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A standard result in linear algebra is that a solution to a system of linear equations of type \(Ax=b\), where A is \(m\times n\), b is \(m \times 1\), exists if and only if the rank of extended matrix [A|b] coincides with rank(A). When a square matrix A is non-singular, this condition holds for all vectors b, however this condition will fail when rank(A) is not full if vector b is linearly independent from columns of A.
Although it is possible that \(f^1_y\) is zero due to a specific steady state calibration.
References
Hespeler, F., & Sorge, M. (2018). Solving rational expectations models with informational subperiods: a comment. Discussion paper.
Kormilitsina, A. (2013). Solving rational expectations models with informational subperiods: A perturbation approach. Computational Economics, 41(4), 525–555.
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Kormilitsina, A. A Reply to Reaction on Kormilitsina (2013): “Solving Rational Expectations Models with Informational Subperiods: A Perturbation Approach”. Comput Econ 53, 1655–1656 (2019). https://doi.org/10.1007/s10614-018-9830-9
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DOI: https://doi.org/10.1007/s10614-018-9830-9