Title: Essays on Simulation-Based Estimation
Description: Complex nonlinear dynamic models with an intractable likelihood or moments are increasingly common in economics. A popular approach to estimating these models is to match informative sample moments with simulated moments from a fully parameterized model using SMM or Indirect Inference. This dissertation consists of three chapters exploring different aspects of such simulation-based estimation methods. The following chapters are presented in the order in which they were written during my thesis. Chapter 1, written with Serena Ng, provides an overview of existing frequentist and Bayesian simulation-based estimators. These estimators are seemingly computationally similar in the sense that they all make use of simulations from the model in order to do the estimation. To better understand the relationship between these estimators, this chapters introduces a Reverse Sampler which expresses the Bayesian posterior moments as a weighted average of frequentist estimates. As such, it highlights a deeper connection between the two class of estimators beyond the simulation aspect.

This Reverse Sampler also allows us to compare the higher-order bias properties of these estimators. We find that while all estimators have an automatic bias correction property (Gourieroux et al., 1993) the Bayesian estimator introduces two additional biases. The first is due to computing a posterior mean rather than the mode. The second is due to the prior, which penalizes the estimates in a particular direction. Chapter 2, also written with Serena Ng, proves that the Reverse Sampler described above targets the desired Approximate Bayesian Computation (ABC) posterior distribution. The idea relies on a change of variable argument: the frequentist optimization step implies a non-linear transformation. As a result, the unweighted draws follow a distribution that depends on the likelihood that comes from the simulations, and a Jacobian term that arises from the non-linear transformation. Hence, solving the frequentist estimation problem multiple times, with different numerical seeds, leads to an optimization-based importance sampler where the weights depend on the prior and the volume of the Jacobian of the non-linear transformation.

In models where optimization is relatively fast, this Reverse Sampler is shown to compare favourably to existing ABC-MCMC or ABC-SMC sampling methods. Chapter 3, relaxes the parametric assumptions on the distribution of the shocks in simulation-based estimation. It extends the existing SMM literature, where even though the choice of moments is flexible and potentially nonparametric, the model itself is assumed to be fully parametric. The large sample theory in this chapter allows for both time-series and short-panels which are the two most common data types found in empirical applications. Using a flexible sieve density reduces the sensitivity of estimates and counterfactuals to an ad hoc choice of distribution such as the Gaussian density. Compared to existing work on sieve estimation, the Sieve-SMM estimator involves dynamically generated data which implies non-standard bias and dependence properties. First, the dynamics imply an accumulation of the bias resulting in a larger nonparametric approximation error than in static models.

To ensure that it does not accumulate too much, a set decay conditions on the data generating process are given and the resulting bias is derived. Second, by construction, the dependence properties of the simulated data vary with the parameter values so that standard empirical process results, which rely on a coupling argument, do not apply in this setting. This non-standard dependent empirical process is handled through an inequality built by adapting results from the existing literature. The results hold for bounded empirical processes under a geometric ergodicity condition. This is illustrated in the paper with Monte-Carlo simulations and two empirical applications.