Abstract This paper calculates response surface models for a large range ofquantiles of the Leybourne (Oxf Bull Econ Stat 57:559–571, 1995) test for the nullhypothesis of a unit root against the alternative of (trend) stationarity. The responsesurface models allow the estimation of critical values for different combinations ofnumber of observations, T , and lag order in the test regressions, p, where the latter canbe either specified by the user or optimally selected using a data-dependent procedure.The results indicate that the critical values depend on the method used to select thenumber of lags. An Excel spreadsheet is available to calculate the p-value associatedwith a test statistic.
Keywords Monte Carlo · Critical values · Lag length · p-values
JEL Classification C12 · C15
1 Introduction
Testing for unit roots has become common practice in macroeconomic analysis.Among the tests available in the literature, the augmented Dickey and Fuller (ADF)
Work on this paper began while Jesús Otero was a Visiting Fellow in the Department of Economics at theUniversity of Warwick. We would like to thank Jürgen Symanzik (Co-editor), an Associate Editor and twoanonymous referees for their constructive comments and suggestions that helped to improve our paper.The usual disclaimer applies.
J. Otero (B)Facultad de Economía, Universidad del Rosario, Bogotá, Colombiae-mail: [email protected]
J. SmithDepartment of Economics, University of Warwick, Coventry, United Kingdome-mail: [email protected]
123
474 J. Otero, J. Smith
test statistic of Dickey and Fuller (1979) continues being one of the most commonlyapplied testing procedures, due to the simplicity of the test. However, a commoncriticism of the ADF unit root test is that it exhibits poor power properties, see e.g.DeJong et al. (1992).
Several authors have proposed alternative tests to the ADF test in an attempt toaddress the low power problem. Elliott et al. (1996) (ERS) use a conditional gener-alised least squares estimator; Perron and Ng (1996) consider modifications of thePhillips and Perron (1988) tests; Park and Fuller (1995) recommend tests based onweighted symmetric least squares estimation; and Leybourne (1995) takes the maxi-mum of two ADF test statistics, calculated using both forward and reversed data. TheLeybourne (1995) tests compare favourably in terms of size and power with respect tothe others, see Leybourne et al. (2005), but also have the advantage of being relativelystraightforward to compute.1 Leybourne (1995) tabulates critical values for the mostcommonly used specifications of the deterministic components in the test regressions,namely constant (but no trend), and constant and trend. The tabulated critical valuesare based on 20,000 replications, for T = 25, 50, 100, 200 and 400 time observations,but do not account for their possible dependence on lag order.
This paper undertakes an extensive set of Monte Carlo simulations that are sum-marised by means of response surface regressions, from which critical values of theLeybourne (1995) unit root tests can be calculated for different specifications of deter-ministic components, number of observations, and lag order, where the latter canbe either fixed by the user or optimally selected using a data-dependent procedure.Response surfaces have been used, among others, by MacKinnon (1991) to calcu-late critical values of the ADF tests for unit roots and cointegration; Cheung and Lai(1995a) for the ADF tests allowing for dependence on lag order; Cheung and Lai(1995b) for the ERS tests also allowing for dependence on lag order; and Harveyand van Dijk (2006) for the Hylleberg et al. (1990) (HEGY) seasonal unit root tests,making allowance for the lag order in the test regression to be chosen optimally usingcommonly applied selection methods.
The plan of the paper is as follows. Section 2 provides a brief overview of theLeybourne (1995) unit root tests. Section 3 discusses the Monte Carlo simulationdesign and the response surface results. Section 4 illustrates the application of thefinite sample critical values and the Excel spreadsheet for calculating p-values todata on quarterly real exchange rates against the US dollar for 17 OECD countries.Section 5 concludes.
2 The Leybourne unit root tests
Leybourne (1995) proposes a unit root testing procedure that is based on applying theADF test to the forward and reverse realisations of a time series. Let yt denote theforward realisation of a time series, and consider the following DF-type regression forthe model that includes a constant and a trend as the deterministic components:
1 Power gains can also be achieved when seasonal and panel unit root tests are developed based on forwardand reverse estimation; see Leybourne and Taylor (2003) and Smith et al. (2004), respectively.
123
Response surface models for the Leybourne unit root tests 475
�yt = α + γ t + βyt−1 + εt , (1)
where � is the first difference operator, and t = 1, . . . , T time observations. In thissetting the null hypothesis to test the presence of a unit root is H0 : β = 0, against thealternative that the series is a stationary process around a trend, that is H1 : β < 0.Let DF f denote the DF test based on the regression t-statistic for β = 0 in (1). Con-sider now the reverse realisation of yt , which is given by zt = yT +1−t . The DF-typeregression applied to zt is:
�zt = α∗ + γ ∗t + β∗zt−1 + ε∗t , (2)
and let DFr denote the DF test based on the regression t-statistic for β∗ = 0 in (2).Within this framework, Leybourne (1995) defines the DFmax test as:
DFmax = max(DF f , DFr
). (3)
As in the ADF test, the test outlined above is not valid in the presence of serialcorrelation. If this is the case, then the test regressions given by (1) and (2) shouldbe augmented by including lags of the dependent variable, so that Eqs. (1) and (2)become:
�yt = α + γ t + βyt−1 +p∑
j=1
δ j�yt− j + εt , (4)
and
�zt = α∗ + γ ∗t + β∗zt−1 +p∑
j=1
δ∗j �zt− j + ε∗
t . (5)
Then, the ADF statistics from the forward and reverse regressions, denoted ADF f (p)
and ADFr (p) respectively, are used to compute:
ADFmax (p) = max(
ADF f (p) , ADFr (p)). (6)
3 Monte Carlo design and main results
Let us assume that yt is generated by the following first-order autoregressive process:
yt = yt−1 + εt , (7)
where εt ∼ N (0, 1) and t = 1, . . . , T + 1. Simulation experiments are carried outfor a total of 53 different sample sizes, with T = 18 (2) 62, 65 (5) 100, 110 (10) 200,220 (20) 300, 350 (50) 500, 600 (100) 800, where e.g. 18 (2) 62 means that we take allsamples from T = 18 up to T = 62 going up in steps of 2, and so on (the same nota-tion is used later on when listing significance levels). The time series yt is generated
123
476 J. Otero, J. Smith
by setting an initial value y−99 = 0 and then the first 100 observations are discarded.Each experiment consists of 50,000 Monte Carlo replications. The number of laggeddifferences of the dependent variable, p, is set equal to p = 0, 1, . . . , 8. For T ≤ 20we use p ≤ 1; for 22 ≤ T ≤ 24 we use p ≤ 2; for 26 ≤ T ≤ 28 we use p ≤ 3; for30 ≤ T ≤ 32 we use p ≤ 4; for 34 ≤ T ≤ 36, we use p ≤ 6; and for T > 36 we useall values of p. Overall, there are 429 different pairings of T and p.
To allow for sampling variability this setup is repeated 25 times, implying that therewill be 25 critical values of the Leybourne test for each combination of number ofobservations, T , and lag truncation, p. Critical values are calculated at each of 221significance levels (l = 0.0001, 0.0002, 0.0005, 0.001 (0.001) 0.01, 0.015 (0.005)
0.990, 0.991 (0.001) 0.999, 0.9995, 0.9998 and 0.9999) of the ADFmax (p) statis-tic for both the model including a constant (but no trend) and for that including aconstant and trend.
Using the simulated critical values we estimate response surface models (seeMacKinnon (1991), Cheung and Lai (1995a,b) and Harvey and van Dijk (2006)),in which the critical values are regressed on an intercept term, and power functionsof
( 1T
)and
( pT
). The chosen functional form includes up to the fourth power of the( 1
T
)and
( pT
)terms, although experimenting with higher powered terms in general
yielded coefficients that did not turn out to be statistically different from zero at the
1% significance level nor led to any increase in the R2
for these models:
CV lT,p = θ l∞ +
4∑
i=1
θ li
(1
T
)i
+4∑
i=1
φli
( p
T
)i + εl , (8)
where CV lT,p is the critical value estimate at significance level l, T refers to the number
of observations on �yt and �zt (which is one less than the total number of availableobservations), and p is the number of lags. Notice that the functional form in (8) issuch that the larger the number of observations, the weaker is the dependence of thecritical values on the lag truncation. Furthermore, as T → ∞ the intercept term, θ l∞,provides an estimate of the corresponding asymptotic critical value.
Table 1 reports response surface regression estimates for 3 of the 221 significancelevels, namely l = 0.01, 0.05 and 0.10. These estimates can be used to obtain criticalvalues for any given T and fixed lag order, p. However, in practice the lag order, p,is rarely fixed by the user, but is rather selected using a data-dependent procedure.We consider two data-dependent procedures to select the lag order, p. Firstly, fol-lowing Hall (1994) and Ng and Perron (1995), we use a general-to-specific (GTS)algorithm in which we start with some upper bound on p, denoted pmax , wherepmax = 0, 1, 2, . . . , 8, estimate Eq. (4) with p = pmax, and test the statistical sig-nificance of δpmax . If this coefficient is statistically significant, using e.g. significancelevels of 5% (referred to as GTS0.05) or 10% (referred to as GTS0.10), one selectsp = pmax. Otherwise, the order of the estimated autoregression in (4) is reduced byone until the coefficient on the last included lag is found to be statistically significant.Secondly, the Akaike and Schwarz information criteria, which we refer to as AIC andSIC, respectively, can also be used to select the optimal lag order. In this approach,the optimal number of lags is determined by varying p in the forward regression
123
Response surface models for the Leybourne unit root tests 477
(4) between pmax and pmin = 0 lags, and choosing the best model according to theinformation criterion that is being used. Then, for both the GTS algorithm and theinformation criteria the same 221 quantiles of the empirical small sample distributionare recorded as before, but the response surface regression given in (8) is estimatedusing pmax instead of p lags. Table 1 also reports the response surface estimates thatare applicable in the case where the lag order is determined optimally using the fourdata-dependent strategies described above.
In total we estimated 2,210 response surfaces (2 models multiplied by the 5 criteriato select p multiplied by the 221 significance levels). Overall the chosen functionalform performed well. The average coefficient of determination was 0.89 and in only218 cases it was below 0.75 (and overwhelming majority of these cases were forthe SIC with a constant and trend and for the AIC with a constant, in both cases atsignificance levels in the middle of the distribution).
Tables 2 and 3 reports the critical values estimated from the response surface mod-els for selected values of T and p. Interestingly, the implied critical values from theresponse surface models exhibit dependence on the method used to select the laglength, and in some cases the differences may be noticeable, especially when T andl are small, and p is large. In particular, for given T the implied critical values fromthe response surfaces decrease (in an absolute sense) in p when the augmentationorder is fixed by the user, while they increase (in an absolute sense) in pmax when itis optimally determined using either GTS0.05, GTS0.10, AIC or SIC. Similar findingswere obtained by Harvey and van Dijk (2006) when they analysed the HEGY test.
As to be expected, the residuals of the estimated response surfaces exhibit heter-oskedasticity. Thus, to assess the robustness of the OLS results we also consideredestimation using the GMM procedure described in MacKinnon (1994) and MacKinnon(1996). For the purposes of our simulation exercise, this procedure amounts to aver-aging the critical values across the 25 replications for each combination of T and p,and scaling all the variables in (8) by the standard error in these replications. Then,the resulting equation using the re-scaled variables can be estimated by OLS. ThisGMM procedure yields very similar results to those obtained when using OLS, andare therefore not reported here.
Finally, to obtain p-values of the ADFmax (p) statistic, we follow MacKinnon(1994) and MacKinnon (1996) by estimating the regression
�−1(l) = γ l0 + γ l
1̂CV l + γ l
2
(̂CV l
)2 + υl , (9)
where �−1 is the inverse of the cumulative standard normal distribution at each of the
221 quantiles, and ̂CV l is the fitted value from (8) at the l quantile. As in Harvey andvan Dijk (2006), Eq. (9) is estimated by OLS using 15 observations, made up of theactual quantile and the seven quantile observations on either side of the desired quan-
tile.2 Approximate p-values of the ADFmax (p) test statistic can then be obtained as:
2 For l ≤ 0.004 and l ≥ 0.996 we use the actual quantile and the 14 observations closest to the desiredquantile, as there will not be seven on either side.
123
478 J. Otero, J. Smith
Tabl
e1
Res
pons
esu
rfac
ees
timat
esfo
rth
eL
eybo
urne
test
Mod
elC
rite
rium
tol
Inte
rcep
t(S
E)
1/T
1/T
21/
T3
1/T
4(p/
T)
(p/
T)2
(p/
T)3
(p/
T)4
R2
choo
sep
Con
stan
tFi
xed
0.01
−3.0
22(0
.000
9)−3
.105
−135
.557
3514
.18
−331
02.8
0.56
919
.158
−141
.122
281.
290
0.87
0.05
−2.4
31(0
.000
6)−0
.297
−121
.780
3460
.65
−324
07.5
0.48
916
.977
−130
.295
300.
238
0.91
0.10
−2.1
24(0
.000
5)0.
696
−115
.560
3304
.17
−308
25.7
0.46
015
.011
−117
.355
289.
622
0.94
GT
S 0.0
50.
01−3
.024
(0.0
008)
−2.1
50−1
90.4
9637
79.1
0−2
2792
.0−4
.018
24.5
10−1
10.5
4218
4.51
90.
98
0.05
−2.4
33(0
.000
4)0.
102
−145
.049
3830
.68
−308
84.2
−2.2
6211
.198
−54.
931
103.
413
0.98
0.10
−2.1
26(0
.000
3)0.
813
−122
.663
3483
.69
−295
93.0
−1.6
417.
968
−43.
569
90.8
830.
98
GT
S 0.1
00.
01−3
.022
(0.0
009)
−1.3
36−3
38.7
8983
43.6
2−6
2758
.1−5
.347
43.9
86−2
19.3
2640
7.67
20.
98
0.05
−2.4
33(0
.000
5)2.
046
−353
.773
9180
.57
−723
83.0
−3.2
2821
.974
−112
.388
222.
895
0.98
0.10
−2.1
27(0
.000
4)2.
839
−325
.800
8546
.26
−679
87.3
−2.4
0316
.154
−90.
389
192.
034
0.97
AIC
0.01
−3.0
20(0
.001
1)0.
725
−636
.094
1756
3.34
−146
006.
0−4
. 447
51.2
68−2
84.0
3657
2.51
50.
94
0.05
−2.4
33(0
.000
8)4.
559
−596
.581
1462
7.42
−108
902.
1−2
.802
37.6
58−2
14.6
3744
5.26
60.
86
0.10
−2.1
26(0
.000
6)4.
585
−483
.566
1162
3.61
−851
52.4
−2.0
2829
.405
−167
.265
352.
395
0.76
SIC
0.01
−3.0
28(0
.001
4)7.
933
−103
3.31
520
837.
39−1
2142
5.7
−1.7
4939
.017
−225
.249
471.
153
0.83
0.05
−2.4
33(0
.000
9)4.
892
−513
.573
9743
.32
−522
67.5
−0.7
4324
.757
−142
.003
295.
369
0.80
0.10
−2.1
25(0
.000
7)4.
086
−371
.433
7287
.16
−420
26.7
−0.4
0318
.625
−108
.197
223.
870
0.82
123
Response surface models for the Leybourne unit root tests 479
Tabl
e1
cont
inue
d
Mod
elC
rite
rium
tol
Inte
rcep
t(S
E)
1/T
1/T
21/
T3
1/T
4(p/
T)
(p/
T)2
(p/
T)3
(p/
T)4
R2
choo
sep
Con
stan
tan
dtr
end
Fixe
d0.
01−3
.673
(0.0
010)
−4.8
05−2
61.5
7664
16.3
5−5
8477
.50.
579
27.2
32−2
08.1
0543
6.58
20.
91
0.05
−3.1
13(0
.000
7)−1
.680
−166
.707
4726
.04
−455
75.9
0.62
022
.575
−172
.512
397.
753
0.92
0.10
−2.8
24(0
.000
6)−0
.309
−144
.275
4304
.05
−419
78.4
0.62
320
.205
−154
.358
376.
795
0.94
GT
S 0.0
50.
01−3
.674
(0.0
009)
−3.3
02−4
13.2
7696
00.6
1−7
2359
.9−6
.677
51.3
21−2
41.2
7442
7.80
90.
99
0.05
−3.1
16(0
.000
5)−0
.275
−263
.343
6016
.01
−426
90.5
−4.2
9925
.216
−111
.980
194.
709
0.99
0.10
−2.8
27(0
.000
4)0.
422
−190
.183
4761
.22
−363
91.3
−3.2
8917
.160
−79.
073
146.
643
0.99
GT
S 0.1
00.
01−3
.672
(0.0
009)
−3.9
77−4
39.4
0011
609.
09−9
5553
.1−8
.168
80.9
62−4
25.9
0182
5.90
00.
98
0.05
−3.1
13(0
.000
7)0.
766
−434
.057
1097
9.75
−841
64.7
−5.8
0646
.432
−222
.802
413.
029
0.98
0.10
−2.8
25(0
.000
6)2.
348
−419
.485
1083
0.92
−844
87.4
−4.6
3033
.059
−156
.693
296.
811
0.98
AIC
0.01
−3.6
59(0
.001
0)−8
.789
−241
. 991
1077
8.79
−117
882.
5−6
.664
69.6
42−3
54.8
1470
4.05
70.
98
0.05
−3.1
12(0
.000
9)2.
951
−770
.885
2187
4.78
−185
518.
1−5
.027
55.2
52−3
14.4
2866
9.11
60.
96
0.10
−2.8
25(0
.000
9)5.
579
−780
.928
2052
6.30
−162
974.
1−4
.170
47.9
65−2
78.3
8459
3.08
80.
93
SIC
0.01
−3.6
85(0
.001
8)15
.191
−216
5.68
353
327.
44−4
0308
8.0
−3.3
4355
.477
− 332
.504
684.
740
0.90
0.05
−3.1
20(0
.001
4)10
.506
−115
8.45
323
788.
33−1
4198
6.6
−2.0
5540
.513
−240
.406
504.
619
0.82
0.10
−2.8
27(0
.001
1)8.
050
−796
.919
1552
5.21
−858
04.7
−1.4
7232
.422
−192
.324
397.
859
0.75
Whi
te’s
hete
rero
sked
astic
cons
iste
ntst
anda
rder
rors
ofθ ∞
are
repo
rted
inpa
rent
hese
s
123
480 J. Otero, J. Smith
Tabl
e2
Lag
orde
ran
dfin
ite-s
ampl
ecr
itica
lval
ues
for
the
mod
elw
ithco
nsta
nt
Cri
teri
umto
choo
sep
Lag
sl=
0.01
l=
0.05
l=
0.10
T=
2550
100
200
400
T=
2550
100
200
400
T=
2550
100
200
400
Fixe
d0
−3.2
2−3
.12
−3.0
6−3
.04
−3.0
3−2
.50
−2.4
6−2
.44
−2.4
4−2
.43
−2.1
5−2
.14
−2.1
3−2
.12
−2.1
2
1−3
.18
−3.1
0−3
.06
−3.0
4−3
.03
−2.4
6−2
.45
−2.4
4−2
.43
−2.4
3−2
.11
−2.1
2−2
.12
−2.1
2−2
.12
2−3
.12
−3.0
7−3
.05
−3.0
3−3
.03
−2.4
1−2
.42
−2.4
3−2
.43
−2.4
3−2
.06
−2.1
0−2
.11
−2.1
2−2
.12
3–
−3.0
4−3
.03
−3.0
3− 3
.03
–−2
.40
−2.4
2−2
.42
−2.4
3–
−2.0
8−2
.10
−2.1
1−2
.12
GT
S 0.0
50
−3.2
3−3
.12
−3.0
6−3
.04
−3.0
3−2
.50
−2.4
6−2
.44
−2.4
4−2
.43
−2.1
4−2
.14
−2.1
3−2
.12
−2.1
2
1−3
.36
−3.1
9−3
.10
−3.0
6−3
.04
−2.5
7−2
.50
−2.4
6−2
.45
−2.4
4−2
.20
−2.1
7−2
.14
−2.1
3−2
.13
2−3
.45
−3.2
4−3
.13
−3.0
8−3
.05
−2.6
3−2
.54
−2.4
8−2
.46
−2.4
5−2
.24
−2.1
9−2
.16
−2.1
4−2
.13
3–
−3.2
9−3
.16
−3.0
9− 3
.06
–−2
.57
−2.5
0−2
.47
−2.4
5–
−2.2
1−2
.17
−2.1
5−2
.14
GT
S 0.1
00
−3.2
4−3
.13
−3.0
6−3
.04
−3.0
3−2
.52
−2.4
7−2
.44
−2.4
3−2
.43
−2.1
6−2
.14
−2.1
2−2
.12
−2.1
2
1−3
.40
−3.2
2−3
.11
−3.0
6−3
.04
−2.6
2−2
.53
−2.4
7−2
.45
−2.4
4−2
.24
−2.1
8−2
.15
−2.1
3−2
.13
2−3
.49
−3.2
8−3
.15
−3.0
9−3
.05
−2.6
8−2
.57
−2.5
0−2
.46
−2.4
5−2
.29
−2.2
2−2
.17
−2.1
4−2
.13
3–
−3.3
3−3
.19
−3.1
1− 3
.06
–−2
.61
−2.5
2−2
.47
−2.4
5–
−2.2
5−2
.18
−2.1
5−2
.14
AIC
0−3
.26
−3.1
4−3
.06
−3.0
3−3
.02
−2.5
5−2
.48
−2.4
3−2
.42
−2.4
3−2
.19
−2.1
5−2
.12
−2.1
1−2
.12
1−3
.37
−3.2
1−3
.10
−3.0
5−3
.03
−2.6
1−2
.52
−2.4
6−2
.44
−2.4
3−2
.23
−2.1
8−2
.14
−2.1
2−2
.12
2−3
.41
−3.2
6−3
.13
−3.0
7−3
.04
−2.6
2−2
.55
−2.4
8−2
.45
−2.4
4− 2
.24
−2.1
9−2
.15
−2.1
3−2
.13
3–
−3.2
8−3
.16
−3.0
9−3
.05
–−2
.55
−2.4
9−2
.46
−2.4
4–
−2.2
0−2
.16
−2.1
4−2
.13
123
Response surface models for the Leybourne unit root tests 481
Tabl
e2
cont
inue
d
Cri
teri
umto
choo
sep
Lag
sl=
0.01
l=
0.05
l=
0.10
T=
2550
100
200
400
T=
2550
100
200
400
T=
2550
100
200
400
SIC
0−3
.34
−3.1
4−3
.03
−3.0
1−3
.01
−2.5
7−2
.47
−2.4
3−2
.42
−2.4
2−2
.20
−2.1
4−2
.11
−2.1
1−2
.12
1−3
.36
−3.1
6−3
.05
−3.0
2−3
.02
−2.5
7−2
.48
−2.4
3−2
.42
−2.4
3−2
.19
−2.1
4−2
.12
−2.1
1−2
.12
2−3
.33
−3.1
6−3
.05
−3.0
3−3
.02
−2.5
3−2
.47
−2.4
3−2
.42
−2.4
3−2
.16
−2.1
3−2
.12
−2.1
1−2
.12
3–
−3.1
4−3
.06
−3.0
3− 3
.03
–−2
.45
−2.4
3−2
.43
−2.4
3–
−2.1
2−2
.11
−2.1
1−2
.12
Ley
bour
ne0
−3.2
5−3
.17
−3.1
1−3
.06
−3.0
4−2
.50
−2.4
8−2
.45
−2.4
4−2
.43
−2.1
5−2
.14
−2.1
4−2
.13
−2.1
3
Ley
bour
necr
itica
lval
ues
are
take
nfr
omL
eybo
urne
(199
5)Ta
ble
1
123
482 J. Otero, J. Smith
Tabl
e3
Lag
orde
ran
dfin
ite-s
ampl
ecr
itica
lval
ues
for
the
mod
elw
ithco
nsta
ntan
dtr
end
Cri
teri
umto
choo
sep
Lag
sl=
0.01
l=
0.05
l=
0.10
T=
2550
100
200
400
T=
2550
100
200
400
T=
2550
100
200
400
Fixe
d0
−4.0
2−3
.83
−3.7
4−3
.70
−3.6
9−3
.26
−3.1
8−3
.14
−3.1
3−3
.12
−2.9
0−2
.86
−2.8
4−2
.83
−2.8
3
1−3
.97
−3.8
1−3
.73
−3.7
0−3
.68
−3.2
1−3
.16
−3.1
3−3
.12
−3.1
2−2
.85
−2.8
4−2
.83
−2.8
3−2
.82
2−3
.89
−3.7
8−3
.72
−3.6
9−3
.68
−3.1
4−3
.13
−3.1
2−3
.12
−3.1
1−2
.78
−2.8
1−2
.82
−2.8
2−2
.82
3–
−3.7
4−3
.70
−3.6
9− 3
.68
–−3
.10
−3.1
1−3
.11
−3.1
1–
−2.7
8−2
.80
−2.8
2−2
.82
GT
S 0.0
50
−4.0
4−3
.84
−3.7
4−3
.70
−3.6
8−3
.27
−3.1
9−3
.14
−3.1
2−3
.12
−2.9
0−2
.86
−2.8
4−2
.83
−2.8
3
1−4
.24
−3.9
5−3
.80
−3.7
3−3
.70
−3.4
1−3
.26
−3.1
8−3
.14
−3.1
3−3
.01
−2.9
2−2
.87
−2.8
4−2
.83
2−4
.35
−4.0
4−3
.85
−3.7
6−3
.72
−3.5
0−3
.32
−3.2
2−3
.16
−3.1
4−3
.09
−2.9
7−2
.90
−2.8
6−2
.84
3–
−4.1
0−3
.90
−3.7
9− 3
.73
–−3
.37
−3.2
5−3
.18
−3.1
5–
−3.0
1−2
.92
−2.8
7−2
.85
GT
S 0.1
00
−4.0
4−3
.85
−3.7
5−3
.70
−3.6
8−3
.29
−3.2
0−3
.14
−3.1
2−3
.11
−2.9
3−2
.87
−2.8
3−2
.82
−2.8
2
1−4
.26
−3.9
8−3
.82
−3.7
4−3
.70
−3.4
6−3
.30
−3.1
9−3
.15
−3.1
3−3
.07
−2.9
5−2
.88
−2.8
4−2
.83
2−4
.36
−4.0
7−3
.88
−3.7
8−3
.72
−3.5
5−3
.37
−3.2
4−3
.17
−3.1
4−3
.15
−3.0
1−2
.91
−2.8
7−2
.84
3–
−4.1
3−3
.93
−3.8
1− 3
.74
–−3
.42
−3.2
8−3
.20
−3.1
6–
−3.0
6−2
.95
−2.8
9−2
.85
AIC
0−4
.01
−3.8
6−3
.76
−3.7
1−3
.68
−3.3
0−3
.22
−3.1
4−3
.11
−3.1
1−2
.95
−2.8
9−2
.83
−2.8
1−2
.82
1−4
.19
−3.9
7−3
.82
−3.7
4−3
.70
−3.4
3−3
.30
−3.1
8−3
.14
−3.1
2−3
.06
−2.9
5−2
.87
−2.8
3−2
.83
2−4
.25
−4.0
4−3
.87
−3.7
7−3
.71
−3.4
8−3
.35
−3.2
2−3
.16
−3.1
3− 3
.10
−2.9
9−2
.89
−2.8
5−2
.84
3–
−4.0
8−3
.91
−3.7
9−3
.73
–−3
.38
−3.2
5−3
.18
−3.1
4–
−3.0
2−2
.92
−2.8
7−2
.84
123
Response surface models for the Leybourne unit root tests 483
Tabl
e3
cont
inue
d
Cri
teri
umto
choo
sep
Lag
sl=
0.01
l=
0.05
l=
0.10
T=
2550
100
200
400
T=
2550
100
200
400
T=
2550
100
200
400
SIC
0−4
.16
−3.8
8−3
.70
−3.6
6−3
.66
−3.3
9−3
.21
−3.1
1−3
.09
−3.1
0−3
.01
−2.8
7−2
.81
−2.8
0−2
.81
1−4
.23
−3.9
3−3
.73
−3.6
7−3
.67
−3.4
3−3
.23
−3.1
2−3
.10
−3.1
1−3
.02
−2.8
9−2
.82
−2.8
1−2
.82
2−4
.22
−3.9
5−3
.75
−3.6
8−3
.67
−3.4
0−3
.24
−3.1
3−3
.11
−3.1
1−3
.00
−2.8
9−2
.83
−2.8
2−2
.82
3–
−3.9
5−3
.76
−3.7
0− 3
.68
–−3
.23
−3.1
4−3
.12
−3.1
1–
−2.8
8−2
.83
−2.8
2−2
.82
Ley
bour
ne0
−3.9
9−3
.84
−3.7
5−3
.72
−3.7
0−3
.26
−3.2
2−3
.16
−3.1
2−3
.09
−2.8
9−2
.87
−2.8
4−2
.83
−2.8
2
Ley
bour
necr
itica
lval
ues
are
take
nfr
omL
eybo
urne
(199
5)Ta
ble
2
123
484 J. Otero, J. Smith
p-value = �(γ̂ l
0 + γ̂ l1 ADFmax (p) + γ̂ l
2 (ADFmax (p))2)
, (10)
where γ̂ l0, γ̂
l1 and γ̂ l
2 are the OLS parameter estimates from (9). An Excel spreadsheetthat calculates critical values (for l = 0.01, 0.05 and 0.010) as well as the p-valueof any ADFmax (p) test statistic is available at http://www2.warwick.ac.uk/fac/soc/economics/staff/academic/jeremysmith/research.
4 Empirical application
Purchasing power parity (PPP) is one of the building blocks of the international eco-nomics literature. According to PPP, exchange rates should equalise relative pricelevels in different countries. Thus, if over a period of time domestic prices have beenrising more than foreign prices, an exchange rate adjustment is called for in order torestore a country’s international competitiveness. However, even if the presence of acompetitive market structure will tend to move the exchange rate towards PPP, suchadjustment may take time to occur because of the presence of transport costs, tariffsand other barriers to trade. This leads to the view that PPP may not hold in the shortrun but in the long run. Essentially, this is a question of whether the real exchangerate, defined as the nominal exchange rate multiplied by the ratio of the foreign pricelevel to the domestic price level, may be characterised as a stationary process.
This section illustrates the use of the finite sample critical values derived earlierand the Excel spreadsheet for calculating p-values to answer this question. The datacorrespond to quarterly real exchange rates against the US dollar for the following17 OECD countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France,Germany, Italy, Japan, Netherlands, New Zealand, Norway, Spain, Sweden, Switzer-land and United Kingdom. Real exchange rates are calculated as the logarithm of theunits of domestic currency per US dollar, plus the logarithm of the consumer priceindex in the US, minus the logarithm of the consumer price index in the domesticcountry. The number of observations is 104, covering the period 1973Q1 to 1998Q4.The data set, which was recently used by Pesaran (2007), was downloaded fromthe Journal of Applied Econometrics Data Archive; the interested reader is thereforereferred to http://econ.queensu.ca/jae/2007-v22.2/pesaran/, where further details onthe construction of the series and their sources can be found.
Table 4 summarises the results of applying the Leybourne unit root test statistic tothe real exchange rate data. Interestingly, the results highlight that in some cases infer-ence may depend upon the procedure used to select the augmentation order of the testregressions. Indeed, notice that in the cases of Finland, Germany, Italy, Netherlandsand the United Kingdom the test statistic rejects the null hypothesis of a unit root atthe 5% significance level when using a fixed lag length of p = 4. By contrast, forthese countries we fail to reject the unit root hypothesis (also at the 5% significancelevel) when using the data-dependent selection procedures. Similar mixed findings areobtained for Austria and Denmark when using a significance level of 10%, that is, thenull hypothesis is rejected when using a fixed lag length of p = 4, but not when thelag order is optimally chosen.
Critical value 1% −3.019 −3.185 −3.213 −3.169 −3.052
Critical value 5% −2.405 −2.517 −2.537 −2.496 −2.424
Critical value 10% −2.091 −2.181 −2.196 −2.160 −2.107
Test regressions include intercept. To estimate critical values and p-values we use T = 103, and set p = 4when the augmentation order is fixed, and pmax = 4 when it is optimally determined
5 Conclusions
This paper uses computer-intensive methods to estimate response surface models forthe critical values of the Leybourne test for the null hypothesis of a unit root against thealternative of (trend) stationarity. The response surface models, which are estimatedfor a total of 221 significance levels, are a function of the number of observations, T ,and lags of the dependent variable used in the test regressions, p. The fit of the 221response surface models is good with almost all models having an R2 in excess of 0.75.The number of lags of the dependent variable that are included in the test regression canbe either specified by the user, or optimally selected using a data-dependent procedure.It turns out that this choice affects the critical values. Models are then estimated toenable the calculation of finite sample probability values of the tests. As an empiricalapplication we consider the question of whether the real exchange rates of a sample of17 OECD countries can be best described as stationary processes. We find that in someinstances inference may depend upon the procedure employed to select the number oflags to include in the test regressions.
123
486 J. Otero, J. Smith
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