William S. Krasker

Efficient Bounded-Influence Regression Estimation

Abstract The least squares estimator for β in the classical linear regression model is strongly efficient under certain conditions. However, in the presence of heavy-tailed errors and/or anomalous data, the least squares efficiency can be markedly reduced. In this article we propose an estimator that limits the influence of any small subset of the data and show that it satisfies a first-order condition for strong efficiency subject to the constraint. We then show that the estimator is asymptotically normal. The article concludes with an outline of an algorithm for computing a bounded-influence regression estimator and with an example comparing least squares, robust regression as developed by Huber, and the estimator proposed in this article.

The ‘peso problem’ in testing the efficiency of forward exchange markets

Abstract In this paper we argue that when there is small probability of an event which would cause a large change in an exchange rate, the standard tests for the efficiency of the corresponding forward exchange market are not always valid. The mark pound forward market during the German hyperinflation is cited as an example. Using data from that hyperinflation, we show that an alternative test can sometimes be constructed in cases where the usual tests are not valid. The results reverse the conclusion of earlier researchers that the mark pound forward market during the hyperinflation was not efficient.

Estimation for dirty data and flawed models

Publisher Summary This chapter focuses on resistant estimation procedures and methods for evaluating the impact of particular data elements on regression estimates. Model builders using macroeconomic time series are often plagued by occasional unusual events, leading them to decrease the weights to be attached to these data in the spirit of resistant estimation. Even when there are good data and theory that correspond reasonably well to the process being modeled, there are episodic model failures. The chapter discusses some model failures that can arise in practice. It describes recent developments in methods for the detection of influential data in regression and discusses several issues about inference in the resistant case and the main theoretical foundations of robust and bounded-influence (BIF) estimation. The chapter presents an example of BIF applied to the Harrison–Rubinfeld large cross-section hedonic price index. The chapter also presents some recent results on instrumental-variables bounded-influence estimation, and discusses resistant estimation for time-series models.

Resistant estimation for simultaneous-equations models using weighted instrumental variables

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 estimators 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

Two-Stage Bounded-Influence Estimators for Simultaneous-Equations Models

This article presents a class of estimators for linear structural models that are robust to heavytailed disturbance distributions, gross errors in either the endogenous or exogenous variables, and certain other model failures. The class of estimators modifies ordinary two-stage least squares by replacing each least squares regression by a bounded-influence regression. Conditions under which the estimators are qualitatively robust, consistent, and asymptotically normal are established, and an empirical example is presented.

Bounding the effects of proxy variables on instrumental-variables coefficients

Abstract We consider instrumental-variables models in which one of the observed variables is a proxy for some unobserved ‘true’ variable. We seek the smallest correlation between the proxy and the unobserved true variable sufficient to guarantee that, regardless of any other correlations involving unobserved variables or disturbances, a coefficients sign has not been reversed by substituting the proxy. We derive exact solutions when the proxy is the dependent variable, the endogenous explanatory variable whose coefficient is of concern, some other endogenous explanatory variable, or a solely instrumental variable. For an exogenous explanatory proxy, we derive either exact solutions or useful lower bounds.

Heterogeneous expectations and capital intensity in an overlapping-generations model

Abstract In this paper we examine the macroeconomic effects of heterogeneous probability beliefs in the context of a modified Diamond-type overlapping-generations model. We derive the effect of an increase in the degree of heterogeneity on the steady-state savings rate, the capital intensity, and welfare.