Hierarchical Clustering As a Novel Solution to the Notorious Multicollinearity Problem in Observational Causal Inference

Yufei Wu, Zhiying Gu, Alex Deng, Jacob Zhu, Linsha Chen

Multicollinearity is a long lasting challenge in observational causal inference, especially in regressions -- highly correlated independent variables make it hard to isolate their individual impacts on outcomes of interest. While common solutions such as shrinkage estimators and principal component regressions are helpful in prediction problems, a crucial limitation hinders their applicability to causal inference problems -- they cannot provide the original causal relationships. To fill the gap, we present an innovative and intuitive solution, by employing hierarchical clustering to aggregate data in a way that effectively alleviates collinearity. This method is generally applicable to causal problems featuring multicollinearity. We use a marketing application to demonstrate how and why it works. Expenditures on different advertising channels often exhibit correlations, making it exceedingly difficult to separately measure their impact. Many previous studies proposed to leverage granular cross-sectional data for better identification but, to our knowledge, none explicitly addressed multicollinearity, which undermines causal identification even with granular data. We propose to hierarchically cluster geographic units based on marketing spend correlation to reduce collinearity, and to implement a Bayesian Marketing Mix Model with cluster-level data. Such clustering happens in two steps -- we first normalize and demean geo-level data to establish a common scale and to eliminate the common trends; we then calculate pairwise distance to summarize marketing spend correlation between geos and cluster the ones with moderate to strong correlation. Both descriptive evidence and regression analysis affirm that such hierarchical clustering effectively mitigates collinearity and facilitates the separate identification of the impact of different marketing channels.