where _γa_ = _σu_<sup>2</sup><sup>_/_(</sup><sup>_σ_</sup> _u_<sup>2+</sup><sup>_σ_</sup> _ε_<sup>2</sup><sup>_/na_)(seee.g.MolinaandRao,2010).4Fornon-normalerrors,wehave</sup> that _E_ [ _ua|e_ ̄ _a_ ] is generally a non-linear function of _e_ ̄ _a_ , and _var_ [ _ua|e_ ̄ _a_ ] will generally be a function ̄ of _ea_ .

# **2.5 Application to small area estimation of poverty and inequality**

For our application let _yah_ measure per capita income (or expenditure) for household _h_ residing in area _a_ , and let _sah_ denote the number of household members for that same household. In vector notation, let _y_ ( _a_ ) and _s_ ( _a_ ) be vectors with elements _yah_ and _sah_ for all households from area _a_ . The objective is to determine the level of welfare for area _a_ which can be expressed as a function of _y_ ( _a_ ) and _s_ ( _a_ ): _W_ ( _y_ ( _a_ ) _, s_ ( _a_ )). The welfare function _W_ is typically non-linear. Popular examples are the share of individuals whose income falls below a pre-specified poverty line (also known as the head-count poverty rate), or the Gini index of income inequality.

Collecting data on income (or expenditure) _yah_ for any given household is generally found to be expensive relative to collecting data on demographics, education, employment status, and housing. This is particularly true for developing countries where much of the income does not come from wage employment. Consequently, income data is often only available in the form of so-called income surveys. The sample size of these surveys is sufficient to estimate national and possibly sub-regional welfare, but too small to estimate welfare directly at the level of much smaller areas (i.e. the target areas _a_ ).

Elbers et al. (2003; henceforward ELL) advocate an approach that combines the income survey with unit record population census data. The census has data on the independent variables _xah_ from equation (1), such as demographics, education, employment and housing, but not the household income variable _yah_ . Crucially, the data on _xah_ are also collected by the income survey. The idea is to use the income survey to estimate the parameters from equation (1), and then use the model to predict income for every household in the census. With these predicted incomes we can subsequently estimate welfare _W_ for each target area _a_ . Standard errors can be obtained by means of simulation which is ideally suited for estimating quantities that are non-linear functions of the random variables at hand, as is the case with measures of poverty and inequality. Let _R_ denote the number of simulations. The estimator then takes the form: where ̃ _y_ (<sup>(</sup> _a_<sup>_r_)</sup> )<sup>denotes the</sup><sup>_r_-th simulated (or predicted) income vector with elements ̃</sup><sup>_y_</sup> _ah_<sup>(</sup><sup>_r_)=</sup><sup>_x_</sup> _ah_<sup>_Tβ_ ̃(</sup><sup>_r_)+</sup> _u_ ̃<sup>(</sup> _a_<sup>_r_)+ ̃</sup><sup>_ε_(</sup> _ah_<sup>_r_).With each simulation, both the model parameters</sup><sup>_β_ ̃(</sup><sup>_r_)and the errors</sup><sup>_u_ ̃(</sup> _a_<sup>_r_)</sup> and _ε_ ̃<sup>(</sup> _ah_<sup>_r_)are</sup> drawn from their estimated distributions.<sup>5</sup> In the end this gives _R_ simulated poverty rates. The

> 4Note that the unconditional variance solves: _var_ [ _ua_ ] = _var_ [ _γae_  ̄ _a_ ]+(1 _− γa_ ) _σu_ 2<sup>=</sup><sup>_γ_</sup> _a_<sup>2(</sup><sup>_σ_</sup> _u_<sup>2+</sup><sup>_σ_</sup> _ε_<sup>2</sup><sup>_/n_</sup> _a_<sup>)+(1</sup><sup>_−γ_</sup> _a_<sup>)</sup><sup>_σ_</sup> _u_<sup>2,</sup> which5Our preferred method is to drawequals _var_ [ _ua_ ] = _γaσu_<sup>2+ (1</sup><sup>_−γ_</sup> _β_  ̃ _a_ (<sup>)</sup> _r_<sup>_σ_</sup> ) _u_<sup>2</sup> by re-estimating the model parameters using the<sup>=</sup><sup>_σ_</sup> _u_<sup>2,since</sup><sup>_σ_</sup> _u_<sup>2+</sup><sup>_σ_</sup> _ε_<sup>2</sup><sup>_/n_</sup> _a_<sup>=</sup><sup>_σ_</sup> _u_<sup>2</sup><sup>_/γ_</sup> _a_<sup>.</sup> _r_ -th bootstrap version of the survey sample. Alternatively, _β_<sup> ̃(</sup><sup>_r_)</sup> may be drawn from its estimated asymptotic distribution. The difference between these two alternatives is expected to be modest, unless the survey sample is particularly small so that finite sample effects may play a role.

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