Theoretical development for overidentified diversified-projection estimation

Develop the theoretical properties of diversified-projection estimation when the number of diversified weights, and hence the rank of the diversified weight matrix, exceeds the true number of factors, thereby extending the exactly identified instrumental-variables formulation to overidentified settings.

Background

The paper’s diversified-projection procedure assumes that the number of factors is known and constructs a diversified weight matrix whose rank equals that number. The authors interpret this setup as an exactly identified instrumental-variables approach: the diversified weights play roles analogous to instruments, while the estimated factors are obtained through cross-sectional weighted averages rather than principal components.

The cited diversified-projection method is robust to overestimating the number of factors for consistent factor estimation, provided that the rank of the diversified weight matrix is at least as large as the true number of factors. The paper reports simulations and empirical analyses examining robustness to overestimating the number of factors, but it does not establish the corresponding theoretical results for the proposed time-varying factor and average-treatment-effect-localization framework. The unresolved problem is therefore to derive formal asymptotic theory for the overidentified version of the method.

References

It is interesting to extend this framework to overidentified IV settings. We examine robustness to over-estimating the number of factors in the simulation and empirical sections, and leave the associated theoretical development for future work.

— Average Treatment Effect Localization: Projection Methods in Synthetic Control  (2609.10617 - Lee, 8 Sep 2026) in Section 2, subsection “Low-rank time-varying factor structure: approximation and estimation”