# Quasi-fixed effects: time-invariant variables in random effects model, or something else?

I am currently studying dynamic panel data models, and a stumbled upon a phenomenon which is referred to as "Quasi fixed effects".

As far as i understand, this boils down to the following idea. Consider latent model for the standard dynamic RE (logit) model for panel data:

$y_{it} = \gamma Y_{i t-1} + X_{it}^\prime \beta + \alpha_{i} + \epsilon_{it},$

where $\alpha_i$ is the normally distributed random individual effect. However, we might think that the the normality assumption on the individual effects cannot be justified (we are expecting more something like fixed effects, but fixed effects in dynamics models is an issue). Therefore, we might allow the individual effect to be correlated with some time-invariant person characteristics (e.g. $z_i$):

$\alpha_i = \lambda z_i + \tilde{\alpha_i},$

where $\tilde{\alpha_i}$ is a pure random effect.

As easily seen, by using this specification for $\alpha_i$, our model specification is now just a dynamic random effects model, including time-invariant variable $z_i$, where $\tilde{\alpha_i}$ is the pure random effect.

Does that mean that "Quasi fixed effects" models, are the same as random effects models with time-invariant variables? Is it just perhaps just a difference from an interpretation point of view, or is there something fundamentally different between the two that i am missing? (And finally, does that mean that we can estimate such models as regular random effects models).

As a reference for the use of "quasi-fixed effects", see: Heitmüller, Axel (2007), The Chicken or the Egg? Endogeneity in Labour Market.