Consider the hybrid model (Alisson 2009):

$ y_{ij} = \beta_W(x_{ij} - \bar{x_i}) + \beta_B\bar{x_i}+\gamma c_i +(u_i +e_{ij})$

Where $\beta_B$ is an estimate of te ‘between effect’ and ‘$\beta_W$ is an estimate of the ‘within effect.’ Estimates are obtained via GLS estimation with an addition of means and deviations from the mean for time varying variables $x_i$, and a vector of time-invariant predictors $c_i$. The terms $u_i$ and $e_{ij}$ are the individual specific and occasion specific disturbances, respectively. They are both assumed to be normally distributed and independent from the predictors but $u_i$ is allowed to depend on $\bar{x_i}$.

Bell and Jones (2012) state that this model is strictly better than both Random and fixed effects models, because it allows for the estimation of time-invariant effects, for the estimation of the portion of covariate effects that is due to “inherent differences” between individuals, all the while being able to estimate the “within effects” as well as Fixed Effects models.

I understand that there are other advantages, namely the possibility of estimating non-linear dependencies of $u_i$ on $\bar{x_i}$, the possibility of including random slopes, and the possibility of extending the model to deal with nonlinear link functions and distributions for which Fixed Effects estimators are inconsistent due to the link function being non-canonical. However, this question concerns mainly the case where the ability of estimating the effects of time-invariant variables and the ‘between effects.’

Wouldn’t these effects still be subject to bias if the assumption that $E(u_i | x_ij, c_i) = 0$ is violated? Since this is the case whenever $\beta_W$ and $\beta_B$ are not equal, it seems to me that this is more the norm than an exception. Moreover, the ‘within effects only use the information on within variation, so they are not more efficient than a normal fixed effects. Thus, what is the inherent advantage of doing this over using a fixed effects estimator? The “within effects” are as (in)efficient as those from A fixed effects regression, and the other effects rely on a somewhat strict assumption on the dependence between $u_i$ and the covariates.

Why do some authors recommend this model so strongly? I’m interested in using them for their ability to estimate effects for time-invariant predictors along with the “within effects,” but am not sure when or if it is safe to interpret those effects in face of the assumptions needed for that.

References Allison 2009

Bell and Jones 2012



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