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A high-dimensional $r$-factor model for an $n$-dimensional vector time seriesis characterised by the presence of a large eigengap (increasing with $n$)between the $r$-th and the $(r+1)$-th largest eigenvalues of the covariancematrix. Consequently, Principal Component (PC) analysis is the most popularestimation method for factor models and its consistency, when $r$ is correctlyestimated, is well-established in the literature. However, popular factornumber estimators often suffer from the lack of an obvious eigengap inempirical eigenvalues and tend to over-estimate $r$ due, for example, to theexistence of non-pervasive factors affecting only a subset of the series. Weshow that the errors in the PC estimators resulting from the over-estimation of$r$ are non-negligible, which in turn lead to the violation of the conditionsrequired for factor-based large covariance estimation. To remedy this, wepropose new estimators of the factor model based on scaling the entries of thesample eigenvectors. We show both theoretically and numerically that theproposed estimators successfully control for the over-estimation error, andinvestigate their performance when applied to risk minimisation of a portfolioof financial time series.

Consistent estimation of high-dimensional factor models when the factor number is over-estimated