Robert Waldmann says to be careful when using income as a control in studies measuring school (or other) achievement:
Robert Waldmann: I just thought of something. It was triggered by this "studies show that race exhibits a substantial, income-independent influence on school achievement," (don't blame Matt he's just the messenger and is making a very good point about something else).
You know what else? Race exhibits a substantial, income-independent influence on consumption -- Black households consume less than White households with the same income.
Few suggest this proves that Blacks are more prudent and patient than Whites. It is similar to the information on school achievement, Blacks have outcomes similar to poorer Whites.
The economics profession's favored explanation for the consumption fact is that consumption doesn't depend only on current income but also on permanent income (not at all on current income if people are not liquidity constrained and rational). Blacks are on average poorer than Whites. A Black household at say the 50th percentile of the income distribution is more likely to have had an unusually good year than a White household with the same income.
The income measure which is likely to explain school achievement and consumption is income averaged over a long period of time (lagged income for school achievement and, I suspect, consumption).
Now it is possible to construct a race specific mapping from current to permanent income (using the PSID say) and use that to see if there is a permanent income independent difference in school achievement. However, just using current income as a control is a mistake.
Another way of looking at is it that current income is permanent income plus noise so the coefficient of say achievement on current income is the coefficient of interest (achievement on permanent income) biased down by errors in measurement of permanent income. Thus one might impose coefficients higher than the estimated coefficients.
Milton Friedman first considered the effect of measurement error on estimates when arguing that the true elasticity of consumption with respect to permanent income is 1 even though the elasticity estimated with cross sectional data is less than one. He thus reconciled the cross sectional estimates with the aggregate time series evidence that the savings rate doesn't increase systematically with economic growth.
[That's not even close to Robert's wonkiest post of the day.]