Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors

Matteo Barigozzi,Marco Lippi,Matteo Luciani

doi:10.3390/econometrics8010003

Abstract

Large-dimensional dynamic factor models and dynamic stochastic general equilibrium models, both widely used in empirical macroeconomics, deal with singular stochastic vectors, i.e., vectors of dimension r which are driven by a q-dimensional white noise, with q < r . The present paper studies cointegration and error correction representations for an I ( 1 ) singular stochastic vector y t . It is easily seen that y t is necessarily cointegrated with cointegrating rank c ≥ r − q . Our contributions are: (i) we generalize Johansen’s proof of the Granger representation theorem to I ( 1 ) singular vectors under the assumption that y t has rational spectral density; (ii) using recent results on singular vectors by Anderson and Deistler, we prove that for generic values of the parameters the autoregressive representation of y t has a finite-degree polynomial. The relationship between the cointegration of the factors and the cointegration of the observable variables in a large-dimensional factor model is also discussed.

Highlights

An r-dimensional stochastic vector yt such that yt = A0 ut + A1 ut−1 + · · ·, where the matrices A j are r × q and ut is a q-dimensional white noise, with q < r, is said to be singular
Several papers have addressed the issue of whether and when an error correction model or an unrestricted vector autoregressive representation (VAR) in the levels should be used for estimation in the case of nonsingular cointegrated vectors: Sims et al (1990) have shown that the parameters of a cointegrated VAR are consistently estimated using an unrestricted VAR in the levels; on the other hand, Phillips (1998) shows that if the variables are cointegrated, the long-run features of the impulse-response functions are consistently estimated only if the unit roots are explicitly taken into account, that is within a VECM specification
If (1 − L)yt has rational spectral density, under assumptions that generalize to the singular case those in Johansen (1995), we show that yt has an error correction representation with c error terms, generalizing the Granger representation theorem to the singular case

Summary

Introduction

(II) Assuming that the parameters of S( L) and B( L) may vary in an open subset of Rλ , see Section 3.2 for the definition of λ, in Proposition 3 we show that all the assumptions used to obtain (4), and the assumption that unity is the only possible zero of B( L), hold for generic values of Usually orthonormality is assumed This is convenient but not necessary in the present paper. 3-dimensional subvectors in the dataset are cointegrated, a kind of regularity that we do not observe in actual large macroeconomic datasets This suggests that an estimation strategy robust to the assumption that the idiosyncratic components can be I (1) has to be preferred (for this aspect we refer to Barigozzi et al 2019). A discussion of some non-uniqueness problems arising with singularity and details on the simulations are collected in the Appendix

Stationary Singular Vectors

Fundamentalness

Permanent and Transitory Shocks

VECMs and Unrestricted VARs in The Levels

Cointegration of the Observable Variables in a DFM

Summary and Conclusions