Estimating σ2 for the Classical Linear Regression Model (CLRM) with the Presence of the Modifiable Areal Unit Problem (MAUP)

In a classical linear regression model (CLRM), the magnitude of disturbances is characterized by σ². When individual observations are aggregated into regions, the modifiable areal unit problem (MAUP) appears. The presence of the MAUP brings significant challenges to estimating σ², as the traditional ordinary least square estimator at the individual level, s², becomes downward biased at the aggregate level. Based on the information available before and after the aggregation process, three estimators of σ² at the aggregate level are proposed in this study: the trace estimator, the harmonic estimator, and the arithmetic estimator. Endorsed by Monte–Carlo simulations, these estimators provide significantly better estimates than directly borrowing s² at the aggregate level, but each achieves a different trade-off between the availability of required information and the accuracy of estimates. The findings provide a solid foundation for inferential statistics, such as constructing confidence intervals and performing hypothesis testing for CLRMs at the aggregate level.