RDP 8706: Numerical Solution of Rational Expectations Models with and Without Strategic Behaviour 3. Numerical Solutions of Rational Expectations Models

This section considers four numerical techniques that solve the multidimensional problem. These are the Blanchard-Kahn solution, Multiple Shooting, Fair-Taylor and the MSG techniques. The multiple shooting and Fair-Taylor techniques are specifically designed to solve the general non-linear problem. For clarity we will concentrate the discussion in the framework of a linear model.

At this point it is worth introducing some terminology.

A model written in minimal state-space representation is in the form:

We can partition the matrix in a convenient way. Let the first n variables be state or predetermined variables whose values are inherited from the past and the remaining m-n variables be “jumping variables”. Jumping variables are variables over which agents form expectations and which can change in response to new information in period t. The model (3.1) can be rewritten as:

This model could easily be solved forward as in the case of standard difference equations given X₀, e₀ and a path for E. That is, given the initial values for the state and jumping variables we can solve forward for X₁ and e₁ and so forth. The problem faced in attempting to numerically solve a rational expectations model is that there are only initial values for the set of state variables X₀ and terminal values for the set of jumping variables e_T or e_∞ (either assumed or from some optimisation solution). To solve the model in period 0, and for every period until T, requires knowledge of e₀ which requires knowledge of the solution in period 1 and so forth. This is called a two-point boundary value problem. Two points on the equilibrium path of the economy are known and these are both needed to define the path between them. Analytical solutions to models containing rational expectations can be found in simple cases by using techniques which solve these types of two-point boundary value problems such as illustrated in section 2. That is, to solve the model requires use of restrictions provided by initial values of state variables and some terminal conditions on jumping variables. The terminal values can be given as a fixed value in some finite terminal period or, in an infinite horizon problem, by a tranversality condition which imposes that in the infinite limit a variable is bounded.

Numerical solutions to more complex systems have been slower to emerge although there are now several techniques commonly used. The solutions provided by these techniques to be discussed can be better understood with the aid of Figure 1. Suppose that the multidimensional system of state and jumping variables can be compressed into points in a two-dimensional space. Each point such as A consists of a set of values for each of the state and jumping variables in the model {x₀, e₀}. Suppose point B summarises the terminal value for the problem which gives {x_T, e_T}. The problem is to find the unique path between A and B. We have initial values for X₀ and terminal values for e_T.

a. Blanchard-Kahn Solution

For a linear system, a general analytical solution is provided by Blanchard and Kahn (1980). This technique is a generalisation of the solution to a difference equation system as derived in section 2 above. It is based on transforming the transition matrix z₁ in (3.1) into its eigenvalue and eigenvector matrices. A solution is obtainable using the Blanchard-Kahn technique if the number of eigenvalues outside the unit circle is equal to the number of jumping variables. A more general analytical solution is provided by Chow and Reny (1984) but will not be discussed here.

b. Multiple Shooting Algorithm

In the case of non-linear systems, the technique of multiple shooting has been applied to the economics literature by Lipton, Poterba, Sachs and Summers (1983). The shooting technique can be described intuitively as follows. Initial values are assumed for the jumping variables . The model is then solved forward until the terminal period (or some finite period that is considered a good approximation to the infinite horizon) is reached. The terminal conditions on the jumping variables e_T, are then compared to the solution values for the jumping variables . If these are not equal, the initial guesses of the jumping variables are updated using some error correction procedure (i.e. Newton's Method). In the case of multiple shooting, the solution interval is divided into sub-intervals, such as shown in figure 1. The object is to solve the model to pass through intermediate points C and D. With the aid of auxiliary variables, the model is then solved, shooting within each sub-interval until convergence of the model solution to the terminal conditions is reached. A problem with this algorithm is that each sub-interval increases the dimensionality of the system to be solved.

c. Fair-Taylor Algorithm

Fair and Taylor (1983) have also developed a technique which has become popular because it tends to find a solution to models at a much lower cost than the multiple shooting algorithm. In the Fair-Taylor technique, an arbitrary terminal period, T, is chosen. Equation (3.3) is rewritten

where is taken as an exogenous guess for the expectation of e.

The paths of expected variables {_te₁,…,_te_T} are guessed and the model is solved assuming these expectations. The solution paths for the expected variables are then compared to the guesses and the guesses are updated using an error correction method. This iterative procedure is repeated until the expected path equals the actual path. The terminal period is then extended and the procedure repeated until the terminal period choice has no effect on the solution path. In our experience, the iteration on terminal period (called type III iteration in Fair and Taylor) is only required at the initial stage of simulating the model. Once a period length is established it rarely needs to be updated.^[4]

d. MSG Algorithm

A fourth solution procedure is used by McKibbin and Sachs in solving the MSG model. It is developed in the next section in more detail. The first step is to linearize the model around an assumed steady state. Assume that in any period, the jumping varables (e) can be written as a function of the inherited state variables (X) current exogenous variables (E), and the future path of exogenous variables, in the following form:

To find the matries H₁, H₂ and C_5t an iterative technique is used which essentially solves the model backwards. First the model is solved in an arbitrary terminal period T. By assuming period T is the last period, the future expectations of variables beyond period T are irrelevant. The model can then be rewritten in period T as:

where the H₁ and H₂ matricies are time subscripted.

Moving back to period T−1, the future jumping variables (e_T) are solved as functions of state variables (which are determined in period T−1) and exogenous variables. The model can now be rewritten in period T−1 given the rule for e_T. This procedure is repeated, solving backwards until the matricies H₁ and H₂ become independent of the terminal period chosen. This process also generates a cumulation rule for all future exogenous variables which is summarized in the matrix C_5t.

Once the rules linking e_t to X_t and current and future exogenous variables are found, the model can be solved in any period t using the additional condition:

This enables the model to be solved forward as a standard difference equation system. This technique is numerically equivalent to the Blanchard-Kahn solution although it is very convenient for implementing dynamic game theory which is discussed below.

Footnote

A copy of this algorithm which solves up to 90 simultaneous equations on a PC is available from Aptech System, P.O. Box 6487 KENT WA 98064, USA. The algorithm is written in GAUSS. [4]