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Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors

Author

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  • Matteo Barigozzi

    (Università di Bologna, Department of Economics, 40126 Bologna, Italy)

  • Marco Lippi

    (Einaudi Institute for Economics and Finance, 00187 Roma, Italy)

  • Matteo Luciani

    (Federal Reserve Board of Governors, Washington, DC 20551, USA)

Abstract

Large-dimensional dynamic factor models and dynamic stochastic general equilibrium models, both widely used in empirical macroeconomics, deal with singular stochastic vectors, i.e., vectors of dimension r which are driven by a q -dimensional white noise, with q < r . The present paper studies cointegration and error correction representations for an I ( 1 ) singular stochastic vector y t . It is easily seen that y t is necessarily cointegrated with cointegrating rank c ≥ r − q . Our contributions are: (i) we generalize Johansen’s proof of the Granger representation theorem to I ( 1 ) singular vectors under the assumption that y t has rational spectral density; (ii) using recent results on singular vectors by Anderson and Deistler, we prove that for generic values of the parameters the autoregressive representation of y t has a finite-degree polynomial. The relationship between the cointegration of the factors and the cointegration of the observable variables in a large-dimensional factor model is also discussed.

Suggested Citation

  • Matteo Barigozzi & Marco Lippi & Matteo Luciani, 2020. "Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors," Econometrics, MDPI, vol. 8(1), pages 1-23, February.
  • Handle: RePEc:gam:jecnmx:v:8:y:2020:i:1:p:3-:d:316273
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    References listed on IDEAS

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    1. Banerjee, Anindya & Marcellino, Massimiliano & Masten, Igor, 2014. "Forecasting with factor-augmented error correction models," International Journal of Forecasting, Elsevier, vol. 30(3), pages 589-612.
    2. Anindya Banerjee & Massimiliano Marcellino & Igor Masten, 2017. "Structural FECM: Cointegration in large‐scale structural FAVAR models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(6), pages 1069-1086, September.
    3. Matteo Barigozzi & Antonio M. Conti & Matteo Luciani, 2014. "Do Euro Area Countries Respond Asymmetrically to the Common Monetary Policy?," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 76(5), pages 693-714, October.
    4. Forni, Mario & Hallin, Marc & Lippi, Marco & Zaffaroni, Paolo, 2015. "Dynamic factor models with infinite-dimensional factor spaces: One-sided representations," Journal of Econometrics, Elsevier, vol. 185(2), pages 359-371.
    5. Matteo Luciani, 2015. "Monetary Policy and the Housing Market: A Structural Factor Analysis," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 30(2), pages 199-218, March.
    6. Domenico Giannone & Lucrezia Reichlin & Luca Sala, 2005. "Monetary Policy in Real Time," NBER Chapters, in: NBER Macroeconomics Annual 2004, Volume 19, pages 161-224, National Bureau of Economic Research, Inc.
    7. Bauer, Dietmar & Wagner, Martin, 2012. "A State Space Canonical Form For Unit Root Processes," Econometric Theory, Cambridge University Press, vol. 28(6), pages 1313-1349, December.
    8. Gregoir, Stéphane, 1999. "Multivariate Time Series With Various Hidden Unit Roots, Part Ii," Econometric Theory, Cambridge University Press, vol. 15(4), pages 469-518, August.
    9. Forni, Mario & Hallin, Marc & Lippi, Marco & Zaffaroni, Paolo, 2017. "Dynamic factor models with infinite-dimensional factor space: Asymptotic analysis," Journal of Econometrics, Elsevier, vol. 199(1), pages 74-92.
    10. Massimo Franchi & Paolo Paruolo, 2019. "A general inversion theorem for cointegration," Econometric Reviews, Taylor & Francis Journals, vol. 38(10), pages 1176-1201, November.
    11. Bai, Jushan & Ng, Serena, 2007. "Determining the Number of Primitive Shocks in Factor Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 52-60, January.
    12. J. B. Taylor & Harald Uhlig (ed.), 2016. "Handbook of Macroeconomics," Handbook of Macroeconomics, Elsevier, edition 1, volume 2, number 2.
    13. Forni, Mario & Lippi, Marco, 2001. "The Generalized Dynamic Factor Model: Representation Theory," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1113-1141, December.
    14. Mario Forni & Marc Hallin & Marco Lippi & Lucrezia Reichlin, 2000. "The Generalized Dynamic-Factor Model: Identification And Estimation," The Review of Economics and Statistics, MIT Press, vol. 82(4), pages 540-554, November.
    15. Sargent, Thomas J, 1989. "Two Models of Measurements and the Investment Accelerator," Journal of Political Economy, University of Chicago Press, vol. 97(2), pages 251-287, April.
    16. 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.
    17. Forni, Mario & Gambetti, Luca, 2010. "The dynamic effects of monetary policy: A structural factor model approach," Journal of Monetary Economics, Elsevier, vol. 57(2), pages 203-216, March.
    18. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
    19. Stock, J.H. & Watson, M.W., 2016. "Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics," Handbook of Macroeconomics, in: J. B. Taylor & Harald Uhlig (ed.), Handbook of Macroeconomics, edition 1, volume 2, chapter 0, pages 415-525, Elsevier.
    20. Phillips, Peter C. B., 1998. "Impulse response and forecast error variance asymptotics in nonstationary VARs," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 21-56.
    21. Amengual, Dante & Watson, Mark W., 2007. "Consistent Estimation of the Number of Dynamic Factors in a Large N and T Panel," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 91-96, January.
    22. Johansen, Soren, 1995. "Likelihood-Based Inference in Cointegrated Vector Autoregressive Models," OUP Catalogue, Oxford University Press, number 9780198774501, Decembrie.
    23. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, vol. 58(1), pages 113-144, January.
    24. Deistler, Manfred & Wagner, Martin, 2017. "Cointegration in singular ARMA models," Economics Letters, Elsevier, vol. 155(C), pages 39-42.
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    Cited by:

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    2. Sergej Gričar & Štefan Bojnec, 2021. "Technical Analysis of Tourism Price Process in the Eurozone," JRFM, MDPI, vol. 14(11), pages 1-25, October.
    3. Donato Ceci & Andrea Silvestrini, 2023. "Nowcasting the state of the Italian economy: The role of financial markets," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 42(7), pages 1569-1593, November.
    4. Barigozzi, Matteo & Lippi, Marco & Luciani, Matteo, 2021. "Large-dimensional Dynamic Factor Models: Estimation of Impulse–Response Functions with I(1) cointegrated factors," Journal of Econometrics, Elsevier, vol. 221(2), pages 455-482.
    5. Feng, Zongbao & Chen, Weiya & Liu, Yang & Chen, Hongyu & Skibniewski, Mirosław J., 2023. "Long-term equilibrium relationship analysis and energy-saving measures of metro energy consumption and its influencing factors based on cointegration theory and an ARDL model," Energy, Elsevier, vol. 263(PD).
    6. Lippi, Marco & Deistler, Manfred & Anderson, Brian, 2023. "High-Dimensional Dynamic Factor Models: A Selective Survey and Lines of Future Research," Econometrics and Statistics, Elsevier, vol. 26(C), pages 3-16.
    7. Poncela, Pilar & Ruiz, Esther & Miranda, Karen, 2021. "Factor extraction using Kalman filter and smoothing: This is not just another survey," International Journal of Forecasting, Elsevier, vol. 37(4), pages 1399-1425.
    8. Juho Koistinen & Bernd Funovits, 2022. "Estimation of Impulse-Response Functions with Dynamic Factor Models: A New Parametrization," Papers 2202.00310, arXiv.org, revised Feb 2022.
    9. Matteo Barigozzi & Marc Hallin, 2023. "Dynamic Factor Models: a Genealogy," Working Papers ECARES 2023-15, ULB -- Universite Libre de Bruxelles.
    10. Li, Y-N. & Chen, J. & Linton, O., 2021. "Estimation of Common Factors for Microstructure Noise and Efficient Price in a High-frequency Dual Factor Model," Cambridge Working Papers in Economics 2150, Faculty of Economics, University of Cambridge.
    11. Tibor Szendrei & Katalin Varga, 2020. "FISS – A Factor-based Index of Systemic Stress in the Financial System," Russian Journal of Money and Finance, Bank of Russia, vol. 79(1), pages 3-34, March.

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