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I was wondering if I can use Akaike or Schwarz criterion even when the residuals that I get from the model when I run the regression are not normal. Is there any normality assumption with these criteria or can I use them all the time regardless?

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    $\begingroup$ I don't know an exact answer to your question, but in general AIC and BIC can be used in many types of models, also those that are not linear and used to model non-normal dependent variables. So I think the normality assumption is not central. $\endgroup$ – tomka Dec 13 '13 at 23:27
  • $\begingroup$ That's what I thought at first. But I was reading this paper arxiv.org/PS_cache/astro-ph/pdf/0701/0701113v2.pdf and if you read on section 2.2, the first paragraph mentions "gaussian assumptions" $\endgroup$ – Wilmer E. Henao Dec 13 '13 at 23:31
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We use the Akaike or Schwarz information criteria to compare a set of "Candidate Models". The candidates means that you have already fitted your regression models and did the model adequacy checking including the normality assumption of your residuals. You are just not sure about the balance between the number of parameters used to fit the model and the closeness of the fit. Even though you may not see a direct normality assumption in the definitions of AIC or BIC criteria, but (as far as I can say) you need to check the normality assumption before comparing your models. Of course, you can still apply these criteria, even for non-normal errors. But how meaningful would be you final model, will be under a big question mark. You can also have a look at this question.

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    $\begingroup$ Would you agree with my answer? I argue that normality is irrelevant unless the likelihood function implies it. Your statement model adequacy checking including the normality assumption of your residuals considers only a special case and is not correct in general -- because likelihood function may assume a non-normal distribution, e.g. a $t$ distribution is common in case of GARCH models. $\endgroup$ – Richard Hardy Jan 14 '16 at 18:18
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AIC and BIC both have consist of two elements: the likelihood and the penalty for the number of parameters. The likelihood need not be normal likelihood, it can be whatever you find reasonable.

However, if you assume the likelihood to be normal (and use it in AIC or BIC), you need normal residuals, too. In other words, the distribution of residuals has to match the distributional assumption used to calculate the likelihood for AIC or BIC.

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