Event Study regression standard errors

Question

When running an 'event study' diff in diff, i.e. :

$y_{i,t} = \sum_{k\neq-1}\beta_k *1\{t=k\} + \lambda_t + \mu_i +error$

where i is a group level(i.e. individual, county etc). and t is time, $\lambda_t$ are time fixed effects and $\mu$ are group fixed effects, and $\beta$ are the event study coefficients, i.e. the diff in diff between event year k relative to event year t=-1, one year before treatment.

When i typically run this specification, I run into errors with my confidence intervals being extremely large for each of the $\beta$'s.

Is there a well defined formula for the standard errors of each coefficient from the treatment effects in the event-study set up? are higher standard errors usually just problem of few observations used to identify each coefficient? I am just curious of the formula so i can think more systematically about what could be driving the high variability of my estimates

Answer 1

Here is a reference on dummy variables that may provide some insight, to quote:

To illustrate dummy variables, consider the simple regression model for a posttest-only two-group randomized experiment. This model is essentially the same as conducting a t-test on the posttest means for two groups or conducting a one-way Analysis of Variance (ANOVA). The key term in the model is β1, the estimate of the difference between the groups. To see how dummy variables work, we’ll use this simple model to show you how to use them to pull out the separate sub-equations for each subgroup. Then we’ll show how you estimate the difference between the subgroups by subtracting their respective equations.

And further:

It should be obvious from the figure that the difference is β1. Think about what this means. The difference between the groups is β1.

One can then use standard least-squares regression theory to supply an estimate of the variance of the respective regression coefficient(s), which is, in the case of the presented example, β1.