  Bayesian variable selection often assumes normality, but the effects of model
misspecification are not sufficiently understood. There are sound reasons
behind this assumption, particularly for large $p$: ease of interpretation,
analytical and computational convenience. More flexible frameworks exist,
including semi- or non-parametric models, often at the cost of some
tractability. We propose a simple extension of the Normal model that allows for
skewness and thicker-than-normal tails but preserves tractability. It leads to
easy interpretation and a log-concave likelihood that facilitates optimization
and integration. We characterize asymptotically parameter estimation and Bayes
factor rates, in particular studying the effects of model misspecification.
Under suitable conditions misspecified Bayes factors are consistent and induce
sparsity at the same asymptotic rates than under the correct model. However,
the rates to detect signal are altered by an exponential factor, often
resulting in a loss of sensitivity. These deficiencies can be ameliorated by
inferring the error distribution from the data, a simple strategy that can
improve inference substantially. Our work focuses on the likelihood and can
thus be combined with any likelihood penalty or prior, but here we focus on
non-local priors to induce extra sparsity and ameliorate finite-sample effects
caused by misspecification. Our results highlight the practical importance of
focusing on the likelihood rather than solely on the prior, when it comes to
Bayesian variable selection. The methodology is available in R package `mombf'.
