Counterfactual and Synthetic Control Method: Causal Inference with Instrumented Principal Component Analysis

The fundamental problem of causal inference lies in the absence of counterfactuals. Traditional methodologies impute the missing counterfactuals implicitly or explicitly based on untestable or overly stringent assumptions. Synthetic control method (SCM) utilizes a weighted average of control units to impute the missing counterfactual for the treated unit. Although SCM relaxes some strict assumptions, it still requires the treated unit to be inside the convex hull formed by the controls, avoiding extrapolation. In recent advances, researchers have modeled the entire data generating process (DGP) to explicitly impute the missing counterfactual. This paper expands the interactive fixed effect (IFE) model by instrumenting covariates into factor loadings, adding additional robustness. This methodology offers multiple benefits: firstly, it incorporates the strengths of previous SCM approaches, such as the relaxation of the untestable parallel trends assumption (PTA). Secondly, it does not require the targeted outcomes to be inside the convex hull formed by the controls. Thirdly, it eliminates the need for correct model specification required by the IFE model. Finally, it inherits the ability of principal component analysis (PCA) to effectively handle high-dimensional data and enhances the value extracted from numerous covariates.