Determining the number of factors after stationary univariate transformations

A very common practice when extracting factors from non-stationary multivariate time series is to differentiate each variable in the system. As a consequence, the ratio between variances and the dynamic dependence of the common and idiosyncratic differentiated components may change with respect to the original components. In this paper, we analyze the effects of these changes on the finite sample properties of several procedures to determine the number of factors. In particular, we consider the information criteria of Bai and Ng (Econometrica 70(1):191–221, 2002), the edge distribution of Onatski (Rev Econ Stat 92(4):1004–1016, 2010) and the ratios of eigenvalues proposed by Ahn and Horenstein (Econometrica 81(3):1203–1227, 2013). The performance of these procedures when implemented to differentiated variables depends on both the ratios between variances and dependencies of the differentiated factor and idiosyncratic noises. Furthermore, we also analyze the role of the number of factors in the original non-stationary system as well as of its temporal and cross-sectional dimensions. Finally, we implement the different procedures to determine the number of common factors in a system of inflation rates in 15 euro area countries.