# Large-dimensional Dynamic Factor Models: Estimation of Impulse–Response Functions with I ( 1 ) cointegrated factors

**Matteo Barigozzi, Marco Lippi, Matteo Luciani – 2021**

Abstract

We study a large-dimensional Dynamic Factor Model where: (i) the vector of factors Ft is I(1) and driven by a number of shocks that is smaller than the dimension of Ft ; and, (ii) the idiosyncratic components are either I(1) or I(0). Under (i), the factors Ft are cointegrated and can be modeled as a Vector Error Correction Model (VECM). Under (i) and (ii), we provide consistent estimators, as both the cross-sectional size n and the time dimension T go to infinity, for the factors, the loadings, the shocks, the coefficients of the VECM and therefore the Impulse–Response Functions (IRF) of the observed variables to the shocks. Furthermore, possible deterministic linear trends are fully accounted for, and the case of an unrestricted VAR in the levels Ft , instead of a VECM, is also studied. The finite-sample properties the proposed estimators are explored by means of a MonteCarlo exercise. Finally, we revisit two distinct and widely studied empirical applications. By correctly modeling the long-run dynamics of the factors, our results partly overturn those obtained by recent literature. Specifically, we find that: (i) oil price shocks have just a temporary effect on US real activity; and, (ii) in response to a positive news shock, the economy first experiences a significant boom, and then a milder recession.

Summary

Common factors among economic time series are often non-stationary and cointegrated, suggesting they should be jointly modeled with a VECM.

Important to keep in mind here that what’s new is the not estimator of the factor loadings (eigenvectors of covariance matrix of differed data, which is standard) but the estimator of the factors themselves, in which the loadings are applied to de-trended data rather than to differenced data. This gives you the factors in levels directly and doesn’t require cumulation at the end.

Note that this method can potentially have issues if every series in the dataset has linear trends.