Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-27T02:08:49.285Z Has data issue: false hasContentIssue false
  • English
  • Français

LONG-RUN COVARIANCE MATRICES FOR FRACTIONALLY INTEGRATED PROCESSES

Published online by Cambridge University Press:  06 September 2007

Peter C.B. Phillips  and
Chang Sik Kim
Peter C.B. Phillips
Affiliation:
Cowles Foundation for Research in Economics, Yale University, University of York and University of Auckland
Chang Sik Kim
Affiliation:
Sungkyunkwan University
Get access Rights & Permissions [Opens in a new window]

Abstract

An asymptotic expansion is given for the autocovariance matrix of a vector of stationary long-memory processes with memory parameters d ∈ [0,½). The theory is then applied to deliver formulas for the long-run covariance matrices of multivariate time series with long memory.Phillips acknowledges partial support from a Kelly Fellowship and from the NSF under grant SES 04-142254.This may be proved directly using a Fourier integral asymptotic expansion when the spectrum of the short-memory component is analytic.

Type
NOTES AND PROBLEMS
Information
Econometric Theory , Volume 23 , Issue 6 , December 2007 , pp. 1233 - 1247
Copyright
© 2007 Cambridge University Press

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.)

References

REFERENCES

Bleistein, N. & R.A. Handelsman (1986) Asymptotic Expansions of Integrals. Dover.
Chan, N.H. & N. Terrin (1995) Inference for long-memory processes with application to fractional unit root autoregression. Annals of Statistics 23, 1662–1683.Google Scholar
Davidson, J. (2004) Convergence to Stochastic Integrals with Fractionally Integrated Integrator Processes: Theory and Applications to Cointegrating Regression. Mimeo, Cardiff University.
Davidson, J. & N. Hashimzade (2008) Alternative frequency and time domain versions of fractional Brownian motion. Econometric Theory (forthcoming).Google Scholar
Erdélyi, A. (1956) Asymptotic Expansions. Dover.
Gradshteyn, I.S. & I.M. Ryzhik (2000) Tables of Integrals, Series and Products, 6th ed., A. Jeffrey & D. Zwillinger (eds.). Academic Press.
Hannan, E.J. (1970) Multiple Time Series. Wiley.
Henry, M. & P. Zaffaroni (2003) The Long Range Dependence Paradigm for Macroeconomics and Finance. Mimeo, Columbia University.
Kim, C. & P.C.B. Phillips (1999) Fully-Modified Estimation of Fractional Cointegration Models. Mimeo, Yale University.
Lieberman, O. & P.C.B. Phillips (2006) A Complete Asymptotic Series for the Autocovariance Function of a Long Memory Process. Mimeo, Yale University.
Marinucci, D. & P.M. Robinson (2000) Weak convergence of multivariate fractional processes. Stochastic Processes and Their Applications 86, 103–120.Google Scholar
Phillips, P.C.B. (1995) Fully modified least squares and vector autoregression. Econometrica 63, 1023–1078.Google Scholar
Phillips, P.C.B. & K. Shimotsu (2004) Local Whittle estimation in nonstationary and unit root cases. Annals of Statistics 32, 656–692.Google Scholar
Robinson, P.M. (1995) Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630–1661.Google Scholar
Robinson, P.M. & J. Hualde (2003) Cointegration in fractional systems with unknown integration orders. Econometrica 71, 1727–1766.Google Scholar
Samorodnitsky, G. & M.S. Taqqu (1994) Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall.
Shimotsu, K. (2006) Gaussian Semiparametric Estimation of Multivariate Fractionally Integrated Processes. Mimeo, Queens University.
Shimotsu, K. & P.C.B. Phillips (2005) Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890–1933.Google Scholar
Sowell, F.B. (1990) The fractional unit root distribution. Econometrica 58, 495–505.Google Scholar
Velasco, C. (2003) Gaussian semiparametric estimation of fractional cointegration. Journal of Time Series Analysis 24, 345–378.Google Scholar