tl;dr: In fixed effects and first difference estimation, does having sets of individuals where the change in $X_{it}$ over time is identical lead to estimation problems?

When using fixed effects (FE) or first difference (FD) to estimate parameters, it is customary to include dummy variables for timepoints in order to control for variables that vary over time but not between individuals. Furthermore, FE and FD removes the effect of variables that vary between individuals but not between them. That is, suppose that the data generating proccess is as follows:

$$Y_{it} = \beta_0 + \beta_1 X_{it} + \beta_2 W_{t} + \beta_3U_i + \varepsilon_{it},$$

where $Y_{it}$ is the outcome of individual $i$ at timepoint $t$, $X_{it}$ is a time-varying variable, $W_t$ is a variable that varies over time but not between individuals and $U_i$ varies between individuals but not over time.

Letting $\Delta$ be the first difference operator such that $\Delta Y_{it} = Y_{it}-Y_{i(t-1)}$, with similar notation for the other variables, regressing $\Delta Y_{it}$ on $\Delta X_{it}$ and the first difference of a set of timepoint dummies, leads to identification of $\beta_1$. Similarly, regressing $Y_{it}$ on $X_{it}$, a set of timepoint dummies and a set of dummies corresponding to the individuals also leads to identification.

However, what happens if there is clustering in $X_{it}$ in the sense that the individuals come from groups that have the same values on $X_{it}$? For instance, if we have individuals from 20 cities and have city-level variables such as crime rate, then $X_{it}=X_{jt}$ if $i$ and $j$ live in the same city. Note that $X_{it}$ and $X_{i(t-1)}$ are different though, so $X_{it}$ varies over time. $Y_{it}$ is different for each individual, same city or not.

Do we still have identification of $\beta_1$ in such a situation? Does the analysis need adjustment for standard errors etc for the fact that $X_{it}$ is grouped?


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