1 The graphical approach appears to have served the time series analysts satisfactorily, but on the...
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1 The graphical approach appears to have served the time series analysts satisfactorily, but on the whole econometricians prefer more formal methods and tests for nonstationarity are no exception. TESTS OF NONSTATIONARITY: INTRODUCTION t t t u Y Y 1 2
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Transcript of 1 The graphical approach appears to have served the time series analysts satisfactorily, but on the...
1
The graphical approach appears to have served the time series analysts satisfactorily, but on the whole econometricians prefer more formal methods and tests for nonstationarity are no exception.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt uYY 12
2
They are often described as tests for unit roots, for reasons related to the theory of difference equations that need not concern us here.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt uYY 12
3
Before embarking on tests of nonstationarity, it is prudent to recognize that we should not have unrealistic expectations concerning what can be achieved. To illustrate the discussion, suppose that we believe a process can be represented by Yt = b2Yt–1 + et.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt uYY 12
4
If b2 = 1, the process is nonstationary. If b2 = 0.99, it is stationary. Can we really expect to be able to discriminate between these two possibilities?
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt uYY 12
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Obviously not. We are seldom able to make such fine distinctions in econometrics. We have to live with the fact that we are unlikely ever to be able to discriminate between nonstationarity and highly autocorrelated stationarity.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt uYY 12
6
We will start with the first-order autoregressive model Yt = b1 + b2Yt–1 + dt + et, including a time trend as well as an autoregressive process.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
General model
7
For the time being we will assume that the disturbance term is iid and we will emphasize this by writing it as et rather than ut. We will relax the iid assumption later on.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
General model
8
We will consider the various special cases that arise from the following possibilities:–1 < b2 < 1 or b2 = 1d = 0 or d ≠ 0
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
General model
11 2 12 0 0
or
or
Alternatives
9
We will exclude b2 > 1 because the explosive process implied by it is seldom encountered in economics. We will also exclude b2 < –1 because it implies an implausible process.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
General model
11 2 12 0 0
or
or
Alternatives
10
ttt tYY 121
This gives us four basic possibilities. For one of them, it is important to separate the two sub-cases b1 = 0 and b1 ≠ 0. This gives us five cases in all. In the table, b1 = * means b1 is unrestricted.
11 2 12 0 0
or
or
12 0
12 01 0
12 01 0
12 0
*1
*1
12 0*1
TESTS OF NONSTATIONARITY: INTRODUCTION
General model
Alternatives
Case (a)
Case (b)
Case (c)
Case (d)
Case (e)
11
Case (a) is a stationary AR(1) process of the kind studied at length in Chapter 11.
ttt YY 121
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
12 0*1 Case (a)
Case (b)
12
01
Case (b) is a random walk.
ttt YY 1
12
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
0
Case (c)
13
Case (c) is a random walk with drift.
ttt YY 11
01
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
12 0
Case (d)
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Case (d) is a stationary autoregressive process around a deterministic trend.
ttt tYY 121
*1
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
12 0
15
Case (e) implies a model that is doubly nonstationary, being a random walk (with drift if b1 ≠ 0) around a deterministic time trend.
ttt tYY 11
*1
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
Case (e) 12 0
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Case (e) can be excluded because it is implausible. Lagging and substituting once, we obtain the equation shown.
ttt
ttt
ttt
Yt
ttY
tYY
121
1211
11
22
1
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 11
*1
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
Case (e) 12 0
17
Lagging and substituting t times, we can express Yt in terms of the initial Y0, the innovations, and a convex quadratic expression for t.
tttt YtY 121 22
t
sst Y
tttY
101 2
1
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 11
*1
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
Case (e) 12 0
18
It is not reasonable to suppose that any (ordinary) time series process can be characterized as a convex quadratic function of time.
TESTS OF NONSTATIONARITY: INTRODUCTION
tttt YtY 121 22
t
sst Y
tttY
101 2
1
ttt tYY 11
*1
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
Case (e) 12 0
19
Our objective is to determine which of cases (a) – (d) provides the best representation of a given process. Two approaches have been advocated.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
12 0
12 01 0
12 01 0
12 0
*1
*1
Case (a)
Case (b)
Case (c)
Case (d)
20
One is to start with the general model at the top of the slide and then conduct a series of tests that might lead to a simplification to one of cases (a) to (d).
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
12 0
12 01 0
12 01 0
12 0
*1
*1
Case (a)
Case (b)
Case (c)
Case (d)
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This has the intellectual appeal of implementing the general-to-specific approach. However, as will be seen, the tests involved often have low power, and this can give rise to ambiguity that cannot be resolved.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
12 0
12 01 0
12 01 0
12 0
*1
*1
Case (a)
Case (b)
Case (c)
Case (d)
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The other approach is pragmatic. One starts by plotting the data and assessing whether there is evidence of a trend. If there is not, then one limits the investigation to cases (a) and (b). If there is, one considers cases (c) and (d).
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
12 0
12 01 0
12 01 0
12 0
*1
*1
Case (a)
Case (b)
Case (c)
Case (d)
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This is the approach that will be adopted in the next two slideshows.
TESTS OF NONSTATIONARITY: INTRODUCTION
ttt tYY 121
11 2 12 0 0
or
or
General model
Alternatives
12 0
12 01 0
12 01 0
12 0
*1
*1
Case (a)
Case (b)
Case (c)
Case (d)
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 13.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.08.22
