Solving dynamic discrete choice models using smoothing and sieve methods

Abstract

We propose to combine smoothing, simulations and sieve approximations to solve for either the integrated or expected value function in a general class of dynamic discrete choice (DDC) models. We use importance sampling to approximate the Bellman operators defining the two functions. The random Bellman operators, and therefore also the corresponding solutions, are generally non-smooth which is undesirable. To circumvent this issue, we introduce a smoothed version of the random Bellman operator and solve for the corresponding smoothed value function using sieve methods. We show that one can avoid using sieves by generalizing and adapting the "self-approximating" method of Rust (1997b) to our setting. We provide an asymptotic theory for the approximate solutions and show that they converge with √ N-rate, where N is number of Monte Carlo draws, towards Gaussian processes. We examine their performance in practice through a set of numerical experiments and find that both methods perform well with the sieve method being particularly attractive in terms of computational speed and accuracy.

A Comment on “Using Randomization to Break the Curse of Dimensionality”

“…Rust (1997a, pp. 35, 47) explained that this “problem can lead to be a poor estimate of the true expected value function” and that it “can be a much more serious problem in higher dimensional problems.” Kristensen, Mogensen, Myun Moon, and Schjerning (2021, p. 329) empirically confirmed this claim, reporting that the “variances of [Rust's] self‐approximating method increase dramatically as the number of state variables increases.”…”

Penalized Sieve Estimation of Structural Models

“…Additional shape constraints, sparsity patterns, and better grid choices are useful in reducing computational burden. See, e.g., Chen (2007) discusses various sieve-based methods and Kristensen et al (2021) discuss various approximation architectures for approximating value functions in dynamic models. Second, there are also many model-specific techniques for approximating functions using a small number of basis functions.…”