In this paper, we propose a new approach to model structural change in cointegrating regressions using penalized regression techniques. First, we consider a setting with known breakpoint candidates and show that a modified adaptive lasso estimator can consistently estimate structural breaks in the intercept and slope coefficient of a cointegrating regression. Second, we extend our approach to a diverging number of breakpoint candidates and provide simulation evidence that timing and magnitude of structural breaks are consistently estimated. Third, we use the adaptive lasso estimation to design new tests for cointegration in the presence of multiple structural breaks, derive the asymptotic distribution of our test statistics and show that the proposed tests have power against the null of no cointegration. Finally, we use our new methodology to study the effects of structural breaks on the long-run PPP relationship.
We thank Robert Jung, Andrew Tremayne, and Jana Mareckova for valuable comments and suggestions. We are also grateful for the valuable remarks and suggestions by the editor, Bruce Mizrach, and an anonymous referee. We thank the organizers and participants of the Junior Research Seminar in Econometrics in Obermarchtal, Economics Brown Bag seminar at the University of Hohenheim, German Statistical Week in Linz, Conference on Decision Sciences in Konstanz, THE Christmas Workshop 2018 in Stuttgart, European Winter Meeting of the Econometric Society in Naples, 12th International Conference on Computational and Financial Econometrics in Pisa, and the 24th Spring Meeting of Young Economists in Brussels. Moreover, we thank Charles Engel and Chang-Jin Kim for making their data available and Daiki Maki for sending us his codes. Finally, we thank Maike Becker for excellent research assistance.
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
Proof of Theorem 1
Next, we define the objective function V T ( b ) by
where b = ( b 1′, b 2′)′, and
is the minimizer of V T with and .
First, we consider the asymptotic counterparts to the least squares terms
We use the decomposition X = ( X 1, X 2) to express the weak convergence result
where U ∼ N(0, ϒσ 2) and .
Under Assumptions 2 and 3, the maximum number of breakpoints is limited and initial least squares estimates are available for the weights of the adaptive lasso estimator. We investigate the consistency of the individual coefficients and distinguish between the true coefficients being zero or nonzero:
If , we have(26)
since (i) , (ii) if the initial estimator is consistent and (iii) as in Zou (2006).
If , we have(27)
since (i) and (ii) the initial least squares estimator is tight and converges to a normal distribution, .
If , we have(28)
since (i) , (ii) if the initial estimator is consistent and (iii) .
If , we have(29)
since (i) , (ii) the least squares estimator is tight and has the following nonstandard distribution(30)
and (iii) .
Thus, V T ( b ) ⇒ V( b ), where
Since V T is a convex function and V has a unique minimum, it follows from Knight and Fu (2000) that
From these results, we can deduce that
where δ 0 denotes the Dirac measure at 0. Correspondingly, we have
It remains to show that coefficients of inactive variables are set to zero with probability approaching one. We begin with a proof of . Consider the event that although . We know from the Karush–Kuhn–Tucker (KKT) optimality conditions that the first order condition for a minimum is given by
since (i) and (ii) is tight. The left hand side of the equation is equivalent to
For the first term, we have the weak convergence
and for the second term, we have the weak convergence of , which depends on the timing of the break fraction τ 1,i relative to all other possible break fractions. Say holds, then
Further, we have already shown the weak convergence of . Hence, the distribution of the first term is tight and
Next, we show that . Again, we consider the event that although . The KKT optimality condition in this case is given by
where the factor T substitutes the factor in Eq. (35). For the right hand side of the equation, we observe that
since is tight. For the left hand side,
we have the weak convergence of the first term using
The expression of the weak convergence result for the second term depends on the timing of the break fraction τ 2,j . Say holds, we have
and as before is tight. Finally, we have shown that
and this completes the proof. □
Proof of Theorem 2
For ease of exposition, we assume that the maximum number of breaks is m* = 2 and the true intercept is known to be μ t = 0 for all t. In this case, we obtain three possible model selection outcomes regarding the number of breaks for the post-lasso regressions:
All coefficients of break indicator regressors are shrunk to zero(47)
One structural break is (falsely) detected(48)
Two structural breaks are (falsely) detected(49)
Note that the specification of indicator terms in Eqs. (48) and (49), i.e., the timing of (falsely) detected breaks, depends on the tuning parameter λ. We continue the proof for the bias-corrected test statistic, Z 2, corresponding to the case of two (falsely) detected structural breaks with unknown timing. The asymptotic distribution of the test statistics for the two remaining cases can be easily deduced from our derivations. We decompose the cumulative sum into S t = (S 1t , S 2t )′. Further, we define the break fraction vector τ 2 = (τ 2,1, τ 2,2)′ as a compact set on (0, 1) × (0, 1) and define the matrix .
Using the result
and the CMT yields the weak convergence of
The result in (51) can straightforwardly be extended to
Define as the post-lasso least squares estimator and set so that
in conformity with S t and define Λ2⋅ = (Λ21, Λ22) and Λ⋅2 = (Λ12, Λ22)′.
For each element of it holds that
and . Since and
for the asymptotic counterpart, we have the identities
Consequently, we can state the following important weak convergence results
Under the null hypothesis, the cointegration residuals can be written as and we can show weak convergence of the sample moments. It holds that
Next, we consider the bias-correction term for the first-order serial correlation coefficient. We denote the kernel weights as w(j/M) = w j and can show that
Hence, we have the weak convergence result
For the long-run variance, we obtain the result
Now, we use the CMT to show that
for each configuration of τ 2. Correspondingly, the test statistics for the remaining model selection outcomes have the asymptotic distributions
respectively. Naturally, the distributions of Z 2 and Z 1 depend on the timing of the breakpoint. Finally, selecting the infimum statistic over all potential model selection outcomes is a continuous transformation so that we can use the CMT to complete the proof. □
Andrews, D. W. K., and W. Ploberger. 1994. “Optimal Tests when a Nuisance Parameter is Present Only Under the Alternative.” Econometrica 62: 1383–414. https://doi.org/10.2307/2951753.Search in Google Scholar
Arai, Y., and E. Kurozumi. 2007. “Testing for the Null Hypothesis of Cointegration with a Structural Break.” Econometric Reviews 26: 705–39. https://doi.org/10.1080/07474930701653776.Search in Google Scholar
Bekiros, S. D., and C. G. H. Diks. 2008. “The Relationship between Crude Oil Spot and Futures Prices: Cointegration, Linear and Nonlinear Causality.” Energy Economics 30: 2673–85. https://doi.org/10.1016/j.eneco.2008.03.006.Search in Google Scholar
Carrion-i Silvestre, J. L., and A. Sanso. 2006. “Testing for the Null Hypothesis of Cointegration with Structural Breaks.” Oxford Bulletin of Economics & Statistics 68: 623–46. https://doi.org/10.1111/j.1468-0084.2006.00180.x.Search in Google Scholar
Castle, J. L., J. A. Doornik, and D. F. Hendry. 2012. “Model Selection when There Are Multiple Breaks.” Journal of Econometrics 169: 239–46. https://doi.org/10.1016/j.jeconom.2012.01.026.Search in Google Scholar
Chan, N. H., C. Y. Yau, and R.-M. Zhang. 2014. “Group LASSO for Structural Break Time Series.” Journal of the American Statistical Association 109: 590–9. https://doi.org/10.1080/01621459.2013.866566.Search in Google Scholar
Davidson, J., and A. Monticini. 2010. “Tests for Cointegration with Structural Breaks Based on Subsamples.” Computational Statistics & Data Analysis 54: 2498–511. https://doi.org/10.1016/j.csda.2010.01.028.Search in Google Scholar
Engle, R. F., and C. W. J. Granger. 1987. “Co-Integration and Error Correction: Representation, Estimation and Testing.” Econometrica 55: 251–76. https://doi.org/10.2307/1913236.Search in Google Scholar
Fan, J., and R. Li. 2001. “Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties.” Journal of the American Statistical Association 96: 1348–60. https://doi.org/10.1198/016214501753382273.Search in Google Scholar
Gregory, A. W., and B. E. Hansen. 1996. “Residual-based Tests for Cointegration in Models with Regime Shifts.” Journal of Econometrics 70: 99–126. https://doi.org/10.1016/0304-4076(69)41685-7.Search in Google Scholar
Gregory, A. W., and B. E. Hansen. 1996a. “Tests for Cointegration in Models with Regime and Trend Shifts.” Oxford Bulletin of Economics & Statistics 58: 555–60.10.1111/j.1468-0084.1996.mp58003008.xSearch in Google Scholar
Gregory, A. W., J. M. Nason, and D. G. Watt. 1996b. “Testing for Structural Breaks in Cointegrated Relationships.” Journal of Econometrics 71: 321–41. https://doi.org/10.1016/0304-4076(96)84508-8.Search in Google Scholar
Grilli, V., and G. Kaminsky. 1991. “Nominal Exchange Rate Regimes and the Real Exchange Rate - Evidence from the United States and Great Britain, 1885-1986.” Journal of Monetary Economics 27: 191–212. https://doi.org/10.1016/0304-3932(91)90041-l.Search in Google Scholar
Hatemi-J, A. 2008. “Tests for Cointegration with Two Unknown Regime Shifts with an Application to Financial Market integration.” Empirical Economics 35: 497–505. https://doi.org/10.1007/s00181-007-0175-9.Search in Google Scholar
Hendry, D. F., S. Johansen, and C. Santos. 2008. “Automatic Selection of indicators in a Fully Saturated Regression.” Computational Statistics 23: 337–9. https://doi.org/10.1007/s00180-008-0112-1.Search in Google Scholar
Herrndorf, N. 1984. “A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables.” Annals of Probability 12: 141–53. https://doi.org/10.1016/0047-259x(84)90073-3.Search in Google Scholar
Horowitz, J., and J. Huang. 2013. “Penalized Estimation of High-Dimensional Models under a Generalized Sparsity Condition.” Statistica Sinica 23: 725–48.10.1920/wp.cem.2012.1712Search in Google Scholar
Karoglou, M., and B. Morley. 2012. “Purchasing Power Parity and Structural instability in the US/UK Exchange Rate.” Journal of International Financial Markets, Institutions and Money 22: 958–72. https://doi.org/10.1016/j.intfin.2012.05.001.Search in Google Scholar
Kejriwal, M., and P. Perron. 2008. “The Limit Distribution of the Estimates in Cointegrated Regression Models with Multiple Structural Changes.” Journal of Econometrics 146: 59–73. https://doi.org/10.1016/j.jeconom.2008.07.001.Search in Google Scholar
Kejriwal, M., and P. Perron. 2010. “Testing for Multiple Structural Changes in Cointegrated Regression Models.” Journal of Business & Economic Statistics 28: 503–22. https://doi.org/10.1198/jbes.2009.07220.Search in Google Scholar
Knight, K., and W. Fu. 2000. “Asymptotics for Lasso-Type Estimators.” Annals of Statistics 28: 1356–78.Search in Google Scholar
Kock, A. B. 2016. “Consistent and Conservative Model Selection with the Adaptive Lasso in Stationary and Nonstationary Autoregressions.” Econometric Theory 32: 243–59. https://doi.org/10.1017/s0266466615000304.Search in Google Scholar
Koo, B., H. Anderson, M. H. Seo, and W. Yao. 2017. “High-dimensional Predictive Regression in the Presence of Cointegration.” Technical report. SSRN Working Paper. 10.2139/ssrn.2851677Search in Google Scholar
Kurozumi, E., and A. Skrobotov. 2017. “Confidence Sets for the Break Date in Cointegrating Regressions.” Oxford Bulletin of Economics & Statistics 80: 514–35. https://doi.org/10.1111/obes.12223.Search in Google Scholar
Mendes, E. F. 2011. “Model Selection Consistency for Cointegrating Regressions.” Technical report. Northwestern University. Search in Google Scholar
Mussa, M. 1986. “Nominal Exchange Rate Regimes and the Behavior of Real Exchange Rates: Evidence and Implications.” In Carnegie-Rochester Conference on Public Policy, Vol. 25, 117–214. 10.1016/0167-2231(86)90039-4Search in Google Scholar
Perron, P. 2006. “Dealing with Structural Breaks.” In Palgrave Handbook of Econometrics – Volume 1: Econometric Theory, edited by H. Hassani, T. Mills, and K. Patterson, 278–352. UK: Palgrave Macmillan.Search in Google Scholar
Perron, P., and Y. Yamamoto. 2016. “On the Usefulness or Lack Thereof of Optimality Criteria for Structural Change Tests.” Econometric Reviews 35: 782–844. https://doi.org/10.1080/07474938.2014.977621.Search in Google Scholar
Phillips, P. C. B., and S. N. Durlauf. 1986. “Multiple Time Series Regression with Integrated Processes.” The Review of Economic Studies 53: 473–95. https://doi.org/10.2307/2297602.Search in Google Scholar
Pötscher, B. M., and U. Schneider. 2009. “On the Distribution of the Adaptive LASSO Estimator.” Journal of Statistical Planning and Inference 139: 2775–90. https://doi.org/10.1016/j.jspi.2009.01.003.Search in Google Scholar
Pretis, F. 2020. “Econometric Modelling of Climate Systems: The Equivalence of Energy Balance Models and Cointegrated Vector Autoregressions.” Journal of Econometrics 214: 256–73. https://doi.org/10.1016/j.jeconom.2019.05.013.Search in Google Scholar
Qian, J., and L. Su. 2016. “Shrinkage Estimation of Regression Models with Multiple Structural Changes.” Econometric Theory 32: 1376–433. https://doi.org/10.1017/s0266466615000237.Search in Google Scholar
Qu, Z., and P. Perron. 2007. “Estimating and Testing Structural Change in Multivariate Regressions.” Econometrica 75: 459–502. https://doi.org/10.1111/j.1468-0262.2006.00754.x.Search in Google Scholar
Schweikert, K. 2019. “Asymmetric Price Transmission in the US and German Fuel Markets: A Quantile Autoregression Approach.” Empirical Economics 56: 1071–95. https://doi.org/10.1007/s00181-017-1376-5.Search in Google Scholar
Stock, J. H., and M. W. Watson. 1993. “A Simple Estimator of Cointegrating Vectors in Higher Order integrated Systems.” Econometrica 61: 783–820. https://doi.org/10.2307/2951763.Search in Google Scholar
Taylor, M. P. 1988. “An Empirical Examination of Long-Run Purchasing Power Parity Using Cointegration Techniques.” Applied Economics 20: 1369–81. https://doi.org/10.1080/00036848800000107.Search in Google Scholar
Taylor, M. P. 2006. “Real Exchange Rates and Purchasing Power Parity: Mean-Reversion in Economic Thought.” Applied Financial Economics 16: 1–17. https://doi.org/10.1080/09603100500390067.Search in Google Scholar
Tibshirani, R. 1996. “Regression Selection and Shrinkage via the Lasso.” Journal of the Royal Statistical Society B (Methodological) 58: 267–88. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x.Search in Google Scholar
Tibshirani, R., M. Saunders, S. Rosset, J. Zhu, and K. Knight. 2005. “Sparsity and Smoothness via the Fused Lasso.” Journal of the Royal Statistical Society - Series B: Statistical Methodology 67: 91–108. https://doi.org/10.1111/j.1467-9868.2005.00490.x.Search in Google Scholar
Wang, H., B. Li, and C. Leng. 2009. “Shrinkage Tuning Parameter Selection with a Diverging Number of Parameters.” Journal of the Royal Statistical Society - Series B: Statistical Methodology 71: 671–83. https://doi.org/10.1111/j.1467-9868.2008.00693.x.Search in Google Scholar
Westerlund, J., and D. L. Edgerton. 2006. “New Improved Tests for Cointegration with Structural Breaks.” Journal of Time Series Analysis 28: 188–224.10.1111/j.1467-9892.2006.00504.xSearch in Google Scholar
Yuan, M., and Y. Lin. 2006. “Model Selection and Estimation in Regression with Grouped Variables.” Journal of the Royal Statistical Society - Series B: Statistical Methodology 68: 49–67. https://doi.org/10.1111/j.1467-9868.2005.00532.x.Search in Google Scholar
Zhang, C. H., and J. Huang. 2008. “The Sparsity and Bias of the Lasso Selection in High-Dimensional Linear Regression.” Annals of Statistics 36: 1567–94. https://doi.org/10.1214/07-aos520.Search in Google Scholar
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