The statistical properties of the innovations in multivariate ARCH processes in high dimensions

The long memory linear ARCH process is extended to a multivariate universe, where the natural cross-product structure of the covariance is generalized by adding two bi-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 measures of quality assessing the independence and unit magnitude of the residual distributions are proposed. The salient statistical 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 to produce uncorrelated residuals, but the mean residual magnitudes are much larger than one. The large magnitudes of the residuals are due to the exponential decay of the covariance eigenvalues, corresponding to directions with very small fluctuations in the historical sample. Because the postulated properties of the innovations cannot be obtained regardless of the parameter values, subsequent inferences reach a fundamental limitation in a large multivariate universe.