Granular difference-in-differences with non-repeating unit of observation

Question

I want to analyze changes in characteristics of job postings around an (exogenous) event. However, rather than conducting the analysis at the job poster level (e.g., a company or geographic area), my regression model is estimated at the job posting level:

$Y_{p,t} = \alpha + \beta_1 \times A_t + \beta_2 \times T_i + \beta_3 \times T_i \times A_t + \epsilon_{p,t}$

where $p$ denotes the unique job post ID, $i$ denotes the state (i.e., geographic area) at which the job posting appears (e.g., California) and $A$ denotes a post event time dummy. In this case, the $T_i$ equals $1$ if $i$ is California; otherwise it is set to 0. $Y$ is some characteristic of the job posting (e.g., word count).

Are there any econometric flaws with this specification? Note in the standard difference-in-differences model, we have a balanced panel structure with pre- and post-event observations for each control and treated unit. However, "job postings" is the unit of observation. Obviously, job postings that appear in the post-(pre-)event period won't have a pre-(post-)event observation. If so, are there solutions to rectify the problem?

Answer 1

I recommend the following notation,

$$ Y_{pst} = \beta_1 T_s + \beta_2 A_t + \beta_3 (T_s \times A_t) + \epsilon_{pst}, $$

where you observe some characteristic of job posting $p$ in state $s$ and time $t$. The subscripts fit well with the particular aspects of your study. Whenever I see the $i$ subscript I think "individual" or the smallest unit of observation. You can keep the original notation, just be very clear about what these variables represent. Your description of $T_s$ and $A_t$ are correct and standard for this particular methodology.

Are there any econometric flaws with this specification?

None at all.

The one aspect of this methodology some people aren't aware of is you don't actually need panel data for it to work. You just need to repeatedly sample from the higher level units (states) before and after the event of interest. The sampling procedure should be the same within each aggregate level unit and time period.

Note in the standard difference-in-differences model, we have a balanced panel structure with pre- and post-event observations for each control and treated unit. However, "job postings" is the unit of observation.

You do not need to observe the same job ads over time. Note, this equation is exploiting group level variation of the sampled job ads pre- versus post-event. We do not care about the differences in say, word count, between the individual ads. Rather, its their average contribution within the higher level entities.

Obviously, job postings that appear in the post-(pre-)event period won't have a pre-(post-)event observation. If so, are there solutions to rectify the problem?

It doesn't matter.

The principal regressors vary at the "group" (i.e., state) level, so the equation is specified properly. You can always aggregate the data up to the state level and rerun this equation if you so desire. It will not influence the identification of $\beta_3$, especially in the absence of company or state level covariates. Note, data aggregation is not really a rectification, just another way of arriving at the same answer.