In regression discontinuity design (RD), researchers use bandwidths around the discontinuity. For agiven bandwidth, one can estimate asymptotic variance based on the assumption that the bandwidthshrinks to zero as sample size increases (the traditional approach) or, alternatively, that the bandwidthis …xed. The main theoretical results for RD rely on the former, while most applications in the literaturetreat the estimates as parametric. This paper develops the “…xed-bandwidth” alternative asymptotictheory for RD designs, bridging the gap between theorists and practitioners while shedding light onimplicit assumptions in both approaches. The …xed-bandwidth approach provides alternative formulas(approximations) for the bias and variance of RD estimators. Simulations indicate that …xed-bandwidthapproximations are usually better than traditional approximations, and improvements are nontrivialwhen there is heteroskedasticity. When there is no heteroskedasticity, both approximations are shown tobe equivalent, under some additional mild conditions. Feasible estimators of …xed-bandwidth standarderrors are easy to implement and improve coverage of con…dence intervals compared to the traditionalapproach, especially in the presence of heteroskedasticity. Fixed-bandwidth approximations are akin totreating RD estimators as locally parametric, providing theoretical justi…cation for the common empiricalpractice of using heteroskedasticity-robust standard errors in RD settings.