many thick-tailed error-term distributions. Other robust estimators include the M and the L-estimators
(Bickel, 1976; Mosteller and Tukey, 1977; Bierens, 1981; and Dielmam and Pfaffenberger, 1982; Joiner
and Hall, 1983). Fully adaptive estimators involve non-parametric estimates of the derivative of the log of
the unknown density. They have the advantage of being asymptotically efficient, as the maximum likelihood
estimator obtained using knowledge of the actual error term distribution. Given that the correct error term
distribution is seldom known, fully adaptive estimation should be preferred when working with large
samples, since it would produce the lowest possible standard errors. Examples of fully adaptive estimators
include Hsieh and Manski’s Adaptive Maximum Likelihood (AML) estimator, which is based on a normal
kernel density; and Newey’s generalized method of moments estimator. For more details about these
techniques please see McDonald and White. The more favorable asymptotic properties of fully adaptive
estimators, however, are meaningless when working with small samples.
Partially adaptive estimators are ML estimators based on specific families of error term
distributions, in hopes that the assumed family is flexible enough to accommodate the shape of the true
unknown distribution of the error. Partially adaptive estimators based on the t distribution (Prucha and
Kelejian), the generalized t (McDonald and Newey), and the generalized beta distribution (McDonald and
White) have been explored in the econometrics literature. Partially adaptive estimators are asymptotically
efficient only if the true error term distribution belongs to the family of the assumed distribution.
However, McDonald and White show that partially adaptive estimators based on assuming a
flexible family of densities that can accommodate a wide variety of distributional shapes can substantially
outperform OLS and all available robust and fully adaptive estimators in small sample applications
involving non-normal errors. In other words, when estimating a multiple regression model with a
continuous but non-normally distributed dependent variable and a small sample size, a partially adaptive
estimator based on a flexible distribution that can reasonably approximate the true underlying error term
distribution would likely produce slope parameter estimators with the lowest possible standard errors,
potentially much lower than OLS.