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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).