I am trying to run a simple regression $$Y = a+bX +e$$ however I want to optimize it on Mean Square Percentage Error and not Mean Square Error as in OLS. Like this: $$argmin: e/Y = (Y-a-bX)/Y$$ instead of $$argmin: e = Y-a-bX$$ The problem is this results in a biased estimator when you just run it through an optimizer. Can someone point me in the right direction please? I tried to work with some kind of weighted regression that but I can't seem to get an unbiased estimator. Much appreciated!
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1$\begingroup$ why do you want an unbiased estimator? I thought you wanted to minimize mean square percentage error. $\endgroup$– spaceisdarkgreenOct 31, 2017 at 22:46
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$\begingroup$ I want both. My variance is down but my predictions are now biased. For my particular situation, I am almost concerned that bias may be worse than high variance even for a minimized MSPE. Is there some method to debias the estimates of a weighted regression, even at the expense of some increased variance? $\endgroup$– user3444632Nov 2, 2017 at 1:37
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$\begingroup$ I want to just either multiply the estimates by mean(sample)/mean(estimates) or add mean(sample)-mean(estimates). I can't have a biased estimator here. Does that mean I just have to forgo MSPE? $\endgroup$– user3444632Nov 2, 2017 at 2:16
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