José António Machado

Goodness of Fit and Related Inference Processes for Quantile Regression

Abstract We introduce a goodness-of-fit process for quantile regression analogous to the conventional R2 statistic of least squares regression. Several related inference processes designed to test composite hypotheses about the combined effect of several covariates over an entire range of conditional quantile functions are also formulated. The asymptotic behavior of the inference processes is shown to be closely related to earlier p-sample goodness-of-fit theory involving Bessel processes. The approach is illustrated with some hypothetical examples, an application to recent empirical models of international economic growth, and some Monte Carlo evidence.

Quantiles for counts

This paper studies the estimation of conditional quantiles of counts. Given the discreteness of the data, some smoothness has to be artificially imposed on the problem. The methods currently available to estimate quantiles of count data either assume that the counts result from the discretization of a continuous process, or are based on a smoothed objective function. However, these methods have several drawbacks. We show that it is possible to smooth the data in a way that allows inference to be performed using standard quantile regression techniques. The performance and implementation of the estimator are illustrated by simulations and an application.

Robust Model Selection and M -Estimation

This paper studies the qualitative robustness properties of the Schwarz information criterion (SIC) based on objective functions defining M -estimators. A definition of qualitative robustness appropriate for model selection is provided and it is shown that the crucial restriction needed to achieve robustness in model selection is the uniform boundedness of the objective function. In the process, the asymptotic performance of the SIC for general M -estimators is also studied. The paper concludes with a Monte Carlo study of the finite sample behavior of the SIC for different specifications of the sample objective function.

Box-Cox quantile regression and the distribution of firm sizes

Using the Box-Cox quantile regression model, we analyse the size distribution of firms in Portuguese manufacturing during the 1980s. Specifically, we estimate the effect of selected industry attributes on the location, scale, skewness and kurtosis of the conditional size distributions of firms. We find that industry attributes affect the size of firms in the same direction across the distribution, but the effects of these variables are typically much greater at the largest quantiles. Over time the distribution shifted towards smaller firms, due mainly to the way the economy responds to industry characteristics rather than to changes of the level of these characteristics. The prediction of lognormality, implied by Gibrats Law, is soundly rejected by the observed distribution of firm sizes. However, we found that, at least in 1983, lognormality is a reasonable description of the conditional size distribution. Copyright

GMM inference when the number of moment conditions is large

Abstract Asymptotic theory typically presumes that the dimensionality of econometric models is independent of the sample size even though this presumption is often quite unrealistic. In GMM estimation, whenever optimal instruments are not available, it can frequently be shown that adding over-identifying restrictions (moment conditions) will increase asymptotic precision. However, the conventional asymptotics which underlies this view insists that the number of moment conditions remain finite even though the number of available moment conditions may grow without bound. We consider the explicit dependence of the number of moment conditions (or instruments), q n , on the sample size, n , and establish that, under conventional regularity conditions for the estimation of a linear model with general heteroskedasticity, q n 3 / n →0 is a sufficient condition for the validity of conventional asymptotic inference about the GMM estimator.

Economic Applications of Quantile Regression

Individual heterogeneity in the returns to schooling: instrumental variables quantile regression using twins data.- Testing for uniform wage trends in West-Germany: A cohort analysis using quantile regressions for censored data.- Quantile regression with sample selection: Estimating womens return to education in the U.S..- Earning functions in Portugal 1982-1994: Evidence from quantile regressions.- Wage inequality in a developing country: decrease in minimum wage or increase in education returns.- How wide is the gap? An investigation of gender wage differences using quantile regression.- The public-private sector wage gap in Zambia in the 1990s: A quantile regression approach.- Asymmetric labor supply.- Quantile regression for duration data: A reappraisal of the Pennsylvania Reemployment Bonus Experiments.- For whom the reductions count: A quantile regression analysis of class size and peer effects on scholastic achievement.- The effects of demographics and maternal behavior on the distribution of birth outcomes.- Nonparametric quantile regression analysis of R & D-sales relationship for Korean firms.- Conditional value-at-risk: Aspects of modeling and estimation.- Portfolio style: Return-based attribution using quantile regression.- Integrated Conditional Moment testing of quantile regression models.

Momentary Lapses: Moment Expansions and the Robustness of Minimum Distance Estimation

This paper explores the robustness of minimum distance (GMM) estimators focusing particularly on the effect of intermediate covariance matrix estimation on final estimator performance. Asymptotic expansions to order O ( n −3/2 ) are employed to construct O ( n −2 ) expansions for the variance of estimators constructed from preliminary least-squares and general M -estimators. In the former case, there is a rather curious robustifying effect due to estimation of the Eicker-White covariance matrix for error distributions with sufficiently large kurtosis.

Exploring Transition Data through Quantile Regression Methods: An Application to U.S. Unemployment Duration

Quantile regression constitutes a natural and flexible framework for the analysis of duration data in general and unemployment duration in particular. Comparison of the quantile regressions for lower and upper tails of the duration distribution shed important insights on the different determinants of short or long-term unemployment. Using quantile regression techniques, we estimate conditional quantile functions of US unemployment duration; then, resampling the estimated conditional quantile process we are able to infer the implied hazard functions. From the economic standpoint, one of the most interesting conclusions pertains the role of “advanced notice of firing”, which was found to impact short durations — low quantiles — but not relatively long durations. On a more general note, the proposed methodology proves to be resilient to several misspecification that typically afflict proportional hazard models such as, neglected heterogeneity and baseline misspecification. Overall, the results provide clear indications of the interest of quantile regression to the analysis of duration data.

The Reservation Wage Unemployment Duration Nexus

A thorny problem in identifying the determinants of reservation wages and particularly the role of continued joblessness in their evolution is the simultaneity issue. We deploy a control function approach to the problem that involves conditioning elapsed duration on completed unemployment duration in the reservation wage equation. Our analysis confirms that the use of elapsed duration alone compounds two separate and opposing influences. Only with the inclusion of completed duration is the negative effect of continued joblessness on reservation wages apparent.

A NOTE ON IDENTIFICATION WITH AVERAGED DATA

In most cases where estimation with averaged data is performed, interest lies on the parameters of a model at the individual level, but grouped data are used because disaggregate data are not observed. In this note we study the conditions under which it is possible to consistently estimate the parameters of the individual data model using averaged data, giving particular attention to the case of endogenous selection into groups.We are grateful to an anonymous referee for helpful comments and suggestions. We also thank Pedro Duarte Neves, Les Godfrey, Paulo Parente, Maximiano Pinheiro, and Pedro Portugal for helpful discussions. The usual disclaimer applies. Machado is consultant for the Research Department of Banco de Portugal. Santos Silva is grateful for the hospitality, working conditions, and financial support provided by Banco de Portugal, which made this work possible. The authors also gratefully acknowledge partial financial support from FundaA§A£o para a CiAancia e Tecnologia, program POCTI, partially funded by FEDER.