Clustered Standard Error: Practical example

Given that there are 100 groups with each 100 observations. In each group, half of the samples are in treatment group. The homoscedastic standard error is 0.04. Given that the intra-cluster correlation is 0.4, what will be the clustered standard error? 

This is a practical problem to understand how clustered standard error works. In case of presence of intra-correlation between groups, the errors terms are no longer independently and identically distributed (`i.i.d`). This is because observations within group share some common shock. We can split variance of the residuals `(\sigma_\varepsilon^2)` into two portions, that is, idiosyncratic variance `(\sigma_\eta^2)` and common variance `(\sigma_\gamma^2)`. The idiosyncratic variances are `i.i.d`, but common variance are same for all members of a group, that is, they are `i.i.d` between groups.

`\sigma_\varepsilon^2 = \sigma_\eta^2+\sigma_\gamma^2`

The formula below provides the theoretical estimate of clustered standard error and we also simulate for the random distribution to corroborate our theoretical estimate. The theoretical estimate will be close to clustered standard error from simulation.

`\text{Clustered SE} = SE_{Homo}\times \text{Design effect}`

`\text{Design effect} = \sqrt{1+(100-1)\times 0.4`

`\text{Design effect} = 6.37`

`\text{Clustered SE} = 0.04\times 6.37 = 0.2548`