Enhancing the Convergence Properties of the BLP (1995) Contraction Mapping
In the wake of Dub´e, Fox and Su (2012), this paper (i) analyzes the consequences for the BLP Contraction Mapping (Berry, Levinsohn and Pakes, 1995) of setting the inner-loop convergence tolerance at the required level of in D 10 14 to avoid propagation of approximation error from inner to outer loop, and (ii) proposes acceleration as a viable alternative within the confines of the Nested-Fixed Point paradigm (Rust, 1987) to enhance the convergence properties of the inner-loop BLP Contraction Mapping. Drawing upon the equivalence between nonlinear rootfinding and fixed-point iter- ation, we introduce and compare two alternative methods specifically designed for handling large-scale nonlinear problems, in particular the derivative-free spectral al- gorithm for nonlinear equations (La Cruz et al., 2006, DF-SANE), and the squared polynomial extrapolation method for fixed-point acceleration (Varadhan and Roland, 2008, SQUAREM). Running a Monte Carlo study with specific scenarios and with Newton-Raphson as a benchmark, we study the characteristics of these algorithms in terms of (i) speed, (ii) robustness, and (iii) quality of approximation. Under the worst of circumstances we find that (i) SQUAREM is faster (up to more than five times as fast) and more robust (up to 14 percentage points better suc- cess rate) than BLP while attaining a comparable quality of approximation, and (ii) also outperforms DF-SANE in nearly all scenarios. Eliminating averaging bias against more robust algorithms, a performance profile subsequently shows that (iii) SQUAREM is both faster, delivers the best quality approximation, and is more robust than BLP, DF-SANE and Newton-Raphson.
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