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Alternative Asymptotics for Cointegration Tests in Large VARs

Author

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  • Alexei Onatski
  • Chen Wang

Abstract

Johansen's (1988,1991) likelihood ratio test for cointegration rank of a vector autoregression (VAR) depends only on the squared sample canonical correlations between current changes and past levels of a simple transformation of the data. We study the asymptotic behavior of the empirical distribution of those squared canonical correlations when the number of observations and the dimensionality of the VAR diverge to infinity simultaneously and proportionally. We find that the distribution weakly converges to the so‐called Wachter distribution. This finding provides a theoretical explanation for the observed tendency of Johansen's test to find “spurious cointegration.”

Suggested Citation

  • Alexei Onatski & Chen Wang, 2018. "Alternative Asymptotics for Cointegration Tests in Large VARs," Econometrica, Econometric Society, vol. 86(4), pages 1465-1478, July.
    • Handle: RePEc:wly:emetrp:v:86:y:2018:i:4:p:1465-1478
    DOI: 10.3982/ECTA14649
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    Cited by:

    1. Matteo Barigozzi & Lorenzo Trapani, 2018. "Determining the dimension of factor structures in non-stationary large datasets," Discussion Papers 18/01, University of Nottingham, Granger Centre for Time Series Econometrics.
    2. Marie Levakova & Susanne Ditlevsen, 2024. "Penalisation Methods in Fitting High‐Dimensional Cointegrated Vector Autoregressive Models: A Review," International Statistical Review, International Statistical Institute, vol. 92(2), pages 160-193, August.
    3. Liang, Chong & Schienle, Melanie, 2019. "Determination of vector error correction models in high dimensions," Journal of Econometrics, Elsevier, vol. 208(2), pages 418-441.
    4. Alexander Chudik & M. Hashem Pesaran & Ron P. Smith, 2025. "Analysis of Multiple Long-Run Relations in Panel Data Models," Working Papers 2523, Federal Reserve Bank of Dallas, revised 29 Sep 2025.
    5. Jesus Gonzalo & Jean-Yves Pitarakis, 2025. "Detecting Sparse Cointegration," Papers 2501.13839, arXiv.org.
    6. Gonzalo, Jesús & Pitarakis, Jean-Yves, 2021. "Spurious relationships in high-dimensional systems with strong or mild persistence," International Journal of Forecasting, Elsevier, vol. 37(4), pages 1480-1497.
    7. Alexander Chudik & M. Hashem Pesaran & Kamiar Mohaddes, 2020. "Identifying Global and National Output and Fiscal Policy Shocks Using a GVAR," Advances in Econometrics, in: Essays in Honor of Cheng Hsiao, volume 41, pages 143-189, Emerald Group Publishing Limited.
    8. Lee, Ji Hyung & Shi, Zhentao & Gao, Zhan, 2022. "On LASSO for predictive regression," Journal of Econometrics, Elsevier, vol. 229(2), pages 322-349.
    9. Casoli, Chiara & Lucchetti, Riccardo (Jack), 2021. "Permanent-Transitory decomposition of cointegrated time series via Dynamic Factor Models, with an application to commodity prices," FEEM Working Papers 312367, Fondazione Eni Enrico Mattei (FEEM).
    10. Ziwei Mei & Zhentao Shi, 2022. "On LASSO for High Dimensional Predictive Regression," Papers 2212.07052, arXiv.org, revised Jan 2024.
    11. Morten {O}rregaard Nielsen & Won-Ki Seo & Dakyung Seong, 2023. "Inference on common trends in functional time series," Papers 2312.00590, arXiv.org, revised Oct 2025.
    12. Onatski, Alexei & Wang, Chen, 2019. "Extreme canonical correlations and high-dimensional cointegration analysis," Journal of Econometrics, Elsevier, vol. 212(1), pages 307-322.
    13. Georg Keilbar & Yanfen Zhang, 2021. "On cointegration and cryptocurrency dynamics," Digital Finance, Springer, vol. 3(1), pages 1-23, March.
    14. Escribano, Alvaro & Peña, Daniel & Ruiz, Esther, 2021. "30 years of cointegration and dynamic factor models forecasting and its future with big data: Editorial," International Journal of Forecasting, Elsevier, vol. 37(4), pages 1333-1337.
    15. Mei, Ziwei & Shi, Zhentao, 2024. "On LASSO for high dimensional predictive regression," Journal of Econometrics, Elsevier, vol. 242(2).
    16. Nielsen, Morten Ørregaard & Seo, Won-Ki & Seong, Dakyung, 2023. "Inference On The Dimension Of The Nonstationary Subspace In Functional Time Series," Econometric Theory, Cambridge University Press, vol. 39(3), pages 443-480, June.
    17. Zhan Gao & Ji Hyung Lee & Ziwei Mei & Zhentao Shi, 2024. "LASSO Inference for High Dimensional Predictive Regressions," Papers 2409.10030, arXiv.org, revised Jan 2026.
    18. Fan, Rui & Lee, Ji Hyung & Shin, Youngki, 2023. "Predictive quantile regression with mixed roots and increasing dimensions: The ALQR approach," Journal of Econometrics, Elsevier, vol. 237(2).
    19. Chudik, A. & Pesaran, M. H. & Smith, R. P., 2025. "Analysis of Multiple Long Run Relations in Panel Data Models with Applications to Financial Ratios," Cambridge Working Papers in Economics 2538, Faculty of Economics, University of Cambridge.
    20. Anna Bykhovskaya & Vadim Gorin, 2022. "Asymptotics of Cointegration Tests for High-Dimensional VAR($k$)," Papers 2202.07150, arXiv.org, revised Nov 2023.
    21. Massimo Franchi & Iliyan Georgiev & Paolo Paruolo, 2024. "Canonical correlation analysis of stochastic trends via functional approximation," Papers 2411.19572, arXiv.org, revised Sep 2025.
    22. Gianluca Cubadda & Marco Mazzali, 2024. "The vector error correction index model: representation, estimation and identification," The Econometrics Journal, Royal Economic Society, vol. 27(1), pages 126-150.
    23. Anna Bykhovskaya & Vadim Gorin, 2020. "Cointegration in large VARs," Papers 2006.14179, arXiv.org, revised Dec 2021.

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