ABSTRACT
The large sample theory is arguably the most popular framework for conducting statistical inference due to its virtually universal applicability and implementational simplicity. There are situations, however, in which the conventional asymptotic methods do not provide satisfactory answers to some important questions. First, a consistent and asymptotically normal estimator may have a signi…cant bias even in very large samples. Second, two asymptotically equivalent estimators may possess very di¤erent statistical properties in …nite samples. Third, an asymptotically standard normal t-statistic may exhibit severe over-or under-rejection, and ignoring this may result in poor inference. These examples show that the standard asymptotic tools are often insu¢ cient for an adequate econometric analysis. While the exact distribution theory is designed to provide help in these sit-
uations, it is valid under some very strong assumptions on the data generating process and is often not available for more general models. An alternative way to address such issues is to go beyond standard (“…rst-order”) asymptotics by using the so-called higher-order asymptotic analysis. This approach has the ability, albeit imperfect, to answer some interesting questions related to the …nite sample moments of estimators and sampling distributions of test statistics. This chapter focuses mainly on two types of asymptotic expansions: stochastic expansions of estimators and Gram–Charlier, Edgeworth and saddlepoint expansions of sampling distributions of test statistics. While the stochastic expansions allow the researcher to analyze …nite sample moments (such as bias and mean squared error) and compare asymptotically equivalent estimators, the Edgeworth-type approximations are useful in capturing certain characteristics of the distributions of test statistics that are not re‡ected in the …rst-order asymptotics. Tools for improved higher-order asymptotic inference have a long history in
statistics but the recent development of new estimators and inference methods in econometrics has led to a renewed interest in higher-order asymptotic analysis for comparing and improving the …nite sample properties of di¤erent
estimators and test statistics. The higher-order expansions also provide theoretical justi…cation for some popular resampling techniques (such as bootstrap and jackknife) and are predominantly used for establishing the higher-order re…nements of the bootstrap.
