## 2.3 Variance Estimation for Decompositions

Some inequality and multidimensional poverty measures can be decomposed. The decomposition methods in the convey library are limited to group decomposition for inequality measures and sub-indices decomposition for poverty measures.

For instance, the generalized entropy index is an inequality measure that can be decomposed into between and within group components. This sheds light on a very simple question: of the overall inequality, how much can be explained by inequalities between groups and within groups? Since this measure is additively decomposable, we can obtain estimates of the “within-inequality” and “between-inequality” that adds up to the “overall inequality” across groups.3 For a more practical approach, see Lima (2013Lima, Luis Cristovao Ferreira. 2013. The Persistent Inequality in the Great Brazilian Cities: The Case of Brasília.” MPRA Papers 50938. University of Brasília. https://mpra.ub.uni-muenchen.de/50938/.). Since those are estimates, it is useful to obtain estimates of their SEs and, more generally, of their variance-covariance matrix. For poverty measures, a sub-indices decomposition example is the decomposition of the FGT (Foster, Greer, and Thorbecke 1984Foster, James, Joel Greer, and Erik Thorbecke. 1984. “A Class of Decomposable Poverty Measures.” Econometrica 52 (3): 761–66. http://www.jstor.org/stable/1913475.) measure into extension, intensity and inequality components. This allows researchers to explain how each components accounts for the difference between this measure over time and across domains.

The Alkire-Foster class of multidimensional poverty indices (not implemented in the convey library) can be decomposed by both dimensions and by groups, showing how much each group and each dimension contributes to poverty overall. Multidimensional poverty techniques can sometimes help economic and policy analysts understand more precisely who is more affected by inequality and poverty, and how those disparities manifest.

The result of decomposition estimation is a vector of component estimates. In this sense, in addition to the variance estimate for each component, the variance estimation should also account for the covariances across components. This is handled directly through each decomposition function, like svygeidec and svyfgtdec. These functions produce estimates of the variance-covariance matrices using either influence functions or replication-based methods. For examples of the decomposition estimation functions, we direct the reader to the section about the specific decomposition function.