RDP 2012-08: Estimation and Solution of Models with Expectations and Structural Changes 2. Solution of Models with Forward-looking Expectations and No Structural Changes

Our solution method is a variant of Binder and Pesaran (1997). Following that paper, a linear rational expectations model of n equations can be written as

where yt is a n × 1 vector of state and jump variables and εt is a l × 1 vector of exogenous variables. With no loss of generality we take the latter to be white noise and to have Il as their covariance matrix. All matrices in Equation (1) conform to the specified dimensions.[2] The formulation can be generalised as in Binder and Pesaran (1997) to allow additional lags of yt as well as conditional expectations at different horizons and from earlier dates.

If it exists and is unique, the solution to Equation (1) will be a VAR of the form

Given that this is the solution and Inline Equation we must have Inline Equation. Substituting this into Equation (1) and re-arranging terms produces

Now Inline Equation and defining Inline Equation, Inline Equation, Equation (3) becomes

Inline Equation

But this must equal Equation (2), establishing the equivalences

Equation (5) implies that

and so determines Q. Equation (4) implies that

where Λ = (IBQ)-1Γ, F = (IBQ)-1B. Thus, once Q is found, it is possible to derive C and G, providing the solution to the model.

Footnote

We may need to make a distinction between the original shocks of a dynamic stochastic model, et, and the shocks εt in Equation (1). Often et are taken to be serially correlated. This can be captured by writting such a system in the form of Equation (1) with lagged values of the endogenous variables included in yt. This means that εt are the innovations to the shock processes et. There may be a numerical advantage to working with et rather than εt, as that reduces the dimension of yt and, consequently, all the matrices involved in finding a solution. But there are conceptual advantages in using the system we work with. Our MATLAB function that computes the Binder Pesaran solution, ‘smatsbp.m’, does allow us to work with an et that is described by a VAR process. [2]