Multivariate Linear Rational Expectations Models: Characterization of the Nature of the Solutions and Their Fully Recursive Computation

This 1997 Econometric Theory paper by Michael Binder and M. Hashem Pesaran is a purely theoretical and computational contribution to solving multivariate linear rational expectations (RE) models – it contains no data and no empirical estimation. The paper studies the general Broze-Gourieroux-Szafarz (1995) formulation of a multivariate linear RE model, in which a G-dimensional vector y_t depends on its own lags, on expectations of its own future values formed at various past dates, and on a shock, and which can be stacked into a canonical companion-form system x_t = A x_{t-1} + B E(x_{t+1}|I_t) + w_t. Extending their own earlier quadratic-determinantal-equation (QDE) method (Binder and Pesaran 1995), the paper characterizes, for this general system, all three possible outcomes for the solution – a unique stable solution, multiple stable solutions, or no stable solution at all – by relating them to the eigenvalue structure of the matrix C solving the quadratic matrix equation BC^2 - C + A = 0. It gives a formal existence condition for a real-valued C (Proposition 1) and a general closed-form solution formula (their equation 14) that nests all three cases according to how many eigenvalues of the associated matrix F lie inside, outside, or on the unit circle, plus a narrower set of sufficient conditions (Proposition 2) under which the stable solution is unique when the coefficient matrices A and B commute. The paper’s central practical contribution is a new “fully recursive” solution method (Proposition 3, Section 4) that computes the solution by backward recursion on matrix inversions alone, without any eigenvalue-eigenvector decomposition; the authors report that in extensive computations across a wide variety of RE models it was typically even faster, often substantially so, than the QDE method, and that it is applicable to high-dimensional systems with coefficient matrices that are highly singular – cases in which eigen-decomposition-based methods such as Blanchard and Kahn (1980) or Uhlig (1997) are less general or inapplicable. The method’s validity is conditional on the recursively defined matrices Q_{N-j} remaining nonsingular over the full backward iteration and on choosing the terminal horizon N large enough that the solution is insensitive to the terminal expectation term, a condition the authors verify case by case rather than prove in general.

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Multivariate Linear Rational Expectations Models: Characterization of the Nature of the Solutions and Their Fully Recursive Computation