IDEAS home Printed from
   My bibliography  Save this article

The statistical properties of the innovations in multivariate ARCH processes in high dimensions


  • Gilles Zumbach


The long memory linear ARCH process is extended to a multivariate universe, where the natural cross-product structure of the covariance is generalized by adding two bi-linear terms with their respective parameter. The residuals of the linear ARCH process are computed using historical data and the (inverse square root of the) covariance matrix. Simple measures of quality assessing the independence and unit magnitude of the residual distributions are proposed. The salient statistical properties of the computed residuals are studied for three data sets of size 54, 55 and 330. Both new terms introduced in the covariance help to produce uncorrelated residuals, but the mean residual magnitudes are much larger than one. The large magnitudes of the residuals are due to the exponential decay of the covariance eigenvalues, corresponding to directions with very small fluctuations in the historical sample. Because the postulated properties of the innovations cannot be obtained regardless of the parameter values, subsequent inferences reach a fundamental limitation in a large multivariate universe.

Suggested Citation

  • Gilles Zumbach, 2013. "The statistical properties of the innovations in multivariate ARCH processes in high dimensions," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 29-44, January.
  • Handle: RePEc:taf:quantf:v:13:y:2013:i:1:p:29-44
    DOI: 10.1080/14697688.2011.589399

    Download full text from publisher

    File URL:
    Download Restriction: Access to full text is restricted to subscribers.

    As the access to this document is restricted, you may want to search for a different version of it.

    More about this item


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:quantf:v:13:y:2013:i:1:p:29-44. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Chris Longhurst). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.