Switchback experiments -- in which treatment is assigned at the level of a cluster crossed with a time period -- are widely used in marketplace and platform settings, yet no closed-form power formula exists for them. We fill this gap by deriving a closed-form, multi-level asymptotic variance approximation for the individual-level OLS estimator, facilitating power budgeting. Using this formula, we reveal a structural floor on statistical power: while idiosyncratic noise vanishes with observation density, macro-level shocks are multiplicatively penalized by cluster size imbalance. We confirm through analytical derivations and Monte Carlo simulations that the formula is exact across typical parameters and serves as a mathematically conservative upper bound in extreme boundary regimes. We study three methodological applications. First, we prove that advanced assignment designs like stratification only partially eliminate the penalty of cluster size imbalance on power. Second, we demonstrate that variance reduction techniques targeting macro-level shocks yield disproportionately greater efficiency gains than those targeting residual noise. Third, we formalize the finite-sample power trade-offs between individual-level and cell-level estimators.

翻译：切换实验（治疗在跨时间段的聚类层面上进行分配）广泛应用于市场平台场景，但尚无闭式功效公式存在。我们通过推导个体层级OLS估计量的多层次渐近方差近似公式来填补这一空白，从而便于功效预算规划。利用该公式，我们揭示了统计功效的结构性下限：当异质性噪声随观测密度消失时，宏观层面冲击会受到聚类规模不平衡的乘数惩罚。通过解析推导和蒙特卡洛模拟，我们证实该公式在典型参数范围内精确成立，并在极端边界区域作为数学保守上界发挥作用。我们研究了三个方法论应用：首先，证明分层等先进分配设计仅能部分消除聚类规模不平衡对功效的惩罚；其次，展示针对宏观层面冲击的方差缩减技术比针对残差噪声的同类技术能产生不成比例的效率提升；第三，形式化个体层级与单元层级估计量之间的有限样本功效权衡。