There is an important difference between correlation and causation. The only reason that higher test scores are correlated with stay-at-home mothers is that this condition tends to be associated with a husband that went to college, has a higher income and can afford to support the family as a single wage earner. Such families tend to value education. However, the issue of causation is NOT that the mother stays home but rather that there is an emphasis on education. In fact, when adjusting for the education of the parents alone, there is very little difference between the education of kids with working mothers and those with moms that stay at home. And, taking it a step further, when adjusting for parents that are actively involved in their kids’ education (through PTA, etc.), even the prior education of the parents drops off in importance. The critical issue of causation is: To what extent do the parents actually value education? (See first and second generation Asians in the US for a good example of this phenomenon.)
My parents got divorced when I was 8 and my brother was 9. She went to work full-time after the divorce. I didn’t see either of them very much growing up after that, but they valued education. My brother ultimately went to a top law school and I’ve got two graduate degrees. It’s anecdotal, but I really don’t think a mom’s staying at home has anything at all to do with success in education or life, for that matter, from a causation standpoint. And the data supports that. Although there is probably a correlation due to the reason stated above.