  We propose a novel estimation approach for the covariance matrix based on the
$l_1$-regularized approximate factor model. Our sparse approximate factor (SAF)
covariance estimator allows for the existence of weak factors and hence relaxes
the pervasiveness assumption generally adopted for the standard approximate
factor model. We prove consistency of the covariance matrix estimator under the
Frobenius norm as well as the consistency of the factor loadings and the
factors.
  Our Monte Carlo simulations reveal that the SAF covariance estimator has
superior properties in finite samples for low and high dimensions and different
designs of the covariance matrix. Moreover, in an out-of-sample portfolio
forecasting application the estimator uniformly outperforms alternative
portfolio strategies based on alternative covariance estimation approaches and
modeling strategies including the $1/N$-strategy.
