practitioners who may wish for more functionality and options may have to program it themselves. In an effort to simplify the process, we have created a family of Stata commands which implement small area estimation methods, and provide users with a valid, modular, and flexible alternative to PovMap. All of the results produced with `sae` have been bench-marked with the well established PovMap software, and are an exact match.<sup>6</sup> In the following section we discuss the linking model and focus on the first stage of the estimation. How the variance of the location effect present in the linking model is accounted for is discussed afterwards. This is initiated by presenting the ELL methodology for decomposing the first stage residuals’s variance parameters. Afterwards, this is followed by Henderson’s Method III decomposition of the residual’s variance parameters and the adaptation of Huang and Hidiroglou’s (2003) GLS and implemented by Van der Weide (2014) in an update to the PovMap software. Then we present Empirical Bayes prediction of the location effects. We finally present the command’s syntax along with examples of its use.

# **2 First Stage: Model**

When survey data is not sufficiently precise for reliable estimates at low geographical levels, small area estimation can be implemented. This commonly relies on borrowing strength from outside sources to generate indirect estimates (Rao and Molina, 2015). For example, when attempting to obtain welfare estimates at low geographical levels, it is possible to borrow strength from census data. Censuses in most cases do not collect sufficient information on incomes and/or expenditures. On the other hand household surveys tend to collect detailed information on incomes and/or expenditures which allows the generation of welfare statistics such as poverty and inequality measures. Although welfare statistics can be obtained using household surveys, these are rarely sufficiently large to allow for statistics corrsponding to small areas (Tarozzi and Deaton, 2009). Small area estimation methods attempt to exploit the large sample size of census data and combine this with the information from household surveys in an attempt to obtain reliable statistics for small areas.

In its essence the small area estimation presented here relies on using household survey data to estimate a joint distribution of the variable of interest and observed correlates, and use the parameters to simulate welfare using census data (Demombynes et al., 2008). Throughout the text we focus on welfare imputation, since this is what the Elbers, Lanjouw, and Lanjouw (2003) methodology focused on, nevertheless it can be applied for other continuous measures aside from welfare.

The first step towards small area welfare simulation is a per-capita<sup>7</sup> welfare model, estimated via ordinary least squares ( _OLS_ ), or weighted least squares ( _WLS_ ): where ln _Y_ is a _N ×_ 1 matrix indicating the household’s welfare measure (usually in logarithms, but not necessarily),<sup>8</sup> _X_ is a _N ×K_ matrix of household characteristics, and _u_ which is a _N ×_ 1 matrix of disturbances. To achieve valid small area estimates it is necessary that the set of explanatory variables, _X,_ used in the

> 6For the 1st stage of the analysis. The second stage relies on Monte-Carlo simulations, and thus differences are likely to arise.

> 7Other household equivalization methods are also feasible, for example adult equivalent household welfare.

> 8Logarithmic transformation is usually preferred since it makes the data more symmetrical (Haslett et al., 2010).

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