The Stacked-Time Simulator in TROLL: A Robust Algorithm for Solving Forward-Looking Models

Peter Hollinger
Intex Solutions, Inc., Needham, MA
peterh@intex.com

Abstract

The TROLL econetric modelling system provides two methods to solve forward- looking ``rational expectations" models with model-consistent endogenous leads.

The traditional Fair-Taylor algorithm solves the model through a specified time horizon by treating the leads as exogenous and then iterating over that process until the leads converge. Fair-Taylor can be considered a sort of ``Gauss- Seidel-over-time". Like Gauss-Seidel, Fair-Taylor may work very well, or it may have trouble converging. Larger shocks, more time periods, or tighter convergence criteria can greatly increase the number of iterations and computer time required to converge or can result in failure to converge. When convergence is successful, the ``solution" values may have errors that are large relative to the convergence criterion. One advantage of Fair-Taylor is that it does not require much computer memory.

The alternative Stacked-Time algorithm ``stacks" all the time periods into one large system of equations and solves them simultaneously using Newton-Raphson. For typical macroeconometric models, Newton-Raphson is usually an efficient and very robust method. With quadratic convergence near the solution, the number of iterations is barely affected by the convergence criteria. The number of iterations also does not seem to change substantially with the number of time periods or the size of the shock, although in some cases large shocks may cause numerical problems such as ``log of a negative number". At convergence, solutions are generally accurate relative to the convergence criterion.

Each iteration in Newton-Raphson does require solving a matrix equation involving the Jacobian matrix, and in a stacked system that can be very large. Typical sizes might be 500 equations by 50 periods = 25000 equations or 3000 equations by 100 periods = 300'000 equations. However, the large stacked Jacobian matrix has a repetitive structure of nonzero blocks along its diagonal, and each of those blocks in turn is sparse. The Stacked-Time simulator in TROLL takes advantage of both the repetitive block structure and of the sparsity within the blocks. The resulting algorithm is generally more accurate and usually faster than Fair-Taylor. It does require more memory than Fair-Taylor, and that is probably the limiting factor for large models with many lags and leads.

This paper compares the two methods in experiments on a variety of models ranging from 50 equations by 50 time periods to 3000 equations by 100 time periods.