Working Papers, Department of Economics, Rutgers, The State University of New Jersey 2013-14
Mild factor loading instability, particularly if sufficiently independent across the different constituent variables, does not affect the estimation of the number of factors, nor subsequent estimation of the factors themselves (see e.g. Stock and Watson (2009)). This result does not hold in the presence of large common breaks in the factor loadings, however. In this case, information criteria overestimate the number of breaks. Additionally, estimated factors are no longer consistent estimators of true factors. Hence, various recent research papers in the diffusion index literature focus on testing the constancy of factor loadings. One reason why this is a positive development is that in applied work, factor augmented forecasting models are used widely for prediction, and it is important to understand when such models are stable. Now, forecast failure of factor augmented models can be due to either factor loading instability, regression coefficient instability, or both. To address this issue, we develop a test for the joint hypothesis of structural stability of both factor loadings and factor augmented forecasting model regression coefficients. The proposed statistic is based on the difference between full sample and rolling sample estimators of the sample covariance of the factors and the variable to be forecasted. Failure to reject the null ensures the structural stability of the factor augmented forecasting model. If the null is instead rejected, one can proceed to disentangle the cause of the rejection as being due to either (or both) of the afore mentioned varieties of instability. Standard inference can be carried out, as the suggested statistic has a chi-squared limiting distribution. We also establish the first order validity of (block) bootstrap critical values. Finally, we provide an empirical illustration by testing for the structural stability of factor augmented forecasting models for 11 U.S. macroeconomic indicators.