Abstract: 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 $(\alpha,\delta)$-fairness consistency, meaning that the quantified unfairness is decreased at most $n^{-\alpha}$ rate with at least probability $1-\delta$, 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 $(\alpha,\delta)$-fairness consistency constraint is $\Theta(\frac{dM}{n})$ provided that $\alpha \le \frac{1}{2}$, where $d$ is the dimensionality, and $M$ is the number of groups induced from the sensitive attributes.