# Multilevel model for causal inference using observational data

I am trying to follow the lecture notes by Imbens/Wooldridge (http://www.nber.org/WNE/lect_10_diffindiffs.pdf) on difference-in-differences estimation.

In page 4, they discuss the general framework to extend the DiD model to many periods and groups, and after presenting this equation they state that it "is an example of a multilevel model".

$y_{igt} = \lambda_t + \alpha_g + X_{gt}\beta + z_{igt}\gamma_{gt} + v_{gt} + u_{igt}$

"where i indexes individual, g indexes group, and t indexes time. This model has a full set of time effects, $\lambda_t$, a full set of group effects, $\alpha_g$, group/time period covariates, $X_{gt}$ (these are the policy variables), individual-specific covariates, $Z_{igt}$, unobserved group/time effects, $v_{gt}$, and individual-specific errors, $u_{igt}$. We are interested in estimating $\beta$."

I am having trouble seeing how a multi-level model would work here. I've been reading Data analysis using regression and multilevel/hierarchical models by Gelman (2006) and explains how a multilevel model can be thought as one regression for each group in one level (e.g. one regression for each school) and depending on the specification the model can have varying intercepts and varying slopes.

So my question is, what would be varying (intercepts/slopes, both?) in the Imbens/Wooldridge equation shown above? are time effects in here $(\lambda_t)$ fixed or random effects?. are group effects in here $(\alpha_g)$ fixed or random effects?.

I would appreciate any comments or examples that would help me get a grasp on this.