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From wikipedia https://en.wikipedia.org/wiki/Bias_of_an_estimator :

because a biased estimator gives a lower value of some loss function (particularly mean squared error) compared with unbiased estimators (notably in shrinkage estimators)

Further evidence is provided in the linked wiki article on shrinkage estimators https://en.wikipedia.org/wiki/Shrinkage_estimator:

A well-known example arises in the estimation of the population variance by sample variance. For a sample size of n, the use of a divisor n − 1 in the usual formula (Bessel's correction) gives an unbiased estimator, while other divisors have lower MSE, at the expense of bias. The optimal choice of divisor (weighting of shrinkage) depends on the excess kurtosis of the population, as discussed at mean squared error: variance, but one can always do better (in terms of MSE) than the unbiased estimator; for the normal distribution a divisor of n + 1 gives one which has the minimum mean square error.

An insight into the tradeoffs of minimizing MSE vs retaining the properites of unbiased estimation and consistency would be helpful. Some practical insights and/or examples into when a biased estimator would be preferred would also be appreciated.

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When you are , you typically aim at minimizing some loss function with your point forecast. One very popular such loss function is the MSE.

Again in forecasting, you often have rather weak signals. For instance, yes, there will be a seasonal pattern to sales of certain products, but the seasonal effect may be weak compared to noise, e.g, for slow movers.

In such a case, it may make sense to use a non-seasonal model. You will miss out on the (present, but weak) seasonal signal, but you may get a lower MSE.

I wrote a little paper illustrating this effect with simulated time series ("Sometimes It's Better to Be Simple than Correct", Kolassa, 2016, Foresight, 40:20-26). Also, here is a little presentation of mine on the paper.

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  • $\begingroup$ I'm not sure if your answer were directly answering the question: it instead in my mind simply changes the use case to "optimize a non-seasonal model". Now starting just with that new non-seasonal model: why would we optimize for MSE instead of unbiased MLE ? $\endgroup$ Aug 24 '19 at 14:10
  • $\begingroup$ That is a valid question. My simple answer: in forecasting, minimizing the MSE is often given as the criterion. My more complex answer: in forecasting, there is no way to assess whether your forecast is truly unbiased, since you typically do not know the data generating process. (But if you do have an unbiased forecast, it will minimize the expected MSE. Which is a good argument for attempting to minimize the MSE, whether or not it results in an unbiased forecast.) $\endgroup$ Aug 24 '19 at 14:51

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