Synthetic control methods are commonly used in panel data settings to evaluate the effect of an intervention. In many of these cases, the treated and control units correspond to spatial units such as regions or neighborhoods. Our approach addresses the challenge of understanding how an intervention applied at specific locations influences the surrounding area. Traditional synthetic control applications may struggle with defining the effective area of impact, the extent of treatment propagation across space, and the variation of effects with distance from the treatment sites. To address these challenges, we introduce Spatial Vertical Regression (SVR) within the Bayesian paradigm. This innovative approach allows us to accurately predict the outcomes in varying proximities to the treatment sites, while meticulously accounting for the spatial structure inherent in the data. Specifically, rooted on the vertical regression framework of the synthetic control method, SVR employs a Gaussian process to ensure that the imputation of missing potential outcomes for areas of different distance around the treatment sites is spatially coherent, reflecting the expectation that nearby areas experience similar outcomes and have similar relationships to control areas. This approach is particularly pertinent to our study on the Florentine tramway's first line construction. We study its influence on the local commercial landscape, focusing on how business prevalence varies at different distances from the tram stops.

翻译：合成控制方法通常用于面板数据设置中以评估干预效果。在许多此类场景中，处理单元与控制单元对应空间单元（如区域或社区）。我们的方法旨在解决理解特定位置干预如何影响周围区域的挑战。传统合成控制应用可能难以界定有效影响区域、干预措施在空间上的传播范围以及作用效果随距离处理点变化的规律。针对这些挑战，我们在贝叶斯范式下引入空间纵向回归（Spatial Vertical Regression, SVR）。这一创新方法能够精确预测处理点邻近区域内不同距离下的结果变量，同时细致考虑数据固有的空间结构。具体而言，SVR基于合成控制方法的纵向回归框架，采用高斯过程确保对处理点周围不同距离区域缺失潜在结果的插补在空间上保持一致性，从而反映邻近区域具有相似结果以及与对照区域存在相似关系的预期。该方法尤为适用于我们对佛罗伦萨有轨电车首条线路建设的研究。我们重点分析了有轨电车停靠站周边不同距离处商业密度的变化，以此评估其对当地商业景观的影响。