In Oster (2019), she discusses how authors typically include controls and examine coefficient stability as a way to test for presence of confounding, and points out that researchers should consider how $R^2$ moves with the addition of controls as well. I have a simple question about one claim early in the paper:
Even under this most optimistic assumption, however, coefficient movements alone are not a sufficient statistic to calculate bias. To illustrate why, consider the case of a researcher estimating wage returns to education with individual ability as the only counfounder, and where there are two orthogonal components of ability, one of which has a higher variance than the other. Assume wages would be fully explained if both ability components were observed but, in practice, the researcher sees only one of the two. The coefficient will appear much more stable if the observed ability control is the lower variance one, but this is not because the bias is smaller but simply because less of the wage outcome is explained by the controls.
Can someone explain why this bolded sentence makes sense? I know if a control has less variance, it's coefficient has higher standard errors, but I dont understand intuitively (or mathematically) why this implies that the $R^2$ would move by less, and therefore the main coefficient of interest would seem more 'stable' to the inclusion of the control. Why does the $R^2$ move by less if the control is lower variance compared to one that has higher variance?
