11-FinEcon-Time Series Page 285 Wednesday, February 4, 2004 12:58 PM
Financial Econometrics: Time Series Concepts, Representations, and Models 285How do we describe a time series? One way to describe a time series
is to determine the mathematical form of the conditional distribution.
This description is called an autopredictive model because the model
predicts future values of the series from past values. However, we can
also describe a time series as a function of another time series. This is
called an explanatory model as one variable is explained by another.
The simplest example is a regression model where a variable is propor-
tional to another exogenously given variable plus a constant term. Time
series can also be described as random fluctuations or adjustments
around a deterministic path. These models are called adjustment mod-
els. Explanatory, autopredictive, and adjustment models can be mixed
in a single model. The data generation process (DGP) of a series is a
mathematical process that computes the future values of the variables
given all information known at time t.
An important concept is that of a stationary time series. A series is
stationary in the “strict sense” if all finite dimensional distributions are
invariant with respect to a time shift. A series is stationary in a “weaker
sense” if only the moments up to a given order are invariant with
respect to a time shift. In this chapter, time series will be considered
(weakly) stationary if the first two moments are time-independent. Note
that a stationary series cannot have a starting point but must extend
over the entire infinite time axis. Note also that a series can be strictly
stationary (that is, have all distributions time-independent, but the
moments might not exist). Thus a strictly stationary series is not neces-
sarily weakly stationary.
A time series can be univariate or multivariate. A multivariate time
series is a time-dependent random vector. The principles of modeling
remain the same but the problem of estimation might become very diffi-
cult given the large numbers of parameters to be estimated.
Models of time series are essential building blocks for financial fore-
casting and, therefore, for financial decision-making. In particular asset
allocation and portfolio optimization, when performed quantitatively,
are based on some model of financial prices and returns. This chapter
lays down the basic financial econometric theory for financial forecasting.
We will introduce a number of specific models of time series and of multi-
variate time series, presenting the basic facts about the theory of these
processes. The next chapter will tackle the problem of model estimation
from empirical data. We will consider primarily models of financial
assets, though most theoretical considerations apply to macroeconomic
variables as well. These models include:■ Correlated random walks. The simplest model of multiple financial
assets is that of correlated random walks. This model is only a rough