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Arbitrary initial conditions and the dimension of indeterminacy in linear rational expectations models

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Abstract

Indeterminate equilibrium rational expectations (RE) models are ubiquitous in both theoretical and applied work in dynamic macroeconomics. The issue of characterizing the exact dimension of indeterminacy—i.e. of deriving the full set of causal and stable solutions to linear RE models—has only recently been addressed in the context of general and multivariate settings. This paper complements existing results by identifying bounds on the observable dimension of indeterminacy of linear RE models in the presence of arbitrary initial conditions. Implications for the estimation of indeterminate equilibrium RE models are discussed.

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Notes

  1. Thus, \(QQ' = I_n = ZZ'\), where the \('\) symbol denotes transposition. The QZ decomposition always exists and is unique up to the ordering of the generalized eigenvalues. The matrices Q, Z, \(\Lambda \) and \(\Omega \) can always be chosen so that the absolute values of the generalized eigenvalues are displayed in descending order. If \(\lambda _{ii}=0\), then the corresponding generalized eigenvalue is infinity.

  2. It is assumed that \(y_0\) is bounded, i.e. they satisfy \(\sup |y_0|<\infty \).

  3. That is, for any structural shocks \(\epsilon _t\), there always exist forecast errors \(\eta _t\) able to keep the system on its stable saddle path, i.e. such that \(Q_{U\bullet } \left( \Psi \epsilon _t + \Pi \eta _t \right) =0\) holds for any time \(t>1\).

  4. E.g. Farmer et al. (2015) and Bianchi and Nicolò (2017) exploit this notion of indeterminacy following Lubik and Schorfheide (2003). See Sorge (2012) for the case of parametric indeterminacy in sunspot-free environments.

  5. See Definition 1 (Regularity) in Farmer et al. (2015, p. 21).

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  1. University of Salerno, University of Göttingen and CSEF, Via Giovanni Paolo II 132, 84084, Fisciano, SA, Italy

    Marco M. Sorge

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Correspondence to Marco M. Sorge.

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I wish to thank Bernd Funovits and two anonymous reviewers for helpful comments and suggestions.

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Sorge, M.M. Arbitrary initial conditions and the dimension of indeterminacy in linear rational expectations models. Decisions Econ Finan 43, 363–372 (2020). https://doi.org/10.1007/s10203-019-00269-4

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