11-FinEcon-Time Series Page 283 Wednesday, February 4, 2004 12:58 PM

CHAPTER

11

Financial Econometrics:

Time Series Concepts,

Representations, and Models

I

n this chapter and the next we introduce models of discrete-time sto-

chastic processes (that is, time series) and address the general problem

of estimating a model from a given set of empirical data. Recall from

Chapter 6 that a stochastic process is a time-dependent random variable.

Stochastic processes explored thus far, for instance Brownian motion and

Itô processes, develop in continuous time. This means that time is a real

variable that can assume any real value. In many 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 ran-

dom variables Xi indexed with the integers ...–n,...,–2,–1,0,1,2,...,n,...

In finance theory, as in the practice of quantitative finance, 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 pro-

cesses 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 at multiples of a single time inter-

val: t = i∆t. For instance, one might consider the closing prices at the

end of each day. The use of fixed spacing is appropriate in many appli-

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