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Encyclopedia of Financial Models, 3 Volume Set by Frank J. Fabozzi

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Time Series Concepts, Representations, and Models

SERGIO M. FOCARDI, PhD

Partner, The Intertek Group

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: A stochastic process is a time-dependent random variable. Stochastic processes such as Brownian motion and Ito processes develop in continuous time. This means that time is a real variable that can assume any real value. In many financial modeling applications, however, it is convenient to constrain time to assume only discrete values. A time series is a discrete-time stochastic process; that is, it is a collection of random variables Xi indexed with the integers …–n, …, –2,–1,0,1,2, …, n, …

In this entry, we introduce models of discrete-time stochastic processes (that is, time series). In financial modeling, both continuous-time and discrete-time models are used. In many instances, continuous-time models allow simpler and more concise expressions as well as more general conclusions, though at the expense of conceptual complication. For instance, in the limit of continuous time, apparently simple processes such as white noise cannot be meaningfully defined. The mathematics of asset management tends to prefer discrete-time processes, while the mathematics of derivatives tends to prefer continuous-time processes.

The first issue to address in financial econometrics is the spacing of discrete points of time. An obvious choice is regular, constant spacing. In this case, the time points are placed ...

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