Bayesian Unconditional Quantile Regression: An Analysis of Recent Expansions in Wage Structure and Earnings Inequality in the US 1992–2009
| Author | Abdoul Aziz Junior Ndoye,Michel Lubrano |
| Date | 01 May 2014 |
| DOI | http://doi.org/10.1111/sjpe.12038 |
| Published date | 01 May 2014 |
BAYESIAN UNCONDITIONAL
QUANTILE REGRESSION: AN
ANALYSIS OF RECENT EXPANSIONS
IN WAGE STRUCTURE AND EARNINGS
INEQUALITY IN THE US 1992–2009
Michel Lubrano* and Abdoul Aziz Junior Ndoye**
ABSTRACT
We develop Bayesian inference for an unconditional quantile regression model.
Our approach provides better estimates in the upper tail of the wage distribution
as well as valid small sample confidence intervals for the Oaxaca–Blinder decom-
position. We analyze the recent changes in the US wage structure using data
from the CPS Outgoing Rotation Group from 1992 to 2009. We find that the
largest part of the recent changes is explained mainly by differences in returns
to education while the decline in the unionization rate has a small impact, and
that earnings inequality is rising more at the top end of the wage distribution.
II
NTRODUCTION
Introduced by Koenker and Bassett (1978), quantile regression models aim at
modeling the effect of the explanatory variables on the conditional distribu-
tion of the outcome variable. Quantile regression have been increasingly used
in empirical labor market studies, to describe parsimoniously the entire wage
conditional distribution (see e.g., Buchinsky, 1994; Chamberlain, 1994; Mach-
ado and Mata, 2001). Several competing methods of estimation in both classi-
cal and Bayesian frameworks have been recently developed (see for instance
Yu and Moyeed, 2001; Kozumi and Kobayashi, 2011, or Kottas and Krnjajic,
2009 for the Bayesian side with a semiparametric approach for the last refer-
ence). As any quantile can be used in any part of the outcome distribution,
the quantile regression models are more flexible and more robust to outliers
than the classical mean regression models.
While the conditional quantile regression models can be useful, they are
very restrictive. First, a change in the distribution of covariates may change
the interpretation of the coefficient estimates. This point is illustrated for
instance in Powell (2011). To overcome this restriction, Firpo et al. (2009)
*Aix-Marseille University (AMSE) and GREQAM-CNRS
**Aix-Marseille University (AMSE) and GREQAM
Scottish Journal of Political Economy, DOI: 10.1111/sjpe.12038, Vol. 61, No. 2, May 2014
©2014 Scottish Economic Society.
129
have proposed a new regression method which evaluates the impact of
changes in the distribution of the explanatory variables on the quantiles of
the unconditional distribution of the outcome variable. Second, the property
that, in the popular Oaxaca–Blinder decomposition method of a simple linear
regression, differences in unconditional means are equal to differences between
conditional means is no longer valid for conditional quantile regressions. As
explained in, e.g., Firpo et al. (2011), with conditional quantile regressions,
the difference in unconditional quantiles is not equal to difference in condi-
tional quantiles. This question has received several answers in the literature,
see e.g., Juhn, Murphy and Pierce (1993), DiNardo et al. (1996), or Machado
and Mata (2005), but none of these methods can be used to decompose gen-
eral distributional measures in the same way that the means can be decom-
posed using the conventional Oaxaca–Blinder method. However, the method
of Melly (2005) and the recentered influence function method of Firpo et al.
(2009) (RIF regression) can perform a detailed decomposition very much in
the spirit of the traditional Oaxaca decomposition for the mean (Firpo et al.,
2011).
In this study, we develop a Bayesian inference method for the RIF regres-
sion model of Firpo et al. (2009) in which we estimate the log wage distribu-
tion by a mixture of normal densities. The mixture of normal densities is
pursued so as to produce a better fit in the tails of the wage distribution
which are essential to have a precise evaluation of higher quantiles. As docu-
mented in Bahadur and Savage (1956), in the presence of a heavy tail distribu-
tion, a nonparametric approach using kernel smoothing can lead to unreliable
inference. As a consequence, the presence of a heavy right-hand tail in the
wage distribution can make less reliable the usual density kernel estimate used
in the RIF-OLS method of Firpo et al. (2009). A semiparametric approach
using mixtures of distributions would provide better estimates of the RIF
regression coefficients for the upper quantiles. Finally, a Bayesian approach
takes a better account of parameter uncertainty of the density estimation in
the first stage of estimation and is pursued so as to propose valid confidence
intervals for the Oaxaca–Blinder decomposition.
We illustrate our approach, analyzing the recent trends in US wage struc-
ture and earnings inequality. The recent rise in earnings dispersion in United
States is remarkable. The literature dealing with the causes of this wage dis-
persion has exponentially increased over the past decades. Several competing
explanations have been offered. Bound and Johnson (1992) attribute the
changes to the skill-biased technological progress which increases the rate of
growth of the relative demand for highly educated and ‘more-skilled’ workers
(see also Mincer, 1993;Katz and Autor, 1999). Murphy and Welch (1992)
stress the impact of the globalization which increases the rate of unskilled
immigration workers and led to a decrease in the growth of the relative supply
of skills (see also Katz and Murphy, 1992). DiNardo et al. (1996) focus on
changes in labor market institutions, in wage setting norms including the
decline in unionization, on the erosion of the real and relative value of the
minimum wage.
130 MICHEL LUBRANO AND ABDOUL AZIZ JUNIOR NDOYE
Scottish Journal of Political Economy
©2014 Scottish Economic Society
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