Meaning of "Pre-Post estimation model"

Question

The model given just looks like a time series panel data with added controls but then I noticed that there was only one constant (so not a Fixed Effects Model) and but the error term changes between restaurant indicitating that it maybe a Random Effects Model.

$$y_{it}= \beta_0 +\text{Grading_Post}_{it}'\beta _1 +X_1'\beta_2 |+\text{Pre_Post}_it+\gamma_i +\delta_i+\epsilon_{it} $$

Where the dependent variable is fines/inspection score

http://www.appam.org/assets/1/7/Impact_paper_9-30-15.pdf

Page 14 has the models

Page 31 onwards has the regressions

I've consulted my book "Principle of Econometrics" but it does not seem to discussed and this book just about discusses everything. Can anyone link me to something can explain this concept better?

edit: I think i get it now this is a panel data however the way the model is written allows for the model to test two different regressions. One is the initial inspection score and the other is final inspection score. Grading_Post uses a dummy variable mechanic where it equal 1 if it's the initial score (if testing for initial score) and 0 otherwise.

Does anyone know what they also mean by "Seasonal Fixed Effects" and "Restaurant Fixed Effects" that has to be panel data right? I think it's an extra intercept that is unique for each restaurant or season.

Answer 1

First, by "pre-post model", they mean that they have panel data that includes observations before the implementation of the new policy and observations after the implementation. This is common in policy analysis, where folks are interested in estimating the causal impact of a policy. It is unfortunate that they confusingly refer to the policy itself a "Post", meaning the restaurant's grade is posted. The estimate of $\beta_1$ gives the effect on the outcome of the new policy. The authors astutely include trend lines to control for the possibility that the outcome was already on the down (or up) swing, independent of the new policy.

Second, the equation you cite is a fixed effect model. As explained in the paper, $\delta_t$ and $\gamma_i$ are season and zip code fixed effects. You are right in your guess--fixed effects adds an intercept for each factor. It is equivalent to a model that has a series of dummy variables.

$y_{it} = \beta_0 + \beta_1X_{it} + \alpha_i + \epsilon_{it}$

Where $\alpha_i$ is seasonal fixed effects. This model is the same as:

$y_{it} = \beta_0 + \beta_1X_{it} + \beta_2SPRING + \beta_3SUMMER + \beta_4FALL + \epsilon_{it}$

With WINTER left out as a reference group.

NOTE: The indexing of the error term is not an indication of a random effects model. Keep in mind that the term "random effects" is used in many different ways! See this and this. But in econometrics, in contrast to fixed effects, it typically means allowing the intercept and slope to vary between groups. While fixed effects estimates use only within-group variation, random effects use within- and between-group variation. This requires the (at times unreasonable) assumption that the group intercepts are uncorrelated with covariates in the model--i.e that variations between groups are not related to the covariates of interest.

Answer 2

The explanation in your link is pretty clear. This is a type of interrupted time series model (ITS). Grading post is a vector of two quantities that estimate a difference in trend pre and post intervention in terms of a gap or instantaneous change in trend and a difference-in-differences or change in trend line. If either of these are significant you'd say the intervention did something.

You can use fixed effects to model cluster level heterogeneity. It works much the same way as random effects. Simply adjust for cluster indicators and estimate them with maximum likelihood. Seasonal fixed effects is a similar thing: create categorical variables for periodic trend (e.g. Fall/Winter/Summer/Spring or Jan/Feb/Mar.../Dec) and adjust for them in the model.