Consistency in the presence of nonstationarity? Single time-series vs two-level panel data

• No cointegration:

  • Single time-series (y_t=\alpha+x_t\beta): Spurious regression, OLS inconsistent (Hamilton, 1994 pg. 592).

  • Two-level panel-data:

    • Model without individual fixed effects (y_it=x_it\beta): Pooled OLS is a consistent estimator of the long-run average regression coefficient (Baltagi, 2006 pg. 291; Phillips and Moon, 1999 pg. 1072).

• Cointegration:

  • Single time-series (y_t=\alpha+x_t\beta): OLS consistent estimator of the linear projection of (Hamilton, 1994 pg. 592).

  • Two-level panel-data:

    • Model with individual fixed effects (y_it=\alpha_i+x_it\beta with x_it=x_it-1+e_it): Panel Fully modified OLS is consistent. Within-OLS is inconsistent. (Baltagi, 2015 pg. 48)

References

Baltagi, B. H. (Ed.). (2006). Panel data econometrics: Theoretical contributions and empirical applications. Emerald Group Publishing.

Baltagi, B. H. (Ed.). (2015). The Oxford handbook of panel data. Oxford Handbooks.

Hamilton, J. D. (1994). Time series analysis. Princeton university press.

Phillips, P. C., & Moon, H. R. (1999). Linear regression limit theory for nonstationary panel data. Econometrica, 67(5), 1057-1111.

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.