RDP 2008-10: Solving Linear Rational Expectations Models with Predictable Structural Changes 2. The Time-invariant Rational Expectations Solution

The method to solve for equilibria in linear rational expectations (LRE) models with predictable structural variations builds on the method proposed in Sims (2002). We begin by introducing notation, then outline the solution in the time-invariant case and establish key results on existence and uniqueness.

2.1 Defining the LRE Model

Define the state vector

where: y_1,t, (n₁ × 1), contains exogenous and possibly some endogenous variables; y_2,t, (n₂ × 1), contains those endogenous variables for which conditional expectations appear in the LRE model; and z_t+1, (k × 1), contains leads of y_2,t so that and k = s × n₂. The dimension of y_t is n × 1, where n = n₁ + n₂ + k.

The LRE model is typically given by n₁ + n₂ equations relating the elements of y_1,t and y_2,t to each other and to

where: ε_t is a l × 1 vector that is a random, exogenous and potentially serially correlated process; and are (n₁ + n₂) × n matrices; is (n₁ + n₂) × 1; and is (n₁ + n₂) × l.

Since we allow z_{t + 1} to potentially contain more than just one lead of y_2,t, we deviate from the terminology of Sims (2002) and define the vector of expectations revisions as follows

where for j ≥ 1. When z_t = y_2,t, η_t becomes a vector of forecast errors . Note that so incorporates y_2,t and the first (s − 1) elements of . So expectation revisions for y_2,t+s do not appear in Equation (2). It is also important to note that the information set in period t contains the value of all variables up to period t – 1 as well as period t shocks.

We augment the system defined by Equation (1) with the k equations from Equation (2) to obtain the following specification

which is equivalent, in the notation of Sims (2002), to

where the matrices Γ₀ and Γ₁ are both n × n, while C is n × 1, Ψ is n × l, and π is n × k. This system contains n equations – the same number as the number of variables in the state vector, y_t . It is worth noting that the vector of expectations revisions, η_t, is determined endogenously as part of the solution.

2.2 Solving the LRE Model

The Generalised Schur (or QZ) decomposition of (Γ₀,Γ₁) yields

where QQ′ = ZZ′ = I and both Λ and Ω are upper triangular. Q,Z,Λ and Ω are, in general, complex-valued. An important property of this decomposition, which always exists, is that it returns the generalised eigenvalues of (Γ₀,Γ₁) as the ratios of the diagonal elements of Ω and Λ, {ω_ii/λ_ii}.

Pre-multiply Equation (4) by Q to get

where w_t = Z^′y_t. Then rearrange the system so that the explosive eigenvalues correspond to the lower right blocks of Λ and Ω and partition w_t as follows

where w_2,t is a m × 1 vector that is associated with the m explosive generalised eigenvalues and w_1,t is (n − m) × 1.

According to this partition, Equation (5) reads

As the lower set of equations is not influenced by w_1,t, the dynamics of w_2,t are isolated as follows

Let and let x_2,t = Q2(C + Ψε_t + πη_t). Since the eigenvalues of Equation (7) are explosive, the equation can be solved forwards

assuming lim _j→∞M^jw_2,t+j = 0. Substituting back the definitions for M and x_2,t and expanding the expression above for w_2,t yields

Equation (8) relates w_2,t to future values of ε_t and η_t. This means that knowing w_2,t requires that all future events be known at time t. Taking expectations (conditional on time t information) does not change the left-hand side of Equation (8), so

since for j ≥ 1. The fact that the right-hand side of Equation (8) never deviates from its expected value implies that expectations revisions must fluctuate as a function of current and future ε_t 's to guarantee that the equality holds.

Taking expectations at time t + 1 gives

Expressions (9) and (10) are equal if and only if the vector of expectations revisions satisfies

The system's stability depends on the existence of expectations revisions η_t to offset the effect that the fundamental shocks ε_t have on w_2,t. To see this, assume for i ≥ 1 and C = 0. The equation for w_2,t becomes

Since this equation has explosive eigenvalues, stability requires that w_2,t = 0 for all t. This means that Q₂Ψε_t + Q₂πη_t = 0 must hold in each period to ensure that the effect on w_2,t of any fundamental shock (ε_t) is offset by revisions to expectations, η_t; if this condition does not hold, w_2,t will behave explosively.

The existence of a stable solution relies on expectations revisions (η_t) to adjust so that the system remains on its stable saddle path (SSP). This means that from any arbitrary starting point, expectations revisions must be able to get the system onto its SSP and then keep it there. Proposition 1 states the condition under which this is possible.

Proposition 1. For any initial starting value y₀, a stable solution exists for the following linear rational expectations system

if and only if rank(Q₂π) = m.

For a proof of Proposition 1, see Appendix A.

Since Q₂π is m × k, rank(Q₂π) ≤ min{m,k}, so the existence of a stable solution requires that m ≤ k; that is, the number of explosive eigenvalues cannot be larger than the dimension of η_t.

Proposition 1 states the condition for existence with arbitrary initial conditions. Should the system already be on its SSP, the rank condition is only sufficient for existence. If initial conditions place the system on its SSP, then the conditions for existence of a stable solution are weaker. Existence, in this case, requires that there is a vector of expectations revisions capable of offsetting the effect of new information on w_2,t. For this to occur, it is both necessary and sufficient that

Regardless of what process ε_t follows, the existence of a rational expectations solution requires solving a system of the form: Q₂πη_t = B_t, where , . The span condition is both necessary and sufficient for the vector B_t to be expressed as a linear combination of the columns of Q₂π and guarantees that a solution exists for η_t.

The kind of parameter variations that we consider in the next section typically alter the SSP of the system. Therefore, it is the rank condition that ensures stability. Announcements about future changes to the structure give rise not only to changes to the SSP, but also to arbitrary ‘initial conditions’ from the perspective of the new SSP. Although the span and rank conditions for existence of a stable solution would typically agree, it is the rank condition which is appropriate if initial conditions are indeed arbitrary.

Existence does not imply uniqueness. In general, it is possible that knowing Q₂πη_t may not be enough to calculate Q₁πη_t, which is needed in order to solve for w_1,t and to completely solve the LRE model. This requires that the row space of Q₁π be contained in the row space of Q₂π, both of which are subspaces of . It turns out that checking the row span condition for the uniqueness of an equilibrium is equivalent to checking the rank of the matrix Q₂π, as the following proposition states.

Proposition 2. Suppose a solution exists for the following linear rational expectations system

then the solution is unique if and only if rank(Q₂π) = k.

For a proof of Proposition 2, see Appendix A.

Since rank(Q₂π) ≤ min{m,k}, this implies that m ≥ k is a necessary condition for a unique solution. For arbitrary initial conditions, existence and uniqueness of a solution requires that m = k.

If a unique solution exists, then there exists a matrix Φ such that

Pre-multiplying Equation (6) by [I_n−m,−Φ] yields

When such a Φ exists, the term involving η_t drops out. Combining Equations (15) and (9), it is not difficult to show that the reduced-form of the LRE model becomes

where