11-FinEcon-Time Series Page 309 Wednesday, February 4, 2004 12:58 PM
Financial Econometrics: Time Series Concepts, Representations, and Models 309xt – (^1) – φ 1 ...φ– p 1 ψ 1 ...ψq – 1 ψq
· ·
· 1 ... 000 ... 0 0
xtp– ·· ·· ·· ·· ·· · · · · · ·
· · · · · · · ·
zt = εt and A = 0 ... 100 ... 0 0
εt – 1
0 ... 000 ... 0 0
·· ··· ··· ··· ··· ··· · · · · · · · · ·
·
εtq–^0 ...^000 ...^10
In general, the number of states will be larger than the number of obser-
vations. However, the number of states can be reduced model reduction
techniques.^3
The connection between ARMA and state-space models has a deep
meaning that will be elucidated after introducing the concept of cointe-
gration and after generalizing the concept of state-space modeling. As
we will see, both cointegration and state-space modeling implement a
fundamental dimensionality reduction which plays a key role in the
econometrics of financial time series.
INTEGRATED SERIES AND TRENDS
This section introduces the fundamental notions of trend stationary
series, difference stationary series, and integrated series. Consider a one-
dimensional time series. A trend stationary series is a series formed by a
deterministic trend plus a stationary process. It can be written as
Xt = ft() ε+ ()t
A trend stationary process can be transformed into a stationary pro-
cess by subtracting the trend. Removing the deterministic trend entails
that the deterministic trend is known. A trend stationary series is an
example of an adjustment model.
Consider now a time series Xt. The operation of differencing a series
consists of forming a new series Yt = ∆Xt = Xt – Xt–1. The operation of
differencing can be repeated an arbitrary number of times. For instance,
differencing twice the series Xt yields the following series:
(^3) The idea of applying model reduction techniques to state-space models was advo-
cated by, among others, Masanao Aoki. See M. Aoki and A. Havenner, “State Space
Modeling of Multiple Time Series,” Econometric Reviews (1991), pp. 10:1–59.