A regression I would like to see

In his column today, David Leonhardt writes the following about college admissions:

But all else equal, a low-income applicant was no more likely to get in than a high-income applicant with the same SAT score. It’s pretty hard to call that meritocracy.

When I first read this pair of sentences, they struck me as odd. But in the context of the whole article, they have some logic. David suggests that high-income applicants' SAT scores are, in some sense, an overstated measure of ability, because these applicants have the benefit of tutors, mulitiple testing opportunities, and so on. As a result, he says that we should correct for this by giving a preference to lower-income applicants.

Maybe David is right, but to convince me, here is what I would like to see.  Regress some measure of college success (such as GPA) on SAT scores and the student's family income.  If David is right, then the coefficient on family income should be negative.  That is, a lower-income student should do better in college, holding reported SAT score constant, because he managed to get that SAT score without all those extra benefits.  This is a regression that some enterprising college admissions committee could easily do.  (Maybe someone has already done it, and I am just not aware of the study.)

If the coefficient does turn out to be significantly negative, that finding would provide strong evidence for the thesis of David's article.  Right now, I would venture to guess that the data would not support David's story, but I am always ready to be proven wrong.

Update:Todd Stinebrickner, an economist at The University of Western Ontario, emails me this comment:

It does seem reasonable to believe that, if a low income student and a high income student have the same SAT scores at the time of college entrance, the low income student was probably born with higher "inherent" ability.  At the same time, SAT scores may not capture all of the educational benefits of being from a high income family that may continue to matter in college. For example, a student's score on the Math SAT may not capture whether the student had the opportunity to take a Calculus course in high school.  This suggests that, from a theoretical standpoint, the effect of family income on college grades conditional on SAT scores is ambiguous.  As part of an ongoing in-depth case study at one particular school (motivated particularly by an interest in college dropout), we discuss this issue and run the type of regression you suggest in Table 3 of a 2003 JHR paper "Understanding educational outcomes of students from low-income families."  It is worth noting that everyone in our sample is of moderate or low family income. Regardless, within the income groups we examine, students from higher income backgrounds have significantly higher grades throughout college conditional on college entrance exam (ACT) scores.

The finding in the last sentence (which I put in bold) is the opposite of what the Leonhardt story suggests. What this means is that if you are a college admissions officer trying to identify the students who will do best in college, as measured by grades, you would give positive rather than negative weight on family income. I am not proposing that they should do this, as colleges have many goals when putting together a class. But it does seem that the hypothesis implicit in Leonhardt's article is not supported by the data.