BLP Estimation Using Laplace Transformation and Overlapping Simulation Draws

We derive the asymptotic distribution of the parameters of the \citet{blp} (BLP) model in a many markets setting which takes into account simulation noise under the assumption of overlapping simulation draws. We show that as long as the number of simulation draws $R$ and the number of markets $T$ approach infinity, our estimator is $\sqrt{m}=\sqrt{min(R,T)}$ consistent and asymptotically normal. We do not impose any relationship between the rates at which $R$ and $T$ go to infinity, thus allowing for the case of $R\ll T$. We provide a consistent estimate of the asymptotic variance which can be used to form asymptotically valid confidence intervals. Instead of directly minimizing the BLP GMM objective function, we propose using Hamiltonian Markov Chain Monte Carlo methods to implement a Laplace-type estimator which is asymptotically equivalent to the GMM estimator