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Given the model

$y =\beta_0 + \beta_1 x_1 + \beta_2 x_2 + u $

where $x_1$ and $x_2$ have completely different scales and units, is it possible to test whether their impact on $y$ is the same? i.e. is there a specific test for this?

I don't think testing $H_0:\beta_1 = \beta_2$ is the right thing to do because of the different interpretations of the estimates.

Any thoughts?

Thanks,

Jona.

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  • $\begingroup$ Can you convert them to standard deviations before running the regression? $\endgroup$ – Dimitriy V. Masterov Apr 29 '15 at 17:52
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One approach is to standardize the covariates first (subtract mean and divide by standard deviation). This puts the 2 covariates on the same scale and their coefficients can be compared directly.

This works best if both covariates are reasonably symmetric without outliers/heavy tails. Strong skewness or outliers in one of the covariates can mess with the estimated standard deviation and make the comparison less meaningful.

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