A. Colin Cameron 741(see McCulloch, Polson and Rossi, 2000). The data augmentation method then
generates draws of unobserved values of the latent variables that can then be
treated as observed values, greatly simplifying analysis. A recent application is
that by Geweke, Gowrisankaran and Town (2003), who model mortality at 114
Los Angeles County hospitals allowing for the complication that better hospitals
may attract more difficult cases, with difficulty depending in part on unobserv-
ables correlated with hospital mortality rates. Recent econometrics books are Koop
(2003), Lancaster (2004), and Koop, Poirier and Tobias (2007).
14.3.5 Quantile regression
Quantiles, such as deciles and quartiles, are often used to summarize the distribu-
tion of variables such as income, earnings and wealth. Quantile regression is an
extension to the regression case where, for example, interest may lie in the different
response of earnings to education at different points of the conditional earnings
distribution.
A leading example is the least absolute deviations (LAD) estimator that minimizes
the sum of absolute residuals
∑N
i= 1 |yi−x′
iβ|. This is a generalization of the median
in the independent and identically distributed (i.i.d.) case, since withx′iβ=βthe
resulting estimate ofβis the sample median.
More generally, conditional quantiles other than the median may be estimated.
Theqth quantile regression estimator̂βqminimizes overβq:
QN(βq)=∑Ni:yi≥x′iβq|yi−xi′βq|+∑Ni:yi<x′iβ( 1 −q)|yi−x′iβq|,where the subscriptqinβqis needed as different choices ofqestimate different
values ofβ. The special caseq=0.5 is the LAD estimator. The objective func-
tion is not differentiable, so linear programming methods are used rather than
more familiar gradient methods. These enable relatively fast computation of̂βq.
The quantile regression estimator is consistent and asymptotically normal. Esti-
mation of the analytical asymptotic variance of̂βqrequires estimation offuq( 0 |x),
the conditional density of the error termuq=y−x′βqevaluated atuq=0. An
easier method is to instead obtain bootstrap standard errors for̂βqusing a paired
bootstrap.
Quantile regression was proposed by Koenker and Bassett (1978). Powell (1984,
1986) adapted the method to permit consistent estimation in censored linear
regression models. With censoring the conditional median can be recovered with-
out the strong distributional assumptions, such as normality, needed to recover the
conditional mean. Buchinsky (1994) provided a much-cited application that doc-
umented recent US changes in the quantiles of the conditional wage distribution.
Such analysis is now easily implemented as Stata includes a quantile regression
command. Koenker and Hallock (2001) provide an early summary of applications.
Existing results specify all quantile regression functions to be linear. Angrist,
Chernozhukov and Ferandez-Val (2006) provide interpretation of the quantile