Palgrave Handbook of Econometrics: Applied Econometrics

David T. Jacho-Chávez and Pravin K. Trivedi 803

of the relationship between outcomeyand regressorsxat different points in the
distribution ofy.
As stated in section 15.3.2, the objective function for QR is in theL 1 -norm.
Unlike the squared error loss function of the OLS framework, in the case of QR the
loss is expressed through an asymmetric absolute loss function. The special case of
median (LAD) regression corresponds toq=0.5.
QRs have a certain robustness property and they also permit more informative
modeling of the data. Specifically, the median regression is more robust to outliers
compared with the mean regression. QR facilitates a richer interpretation of the
data because it permits the study of the impact of regressors on both the location
and scale parameters of the model, while avoiding parametric assumptions about
data.
Other attractive properties of QR are as follows: (i) it is consistent under weaker
stochastic assumptions than least squares estimation; (ii) it is based on weaker dis-
tributional assumptions; (iii) it has an equivariance to monotone transformations
property, which implies that it does not run into the retransformation problem.
The quantiles of a transformed variabley, denotedh(y), wherehis a monotonic
function, equal the transforms of the quantiles ofy:Qh(y)(τ)=h(Qy(τ)). Hence, if
the quantile model is expressed in terms ofh(y), e.g., ln(y), then one can use the
inverse transformation to translate the results back toy. This is not possible for the

least squares estimator, i.e., ifE[h(y)]=x$β,thenE[y|x] =h−^1 (x$β).
Impediments to the use of QR are twofold. First, there are computational hur-
dles. Because the objective function is not differentiable, the gradient optimization
methods mentioned in section 15.3 cannot be used and, instead, linear program-
ming methods are applied. There is no closed-form solution for̂βqand hence
the asymptotic distribution of̂βqcannot be obtained using standard methods. An

analytical expression for the asymptotic variance of̂βqcomes from the result that:

√

n(̂βq−βq)→d N

[

0 ,q( 1 −q)A−^1 BA−^1

]

, (15.32)

wheren−^1

∑n

i= 1 xix

$

i

p

→A,n−^1

∑n

i= 1 fuq(ξ

(

q

)

|xi)xix$i

p

→B,

p
→represents conver-
gence in probability, andfuq(ξ

(

q

)

|x)is the conditional density of the error term

uq=y−x$βqevaluated atξ

(

q

)

, i.e., theq-quantile ofuq. Estimation of the vari-

ance of̂βqis complicated by the need to estimatefuq(ξ

(

q

)

|x). It is easier to obtain

standard errors for̂βqusing the computationally more intensive bootstrap method,
as is done in the example that follows.
Buchinsky (1995) evaluated a number of variance estimators for the QR in a
Monte Carlo setting and recommended the use of a “design matrix” (or paired)
bootstrap estimator. Application of the bootstrap variance is shown in algorithm
15.5.1.0.1.
Computational advances have made the application of QR more accessible to
users and many popular packages include it. As an illustration we report a regression
analysis of total medical expenditure (TOTEXP) by the Medicare elderly. The data