Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T18:29:39.889Z Has data issue: false hasContentIssue false
  • English
  • Français

REPRESENTATION OF I(1) AND I(2) AUTOREGRESSIVE HILBERTIAN PROCESSES

Published online by Cambridge University Press:  22 November 2019

Brendan K. Beare  and
Won-Ki Seo
Brendan K. Beare*
Affiliation:
University of Sydney
Won-Ki Seo
Affiliation:
Queen’s University
*
Address correspondence to Brendan K. Beare, School of Economics, University of Sydney, Sydney, Australia; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop versions of the Granger–Johansen representation theorems for I(1) and I(2) vector autoregressive processes that apply to processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. An I(1) or I(2) solution to an autoregressive law of motion is obtained when the inverse of the autoregressive operator pencil has a pole of first or second order at one. We obtain a range of necessary and sufficient conditions for such a pole to be of first or second order. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 5 , October 2020 , pp. 773 - 802
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We thank Massimo Franchi, Peter Phillips and seminar participants at the Einaudi Institute for Economics and Finance, UC San Diego, the Université libre de Bruxelles, and the 2019 NBER-NSF Time Series Conference in Hong Kong for helpful discussions. Beare also thanks Phil Roberts for his feedback and encouragement. An earlier version of this article titled “Representation of I(1) autoregressive Hilbertian processes” was posted on the arXiv.org preprint repository in January 2017.

References

REFERENCES

Al Sadoon, M. (2018) The linear systems approach to rational expectations models. Econometric Theory 34, 628–658.CrossRefGoogle Scholar
Beare, B.K. (2017) The Chang-Kim-Park model of cointegrated density-valued time series cannot accomodate a stochastic trend. Econ Journal Watch 14, 133–137.Google Scholar
Beare, B.K. & Seo, W.-K. (2017) Representation of I(1) autoregressive Hilbertian processes. ArXiv e-prints 1701.08149v1 [math.ST].Google Scholar
Beare, B.K., Seo, J., & Seo, W.-K. (2017) Cointegrated linear processes in Hilbert space. Journal of Time Series Analysis 38, 1010–1027.CrossRefGoogle Scholar
Ben-Israel, A. & Greville, T.N.E. (2003) Generalized Inverses: Theory and Applications, 2nd ed. Springer.Google Scholar
Bosq, D. (2000) Linear Processes in Function Spaces. Springer.CrossRefGoogle Scholar
Cerovecki, C. & Hörmann, S. (2017) On the CLT for discrete Fourier transforms of functional time series. Journal of Multivariate Analysis 154, 282–295.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 1050–1063.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1988) Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367–401.CrossRefGoogle Scholar
Chang, Y., Hu, B., & Park, J.-Y. (2016) On the error correction model for functional time series with unit roots. Mimeo, Indiana University.Google Scholar
Chang, Y., Kim, C.-S., & Park, J.-Y. (2016) Nonstationarity in time series of state densities. Journal of Econometrics 192, 152–167.CrossRefGoogle Scholar
Cheng, X. & Phillips, P.C.B. (2009) Semiparametric cointegrating rank selection. Econometrics Journal 12, S83–S104.CrossRefGoogle Scholar
Cheng, X. & Phillips, P.C.B. (2012) Cointegrating rank selection in models with time-varying variance. Journal of Econometrics 169, 155–165.CrossRefGoogle Scholar
Davidson, J.E.H., Hendry, D.F., Srba, F., & Yeo, S. (1978) Econometric modelling of the aggregate time-series relationship between consumers’ expenditure and income in the United Kingdom. Economic Journal 88, 661–692.CrossRefGoogle Scholar
Engle, R.F. & Granger, C.W.J. (1987) Co-integration and error correction: Representation, estimation and testing. Econometrica 55, 251–276.CrossRefGoogle Scholar
Engsted, T. & Johansen, S. (1999) Granger’s representation theorem and multicointegration. In Engle, R.F. & White, H. (eds.), Cointegration, Causality and Forecasting: Festschrift in Honour of Clive Granger, pp. 200-211. Oxford University Press.Google Scholar
Faliva, M. & Zoia, M.G. (2002) Matrix polynomials and their inversion: The algebraic framework of unit-root econometrics representation theorems. Statistica 62, 187–202.Google Scholar
Faliva, M. & Zoia, M.G. (2009) Dynamic Model Analysis: Advanced Matrix Methods and Unit-root Econometrics Representation Theorems, 2nd ed. Springer.CrossRefGoogle Scholar
Faliva, M. & Zoia, M.G. (2011) An inversion formula for a matrix polynomial about a (unit) root. Linear and Multilinear Algebra 59, 541–556.CrossRefGoogle Scholar
Franchi, M. (2007) The integration order of vector autoregressive processes. Econometric Theory 23, 546–553.CrossRefGoogle Scholar
Franchi, M. & Paruolo, P. (2011) Inversion of regular analytic matrix functions: Local Smith form and subspace duality. Linear Algebra and its Applications 435, 2896–2912.CrossRefGoogle Scholar
Franchi, M. & Paruolo, P. (2016) Inverting a matrix function around a singularity via local rank factorization. SIAM Journal of Matrix Analysis and Applications 37, 774–797.CrossRefGoogle Scholar
Franchi, M. & Paruolo, P. (2019a) Cointegration in functional autoregressive processes. Econometric Theory, doi: https://doi.org/10.1017/S0266466619000306.Google Scholar
Franchi, M. & Paruolo, P. (2019b) A general inversion theorem for cointegration. Econometric Reviews, in press.CrossRefGoogle Scholar
Gohberg, I., Goldberg, S., & Kaashoek, M.A. (1990) Classes of Linear Operators, vol. 1. Birkhäuser.CrossRefGoogle Scholar
Granger, C.W.J. (1981) Some properties of times series data and their use in econometric model specification. Journal of Econometrics 16, 121–130.CrossRefGoogle Scholar
Granger, C.W.J. (1983) Cointegrated variables and error-correcting models. Mimeo, University of California, San Diego.Google Scholar
Granger, C.W.J. (1986) Developments in the study of cointegrated economic variables. Oxford Bulletin of Economics and Statistics 48, 213–228.CrossRefGoogle Scholar
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, S145–S159.CrossRefGoogle Scholar
Granger, C.W.J. & Lee, T.H. (1990) Multicointegration. In Rhodes, G.F. & Fomby, T.B. (eds.), Advances in Econometrics, Vol. 8: Co-integration, Spurious Regressions, and Unit Roots, pp. 71–84. JAI Press.Google Scholar
Hörmann, S. & Kokoszka, P. (2012) Functional time series. In Rao, T.S., Rao, S.S., & Rao, C.R. (eds.), Handbook of Statistics, Volume 30: Time Series Analysis—Methods and Applications, chapter 7. pp. 157–186. North-Holland.CrossRefGoogle Scholar
Howland, J.S. (1971) Simple poles of operator-valued functions. Journal of Mathematical Analysis and Applications 36, 12–21.CrossRefGoogle Scholar
Hu, B. & Park, J.-Y. (2016) Econometric analysis of functional dynamics in the presence of persistence. Mimeo, Indiana University.Google Scholar
Johansen, S. (1988) The mathematical structure of error correction models. Contemporary Mathematics 80, 359–386.CrossRefGoogle Scholar
Johansen, S. (1991) Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551–1580.CrossRefGoogle Scholar
Johansen, S. (1992) A representation of vector autoregressive processes integrated of order 2. Econometric Theory 8, 188–202.CrossRefGoogle Scholar
Johansen, S. (1996) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press.Google Scholar
Johansen, S. (2009) Representation of cointegrated autoregressive processes with application to fractional processes. Econometric Reviews 28, 121–145.CrossRefGoogle Scholar
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
La Cour, L. (1998) A parametric characterization of integrated vector autoregressive (VAR) processes. Econometric Theory 14, 187–199.CrossRefGoogle Scholar
Markus, A.S. (2012) Introduction to the Spectral Theory of Polynomial Operator Pencils. American Mathematical Society.CrossRefGoogle Scholar
Nielsen, M. Ø., Seo, W.-K., & Seong, D. (2019) Variance ratio test for the number of stochastic trends in functional time series. Queen’s Economics Department Working paper No. 1420.Google Scholar
Park, J.-Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated processes: Part I. Econometric Theory 4, 468–497.CrossRefGoogle Scholar
Park, J.-Y. & Phillips, P.C.B. (1989) Statistical inference in regressions with integrated processes: Part II. Econometric Theory 5, 95–131.CrossRefGoogle Scholar
Phillips, P.C.B. (1986) Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311–340.CrossRefGoogle Scholar
Phillips, P.C.B. (1988) Regression theory for near-integrated time series. Econometrica 56, 1021–1043.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) Optimal inference in cointegrated systems. Econometrica 59, 283–306.CrossRefGoogle Scholar
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 473–495.CrossRefGoogle Scholar
Phillips, P.C.B. & Hansen, B.E. (1990) Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125.CrossRefGoogle Scholar
Phillips, P.C.B. & Kheifets, I. (2019) On multicointegration. Mimeo, Yale University.Google Scholar
Phillips, P.C.B. & Park, J.-Y. (1988) Asymptotic equivalence of ordinary least squares and generalized least squares in regressions with integrated regressors. Journal of the American Statistical Association 83, 111–115.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Sargan, J.D. (1964) Wages and prices in the United Kingdom: A study in econometric methodology. In Hart, P.E., Mills, G., & Whitacker, J.K. (eds.), Econometric Analysis for National Economic Planning, pp. 23–63. Butterworths.Google Scholar
Schumacher, J.M. (1991) System-theoretic trends in econometrics. In Antoulas, A.C. (ed.), Mathematical System Theory: The Influence of R. E. Kalman, 559–577. Springer.CrossRefGoogle Scholar
Seo, W.-K. (2018) Cointegration and representation of integrated autoregressive processes in function space. ArXiv e-prints 1712.08748v3 [math.FA].Google Scholar
Seo, W.-K. & Beare, B.K. (2019) Cointegrated linear processes in Bayes Hilbert space. Statistics and Probability Letters 147, 90–95.CrossRefGoogle Scholar
Steinberg, S. (1968) Meromorphic families of compact operators. Archive for Rational Mechanics and Analysis 31, 372–379.CrossRefGoogle Scholar
Yoo, B.-S. (1987) Co-integrated time series: Structure, forecasting and testing. Doctoral thesis, University of California, San Diego.Google Scholar