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Short communication
Unit root and stationarity tests’ wedding
a ,
*
b b˜
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Josep Lluıs Carrion-i-Silvestre
, Andreu Sanso-i-Rossello , Manuel Artıs Ortuno
a
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Departament d’Econometria, Estadıstica i Economia Espanyola, Universitat de Barcelona, Avd. Diagonal, 690-08034
Barcelona, Spain
b
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Analisi Quantitativa Regional(AQR), Departament d’Econometria, Estadıstica i Economia Espanyola, Universitat de
Barcelona, Barcelona, Spain
Received 25 October 1999; accepted 15 July 2000
Abstract
In this paper the joint confirmation hypothesis (JCH) of unit root for the simultaneous use of the DF and KPSS tests, on the one hand, and the PP and KPSS tests, on the other, is studied. Critical values to be applied when testing this confirmation hypothesis are computed. The performance of this approach is analysed through a Monte Carlo experiment which shows that the use of these critical values produces a better characterization of the DGP when there is a unit root. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Unit root; Dickey–Fuller; Phillips–Perron; KPSS; Joint confirmation hypothesis; Monte Carlo experiments
JEL classification: C12; C15; C22
1. Introduction
Since the 1980s many empirical applications have focused on testing the unit root hypothesis in certain macroeconomic aggregates. Another way to analyse the stochastic properties of time series is through the stationarity tests. Unlike unit root tests, stationarity tests specify the null of the stationary process against the alternative of I(1). Papers by Amano and Van Norden (1992), Kwiatkowski et al. (1992) and Cheung and Chinn (1996), among others, have pointed out the complementarity of the results of the unit root and stationarity tests statistics.
In this paper we try to establish how we might confirm the reliability of the conclusions that are separately obtained by means of the application of the unit root and stationarity tests statistics. Our analysis follows that of Charemza and Syczewska (1998) who, via Monte Carlo experiments, computed the critical values needed for the joint confirmation hypothesis of stationarity when
*Corresponding author: Tel.:134-93-402-1826; fax: 134-93-402-1821.
E-mail address: [email protected] (J.L. Carrion-i-Silvestre).
applying the standard Fuller (1976) and Dickey and Fuller (1979) — hereafter DF — and Kwiatkowski et al. (1992) — hereafter KPSS — test statistics. However, priors that are introduced in the approximation of Charemza and Syczewska — because of the dependency of the critical values on the value of the autoregressive parameter — advise us against specifying the joint confirmation hypothesis of stationarity but recommend specification of the joint confirmation hypothesis of the unit root.
The paper is organized as follows. Section 2 formulates the joint confirmation hypothesis of unit root. Section 3 presents the critical values that have been obtained through a Monte Carlo experiment. In Section 4 we conduct a set of simulation experiments to assess the performance of our proposal. Finally, Section 5 concludes.
2. The unit root confirmation hypothesis
Following the notation of Charemza and Syczewska (1998), we define the joint confirmation hypothesis of unit root as: null hypothesis and i51 for the alternative, ß is the vector of the DGP parameters, and T is the
PJC
˜ sample size. fD,K(?) is the joint density function, PJC is the probability of joint confirmation and (zD ,
PJC
˜zK ) are the critical values associated with the joint hypothesis at a PJC confidence level. Notice that for notational convenience, we focus on the DF–KPSS wedding but it can be easily generalized to the joint application of Phillips (1987) and Phillips and Perron (1988) — hereafter PP — and KPSS test
PJC PJC
˜ ˜
statistics. There exists an infinite number of couples of critical values (zD , zK ) for each value of the PJC, so that in practice we would not be able to compute the PJCs if we did not impose some kind of restriction. A way around this difficulty is to notice that there only exists one pair of critical values that satisfies the following restriction:
3. Tabulation of critical values for the joint hypothesis
Unlike the JCH of stationarity suggested by Charemza and Syczewska (1998), the JCH of a unit root only depends on the elements that are presumed to be in the deterministic component of the time series, that is, the no-constant, no-trend and with-trend statistics. For the JCH of a unit root the DGP of interest is given by:
yt5b1yt211´t. (3)
Using this DGP, the empirical distributions of the no-constant, no-trend and with-trend statistics were
1
obtained for the DF, PP and KPSS test statistics through a Monte Carlo experiment. For the sample sizes T5h50, 100, 150, 300j, y050 and a number of replications n550 000, couples of DF–KPSS and PP–KPSS test statistics were computed. The DGP used in the simulation experiment was that of 3 with drift (b51) and without drift (b50). After sorting the observations according to the DF or PP, 250 fractiles were computed. For each fractile of the DF or PP, 250 fractiles were obtained for the KPSS test statistic. As a result, we were able to construct a 2503250 table of joint frequencies. The next step involved obtaining the cumulative frequencies for the joint distribution function so that the critical values could be tabulated. The bandwidth for the spectral window of the KPSS and PP statistics was chosen following the automatic selection procedure of Newey and West (1994) for the Bartlett (B) and Quadratic spectral (QS) kernels. The DF test statistic was also computed using an automatic selection method for the autoregressive correction based on the t-sig criteria (see Ng and Perron, 1995). Critical values are shown in Table 1 for the DF–KPSS wedding and in Table 2 for the PP–KPSS wedding.
4. The performance of the procedure
In this section we describe a set of simulation experiments undertaken to check the performance of of the critical values of the JCH when choosing between I(1) or I(0) processes. The DGPs used in the simulation experiment are given by the following ARMA(1,1) model:
yt 5 f ts d1ryt211ut
ut 5 ´t1u´t21
wherer5h0.5, 0.85, 0.9, 1j,u5h20.8,20.5, 0j, ´ |iid N(0,1) and, for simplicity, the initial value
t
for the process was set equal to zero, y 50. When r±1, f(t)5h0,1,110.2t , while whenj r51,
0
f(t)5h0,0,0.2 , for the no-constant, no-trend and with-trend test statistics, respectively. This Montej
Carlo simulation experiment is similar to that of Amano and Van Norden (1992). Up to five lag-differences of the endogenous variable are allowed when computing the DF unit root test. The actual number of lag-differences included in the regression is fixed depending on the individual significance of the last lag. The PP and KPSS were computed using the Bartlett and the Quadratic spectral kernels where the bandwidth for the spectral windows was selected using the automatic
1
Table 1
a
Critical values for the joint confirmation hypothesis of unit root for the DF–KPSS wedding
PJC No constant No trend With trend
ADF KPSS(B) KPSS(Q) ADF KPSS(B) KPSS(Q) ADF KPSS(B) KPSS(Q)
T550 0.99 22.909 0.113 0.104 23.888 0.089 0.081 24.579 0.053 0.048 0.97 22.515 0.148 0.137 23.522 0.107 0.098 24.232 0.059 0.054 0.95 22.294 0.176 0.163 23.307 0.121 0.111 24.020 0.063 0.057 0.90 21.969 0.224 0.209 22.968 0.152 0.140 23.687 0.071 0.065 0.85 21.739 0.272 0.255 22.757 0.176 0.163 23.475 0.078 0.071
T5100 0.99 22.809 0.141 0.140 23.738 0.097 0.093 24.365 0.052 0.048 0.97 22.417 0.190 0.189 23.385 0.124 0.121 24.025 0.060 0.056 0.95 22.209 0.231 0.231 23.188 0.142 0.139 23.826 0.066 0.062 0.90 21.915 0.305 0.306 22.879 0.186 0.183 23.516 0.076 0.073 0.85 21.695 0.381 0.382 22.675 0.222 0.219 23.323 0.085 0.082
T5150 0.99 22.735 0.163 0.174 23.649 0.112 0.113 24.258 0.057 0.055 0.97 22.389 0.228 0.248 23.318 0.147 0.149 23.922 0.067 0.065 0.95 22.189 0.270 0.295 23.126 0.171 0.177 23.740 0.073 0.072 0.90 21.899 0.355 0.392 22.833 0.222 0.232 23.444 0.087 0.086 0.85 21.680 0.443 0.490 22.641 0.266 0.281 23.261 0.097 0.097
T5300 0.99 22.694 0.246 0.280 23.592 0.149 0.166 24.159 0.070 0.076 0.97 22.357 0.339 0.389 23.270 0.200 0.225 23.851 0.084 0.092 0.95 22.159 0.414 0.476 23.100 0.238 0.269 23.670 0.095 0.104 0.90 21.875 0.559 0.647 22.812 0.316 0.358 23.391 0.114 0.127 0.85 21.672 0.699 0.818 22.621 0.383 0.436 23.217 0.130 0.145
a
Note: critical values have been obtained through a Monte Carlo experiment win n550 000 replications of the DGP
y 5b 1y 1´ on y 50, whereb 51 for DGP with drift andb 50 for the DGP without the drift,´|iidN (0,1). ADF is
t t21 t 0 t
the DF test statistic with five lags for the initial parametric correction. KPSS ( j ) is the KPSS statistic with four lags for the initial bandwidth; j5B for the Bartlett kernal, j5QS for the quadratic kernal. PJC denotes the probability of joint confirmation.
procedure described in Newey and West (1994) assuming l54 for the initial bandwidth. Then
n52000 replications were conducted for a time series with a sample size of T5100 observations. After computing the unit root and stationarity tests, each time series was classified in one of the following outcomes: (i) when the unit root test does not reject its null hypothesis and the stationarity test does,the process is thought to be I(1); (ii) when the unit root test rejects its null hypothesis but the stationarity test does not, the process is thought to be I(0); (iii) when there are two different situations and the unit root and stationarity tests can lead to inconclusive results (see Cheung and Chinn, 1996). Up to three sets of critical values were considered in these simulation experiments. The first were derived from the marginal distributions of the test statistics, reported in MacKinnon et al. (1991); Shepton (1995) and Hobijn et al. (1998). The second corresponded to the JCH of stationarity of Charemza and Syczewska (1998). Finally, the third set was the JCH of unit root computed in this paper.
For reasons of space, here we shall only report the results of the simulation experiments for the
no-trend test statistic, though a copy of the entire experiment is available on request. The results are
Table 2
a
Critical values for the joint confirmation hypothesis of unit root for the PP–KPSS wedding
PJC No constant No trend With trend
Bartlett Quadratic Bartlett Quadratic Bartlett Quadratic
PP KPSS PP KPSS PP KPSS PP KPSS PP KPSS PP KPSS
T550 0.99 22.718 0.109 22.729 0.101 23.76 0.09 23.78 0.08 24.23 0.05 24.25 0.05 0.97 22.384 0.190 22.402 0.190 23.280 0.126 23.297 0.122 23.853 0.060 23.879 0.056 0.95 22.175 0.230 22.200 0.230 23.093 0.145 23.118 0.142 23.681 0.066 23.706 0.062 0.90 21.887 0.303 21.903 0.304 22.851 0.184 22.868 0.181 23.423 0.076 23.451 0.072 0.85 21.686 0.379 21.692 0.381 22.663 0.221 22.681 0.219 23.238 0.085 23.268 0.081
T5150 0.99 22.724 0.161 22.741 0.170 23.603 0.114 23.607 0.116 24.137 0.056 24.181 0.055 0.97 22.362 0.222 22.377 0.240 23.288 0.147 23.300 0.150 23.827 0.067 23.858 0.065 0.95 22.171 0.269 22.184 0.295 23.112 0.169 23.128 0.175 23.657 0.073 23.682 0.072 0.90 21.892 0.361 21.903 0.397 22.851 0.213 22.861 0.223 23.421 0.085 23.422 0.087 0.85 21.681 0.448 21.686 0.496 22.648 0.257 22.665 0.270 23.219 0.098 23.242 0.098
T5300 0.99 22.704 0.247 22.716 0.283 23.553 0.152 23.574 0.170 24.110 0.070 24.137 0.076 0.97 22.354 0.347 22.361 0.397 23.257 0.204 23.276 0.229 23.822 0.085 23.839 0.093 0.95 22.160 0.427 22.168 0.492 23.081 0.239 23.092 0.270 23.657 0.095 23.672 0.104 0.90 21.883 0.565 21.888 0.655 22.800 0.320 22.813 0.361 23.374 0.114 23.393 0.127 0.85 21.678 0.703 21.681 0.820 22.630 0.386 22.636 0.439 23.199 0.130 23.212 0.145
a
Note: critical values have been obtained through a Monte Carlo experiment win n550 000 replications of the DGP
yt5b 1yt211´ton y050, whereb 51 for DGP with drift andb 50 for the DGP without the drift,´t|iidN (0,1). For PP
and KPSS tests it has been fixed l54 lags for the initial bandwidth. PJC denotes the probability of joint confirmation.
Similar conclusions were obtained for both the DF–KPSS wedding and the PP–KPSS wedding and they were found not to differ across no-constant, no-trend and with-trend test statistics. Likewise, the
results were similar for both kernel types.
Table 3
a
Performance of the different sets of critical values for the joint application of DF and KPSS (no-trend ) test statistics
DGP: yt5f ts d1ryt211u ; ut t5´ 1 u´t t21,
´t|iid N(0,1); y050
r,u I(1)–I(1) I(0)–I(0) I(0)–I(1) I(1)–I(0)
s d
0.5,20.8 Marginal 0 0.9480 0.0520 0
s d
JCH ss d 0 0.9015 0 0.0985
JCH us d 0 0.4710 0.5290 0
0.5,20.5 Marginal 0.0015 0.9230 0.0670 0.0085
s d
JCH ss d 0.0060 0.6975 0.0045 0.2920 JCH us d 0.0055 0.5825 0.4080 0.0040
0.5,0 Marginal 0.0070 0.8865 0.0810 0.0255
s d
JCH ss d 0.0135 0.3750 0.0035 0.6080 JCH us d 0.0225 0.5575 0.4105 0.0095
0.85,20.8 Marginal 0.0015 0.1685 0.8295 0.0005
s d
JCH ss d 0.1705 0.0490 0.0030 0.7775
JCH us d 0.0020 0 0.9980 0
0.85,20.5 Marginal 0.0275 0.4775 0.4745 0.0205
s d
JCH ss d 0.1730 0.0100 0 0.8170
JCH us d 0.0430 0.0555 0.8990 0.0025
0.85,0 Marginal 0.1205 0.5660 0.2080 0.1055
s d
JCH ss d 0.1280 0 0 0.8720
JCH us d 0.2010 0.2010 0.578 0.0200
0.9,20.8 Marginal 0.0000 0.0050 0.9950 0
s d
JCH ss d 0.4880 0.0395 0.0285 0.4440
JCH us d 0 0 1 0
0.9,20.5 Marginal 0.0365 0.1790 0.7825 0.0020
s d
JCH ss d 0.3935 0.0040 0.0020 0.6005
JCH us d 0.0355 0.0025 0.9620 0
0.9,0 Marginal 0.2310 0.3480 0.3300 0.0910
s d
JCH ss d 0.2805 0 0 0.7195
JCH us d 0.3065 0.0710 0.6100 0.0125
1,20.8 Marginal 0.4815 0.2300 0.1700 0.1185
s d
JCH ss d 0.4430 0.0910 0.0085 0.4575 JCH us d 0.5855 0.0705 0.3330 0.0110
1,20.5 Marginal 0.6405 0.0835 0.0360 0.2400
s d
JCH ss d 0.5030 0.0050 0 0.4920
JCH us d 0.8415 0.0345 0.0865 0.0375
1,0 Marginal 0.6905 0.0400 0.0220 0.2475
s d
JCH ss d 0.5285 0 0 0.4715
JCH us d 0.8950 0.0155 0.0470 0.0425
a
Notes: the third column shows the frequency of not to reject the null (no RH ) of the DF test and reject the null (RH ) of0 0
the KPSS test; the fourth column is the frequency of RH for the DF and no RH for the KPSS; the fifth column is the0 0 frequency of RH for the DF and RH for the KPSS; The sixth column in the no RH for the DF and no RH for the KPSS.0 0 0 0
highest frequency was that of the first inconclusive case — this situation might have been produced by a misspecification error of the functional form.
Finally, notice that the classification arising from the critical values of the JCH of stationarity was, except for r50.9 andu5 20.8, always surpassed by both the marginal and the JCH of unit root critical values sets. For the DGP defined by r50.9 andu5 20.8 the critical values of the JCH of stationarity classified the process as stationary more frequently (but only about a 4% of the cases) than the other two sets of critical values. However, with the critical values of the JCH of stationarity the process was classified incorrectly as I(1) about a 48% of the cases whereas this percentage was 0% for the other two sets of critical values. Moreover, these other two sets tended to be inconclusive, suggesting that there might be a misspecification error.
5. Conclusions
The critical values for the JCH of unit root demonstrated certain advantages over the marginal critical values. First, for the DGPs containing a unit root, the frequency of the I(1) classification using the JCH of unit root critical values was higher than that corresponding to the JCH of stationarity and marginal critical values. Second, in those situations where time series were not classified as an I(1) process, the category that recorded a high frequency was the one that might have been produced by misspecification errors in the functional form. However, when the DGP was stationary the best classification was achieved using the marginal critical values. The marginal critical values set outperformed the other two sets of critical values when the DGP was stationary since in almost all cases the stochastic process was classified as I(0). Third, it should be noticed that the classification given by the critical values of the JCH of stationarity given by Charemza and Syczewska (1998) were almost always surpassed by the marginal or the JCH of unit root critical values sets — there was only one exception for stationary DGP where this set of critical values surpassed the others.
Hence, our results suggest that critical values associated with JCH of unit root can lead to a better characterization of stochastic processes when there is a unit root affecting time series behaviour, so they can be used as a confirmation of the unit root hypothesis in applied research.
Acknowledgements
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We are grateful to Antonio Aguirre, Niels Haldrup, Antonio Montanes and an anonymous referee for helpful criticism and comments. Research supported by CICYT SEC99-0693.
References
Amano, R., Van Norden, S., 1992. Unit-Root Tests and the Burden of proof, Bank of Canada.
´ `
Carrion, J.Ll., 1999. Integracio I Estacionarietat de Series Temporals Amb Ruptures Estructurals, Departament d’E-´
conometria, Estadıstica i Economia Espanyola. Universitat de Barcelona.
Cheung, Y.-W., Chinn, M.D., 1996. Further investigation of the Uncertain Unit Root in GNP, 288, University of California, Santa Cruz.
Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 423–431.
Fuller, W.A., 1976. In: Introduction to Time Series Analysis. Wiley, New York.
Hobijn, B., Franses, P.H.B., Ooms, M., 1998. Generalizations of the KPSS-test for Stationarity, 9802. Econometric Institute, Erasmus University Rotterdam.
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P.J., Shin, Y., 1992. Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root. Journal of Econometrics 54, 159–178.
MacKinnon, J.G., Engle, R.F., Granger, C.W.J., 1991. Critical Values for Cointegration Tests, Long-Run Economic Relationships. Readings in Cointegration. Oxford University Press, Oxford, pp. 267–276.
Newey, W.K., West, K.D., 1994. Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61, 631–653.
Ng, S., Perron, P., 1995. Unit root test in ARMA models with data-dependent methods for the selection of the truncation lag. Journal of the American Statistical Association 90, 268–281.
Phillips, P.C.B., 1987. Time series regression with a unit root. Econometrica 55 (2), 277–301.
