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Inference on co-integration parameters in heteroskedastic vector autoregressions

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  • Boswijk, H. Peter
  • Cavaliere, Giuseppe
  • Rahbek, Anders
  • Taylor, A.M. Robert
Abstract
We consider estimation and hypothesis testing on the coefficients of the co-integrating relations and the adjustment coefficients in vector autoregressions driven by shocks which display both conditional and unconditional heteroskedasticity of a quite general and unknown form. We show that the conventional results in Johansen (1996) for the maximum likelihood estimators and associated likelihood ratio tests derived under homoskedasticity do not in general hold under heteroskedasticity. As a result, standard confidence intervals and hypothesis tests on these coefficients are potentially unreliable. Solutions based on Wald tests (using a “sandwich” estimator of the variance matrix) and on the use of the wild bootstrap are discussed. These do not require the practitioner to specify a parametric model for volatility. We establish the conditions under which these methods are asymptotically valid. A Monte Carlo simulation study demonstrates that significant improvements in finite sample size can be obtained by the bootstrap over the corresponding asymptotic tests in both heteroskedastic and homoskedastic environments. An application to the term structure of interest rates in the US illustrates the difference between standard and bootstrap inferences regarding hypotheses on the co-integrating vectors and adjustment coefficients.

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  • Boswijk, H. Peter & Cavaliere, Giuseppe & Rahbek, Anders & Taylor, A.M. Robert, 2016. "Inference on co-integration parameters in heteroskedastic vector autoregressions," Journal of Econometrics, Elsevier, vol. 192(1), pages 64-85.
    • Handle: RePEc:eee:econom:v:192:y:2016:i:1:p:64-85
    DOI: 10.1016/j.jeconom.2015.07.005
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    1. Marianne Sensier & Dick van Dijk, 2004. "Testing for Volatility Changes in U.S. Macroeconomic Time Series," The Review of Economics and Statistics, MIT Press, vol. 86(3), pages 833-839, August.
    2. MacKinnon, James G & Haug, Alfred A & Michelis, Leo, 1999. "Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 14(5), pages 563-577, Sept.-Oct.
    3. Cavaliere, Giuseppe & Rahbek, Anders & Taylor, A.M. Robert, 2010. "Testing for co-integration in vector autoregressions with non-stationary volatility," Journal of Econometrics, Elsevier, vol. 158(1), pages 7-24, September.
    4. Davidson, Russell & Flachaire, Emmanuel, 2008. "The wild bootstrap, tamed at last," Journal of Econometrics, Elsevier, vol. 146(1), pages 162-169, September.
    5. Johansen, Soren, 1992. "Determination of Cointegration Rank in the Presence of a Linear Trend," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 54(3), pages 383-397, August.
    6. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2009. "Heteroskedastic Time Series With A Unit Root," Econometric Theory, Cambridge University Press, vol. 25(5), pages 1228-1276, October.
    7. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
    8. Diebold, Francis X. & Li, Canlin, 2006. "Forecasting the term structure of government bond yields," Journal of Econometrics, Elsevier, vol. 130(2), pages 337-364, February.
    9. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2007. "Testing for unit roots in time series models with non-stationary volatility," Journal of Econometrics, Elsevier, vol. 140(2), pages 919-947, October.
    10. Giuseppe Cavaliere & Anders Rahbek & A. M. Robert Taylor, 2014. "Bootstrap Determination of the Co-Integration Rank in Heteroskedastic VAR Models," Econometric Reviews, Taylor & Francis Journals, vol. 33(5-6), pages 606-650, August.
    11. Chung, Heetaik & Park, Joon Y., 2007. "Nonstationary nonlinear heteroskedasticity in regression," Journal of Econometrics, Elsevier, vol. 137(1), pages 230-259, March.
    12. Hansen, Bruce E., 1992. "Convergence to Stochastic Integrals for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 8(4), pages 489-500, December.
    13. Pieter Omtzigt & Stefano Fachin, 2006. "The Size and Power of Bootstrap and Bartlett-Corrected Tests of Hypotheses on the Cointegrating Vectors," Econometric Reviews, Taylor & Francis Journals, vol. 25(1), pages 41-60.
    14. Giuseppe Cavaliere & Heino Bohn Nielsen & Anders Rahbek, 2015. "Bootstrap Testing of Hypotheses on Co‐Integration Relations in Vector Autoregressive Models," Econometrica, Econometric Society, vol. 83, pages 813-831, March.
    15. Bruce E. Hansen, 2000. "Sample Splitting and Threshold Estimation," Econometrica, Econometric Society, vol. 68(3), pages 575-604, May.
    16. Margaret M. McConnell & Gabriel Perez-Quiros, 2000. "Output fluctuations in the United States: what has changed since the early 1980s?," Proceedings, Federal Reserve Bank of San Francisco, issue Mar.
    17. Goncalves, Silvia & Kilian, Lutz, 2004. "Bootstrapping autoregressions with conditional heteroskedasticity of unknown form," Journal of Econometrics, Elsevier, vol. 123(1), pages 89-120, November.
    18. Kristensen, Dennis & Rahbek, Anders, 2013. "Testing And Inference In Nonlinear Cointegrating Vector Error Correction Models," Econometric Theory, Cambridge University Press, vol. 29(6), pages 1238-1288, December.
    19. Omtzigt Pieter & Fachin Stefano, 2002. "Bootstrapping and Bartlett corrections in the cointegrated VAR model," Economics and Quantitative Methods qf0212, Department of Economics, University of Insubria.
    20. Russell Davidson & James MacKinnon, 2000. "Bootstrap tests: how many bootstraps?," Econometric Reviews, Taylor & Francis Journals, vol. 19(1), pages 55-68.
    21. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, vol. 64(2), pages 413-430, March.
    22. Cavaliere, Giuseppe & Rahbek, Anders & Taylor, A.M. Robert, 2010. "Cointegration Rank Testing Under Conditional Heteroskedasticity," Econometric Theory, Cambridge University Press, vol. 26(6), pages 1719-1760, December.
    23. Park, Joon Y., 2002. "Nonstationary nonlinear heteroskedasticity," Journal of Econometrics, Elsevier, vol. 110(2), pages 383-415, October.
    24. Anders Rygh Swensen, 2006. "Bootstrap Algorithms for Testing and Determining the Cointegration Rank in VAR Models -super-1," Econometrica, Econometric Society, vol. 74(6), pages 1699-1714, November.
    25. Engle, Robert & Granger, Clive, 2015. "Co-integration and error correction: Representation, estimation, and testing," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 39(3), pages 106-135.
    26. Deo, Rohit S., 2000. "Spectral tests of the martingale hypothesis under conditional heteroscedasticity," Journal of Econometrics, Elsevier, vol. 99(2), pages 291-315, December.
    27. Campbell, John Y & Shiller, Robert J, 1987. "Cointegration and Tests of Present Value Models," Journal of Political Economy, University of Chicago Press, vol. 95(5), pages 1062-1088, October.
    28. Godfrey, L.G. & Tremayne, A.R., 2005. "The wild bootstrap and heteroskedasticity-robust tests for serial correlation in dynamic regression models," Computational Statistics & Data Analysis, Elsevier, vol. 49(2), pages 377-395, April.
    29. Donald W. K. Andrews & Moshe Buchinsky, 2000. "A Three-Step Method for Choosing the Number of Bootstrap Repetitions," Econometrica, Econometric Society, vol. 68(1), pages 23-52, January.
    30. Hansen, Bruce E, 1995. "Regression with Nonstationary Volatility," Econometrica, Econometric Society, vol. 63(5), pages 1113-1132, September.
    31. He, Changli & Terasvirta, Timo, 1999. "Properties of moments of a family of GARCH processes," Journal of Econometrics, Elsevier, vol. 92(1), pages 173-192, September.
    32. H. Peter Boswijk & Jurgen A. Doornik, 2004. "Identifying, estimating and testing restricted cointegrated systems: An overview," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 58(4), pages 440-465, November.
    33. Giuseppe Cavaliere & A. M. Robert Taylor & Carsten Trenkler, 2015. "Bootstrap Co-integration Rank Testing: The Effect of Bias-Correcting Parameter Estimates," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 77(5), pages 740-759, October.
    34. Chang-Jin Kim & Charles R. Nelson, 1999. "Has The U.S. Economy Become More Stable? A Bayesian Approach Based On A Markov-Switching Model Of The Business Cycle," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 608-616, November.
    35. Hansen, Bruce E., 1992. "Heteroskedastic cointegration," Journal of Econometrics, Elsevier, vol. 54(1-3), pages 139-158.
    36. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-1580, November.
    37. Fama, Eugene F & Bliss, Robert R, 1987. "The Information in Long-Maturity Forward Rates," American Economic Review, American Economic Association, vol. 77(4), pages 680-692, September.
    38. Johansen, Soren, 1995. "Identifying restrictions of linear equations with applications to simultaneous equations and cointegration," Journal of Econometrics, Elsevier, vol. 69(1), pages 111-132, September.
    39. Saikkonen, Pentti, 1995. "Problems with the Asymptotic Theory of Maximum Likelihood Estimation in Integrated and Cointegrated Systems," Econometric Theory, Cambridge University Press, vol. 11(5), pages 888-911, October.
    40. Giuseppe Cavaliere & Anders Rahbek & A. M. Robert Taylor, 2012. "Bootstrap Determination of the Co‐Integration Rank in Vector Autoregressive Models," Econometrica, Econometric Society, vol. 80(4), pages 1721-1740, July.
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    9. Boldea, Otilia & Cornea-Madeira, Adriana & Hall, Alastair R., 2019. "Bootstrapping structural change tests," Journal of Econometrics, Elsevier, vol. 213(2), pages 359-397.
    10. Georgiev, Iliyan & Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2018. "Testing for parameter instability in predictive regression models," Journal of Econometrics, Elsevier, vol. 204(1), pages 101-118.
    11. Graziano Moramarco, 2020. "Measuring Global Macroeconomic Uncertainty," Working Papers wp1148, Dipartimento Scienze Economiche, Universita' di Bologna.
    12. Boswijk, H. Peter & Cavaliere, Giuseppe & Georgiev, Iliyan & Rahbek, Anders, 2021. "Bootstrapping non-stationary stochastic volatility," Journal of Econometrics, Elsevier, vol. 224(1), pages 161-180.
    13. Shuping Shi & Peter C. B. Phillips & Stan Hurn, 2018. "Change Detection and the Causal Impact of the Yield Curve," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(6), pages 966-987, November.
    14. Motegi, Kaiji & Iitsuka, Yoshitaka, 2023. "Inter-regional dependence of J-REIT stock prices: A heteroscedasticity-robust time series approach," The North American Journal of Economics and Finance, Elsevier, vol. 64(C).
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    16. Canepa Alessandra, 2022. "Small Sample Adjustment for Hypotheses Testing on Cointegrating Vectors," Journal of Time Series Econometrics, De Gruyter, vol. 14(1), pages 51-85, January.
    17. Takamitsu Kurita & Bent Nielsen, 2019. "Partial Cointegrated Vector Autoregressive Models with Structural Breaks in Deterministic Terms," Econometrics, MDPI, vol. 7(4), pages 1-35, October.
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    More about this item

    Keywords

    Co-integration; Adjustment coefficients; (un)Conditional heteroskedasticity; Heteroskedasticity-robust inference; Wild bootstrap;
    All these keywords.

    JEL classification:

    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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