Resistant Estimation for Simultaneous-Equations Models Using Weighted Instrumental Variables - Krasker, William S.; Welsch, Roy E. (Econometrica, 1985)
IN THIS PAPER we present a weighted-instrumental-variables estimator that is resistant2 to heavy-tailed errors, aberrant data in either the endogenous or exogenous variables, and certain other model failures. The estimator is analogous to the weighted-least-squares approach to robustness proposed by Krasker and Welsch [21] for ordinary regression. We will discuss the theory that motivated this estimator, derive some of its properties, describe our computational algorithm, and, using an empirical example, illustrate the estimator's utility as a tool for data analysis in structural models. The evolution of robust estimators for simultaneous-equations models has closely resembled the development of robust methods for ordinary regression. In both cases, research focused at first on the well-documented effects of long-tailed error distributions on the classical procedures. For ordinary regression models, statisticians have studied the properties of least-absolute-deviations (LAD) estimators (see Bassett and Koenker [2] and Amemiya [1]) and maximum-likelihood-type estimators (called M-estimators; see Huber [14]). For simultaneous-equations models, LAD can be generalized in several different ways to modify two-stage least squares or instrumental variables; Amemiya [1] has presented these estimators in a unified framework, and Powell [31] has proven their asymptotic normality under weak assumptions. Fair [8] has compared two-stage LAD estimates of a U.S. macroeconomic model with two-stage least squares and full-information maximum likelihood estimates. The M-estimator concept can also be generalized to simultaneous equations. For example, Prucha and Kelejian [32] have considered maximum-likelihood estimation under the assumption that the disturbance vector is multivariate student t. Although LAD and M-estimators maintain a high efficiency when the error distribution is heavy-tailed, they are not robust in the stronger sense of Hampel [10] which, roughly speaking, requires an estimator to have a limited sensitivity to any small fraction of the data.3 LAD and M-estimators fail under this criterion because a single observation whose values for the right-side variables are highly anomalous can have an arbitrarily large effect on LAD estimates or M-estimates, just as on the classical procedures. Consequently, these estimators do not provide insurance against the sort of gross errors that occur in some data sets; nor do they serve as reliable diagnostics for departures from linearity occurring in extreme regions of the space of right-side variables. For linear models, the estimators that satisfy the strong Hampel robustness criterion have become known as bounded-influence estimators. Several such estimators have been