We investigate the minimax optimal error of a fair regression problem under a linear model employing demographic parity as a fairness constraint. As a tractable demographic parity constraint, we introduce (α,δ)-fairness consistency, meaning that the quantified unfairness is decreased at most n^-α rate with at least probability 1-δ, where n is the sample size. In other words, the consistently fair algorithm eventually outputs a regressor satisfying the demographic parity constraint with high probability as n tends to infinity. As a result of our analyses, we found that the minimax optimal error under the (α,δ)-fairness consistency constraint is Θ(dM/n) provided that α<1/2, where d is the dimensionality, and M is the number of groups induced from the sensitive attributes.