Bank, and the EB approach by MR (2010).<sup>2</sup> This paper focuses on unit-level models for small area estimation; in particular, on the traditional ELL (2003) approach, the latest additions to the World Bank toolkit by Van der Weide (2014), and the original EB approach introduced by MR (2010). It sheds light on the nuances of the traditional approach by ELL (2003) and the EB addition by Van der Weide (2014), and it updates these methods in line with the original EB approach by MR (2010). The paper goes in depth into the differences between the estimation of the noise in the ELL approach (as implemented in PovMap and the sae Stata command), and the estimation of the MSE according to MR (2010). Then, the paper proposes: 1) an adaptation of the Monte Carlo simulation procedure by MR (2010) for calculation of the extended EB estimators that incorporate heteroscedasticity and survey weights as proposed by Van der Weide (2014); and 2) an adaptation of the parametric bootstrap method considered by MR (2010), which comes from González-Manteiga et al. (2008), for estimation of the corresponding MSE of the extended EB estimators. According to our simulation results, the adapted procedures for calculation of extended EB point estimates and corresponding MSEs represent a substantial improvement over the current approaches.

The paper reviews the current methods in chronological order. It begins describing the traditional ELL method in Section 2, then moves on to the EB method proposed by MR (2010) in Section 3 and finishes with the additions by Van der Weide (2014) in Section 4. It discusses these methods in Section 5. Then, in Section 6 it describes the proposed computational approaches for calculation of the extended EB estimators and for estimation of their MSEs. In Section 7, several simulation exercises, imitating those performed by MR (2010) but extending them to more realistic scenarios, are conducted to compare the different methods. Finally, conclusions are presented in Section 8.

# **2 Traditional ELL approach**

The Elbers, Lanjouw and Lanjouw (2002, 2003) methodology is the de facto small area estimation approach utilized by the World Bank. The methodology has been widely applied across the globe to produce poverty maps conducted by the institution.<sup>3</sup> The popularity of the methodology can be attributed, to some degree, to the availability of the PovMap software (Zhao, 2006),<sup>4</sup> which was programmed in C and offers users a simple point and click interface. The PovMap software is also incredibly computationally efficient and fast, allowing users, even with limited computing power, to work with census data without facing memory limitations. In 2018, a Stata version of the PovMap software was released (Nguyen et al., 2018). The Stata command “sae” replicates most of the procedures and methods of the original PovMap software, and has become popular because it allows for expansion of the methods available to users.<sup>5</sup> The ELL method assumes that the welfare _ych_ for each household _h_ within each location _c_ in the population is linearly related to a 1 _× K_ vector of characteristics (or correlates) _xch_ for that household, according to the nested error model: where _ηc_ and _ech_ are respectively location and household-specific idiosyncratic errors, assumed to be

> 2A factor that likely contributes to these being well known is the availability of software, that can be easily used to apply these approaches; PovMap (Zhao, 2006) for ELL and R’s sae package (Molina and Marhuenda, 2015)

> 3Poverty mapping is the common name within the World Bank for SAE methodology, where the obtained estimates are mapped for illustrative purposes.

> 4Downloadable from: http://iresearch.worldbank.org/PovMap/PovMap2/setup.zip

> 5Users should be aware that results from Stata and PovMap differ slightly due to the use of different random number generators

2