Clustered standard errors and time dummies in panel data

Question

Assume a simple linear regression model, I have $i$ firms and $t=17$ periods

$$Y_{it}=\alpha + \beta_2 T_2 + \beta_3 T_3 + \cdots + \beta_{16} T_{16} + \gamma_i + \varepsilon_{it}$$

In this case, $t=1$ is the base period and $T_t$ are dummies for each other periods. $\gamma_i$ represents firm fixed effects.

When I run this regression using clustered standard errors at the firm level, the 95% confidence interval (CI) of the estimates of the coefficients of the time dummies increases with $t$, i.e. CI of $\beta_3$ is higher than CI of $\beta_2$, CI of $\beta_4$ is higher than CI of $\beta_3$, and it keeps on increasing until 16. Which is not the case when I don't cluster the standard errors, the CI intervals remain similar for the different $\beta$s.

I suspect this is logical, but could it be possible to have a simple explanation for this behaviour of the clustered standard errors?

Answer 1

"Any piece of information would be useful!"

First, this seems to be somehow data-dependent. If I consider instead of $Y_{i,t}$ a random variable, I do not get an increasing pattern over time with clustered and robust standard errors.

Second, if three points make a good start for making generalisations, this may be "quite general". This indeed also happens in two other panels on which I made this test (that is: there is a pattern of increase in the standard errors over time when the standard errors are robust and clustered, which is obviously not present otherwise*). *since the standard errors are constant for the time dummies in such a specification.

Hence, I would guess this implies some meaningful increase in variance over time. Maybe you have selected some units at time $T_1$, and their path is progressively diverging regarding variable $Y_{i,t}$?

code on Stata is here: https://sites.google.com/view/acazenave-lacroutz/stack_ansb