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In econometrics, should two subgroups of the data (for dummy=0 and dummy=1) have some variation, that can be explained also with a constant difference, how would one test for it? And is it a problem (multicollinearity perhaps)?

Example being, that income is a function of dummy female (which is 0 for male and 1 for female observations), but there is a constant difference between both groups.

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    $\begingroup$ Your first sentence is not very clear. Are you asking how to test whether the difference in outcome variable between, say males and females, is constant? If you think that the difference depends on the value of some other variable, you can add an interaction term to the regression, interacting gender with this other variable, and then test the significance of the estimated coefficient. $\endgroup$ – AlexK May 25 at 23:14
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I concur with @AlexK that your question is not very clear. The two groups will give you two distributions of income (one for males, one for females). Depending on your research interests, you can compare the groups with respect to:

  1. The centers of the their respective distributions of income; AND/OR
  2. The spreads of their respective distributions of income.

How you define the centers and/or spreads of the distributions will ultimately depend on the nature of the income data. Generally, incomes tend to follow a right-skewed distribution, so you may find it meaningful to define the center of the income distribution for males/females as the median and the spread as the interquartile range. With these definitions in place, you could use quantile regression to estimate the first quartile, the median and the third quartiles of the income distribution, conditional on gender. The interquartile range is the difference between the third and first quantiles.

For income distributions that would look approximately normal for each gender (I doubt you could find such distributions, but one never knows), you could define the center and spread as the mean and standard deviations (or variances) of those distributions. You could then use a t-test to compare the distribution means and/or an F-test to compare the distribution variances. Alternatively, you could use generalized least squares regression to simultaneously compare the means and standard deviations of the two income distributions (if the distributions have unequal spreads).

To sum up, what you compare and how you compare it will depend on the shape of the underlying income distributions.

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