Methodology (stat.ME)

Abstract: In this article, we consider a $n$-dimensional random walk $X_t$, whose error terms are linear processes generated by $n$-dimensional noise vectors, and each component of these noise vectors is independent and may not be identically distributed with uniformly bounded 8th moment and densities. Given $T$ observations such that the dimension $n$ and sample size $T$ going to infinity proportionally, define $\boldsymbol{X}$ and $\hat{\boldsymbol{R}}$ as the data matrix and the sample correlation matrix of $\boldsymbol{X}$ respectively. This article establishes the central limit theorem (CLT) of the first $K$ largest eigenvalues of $n^{-1}\hat{\boldsymbol{R}}$. Subsequently, we propose a new unit root test for the panel high-dimensional nonstationary time series based on the CLT of the largest eigenvalue of $n^{-1}\hat{\boldsymbol{R}}$. A numerical experiment is undertaken to verify the power of our proposed unit root test.

Abstract: Granger causal inference is a contentious but widespread method used in fields ranging from economics to neuroscience. The original definition addresses the notion of causality in time series by establishing functional dependence conditional on a specified model. Adaptation of Granger causality to nonlinear data remains challenging, and many methods apply in-sample tests that do not incorporate out-of-sample predictability leading to concerns of model overfitting. To allow for out-of-sample comparison, we explicitly define a measure of functional connectivity using permutations of the covariate set. Artificial neural networks serve as featurizers of the data to approximate any arbitrary, nonlinear relationship, and under certain conditions on the featurization process and the model residuals, we prove consistent estimation of the variance for each permutation. Performance of the permutation method is compared to penalized objective, naive replacement, and omission techniques via simulation, and we investigate its application to neuronal responses of acoustic stimuli in the auditory cortex of anesthetized rats. We contend that targeted use of the Granger causal framework, when prior knowledge of the causal mechanisms in a dataset are limited, can help to reveal potential predictive relationships between sets of variables that warrant further study.