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  • Massimo Franchi (a1) and Paolo Paruolo (a2)


This article defines the class of ${\cal H}$ -valued autoregressive (AR) processes with a unit root of finite type, where ${\cal H}$ is a possibly infinite-dimensional separable Hilbert space, and derives a generalization of the Granger–Johansen Representation Theorem valid for any integration order $d = 1,2, \ldots$ . An existence theorem shows that the solution of an AR process with a unit root of finite type is necessarily integrated of some finite integer order d, displays a common trends representation with a finite number of common stochastic trends, and it possesses an infinite-dimensional cointegrating space when ${\rm{dim}}{\cal H} = \infty$ . A characterization theorem clarifies the connections between the structure of the AR operators and (i) the order of integration, (ii) the structure of the attractor space and the cointegrating space, (iii) the expression of the cointegrating relations, and (iv) the triangular representation of the process. Except for the fact that the dimension of the cointegrating space is infinite when ${\rm{dim}}{\cal H} = \infty$ , the representation of AR processes with a unit root of finite type coincides with the one of finite-dimensional VARs, which can be obtained setting ${\cal H} = ^p $ in the present results.


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*Address correspondence to Paolo Paruolo, European Commission, Joint Research Centre, Via E. Fermi 2749, I-21027 Ispra (VA), Italy; e-mail:


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The article benefited from useful comments from the Editor, Peter C.B. Phillips, three anonymous referees and conference and seminar participants at 2018 NBER-NSF Time Series Conference, University of California San Diego, University of Bologna and ICEEE 2019, University of Lecce. The idea of the present article was conceived while the first author was visiting the Department of Economics, Indiana University, in January 2017; the hospitality of Yoosoon Chang and Joon Park is gratefully acknowledged. During the revision of the article in September 2018, the first author visited the Department of Economics, University of California San Diego, and the hospitality of Brendan K. Beare is gratefully acknowledged. The first author acknowledges partial financial support from MIUR PRIN grant 2010J3LZEN. The views expressed in this article do not necessarily reflect those of the authors’ institutions.



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Beare, B. (2017) The Chang-Kim-Park model of cointegrated density-valued time series cannot accommodate a stochastic trend. Econ Journal Watch 14, 133137.
Beare, B., Seo, J., & Seo, W. (2017) Cointegrated linear processes in Hilbert space. Journal of Time Series Analysis 38, 10101027.
Beare, B. & Seo, W. (2019) Representation of I(1) and I(2) autoregressive Hilbertian processes. Econometric Theory,
Ben-Israel, A. & Greville, T. (2003) Generalized Inverses: Theory and Applications, 2nd ed. Springer.
Bosq, D. (2000) Linear Processes in Function Spaces. Springer-Verlag.
Bourguignon, F., Ferreira, F.H., & Lustig, N. (2005) The Microeconomics of Income Distribution Dynamics in East Asia and Latin America. World Bank and Oxford University Press
Chang, Y., Hu, B., & Park, J. (2016a). On the Error Correction Model for Functional Time Series with Unit Roots. Mimeo, Indiana University.
Chang, Y., Kim, C., & Park, J. (2016b) Nonstationarity in time series of state densities. Journal of Econometrics 192, 152167.
Cochrane, J.H. & Piazzesi, M. (2005) Bond risk premia. American Economic Review 95(1), 138160.
Franchi, M. & Paruolo, P. (2016) Inverting a matrix function around a singularity via local rank factorization. SIAM Journal of Matrix Analysis and Applications 37, 774797.
Franchi, M. & Paruolo, P. (2019) A general inversion theorem for cointegration. Econometric Reviews 38, 11761201.
Gabrys, R., Hörmann, S., & Kokoszka, P. (2013) Monitoring the intraday volatility pattern. Journal of Time Series Econometrics 5, 87116.
Gohberg, I., Goldberg, S., & Kaashoek, M. (1990) Classes of Linear Operators, vol. 1. Operator Theory. Birkhäuser Verlag.
Gohberg, I., Goldberg, S., & Kaashoek, M. (2003) Basic Classes of Linear Operators. Birkhäuser.
Granger, C.W.J. & Lee, T.H. (1989) Investigation of production, sales and inventory relationships using multicointegration and non-symmetric error correction models. Journal of Applied Econometrics 4, S145S159.
Gregoir, S. (1999) Multivariate time series with various hidden unit roots, Part I. Econometric Theory 15, 435468.
Hörmann, S., Horváth, L., & Reeder, R. (2013) A functional version of the ARCH model. Econometric Theory 29, 267288.
Hörmann, S. & Kokoszka, P. (2012) Functional time series. Handbook of Statistics 30, 157186.
Horn, R. & Johnson, C. (2013) Matrix Analysis. Cambridge University Press.
Horváth, L. & Kokoszka, P. (2012) Inference for Functional Data with Application. Springer.
Howlett, P., Avrachenkov, K., Pearce, C., & Ejov, V. (2009) Inversion of analytically perturbed linear operators that are singular at the origin. Journal of Mathematical Analysis and Applications 353, 6884.
Hu, B. & Park, J. (2016). Econometric Analysis of Functional Dynamics in the Presence of Persistence. Mimeo, Indiana University.
Johansen, S. (1996) Likelihood-based Inference in Cointegrated Vector Auto-Regressive Models. Oxford University Press.
Kargin, V. & Onatski, A. (2008) Curve forecasting by functional autoregression. Journal of Multivariate Analysis 99, 25082526.
Kokoszka, P. & Reimherr, M. (2017) Introduction to Functional Data Analysis. Chapman and Hall.
Petersen, A. & Müller, H.-G. (2016) Functional data analysis for density functions by transformation to a Hilbert space. Annals of Statistics 44(1), 183218.
Phillips, P. (1991a) Optimal inference in cointegrated systems. Econometrica 59, 283306.
Phillips, P. (1991b). Spectral regression for cointegrated time series. In Barnett, W. (ed.), Nonparametric and Semiparametric Methods in Economics and Statistics, pp. 413435. Cambridge University Press.
Piketty, T. (2014) Capital in the Twenty-First Century. The Belknap Press of Harvard University Press.
Seo, W. & Beare, B. (2019) Cointegrated linear processes in Bayes Hilbert space. Statistics and Probability Letters 147, 9095.
Stock, J. & Watson, M. (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783820.


  • Massimo Franchi (a1) and Paolo Paruolo (a2)


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