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When I run a multiple regression for forecasting purpose, is it better to drop insignificant covariates or not?
What is the effect of insignificant variables for the forecast accuracy?

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Whether a covariate is significant or not is a function of a number of things such as sample size, adequate number of subjects with differing covariate values and so on, but of course also the size of the effect associated with it. Only the last one of these things really tells you something about whether it is a useful variable for forecasting. In general, signficance is not a good criterion (and was never designed to be) for whether including or excluding a variable will lead to better forecasts.

Things like cross-validation or e.g. AIC based model weights are actually what you would want for situations with multiple competing models. When it's so clear-cut that the model with [or without] a covariate essentially never gets selected in any of the cross-validated samples/gets zero weight in the model averaging then you essentially have the selection of a single model.

However, there will very often be considerable remaining uncertainty about model choice so that for optimizing forecasting selecting a single model is usually a bad idea.

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