Abstract
Gini covariance plays a vital role in analyzing the relationship between random variables with heavy-tailed distributions. In this papaer, with the existence of a finite second moment, we establish the Gini–Yule–Walker equation to estimate the transition matrix of high-dimensional periodic vector autoregressive (PVAR) processes, the asymptotic results of estimators have been established. We apply this method to study the Granger causality of the heavy-tailed PVAR process, and the results show that the robust transfer matrix estimation induces sign consistency in the value of Granger causality. Effectiveness of the proposed method is verified by both synthetic and real data.
Highlights
The heavy-tailed distribution is graphically thicker in the tails and sharper in the peaks than the normal distribution
Periodic vector autoregressive (PVAR) models extend the classical vector autoregressive (VAR) models by allowing the parameters to vary with the cyclicality
We developed a Gini–Yule–Walker equation for modeling and estimating the heavy-tailedness data and the possible presence of outliers in high dimensions
Summary
The heavy-tailed distribution is graphically thicker in the tails and sharper in the peaks than the normal distribution. PVAR models have been extensively studied under the Gaussian assumption, the Gaussian PVAR models assume that the latent innovations are independent identity distribution Gaussian random vectors Under this model, there are two kinds of methods to estimate the transition matrix under high dimensional setting, one is the Lassobased estimation procedures, see [5,6,7,8], and the other is Dantzig-selector-type estimators, see [9,10,11,12]. In this paper, relaxing the strong assumption of the existence of higher order moments of the regressors, we use a non-parametric method to estimate the Gini covariance matrix, establish the Gini–Yule– Walker equation to estimate the sparse transition matrix of stationary PVAR processes. We establish the Gini–Yule–Walker equation, obtain simple non-parametric estimators for Gini covariance matrix and investigate the convergence rate of the sample Gini covariance matrix
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