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 <i>r</i> which are driven by a <i>q</i>-dimensional white noise, with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>&lt;</mo> <mi>r</mi> </mrow> </semantics> </math> </inline-formula>. The present paper studies cointegration and error correction representations for an <inline-formula> <math display="inline"> <semantics> <mrow> <mi>I</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> singular stochastic vector <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="bold">y</mi> <mi>t</mi> </msub> </semantics> </math> </inline-formula>. It is easily seen that <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="bold">y</mi> <mi>t</mi> </msub> </semantics> </math> </inline-formula> is necessarily cointegrated with cointegrating rank <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mo>&#8805;</mo> <mi>r</mi> <mo>&#8722;</mo> <mi>q</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> <mi>I</mi> <mo>(</mo> <mn>1</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> <mi>t</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> <mi>t</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.

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