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Inference in Heavy-Tailed Nonstationary Multivariate Time Series

Barigozzi, Matteo; Cavaliere, Giuseppe; Trapani, Lorenzo


Matteo Barigozzi

Giuseppe Cavaliere

Lorenzo Trapani


We study inference on the common stochastic trends in a nonstationary, N-variate time series yt, in the possible presence of heavy tails. We propose a novel methodology which does not require any knowledge or estimation of the tail index, or even knowledge as to whether certain moments (such as the variance) exist or not, and develop an estimator of the number of stochastic trends m based on the eigenvalues of the sample second moment matrix of yt. We study the rates of such eigenvalues, showing that the first m ones diverge, as the sample size T passes to infinity, at a rate faster by (Formula presented.) than the remaining N–m ones, irrespective of the tail index. We thus exploit this eigen-gap by constructing, for each eigenvalue, a test statistic which diverges to positive infinity or drifts to zero according to whether the relevant eigenvalue belongs to the set of the first m eigenvalues or not. We then construct a randomized statistic based on this, using it as part of a sequential testing procedure, ensuring consistency of the resulting estimator of m. We also discuss an estimator of the common trends based on principal components and show that, up to a an invertible linear transformation, such estimator is consistent in the sense that the estimation error is of smaller order than the trend itself. Importantly, we present the case in which we relax the standard assumption of iid innovations, by allowing for heterogeneity of a very general form in the scale of the innovations. Finally, we develop an extension to the large dimensional case. A Monte Carlo study shows that the proposed estimator for m performs particularly well, even in samples of small size. We complete the article by presenting two illustrative applications covering commodity prices and interest rates data. Supplementary materials for this article are available online.


Barigozzi, M., Cavaliere, G., & Trapani, L. (2022). Inference in Heavy-Tailed Nonstationary Multivariate Time Series. Journal of the American Statistical Association,

Journal Article Type Article
Acceptance Date Sep 21, 2022
Online Publication Date Nov 4, 2022
Publication Date Nov 4, 2022
Deposit Date Sep 23, 2022
Publicly Available Date Nov 5, 2023
Journal Journal of the American Statistical Association
Print ISSN 0162-1459
Electronic ISSN 1537-274X
Publisher Taylor and Francis
Peer Reviewed Peer Reviewed
Keywords Statistics, Probability and Uncertainty; Statistics and Probability
Public URL
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