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).