Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors

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 yt. It is easily seen that yt 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 yt 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 yt 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.