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In this paper, we analyze and compare the finite sample properties of alternative factor extraction procedures in the context of non-stationary Dynamic Factor Models (DFMs). On top of considering procedures already available in the literature, we extend the hybrid method based on the combination of principal components and Kalman filter and smoothing algorithms to non-stationary models. We show that if the idiosyncratic noises are stationary, procedures based on extracting the factors using the non-stationary original series work better than those based on differenced variables. We apply the methodology to the analysis of cross-border risk sharing by fitting nonstationary DFM to aggregate Gross Domestic Product and consumption of a set of 21 industrialized countries from the Organization for Economic Cooperation and Development (OECD). The goal is to check if international risk sharing is a short-or long-run issue. Keywords Consumption smoothing • Non-stationary Dynamic Factor Models • Kalman filter • Principal components • Risk sharing Financial support from the Spanish Government Projects ECO2015-70331-C2-1-R and ECO2015-70331-C2-2-R (MINECO/FEDER) is gratefully acknowledged. This paper was started while Pilar Poncela was still at Universidad Autónoma de Madrid. We are very grateful for the detailed comments of an anonymous referee which have been very useful to improve the presentation of this paper. The views expressed in this paper are those of the authors and should not be attributed neither to the European Commission nor to INEGI.

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) 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.

In this paper we propose to use the common trends of the Mexican economy in order to predict economic activity one and two steps ahead. We exploit the cointegration properties of the macroeconomic time series, such that, when the series are I(1) and cointegrated, there is a factor representation, where the common factors are the common trends of the macroeconomic variables. Thus, we estimate a large non-stationary dynamic factor model using principal components (PC) as suggested by Bai (J Econom 122(1): 2004), where the estimated common factors are used in a factor-augmented vector autoregressive model to forecast the Global Index of Economic Activity. Additionally, we estimate the common trends through partial least squares. The results indicate that the common trends are useful to predict Mexican economic activity, and reduce the forecast error with respect to benchmark models, mainly when estimated using PC.

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