Jackson Bunting

A panel dataset satisfies marginal homogeneity if the time-specific marginal distributions are homogeneous or time-invariant. Marginal homogeneity is relevant in economic settings such as dynamic discrete games. In this paper, we propose several tests for the hypothesis of marginal homogeneity and investigate their properties. We consider an asymptotic framework in which the number of individuals n in the panel diverges, and the number of periods T is fixed. We implement our tests by comparing a studentized or non-studentized T-sample version of the Cramer-von Mises statistic with a suitable critical value. We propose three methods to construct the critical value: asymptotic approximations, the bootstrap, and time permutations. We show that the first two methods result in asymptotically exact hypothesis tests. The permutation test based on a non-studentized statistic is asymptotically exact when T=2, but is asymptotically invalid when T>2. In contrast, the permutation test based on a studentized statistic is always asymptotically exact. Finally, under a time-exchangeability assumption, the permutation test is exact in finite samples, both with and without studentization.

The literature on dynamic discrete games often assumes that the conditional choice probabilities and the state transition probabilities are homogeneous across markets and over time. We refer to this as the "homogeneity assumption" in dynamic discrete games. This assumption enables empirical studies to estimate the game's structural parameters by pooling data from multiple markets and from many time periods. In this paper, we propose a hypothesis test to evaluate whether the homogeneity assumption holds in the data. Our hypothesis test is the result of an approximate randomization test, implemented via a Markov chain Monte Carlo (MCMC) algorithm. We show that our hypothesis test becomes valid as the (user-defined) number of MCMC draws diverges, for any fixed number of markets, time periods, and players. We apply our test to the empirical study of the U.S. Portland cement industry in Ryan (2012).

Many estimators of dynamic discrete choice models with persistent unobserved heterogeneity have desirable statistical properties but are computationally intensive. In this paper we propose a method to quicken estimation for a broad class of dynamic discrete choice problems by exploiting semiparametric index restrictions. Specifically, we propose an estimator for models whose reduced form parameters are invertible functions of one or more linear indices (Ahn, Ichimura, Powell, and Ruud 2018), a property we term index invertibility. We establish that index invertibility implies a set of equality constraints on the model parameters. Our proposed estimator uses the equality constraints to decrease the dimension of the optimization problem, thereby generating computational gains. Our main result shows that the proposed estimator is asymptotically equivalent to the unconstrained, computationally heavy estimator. In addition, we provide a series of results on the number of independent index restrictions on the model parameters, providing theoretical guidance on the extent of computational gains. Finally, we demonstrate the advantages of our approach via Monte Carlo simulations.

In dynamic discrete choice (DDC) analysis, it is common to use mixture models to control for unobserved heterogeneity. However, consistent estimation typically requires both restrictions on the support of unobserved heterogeneity and a high-level injectivity condition that is difficult to verify. This paper provides primitive conditions for point identification of a broad class of DDC models with multivariate continuous permanent unobserved heterogeneity. The results apply to both finite- and infinite-horizon DDC models, do not require a full support assumption, nor a long panel, and place no parametric restriction on the distribution of unobserved heterogeneity. In addition, I propose a seminonparametric estimator that is computationally attractive and can be implemented using familiar parametric methods.

We provide semiparametric identification results for a broad class of learning models in which continuous outcomes depend on three types of unobservables: i) known heterogeneity, ii) initially unknown heterogeneity that may be revealed over time, and iii) transitory uncertainty. We consider a common environment where the researcher only has access to a short panel on choices and realized outcomes. We establish identification of the outcome equation parameters and the distribution of the three types of unobservables, under the standard assumption that unknown heterogeneity and uncertainty are normally distributed. We also show that, absent known heterogeneity, the model is identified without making any distributional assumption. We then derive the asymptotic properties of a sieve MLE estimator for the model parameters, and devise a tractable profile likelihood based estimation procedure. Monte Carlo simulation results indicate that our estimator exhibits good finite-sample properties.

We study the asymptotic properties of a class of estimators of the structural parameters in dynamic discrete choice games. We consider K-stage policy iteration (PI) estimators, where K denotes the number of policy iterations employed in the estimation. This class nests several estimators proposed in the literature such as those in Aguirregabiria and Mira (2002, 2007), Pesendorfer and Schmidt-Dengler (2008), and Pakes et al. (2007). First, we establish that the K-PML estimator is consistent and asymptotically normal for all K. This complements findings in Aguirregabiria and Mira (2007), who focus on K=1 and K large enough to induce convergence of the estimator. Furthermore, we show under certain conditions that the asymptotic variance of the K-PML estimator can exhibit arbitrary patterns as a function of K. Second, we establish that the K-MD estimator is consistent and asymptotically normal for all K. For a specific weight matrix, the K-MD estimator has the same asymptotic distribution as the K-PML estimator. Our main result provides an optimal sequence of weight matrices for the K-MD estimator and shows that the optimally weighted K-MD estimator has an asymptotic distribution that is invariant to K. The invariance result is especially unexpected given the findings in Aguirregabiria and Mira (2007) for K-PML estimators. Our main result implies two new corollaries about the optimal 1-MD estimator (derived by Pesendorfer and Schmidt-Dengler (2008)). First, the optimal 1-MD estimator is optimal in the class of K-MD estimators. In other words, additional policy iterations do not provide asymptotic efficiency gains relative to the optimal 1-MD estimator. Second, the optimal 1-MD estimator is more or equally asymptotically efficient than any K-PML estimator for all K. Finally, the appendix provides appropriate conditions under which the optimal 1-MD estimator is asymptotically efficient.