Today the biggest handicap associated with education level is the lack of marriage / stable families in the lower class.
So if you have college educated parents, and they mostly marry others with bachelors degrees, the illegitimacy rate is less than 5%, divorce rate is less than 25% for those under 40, and a very high marriage rate.
If you are a high school drop out, the marriage rate is less than half, the odds of divorce for those who marry is over 50%, and the odds the child is illegitimate is over 50%, In short, most kids out of wedlock, if they do marry even after having kids, rarely stay together.
And regardless of income level, living together without marriage is an 80% chance of separating before the kid is 18, whereas when married, the odds of separating before kids are 18 is less than a third. (Most divorces don't involve children.) Hat tip Chanti Feldman's research.
Poverty is directly tied to single mother families and family instability, because a mother with two or three kids by as many guys lacks reliable support and reinforcement with the kids and lacks economies of scale with a committed man living with her. And his attention is divided among his children by various relationships, and he is less likely to save for retirement and kids' college even if you live together.
So most of the poverty guarantee we see for the lower class is the breakdown of the family and the financial handicaps that causes.