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So I was reading about the consequences of various loss functions on what regressions result in, i.e. the L1 norm giving conditional median estimation and L2 giving conditional mean, etc.

What happens if you minimize the sum of $\left(\frac{\hat{y}}{y}-1 \right)^2$ over all observations? Provided no y != 0, I get Monte Carlo results that are very close to standard OLS results.

I SHOULD be doing my time series homework but instead I've been tinkering around.

Not a super serious question, just curiosity :)

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  • $\begingroup$ This is probably better asked on stats.stackexchange.com $\endgroup$ – shadowtalker Jan 31 '15 at 20:16
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Is it the case that since you can rearrange this expression to $\Sigma (\frac{1}{y^2})(\hat{y}-y)^2$, that you are really weighting the standard L2 norm with weights that are higher for lower absolute values of y?

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  • $\begingroup$ Yes, check out weighted least squares. You could do this if you cared more about getting small absolute values of y predicted correctly or if you though that there was measurement error in y that was larger for large values of y. $\endgroup$ – BKay Jan 30 '15 at 16:38

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