Studies have found correlations between breastfeeding and later outcomes. But those studies have a bias. For example, mothers who graduated college are more likely to breastfeed, and their children are more likely to go to college.
The only way to factor out the biases completely is with a twin study: same mother, same environment. That's what they did with the "Twins Early Development Study." This study found no correlation between breastfeeding and IQ after the age of 2.
There’s a good study out there that used parents that couldn’t breastfeed. Some used donated milk and the other group used formula. There was a correlation for IQ, though it was pretty small.
In the PROBIT trial, "13 percent of the children of mothers in the group that wasn’t encouraged to breastfeed had at least one diarrhea episode, versus only 9 percent of those whose mothers were encouraged. The rate of rashes and eczema was also lower in the group whose mothers were encouraged to breastfeed: 3 percent versus 6 percent."
"In this particular study, we cannot reject the possibility that breastfeeding could matter in either direction—that it could decrease or increase respiratory infections. What we can say is that the data doesn’t support the claim of a reduction in respiratory infections as a result of breastfeeding."
"These researchers have continued to follow the children in the trial through the age of seven. They find no evidence of any long-term health impacts: no change in allergies or asthma, cavities, height, blood pressure, weight, or indicators for being overweight or obese."
Quotes from Crib sheet by Emily Oster. On the other hand, one compelling reason to breastfeed is for the mother, not the child. Breastfeeding lowers risk of breast cancer.
The only way to factor out the biases completely is with a twin study: same mother, same environment. That's what they did with the "Twins Early Development Study." This study found no correlation between breastfeeding and IQ after the age of 2.