Baggerly (1998) showed that empirical likelihood is the only member in the Cressie-Read power divergence family to be Bartlett correctable. This paper strengthens Baggerly’s result by showing that in a generalized class of the power divergence family, which includes the Cressie-Read family and other nonparametric likelihood such as Schennach’s (2005, 2007) exponentially tilted empirical likelihood, empirical likelihood is still the only member to be Bartlett correctable.
This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency ﬁnancial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a ﬁxed time interval increases to inﬁnity. Our large deviation result can be used to evaluate tail probabilities of the realized volatility. We also derive a moderate deviation rate function for a standardized realized volatility statistic. The moderate deviation result is useful for assessing the validity of normal approximations based on the central limit theorem. In particular, it clariﬁes that there exists a trade-oﬀ between the accuracy of the normal approximations and the path regularity of an underlying volatility process. Our large and moderate deviation results complement the existing asymptotic theory on high frequency data. In addition, the paper contributes to the literature of large deviation theory in that the theory is extended to a high frequency data environment.
This paper studies robustness of bootstrap inference methods for instrumental variable regression models. In particular, we compare the uniform weight and implied probability bootstrap approximations for parameter hypothesis test statistics by applying the breakdown point theory, which focuses on behaviors of the bootstrap quantiles when outliers take arbitrarily large values. The implied probabilities are derived from an information theoretic projection from the empirical distribution to a set of distributions satisfying orthogonality conditions for instruments. Our breakdown point analysis considers separately the eﬀects of outliers in dependent variables, endogenous regressors, and instruments, and clariﬁes the situations where the implied probability bootstrap can be more robust than the uniform weight bootstrap against outliers. Eﬀects of tail trimming introduced by Hill and Renault (2010) are also analyzed. Several simulation studies illustrate our theoretical ﬁndings.
This paper studies robustness of bootstrap inference methods under moment conditions. In particular, we compare the uniform weight and implied probability bootstraps by analyzing behaviors of the bootstrap quantiles when outliers take arbitrarily large values, and derive the breakdown points for those bootstrap quantiles. The breakdown point properties characterize the situation where the implied probability bootstrap is more robust than the uniform weight bootstrap against outliers. Simulation studies illustrate our theoretical ﬁndings.
This paper studies second-order properties of the empirical likelihood overidentifying restriction test to check the validity of moment condition models. We show that the empirical likelihood test is Bartlett correctable and suggest second-order reﬁnement methods for the test based on the empirical Bartlett correction and adjusted empirical likelihood. Our second-order analysis supplements the one in Chen and Cui (2007) who considered parameter hypothesis testing for overidentiﬁed models. In simulation studies we ﬁnd that the empirical Bartlett correction and adjusted empirical likelihood assisted by bootstrapping provide reasonable improvements for the properties of the null rejection probabilities.
This paper proposes a simple, fairly general, test for global identiﬁcation of unconditional moment restrictions implied from point-identiﬁed conditional moment restrictions. The test is based on the Hausdorﬀ distance between an estimator that is consistent even under global identiﬁcation failure of the unconditional moment restrictions, and an estimator of the identiﬁed set of the unconditional moment restrictions. The proposed test has a chi-squared limiting distribution and is also able to detect weak identiﬁcation alternatives. Some Monte Carlo experiments show that the proposed test has competitive ﬁnite sample properties already for moderate sample sizes.
This paper studies the Hodges and Lehmann (1956) optimality of tests in a general setup. The tests are compared by the exponential rates of growth to one of the power functions evaluated at a ﬁxed alternative while keeping the asymptotic sizes bounded by some constant. We present two sets of suﬀicient conditions for a test to be Hodges-Lehmann optimal. These new conditions extend the scope of the Hodges-Lehmann optimality analysis to setups that cannot be covered by other conditions in the literature. The general result is illustrated by our applications of interest: testing for moment conditions and overidentifying restrictions. In particular, we show that (i) the empirical likelihood test does not necessarily satisfy existing conditions for optimality but does satisfy our new conditions; and (ii) the generalized method of moments (GMM) test and the generalized empirical likelihood (GEL) tests are Hodges-Lehmann optimal under mild primitive conditions. These results support the belief that the Hodges-Lehmann optimality is a weak asymptotic requirement.
This paper is concerned with robust estimation under moment restrictions. A moment restriction model is semiparametric and distribution-free, therefore it imposes mild assumptions. Yet it is reasonable to expect that the probability law of observations may have some deviations from the ideal distribution being modeled, due to various factors such as measurement errors. It is then sensible to seek an estimation procedure that are robust against slight perturbation in the probability measure that generates observations. This paper considers local deviations within shrinking topological neighborhoods to develop its large sample theory, so that both bias and variance matter asymptotically. The main result shows that there exists a computationally convenient estimator that achieves optimal minimax robust properties. It is semiparametrically eﬀicient when the model assumption holds, and at the same time it enjoys desirable robust properties when it does not.
Suppose that the econometrician is interested in comparing two misspeciﬁed moment restriction models, where the comparison is performed in terms of some chosen measure of ﬁt. This paper is concerned with describing an optimal test of the Vuong (1989) and Rivers and Vuong (2002) type null hypothesis that the two models are equivalent under the given measure of ﬁt (the ranking may vary for diﬀerent measures). We adopt the generalized Neyman-Pearson optimality criterion, which focuses on the decay rates of the type I and II error probabilities under ﬁxed non-local alternatives, and derive an optimal but practically infeasible test. Then, as an illustration, by considering the model comparison hypothesis deﬁned by the weighted Euclidean norm of moment restrictions, we propose a feasible approximate test statistic to the optimal one and study its asymptotic properties. Local power properties, one-sided test, and comparison under the generalized empirical likelihood-based measure of ﬁt are also investigated. A simulation study illustrates that our approximate test is more powerful than the Rivers-Vuong test.
We present a simple way to estimate the eﬀects of changes in a vector of observable variables X on a limited dependent variable Y when Y is a general nonseparable function of X and unobservables, and X is independent of the unobservables. We treat models in which Y is censored from above, below, or both. The basic idea is to ﬁrst estimate the derivative of the conditional mean of Y given X at x with respect to x on the uncensored sample without correcting for the eﬀect of x on the censored population. We then correct the derivative for the eﬀects of the selection bias. We discuss nonparametric and semiparametric estimators for the derivative. We also discuss the cases of discrete regressors and of endogenous regressors in both cross section and panel data contexts.