Do We Exploit all Information for Counterfactual Analysis? Benefits of Factor Models and Idiosyncratic Correction
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The measurement of treatment (intervention) effects on a single (or just a few) treated unit(s) based on counterfactuals constructed from artificial controls has become a popular practice in applied statistics and economics since the proposal of the synthetic control method. In high-dimensional setting, we often use principal component or (weakly) sparse regression to estimate counterfactuals. Do we use enough data information? To better estimate the effects of price changes on the sales in our case study, we propose a general framework on counterfactual analysis for high dimensional dependent data. The framework includes both principal component regression and sparse linear regression as specific cases. It uses both factor and idiosyncratic components as predictors for improved counterfactual analysis, resulting a method called Factor-Adjusted Regularized Method for Treatment (FarmTreat) evaluation. We demonstrate convincingly that using either factors or sparse regression is inadequate for counterfactual analysis in many applications and the case for information gain can be made through the use of idiosyncratic components. We also develop theory and methods to formally answer the question if common factors are adequate for estimating counterfactuals. Furthermore, we consider a simple resampling approach to conduct inference on the treatment effect as well as bootstrap test to access the relevance of the idiosyncratic components. We apply the proposed method to evaluate the effects of price changes on the sales of a set of products based on a novel large panel of sale data from a major retail chain in Brazil and demonstrate the benefits of using additional idiosyncratic components in the treatment effect evaluations.
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