11-FinEcon-Time Series Page 313 Wednesday, February 4, 2004 12:58 PM
Financial Econometrics: Time Series Concepts, Representations, and Models 313correspond to different confidence intervals. The theoretical distribution
of ρ is determined computing a functional of the Brownian motion.
Given a sample of a series, for instance a series of log prices, appli-
cation of the DF test entails computing the autoregressive parameter ρ
on the given sample and comparing it with the known critical values for
different confidence intervals. The strict hypothesis of random walk is
too strong for most econometric applications. The DF test was extended
to cover the case of correlated residuals that are modeled as a linear
model. In the latter case, the DF test is called the Augmented Dickey
Fuller or ADF test. The Phillips and Perron test is the DF test in the gen-
eral case of autocorrelated residuals.SUMMARY
■ A time series is a discrete-time stochastic process, that is, a denumera-
ble collection of random variables indexed by integer numbers.
■ Any stationary time series admits an infinite moving average represen-
tation, that is to say, it can be represented as an infinite sum of white
noise terms with appropriate coefficients.
■ A time series is said to be invertible if it can also be represented as an
infinite autoregression, that is, an infinite sum of all past terms with
appropriate coefficients.
■ ARMA models are parsimonious representations that involve only a
finite number of moving average and autoregressive terms.
■ An ARMA model is stationary if all the roots of the inverse characteris-
tic equation of the AR or the MA part have roots with modulus strictly
greater than one.
■ A process is said to be integrated of order p if it becomes stationary
after differencing p times.
■ A state-space model is a regression of observable variables over an
ARMA model of lower dimensionality.
■ Every ARMA process admits a state-space representation.