Essays in econometric theory - Casalecchi, Alessandro Ribeiro de Carvalho (2017)
The two papers in this work, chapters 2 and 3, regard hypothesis testing but address different issues. Chapter 2, entitled 'Improvements for external validity tests in fuzzy regression discontinuity designs', shows conditions --- assumptions of continuity, strict monotonicity and pointwise convergence --- under which two-sample goodness-of-fit (GOF) tests can be used to test for external validity in treatment-control models that suffer from imperfect compliance of units with respect to the assigned treatment. Imperfect compliance allows researchers to estimate only treatment effects for the subpopulation of compliers, and the validity of these estimates for other subpopulations (always-takers and never-takers) remains an open problem. Under the conditions in Chapter 2, the use of GOF tests in place of mean difference tests represents an improvement over other external validity tests in the literature, since more alternative hypotheses are detectable by the test statistic. We suggested to combine two GOF test statistics (one for the treated and one for the untreated) in a multiple test instead of a joint test. Chapter 3, entitled 'Higher-order UMP tests', suggests a strategy to choose among candidate test statistics, according to a power criterion, when their power performances are not distinguishable by usual methods of asymptotic comparison like local power analysis. We propose the use of higher-order asymptotic expansions, like Edgeworth expansions, to approximate the sample densities of the candidate test statistics and verify which of them has the monotone likelihood ratio property. This property implies, by the Karlin-Rubin Theorem, that the test is uniformly most powerful (UMP) --- at least to an order of approximation --- if the statistic is sufficient for the relevant parameter. When the statistics under study are not sufficient, we argue that they can often be made sufficient for a desired parametric family after appropriate reparameterization. We applied the method to search for the power-optimal bandwidth for the kernel density estimator in simulated data sets, and concluded that the order of approximation that we used (second order) is still too low to allow us to distinguish among bandwidths.