Spurious relationships in high-dimensional systems with strong or mild persistence

This paper is concerned with the interactions of persistence and dimensionality in the context of the eigenvalue estimation problem of large covariance matrices arising in cointegration and principal component analysis. Following a review of the early and more recent developments in this area, we investigate the behaviour of these eigenvalues in a vector autoregression setting that blends pure unit root, local to unit root and mildly integrated components. Our results highlight the seriousness of spurious relationships that may arise in such big data environments even when the degree of persistence of the variables involved is mild and affects only a small proportion of a large data matrix, with important implications for forecasts based on principal component regressions and related methods. We argue that, prior to principal component analysis, first-differencing may be suitable even in stationary or nearly stationary environments.