction of a minute. We also expect that although the number of unemployment durations experienced by one person is theoretically unlimited, their total length cannot

exceed the maximum time during which the individual is able to actively partici­pate in the labor force. In practice we do not observe more than a few unem­ployment spells per person, on average. For this reason researchers are mainly interested in cross section studies of unemployment durations based on a large number of individuals, rather than in investigating personal duration patterns. This is not the case of intertrade durations where each stock generates a series of durations consisting of thousands of observations per month. Such duration data are primarily interesting from the point of view of their dynamics and fall into a distinct category of duration time series. Therefore it is important to distinguish between duration models applied to panel and time series data.

The dynamics of durations is often related to transitions of a stochastic process between different admissible states. In this framework, a duration may be viewed as the sojourn time spent in a given state (unemployment) before exiting this state to enter into another state (employment). Besides the randomness related to the exit time, its destination may also be stochastic. There exist durations which may be terminated by events admitting several various states. An example of such a duration is the length of a hospital stay, which may end up in a recovery or a death of the patient. These durations data are called transition data.

An important issue in duration analysis concerns the measurement, or more precisely the choice of the time scale of reference. Several economic applications require time scales different from the conventional calendar time. The change of the time scale is called time deformation. The operational time unit is often selected with respect to some exogenous variables which may effect the speed of the time flow. Intuitively, an individual has a different perception of the time flow during busy working hours, and quiet periods of leisure. For objects like machines and instruments a typical determinant of the time speed is the depre­ciation rate. Accordingly, the time measuring the lifetime of a car flows at a different speed for a new car which leaves the assembly line, from an old car which has accumulated 100,000 km on the odometer. For this reason, a natural time scale of reference seems to be in this example the calendar time discounted by the mileage. As another example illustrating the economic sense of opera­tional time, imagine an efficient stock trader who instead of measuring his time spent on the market floor in minutes is using instead time units necessary to trade, say, 1,000 shares or to make transactions worth 1,000 dollars. Obviously, deformed time does not have equal, unitary increments, but it resembles the calendar time in that it cannot stop or reverse its direction.

Finally, note that in everyday life we often encounter durations arising as conditions specified by various contracts, such as lease agreements, rentals, or credit terms. As such these predetermined durations are not of interest to ana­lysts, who examine durations between events which are intrinsically random. However the randomness reappears if a side of the contract is allowed to quit by early termination, i. e. when, for example, borrowers have the option to prepay the outstanding credit balances. Given that not all individuals display the same behavior, this population is considered by a duration analyst as a heterogenous one.

This chapter is organized as follows. In Section 2, we discuss the standard characterizations of duration variables and present the main duration distribution
families. In Section 3, we introduce individual heterogeneity in parametric dura­tion models. This heterogeneity is partly observed through individual explanatory variables and partly unobserved. We discuss the effect of unobserved heterogeneity in terms of negative duration dependence. Section 4 covers semiparametric models with a parametric effect of observed explanatory variables, and an unspecified baseline duration distribution. Finally, we introduce in Section 5 dynamic models for the analysis of time series of durations which are especially useful for applica­tions to financial transactions data.

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