Unit Roots and Cointegration

  • Stephan SmeekesEmail author
  • Etienne Wijler
Part of the Advanced Studies in Theoretical and Applied Econometrics book series (ASTA, volume 52)


In this chapter we investigate how the possible presence of unit roots and cointegration affects forecasting with Big Data. As most macroeoconomic time series are very persistent and may contain unit roots, a proper handling of unit roots and cointegration is of paramount importance for macroeconomic forecasting. The high-dimensional nature of Big Data complicates the analysis of unit roots and cointegration in two ways. First, transformations to stationarity require performing many unit root tests, increasing room for errors in the classification. Second, modelling unit roots and cointegration directly is more difficult, as standard high-dimensional techniques such as factor models and penalized regression are not directly applicable to (co)integrated data and need to be adapted. In this chapter we provide an overview of both issues and review methods proposed to address these issues. These methods are also illustrated with two empirical applications.


  1. Ahn, S. C., & Horenstein, A. R. (2013). Eigenvalue ratio test for the number of factors. Econometrica, 81(3), 1203–1227.Google Scholar
  2. Alessi, L., Barigozzi, M., & Capasso, M. (2010). Improved penalization for determining the number of factors in approximate factor models. Statistics & Probability Letters, 80(23), 1806–1813.CrossRefGoogle Scholar
  3. Bai, J. (2004). Estimating cross-section common stochastic trends in nonstationary panel data. Journal of Econometrics, 122(1), 137–183.CrossRefGoogle Scholar
  4. Bai, J., & Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica, 70(1), 191–221.CrossRefGoogle Scholar
  5. Bai, J., & Ng, S. (2004). A panic attack on unit roots and cointegration. Econometrica, 72(4), 1127–1177.CrossRefGoogle Scholar
  6. Banerjee, A., Marcellino, M., & Masten, I. (2014). Forecasting with factor-augmented error correction models. International Journal of Forecasting, 30(3), 589–612.CrossRefGoogle Scholar
  7. Banerjee, A., Marcellino, M., & Masten, I. (2016). An overview of the factor augmented error-correction model. In E. Hillebrand & S. J. Koopman (Eds.), Dynamic factor models (Chap. 1, Vol. 35, pp. 3–41). Advances in Econometrics. Bingley: Emerald Group Publishing Limited.Google Scholar
  8. Banerjee, A., Marcellino, M., & Masten, I. (2017). Structural FECM: Cointegration in large-scale structural FAVAR models. Journal of Applied Econometrics, 32(6), 1069–1086.CrossRefGoogle Scholar
  9. Barigozzi, M., Lippi, M., & Luciani, M. (2017). Dynamic factor models, cointegration, and error correction mechanisms (arXiv e-prints No. 1510.02399).Google Scholar
  10. Barigozzi, M., Lippi, M., & Luciani, M. (2018). Non-stationary dynamic factor models for large datasets (arXiv e-prints No. 1602.02398).Google Scholar
  11. Barigozzi, M., & Trapani, L. (2018). Determining the dimension of factor structures in non-stationary large datasets (arXiv e-prints No. 1806.03647).Google Scholar
  12. Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of the Royal Statistical Society: Series B, 57(1), 289–300.Google Scholar
  13. Bernanke, B., Boivin, J., & Eliasz, P. S. (2005). Measuring the effects of monetary policy: A factor-augmented vector autoregressive (FAVAR) approach. The Quarterly Journal of Economics, 120(1), 387–422.Google Scholar
  14. Callot, L. A., & Kock, A. B. (2014). Oracle efficient estimation and forecasting with the adaptive lasso and the adaptive group lasso in vector autoregressions. Essays in Nonlinear Time Series Econometrics, 238–268.Google Scholar
  15. Cavaliere, G. (2005). Unit root tests under time-varying variances. Econometric Reviews, 23(3), 259–292.CrossRefGoogle Scholar
  16. Cavaliere, G., Phillips, P. C. B., Smeekes, S., & Taylor, A. M. R. (2015). Lag length selection for unit root tests in the presence of nonstationary volatility. Econometric Reviews, 34(4), 512–536.CrossRefGoogle Scholar
  17. Cavaliere, G., & Taylor, A. M. R. (2008). Bootstrap unit root tests for time series with nonstationary volatility. Econometric Theory, 24(1), 43–71.CrossRefGoogle Scholar
  18. Cavaliere, G., & Taylor, A. M. R. (2009). Bootstrap M unit root tests. Econometric Reviews, 28(5), 393–421.CrossRefGoogle Scholar
  19. Cheng, X., & Phillips, P. C. B. (2009). Semiparametric cointegrating rank selection. Econometrics Journal, 12(suppl1), S83–S104.CrossRefGoogle Scholar
  20. Choi, I. (2015). Almost all about unit roots: Foundations, developments, and applications. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  21. Chortareas, G., & Kapetanios, G. (2009). Getting PPP right: Identifying mean-reverting real exchange rates in panels. Journal of Banking and Finance, 33(2), 390–404.CrossRefGoogle Scholar
  22. Christoffersen, P. F., & Diebold, F. X. (1998). Cointegration and long-horizon fore-casting. Journal of Business & Economic Statistics, 16(4), 450–456.Google Scholar
  23. Clements, M. P., & Hendry, D. F. (1995). Forecasting in cointegrated systems. Journal of Applied Econometrics, 10(2), 127–146.CrossRefGoogle Scholar
  24. De Mol, C., Giannone, D., & Reichlin, L. (2008). Forecasting using a large number of predictors: Is Bayesian shrinkage a valid alternative to principal components? Journal of Econometrics, 146, 318–328.CrossRefGoogle Scholar
  25. Dickey, D. A., & Fuller, W. A. (1979). Distribution of estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74(366a), 427–431.CrossRefGoogle Scholar
  26. Diebold, F. X., & Kilian, L. (2000). Unit-root tests are useful for selecting forecasting models. Journal of Business & Economic Statistics, 18(3), 265–273.Google Scholar
  27. Elliott, G., Rothenberg, T. J., & Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813–836.CrossRefGoogle Scholar
  28. Enders, W. (2008). Applied econometric time series (4th ed.). New Delhi: Wiley.Google Scholar
  29. Forni, M., Hallin, M., Lippi, M., & Reichlin, L. (2005). The generalized dynamic factor model: One-sided estimation and forecasting. Journal of the American Statistical Association, 100(471), 830–840.CrossRefGoogle Scholar
  30. Franses, P. H., & McAleer, M. (1998). Testing for unit roots and non-linear transformations. Journal of Time Series Analysis, 19(2), 147–164.CrossRefGoogle Scholar
  31. Friedman, J., Hastie, T., & Tibshirani, R. (2010a). A note on the group lasso and a sparse group lasso (arXiv e-prints No. 1001.0736).Google Scholar
  32. Friedman, J., Hastie, T., & Tibshirani, R. (2010b). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33(1), 1–22.CrossRefGoogle Scholar
  33. Friedrich, M., Smeekes, S., & Urbain, J.-P. (2018). Autoregressive wild bootstrap inference for nonparametric trends (arXiv e-prints No. 1807.02357).Google Scholar
  34. Gonçalves, S., & Perron, B. (2014). Bootstrapping factor-augmented regression models. Journal of Econometrics, 182(1), 156–173.CrossRefGoogle Scholar
  35. Hallin, M., & Liška, R. (2007). Determining the number of factors in the general dynamic factor model. Journal of the American Statistical Association, 102(478), 603–617.CrossRefGoogle Scholar
  36. Hamilton, J. D. (1994). Time series analysis. Princeton: Princeton University Press.Google Scholar
  37. Hanck, C. (2009). For which countries did PPP hold? A multiple testing approach. Empirical Economics, 37(1), 93–103.CrossRefGoogle Scholar
  38. Harvey, D. I., Leybourne, S. J., & Taylor, A. M. R. (2009). Unit root testing in practice: Dealing with uncertainty over the trend and initial condition. Econometric Theory, 25(3), 587–636.CrossRefGoogle Scholar
  39. Harvey, D. I., Leybourne, S. J., & Taylor, A. M. R. (2012). Testing for unit roots in the presence of uncertainty over both the trend and initial condition. Journal of Econometrics, 169(2), 188–195.CrossRefGoogle Scholar
  40. Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: Principles and practice. OTexts.Google Scholar
  41. Johansen, S. (1995a). A statistical analysis of cointegration for i(2) variables. Econometric Theory, 11(1), 25–59.CrossRefGoogle Scholar
  42. Johansen, S. (1995b). Likelihood-based inference in cointegrated vector autoregressive models. Oxford: Oxford University Press.CrossRefGoogle Scholar
  43. Justiniano, A., & Primiceri, G. (2008). The time-varying volatility of macroeconomic fluctuations. American Economic Review, 98(3), 604–641.CrossRefGoogle Scholar
  44. Klaassen, S., Kueck, J., & Spindler, M. (2017). Transformation models in high-dimensions (arXiv e-prints No. 1712.07364).Google Scholar
  45. Kock, A. B. (2016). Consistent and conservative model selection with the adaptive lasso in stationary and nonstationary autoregressions. Econometric Theory, 32, 243–259.CrossRefGoogle Scholar
  46. Kramer, W., & Davies, L. (2002). Testing for unit roots in the context of misspecified logarithmic random walks. Economics Letters, 74(3), 313–319.CrossRefGoogle Scholar
  47. Liang, C., & Schienle, M. (2019). Determination of vector error correction models in high dimensions. Journal of Econometrics, 208(2), 418–441.CrossRefGoogle Scholar
  48. Liao, Z., & Phillips, P. C. B. (2015). Automated estimation of vector error correction models. Econometric Theory, 31(3), 581–646.CrossRefGoogle Scholar
  49. Marcellino, M., Stock, J. H., & Watson, M. W. (2006). A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series. Journal of Econometrics, 135(2), 499–526.CrossRefGoogle Scholar
  50. McCracken, M. W., & Ng, S. (2016). FRED-MD: A monthly database for macroeconomic research. Journal of Business & Economic Statistics, 34(4), 574–589.CrossRefGoogle Scholar
  51. Meier, L., Van De Geer, S., & Bühlmann, P. (2008). The group lasso for logistic regression. Journal of the Royal Statistical Society: Series B, 70(1), 53–71.CrossRefGoogle Scholar
  52. Moon, H. R., & Perron, B. (2012). Beyond panel unit root tests: Using multiple testing to determine the non stationarity properties of individual series in a panel. Journal of Econometrics, 169(1), 29–33.CrossRefGoogle Scholar
  53. Müller, U. K., & Elliott, G. (2003). Tests for unit roots and the initial condition. Econometrica, 71(4), 1269–1286.CrossRefGoogle Scholar
  54. Ng, S. (2008). A simple test for nonstationarity in mixed panels. Journal of Business and Economic Statistics, 26(1), 113–127.CrossRefGoogle Scholar
  55. Onatski, A. (2010). Determining the number of factors from empirical distribution of eigenvalues. The Review of Economics and Statistics, 92(4), 1004–1016.CrossRefGoogle Scholar
  56. Onatski, A., & Wang, C. (2018). Alternative asymptotics for cointegration tests in large VARs. Econometrica, 86(4), 1465–1478.CrossRefGoogle Scholar
  57. Palm, F. C., Smeekes, S., & Urbain, J.-P. (2008). Bootstrap unit root tests: Comparison and extensions. Journal of Time Series Analysis, 29(1), 371–401.CrossRefGoogle Scholar
  58. Palm, F. C., Smeekes, S., & Urbain, J.-P. (2011). Cross-sectional dependence robust block bootstrap panel unit root tests. Journal of Econometrics, 163(1), 85–104.CrossRefGoogle Scholar
  59. Pantula, S. G. (1989). Testing for unit roots in time series data. Econometric Theory, 5(2), 256–271.CrossRefGoogle Scholar
  60. Pedroni, P., Vogelsang, T. J., Wagner, M., & Westerlund, J. (2015). Nonparametric rank tests for non-stationary panels. Journal of Econometrics, 185(2), 378–391.CrossRefGoogle Scholar
  61. Rho, Y., & Shao, X. (2019). Bootstrap-assisted unit root testing with piecewise locally stationary errors. Econometric Theory, 35(1), 142–166.CrossRefGoogle Scholar
  62. Romano, J. P., Shaikh, A. M., & Wolf, M. (2008a). Control of the false discovery rate under dependence using the bootstrap and subsampling. Test, 17(3), 417–442.CrossRefGoogle Scholar
  63. Romano, J. P., Shaikh, A. M., & Wolf, M. (2008b). Formalized data snooping based on generalized error rates. Econometric Theory, 24(2), 404–447.CrossRefGoogle Scholar
  64. Romano, J. P., & Wolf, M. (2005). Stepwise multiple testing as formalized data snooping. Econometrica, 73(4), 1237–1282.CrossRefGoogle Scholar
  65. Schiavoni, C., Palm, F., Smeekes, S., & van den Brakel, J. (2019). A dynamic factor model approach to incorporate big data in state space models for official statistics (arXiv e-print No. 1901.11355).Google Scholar
  66. Schwert, G. W. (1989). Tests for unit roots: A Monte Carlo investigation. Journal of Business and Economic Statistics, 7(1), 147–159.Google Scholar
  67. Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association, 105(489), 218–235.CrossRefGoogle Scholar
  68. Simon, N., Friedman, J., Hastie, T., & Tibshirani, R. (2013). A sparse-group lasso. Journal of Computational and Graphical Statistics, 22(2), 231–245.CrossRefGoogle Scholar
  69. Smeekes, S. (2015). Bootstrap sequential tests to determine the order of integration of individual units in a time series panel. Journal of Time Series Analysis, 36(3), 398–415.CrossRefGoogle Scholar
  70. Smeekes, S., & Taylor, A. M. R. (2012). Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory, 28(2), 422–456.CrossRefGoogle Scholar
  71. Smeekes, S., & Urbain, J.-P. (2014a). A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing (GSBE Research Memorandum No. RM/14/008). Maastricht University.Google Scholar
  72. Smeekes, S., & Urbain, J.-P. (2014b). On the applicability of the sieve bootstrap in time series panels. Oxford Bulletin of Economics and Statistics, 76(1), 139–151.CrossRefGoogle Scholar
  73. Smeekes, S., & Wijler, E. (2018a). An automated approach towards sparse single-equation cointegration modelling (arXiv e-print No. 1809.08889).Google Scholar
  74. Smeekes, S., & Wijler, E. (2018b). Macroeconomic forecasting using penalized regression methods. International Journal of Forecasting, 34(3), 408–430.CrossRefGoogle Scholar
  75. Stock, J. H., & Watson, M. W. (2002a). Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association, 97(460), 1167–1179.CrossRefGoogle Scholar
  76. Stock, J. H., & Watson, M. W. (2002b). Macroeconomic forecasting using diffusion indexes. Journal of Business & Economic Statistics, 20(2), 147–162.CrossRefGoogle Scholar
  77. Stock, J. H., & Watson, M. W. (2003). Has the business cycle changed and why? In M. Gertler & K. Rogoff (Eds.), NBER macroeconomics annual 2002 (Chap. 4, Vol. 17, pp. 159–230). Cambridge: MIT Press.Google Scholar
  78. Stock, J. H., & Watson, M. W. (2012). Generalized shrinkage methods for forecasting using many predictors. Journal of Business & Economic Statistics, 30, 481–493.CrossRefGoogle Scholar
  79. Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society (Series B), 58(1), 267–288.Google Scholar
  80. Trapani, L. (2013). On bootstrapping panel factor series. Journal of Econometrics, 172(1), 127–141.CrossRefGoogle Scholar
  81. Trapletti, A., & Hornik, K. (2018). Tseries: Time series analysis and computational finance. R package version 0.10-46. Retrieved from
  82. Wilms, I., & Croux, C. (2016). Forecasting using sparse cointegration. International Journal of Forecasting, 32(4), 1256–1267.CrossRefGoogle Scholar
  83. Zhang, R., Robinson, P., & Yao, Q. (2018). Identifying cointegration by eigenanalysis. Journal of the American Statistical Association, 114, 916–927.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Quantitative Economics, School of Business and EconomicsMaastricht UniversityMaastrichtThe Netherlands

Personalised recommendations