This paper studies the determinants of children’s scores on tests of cognitive achievement in math and reading. Using rich longitudinal data on test scores, home environments, and schools, we implement alternative specifications for the cognitive achievement production function that allow achievement to depend on the entire history of lagged home and school inputs as well as on parents’ ability and unobserved endowments. We use cross‐validation methods to select among competing specifications and find support for a variant of a value‐added model of the production function augmented to include information on lagged inputs. Using this specification, we study the sources of test score gaps between black, white, and Hispanic children. The estimated model captures key patterns in the data, such as the widening of minority‐white test score gaps with age and differences in the gap pattern between Hispanics and blacks. We find that differences in mother’s “ability,” as measured by AFQT, account for about half of the test score gap. Home inputs also account for a significant proportion. Equalizing home inputs at the average levels of white children would close the black‐white and the Hispanic‐white test score gaps in math and reading by about 10–20 percent.