  Despite the popularity of sparse factor models, little attention has been
given to formally address identifiability of these models beyond standard
rotation-based identification such as the positive lower triangular constraint.
To fill this gap, we provide a counting rule on the number of nonzero factor
loadings that is sufficient for achieving uniqueness of the variance
decomposition in the factor representation. Furthermore, we introduce the
generalised lower triangular representation to resolve rotational invariance
and show that within this model class the unknown number of common factors can
be recovered in an overfitting sparse factor model. By combining point-mass
mixture priors with a highly efficient and customised MCMC scheme, we obtain
posterior summaries regarding the number of common factors as well as the
factor loadings via postprocessing. Our methodology is illustrated for monthly
exchange rates of 22 currencies with respect to the euro over a period of eight
years and for monthly log returns of 73 firms from the NYSE100 over a period of
20 years.
