发表机构
University of Calgary（卡尔加里大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究合成控制中的谱截断，提出谱合成控制及混合估计器，证明相关理论，通过在11种数据生成机制下评估，发现截断谱SC的RMSE高于原始路径SC，混合估计器多数选原始路径匹配，结果受预处理影响。
AI 中文摘要
合成控制（SC）将处理单元的预处理轨迹与供体单元的加权组合相匹配。我们研究谱合成控制（Spectral SC），它在供体面板的主导时间奇异向量定义的坐标中匹配处理单元，以及一种混合估计器，该估计器在保留和丢弃方向上分别设置可调权重，将原始路径SC和截断谱SC作为端点嵌套。我们证明，在满秩时，该族精确地简化为原始路径SC；当供体数量\(N_0\)大于保留维度数\(K\)加1时，在\(K\)个保留维度上与\(N_0\)个供体的精确平衡是欠定的，仿射解集维度为\(N_0 - K - 1\)；并且谱不平衡通过有限样本最佳线性预测器分解映射到处理效应偏差。我们在11种数据生成机制中评估这些估计器，每种机制使用400次重复，并使用仅供体的安慰剂验证来选择正则化和混合权重。在每种机制中，截断谱SC的均方根误差（RMSE）显著高于调整后的原始路径SC，配对差异等于4到11个蒙特卡罗标准误差。混合估计器在大多数重复中选择原始路径匹配，并且在大多数机制中与调整后的SC在统计上无法区分。结果对预处理高度敏感。使用原始输入时，性能差距很大；按照我们边界背后的假设，在谱分解之前去除单位和时间固定效应后，差距几乎消失，安慰剂验证开始倾向于截断。我们将这些发现用于诊断，而不是作为谱合成控制应取代原始路径合成控制的证据。基估计噪声、平衡欠定性和固定效应污染决定了谱匹配何时有用。
英文摘要
Synthetic control (SC) matches a treated unit's pre-treatment trajectory to a weighted combination of donor units. We study Spectral SC, which instead matches the treated unit in coordinates defined by the leading temporal singular vectors of the donor panel, and a hybrid estimator that places separately tunable weight on retained and discarded directions, nesting raw-path SC and truncated Spectral SC as endpoints. We prove that the family reduces exactly to raw-path SC at full rank, that exact balance on $K$ retained dimensions with $N_0$ donors is underdetermined whenever $N_0>K+1$, with an affine solution set of dimension $N_0-K-1$, and that spectral imbalance maps to treatment-effect bias through a finite-sample best-linear-predictor decomposition. We evaluate the estimators across eleven data-generating regimes, using $400$ replications per regime and donor-only placebo validation to select regularization and the mixing weight. Truncated Spectral SC has significantly higher RMSE than tuned raw-path SC in every regime, with paired differences equal to $4$ to $11$ Monte Carlo standard errors. The hybrid estimator selects raw-path matching in most replications and is statistically indistinguishable from tuned SC in most regimes. The result is highly sensitive to preprocessing. With raw inputs, the performance gap is large; after removing unit and time fixed effects before spectral decomposition, as suggested by the assumptions behind our bound, the gap nearly disappears and placebo validation begins to favor truncation. We interpret these findings diagnostically rather than as evidence that Spectral SC should replace raw-path SC. Basis-estimation noise, balancing underdetermination, and fixed-effects contamination determine when spectral matching can help.
