Power of Unit Root Tests Against Nonlinear and Noncausal Alternatives with an Application to the Brent Crude Oil Price

Frédérique Bec; Alain Guay; Heino Bohn Nielsen; Sarra Saïdi

doi:10.1515/snde-2022-0084

Article

Power of Unit Root Tests Against Nonlinear and Noncausal Alternatives with an Application to the Brent Crude Oil Price

Frédérique Bec , Alain Guay , Heino Bohn Nielsen and Sarra Saïdi

Published/Copyright: December 15, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Studies in Nonlinear Dynamics & Econometrics Volume 29 Issue 1

Abstract

The increasing sophistication of economic and financial time series modelling creates a need for a test of the time dependence structure of the series which does not require a proper specification of the alternative. Indeed, the latter is unknown beforehand. Yet, the stationarity has to be established before proceeding to the estimation and testing of causal/noncausal or linear/nonlinear models as their econometric theory has been developed under the maintained assumption of stationarity. In this paper, we propose a new unit root test statistics which is both asymptotically consistent against all stationary alternatives and still keeps good power properties in finite sample. A large simulation study is performed to assess the power of our test compared to existing unit root tests built specifically for various kinds of stationary alternatives, when the true DGP is either causal or noncausal, linear or nonlinear stationary. Based on various sample sizes and degrees of persistence, it turns out that our new test performs very well in terms of power in finite sample, no matter the alternative under consideration. The proposed approach is illustrated using recent Brent crude oil price data.

Keywords: unit root test; threshold autoregressive model; noncausal model; oil price

JEL Classification: C12; C22; C46

Corresponding author: Frédérique Bec, Thema, CY Cergy Paris University, Cergy and CREST, Palaiseau, France, E-mail: frederique.bec@cyu.fr

Funding source: Labex MMEDII (ANR11-LBX-0023-01)

Funding source: Institute for Advanced Studies at CY Cergy Paris University

Acknowledgment

The authors are grateful to the Editor, Brice Mizrach, and the anonymous reviewer for their helpful comments. Previous versions of this work have also benefited from fruitful remarks by participants at the QFFE 2023 conference in Marseille and the CEF 2023 conference in Nice.

Research funding: F. Bec and S. Saïdi acknowledge financial support from the Labex MMEDII (ANR11-LBX-0023-01). H. B. Nielsen acknowledges financial support from the Institute for Advanced Studies at CY Cergy Paris University.

Appendix

Proof of Theorem 1

This directly results from the fact that the statistic ADF_T diverges with exact order T in probability for any {y _t} in H ₁ and the TAR specification (2) is equivalent to the autoregressive linear model when the threshold is λ _T = |y|₍₀₎.^[11] In this case, the lower and the central regimes vanish and the SETAR shrinks to the auxiliary model of the ADF test: Δy _t = a(L)Δy _t−1 + μ + ρy _t−1 + u _t. □

Before turning to the next theorem, let us first define for the outer regime:

(7) ξ O ( λ ) = ∫ 0 1 W ( v ) I I out ( λ ) W ( v ) d W ( v ) − ∫ 0 1 W ( v ) I I out ( λ ) W ( v ) d v ∫ 0 1 I I out ( λ ) W ( v ) d v ∫ 0 1 I I ℓ ( λ ) − I I u ( λ ) W ( v ) d W ( v ) ∫ 0 1 W 2 ( v ) I I out ( λ ) W ( v ) d v − ∫ 0 1 W ( v ) I I out ( λ ) W ( v ) d v 2 ∫ 0 1 I I out ( λ ) W ( v ) d v 1 / 2 ,

where I I denotes the indicator function which takes value one if y _t ∈ I and zero otherwise. I _out(λ) = I _ℓ(λ) ∪ I _u(λ), with I ℓ ( λ ) = − ∞ , − λ and I u ( λ ) = λ , + ∞ .

Theorem A.1

Consider the TAR specification (2) with ρ ₂ = 0. Let Λ_T be as in (3) and assume that Assumption E(s) given in Section 7 of Bec, Guay, and Guerre (2008a) for s > 4 holds. Then, under H ₀, t T i ( Λ T ) converges in distribution to inf λ ∈ Λ ξ O ( λ / σ ) , which has a pivotal distribution.

Proof of Theorem A.1

Theorem A.1 follows directly from Theorem 2, p.99 of Bec, Guay, and Guerre (2008a) but without the inner regime. In particular, it can be seen that for λ ̲ T = | y | ( 0 ) ,

ξ O ( λ ̲ T ) = ∫ 0 1 W ( v ) d W ( v ) − W ( 1 ) ∫ 0 1 W ( v ) d v ∫ 0 1 W 2 ( v ) d v − ∫ 0 1 W ( v ) d v 2 1 / 2

which is the limit distribution of the ADF_T statistic. □

Theorem A.2

Consider the TAR specification (2) with ρ ₂ = 0. Let Λ_T be as in (3) and assume that Assumption E(s) given in Section 7 of Bec, Guay, and Guerre (2008a) for s > 4 holds. Then, under H ₀,

(8) t T a ( Λ T ) ⇒ ∫ 0 1 ξ O ( λ / σ ) d λ and t T e ( Λ T ) ⇒ ∫ 0 1 exp ξ O ( λ / σ ) 2 d λ

where ⇒ means convergence in distribution.

Proof of Theorem A.2

This result is directly obtained through the application of the continuous mapping theorem, see Pollard (1984). □

Table A.1:

Empirical critical values of t _all statistics.

	T = 100			T = 250			T = 500
	1 %	5 %	10 %	1 %	5 %	10 %	1 %	5 %	10 %
t all i	−4.271	−3.715	−3.433	−4.312	−3.788	−3.526	−4.377	−3.858	−3.607
t all a	−3.278	−2.750	−2.476	−3.229	−2.726	−2.472	−3.226	−2.730	−2.481
t all e	0.208	0.268	0.305	0.212	0.269	0.304	0.212	0.268	0.302
	T = 1000			T = 10,000
	1 %	5 %	10 %	1 %	5 %	10 %
t all i	−4.434	−3.921	−3.679	−4.584	−4.112	−3.876
t all a	−3.242	−2.738	−2.490	−3.238	−2.745	−2.498
t all e	0.210	0.266	0.299	0.209	0.265	0.298

Note: Empirical critical values computed from 100,000 replications.

Figure A.1:

Power of t _all unit root tests as a function of (1 − ρ) for T = 250.

Figure A.2:

Power of t _b unit root tests as a function of (1 − ρ) for T = 250.

Figure A.3:

Power of t _ks unit root tests as a function of (1 − ρ) for T = 250.

Figure A.4:

Power of t ^e unit root tests as a function of (1 − ρ) for T = 250.

Figure A.5:

Power of W _b unit root tests as a function of (1 − ρ) for T = 250.

References

Arltová, M., and D. Fedorová. 2016. “Selection of Unit Root Test on the Basis of Length of the Time Series and Value of AR(1) Parameter.” Statistika – Statistics and Economics Journal 96: 47–64.Search in Google Scholar

Balke, N., and T. Fomby. 1997. “Threshold Cointegration.” International Economic Review 38: 627–45. https://doi.org/10.2307/2527284.Search in Google Scholar

Bec, F., M. Ben Salem, and M. Carrasco. 2004. “Test for Unit-Root versus Threshold Specification with an Application to the PPP.” Journal of Business & Economic Statistics 22: 382–95. https://doi.org/10.1198/073500104000000389.Search in Google Scholar

Bec, F., M. Ben Salem, and M. Carrasco. 2010. “Detecting Mean Reversion in Real Exchange Rates from a Multiple Regime STAR Model.” Annals of Economics and Statistics 99/100: 395–427. https://doi.org/10.2307/41219172.Search in Google Scholar

Bec, F., H. Bohn Nielsen, and S. Saïdi. 2020. “Mixed Causal–Noncausal Autoregressions: Bimodality Issues in Estimation and Unit Root Testing.” Oxford Bulletin of Economics & Statistics 82: 1413–28. https://doi.org/10.1111/obes.12372.Search in Google Scholar

Bec, F., A. Guay, and E. Guerre. 2008a. “Adaptive Consistent Unit Root Tests Based on Autoregressive Threshold Model.” Journal of Econometrics 142: 94–133. https://doi.org/10.1016/j.jeconom.2007.05.011.Search in Google Scholar

Bec, F., A. Rahbek, and N. Shephard. 2008b. “The Autoregressive Conditional Root (ACR) Model.” Oxford Bulletin of Economics & Statistics 70 (5): 583–618. https://doi.org/10.1111/j.1468-0084.2008.00512.x.Search in Google Scholar

Bierens, H. 1997a. “Testing the Unit Root with Drift Hypothesis against Nonlinear Trend Stationarity, with an Application to the US Price Level and Interest Rate.” Journal of Econometrics 81: 29–64. https://doi.org/10.1016/s0304-4076(97)00033-x.Search in Google Scholar

Bierens, H. 1997b. “Nonparametric Cointegration Analysis.” Journal of Econometrics 77: 379–404. https://doi.org/10.1016/s0304-4076(96)01820-9.Search in Google Scholar

Breidt, F., and R. Davis. 1992. “Time-Reversibility, Identifiability and Independence of Innovations for Stationary Time Series.” Journal of Time Series Analysis 13: 377–90. https://doi.org/10.1111/j.1467-9892.1992.tb00114.x.Search in Google Scholar

Breidt, F., R. Davis, and W. Dunsmuir. 1992. “On Backcasting in Linear Time Series Models.” In New Directions in Time Series Analysis. Part I, edited by D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt, and M. Taqqu, 25–40. New-York: Springer-Verlag.Search in Google Scholar

Breidt, F., R. A. Davis, K. S. Lh, and M. Rosenblatt. 1991. “Maximum Likelihood Estimation for Noncausal Autoregressive Processes.” Journal of Multivariate Analysis 36: 175–98. https://doi.org/10.1016/0047-259x(91)90056-8.Search in Google Scholar

Breitung, Jorg. 2002. “Nonparametric Tests for Unit Roots and Cointegration.” Journal of Econometrics 108: 343–63. https://doi.org/10.1016/s0304-4076(01)00139-7.Search in Google Scholar

Cambanis, S., and I. Fakhre-Zakeri. 1994. “On Prediction of Heavy-Tailed Autoregressive Sequences: Forward versus Reversed Time.” Teoriya Veroyatnostei i ee Primeneniya 39: 294–312. https://doi.org/10.1137/1139013.Search in Google Scholar

Cambanis, S., and I. Fakhre-Zakeri. 1996. “Forward and Reversed Time Prediction of Autoregressive Sequences.” Journal of Applied Probability 33: 1053–60. https://doi.org/10.1017/s0021900200100476.Search in Google Scholar

Choi, I. 2015. Almost All About Unit Roots: Foundations, Developments, and Applications. Cambridge: Cambridge University Press.10.1017/CBO9781316157824Search in Google Scholar

Dickey, D., and W. Fuller. 1979. “Distribution of the Estimators for Autoregressive Time Series with a Unit Root.” Journal of the American Statistical Association 74: 427–31. https://doi.org/10.2307/2286348.Search in Google Scholar

Elliott, G., T. Rothenberg, and J. Stock. 1996. “Efficient Tests for an Autoregressive Unit Root.” Econometrica 64: 813–36. https://doi.org/10.2307/2171846.Search in Google Scholar

Enders, W., and C. W. J. Granger. 1998. “Unit-Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates.” Journal of Business & Economic Statistics 16: 304–11. https://doi.org/10.2307/1392506.Search in Google Scholar

Findley, D. 1986. “On Bootstrap Estimates of Forecast Mean Square Errors for Autoregressive Processes.” In Computer Science and Statistics: The Interface, edited by D. Allen, 11–7. Amsterdam: North-Holland.Search in Google Scholar

Fries, S. 2021. “Conditional Moments of Noncausal Alpha-Stable Processes and the Prediction of Bubble Crash Odds.” Journal of Business & Economic Statistics 40: 1596–616, https://doi.org/10.1080/07350015.2021.1953508.Search in Google Scholar

Fries, S., and J.-M. Zakoian. 2019. “Mixed Causal-Noncausal AR Processes and the Modelling of Explosive Bubbles.” Econometric Theory 35: 1234–70. https://doi.org/10.1017/s0266466618000452.Search in Google Scholar

Gouriéroux, C., and J. Jasiak. 2016. “Filtering, Prediction and Simulation Methods for Noncausal Processes.” Journal of Time Series Analysis 37: 405–30. https://doi.org/10.1111/jtsa.12165.Search in Google Scholar

Gouriéroux, C., and C. Robert. 2006. “Stochastic Unit Root Models.” Econometric Theory 22: 1052–90. https://doi.org/10.1017/s0266466606060518.Search in Google Scholar

Gouriéroux, C., and J.-M. Zakoian. 2015. “On Uniqueness of Moving Average Representation of Heavy Tailed Stationary Processes.” Journal of Time Series Analysis 36: 876–87. https://doi.org/10.1111/jtsa.12139.Search in Google Scholar

Gouriéroux, C., and J.-M. Zakoian. 2017. “Local Explosion Modelling by Noncausal Process.” Journal of the Royal Statistical Society B 79: 737–56. https://doi.org/10.1111/rssb.12193.Search in Google Scholar

Gourieroux, C., J. Jasiak, and M. Tong. 2021. “Convolution-Based Filtering and Forecasting: An Application to WTI Crude Oil Prices.” Journal of Forecasting 40 (7): 1230–44. https://doi.org/10.1002/for.2757.Search in Google Scholar

Hecq, A., and E. Voisin. 2021. “Forecasting Bubbles with Mixed Causal-Noncausal Autoregressive Models.” Econometrics and Statistics 20: 29–45. https://doi.org/10.1016/j.ecosta.2020.03.007.Search in Google Scholar

Hecq, A., J. V. Issler, and S. Telg. 2020. “Mixed Causal–Noncausal Autoregressions with Exogenous Regressors.” Journal of Applied Econometrics 35: 328–43. https://doi.org/10.1002/jae.2751.Search in Google Scholar

Hencic, A., and C. Gouriéroux. 2015. “Noncausal Autoregressive Model in Application to Bitcoin/USD Exchange Rates.” In Studies in Computational Intelligence: Econometrics of Risk, Vol. 583, 17–40. Cham: Springer.10.1007/978-3-319-13449-9_2Search in Google Scholar

Kapetanios, G., and Y. Shin. 2006. “Unit Root Tests in Three-Regime SETAR Models.” The Econometrics Journal 9: 252–78. https://doi.org/10.1111/j.1368-423x.2006.00184.x.Search in Google Scholar

Kapetanios, G., Y. Shin, and A. Snell. 2003. “Testing for a Unit Root in the Nonlinear STAR Framework.” Journal of Econometrics 112: 359–79. https://doi.org/10.1016/s0304-4076(02)00202-6.Search in Google Scholar

Kwiatkowski, D., P. C. Phillips, P. Schmidt, and Y. Shin. 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: 159–78. https://doi.org/10.1016/0304-4076(92)90104-y.Search in Google Scholar

Lanne, M., and P. Saikkonen. 2011. “Noncausal Autoregressions for Economic Time Series.” Journal of Time Series Econometrics 3: 1–32. https://doi.org/10.2202/1941-1928.1080.Search in Google Scholar

Lanne, M., and P. Saikkonen. 2013. “Noncausal Vector Autoregression.” Econometric Theory 29: 447–81. https://doi.org/10.1017/s0266466612000448.Search in Google Scholar

Lanne, M., A. Luoma, and J. Luoto. 2012. “Bayesian Model Selection and Forecasting in Noncausal Autoregressive Models.” Journal of Applied Econometrics 27: 812–30. https://doi.org/10.1002/jae.1217.Search in Google Scholar

Lawrance, A. 1991. “Directionality and Reversibility in Time Series.” International Statistical Review 59: 67–79. https://doi.org/10.2307/1403575.Search in Google Scholar

Lo, M. C., and E. Zivot. 2001. “Threshold Cointegration and Nonlinear Adjustment to the Law of One Price.” Macroeconomic Dynamics 5: 533–76. https://doi.org/10.1017/s1365100501023057.Search in Google Scholar

Michael, P., A. Nobay, and D. Peel. 1997. “Transactions Costs and Nonlinear Adjustment in Real Exchange Rates: An Empirical Investigation.” Journal of Political Economy 105: 862–79. https://doi.org/10.1086/262096.Search in Google Scholar

Moshiri, S., and F. Foroutan. 2006. “Forecasting Nonlinear Crude Oil Futures Prices.” Energy Journal 27: 81–96. https://doi.org/10.5547/issn0195-6574-ej-vol27-no4-4.Search in Google Scholar

Ng, S., and P. Perron. 1995. “Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag.” Journal of the American Statistical Association 90: 268–81. https://doi.org/10.1080/01621459.1995.10476510.Search in Google Scholar

Ng, S., and P. Perron. 2001. “Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power.” Econometrica 69: 1519–54. https://doi.org/10.1111/1468-0262.00256.Search in Google Scholar

Park, J. 1990. “Testing for Unit Roots and Cointegration by Variable Addition.” In Advances in Econometrics: Cointegration, Spurious Regressions and Unit Roots, edited by T. Fomby, and F. Rhodes. Greenwich: Jai Press.Search in Google Scholar

Park, J., and B. Choi. 1988. A New Approach to Testing for A Unit Root. CAE Working Paper No. 88-23. Ithaca, NY: Cornell University.Search in Google Scholar

Phillips, P., and P. Perron. 1988. “Testing for a Unit Root in Time Series Regression.” Biometrika 75: 335–46. https://doi.org/10.1093/biomet/75.2.335.Search in Google Scholar

Pollard, D. 1984. Convergence of Stochastic Processes. New York: Springer-Verlag.10.1007/978-1-4612-5254-2Search in Google Scholar

Reitz, S., and U. Slopek. 2009. “Non-Linear Oil Price Dynamics: A Tale of Heterogeneous Speculators?” German Economic Review 10: 270–83. https://doi.org/10.1111/j.1468-0475.2008.00456.x.Search in Google Scholar

Rosenblatt, M. 1993. “A Note on Prediction and an Autoregressive Sequence.” In Stochastic Processes, A Festschrift in Honour of Gopinath Kallia, edited by S. Cambanis, J. Ghosh, R. Karandikar, and P. Sen, 291–5. New-York: Springer-Verlag.Search in Google Scholar

Rosenblatt, M. 2000. Gaussian and Nno-Gaussian Linear Time Series and Random Fields. New-York: Springer-Verlag.10.1007/978-1-4612-1262-1Search in Google Scholar

Saikkonen, P., and R. Sandberg. 2016. “Testing for a Unit Root in Noncausal Autoregressive Models.” Journal of Time Series Analysis 37: 99–125. https://doi.org/10.1111/jtsa.12141.Search in Google Scholar

Weiss, G. 1975. “Time-Reversibility of Linear Stochastic Processes.” Journal of Applied Probability 12: 821–36. https://doi.org/10.1017/s0021900200048804.Search in Google Scholar

Zimmerman, D., and B. Zumbo. 1993. “The Relative Power of Parametric and Nonparametric Statistical Methods.” In A Handbook for Data Analysis in the Behavioral Sciences: Methodological Issues, edited by G. Keren, and C. Lewis, 481–517. Mahwah: Lawrence Erlbaum Associates, Inc.Search in Google Scholar

Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2022-0084).

Received: 2022-09-21

Accepted: 2023-11-20

Published Online: 2023-12-15

You are currently not able to access this content.

Supplementary Material Details

Articles in the same Issue

https://doi.org/10.1515/snde-2022-0084

Keywords for this article

unit root test; threshold autoregressive model; noncausal model; oil price