The HPQUANTSELECT Procedure
Quasi-likelihood Information Criteria
Given the quantile level 
, assume that the distribution of 
conditional on 
follows the linear model
where 
for 
are iid in distribution F. Further assume that F is an asymmetric Laplace distribution whose density function is
where 
is the scale parameter. Then, the negative log-likelihood function is
Under these settings, the maximum likelihood estimate (MLE) of 
is the same as the relevant level- quantile regression solution 
, and the MLE for is
where 
equals the level- average check loss 
for the quantile regression solution.
Because the general form of Akaike’s information criterion (AIC) is 
, the quasi-likelihood AIC for quantile regression is
where p is the degrees of freedom for the fitted model.
Similarly, the quasi-likelihood AICC (corrected AIC) and SBC (Schwarz Bayesian information criterion) can be formulated as follows:
In fact, the quasi-likelihood AIC, AICC, and SBC are fairly robust, and you can use them to select effects for data sets without the iid assumption in asymmetric Laplace distribution. For a simulation study that applies SBC for effect selection, see Simulation Study. The study generates a data set by using a naive instrumental model (Chernozhukov and Hansen 2008).