I don't understand when one should check the assumptions of the model and when not in regression analysis. If the statistician is only interested in forecasting accurately the response variable, does he still need to do diagnostics? Suppose to have two linear regression models. The first one has a low prediction error but the residuals l and the second one has a higher prediction error but has good residuals. Which model should one choose?
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If the assumptions of a model are met, then the model estimates are usually optimal.
This does not mean that if the assumptions are not met that the model is 'not good'. If an assumption is violated, this can however point into the direction of a better model.
See for example: Statistic test when bias is not random The residuals are all positive on the right side of the graph. In this case, adding a quadratic component could solve this violation, and give a better model. The deviation from normality of the residuals pointed to a better model.
