Complementarities in the Production of Child Health

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 (1997) to our setting. We provide an asymptotic theory for the approximate solutions and show that they converge with root-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.