More powerful cointegration tests with non-normal errors

Hyejin Lee; Junsoo Lee; Kyungso Im

doi:10.1515/snde-2013-0060

Published by De Gruyter December 13, 2014

More powerful cointegration tests with non-normal errors

Hyejin Lee , Junsoo Lee and Kyungso Im

From the journal Studies in Nonlinear Dynamics & Econometrics

https://doi.org/10.1515/snde-2013-0060

Showing a limited preview of this publication:

Abstract

In this paper, we suggest new cointegration tests that can become more powerful in the presence of non-normal errors. Non-normal errors will not pose a problem in usual cointegration tests even when they are ignored. However, we show that they can become useful sources to improve the power of the tests when we use the “residual augmented least squares” (RALS) procedure to make use of nonlinear moment conditions driven by non-normal errors. The suggested testing procedure is easy to implement and it does not require any non-linear estimation techniques. We can exploit the information on the non-normal error distribution that is already available but ignored in the usual cointegration tests. Our simulation results show significant power gains over existing cointegration tests in the presence of non-normal errors.

Keywords: cointegration; non-normal errors; nonlinear processes; RALS

JEL classification: C12; C15; C22

Corresponding author: Hyejin Lee, Department of Economics, Finance, and Legal Studies, University of Alabama, Tuscaloosa, AL 35487, USA, e-mail: hlee57@cba.ua.edu

Appendix

Proof of Theorem 1

To show the asymptotic distribution of the RALS ECM test (tECM∗), we express the RALS based conditional error correction model alternatively as done in Zivot (2000)

a(L)Δα′yt=δ1α′yt−1+b(L)′Δy2t+w^′tγ+νt,

where y_t=y_1t–βy_2t, a(L)=1–C₁₁(L)L, b(L)=(ϕ–β)+[C₁₂(L)+C₁₁(L)β]L, and et=b(L)′Δy2t+w^′tγ+νt. The OLS estimator of δ₁ is characterized by

T(δ^1−δ1)=1T∑t=2Tyt−1νt−∑t=2Tyt−1Q′t(1T∑t=2TQtQ′t)−11T∑t=2TQtνt1T2∑t=2Tyt−12−1T2∑t=2Tyt−1Q′t(1T∑t=2TQtQ′t)−11T∑t=2TQtyt−1.

Here, we let Q denote a set of variables appear on the right hand side including stationary covariate terms. That is, Qt=(C′˜t, w′^t)′ with C_t=(Δy_{1, t–1}, …, Δy_{1, t–m}, Δy_{2, t}, Δy_{2, t–1}, …, Δy_{2, t–p}) and C˜t=Ct−T−1∑t=2TCt. Since 1T∑t=2Tw^tC′t=op(1), and 1T∑t=2TC˜tνt=op(1), we have

T(δ^1−δ1)=1T∑t=2Tyt−1νt−∑t=2Tyt−1w^′t(1T∑t=2Tw^tw^′t)−11T∑t=2Tw^′tνt1T2∑t=2Tyt−12+op(1).

It can be rewritten as

T(δ^1−δ1)=1T∑t=2Tyt−1νt1T2∑t=2Tyt−12+op(1).

since 1T∑t=2Tyt−1w^t=Op(1) and 1T∑t=2Tw^tνt=op(1).

According to lemma in Hansen (1995), we have the following results:

(A.1)

1T2∑t=2Tyt−12⇒a(1)−2σe2∫01(w1c)2 (A.1)

(A.2)

1T∑t=2Ty1, t−1νt⇒a(1)−1σeσν(ρ∫01w1cdw1+(1−ρ2)(1/2)∫01w1cdw2) (A.2)

where w1c is a projection residual of a Brownian motion, ρ=σνeσνσe, and a(1)=1–a₁–a₂–…–a_p is given from a p–th order polynomial in the lag operator, a(L)=1–a₁L–a₂L²–…–a_pL^p. We wish to note that the RALS term w^′t is a nonlinear function of stationary process. As such, including this term does not change the above results. Then, we can show

(A.3)

T(δ^1−δ1)=a(1)R(ρ∫01w1cdw1∫01(w1c)2+(1−ρ2)(1/2)∫01w1cdw2∫01(w1c)2) (A.3)

where R=σνσe. The test statistic under the null of δ₁=0 can be obtained as

(A.4)

t(δ^1)=δ^1⋅σν−1⋅T(1T2∑t=2Tyt−12−1T2∑t=2Tyt−1Q′t(∑t=2TQtQ′)−1∑t=2TQtyt−1)12=Tδ^1⋅σν−1(1T2∑t=2Tyt−12−1T2∑t=2Tyt−1w^′t(∑t=2Tw^tw^′t)−1∑t=2Tw^tyt−1)12=σν−1(1T2∑t=2Tyt−12)1/2⋅Tδ^1+op(1). (A.4)

Substituting (A.1) and (A.3) into (A.4) yields

(A.5)

t(δ^1)⇒σν−1(a(1)−2σe2∫01(w1c)2)12×[−ca(1)+a(1)R(ρ∫01w1cdw1∫01(w1c)2+(1−ρ2)12∫01w1cdw2∫01(w1c)2)]=−cR(∫01(w1c)2)1/2+ρ∫01w1cdw1(∫01(w1c)2)1/2+(1−ρ2)1/2∫01w1cdw2(∫01(w1c)2)1/2 (A.5)

The null of cointegration holds when c=0 in the local departures from the null hypothesis, Tδ=–ca(1). Therefore, we have

t(δ^1)=ρ⋅tECM+(1−ρ2)⋅Z

since w1c and w₂ are independent Brownian motions and the ratio in the last term of (A.5) is normally distributed.

References

Banerjee, A., and G. W. Smith. 1986. “Exploring Equilibrium Relationship in Econometrics through Static Models: Some Monte Carlo Evidence.” Oxford Bulletin of Economics and Statistics 48 (3): 253–277.10.1111/j.1468-0084.1986.mp48003005.xSearch in Google Scholar

Banerjee, A., J. J. Dolado, and R. Mestre. 1998. “Error-Correction Mechanism Tests for Cointegration in a Single-Equation Framework.” Journal of Time Series Analysis 19 (3): 267–283.10.1111/1467-9892.00091Search in Google Scholar

Boswijk, H. P. 1994. “Testing for an Unstable Root in Conditional and Structural Error Correction Models.” Journal of Econometrics 63 (1): 37–60.10.1016/0304-4076(93)01560-9Search in Google Scholar

Boswijk, H. P., and P. H. Franses. 1992. “Dynamic Specification and Cointegration.” Oxford Bulletin of Economics and Statistics 54 (3): 369–381.10.1111/j.1468-0084.1992.tb00007.xSearch in Google Scholar

Enders W., K. S. Im, J. Lee, and M. C. Strazicich. 2010. “IV Threshold Cointegration Tests and the Taylor Rule.” Economic Modeling 27 (6): 1463–1472.10.1016/j.econmod.2010.07.013Search in Google Scholar

Engle, R. F., and C. W. J. Granger. 1987. “Co-Integration and Error Correction: Representation, Estimation, and Testing.” Econometrica 55 (2): 251–276.10.2307/1913236Search in Google Scholar

Ericsson, N. R., and J. G. MacKinnon. 2002. “Distributions of Error Correction Tests for Cointegration.” Econometrics Journal 5: 285–318.10.1111/1368-423X.00085Search in Google Scholar

Hansen, B. 1995. “Rethinking the Univariate Approach to Unit Root Testing.” Econometric Theory 11: 1148–1171.10.1017/S0266466600009993Search in Google Scholar

Harbo, I., S. Johansen, B. Nielsen, and A. Rahbek. 1998. “Asymptotic Inference on Cointegrating Rank in Partial Systems.” Journal of Business & Economic Statistics 16 (4): 388–399.Search in Google Scholar

Im, K., and P. Schmidt. 2008. “More Efficient Estimation under Non-normality When Higher Moments Do Not Depend on the Regressors, Using Residual-augmented Least Squares.” Journal of Econometrics 144: 219–233.10.1016/j.jeconom.2008.01.003Search in Google Scholar

Im, K., J. Lee, and M. A. Tieslau. 2014. “More Powerful Unit Root Tests with Non-normal Errors.” Festschrift in Honor of Peter Schmidt, New York: Springer, 315–342.Search in Google Scholar

Kremers, J. J. M., N. R. Ericsson, and J. J. Dolado. 1992. “The Power of Cointegration Tests.” Oxford Bulletin of Economics and Statistics 54 (3): 325–348.10.1111/j.1468-0084.1992.tb00005.xSearch in Google Scholar

Lee, H., and J. Lee. 2014. “More Powerful Engle-Granger Cointegration Tests.” Journal of Statistical Computation and Simulation forthcoming.10.1080/00949655.2014.957206Search in Google Scholar

Li, J., and J. Lee. 2010. “ADL tests for Threshold Cointegration.” Journal of Time Series Analysis 31 (4): 241–254.10.1111/j.1467-9892.2010.00659.xSearch in Google Scholar

Zivot, E. 2000. “The Power of Single Equation Tests for Cointegration when the Cointegrating Vector is Prespecified.” Econometric Theory 16: 407–439.10.1017/S0266466600163054Search in Google Scholar

Supplemental Material

The online version of this article (DOI: 10.1515/snde-2013-0060) offers supplementary material, available to authorized users.

Published Online: 2014-12-13

Published in Print: 2015-9-1

More powerful cointegration tests with non-normal errors

Abstract

Appendix

References

Supplemental Material

Journal and Issue

Articles in the same Issue