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.