  In the paper, we suggest three tests on the validity of a factor model which
can be applied for both small dimensional and large dimensional data. Both the
exact and asymptotic distributions of the resulting test statistics are derived
under classical and high-dimensional asymptotic regimes. It is shown that the
critical values of the proposed tests can be calibrated empirically by
generating a sample from the inverse Wishart distribution with identity
parameter matrix. The powers of the suggested tests are investigated by means
of simulations. The results of the simulation study are consistent with the
theoretical findings and provide general recommendations about the application
of each of the three tests. Finally, the theoretical results are applied to two
real data sets, which consist of returns on stocks from the DAX index and on
stocks from the S&P 500 index. Our empirical results do not support the
hypothesis that all linear dependencies between the returns can be entirely
captured by the factors considered in the paper.
