摘要：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 ir/i which are driven by a iq/i-dimensional white noise, with inline-formula math display="inline" semantics mrow miq/mi molt;/mo mir/mi /mrow /semantics /math /inline-formula. The present paper studies cointegration and error correction representations for an inline-formula math display="inline" semantics mrow miI/mi mo(/mo mn1/mn mo)/mo /mrow /semantics /math /inline-formula singular stochastic vector inline-formula math display="inline" semantics msub mi mathvariant="bold"y/mi mit/mi /msub /semantics /math /inline-formula. It is easily seen that inline-formula math display="inline" semantics msub mi mathvariant="bold"y/mi mit/mi /msub /semantics /math /inline-formula is necessarily cointegrated with cointegrating rank inline-formula math display="inline" semantics mrow mic/mi mo#8805;/mo mir/mi mo#8722;/mo miq/mi /mrow /semantics /math /inline-formula. Our contributions are: (i) we generalize Johansen#8217;s proof of the Granger representation theorem to inline-formula math display="inline" semantics mrow miI/mi mo(/mo mn1/mn mo)/mo /mrow /semantics /math /inline-formula singular vectors under the assumption that inline-formula math display="inline" semantics msub mi mathvariant="bold"y/mi mit/mi /msub /semantics /math /inline-formula 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 inline-formula math display="inline" semantics msub mi mathvariant="bold"y/mi mit/mi /msub /semantics /math /inline-formula 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.
