Testing a Large Number of Hypotheses in Approximate Factor Models

We propose a method to test hypotheses in approximate factor models when the number of restrictions under the null hypothesis grows with the sample size. We use a simple test statistic, based on the sums of squared residuals of the restricted and the unrestricted versions of the model, and derive its asymptotic distribution under different assumptions on the covariance structure of the error term. We show how to standardize the test statistic in the presence of both serial and cross-section correlation to obtain a standard normal limiting distribution. We provide estimators for those quantities that are easy to implement. Finally, we illustrate the small sample performance of these testing procedures through Monte Carlo simulations and apply them to reconsider Reis and Watson (2010)'s hypothesis of existence of a pure inflation factor in the US economy.