Quasi-likelihood Information Criteria

Given the quantile level $\tau $, assume that the distribution of $Y_ i$ conditional on $\mb{x}_ i$ follows the linear model

\[  Y_ i = \mb{x}_ i^{\prime }\bbeta + \epsilon _ i \]

where $\epsilon _ i$ for $i=1,\ldots ,n$ are iid in distribution F. Further assume that F is an asymmetric Laplace distribution whose density function is

\[ f_\tau (r)={\tau (1-\tau )\over \sigma }\exp \left(-{\rho _\tau (r)\over \sigma }\right) \]

where $\sigma $ is the scale parameter. Then, the negative log-likelihood function is

\[ l_\tau (\bbeta ,\sigma )=n\log (\sigma )+\sigma ^{-1}\sum _{i=1}^ n\rho _\tau (y_ i-\mb{x}_ i’\bbeta )-n\log (\tau (1-\tau )) \]

Under these settings, the maximum likelihood estimate (MLE) of $\bbeta $ is the same as the relevant level-$\tau $ quantile regression solution $\hat{\bbeta }(\tau )$, and the MLE for $\sigma $ is

\[ \hat{\sigma }(\tau )=n^{-1}\sum _{i=1}^ n \rho _\tau \left(y_ i-\mb{x}_ i^{\prime }\hat{\bbeta }(\tau )\right) \]

where $\hat{\sigma }(\tau )$ equals the level-$\tau $ average check loss $\mbox{ACL}(\tau )$ for the quantile regression solution.

Because the general form of Akaike’s information criterion (AIC) is $\mbox{AIC}=(-2l+2p)$, the quasi-likelihood AIC for quantile regression is

\[ \mbox{AIC}(\tau )=2n\ln \left( \mbox{ACL}(\tau ) \right) + 2p \]

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:

\[ \mbox{AICC}(\tau )=2n\ln \left(\mbox{ACL}(\tau )\right)+{2pn\over n-p-1} \]
\[ \mbox{SBC}(\tau )=2n\ln \left(\mbox{ACL}(\tau )\right)+p\ln (n) \]

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).