Powell’s Least Absolute Deviations Estimator
Powell (1981, 1983) proposed the least absolute deviations (LAD) estimator (see Section 4.6) for censored and truncated regression models, proved its consistency under general distributions, and derived its asymptotic distribution. The intuitive appeal for the LAD estimator in a censored regression model arises from the simple fact that in the i. i.d. sample case the median (of which the LAD estimator is a generalization) is not affected by censoring (more strictly, left censoring below the mean), whereas the mean is. In a censored regression model the LAD estimator is defined as that which minimizes 2"_1|yl — max (0, x’tP)|. The motivation for the LAD estimator in a truncated regression model is less obvious. Powell defined the LAD estimator in the truncated case as that which minimizes SjLJy,- — max (2~* , х,’Д)|. In the censored case the limit distribution of ‘fn(fi-fi), where fi is the LAD estimator, is normal with zero mean and variance-covariance matrix [4/(0)2 lim,,^ n-l2jLi*(x|/I> 0)xIx|]_1, where/is the density of the errorterm and/ is the indicator function taking on unity if xj/f > 0 holds and 0 otherwise. In the truncated case the limit distribution of — is normal with zero mean and variance-covariance matrix 2-1A-1BA_l, where
A = lim n~l £ *(x’/?>0)[/(0)
В = lim n~l Y XWP > O)[F(x;/0 – ДОІДх’ДГЧх’,
where F is the distribution function of the error term.
Powell’s estimator is attractive because it is the only known estimator that is consistent under general nonnormal distributions. However, its main drawback is its computational difficulty. Paarsch (1984) conducted a Monte Carlo study to compare Powell’s estimator, the Tobit MLE, and Heckman’s two – step estimator in the standard Tobit model with one exogenous variable under situations where the error term is distributed as normal, exponential, and Cauchy. Paarsch found that when the sample size is small (SO) and there is much censoring (50% of the sample), the minimum frequently occurred at the boundary of a wide region over which a grid search was performed. In laige samples Powell’s estimator appears to perform much better than Heckman’s estimator under any of the three distributional assumptions and much better than the Tobit MLE when the errors are Cauchy.
Another problem with Powell’s estimator is finding a good estimator of the asymptotic variance-covariance matrix that does not require the knowledge of the true distribution of the error. Powell (1983) proposed a consistent estimator.
Powell observed that his proof of the consistency and asymptotic normality of the LAD estimator generally holds even if the errors are heteroscedastic. This fact makes Powell’s estimator even more attractive because the usual estimators are inconsistent under heteroscedastic errors, as noted earlier.
Another obvious way to handle nonnormality is to specify a nonnormal distribution for the щ in (10.2.3) and use the MLE. See Amemiya and Boskin (1974), who used a lognormal distribution with upper truncation to analyze the duration of welfare dependency.