Factor Models with Real Data: A Robust Estimation of the Number of Factors

Factor models are a very efficient way to describe high-dimensional vectors of data in terms of a small number of common relevant factors. This problem, which is of fundamental importance in many disciplines, is usually reformulated in mathematical terms as follows. We are given the covariance matrix $Sigma$ of the available data. $Sigma$ must be additively decomposed as the sum of two positive semidefinite matrices $D$ and $L$: $D$—that accounts for the idiosyncratic noise affecting the knowledge of each component of the available vector of data—must be diagonal and $L$ must have the smallest possible rank in order to describe the available data in terms of the smallest possible number of independent factors. In practice, however, the matrix $Sigma$ is typically unknown and therefore it must be estimated from the data so that only an approximation of $Sigma$ is actually available. This paper discusses the issues that arise from this uncertainty and provides a strategy to deal with the problem of robustly estimating the number of factors.