The generalized dynamic factor model consistency and rates

A factor model generalizing those proposed by Geweke (in: D.J. Aigner and A.S. Goldberger, Latent Variables in Socio-Economic Models, North-Holland, Amsterdam, 1977), Sargent and Sims (New Methods in Business Research, Federal Reserve Bank of Minneapolis, Minneapolis, 1977), Engle and Watson (J. Amer. Statist. Assoc. 76 (1981) 774) and Stock and Watson (J. Business. Econom. Statist. 20 (2002) 147) has been introduced in Forni et al. (Rev. Econ. Statist. 80 (2000) 540), where consistent (as the number n of series and the number T of observations both tend to infinity along appropriate paths (n,T(n))) estimation methods for the common component are proposed. Rates of convergence associated with these methods are obtained here as functions of the paths (n,T(n)) along which n and T go to infinity. These results show that, under suitable assumptions, consistency requires T(n) to be at least of the same order as n, whereas an optimal rate of n is reached for T(n) of the order of n2. If convergence to the space of common components is considered, consistency holds irrespective of the path (T(n) thus can be arbitrarily slow); the optimal rate is still n, but only requires T(n) to be of the order of n.