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Abstract:
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Several choices have to be made such as 'which and how many estimators to average over' and 'which weights to use'. Data-driven weights can be chosen by minimizing an estimator of the mean squared error. In general those weights are not unique. We prove that there are multiple weight vectors which yield equal model averaged estimators in linear regression. A restriction to singleton models results in a drastic reduction in the computational cost. If we take into account that the weights are random variables rather than fixed during selection, we show that the averaged estimator is biased even when the original estimators are unbiased and that its variance is larger than in the fixed weights case. This relates to the 'forecast combination puzzle'; there is no guarantee that the weighted averaged forecast will improve on the original forecasts. The distribution of model averaged estimators is, in general, hard to obtain. We work out the special case of an estimator after model selection by the Akaike information criterion AIC. We exploit the overselection properties of AIC to construct valid confidence regions that take the model selection uncertainty into account.
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