Testing for Unit Roots in Dynamic Panels with Smooth Breaks and Cross-sectionally
Dependent Errors
Technical Annex and Further Results
Tolga Omay (Corresponding Author)
Department of Management, Türk Hava Kurumu University, Okul Sokak, Etimesgut, Ankara,
Turkey,
[email protected]
Mübariz Hasanov
Department of Banking and Finance, Okan University, Tuzla Kampusu, Istanbul, Turkey
[email protected]
Yongcheol Shin
Department of Economics and Related Studies, University of York, Heslington, York, YO10
5DD, UK
[email protected]
1. Density Functions of the LNV-type t-statistics.
Leybourne et al. (1998) have clearly indicated that the analytical representation of the
invariance property is impossible under the proposed de-trending methods. They note that:
“As NLS estimation of the parameters  and
 does not admit closed-form solutions,
it would be extremely difficult to subsequently establish any analytical relationship
between the v̂ and yt . This, of course, makes the determination of the null asymptotic
distribution of the test statistics s , s    and s by analytical means more or less
intractable. … the above nonlinear models are in fact linear in the parameters  1 , 1 ,
 2 ,  2 . It is possible to obtain substantial economy in NLS concentrating sum of
squares function. Hence the NLS estimation problem reduces to minimizing the sum of
squares function with respect to just two parameters ˆ and ˆ . … Furthermore, this
linearity property in the intercept and trend terms ensures that the residual v̂ from all
the models are invariant to the choice of starting value 0   and Model A, B and C
are invariant to both starting value and drift term  .”
Leybourne et al. (1998) set the starting values for the parameters ˆ and ˆ in the NLS iteration
at 1.0 and 0.5, respectively, and found that the solutions at convergence were not sensitive at all
to these choices. Sollis (2004) also used similar procedures for their proposed unit root tests.
Previous researchers provided only a general explanation for the insensitivity of the results
without providing any evidence. In this appendix, we carry out a simulation exercise to see
whether the derived test statistics obtained after non-linear de-trending have the suggested
invariance properties as argued in Leybourne et al. (1998).
We apply a Monte-Carlo simulation to see establish the invariance of the critical values with
respect to
 ,  ,  and  . For this purpose, we have used  ,  ; 1.0,0.5 , 1.5,0.6 and
0.5,0.4
for generating the critical values. The density function of the simulated critical
values for t=100 and 1000 trial are given below:
1
As it can be seen from the generated density functions obtained for different initial values of 
and
 , critical values at convergence are clearly not sensitive to the choices of initial values as
stated by Leybourne et al. (1998). In fact, density functions of the test statistics with different
pairs of transition parameters overlap each other. In order to ascertain that test statistics are in
fact invariant to parameters  ,  ,  and  , we repeat the simulation experiment with 10000
replications. After carrying out this new simulation experiment we have obtained the same
density functions for the LNV t-test with different initial values of  and  .
From the figures given above it can be clearly seen that the critical values are invariant to the
parameters
 ,  ,  and  in NLS estimation with respect to the choice of different initial
values.
2. Power Distortion of IPS test in the Presence of Nonlinear Trend Function
Simulation results presented in the paper suggest that power of the conventional IPS (Im et. al
2003) test in presence of a nonlinear trend function first decreases, then increases with time
dimension. This can be explained by the dependence of the curvature of the transition function
on time dimension. In particular, as the sample size is small, moderate and sufficiently large, the
transition function resembles a straight line, an S-shaped curve and a broken straight line,
2
respectively. To see why, consider Model A with a gradual change in mean and no trend.
Structural change in this case is governed by the nonlinear transition function,
t  1  2 St  ,  . The time required for the mean to change from 1 to 1   2 , depends
on the smoothness parameter  , but not on the time dimension (T). For fixed values of  , 
denote the time required for transition from 1 to 1   2 by to T . The transition will be
incomplete if T is sufficiently small, i.e., T  T , in which case the transition function looks
like a straight line. On the other hand, if T is sufficiently large such that T  T  k  0 , then
transition will be complete. Further, if k is sufficiently large,  k / 2  observations are located
to the left of 1 (the start of transition) and the remaining  k / 2  observations to the right of
1   2 (the end of transition), in which case the transition function becomes an S-shaped
curve. If the time dimension tends to infinity, and thus k   , the time required for structural
change, T will be negligible relative to T . In this case, the transition function will resemble a
broken straight line.
Figure A2. Smooth trend function for different time dimensions
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
T= 20
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
T=100
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
5
10
15
20
25
30
35
40
45
50
55
60
65
T= 70
70
25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500
T=500
In all cases, 𝛾 = 0.25 and 𝜏 = 0.5
3
In Figure A2, we plot the figures of the transition function Si ,t  i , i  for different time
dimensions. For the small T=20, the transition function looks like a straight line. As time
dimension increases (T=70 or T=100), structural change occurs gradually whilst structural
change becomes abrupt for very large T=500.
3. Further Power Analysis of CCE Test Statistics
Simulation results presented in the paper implied that the CADF test (Pesaran, 2007) has
relatively good power properties in the case of homogeneous trend specification. In order
elaborate the properties of the CCE based stationarity tests for relatively homogeneous trend
specifications further, we present results of additional simulation exercises below. Table A1
presents the powers of Ct and CADF test statistics for different combinations of
N  5, 25,50,100 and T  30,50, 70,100 . In order to make the power comparison
informative, we consider three ranges of the smooth transition parameters, namely
 i 0.005,0.015 corresponding to the rather gradual shift in mean,  i 0.3,0.4 to the
moderate shift, and  i  0.5,1.0 to the sharp break.
Table A1. Power Analysis of CCE test statistics
Ct
T/N
CADF
5
Ct
CADF
25
Ct
CADF
50
Ct
CADF
100
Low cross-section dependence: i ~ iid .U  0.0,0.2
 i 0.005,0.015
30
50
70
100
0.042
0.183
0.495
0.884
0.122
0.424
0.809
0.984
0.047
0.593
0.998
1.000
0.272
0.969
1.000
1.000
0.037
0.890
1.000
1.000
0.322
1.000
1.000
1.000
0.017
0.990
1.000
1.000
0.357
1.000
1.000
1.000
30
50
70
100
0.249
0.444
0.658
0.942
0.187
0.344
0.503
0.690
0.814
0.982
1.000
1.000
0.812
0.980
0.994
1.000
0.971
1.000
1.000
1.000
0.970
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
30
50
70
100
0.202
0.270
0.527
0.890
0.160
0.226
0.282
0.491
0.535
0.860
1.000
1.000
0.727
0.856
0.935
0.996
0.788
0.996
1.000
1.000
0.975
0.994
1.000
1.000
0.952
1.000
1.000
1.000
1.000
1.000
1.000
1.000
 i 0.3,0.4
 i  0.5,1.0
4
High cross-section dependence: i ~ iid .U  1.0,3.0
 i 0.005,0.015
30
50
70
100
0.038
0.136
0.324
0.792
0.119
0.441
0.762
0.987
0.007
0.272
0.886
1.000
0.199
0.952
1.000
1.000
0.008
0.424
1.000
1.000
0.258
1.000
1.000
1.000
0.005
0.580
1.000
1.000
0.288
1.000
1.000
1.000
30
50
70
100
0.141
0.266
0.465
0.816
0.106
0.194
0.251
0.458
0.269
0.650
0.944
0.998
0.122
0.202
0.322
0.625
0.414
0.843
0.982
1.000
0.125
0.226
0.358
0.686
0.540
0.920
0.991
1.000
0.135
0.225
0.384
0.719
30
50
70
100
0.100
0.170
0.345
0.762
0.090
0.134
0.157
0.277
0.143
0.442
0.873
0.997
0.113
0.124
0.170
0.402
0.179
0.608
0.957
1.000
0.096
0.143
0.185
0.413
0.227
0.757
0.980
1.000
0.091
0.133
0.184
0.475
 i 0.3,0.4
 i  0.5,1.0
Note: Threshold value (transition midpoint) has been drawn from
 i 0.4,0.6 .
Nominal sizes of
both tests are set to 0.05 for power simulations. The results are based on 5000 replications.
Simulation results suggest that the CADF tests are outperformed by the Ct test except for
the very small values of  , namely for  i 0.005,0.015 . As noted in the paper for the IPS
test in the model with no cross-section dependence, a rather gradual change in mean resembles a
linear trend in which case we expect the CADF test to be more powerful than the Ct test
because the latter estimates two more parameters. On the other hand, as  rises, the Ct test
tends to outperform the CADF test. For example, when the range of  i increases from
0.005,0.015 to 0.3,0.4 , the power of the Ct
test significantly increases while the power
of the CADF test greatly declines. For  i  0.5,1.0 , the transition between means becomes
almost an instantaneous structural break. Yet, when the range of  i changes from 0.3,0.4 to
0.5,1.0 , the power of both tests drop though the Ct
test still outperforms the CADF test.
A notable finding is that the power of the CADF test does not decline as sharply as the IPS
test. Rather, it continues to be fairly powerful even for both gradual and abrupt breaks. This
suggests that the CADF test, mainly designed to control for CSD, may be effectively powerful
against the gradually changing stationary processes. Smith and Fuertes (2010) also notice “that
apparent structural changes may result from having left out an unobserved global variable”. If
5
all of the cross-section entities share the same deterministic components, then such components
can be regarded as a common global factor. In fact, in the special case where the coefficients on
deterministic trends are homogenous, then the cross-section averages of the series of interest can
serve as a good proxy for such common trends. The below figure plots the common nonlinear
trend function along with the augmentation terms of the CCE estimator1:
̅̅̅̅̅̅
Figure A3. Homogenous SMT function against the ∆𝒚̅𝒕 and 𝒚
𝒕−𝟏
∆𝑦̅𝑡 , i  (0.0,0.2) ,   0.5 ,   0.5
yt 1 , i  (0.0,0.2) ,   0.5 ,   0.5
5.0
6
4
2.5
2
0.0
0
-2
-2.5
-4
-5.0
-6
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95 100
6
5
10
15
20
25
30
35
40
45
50
55
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95 100
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-8
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
60
65
70
75
80
85
90
95 100
From the above Figure A3 one can clearly see that the yt 1 used in the CCE estimator mimics
the gradually changing trend function quite well, and hence, this approximation method is
reasonably effective in capturing common trends. Note, however, that the ability of the crosssection averages to imitate common trends crucially hinges on the assumption of homogeneity
of the trend functions. Heterogeneity of the transition parameters, and hence, of the trend
functions reduces the fit of cross-section averages to actual trend functions significantly. Figure
1
It is straightforward to show that the approximation of the unobserved factors a-la Pesaran (2006, 2007), i.e. mean
of the series over cross-section entities, will include the common deterministic trend function. Here, we prefer to
present visual illustration as graphical representation is more clear and understandable.
6
A4 below plots graph of cross-section averages of series along with randomly chosen transition
functions.
̅̅̅̅̅̅
Figure A4. Heterogeneous SMT function against the ∆𝒚̅𝒕 and 𝒚
𝒕−𝟏
yt , i  (0.0,0.2) ,  i  (0.2, 0.8) ,   0.5
7.5
yt 1 , i  (0.0,0.2) ,  i  (0.2, 0.8) ,   0.5
8
6
5.0
4
2
2.5
0
-2
0.0
-4
-2.5
-6
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
2.5
6
0.0
4
-2.5
2
-5.0
0
-7.5
-2
-10.0
-4
-12.5
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95 100
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95 100
-6
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
As the figure suggests, cross-section averages of level and differenced series share almost no
similar pattern with the gradually changing trend functions. Hence, one can reasonably deduce
that the relatively good power performance of the
CADF test depends crucially on the
homogeneity assumption. In fact, simulation results presented in the paper clearly show that the
power of the CADF test declines rapidly as the deterministic components become more
heterogeneous.
7
4. Power Analysis of Bootstrap CCE Test Statistics
Simulation results have revealed that the CADF test performs quite satisfactorily in the case of
homogeneous trend functions but power of the test drops significantly when we allow for
heterogeneity in the transition functions. As we have already noted, relatively better
performance of the CADF test in the case of homogeneous trend setting can be attributed to the
fact that cross-section averages of lagged series imitates the gradually changing trend function
quite well. Also note that homogeneous nonlinear trend function can also be considered as a
common factor affecting individual cross-section units (see also Smith and Fuertes 2010). In
this case, the panel members will share two common factors, one in the deterministic trend and
one imposed in the error terms. It has been documented that multifactor error structure distorts
small sample performance of the CADF
test seriously (see, for example, Gengenbach et al.
2009). It can be argued that the CCE approach tackles only one of the common factors (in our
case, the nonlinear trend function). This calls for additional remedies2. Therefore, we also
consider bootstrapping the CCE tests as a natural extension, to see whether the combination of
two different approaches, i.e., the bootstrap and CCE approaches will bring power gains over
CCE or bootstrap tests.
For this purpose, we use the same DGP given in eq. (20) together with the error structure as in
eq. (18)-(19) in the paper. Again, we generate i ~ iid .U  0.0,0.2 and i ~ iid .U  1.0,3.0
respectively for both the low and the high levels of CSD. For completeness of analysis, we
consider bootstrap versions of both Ct and
CADF tests, which we denote BCt and
BCADF , respectively. Powers of the BCt and BCADF tests are provided below in Table
A2.
2
Pesaran et al. (2013), for example, in addition to cross-section averages of the series of interest, made use of the
information in additional variables that are assumed to share the same common factors as the series under
investigation.
8
We find that the bootstrap CCE tests display almost the same pattern in terms of power
performance as the CCE estimator. In particular, the simulation results show that the BCt
test has better power than the BCADF test for moderate and sharp breaks whereas the latter
test outperforms the former only in the case of rather smooth breaks, i.e., for
. Note that the CADF test also dominated the Ct test in the same interval
where the gradually changing trend function resembles a straight line in short time spans.
Table A2. Power Analysis of bootstrap CCE tests
BCt
BCADF
BCt
BCADF
T/N 5
25
Low cross-section dependence:
BCt
BCADF
50
BCt
BCADF
100
30
50
70
100
0.126
0.280
0.551
0.902
0.236
0.577
0.872
0.995
0.152
0.790
0.994
1.000
0.430
0.992
1.000
1.000
0.228
0.932
1.000
1.000
0.592
1.000
1.000
1.000
0.280
0.996
1.000
1.000
0.772
1.000
1.000
1.000
30
50
70
100
0.472
0.554
0.712
0.946
0.110
0.220
0.364
0.648
0.954
0.988
1.000
1.000
0.128
0.528
0.850
0.986
0.996
1.000
1.000
1.000
0.136
0.732
0.990
1.000
1.000
1.000
1.000
1.000
0.188
0.920
1.000
1.000
30
0.303
0.071
0.637
50
0.383
0.154
0.826
70
0.580
0.275
0.985
100
0.903
0.475
1.000
High cross-section dependence:
0.101
0.318
0.779
0.907
0.858
0.982
1.000
1.000
0.114
0.722
0.955
1.000
0.918
1.000
1.000
1.000
0.141
0.931
1.000
1.000
30
50
70
100
0.118
0.377
0.639
0.854
0.198
0.386
0.741
0.979
0.640
0.864
0.972
0.998
0.034
0.608
0.984
1.000
0.732
0.936
0.998
1.000
0.018
0.680
1.000
1.000
0.800
0.972
1.000
1.000
0.006
0.684
1.000
1.000
30
50
70
100
0.227
0.308
0.477
0.735
0.066
0.098
0.174
0.388
0.308
0.484
0.694
0.958
0.012
0.042
0.070
0.182
0.346
0.494
0.766
0.991
0.002
0.019
0.061
0.161
0.364
0.500
0.822
0.995
0.003
0.000
0.033
0.183
30
50
70
100
0.148
0.179
0.353
0.739
0.055
0.085
0.149
0.260
0.130
0.223
0.552
0.943
0.016
0.044
0.103
0.223
0.178
0.295
0.627
0.979
0.009
0.011
0.060
0.082
0.217
0.360
0.753
0.987
0.000
0.015
0.021
0.081
Note: Threshold value (transition midpoint) has been drawn from  i   0.4, 0.6  . Nominal sizes of both
tests are set to 0.05 for power simulations. The results are based on 5000 replications .
9
The simulation results given in the Table A2 further reveal that bootstrapping of the CCE tests
bring almost no power gains over either CCE or bootstrap tests. In fact, comparison across
Tables 4.2 and 4.3 given in the paper, shows that the bootstrap test Bt and CCE test Ct
have better power performances in most of the cases when compared to the bootstrap CCE test
BCt . We find that the BCt test has relatively better power properties only for the high CSD
case with rather gradual transition function, i.e., for the smoothness parameter in the interval
. In other cases, however, both the Bt and Ct tests dominate the BCt
test in terms of power performance. This finding suggests that both of the remedies are
reasonably efficient in tackling the CSD problem and their combination will bring no power
gains in panels with a trend break.
10
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