A Test of the Conditional Independence Assumption In Sample Selection Models

Abstract

Identification in most sample selection models depends on the independence of the regressors and the error terms conditional on the selection probability. All quantile and mean functions are parallel in these models; this implies that quantile estimators cannot reveal any-per assumption non-existing-heterogeneity. Quantile estimators are nevertheless useful for testing the conditional independence assumption because they are consistent under the null hypothesis. We propose tests of the Kolmogorov-Smirnov type based on the conditional quantile regression process. Monte Carlo simulations show that their size is satisfactory and their power sufficient to detect deviations under plausible data-generating processes. We apply our procedures to female wage data from the 2011 Current Population Survey and show that homogeneity is clearly rejected. assumption is an identifying assumption; its violation leads to the inconsistency of most sample selection estimators (including those only concerned with mean effects). To the best of our knowledge, ours is the first test of this identifying assumption.To implement this testing idea we use the sample selection correction for quantile regression proposed by Buchinsky (1998aBuchinsky ( , 2001. These papers extended the series estimator of Newey (2009) to the estimation of quantiles. The estimator has been applied to many data-among others in both original articles-to analyze the heterogeneous effects of the explanatory variables on the distribution of the outcome. Another contribution of this paper is to show that this estimator is consistent only in two cases: when all quantile regression slopes are equal or when selection is random. The reason is that Buchinsky (1998a) assumes that the error terms are independent of the regressors given the selection probability. This implies that all quantile slope coefficients and the mean coefficients are identical; i.e. it excludes heterogeneous effects even though their analysis has been the main motivation for using quantile regression in recent applications.The Buchinsky estimator nevertheless remains useful for two reasons, which were the initial motivations for quantile regression: it is robust and can be used as the basis for a test of independence. Koenker and Bassett (1978) also assume independence in their seminal paper and motivate quantile regression purely from a robustness and efficiency point of view in the presence of non-Gaussian errors. Similarly, the estimator of Buchinsky has a bounded influence function (in the direction of the dependent variable), so that it is more robust than mean regression. Under the null hypothesis of conditional independence the slope coefficients are constant as a function of the quantile, and the procedure proposed by Buchinsky (1998a) consistently estimates them. Under the alternative hypothesis, the estimates do not converge to the true values but will be a non-trivial function of the quantile. These two properties justify testing the conditional independence assumption by testing whether th...

Quantile Selection Models With an Application to Understanding Changes in Wage Inequality

“…In contrast, in our framework, conditional quantiles of participants are non-additive. Huber and Melly (2015) made a related point in a testing context. Correcting for sample selection thus requires shifting the percentile ranks of individual observations.…”

“…Buchinsky (1998Buchinsky ( , 2001 proposed an additive approach to correct for sample selection in quantile regression. Huber and Melly (2015) considered a more general, non-additive quantile model, as we do; they focused on testing for additivity. In contrast, our focus is on providing a practical estimation method.…”

Robust Inference in Sample Selection Models

“…Moreover, when an exclusion restriction is available, we add the quantile regression estimator. This is the estimator that was proposed by Buchinsky (1998) and extended by Huber and Melly (2015), which is a combination of a semiparametric binary regression as in Klein and Spady (1993) in the first stage and quantile regression in the second stage. It is computed by using the code kindly provided to us by M. Huber.…”

“…In recent decades, robust estimators and tests have been developed for large classes of models in both the statistical and the econometric literature; see for instance, Huber (1981), Ronchetti (2009), Hampel et al (1986) and Maronna et al (2006) in the statistical literature and Peracchi (1990Peracchi ( , 1991 and Ronchetti and Trojani (2001) in the econometric literature. In particular, the quantile regression approach (Koenker, 2005) has proved fruitful as a specific way to robustify classical procedures and, in the framework of sample selection models, it has been proposed by Buchinsky (1998) and Huber and Melly (2015). Details about this approach are provided in Section 4.…”

Accounting for the permanent vs temporary wage gaps among young adults: Three European countries in perspective

“…The alternative approach taken by Buchinsky (2001), also developed for quantile regression, does not really correct such bias when the effect of covariates is quantile-specific. In fact, as shown in Huber and Melly (2015), based on an argument by Angrist (1997), "this estimator is consistent only ... when all quantile regression slopes are equal or when selection is random" (Huber andMelly, 2015, p. 1145). No panel data were used in our study and no convincing instruments were available that were comparable to the randomly assigned treatment group in the Job Training Partnership Act (JTPA) programme 6 used by Abadie, Angrist and Imbens (2002) or the staggered and region-specific implementation of FTC reforms in Italy used by Bosio (2014).…”