Tests for Overidentifying Restrictions in Factor-Augmented VAR Models

This paper develops tests for overidentifying restrictions in Factor-Augmented Vector Autoregressive (FAVAR) models. The FAVAR combines a high-dimensional factor model and a conventional VAR for the latent factors. The identification of structural shocks in FAVAR can lead to restrictions on the factor loadings of many variables, so it can involve infinitely many identifying restrictions as the number of cross sections goes to infinity. Our focus is to test the joint null hypothesis that all the restrictions are satisfied. Conventional tests cannot be used due to the large dimension. We transform the infinite-dimensional problem into a finite-dimensional one by combining the individual statistics across the cross section dimension. We find the limit distribution of our joint test statistic under the null hypothesis and prove that it is consistent against the alternative that a fraction of or all identifying restrictions are violated. The Monte Carlo results show that the joint test statistic has good finite-sample size and power. We implement our tests to an updated version of Stock and Watson’s (2005) data set. The proposed test rejects the null hypotheses that the number of fast shocks is two or more, but does not reject the null that there is only one fast shock, which is the monetary policy shock. This result is further confirmed by the impulse responses of major macroeconomic variables to the monetary policy shock: the impulse responses based on one fast shock are generally more economically plausible than those based on two or more fast shocks; and the price puzzle is either considerably reduced or entirely solved for all price indexes when only one fast shock is used.