Cluster-Robust Heteroskedasticity-Consistent Standard Errors

This repository explains and illustrates a variety of Cluster-Robust Heteroskedasticity-Consistent Standard Error estimators. It was created by Düzgün Dilsiz (University of Basel) for teaching purposes. Note that there is also a website version.

Heteroskedasticity-Consistent (HC) Standard Errors (SE) allow different units having different variance of the error term, rather than assuming all units having the same variance of the error term. In addition to this, cluster-robust HC SE allow for correlation within a cluster. In the following the estimated asymptotic variance for the panel data fixed effects estimator is presented for the standard small sample size correction (Stata and R), and the HC0-HC3 estimators, as computed by the function vcovHC (plm package) in R.

STATA SSS: $\widehat{Avar}\left[\widehat \beta{FE}\right]$ = $\frac{N}{N-1}$ $\frac{NT-1}{NT-K-1}$ $\left(\ddot X' \ddot X \right)^{-1}$ $\Sigma{i=1}^{N}{\ddot X{i}'\widehat{\ddot u{i}} \widehat{\ddot u{i}'}\ddot X{i}}$ $\left(\ddot X' \ddot X\right)^{-1}$

R SSS: $\widehat{Avar}\left[\widehat \beta{FE}\right]$ = $\frac{N}{N-1}$ $\frac{NT-1}{NT-K}$ $\left(\ddot X' \ddot X \right)^{-1}$ $\Sigma{i=1}^{N}{\ddot X{i}'\widehat{\ddot u{i}} \widehat{\ddot u{i}'}\ddot X{i}}$ $\left(\ddot X' \ddot X\right)^{-1}$

R HC0: $\widehat{Avar}\left[\widehat \beta{FE}\right]$ = $\left(\ddot X' \ddot X \right)^{-1}$ $\Sigma{i=1}^{N}{\ddot X{i}'\widehat{\ddot u{i}} \widehat{\ddot u{i}'}\ddot X{i}}$ $\left(\ddot X' \ddot X\right)^{-1}$

R HC1: $\widehat{Avar}\left[\widehat \beta{FE}\right]$ = $\frac{NT}{NT-K}$ $\left(\ddot X' \ddot X \right)^{-1}$ $\Sigma{i=1}^{N}{\ddot X{i}'\widehat{\ddot u{i}} \widehat{\ddot u{i}'}\ddot X{i}}$ $\left(\ddot X' \ddot X\right)^{-1}$

R HC2: $\widehat{Avar}\left[\widehat \beta{FE}\right]$ = $\left(\ddot X' \ddot X \right)^{-1}$ $\Sigma{i=1}^{N}{\ddot X{i}'\tilde{\ddot u{i}} \tilde{\ddot u{i}}'\ddot X{i}}$ $\left(\ddot X' \ddot X\right)^{-1}$

R HC3: $\widehat{Avar}\left[\widehat \beta{FE}\right]$ = $\left(\ddot X' \ddot X \right)^{-1}$ $\Sigma{i=1}^{N}{\ddot X{i}'\tilde{\ddot u{i}} \tilde{\ddot u{i}}'\ddot X{i}}$ $\left(\ddot X' \ddot X\right)^{-1}$

where $\tilde{\ddot u{it}}=\widehat{\ddot u{i}}(1-\ddot h{it})^{-\delta{m}}$ where m=0.5 for HC2 and $m=1$ for HC3, and $\ddot h{it}$ indicating the diagonal elements of $\ddot H{jj}$ of the hat matrix $\ddot H = \ddot X\left(\ddot X' \ddot X \right)^{-1}\ddot X'$.

In the code, you find a small example with crime data from the United States where all the estimators are computed and it is shown that the results are identical to when using the provided functions directly.

File structure

This repository consists of 2 folders: * code: provides the R-code for cluster-robust HC0-HC3 standard error estimators with examples * pictures: resources for website version

License

This work is licensed under a Creative Commons Attribution 4.0 International License.