Skip to main content
    • Aa
    • Aa


  • James Davidson (a1) and Nigar Hashimzade (a2)

This paper considers the asymptotic distribution of the sample covariance of a nonstationary fractionally integrated process with the stationary increments of another such process—possibly itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analyzed in a previous paper (Davidson and Hashimzade, 2008), and the construction derived from moving average representations in the time domain. Depending on the values of the long memory parameters and choice of normalization, the limiting integral is shown to be expressible as the sum of a constant and two Itô-type integrals with respect to distinct Brownian motions. In certain cases the latter terms are of small order relative to the former. The mean is shown to match that of the harmonic representation, where the latter is defined, and satisfies the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulas are valid for the full range of the long memory parameters and that they extend to non-Gaussian processes.

Corresponding author
*Address correspondence to James Davidson, Department of Economics, University of Exeter, Exeter EX4 4PU, United Kingdom; e-mail:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

J Davidson . (1994) Stochastic Limit Theory. Oxford University Press.

J Davidson . (2006) Alternative bootstrap procedures for testing cointegration in fractionally integrated processes. Journal of Econometrics 133, 741777.

J. Davidson & R.M. de Jong (2000) The functional central limit theorem and convergence to stochastic integrals II: Fractionally integrated processes. Econometric Theory 16, 643666.

J. Davidson & N. Hashimzade (2008) Alternative frequency and time domain versions of fractional Brownian motion. Econometric Theory 24, 256293.

R.M. De Jong & J. Davidson (2000) The functional central limit theorem and convergence to stochastic integrals I: the weakly dependent case. Econometric Theory 16, 621642.

S Johansen . (1991) Estimation and hypothesis testing of cointegration in Gaussian vector autoregressive models. Econometrica 59, 15511580.

T.G. Kurtz & P.E. Protter (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 10351070.

T.G. Kurtz & P.E. Protter (1996) Weak convergence of stochastic integrals and differential equations. In K. Graham , T.G. Kurtz , S. Meleard , P.E. Protter , M. Pulverenti , & D. Talay (eds.), Probabilistic Models for Nonlinear Partial Differential Equations: Lectures Given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Lecture Notes in Mathematics 1627, pp. 141. Springer Verlag.

M. Lettau & S. Ludvigson (2001) Consumption, aggregate wealth and expected stock returns. Journal of Finance 56, 815849.

B.B. Mandelbrot & J.W. van Ness (1968) Fractional Brownian motions, fractional noises and applications. SIAM Review 10, 422437.

P.C.B Phillips . (1987) Time series regression with a unit root. Econometrica 55, 277301.

P.C.B. Phillips & B.E. Hansen (1990) Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99125.

F Sowell . (1990) The fractional unit root distribution. Econometrica 58, 495505.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 12 *
Loading metrics...

Abstract views

Total abstract views: 88 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd May 2017. This data will be updated every 24 hours.