Signal Processing - May 2017 - 28

E 6Yt +1 Yt, X t, X (t - 1), g@ = E 6Yt +1 Yt@ .

S&P 500 Daily 2000-2016

S&P 500 Index

1,000

S&P 500 Returns

0.1

2,000

Year

FIGURE 8. The S&P 500 daily prices and returns (2000-2016). When the
price is low, the return volatility is high.

with at least one unit root. The integration order is a property of
the process and represents a trend of the time series. In SP
research, researchers do not care about this trend because they are
working on stationary signals. However, a serious problem occurs
when examining relationships of multiple nonstationary integrated
economic time series.
Consider the following simple linear regression:

If we further agree that for X t to be the cause of Y (t + 1), it
must contain unique information about Yt +1, then X t is said
to Granger-cause Yt +1 if for some A,
P ^Yt +1 ! A X t, X t h = P ^Yt +1 ! A X t - X t h,
where P is the probability and X t represents all knowledge in
the universe available at time t. Note that Granger causality is
only one of many definitions on causality, but it is statistically
testable using a time-series model (VAR or ARDL model),
making it instrumental in causality analysis.

Generalized AR conditional heteroskedasticity models
As we already discussed, volatility is a fundamental risk quantity that needs to be estimated in finance, especially in risk
modeling and option pricing. As has been observed, the historical volatility of a time series changes over time. See Figure 8
for S&P 500 daily prices and returns.
The time-varying nature of volatility is called heteroskedasticity. Robert F. Engle invented the AR conditional heteroskedasticity (ARCH) model to capture the time-varying dynamics
of volatility, winning the 2003 Nobel Memorial Prize in Economic Sciences. The qth order ARCH(q) model for a zeromean normally distributed asset return time series, Yt, with
time-varying volatility v t is specified as

Yt = c + bX t + f t .
When X t and Yt are both unit-root processes with different
orders, i.e., when f t is also a unit-root process, regression
results and statistics become spurious or meaningless. Regression results are only meaningful when X t and Yt have a common trend (i.e., the same integration order) or are cointegrated.
Clive W.J. Granger's finding of such spurious regressions [46]
invalidated many empirical economic studies before the 1970s
and, along with his work on cointegration [47], won him the
2003 Nobel Memorial Prize in Economic Sciences. For cointegrated time series, there must exist a linear combination of
them that is stationary. The Engle-Granger test [47] applies
the Dickey-Fuller unit-root test to examine the cointegration
of multiple time series.
The time-series model helps capture correlations among
time series but does not find causal relationships. In SP, inputs
cause outputs because information flows are obvious in a physical system. However, in analyzing economic data or any data
from nonphysical systems, such as a social network, causal
relationships are not obvious and cannot be taken for granted.
Yet identifying such causal relationships is of great importance
to discover information hidden in the data and is necessary for
decision making in many big data applications.
The causality relationship is always difficult to define and
quantify. Granger gives a definition from the time-series perspective [48], [49]. If we agree that the cause must occur before
the effect, the Granger noncausality (or strong exogeneity) can be
defined if the following equation holds for the conditional mean:
28

Yt ~ N ^0, v 2t h,
q

vt = a0 +

/ a i Y 2t -i .

i =1

It is indeed an MA model for time-varying variance.
Adding an AR term for v t, the generalized ARCH
(GARCH) model [50], GARCH(p, q), is defined as
Yt ~ N ^0, v 2t h,
2

vt = a0 +

i =1

/ a i Y 2t -i + / a i v t2-i .

Many variations of the GARCH model have been subsequently developed and widely used in risk models, high-frequency
volatility models, large-scale multivariate ARCH models, and
derivative pricing models [51].

Relationships between SP and econometric models
The time-series analysis has achieved great success in economics, finance, and business studies. It is encouraging for SP
researchers, as the time-series models and basic concepts are
essentially similar to SP models. Indeed, the Granger causality concept was partly inspired by Nobert Wiener [48], [52].
Many other models used in econometrics [14], [53], including
spectrum analysis, Kalman filtering, Markov models, maximum likelihood and Bayes methods, particle filtering, and

IEEE Signal Processing Magazine

May 2017

Table of Contents for the Digital Edition of Signal Processing - May 2017