Paul Rilstone

The second-order bias and mean squared error of nonlinear estimators

Abstract Despite the now widespread use of nonlinear estimators, their finite-sample properties have received very little attention in either the statistics or econometrics literature. We partially redress this problem by deriving and examining the second-order bias and mean squared error of a fairly wide class of nonlinear estimators which includes Nonlinear Least Squares, Maximum Likelihood, and many Generalized Method of Moments estimators as special cases. A number of examples are provided. The results from a Monte Carlo exercise demonstrate how the results can be applied for improved inferences.

Semiparametric estimation of count regression models1

This paper develops a semiparametric estimation approach for mixed count regression models based on series expansion for the unknown density of the unobserved heterogeneity. We use the generalized Laguerre series expansion around a gamma baseline density to model unobserved heterogeneity in a Poisson mixture model. We establish the consistency of the estimator and present a computational strategy to implement the proposed estimation techniques in the standard count model as well as in truncated, censored, and zero-inflated count regression models. Monte Carlo evidence shows that the finite sample behavior of the estimator is quite good. The paper applies the method to a model of individual shopping behavior.

Using Bootstrapped Confidence Intervals for Improved Inferences with Seemingly Unrelated Regression Equations

The usual standard errors for the regression coefficients in a seemingly unrelated regression model have a substantial downward bias. Bootstrapping the standard errors does not seem to improve inferences. In this paper, Monte Carlo evidence is reported which indicates that bootstrapping can result in substantially better inferences when applied to t -ratios rather than to standard errors.

Environmental Inspections and Emissions of the Pulp and Paper Industry: The Case of Quebec

Since the early 1970s, industrial countries have enacted (or amended) many environmental laws and regulations to control and improve air and water quality. Developing countries are increasingly enacting similar legislation. But imposing a ceiling on a plants emissions does not guarantee reduced emissions or an improved environment. Ensuring the attainment of the regulations objectives requires monitoring the behavior of the regulated facility and enforcing environmental standards. Most of the literature in environmental economics is theoretical and simply assumes that polluters comply with regulations. Although monitoring and enforcement problems are clearly a pitfall of environmental regulation, little empirical work has been done about the effect of current monitoring strategies on pollution emissions. The authors supply an empirical framework for measuring the impact of environmental inspections on plant emissions. They apply it to pulp and paper plants in Quebec for which reliable data were available. The results suggest that both inspection and the threat of inspections reduce pollution emissions. They also show that a plants decision whether to report its emissions levels to the regulator is not random. Inspections improve the frequency of reporting.

The Third-Order Bias of Nonlinear Estimators

The third-order bias of nonlinear estimators is derived and illustrated using a variety of estimators popular in applied econometrics. A simulation using the exponential regression model indicates that the third-order analytical correction reduces bias substantially compared to higher-order bootstrap and Jackknife corrections, particularly in very small samples.

EFFICIENT SEMIPARAMETRIC ESTIMATION OF DURATION MODELS WITH UNOBSERVED HETEROGENEITY

This paper develops a new semiparametric approach for the estimation of hazard functions in the presence of unobserved heterogeneity. The hazard function is specified parametrically, whereas the distribution of the unobserved heterogeneity is indirectly estimated using the method of kernels. The semiparametric efficiency bounds are derived. The estimator obtains these bounds in large samples.The authors thank Yongmiao Chen, James Heckman, Hidehiko Ichimura, Tony Lancaster, Qi Li, Adrian Pagan, Barry Smith, two anonymous referees, and the co-editor for helpful input. We particularly thank Steven Stern, who prompted us toward this line of research. Any errors are those of the authors. Research funding for Rilstone was provided by the Social Sciences and Humanities Research Council of Canada.

EDGEWORTH AND SADDLEPOINT EXPANSIONS FOR NONLINEAR ESTIMATORS

Simple methods are developed for deriving Edgeworth, saddlepoint, and related expansions for the estimators of multivariate and nonlinear models. Illustrations are provided. Simulations are reported indicating the methods work well compared to standard asymptotic and bootstrapped approaches.

Edgeworth expansions for GEL estimators

Finite sample approximations for the distribution functions of Generalized Empirical Likelihood (GEL) are derived using Edgeworth expansions. The analytical results obtained are shown to apply to most of the common extremum estimators used in applied work in an i.i.d. sampling context. The GEL estimators considered include the Continuous Updating, Empirical Likelihood and Exponential Tilting estimators. These estimators are popular alternatives to Generalized Method of Moment (GMM) estimators and their finite sample properties are examined. In a Monte Carlo Experiment, higher order analytical corrections provided by Edgeworth approximations work well in comparison to first order approximations and improve inferences in finite samples.

NONPARAMETRIC IDENTIFICATION OF LATENT COMPETING RISKS MODELS

This paper shows that nonparametric identification of latent competing risks models is possible without the usual conditional independence and exclusion restrictions.The authors thank Jinyong Hahn, James Heckman, and Shinichi Sakata for useful discussions on the subject matter of this paper. Any errors are the fault of the authors. Research funding for Rilstone was provided by the Social Sciences and Humanities Research Council of Canada.

Saddlepoint expansions for GEL estimators

A simple saddlepoint (SP) approximation for the distribution of generalized empirical likelihood (GEL) estimators is derived. Simulations compare the performance of the SP and other methods such as the Edgeworth and the bootstrap for special cases of GEL: continuous updating, empirical likelihood and exponential tilting estimators.