Accuracy of Simulations for Stochastic Dynamic Models

Abstract

This paper is concerned with accuracy properties of simulations of approximate solutions for stochastic dynamic models. Our analysis rests upon a continuity property of invariant distributions and a generalized law of large numbers. We then show that the statistics generated by any sufficiently good numerical approximation are arbitrarily close to the set of expected values of the model's invariant distributions. Also, under a contractivity condition on the dynamics we establish error bounds. These results are of further interest for the comparative study of stationary solutions and the estimation of structural dynamic models.

Finite state Markov-chain approximations to highly persistent processes

“…See, for instance, den Haan (2010). 19 Note that this result is not due to the coarseness of the asset grids as doubling their size does not improve the accuracies of these statistics.…”

“…This problem remains even when a 25-state grid for e t is used and arises from errors in the approximation of the policy function that occur when the domain for e t is discretized. 19 The table also shows that the choice of discretization method is important when using the baseline approach. Moreover, under this approach, methods that generate relatively more accurate approximations for the persistence and the standard deviation of the AR(1) process also tend to yield relatively more accurate solutions.…”

Analysis of Numerical Errors

“…The above one-period approximation error (27) is just a first step to control the cumulative error of numerical simulations. Following Santos and Peralta-Alva (2005), our goal now is to present some regularity conditions so that the error from the simulated statistics converges to zero as the approximated equilibrium function approaches the exact equilibrium function. The following example illustrates that certain convergence properties may not always hold.…”

“…The following contraction property is taken from Stenflo (2001) Condition C may arise naturally in growth models [Schenk-Hoppe and Schmalfuss (2001)], in learning models [Ellison and Fudenberg (1993)], and in certain types of stochastic games [Sanghvi and Sobel (1976)]. Using Condition C, the following bounds for the approximation error of the simulated moments are established in Santos and Peralta-Alva (2005)…”

Analysis of Numerical Errors

“…Using Condition C, the following bounds for the approximation error of the simulated moments are established in Santos and Peralta-Alva (2005) …”