Signal Processing - May 2017 - 27
U.S. GDP (in Billions of Dollars) 1940-2015
2
Unit-Root Test
× 104
AR(1) unit-root (random-walk) test:
1.5
Yt = c + zYt -1 + f t , f t ~N (0, v 2), Cov (f t , f s) = 0, 6t ! s.
1
0.5
0
1940 1950 1960 1970 1980 1990 2000 2010 2020
1,000
First-Order Difference
The null hypothesis is H 0 : z = 1(unit root) . There is no
analytical closed-form expression for the distribution of
z estimate. The Dickey-Fuller test [45] uses empirical
distribution of the test statistics:
DFt =
500
0
-500
1940 1950 1960 1970 1980 1990 2000 2010 2020
zt - 1
,
SE (zt )
where zt is the OLS estimate.
The augmented Dickey-Fuller test tests unit root for the
AR(p) process:
p -1
1,000
Yt = c + zYt -1 + / d i DYt -i + f t ,
Second-Order Difference
i =1
500
for the null hypothesis H 0 : z = 1(unit root) . The test statistics are the same as above.
0
-500
-1,000
1940 1950 1960 1970 1980 1990 2000 2010 2020
FIGURE 7. The U.S. GDP time series and its differences. This time series is
an I(2) process.
Nonstationary, integrated processes, and unit roots
The subtle differences in econometric time-series model definitions from SP reveal the different applications and assumptions in economics. A main difference is that in SP, the
distinction between the system/model and the process/signal is
clear because researchers can actively control or at least simulate the signal. In economics, the model is often a characteristic of the time series, which oftentimes can only be observed
but not controlled or changed. In SP systems, there is no constant term c, as the output in a linear SP system is zero with
zero input. In economic time series, the constant term c actually defines the stable status of the variable. For example, for
AR( p) time series, we have
E [Yi] =
c
p
1 - / zi
hopes that its sales grow over time rather than die out over
time, as does a country in terms of its gross domestic product (GDP). The analysis of nonstationary processes generates
many research problems and tools.
A basic nonstationary evolving process is a random-walk
process with drift:
Yt = c + Yt -1 + f t .
Apparently, this process has one root on the unit circle, i.e.,
it has a unit root. Its first-order difference is stationary, and
thus it is a process of integrated order 1, denoted by I(1). An
integrated process with order d, I(d) process, is a nonstationary (unit-root) process whose dth-order difference is stationary. A unit-root process is also called an evolving process. See
Figure 7 for the U.S. GDP time series and its differences.
The related AR integrated moving-average (ARIMA),
ARIMA( p, d, q), model is defined as
.
i =1
A more startling difference is that in SP, only stable processes and systems are investigated, i.e., the roots/poles must
be within the unit circle. However, economic time series are
often nonstable (the econometrics term is nonstationary or
evolving). The nonstationarity (i.e., evolution) is often a desired
property for economic variables. For example, a company
p
q
i =1
j =1
D d Yt = c + / z i D d Yt -i + f t + / i j f t -j ,
where D d is the dth difference operator.
The first analysis step for an economic time series is often
to determine whether it has a unit root, i.e., to conduct the unitroot test; see "Unit-Root Test."
Cointegration and causality
As we have shown, in contrast with the signals studied in SP,
many economic time series are nonstationary integrated processes
IEEE Signal Processing Magazine
|
May 2017
|
27
Table of Contents for the Digital Edition of Signal Processing - May 2017
Signal Processing - May 2017 - Cover1
Signal Processing - May 2017 - Cover2
Signal Processing - May 2017 - 1
Signal Processing - May 2017 - 2
Signal Processing - May 2017 - 3
Signal Processing - May 2017 - 4
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Signal Processing - May 2017 - 111
Signal Processing - May 2017 - 112
Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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