In linear regression, if the assumptions of normally distributed residuals and homogenous residuals are broken, incorrect standard errors can be calculated. This can lead to some predictors appearing statistically significant when they have no effect on the response variable.

But in information theoretic model selection, there is no concept of 'statistically significant predictors'. Basically, the models that result in the smallest residual variation are considered 'the best'.

So, does this mean that in information theoretic model selection, incorrect standard errors are not a problem?

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    $\begingroup$ What do you mean by information theoretic model selection? Would model selection by AIC fit into this? If so, non-normally distributed residuals should be accounted for by using a non-normal likelihood function as a component in AIC (and smallest residual variation would not play a role). $\endgroup$ – Richard Hardy Feb 17 '15 at 11:12
  • $\begingroup$ Yes I'm referring to model selection using AIC. But if the only issue with non-normally distributed residuals is incorrect standard errors, I cannot see why non-normally distributed residuals would be a problem when ranking models using AIC. Why are non-normally distributed residuals (and therefore incorrect standard errors) a problem when ranking models using AIC? $\endgroup$ – luciano Feb 17 '15 at 19:55
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    $\begingroup$ If you estimate a model by OLS in presence of non-normal residuals and obtain this model's AIC value, this will be a problem because this will not be a correct AIC value; you will be implicitly assuming normality in AIC calculation. At the same time, you will have incorrect standard errors. On the other hand, if you estimate your model by maximum likelihood using the correct distribution, you will avoid both incorrect standard errors and incorrect AIC. I am thinking if you can have incorrect standard errors but correct AIC simultaneously... $\endgroup$ – Richard Hardy Feb 17 '15 at 20:22
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    $\begingroup$ Its always good to remember, AIC will tell you the best model among those you are working with; it doesn't tell you whether that model is actually any good. $\endgroup$ – N Brouwer Feb 18 '15 at 5:35

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