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.