Essays on regression models for double bounded and extreme-value random variables : improved testing inferences and empirical analyses

Abstract

Beta regressions are commonly used with responses that assume values in the standard unit interval, such as rates, proportions and concentration indices. Hypothesis testing inferences on the model parameters are typically performed using the likelihood ratio test. It delivers accurate inferences when the sample size is large, but can otherwise lead to unreliable conclusions. It is thus important to develop alternative tests with superior finite sample behavior. We derive the Bartlett correction to the likelihood ratio test under the more general formulation of the beta regression model, i.e. under varying precision. The model contains two submodels, one for the mean response and a separate one for the precision parameter. Our interest lies in performing testing inferences on the parameters that index both submodels. We use three Bartlett-corrected likelihood ratio test statistics that are expected to yield superior performance when the sample size is small. We present Monte Carlo simulation evidence on the finite sample behavior of the Bartlett-corrected tests relative to the standard likelihood ratio test and to two improved tests that are based on an alternative approach. The numerical evidence shows that one of the Bartlett-corrected typically delivers accurate inferences even when the sample is quite small. An empirical application related to behavioral biometrics is presented and discussed. We also address the issue of performing testing inference in a general extreme value regression model when the sample size is small. The model contains separate submodels for the location and dispersion parameters. It allows practitioners to investigate the impacts of different covariates on extreme events. Testing inferences are frequently based on the likelihood test, including those carried out to determine which independent variables are to be included into the model. The test is based on asymptotic critical values and may be considerably size-distorted when the number of data points is small. In particular, it tends to be liberal, i.e., it yields rates of type I errors that surpass the test’s nominal size. We derive the Bartlett correction to the likelihood ratio test and use it to define three Bartlett-corrected test statistics. Even though these tests also use asymptotic critical values, their size distortions vanish faster than that of the unmodified test and thus they yield better control of the type I error frequency. Extensive Monte Carlo evidence and an empirical application that uses COVID-19 related data are presented and discussed.