Skip to main content Accessibility help
×
Home

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

  • Brendan K. Beare (a1) and Won-Ki Seo (a2)

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.

Copyright

Corresponding author

*Address correspondence to Brendan K. Beare, School of Economics, University of Sydney, Sydney, Australia; e-mail: brendan.beare@sydney.edu.au.

Footnotes

Hide All

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.

Footnotes

References

Hide All
Al Sadoon, M. (2018) The linear systems approach to rational expectations models. Econometric Theory 34, 628658.
Beare, B.K. (2017) The Chang-Kim-Park model of cointegrated density-valued time series cannot accomodate a stochastic trend. Econ Journal Watch 14, 133137.
Beare, B.K. & Seo, W.-K. (2017) Representation of I(1) autoregressive Hilbertian processes. ArXiv e-prints 1701.08149v1 [math.ST].
Beare, B.K., Seo, J., & Seo, W.-K. (2017) Cointegrated linear processes in Hilbert space. Journal of Time Series Analysis 38, 10101027.
Ben-Israel, A. & Greville, T.N.E. (2003) Generalized Inverses: Theory and Applications, 2nd ed. Springer.
Bosq, D. (2000) Linear Processes in Function Spaces. Springer.
Cerovecki, C. & Hörmann, S. (2017) On the CLT for discrete Fourier transforms of functional time series. Journal of Multivariate Analysis 154, 282295.
Chan, N.H. & Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 10501063.
Chan, N.H. & Wei, C.Z. (1988) Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367401.
Chang, Y., Hu, B., & Park, J.-Y. (2016) On the error correction model for functional time series with unit roots. Mimeo, Indiana University.
Chang, Y., Kim, C.-S., & Park, J.-Y. (2016) Nonstationarity in time series of state densities. Journal of Econometrics 192, 152167.
Cheng, X. & Phillips, P.C.B. (2009) Semiparametric cointegrating rank selection. Econometrics Journal 12, S83S104.
Cheng, X. & Phillips, P.C.B. (2012) Cointegrating rank selection in models with time-varying variance. Journal of Econometrics 169, 155165.
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, 661692.
Engle, R.F. & Granger, C.W.J. (1987) Co-integration and error correction: Representation, estimation and testing. Econometrica 55, 251276.
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.
Faliva, M. & Zoia, M.G. (2002) Matrix polynomials and their inversion: The algebraic framework of unit-root econometrics representation theorems. Statistica 62, 187202.
Faliva, M. & Zoia, M.G. (2009) Dynamic Model Analysis: Advanced Matrix Methods and Unit-root Econometrics Representation Theorems, 2nd ed. Springer.
Faliva, M. & Zoia, M.G. (2011) An inversion formula for a matrix polynomial about a (unit) root. Linear and Multilinear Algebra 59, 541556.
Franchi, M. (2007) The integration order of vector autoregressive processes. Econometric Theory 23, 546553.
Franchi, M. & Paruolo, P. (2011) Inversion of regular analytic matrix functions: Local Smith form and subspace duality. Linear Algebra and its Applications 435, 28962912.
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. (2019a) Cointegration in functional autoregressive processes. Econometric Theory, doi: https://doi.org/10.1017/S0266466619000306.
Franchi, M. & Paruolo, P. (2019b) A general inversion theorem for cointegration. Econometric Reviews, in press.
Gohberg, I., Goldberg, S., & Kaashoek, M.A. (1990) Classes of Linear Operators, vol. 1. Birkhäuser.
Granger, C.W.J. (1981) Some properties of times series data and their use in econometric model specification. Journal of Econometrics 16, 121130.
Granger, C.W.J. (1983) Cointegrated variables and error-correcting models. Mimeo, University of California, San Diego.
Granger, C.W.J. (1986) Developments in the study of cointegrated economic variables. Oxford Bulletin of Economics and Statistics 48, 213228.
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.
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. 7184. JAI Press.
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. 157186. North-Holland.
Howland, J.S. (1971) Simple poles of operator-valued functions. Journal of Mathematical Analysis and Applications 36, 1221.
Hu, B. & Park, J.-Y. (2016) Econometric analysis of functional dynamics in the presence of persistence. Mimeo, Indiana University.
Johansen, S. (1988) The mathematical structure of error correction models. Contemporary Mathematics 80, 359386.
Johansen, S. (1991) Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 15511580.
Johansen, S. (1992) A representation of vector autoregressive processes integrated of order 2. Econometric Theory 8, 188202.
Johansen, S. (1996) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press.
Johansen, S. (2009) Representation of cointegrated autoregressive processes with application to fractional processes. Econometric Reviews 28, 121145.
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 10351070.
La Cour, L. (1998) A parametric characterization of integrated vector autoregressive (VAR) processes. Econometric Theory 14, 187199.
Markus, A.S. (2012) Introduction to the Spectral Theory of Polynomial Operator Pencils. American Mathematical Society.
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.
Park, J.-Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated processes: Part I. Econometric Theory 4, 468497.
Park, J.-Y. & Phillips, P.C.B. (1989) Statistical inference in regressions with integrated processes: Part II. Econometric Theory 5, 95131.
Phillips, P.C.B. (1986) Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311340.
Phillips, P.C.B. (1988) Regression theory for near-integrated time series. Econometrica 56, 10211043.
Phillips, P.C.B. (1991) Optimal inference in cointegrated systems. Econometrica 59, 283306.
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 473495.
Phillips, P.C.B. & Hansen, B.E. (1990) Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99125.
Phillips, P.C.B. & Kheifets, I. (2019) On multicointegration. Mimeo, Yale University.
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, 111115.
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 9711001.
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. 2363. Butterworths.
Schumacher, J.M. (1991) System-theoretic trends in econometrics. In Antoulas, A.C. (ed.), Mathematical System Theory: The Influence of R. E. Kalman, 559577. Springer.
Seo, W.-K. (2018) Cointegration and representation of integrated autoregressive processes in function space. ArXiv e-prints 1712.08748v3 [math.FA].
Seo, W.-K. & Beare, B.K. (2019) Cointegrated linear processes in Bayes Hilbert space. Statistics and Probability Letters 147, 9095.
Steinberg, S. (1968) Meromorphic families of compact operators. Archive for Rational Mechanics and Analysis 31, 372379.
Yoo, B.-S. (1987) Co-integrated time series: Structure, forecasting and testing. Doctoral thesis, University of California, San Diego.

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

  • Brendan K. Beare (a1) and Won-Ki Seo (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed