Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors
- A generalization of Johansen’s proof of the Granger Representation Theorem (from MA to AR), this is Proposition 2. Consider an singular vector , with dimension r, rank , and cointegrating rank . Assuming that has an ARMA structure, and that some simple additional conditions hold, has a representation as a vector error correction mechanism (VECM) with c error correction terms:
- Assuming that the parameters of and may vary in an open subset of , see Section 3.2 for the definition of , in Proposition 3 we show that all the assumptions used to obtain (4), and also the assumption that unity is the only possible zero of , hold for generic values of the parameters. This implies that the matrices and are generically of finite degree, which is obviously not the case for nonsingular vectors.2
2. Stationary and Singular Vectors
2.1. Stationary Singular Vectors
- a non-singular q-dimensional white-noise process ,
- an stable polynomial matrix , with ,
- an matrix whose rank is q for all z with the exception of a finite subset of , such that
- is an polynomial matrix of degree .
- is an polynomial matrix of degree . .
- Denoting by the vector containing the coefficients of the entries of and , we assume that , where Π is an open subset of such that for ,(1) is stable,(2) with the exception of a finite subset of .
- Suppose that is an matrix polynomial in L. If is zeroless then has an finite-degree stable left inverse, i.e., there exists a finite-degree polynomial matrix such that:(a),(b) implies ,(c). Let be the stationary solution of and suppose that is zeroless. Then has a finite vector autoregressive representation (VAR) , where and is a finite-degree left inverse of .
- Assume that is the stationary solution of , where belongs to a rational reduced-rank family of filters with parameter set Π. For generic values of the parameters in Π, is zeroless so that has a finite VAR representation.
2.3. Singular Vectors
3. Representation Theory for Singular Vectors
3.1. The Granger Representation Theorem (MA to AR)
3.2. Generically, Is a Finite-Degree Polynomial
- The matrix has the parameterization
- is an polynomial matrix of degree . .
- Denoting by the vector containing the coefficients of the matrices , , , and , we assume that , where Π is an open subset of such that for :(1) is stable,(2) with the exception of a finite subset of ,(3).
3.3. Permanent and Transitory Shocks
3.4. VECMs and Unrestricted VARs in The Levels
- For each replication, we estimate a (misspecified) VAR in differences (DVAR), a VAR in the levels (LVAR) and a VECM, as in Johansen (1988, 1991), assuming known c, the degree of and that of . For the VAR in differences the impulse-response functions for are cumulated to obtain impulse-response function for . The root mean square error between estimated and actual impulse-response functions is computed for each replication using all 12 impulse-responses and averaged over all replications.
4. Cointegration of the Observable Variables in a DFM
- is cointegrated only if and are both cointegrated.
- If then is cointegrated. If and rank then is cointegrated.
- Let and be the cointegrating spaces of and respectively. The vector is cointegrated if and only if the intersection of and contains non-zero vectors. In particular, (a) if and then is cointegrated, (b) if and is stationary then is cointegrated.
5. Summary and Conclusions
Conflicts of Interest
Appendix A. Proofs
Appendix A.1. Assumption 3 Holds Generically
- Let be such that none of the leading coefficients of the polynomials e, and vanishes. Of course .
- Let be a root of . If then is not a root of for all . Suppose that is a root of , for some j. As the parameters of the polynomials and are free to vary in , then, generically in , . Iterating for all roots of , generically in , and have no roots in common. Moreover, generically in , . Thus, there exists such that (a) , (b) and have no roots in common.
- Now let , so that
Appendix A.2. if R>Q and C≤Q, Assumptions 5 and 6 Do Not Imply That e t Is a Non-Cointegrated I(0) Process.
Appendix B. Non Uniqueness
Appendix B.1. Alternative Representations with Different Numbers of Error Terms
Appendix B.2. Uniqueness of Impulse-Response Functions
Appendix C. Data Generating Process for the Simulations
- Amengual, Dante, and Mark W. Watson. 2007. Consistent estimation of the number of dynamic factors in a large N and T panel. Journal of Business and Economic Statistics 25: 91–96. [Google Scholar] [CrossRef]
- Anderson, Brian DO, and Manfred Deistler. 2008a. Generalized linear dynamic factor models–A structure theory. Paper presented at IEEE Conference on Decision and Control, Cancun, Mexico, December 9–11. [Google Scholar]
- Anderson, Brian DO, and Manfred Deistler. 2008b. Properties of zero-free transfer function matrices. SICE Journal of Control, Measurement and System Integration 1: 284–92. [Google Scholar] [CrossRef][Green Version]
- Anderson, Brian DO, Manfred Deistler, Weitian Chen, and Alexander Filler. 2012. Autoregressive models of singular spectral matrices. Automatica 48: 2843–49. [Google Scholar] [CrossRef][Green Version]
- Bai, Jushan, and Serena Ng. 2007. Determining the number of primitive shocks in factor models. Journal of Business and Economic Statistics 25: 52–60. [Google Scholar] [CrossRef][Green Version]
- Banerjee, Anindya, Massimiliano Marcellino, and Igor Masten. 2014. Forecasting with factor-augmented error correction models. International Journal of Forecasting 30: 589–612. [Google Scholar] [CrossRef][Green Version]
- Banerjee, Anindya, Massimiliano Marcellino, and Igor Masten. 2017. Structural FECM: Cointegration in large–scale structural FAVAR models. Journal of Applied Econometrics 32: 1069–86. [Google Scholar] [CrossRef][Green Version]
- Barigozzi, Matteo, Antonio M. Conti, and Matteo Luciani. 2014. Do euro area countries respond asymmetrically to the common monetary policy? Oxford Bulletin of Economics and Statistics 76: 693–714. [Google Scholar] [CrossRef]
- Barigozzi, Matteo, Marco Lippi, and Matteo Luciani. 2019. Large-dimensional dynamic factor models: Estimation of impulse-response functions with I(1) cointegrated factors. arXiv arXiv:1602:02398. [Google Scholar]
- Bauer, Dietmar, and Martin Wagner. 2012. A State Space Canonical Form For Unit Root Processes. Econometric Theory 28: 1313–49. [Google Scholar] [CrossRef]
- Brockwell, Peter J., and Richard A. Davis. 1991. Time Series: Theory and Methods, 2nd ed. New York: Springer. [Google Scholar]
- Canova, Fabio. 2007. Methods for Applied Macroeconomics. Princeton: Princeton University Press. [Google Scholar]
- Chen, Weitian, Brian DO Anderson, Manfred Deistler, and Alexander Filler. 2011. Solutions of Yule-Walker equations for singular AR processes. Journal of Time Series Analysis 32: 531–38. [Google Scholar] [CrossRef]
- Deistler, Manfred, Brian DO Anderson, A. Filler, Ch Zinner, and W. Chen. 2010. Generalized linear dynamic factor models: An approach via singular autoregressions. European Journal of Control 16: 211–24. [Google Scholar] [CrossRef]
- Deistler, Manfred, and Martin Wagner. 2017. Cointegration in singular ARMA models. Economics Letters 155: 39–42. [Google Scholar] [CrossRef][Green Version]
- Forni, Mario, and Luca Gambetti. 2010. The dynamic effects of monetary policy: A structural factor model approach. Journal of Monetary Economics 57: 203–16. [Google Scholar] [CrossRef]
- Forni, Mario, Domenico Giannone, Marco Lippi, and Lucrezia Reichlin. 2009. Opening the Black Box: Structural Factor Models versus Structural VARs. Econometric Theory 25: 1319–47. [Google Scholar] [CrossRef][Green Version]
- Forni, Mario, Marc Hallin, Marco Lippi, and Lucrezia Reichlin. 2000. The Generalized Dynamic Factor Model: Identification and Estimation. The Review of Economics and Statistics 82: 540–54. [Google Scholar] [CrossRef]
- Forni, Mario, Marc Hallin, Marco Lippi, and Paolo Zaffaroni. 2015. Dynamic factor models with infinite-dimensional factor spaces: One-sided representations. Journal of Econometrics 185: 359–71. [Google Scholar] [CrossRef][Green Version]
- Forni, Mario, and Marco Lippi. 2001. The Generalized Dynamic Factor Model: Representation Theory. Econometric Theory 17: 1113–41. [Google Scholar] [CrossRef][Green Version]
- Franchi, Massimo, and Paolo Paruolo. 2019. A general inversion theorem for cointegration. Econometric Reviews 38: 1176–201. [Google Scholar] [CrossRef][Green Version]
- Franklin, J. N. 2000. Matrix Theory, 2nd ed. New York: Dover Publications. [Google Scholar]
- Giannone, Domenico, Lucrezia Reichlin, and Luca Sala. 2005. Monetary policy in real time. In NBER Macroeconomics Annual 2004. Edited by Mark Gertler and Kenneth Rogoff. Cambridge: MIT Press, chp. 3. pp. 161–224. [Google Scholar]
- Gregoir, Stéphane. 1999. Multivariate Time Series With Various Hidden Unit Roots, Part I. Econometric Theory 15: 435–68. [Google Scholar] [CrossRef]
- Johansen, Søren. 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12: 231–54. [Google Scholar] [CrossRef]
- Johansen, Søren. 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59: 1551–80. [Google Scholar] [CrossRef]
- Johansen, Søren. 1995. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, 1st ed. Oxford: Oxford University Press. [Google Scholar]
- Lancaster, Peter, and Miron Tismenetsky. 1985. The Theory of Matrices, 2nd ed. New York: Academic Press. [Google Scholar]
- Luciani, Matteo. 2015. Monetary policy and the housing market: A structural factor analysis. Journal of Applied Econometrics 30: 199–218. [Google Scholar] [CrossRef][Green Version]
- Phillips, Peter C.B. 1998. Impulse response and forecast error variance asymptotics in nonstationary VARs. Journal of Econometrics 83: 21–56. [Google Scholar] [CrossRef][Green Version]
- Rozanov, Yu. A. 1967. Stationary Random Processes. San Francisco: Holden-Day. [Google Scholar]
- Sargent, Thomas J. 1989. Two Models of Measurements and the Investment Accelerator. Journal of Political Economy 97: 251–87. [Google Scholar] [CrossRef]
- Sims, Christopher, James H. Stock, and Mark W. Watson. 1990. Inference in linear time series models with some unit roots. Econometrica 58: 113–44. [Google Scholar] [CrossRef]
- Stock, James H., and Mark W. Watson. 1988. Testing for common trends. Journal of the American Statistical Association 83: 1097–107. [Google Scholar] [CrossRef]
- Stock, James H., and Mark W. Watson. 2002a. Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association 97: 1167–79. [Google Scholar] [CrossRef][Green Version]
- Stock, James H., and Mark W. Watson. 2002b. Macroeconomic forecasting using diffusion indexes. Journal of Business and Economic Statistics 20: 147–62. [Google Scholar] [CrossRef][Green Version]
- Stock, James H., and Mark W. Watson. 2016. Dynamic factor models, factor-augmented vector autoregressions, and structural vector autoregressions in macroeconomics. In Handbook of Macroeconomics. Edited by John B. Taylor and Harald Uhlig. Amsterdam: North Holland, Elsevier, vol. 2A, chp. 8. pp. 415–525. [Google Scholar]
- Van der Waerden, Bartel Leendert. 1953. Modern Algebra, 2nd ed. New York: Frederick Ungar, vol. I. [Google Scholar]
- Watson, Mark W. 1994. Vector autoregressions and cointegration. In Handbook of Econometrics. Edited by Robert F. Engle and Daniel L. McFadden. Amsterdam: North Holland, Elsevier, vol. 4, chp. 47. pp. 2843–915. [Google Scholar]
Usually orthonormality is assumed. This is convenient but not necessary in the present paper.
To our knowledge, the present paper is the first to study cointegration and error correction representations for singular vectors, the factors of dynamic factor models in particular. An error correction model in the DFM framework is studied in (Banerjee et al. 2014, 2017). However, their focus is on the relationship between the observable variables and the factors. Their error correction term is a linear combination of the variables and the factors , which is stationary if the idiosyncratic components are stationary (so that the x’s and the factors are cointegrated). Because of this and other differences their results are not directly comparable to those in the present paper.
In the square case, r = q, Assumption 3 holds if and only if M(z) is unimodular.
If is a zero of , multiply by an invertible matrix such that is a zero of, say, the first row of . Then multiply by the diagonal matrix with in position and unity elsewhere on the main diagonal. Iterating, all the zeros of are removed.
© 2020 by Matteo Barigozzi and Marco Lippi. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Barigozzi, M.; Lippi, M.; Luciani, M. Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors. Econometrics 2020, 8, 3. https://doi.org/10.3390/econometrics8010003
Barigozzi M, Lippi M, Luciani M. Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors. Econometrics. 2020; 8(1):3. https://doi.org/10.3390/econometrics8010003Chicago/Turabian Style
Barigozzi, Matteo, Marco Lippi, and Matteo Luciani. 2020. "Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors" Econometrics 8, no. 1: 3. https://doi.org/10.3390/econometrics8010003