Chapter 15 Approximating the distributions of econometric estimators and test statistics

Approximate distribution theory derives results from assumptions on the stochastic process generating the data. The quality of the approximation is not better than the quality of the specifications on which it is based. The models used by econometricians are, at best, crude and rather arbitrary. As most of the approximation methods employ information on the first four moments of the data whereas the usual asymptotic theory typically requires information only on the first two moments, some loss in robustness must be expected. However, if a rough idea about the degree of skewness and kurtosis is available, that information can be often exploited to obtain considerably improved approximations to sample statistics. The chapter discusses that sophisticated approximation theory is most appropriate in situations where the econometrician is able to make correct and detailed assumptions about the process being studied. In current practice, applied econometricians occasionally draw incorrect conclusions on the basis of alleged asymptotic properties of their procedures. In recent years, an extraordinary fondness for asymptotic theory has developed among econometricians. Considerable effort is devoted to showing that some new estimator or test is asymptotically normal and efficient. The assertion that a given estimator is approximately normal suggests that the speaker believes that it would be sensible to treat the estimator as though it were really normal. Accurate and convenient approximations for the distributions of econometric estimators and test statistics are of great value.