Multivariate Linear Rational Expectations Models: Characterization of the Nature of the Solutions and Their Fully Recursive Computation
📄 Summarized from the full manuscript · Human-reviewed for faithfulness before publication
In brief
Models in which people's expectations of the future shape today's outcomes may have exactly one sensible solution, many, or none. This purely mathematical paper sets out, for a general class of such models, which of the three you face, tying the answer to properties of a single matrix solving a quadratic equation. Its practical contribution is a computing method that works backward through repeated matrix inversions, avoiding the decomposition step rival methods need; the authors report it was usually faster and worked on large or awkward systems where standard techniques stall. It matters because nearly every quantitative macroeconomic model must be solved this way first.
What this paper finds — and why it matters
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.
Summary of a classic paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.
Questions & answers
Q1. What problem is this paper solving, and why does it matter for applied macroeconometrics?
The paper solves the general problem of characterizing and computing the full solution space of a multivariate linear rational expectations (RE) model – including cases where a unique stable solution does not exist – and does so for high-dimensional systems whose coefficient matrices may be highly singular. Many macro and monetary models (dynamic adjustment-cost models, multivariate expectations-formation systems) reduce to a stacked linear RE system, and applied researchers need both a theoretical account of when the solution is unique, multiple, or absent, and a practical way to compute it that does not break down as the system grows large or degenerate (Abstract; Section 1, pp. 877-879).
Q2. What is the general model being solved, and how is it reduced to a tractable canonical form?
The starting point is the Broze-Gourieroux-Szafarz (1995) general multivariate linear RE model, A_00 y_t = sum of lagged y’s, lagged expectations of future y’s, and a shock u_t, with A_00 normalized to the identity (Eq. 1, p. 877). By stacking y_t together with its current and expected future values into a vector x_t of dimension m = K(H+1)G, the system is rewritten in the canonical companion form x_t = A x_{t-1} + B E(x_{t+1}|I_t) + w_t (Eq. 2, p. 879), where A and B are m x m matrices constructed from the original coefficient matrices. This canonical form is the object the rest of the paper analyzes.
Q3. How does the quadratic determinantal equation (QDE) method characterize the nature of the solution?
The QDE method transforms the canonical system via X_t = x_t - C x_{t-1}, where C must solve the quadratic matrix equation P(C) = BC^2 - C + A = 0 (Eq. 4, p. 880); finding the real-valued solutions of this matrix equation is what pins down whether the model has a unique stable solution, multiple stable solutions, or none. Proposition 1 (p. 881) gives the existence condition: forming the polynomial equation phi(lambda_c) = det(Blambda_c^2 - Ilambda_c + A) = 0 and taking its real-valued Jordan-form solutions {J_1,…,J_l}, at least one real-valued C exists if and only if at least one of the associated matrices M_i (Eq. 7) has a nonempty null space.
Q4. Once a solution matrix C is found, what does the general solution for x_t look like, and how does it nest the different solution cases?
The general solution (Eq. 14, pp. 882-883) writes x_t as C x_{t-1} plus a moving sum of expected future shocks weighted by matrices built from the eigen-decomposition of F = (I - BC)^{-1} B, plus (in the non-unique case) free martingale processes. The eigenvalues of F split into three blocks – inside the unit circle (dimension m_1), outside it (m_2), and exactly zero (m_3) – and the formula encompasses the unique stable solution (when m_2 = 0, so there is no free “bubble” martingale component), the multiple-stable-solutions case (m_2 > 0, with an indeterminate martingale process M_t^b of order m_2), and the no-stable-solution case, all within one formula (Section 2, pp. 882-883).
Q5. Under what conditions is the stable solution unique, and how restrictive is that condition?
Proposition 2 (p. 883) gives sufficient conditions for a unique real-valued stable solution when A and B commute and their nonzero eigenvalues satisfy an explicit inequality (Eq. 15) together with further eigenvalue/stability conditions; under these conditions the solution simplifies to x_t = C x_{t-1} + sum_{i=0}^infinity F^i E(W_{t+i}|I_t) (Eq. 16). The authors note this directly extends Proposition 3 of their own 1995 paper, but flag that the commutativity requirement on A and B – while satisfied by dynamic adjustment-cost models – is restrictive in general (Section 2, pp. 883-884).
Q6. What is the paper’s main new contribution beyond the 1995 QDE method, and how does it work?
The main novel contribution is a “fully recursive” solution method (Proposition 3, Section 4, pp. 885-886) that computes the solution purely by backward recursion on matrix inversions, avoiding any eigenvalue-eigenvector decomposition. Starting from terminal conditions Q_N = I_m and R_{t+N} built from the terminal expectation, the method iterates backward for j = 1,…,N via Q_{N-j} = I - B*Q_{N-j+1}^{-1}*A (Eq. 26) and a corresponding recursion for R_{t+N-j} (Eq. 27), yielding the solution x_t = Q_0^{-1} A x_{t-1} + Q_0^{-1} R_t (Eq. 28). The authors describe it as “straightforward to implement, fast, and applicable to high-dimensional problems possibly involving coefficient matrices with a high degree of singularity” (Abstract, p. 877).
Q7. Is the fully recursive method actually faster than the QDE method, and are there caveats to its validity?
In the authors’ extensive computations across a wide variety of RE models, the fully recursive method was found to be “typically even faster, often substantially so, than the QDE method” (Footnote 9, p. 886), but its validity requires that Q_{N-j} remain nonsingular for every j = 1,…,N, and that the terminal horizon N be chosen large enough that the solution is insensitive to the assumed terminal expectation E(x_{t+N+1}|I_t). The authors report that in practice no economic model they encountered had a singular Q_{N-j}, and that the standard practical check is to solve for several values of N and confirm the solution has converged (Footnote 8, p. 886). In the case of multiple stable solutions, the method handles each one by supplying the solution corresponding to each valid terminal condition rather than picking a single one automatically (Section 4, pp. 885-886).
Q8. How does this method relate to competing approaches such as Blanchard-Kahn, King-Watson, and Uhlig?
The paper positions its approach as more general than Blanchard and Kahn (1980), which rules out certain singularities in the coefficient matrices and is therefore not applicable in the fully general case, and than King and Watson (1996), which does not require Blanchard-Kahn nonsingularity but focuses only on the regular (unique-solution) case. Uhlig (1997) rewrites the QDE as a standard eigenvalue-eigenvector problem but only when B is nonsingular, which the authors describe as less general than their QDE approach; BGS (1995) itself fully characterizes the dimension of the solution space using matrix polynomial techniques but does not provide a computational implementation for models with both lagged dependent variables and future expectations (Section 1, p. 878; Section 3, p. 885; Footnote 2, p. 879).
Q9. What are the paper’s own stated limitations, and when should a researcher still prefer the older QDE method?
The authors are explicit that the fully recursive method is best suited to numerical computation in the unique-stable-solution case, and that when multiple stable solutions exist, the QDE method (or BGS’s approach) remains preferable for determining the dimension of the solution space (Section 5, pp. 886-887). The recursive method’s own validity also rests on two conditions that must be checked rather than assumed: nonsingularity of Q_{N-j} throughout the backward iteration, and a large-enough terminal horizon N that the solution has effectively converged with respect to the terminal expectation (Footnote 8, p. 886; Proposition 3, p. 886).
Key terms in this paper
Definitions below follow the paper's own usage.
- Quadratic determinantal equation (QDE) method
- the technique, introduced in Binder and Pesaran (1995) and extended here, of transforming the canonical RE system via X_t = x_t - C x_{t-1}, where C solves the quadratic matrix equation BC^2 - C + A = 0; the real-valued solutions of this equation characterize whether the model has a unique stable solution, multiple stable solutions, or none.
- Fully recursive solution method
- the paper's new numerical algorithm (Proposition 3) that computes the RE model's solution by backward recursion on matrix inversions alone -- no eigenvalue-eigenvector decomposition -- starting from terminal conditions Q_N = I_m and iterating Q_{N-j} = I - B*Q_{N-j+1}^{-1}*A backward to j=1; valid whenever every Q_{N-j} in the recursion is nonsingular.
- Canonical companion form
- the representation x_t = A x_{t-1} + B E(x_{t+1}|I_t) + w_t obtained by stacking the original vector y_t and its current/expected-future values into a single vector x_t of dimension m = K(H+1)G, which converts the general BGS (1995) multi-lag, multi-horizon RE model into a first-order system amenable to the QDE and recursive methods.
- Solution multiplicity (unique vs. multiple stable solutions vs. no stable solution)
- the three mutually exclusive outcomes the general solution formula (Eq. 14) nests, determined by how many eigenvalues of F = (I - BC)^{-1}B lie inside (m_1), outside (m_2), or exactly on (m_3) the unit circle; m_2 = 0 gives the unique stable solution, while m_2 > 0 introduces a free martingale component and hence indeterminacy.
- Commutativity condition (Proposition 2)
- the sufficient condition -- that A and B commute and their nonzero eigenvalues satisfy a specific inequality (Eq. 15) plus further stability restrictions -- under which the stable solution is proved unique; satisfied by dynamic adjustment-cost models but noted by the authors as restrictive for general multivariate RE systems.