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ASYMPTOTIC THEORY FOR SPECTRAL DENSITY ESTIMATES OF GENERAL MULTIVARIATE TIME SERIES

  • Wei Biao Wu (a1) and Paolo Zaffaroni (a2)
Abstract

We derive uniform convergence results of lag-window spectral density estimates for a general class of multivariate stationary processes represented by an arbitrary measurable function of iid innovations. Optimal rates of convergence, that hold as both the time series and the cross section dimensions diverge, are obtained under mild and easily verifiable conditions. Our theory complements earlier results, most of which are univariate, which primarily concern in-probability, weak or distributional convergence, yet under a much stronger set of regularity conditions, such as linearity in iid innovations. Based on cross spectral density functions, we then propose a new test for independence between two stationary time series. We also explain the extent to which our results provide the foundation to derive the double asymptotic results for estimation of generalized dynamic factor models.

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Copyright
Corresponding author
*Address correspondence to Paolo Zaffaroni Imperial College Business School, Exhibition Road, SW7 2AZ London, UK; e-mail: p.zaffaroni@imperial.ac.uk.
Footnotes
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We thank the Co-Editor (Benedikt Potscher) and two anonymous referees for their time and their comments that greatly improved the paper. We also thank Marco Lippi for encouraging us to write this paper and his suggestions on how to improve part of the proofs.

Footnotes
References
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Econometric Theory
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