When clustered errors and how? A short guide

1. (Abadie et al, WP 2017) When should we cluster s.e.? We should cluster if:

   1. Cluster sampling.

   2. Random sampling and cluster assignment (fixed within clusters).

2. (Cameron et al, JHR 2015) Types of clustering:

   1. One-way clustering:

      1. Number of clusters is large: the cluster-robust variance estimator is unbiased (Cameron et al, JHR 2015 ).

      2. Number of clusters is small: wild cluster bootstrap-t (Cameron et al, RES 2008; Djogbenou et al, JE 2019).

   2. Multi-way clustering:

      1. Nested multi-way clustering: cluster at the highest level ((Cameron et al, JHR 2015) ).

         1. Number of clusters is large: the cluster-robust variance estimator is unbiased (Cameron et al, JHR 2015 ).

         2. Number of clusters is small: wild cluster bootstrap-t (Cameron et al, RES 2008; Djogbenou et al, JE 2019).

      2. Non-nested multi-way clustering: appropriate when the errors are correlated within non-nested clusters. For instance, errors are correlated within villages but also due to a common factor across villages.

         1. Number of clusters in all dimensions is large: multi-way cluster robust variance estimator (Cameron et al, JBES 2011).

         2. Number of clusters in either dimension is small: Wild cluster bootstrap-t (MacKinnon et al, JBES, 2021).

References:

1. Abadie, A., Athey, S., Imbens, G. W., & Wooldridge, J. (2017). When should you adjust standard errors for clustering? (No. w24003). National Bureau of Economic Research.

2. Cameron, A. C., Gelbach, J. B., & Miller, D. L. (2008). Bootstrap-based improvements for inference with clustered errors. The review of economics and statistics, 90(3), 414-427.

3. Cameron, A. C., Gelbach, J. B., & Miller, D. L. (2011). Robust inference with multiway clustering. Journal of Business & Economic Statistics, 29(2), 238-249.

4. Cameron, A. C., & Miller, D. L. (2015). A practitioner’s guide to cluster-robust inference. Journal of human resources, 50(2), 317-372.

5. Djogbenou, A. A., MacKinnon, J. G., & Nielsen, M. Ø. (2019). Asymptotic theory and wild bootstrap inference with clustered errors. Journal of Econometrics, 212(2), 393-412.

6. MacKinnon, J. G., Nielsen, M. Ø., & Webb, M. D. (2021). Wild bootstrap and asymptotic inference with multiway clustering. Journal of Business & Economic Statistics, 39(2), 505-519.