# Inference in Heavy-Tailed Nonstationary Multivariate Time Series

**Matteo Barigozzi, Giuseppe Cavaliere, Lorenzo Trapani – 2022**

Abstract

We study inference on the common stochastic trends in a non-stationary, 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 O (T ) 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 randomised 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 i.i.d. 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 paper by presenting two illustrative applications covering commodity prices and interest rates data.

Summary

Method for estimating the number of common stochastic trends in nonstationary, fat-tailed data that doesn’t require knowing the tail index.

https://www.youtube.com/watch?v=ooTqSSs56j0

The key idea of this paper analytically is to scale the covariance matrix of the data in levels by the covariance matrix of the data in differences. Doing so ensures that eigen-analysis is scale-free and independent of the fat-tailedness of the data. The scaling is important, otherwise you’re simply in the same realm analyzed by @onatskiSpuriousFactorAnalysis2021 where the first few eigenvalues are destined to be large even if there are no true factors.

PCA is always superconsistent in the presence of integrated processes, regardless of how many individual time series you have.

Interesting implicit test of stationarity – if the test cannot find at least one common trend, the data is by definition stationary.

The one uncertainty I have about this paper is whether or not you need to standardize the variables before taking the eigenvalues. The paper doesn’t say that you have to, perhaps because multiplying by the inverse of the covariance of the differences takes care of this? Also have some questions about detrending the variables.

## References

@penaNonstationaryDynamicFactor2006 @zhangIdentifyingCointegrationEigenanalysis2019 @onatskiSpuriousFactorAnalysis2021