Fixed effects, lagged dependent variables, and bracketing: Cautionary remarks

We investigate a bracketing property that purports to yield upper- and lower bounds on the treatment effects obtained from a fixed effects- and lagged dependent variable model. Referencing both analytical results and a Monte Carlo simulation, we explore the conditions under which the bracketing property holds, confirming this to be the case when the data generating process (DGP) is characterized by either unobserved heterogeneity or feedback effects from a lagged dependent variable (LDV). However, when the DGP is characterized by both features simultaneously, we find that bracketing of the treatment effect only holds under certain conditions - but not in general. Practitioners can nevertheless obtain the lower bound estimate by referencing a model that includes both fixed effects and a LDV. While the Nickell bias in the coefficient of the LDV is known to be of order 1/T, we show that the Nickell-type bias in the estimator of the treatment effect is of order 1/T squared.