Inference on multivariate ARCH processes with large sizes

TL;DR

This paper develops a multivariate ARCH model with long memory to analyze large covariance matrices, proposing new residual quality measures and revealing limitations in inference accuracy for high-dimensional data.

Contribution

It introduces a generalized covariance structure with additional linear terms in multivariate ARCH processes and assesses residual properties for large datasets.

Findings

Adding new linear terms improves residual independence

Residual magnitudes deviate significantly from unity

Inference limitations increase with data dimensionality

Abstract

The covariance matrix is formulated in the framework of a linear multivariate ARCH process with long memory, where the natural cross product structure of the covariance is generalized by adding two linear terms with their respective parameter. The residuals of the linear ARCH process are computed using historical data and the (inverse square root of the) covariance matrix. Simple measure of qualities assessing the independence and unit magnitude of the residual distributions are proposed. The salient properties of the computed residuals are studied for three data sets of size 54, 55 and 330. Both new terms introduced in the covariance help in producing uncorrelated residuals, but the residual magnitudes are very different from unity. The large sizes of the inferred residuals are due to the limited information that can be extracted from the empirical data when the number of time series is large, and denotes a fundamental limitation to the inference that can be achieved.