In this paper we present explanation on the phenomenon pointed out in Cook and Manning (2002) on the unusual behaviour of the Dickey-Fuller test in the presence of trend misspecification. It appears that the rejection frequency of the unit root tests in the presence of trend misspecification is very sensitive to the number of the initial observations that need to be discarded. Based on the evidence from the Monte Carlo simulations, we show that for the DGP in Cook and Manning (2002), the unusual behaviour of the Dickey-Fuller test disappears as the number of the discarded initial observations becomes sufficiently large.

Boriss Siliverstovs

English

Research Papers in Economics

Unusual behaviour of Dickey−Fuller tests in the presence of
trend misspecification: comment 
Boriss Siliverstovs
DIW Berlin
Abstract
In this paper we present explanation on the phenomenon pointed out in Cook and Manning
(2002) on the unusual behaviour of the Dickey−Fuller test in the presence of trend
misspecification. It appears that the rejection frequency of the unit root tests in the presence
of trend misspecification is very sensitive to the number of the initial observations that need
to be discarded. Based on the evidence from the Monte Carlo simulations, we show that for
the DGP in Cook and Manning (2002), the unusual behaviour of the Dickey−Fuller test
disappears as the number of the discarded initial observations becomes sufficiently large.
Citation: Siliverstovs, Boriss, (2003) "Unusual behaviour of Dickey−Fuller tests in the presence of trend misspecification:
comment." Economics Bulletin, Vol. 3, No. 29 pp. 1−7
Submitted: November 6, 2003.  Accepted: November 26, 2003.
URL: http://www.economicsbulletin.com/2003/volume3/EB−03C10008A.pdf1 Introduction.
The literature on unit root testing in economic time series is already extensive and
well established. Nevertheless, it is too early to say whether it is entirely complete.
Although the properties of the most popular unit root tests have been well described
and understood, there are still nuances that remain to be explained and possibly
discovered by future research.
One example is the recent paper by Cook and Manning (2002), where the be-
haviour of the two unit root tests – the ττ− and K−tests suggested in Fuller (1976)
– is compared under trend misspeciﬁcation. The authors point out that the former
test exhibits unusual behaviour whereas the latter test performs as expected.
In this paper, we address the issue and provide an explanation for this seemingly
puzzling ﬁnding. In particular, we extend the Monte Carlo results of Cook and
Manning (2002) by examining to what extent the rejection frequency of the examined
unit root tests depends on the number of observations that are discarded to lessen
the inﬂuence of the initial values.
We ﬁnd that the unusual behaviour of the ττ-test eventually disappears as one
discards an increasing number of initial observations. Moreover, for a rather large
number of the discarded observations, the rejection frequency of the ττ-a n dK-tests
is quite similar. This is rather remarkable since when rather small number of initial
observations is discarded, their behaviour varies considerably, as reported in Cook
and Manning (2002).
The rest of the paper is organized as follows. Section 2 discusses the motivation
behind Cook and Manning (2002) and explains their Monte Carlo setup. Section 3
presents the Monte Carlo setup used in the simulations and presents the obtained
results. The ﬁnal section concludes.
All graphics as well as the computational results were obtained using the pro-
grams Ox 3.30 Professional and Givewin 2.10, see Doornik (2001) and Hendry and
Doornik (2001), respectively.
2 Results of Cook and Manning (2002).
Consider the unit root process with drift
yt = µ1 + yt−1 + et = y0 + µ1t +
t 
i=1
ei,e i ∼ i.i.d.(0,σ
2)( 1 )
As seen from the corresponding MA-representation, this process exhibits trending
behaviour. Hence it has been suggested in the literature (e.g. see Banerjee et al.,
1994; Harris, 1995) that the following auxiliary Dickey-Fuller (DF) regression be
used for unit root testing
yt = a + bt + φyt−1 + et (2)
1in order to gain power against the trend-stationarity of a process. Observe that by
using equation (2) we ensure similar behaviour of the time series both under the null
and alternative hypotheses.
Fuller (1976) suggests the following ways of testing for unit root hypothesis (φ =
1). The ﬁrst is ττ−test in the form of t−ratio statistic
ττ =
 φ − 1
se( φ)
(3)
and the K−test in the coeﬃcient form
K = T( φ − 1). (4)
Under the null hypothesis of unit root, both tests have nonstandard Dickey-Fuller
distributions, tabulated in Fuller (1976). As usual, the rejection/acceptance decision
of the null hypothesis of unit root is based on comparing the obtained values of the
respective test statistics with the tabulated critical values for a chosen signiﬁcance
level.
However, since in reality the true DGP remains unknown, there always exists a
possibility for running the DF-regression (2) under a false assumption that the data
have been generated in accordance with equation (1). Cook and Manning (2002)
investigate the power properties of the unit root test based on equation (2) when
the data have been generated by the following DGP
yt = α + βt+ ρyt−1 + ξt,t=1 ,...,T. (5)
It is easy to verify that under the null hypothesis of unit root there is a quadratic
deterministic trend in the MA-form of equation (5). Hence, an appropriate set of
deterministic terms in the DF-regression should include a quadratic trend as well.
In the opposite case, the misspeciﬁcation of the test regression is apparent.
Cook and Manning (2002) study the consequences of such trend misspeciﬁca-
tion. They report the empirical rejection frequency of the ττ− and K−forms of the
Dickey-Fuller tests using the relevant critical values that correspond to the nomi-
nal 5% signiﬁcance level. Although the authors indicate that in their Monte Carlo
experiments they have tried diﬀerent parameter combinations, they speciﬁcally re-
port the results of only one particular parameter set {α,β,T} = {0,0.08,250} and
ρ = {0.9600,0.9605,..,0.9995,1}, which illustrate their main ﬁnding. In particular
they consider the initial value y0 = 0 and discard ﬁrst N = 100 observations to lessen
its inﬂuence on the obtained results. Observe that the error term is ξt ∼ i.i.d.N(0,1).
Figure 1 replicates their results. Notice that in the presence of the trend misspeci-
ﬁcation it is expected that the DF tests will be biased towards nonrejection of the
hypothesis of the unit root especially for values of ρ close to one. The behaviour of
the K−test conforms with the expectations, whereas the DF ττ-test displays rather
20.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
0.25
0.50
0.75
1.00 PowerADF ×  Rho 
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
0.25
0.50
0.75
1.00
PowerK ×  Rho 
Figure 1: Upper panel: rejection frequencies of ττ− test; Lower panel: rejec-
tion frequencies of K−test. Replicated results of Cook and Manning (2002):
{α,β,T} = {0,0.08,250} and ρ = {0.9000,0.9005,..,0.9995,1},ﬁ r s tN = 100 ob-
servations discarded. Each curve has been obtained using 5000 replications.
unusual behaviour. As seen, the rejection frequency after the initial decline rises
sharply and then falls rapidly towards zero at ρ =1 .
In the subsequent section we provide an explanation of this apparently aberrant
behaviour. The main argument is that this puzzling ﬁnding is caused by the fact
that the removal of the ﬁrst N = 100 observations as done by Cook and Manning
(2002) is not enough to remove the inﬂuence of the initial observation y0 =0a s
claimed by the authors. We show that once one removes enough initial observations
in the starting period, then this puzzling result disappears. In the limit for the large
number of discarded initial observations, the rejection frequencies of both ττ-a n d
K-tests are very similar.
3 Our results.
In this section, we extend the Monte Carlo results of Cook and Manning (2002) by
considering the rejection frequencies of the ττ-a n dK-tests for diﬀerent number of
startup observations. Note that otherwise the DGP is the same as in Cook and
Manning (2002), see above. To this end we consider the following numbers of initial
N observations removed N = {50,100,200,10000}. The results of the Monte Carlo
simulations are presented in Figure 2. As seen in Figure 1, the unusual behaviour
30.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
0.25
0.50
0.75
1.00
N = 50 
N = 200 
N = 100 
N = 10000 
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
0.25
0.50
0.75
1.00
N = 50 
N = 200 
N = 100 
N = 10000 
Figure 2: Upper panel: rejection frequencies of ττ− test; Lower panel: rejection
frequencies of K−test. DGP of Cook and Manning (2002): {α,β,T} = {0,0.08,250}
and ρ = {0.9000,0.9005,..,0.9995,1},ﬁ r s tN = {50,100,200,10000} observations
discarded. Each curve has been obtained using 5000 replications.
of the Dickey-Fuller ττ-test fades away as one eliminates more and more initial
observations. Also notice that the rejection frequencies of the ττ-a n dK-tests, so
diﬀerent for rather small values of N, become very similar for rather large values of
N,e . g .N = 10000.
Observe that in the example of Cook and Manning (2002), the trend misspec-
iﬁcation is introduced by omitting the quadratic deterministic trend from the test
regression. It also possible to construct a case where the trend misspeciﬁcation oc-
curs due to the absence of a linear deterministic trend in the DF regression. That
is, the testing regression is given by
yt = a + φyt−1 + et, (6)
but the data are generated as follows
yt = α + ρyt−1 + ξt. (7)
Observe that similarly to the case considered in Cook and Manning (2002), equation
(6) is misspeciﬁed as it omits a linear deterministic trend that is present in the MA-
form of equation (7) under the null hypothesis of unit root ρ =1 .
In order to investigate the properties of these two unit root tests under this mis-
speciﬁcation type, we chose the following parameter values {α,β,T} = {2.0,0,250}
40.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
0.25
0.50
0.75
1.00
N = 50 
N = 200 
N = 100 
N = 10000 
0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
0.25
0.50
0.75
1.00
N = 50 
N = 200 
N = 100 
N = 1000 
Figure 3: Upper panel: rejection frequencies of ττ− test; Lower panel: re-
jection frequencies of K−test. DGP: {α,β,T} = {2.0,0,250} and ρ =
{0.9000,0.9005,..,0.9995,1},ﬁ r s tN = {50,100,200,10000} observations discarded.
Each curve has been obtained using 5000 replications.
and ρ = {0.9000,0.9005,..,0.9995,1} with the following values of the discarded
startup period N = {50,100,200,10000}. Figure 3 displays the rejection frequen-
cies obtained for both ττ-a n dK−tests.
As seen in Figure 3, the qualitatively similar picture to that displayed in Figure
2 emerges here as well. That is, for rather small values of N, the behaviour of the
ττ−test is counterintuitive and diﬀers markedly from that of the K−tests. However,
this diﬀerence is decreasing as the value of N grows, and eventually for rather large
values of N, the behaviour of both tests is very similar.
What can we learn from this Monte Carlo evidence? We observe the unusual
behaviour of the ττ-test for unit roots only when a relatively small number of initial
observations is discarded. In this case, when a situation with trend misspeciﬁcation
as described in Cook and Manning (2002) – see equations (2) and (5) as well as
equations (6) and (7) – arises, the two unit root tests suggested in Fuller (1976)
display remarkably diﬀerent behaviour. However, when there is a suﬃcient number
of initial observations being discarded, the behaviour of these two unit root tests is
rather similar.
54 Conclusions.
In this paper, we analyze the results reported in Cook and Manning (2002) on the
unusual behaviour of the unit root test in the presence of trend misspeciﬁcation. We
are able to replicate the results of their study when the quadratic deterministic trend
has been omitted from the testing regression for unit roots. Furthermore, we provide
evidence of the impact of trend misspeciﬁcation when the linear deterministic trend
has been omitted from the testing regression for unit roots. The results of the latter
case fully mirror the results obtained for the former case.
Thus, when the number of the initial observations that are discarded is rather
small, the ττ-test exhibits very unusual behaviour. For the range of values of the
autoregressive parameter that are considerably smaller than one, the rejection fre-
quency of the null hypothesis of unit root declines as this parameter approaches
unity. Then for the range of values that are quite close to unity, the rejection fre-
quency of the ττ-test surprisingly increases, such that it exhibits an unusually high
rejection rate of the null hypothesis. On the other hand, the rejection frequency
of the K−test declines monotonically as the value of the autoregressive parameter
approaches unity.
As the main contribution of our paper, we show that the reason for these two
unit root tests exhibiting such remarkably diﬀerent behaviour in the Monte Carlo
setup used by Cook and Manning (2002) is that an insuﬃcient number of initial
observations has been discarded. We show that when a suﬃciently large number
of initial observations have been discarded, there is no evidence for the abnormal
behaviuor of the ττ-test recorded in Cook and Manning (2002). In fact, under these
conditions, the rejection frequency of both the ττ-a n dK−tests steadily declines as
the autoregressive parameter value approaches unity.
References
Banerjee, A., J. Dolado, D. Hendry, and J. W. Galbraith (1994). Co-Integration, Er-
ror Correction, and the Econometric Analysis of Non-Stationary Data. Advanced
Texts in Econometrics. Oxford University Press.
Cook, S. and N. Manning (2002). Unusual behaviour of Dickey-Fuller tests in the
presence of trend misspeciﬁcation. Economics Bulletin 3(8), 1–7.
Doornik, J. A. (2001). Ox: An Object-Oriented Matrix Language (4th ed.). London:
Timberlake Consultants Press.
Fuller, W. A. (1976). Introduction to Statistical Time Series. New York: John Wiley.
Harris, R. I. D. (1995). Using Cointegration Analysis in Econometric Modelling.
Prentice Hall PTR.
6Hendry, D. F. and J. A. Doornik (2001). GiveWin: An Interface to Empirical
Modelling (3rd ed.). London: Timberlake Consultants Press.
7

Co-Integration, Error Correction, and the Econometric Analysis of Non-Stationary Data. Advanced Texts in Econometrics.

Introduction to Statistical Time Series.

Unusual behaviour of Dickey-Fuller tests in the presence of trend misspeciﬁcation.

Using Cointegration Analysis in Econometric Modelling.

Unusual behaviour of Dickey-Fuller tests in the presence of trend misspecification: comment

http://www.accessecon.com/pubs/EB/2003/Volume3/EB-03C10008A.pdf

Unusual behaviour of Dickey-Fuller tests in the presence of trend misspecification: comment

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics