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Suppose we have two estimation techniques:

  • Linear Least Squares, which aims to minimize squared residuals
  • Least Absolute Deviation, which aims to minimize absolute residuals

We have a model, which purpose is prediction. More specifically, we want the model to perform good based on the prediction evaluation measure: Mean Absolute Prediction Error (MAPE). Assuming we have no information of the data, can we say something about which estimation technique will likely perform better?

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If the probability density of your future distribution is positively skewed, then typically (though not always; von Hippel, 2005) the median will be lower than its mean. So a technique that aims at the median as a point forecast will be biased low. Since the MAPE usually prefers a low biased prediction, such a technique will usually perform better in terms of the MAPE in such a situation.

Note that there are a couple of caveats in this description. The chain of reasoning is not perfect, because you can find pathological counter-examples for at least two steps. Nevertheless, it should work in most practical cases.

You may be better off using a custom optimization routine that directly attempts to minimize the MAPE. The problem being, of course, that the MAPE is not differentiable at perfect forecasts. Alternatively, you could try to estimate full predictive densities and then output the (-1)-median of this density as a point forecast, which is the functional that minimizes the MAPE in expectation (Gneiting, 2011, p. 752 with $\beta=-1$).

You may be interested in What are the shortcomings of the Mean Absolute Percentage Error (MAPE)?

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  • $\begingroup$ The Mean Absolute Prediction Error is something else than the Mean Absolute Percentage Error. Mean Absolute Prediction Error is just the mean of the absolute prediction errors. $\endgroup$ – K. Roelofs Jan 8 '19 at 14:44
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    $\begingroup$ Ah. Sorry. I misread. (So did three upvoters.) Question: since you already consider techniques that minimize the mean absolute error, what else are you looking for? Of course a technique can overfit in minimizing the MAE just as it can overfit in minimizing the MSE, so is control of overfitting your question? If so, the very same methods apply to MAE optimizers: cross-validation and regularization. What am I missing? $\endgroup$ – Stephan Kolassa Jan 8 '19 at 14:50

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