# A New Estimator for Standard Errors with Few Unbalanced Clusters

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## Abstract

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## 1. Introduction

## 2. Basic Theory: CRVE, CR2VE and CR3VE

## 3. From CR3VE to CR3VE-$\lambda $

## 4. Monte Carlo Simulations

## 5. A Note on Future Research

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Estimated bias of $se(\widehat{\beta})$ based on different methods: 100,000 Monte Carlo simulations.

Std. Deviation Cluster Size | |||||
---|---|---|---|---|---|

Balanced | wa145 | wa289 | wa520 | wa572 | |

4 clusters | |||||

$\widehat{\mathrm{E}}[\mathrm{sd}(\widehat{\beta})]$ | w0.1978 | w0.1967 | w0.1929 | w0.1790 | w0.1745 |

$\widehat{\mathrm{Bias}}[{se}_{UN}(\widehat{\beta})]$ | −0.1820 | −0.1809 | −0.1769 | −0.1628 | −0.1581 |

$\widehat{\mathrm{Bias}}[{se}_{CRVE}(\widehat{\beta})]$ | −0.1293 | −0.1271 | −0.1207 | −0.1069 | −0.1043 |

$\widehat{\mathrm{Bias}}[{se}_{CR2VE}(\widehat{\beta})]$ | −0.0667 | −0.0663 | −0.0644 | −0.0605 | −0.0599 |

$\widehat{\mathrm{Bias}}[{se}_{CR3VE}(\widehat{\beta})]$ | w0.0191 | w0.0192 | w0.0188 | w0.0164 | w0.0157 |

$\widehat{\mathrm{Bias}}[{se}_{CR3VE-\lambda}(\widehat{\beta})]$ | w0.0191 | w0.0184 | w0.0157 | w0.0066 | w0.0040 |

6 clusters | |||||

$\widehat{E}[\mathrm{sd}(\widehat{\beta})]$ | w0.1839 | w0.1837 | w0.1829 | w0.1811 | w0.1807 |

$\widehat{\mathrm{Bias}}[{se}_{UN}(\widehat{\beta})]$ | −0.1709 | −0.1707 | −0.1699 | −0.1679 | −0.1675 |

$\widehat{\mathrm{Bias}}[{se}_{CRVE}(\widehat{\beta})]$ | −0.0775 | −0.0774 | −0.0792 | −0.0844 | −0.0868 |

$\widehat{\mathrm{Bias}}[{se}_{CR2VE}(\widehat{\beta})]$ | −0.0301 | −0.0300 | −0.0325 | −0.0386 | −0.0413 |

$\widehat{\mathrm{Bias}}[{se}_{CR3VE}(\widehat{\beta})]$ | w0.0198 | w0.0208 | w0.0199 | w0.0202 | w0.0195 |

$\widehat{\mathrm{Bias}}[{se}_{CR3VE-\lambda}(\widehat{\beta})]$ | w0.0198 | w0.0204 | w0.0182 | w0.0142 | w0.0120 |

**Table 2.**Estimated bias of $se(\widehat{\gamma})$ based on different methods: 100,000 Monte Carlo simulations.

Std. Deviation Cluster Size | |||||
---|---|---|---|---|---|

Balanced | wa145 | wa289 | wa520 | wa572 | |

4 clusters | |||||

$\widehat{\mathrm{E}}[\mathrm{sd}(\widehat{\gamma})]$ | w1.0209 | w1.0250 | w1.0369 | w1.0847 | w1.1066 |

$\widehat{\mathrm{Bias}}[{se}_{UN}(\widehat{\gamma})]$ | −0.9805 | −0.9843 | −0.9957 | −1.0416 | −1.0623 |

$\widehat{\mathrm{Bias}}[{se}_{CRVE}(\widehat{\gamma})]$ | −0.4700 | −0.4790 | −0.5038 | −0.6066 | −0.6533 |

$\widehat{\mathrm{Bias}}[{se}_{CR2VE}(\widehat{\gamma})]$ | −0.1868 | −0.1953 | −0.2181 | −0.3191 | −0.3703 |

$\widehat{\mathrm{Bias}}[{se}_{CR3VE}(\widehat{\gamma})]$ | w0.1005 | w0.1000 | w0.1023 | w0.1054 | w0.1068 |

$\widehat{\mathrm{Bias}}[{se}_{CR3VE-\lambda}(\widehat{\gamma})]$ | w0.1005 | w0.0960 | w0.0856 | w0.0410 | w0.0225 |

6 clusters | |||||

$\widehat{E}[\mathrm{sd}(\widehat{\gamma})]$ | w0.8306 | w0.8355 | w0.8506 | w0.9059 | w0.9276 |

$\widehat{\mathrm{Bias}}[{se}_{UN}(\widehat{\gamma})]$ | −0.7965 | −0.8013 | −0.8163 | −0.8706 | −0.8919 |

$\widehat{\mathrm{Bias}}[{se}_{CRVE}(\widehat{\gamma})]$ | −0.2478 | −0.2531 | −0.2786 | −0.3628 | −0.3953 |

$\widehat{\mathrm{Bias}}[{se}_{CR2VE}(\widehat{\gamma})]$ | −0.0837 | −0.0861 | −0.1057 | −0.1653 | −0.1894 |

$\widehat{\mathrm{Bias}}[{se}_{CR3VE}(\widehat{\gamma})]$ | w0.0524 | w0.0556 | w0.0514 | w0.0564 | w0.0610 |

$\widehat{\mathrm{Bias}}[{se}_{CR3VE-\lambda}(\widehat{\gamma})]$ | w0.0524 | w0.0537 | w0.0436 | w0.0265 | w0.0223 |

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**MDPI and ACS Style**

Niccodemi, G.; Wansbeek, T.
A New Estimator for Standard Errors with Few Unbalanced Clusters. *Econometrics* **2022**, *10*, 6.
https://doi.org/10.3390/econometrics10010006

**AMA Style**

Niccodemi G, Wansbeek T.
A New Estimator for Standard Errors with Few Unbalanced Clusters. *Econometrics*. 2022; 10(1):6.
https://doi.org/10.3390/econometrics10010006

**Chicago/Turabian Style**

Niccodemi, Gianmaria, and Tom Wansbeek.
2022. "A New Estimator for Standard Errors with Few Unbalanced Clusters" *Econometrics* 10, no. 1: 6.
https://doi.org/10.3390/econometrics10010006