Dougherty: Introduction to Econometrics 3e
Study Guide
Chapter 13
Introduction to nonstationary time series
Overview
This chapter begins by defining the concepts of stationarity and nonstationarity as applied to univariate time
series and, in the case of nonstationary series, the concepts of difference-stationarity and trend-stationarity.
It
next describes the consequences of nonstationarity for models fitted using nonstationary time-series data and
gives an account of the Granger–Newbold Monte Carlo experiment with random walks.
Next the two main
methods of detecting nonstationarity in time series are described, the graphical approach using correlograms and
the more formal approach using Augmented Dickey–Fuller unit root tests.
This leads to the topic of
cointegration.
The chapter concludes with a discussion of methods for fitting models using nonstationary time
series: detrending, differencing, and error-correction models.
Further material
Generalization of the Augmented Dickey Fuller test for unit roots
In Section 13.3 it was shown how a Dickey–Fuller test could be used to detect a unit root in the process
t
t
t
t
X
X
ε
γ
β
β
+
+
+
=
−
1
2
1
and an Augmented Dickey–Fuller test could be used for the same purpose when the process included an
additional lagged value of
X
:
t
t
t
t
t
X
X
X
ε
γ
β
β
β
+
+
+
+
=
−
−
2
3
1
2
1
In principle the process may have further lags, the general form being
t
p
s
s
t
s
t
t
X
X
ε
γ
β
β
+
+
+
=
∑
=
−
+
1
1
1
One condition for stationarity is that the sum of the coefficients of the lagged
X
variables should be less than one.
Writing our test statistic
θ
as
1
1
1
−
=
∑
=
+
p
s
s
β
θ
the null hypothesis of nonstationarity is
H
0
:
θ
= 0 and the alternative hypothesis of stationarity is
H
1
:
θ
< 0.
Two
issues now arise.
One is how to reparameterize the specification so that we can obtain a direct estimate of
θ
.
The other is how to determine the appropriate value of
p
.
From the definition of
θ
, we have
∑
=
+
−
+
=
p
s
s
2
1
2
1
β
θ
β
Substituting this into the original specification, we have
t
p
s
s
t
s
t
p
s
s
t
t
X
X
X
ε
γ
β
β
θ
β
+
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+
+
=
∑
∑
=
−
+
−
=
+
2
1
1
2
1
1
1
© Christopher Dougherty, 2007
The material in this book has been adapted and developed from material originally produced for the degrees and diplomas by distance
learning offered by the University of London External System (
www.londonexternal.ac.uk
)

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