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MANUEL S. SANTOS
Euler equation residuals may be compatible with large deviations from the
optimal policy; moreover, for a given numerical solution these residuals can be
computed numerically at a relatively low cost without resorting to formal
statistical techniques.
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Aware of these criticisms, Judd 1992 suggested an alternative test that
entails numerical computation of the Euler equation residuals over the whole
state space. But from these computations, none of these papers has attempted
to infer the size of the approximation errors of the computed value and policy
functions, without specific knowledge of the true functions.
The remaining sections proceed as follows. Section 2 offers a brief introduc-
tion on existing approaches for checking accuracy of numerical solutions, and
draws attention to some of the pitfalls associated with the current application of
these techniques. Section 3 presents the model and our main results. Section 4
focuses on some simple examples and numerical experiments that illustrate the
nature of our theoretical findings along with some other relevant implementa-
tional issues. Some concluding remarks follow in the final section.
2. TESTING ACCURACY OF NUMERICAL SOLUTIONS
As a framework for discussion of the present results, we begin with a brief
introduction to some simple methods for testing accuracy of numerical solutions
and for comparing the performance of alternative algorithms. The current
literature is not particularly helpful for making these critical comparisons:
Available tests for checking accuracy of numerical solutions cannot generally
assess with the required precision the approximation error of a numerical
solution. Thus, sometimes a typical evaluation procedure is to compare the
outcome of a computational method against the solution of a slower, more
reliable algorithm. This is, however, a rather awkward accuracy check, which
may be very costly to implement and is not always feasible.
One widely used strategy is to test the outcome of a computational algorithm
in a particular case where the model displays an analytical solution. The
problem with this approach is that for alternative parameterizations of the
model the approximation error of the computed value and policy functions may
change substantially, and correspondingly, this initial test may become ineffec-
tive. As shown below, changes in the curvature of the utility function and in the
discount factor may affect considerably the accuracy of the algorithm.
Another commonly employed stopping rule is to fix a tolerance level )0 for
the error obtained after successive approximate solutions. For instance, the
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algorithm may be instructed to stop when under a given metric, Wn yWnq1 -,
where Wn and Wnq1 are the functions obtained at iterations n and nq1,
respectively. If the algorithm is generated by a contractive operator with
modulus , and W is the true solution, then it is well known that the
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approximation error, Wn yW -r 1y . But if the algorithm does not
satisfy the contraction property, then one generally cannot infer the magnitude
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of the error Wn yW from these differences from successive approximations.