A Technique for Solving Rational-Expectations Models

I consider a method using loop variables that can limit the size of the of a rational-expectations model and hopefully speed up the process of solving it. We can generally write such a model using a "Blanchard-Kahn" specification f_t(y_t, y_{t-1}, y_{t+1}, x_t) = 0 , where y is endogenous, x is exogenous, t is time and y_0 and y_{T+1} are known. Two methods are generally considered for solving such a system: Fair-Taylor and Stacked-Time. Let us consider the model (without rational expectations) y_t = f_t(y(t), y(t-1), x(t)) . Considering a particular ordering of the equations, the loop variables are the ones used through their present value before they are computed. Computing the value of y associated through g to a starting value of y_b can be done using a Gauss-Seidel iteration. This allows computation, through finite differences, of the Jacobian of y_b = g(y_b) and the application of Newton-Raphson method to a problem with the size of the number of loop variables. This technique is easily applied to rational-expectations models. In the full model, the loop variables are actually the union of the original loop variables and the leads so that, if we consider computing (not solving) the whole set of equations in one pass, only loop variables and leads affect the result. Computing the Jacobian of the whole model will be limited, and to do the Newton-Raphson process, we need realize (at most) T x n_b + (T-1) x n_l + 1 iterations and invert a matrix of dimension T x n_b + (T-1) x n_l . To evaluate the efficiency (speed and convergence probability) of this method, we use a small macro economic model of the French economy. The initial version does not use rational expectations and contains three loop variables, associated with the Keynesian loop, the price-wage loop, and the exchange rate loop. In this version, we introduce rational expectations in the investment equation (where firms are supposed to know the next production level) and in the consumption equation (where households know in advance their future revenue). We produce simulations over 20 to 100 periods to compare our method with the above in terms of speed and convergence reliability.