metadata: authors: 0: author: Elbers,Chris T.M.
1: author: Van Der Weide,Roy
count: World
docna: 0: docna: Estimation of normal mixtures in a nested
            error model with an application to small area estimation of
            poverty and inequality
docty: Policy Research Working Paper
subtopic: Urban Housing and Land Settlements,Urban Governance and Management,Urban Housing,Demographics,Educational Sciences,Poverty Reduction Strategies,Municipal Management and Reform,Inequality
teratopic: Poverty Reduction
abstracts: cdata!: This paper proposes a method for
            estimating distribution functions that are associated with
            the nested errors in linear mixed models. The estimator
            incorporates Empirical Bayes prediction while making minimal
            assumptions about the shape of the error distributions. The
            application presented in this paper is the small area
            estimation of poverty and inequality, although this denotes
            by no means the only application. Monte-Carlo simulations
            show that estimates of poverty and inequality can be
            severely biased when the non-normality of the errors is
            ignored. The bias can be as high as 2 to 3 percent on a
            poverty rate of 20 to 30 percent. Most of this bias is
            resolved when using the proposed estimator. The approach is
            applicable to both survey-to-census and survey-to-survey prediction.
repnme: repnme: Estimation of normal mixtures in a nested
            error model with an application to small area estimation of
            poverty and inequality
display_title: Estimation of normal mixtures in a
            nested error model with an application to small area
            estimation of poverty and inequality
listing_relative_url: /research/2014/07/19756129/estimation-normal-mixtures-nested-error-model-application-small-area-estimation-poverty-inequality
topicv3: Poverty m1326687 695,Demographic Change m1327040 1448,Education m1327048 470,Land and Housing m1327648 817,Income Inequality m1503891 2668
publishtoextweb_dt: 2014-07-01T00:00:00Z
historic_topic: Poverty Reduction
no_of_pages: 33
author_keys: 0: author_key: 606516
1: author_key: 677092
model_extractions: WPS6962 Policy Research Working Paper 6962 Estimation of Normal Mixtures in a Nested Error Model with an Application to Small Area Estimation of Poverty and Inequality _Chris Elbers Roy van der Weide_ Development Research Group Poverty and Inequality Team July 2014
Policy Research Working Paper 6962

# **Abstract**

This paper proposes a method for estimating distribution functions that are associated with the nested errors in linear mixed models. The estimator incorporates Empirical Bayes prediction while making minimal assumptions about the shape of the error distributions. The application presented in this paper is the small area estimation of poverty and inequality, although this denotes by no means the only application. Monte-Carlo simulations show that estimates of poverty and inequality can be severely biased when the non-normality of the errors is ignored. The bias can be as high as 2 to 3 percent on a poverty rate of 20 to 30 percent. Most of this bias is resolved when using the proposed estimator. The approach is applicable to both survey-to-census and survey-to-survey prediction.

This paper is a product of the Poverty and Inequality Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at rvanderweide@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
Estimation of normal mixtures in a nested error model with an application to small area estimation of poverty and inequality Chris Elbers and Roy van der Weide<sup>1</sup> Keywords: Normal mixtures, linear mixed models, small area estimation, Empirical Bayes, poverty, inequality JEL Classification: I32, C31, C42

> 1 Chris Elbers (c.t.m.elbers@vu.nl) is at VU University Amsterdam and the Tinbergen Institute. Roy van der Weide (rvanderweide@worldbank.org) is at the World Bank. We gratefully acknowledge financial support from VU University Amsterdam and the World Bank's Knowledge for Change Program (KCP II). A thank you also goes to Peter Lanjouw for providing comments on earlier versions of this paper.
# **1 Introduction**

We propose a method for estimating distribution functions that are associated with the nested errors in linear mixed models. The proposed estimator is accommodating Empirical Bayes prediction and makes minimal assumptions about the shape of the error distributions. Our objective is to accurately predict nonlinear functions of the dependent variable, where the shape of the error distribution functions potentially plays an important role. The application we have in mind is the small area estimation of poverty and inequality, although this denotes by no means the only application. This particular application has been made popular by the work of Elbers, Lanjouw and Lanjouw (2003; henceforward ELL). The approach they put forward has since been applied to obtain maps of poverty and inequality in over 60 countries worldwide. The highly disaggregated estimates of poverty and inequality have in turn inspired a range of other applications, for example: Demombynes and Ozler (2005) investigate whether inequality at the small area level has an impact on local crime rates, and find that it does using data from South Africa; Elbers et al. (2007) conduct an empirical experiment in order to estimate by how much one could potentially lower the costs of getting resources to the poor if one had access to a poverty map, and conclude that the gains can be substantial; Araujo et al. (2008) examine whether villages with higher levels of inequality are less likely to invest in public goods that would benefit the poor. Using data from Ecuador they find empirical support for this hypothesis which they attribute to elite capture. Recently, Fujii (2010) modified the approach to make it better suited for the small area estimation of child malnutrition outcomes with an application to Cambodia.

## **1.1 Problem statement**

Consider the following standard linear mixed model for log income of household _h_ residing in area _a_ : Let _yd_ denote a vector of log incomes for all households from domain _d_ (where “domain” typically refers to a small geographic area). We are interested in estimating _W_ ( _yd_ ), where _W_ is some (possibly nonlinear) function of _yd_ . It is assumed that we have data on _x_ for the entire population, while data on both _x_ and _y_ is available for a sample _S_ of households only. ELL propose to estimate _W_ by: where the expectation is taken over the unobserved errors _ua_ and _εah_ . Because _W_ is nonlinear, the shape of the error distributions will matter for the expected value of _W_ . Estimation of _β_ is obviously of primary importance when estimating _E_ [ _W_ ], but in this paper we will concentrate on the error distributions and assume – except for the application in section 6 – that _β_ is known. We argue that even if _β_ is estimated perfectly, getting the error distribution wrong still has the 2
potential to introduce a significant bias.

ELL felt their approach would be most convincing if they make minimal assumptions about the errors _ua_ and _εah_ . They proceed by first obtaining estimates of the area errors _ua_ and _εah_ , ˆ ˆ which then allows them to sample from the empirical errors _ua_ and _εah_ . _ua_ is estimated as the simple area average of the total residuals (appropriately re-scaled so that the sample variance of _u_ ˆ _a_ equals the estimate of _σu_<sup>2thatcorrectsforthecontributionoftheareaaverageof</sup><sup>_εah_).</sup> The estimate for _εah_ is obtained by subtracting _u_ ˆ _a_ from the total residual (re-scaled so that its sample variance matches _σε_<sup>2).1</sup> This non-parametric approach to estimating the error distributions has two limitations. Firstly, this procedure does not adequately account for the fact that _u_ ˆ _a_ equals a sum of _ua_ ̄ and _εa_ . This latter term is often large enough for it to affect the ability of the empirical ˆ distribution of _ua_ to reproduce the shape of the actual distribution of _ua_ . Simply ignoring this “contamination” may result in a significant bias. If the empirical distribution of _ua_ is biased, then this bias will also have implications for the empirical distribution of _εah_ . This is also referred to as a “convolution problem”, where the objective is to estimate the distribution function of a random variable that is observed with error.

Secondly, the approach adopted by ELL does not easily lend itself for Empirical Bayes (EB) estimation where the distribution of _ua_ is tightened by conditioning on household data _y_ and _x_ available for domain _d_ (i.e. in the event that some households in domain _d_ are included in the sample). Working out the conditional distribution is not a trivial exercise without making further distributional assumptions. Consequently ELL decided to forego EB estimation altogether. In doing so, they have accepted a certain loss in efficiency by not fully utilizing all available information. Molina and Rao (2010) recently picked up on this and put forward an alternative approach that does implement EB estimation. They take ELL as a point of departure but then assume that both _ua_ and _εah_ are normal distributed, in which case the conditional distribution too will be normal distributed. Where ELL accept a loss in precision by not implementing EB estimation, Molina and Rao (2010) accept a loss in precision that might stem from a misspecification of the error distribution functions. The data at hand will ultimately determine which of the two will be accepting the larger loss. ELL are arguably most interested in estimating poverty and inequality in developing countries where the number of small areas (or domains) that are covered by the income surveys are often small, think of 5 to 25 percent of all domains in the population. In this case the benefits of EB estimation will be modest (as survey data are available for only few areas). However, in more developed countries, or countries where travel costs that are incurred when covering all small areas are manageable, income surveys often cover a much larger number of the domains. In fact, there are numerous examples where surveys cover between 50 and 100 percent of all domains in the country. In those instances, there may be clear benefits to adopting EB estimation.

The approach presented in this paper improves on both ELL and Molina and Rao (2010). Like ELL we make no restrictive assumptions about the error distributions. Our estimator

> 1Note that ELL allows for heteroskedasticity, so that _σε_ 2<sup>canbehousehold-specific.</sup>

3
for the distribution functions will generally be more accurate than the estimator adopted by ELL however, as we explicitly account for the nested error structure (that is responsible for the “convolution problem”). Unlike ELL we also accommodate EB estimation. We achieve this by fitting finite normal mixtures (NM) to the error distribution functions. Normal mixtures are extremely flexible; they are able to fit any well-behaved distribution function, and are ideally suited for accommodating EB estimation. If the marginal distributions of _ua_ and _εah_ can be described by normal mixtures, then the conditional distribution too can be described by a normal mixture with known parameters (that are functions of these parameters and of the data on which is being conditioned). Estimation of the normal mixtures for _ua_ and _εah_ is complicated by the fact that neither _ua_ nor _εah_ are observed. Our estimator for the NM parameters may be viewed as a modified version of the EM algorithm.

Monte Carlo simulations indicate that estimates of poverty and inequality can be severely biased when ignoring the non-normality of the errors. The bias can be as high as 2 to 3 percent on a poverty rate of 20 to 30 percent. Most of this bias is resolved when implementing our estimator. This is confirmed by an empirical application to US data.

# **1.2 Normal mixtures in nested-error models**

There are a number of other studies that have explored different ways of relaxing the normality assumption in mixed linear models. Verbeke and Lesaffre (1996) is an early example that also considers normal mixtures as a “non-parametric” representation of the error distribution function. However, they impose a number of important restrictions. First, only the area random effects _ua_ are allowed to be non-normal; a normal-mixture is fitted to the distribution of _ua_ under the assumption that _εah_ is normally distributed. Second, it is assumed that the “component distributions” that make up the normal mixture share a common variance, which noticeably simplifies estimation but at the same time significantly limits the flexibility of the normal mixture to fit any given distribution function. This approach has also been followed by Cordy and Thomas (1997) who work with the same setup and adopt the same set of restrictions. There is another strand of the literature that permits both error terms to be non-normal distributed by imposing an alternative parametric family for the distribution functions. See for example Zhou and He (2008) who fit skewed t-distributions to the nested errors of the linear mixed model. Recently, there have also been efforts to explore the impact of misspecifications in the error distributions for Empirical Bayes predictions derived from linear mixed models, see for example Skrondal and Rabe-Hesketh (2009) and McCulloch and Neuhaus (2011). They conclude that the bias is reasonably small. It should be noted however that those studies focus on prediction of the dependent variable itself, in which case the Empirical Bayes estimates of the area random effects _ua_ denote the only source of bias. Misspecifications in the error distributions become considerably more important when predicting nonlinear functions of the dependent variable, such as measures of poverty and inequality, as we will show in this paper.

The remainder of this paper is organized as follows. In Section 2 we briefly discuss how the error distribution functions associated with the errors in a linear mixed model will matter for 4
prediction, with an application to poverty and inequality measurement. In this section we will also introduce normal mixtures as a flexible “non-parametric” representation of any given wellbehaved distribution function, and demonstrate the implications for EB estimation. Estimation of the normal mixture distributions to both errors from the linear mixed model is presented in Section 3. A modest Monte Carlo simulation study followed by an equally modest empirical application is provided in Sections 4 and 5, respectively. Finally, Section 6 concludes.

# **2 Estimation of poverty and inequality: Distributions matter**

## **2.1 Linear mixed model for income**

Suppose that at the household level the data generating process (DGP) satisfies the equation already mentioned above: where _xah_ denotes a vector with independent variables, and where _ua_ and _εah_ denote zero expectation error terms that are independent of each other. The subscripts indicate target area (or domain) _a_ and household _h_ .<sup>2</sup> In this paper we assume that errors are homoskedastic, so that for each household _h_ and area _a_ we have: _var_ [ _yah|xah_ ] = _σu_<sup>2+</sup><sup>_σ_</sup> _ε_<sup>2.Throughoutthepaper</sup> it is assumed that consistent estimators for the variance parameters are available, which we shall denote by _σ_ ˆ _u_<sup>2and</sup><sup>_σ_ˆ</sup> _ε_<sup>2.3Wewillnotmakeanyassumptionsabouttheshapeoftheerror</sup> distributions.

Let _A_ denote the total number of areas covered by the income survey and let _na_ denote the number of households that have been sampled in area _a_ , so that _n_ =<sup>�</sup><sup>_A_</sup> _a_ =1<sup>_na_denotesthetotal</sup> sample size of the survey. We shall denote the total household error by: _eah_ = _yah − x_<sup>_T_</sup> _ah_<sup>_β_,and</sup> ̄ ̄ ̄ its area _a_ average by _ea_ = _ya − x_<sup>_T_</sup> _a_<sup>_β_where</sup><sup>_e_ ̄</sup><sup>_a_=�</sup> _h_<sup>_eah/na_.Weshallalsousethenotation</sup> _ea_ = ( _ea,_ 1 _, . . . , ea,na_ ) which denotes the vector of length _na_ with residuals for all households ̄ from area _a_ . With a slight abuse of terminology we will at times refer to the errors _eah_ and _ea_ as data (as if we know the parameter vector _β_ ).

Let the probability distribution functions for _ua_ and _εah_ be denoted by _Fu_ and _Gε_ . We will propose a computationally attractive method for estimating these distribution functions, where we make no restrictive assumptions concerning their functional form, in particular allowing these functions to be other than normal distribution functions. Another appealing feature of our non-parametric estimator for the error distribution functions is that it can easily accommodate Empirical Bayes estimation.

> 2These areas may refer to geographic areas such as districts or municipalities, but also to non-geographic domains such as ethnic groups or age groups, say.

> 3A commonly used estimator for the variance parameters from a nested error model is Henderson’s method III estimator (see Henderson, 1953; and Searle et al., 1992), which may be viewed as a method of moments estimator which does not require any assumptions about the shape of the error distributions. Alternative estimators are restricted maximum likelihood, minimum norm quadratic unbiased estimation (MINQUE; see e.g. Westfall, 1987; Searle et al., 1992), and so-called spectral decomposition estimation (see e.g. Wang and Yin, 2002; and Wu et al., 2009).

5
# **2.2 Empirical Bayes estimation**

Empirical Bayes (EB) estimation, also known as Empirical Best estimation, tightens the error distributions by conditioning on all available data in the survey for the purpose of prediction. The ‘observation’ _ea_ (provided that area _a_ is covered by the survey) clearly carries information ̄ about the area random effect _ua_ . In the extreme, for example, where _na_ tends to infinity _ea_ will perfectly reveal _ua_ .

In effect EB estimates are obtained by integrating out area errors _ua_ using probability density _p_ ( _ua|ea_ ), i.e. the probability density of _ua_ conditional on _ea_ observed from the survey. (Non-EB estimates are obtained by using the unconditional density _p_ ( _ua_ ).) The challenge is to work out _p_ ( _ua|ea_ ) along with the marginals _p_ ( _ua_ ) and _p_ ( _εah_ ) without imposing restrictions about their form so that the resulting conditional density _p_ ( _ua|ea_ ) can also take on any form. Currently, the literature on EB estimation avoids this challenge by assuming normally distributed marginals, in which case the conditional distribution will be normal too.

For normally distributed errors it can be shown that _p_ ( _ua|ea_ ) can be written as a function ̄ of _ea_ , the mean of sample errors from domain _a_ . This is not true for general error distributions. However, conditioning on the full vector _ea_ becomes computationally intractable. Therefore we will condition on _e_ ̄ _a_ rather than the full vector _ea_ even in the case of non-normal errors.

A second simplification concerns the known regression residuals for sample households. When conditioning on _ea_ , we ignore that for these households the total error _eah_ = _ua_ + _εah_ is known (since it is one of the components of _ea_ ). Instead we will assume that _εah_ is independent of _ea_ even for sample households. The reason is that in practice sample household _h_ cannot be traced among households in the target domain _a_ (households for which only data on _x_ is available). In most practical settings, the error thus introduced will be negligible. In fact, this error tends to zero as the size of the sample relative to the population size tends to zero.

# **2.3 Distributions matter**

Let _y_ ( _a_ ) and _e_ ( _a_ ) denote vectors of length _Na_ with elements _yah_ and _eah_ for all households from the population. Similarly, _x_ ( _a_ ) will denote a matrix with rows given by _x_<sup>_T_</sup> _ah_<sup>for all</sup><sup>_Na_households.</sup> _ya_ , _ea_ and _xa_ will denote the survey sample analogues. As mentioned above our objective is to estimate: where the function _W_ will generally be non-linear.

Non-normality of the errors _u_ and _ε_ will affect the estimates of _E_ [ _W_ ] via two different channels. Firstly, when the function _W_ is indeed non-linear, the expected value _E_ [ _W_ ] will be a function of the higher moments of the distributions of _u_ and _ε_ . Getting these moments wrong, in other words getting the distributions wrong, is then likely to introduce bias. Secondly, if 6
the distributions of _u_ and _ε_ are wrong then the conditional density _p_ ( _u|ea_ ), which concerns EB estimation, will also be wrong. This will affect all moments of the conditional distribution of _u_ , including the first, and hence has the potential to introduce bias in the estimate of _E_ [ _W_ ] even if _W_ is linear. (Note that in the case of linear _W_ , non-EB estimates of _E_ [ _W_ ] are unbiased even if the distributions for _u_ and _ε_ are wrong.) Recently, Skrondal and Rabe-Hesketh (2009) and McCulloch and Neuhaus (2011) have explored the magnitude of the bias that is introduced by getting the first moment of _p_ ( _u|ea_ ) wrong, in the case of linear _W_ , and found it to be modest. Our estimator would therefore be most relevant for the case of nonlinear _W_ .

# **2.4 Empirical Bayes estimation with normal mixtures**

Let us assume that _Fu_ and _Gε_ can be represented by mixture distributions: where the _Fi_ ’s and _Gj_ ’s denote a basis of distribution functions which we will also refer to as components or component distribution functions. The _πi_ ’s and _λj_ ’s denote unknown nonnegative probabilities that satisfy<sup>�</sup> _i_<sup>_πi_=1and�</sup> _j_<sup>_λj_=1,whichwewillalsorefertoas</sup> mixing probabilities. _mu_ and _mε_ denote the number of components used to represent _Fu_ and _Gε_ , respectively. We will denote the probability density functions associated with _Fi_ and _Gj_ by respectively _fi_ and _gj_ .

Mixture distributions are remarkably well equipped to fit any well-behaved distribution function. For example, kernel density estimators are closely related to mixture distributions. We will be working with normal component distributions, so that the mixture distributions are normal mixtures.

**Assumption 1** _The components Fi are normal distribution functions with mean μi and variance σi_<sup>2</sup><sup>_.Similarly,components Gjare normal distribution functions with mean νjand variance_</sup> _ωj_<sup>2</sup><sup>_._</sup>

Note that the modeler is at liberty to work with a different basis of component distributions. This choice does not have real implications for the ability of the mixture to fit a given distribution function.

If _p_ ( _ua_ ) and _p_ ( _εah_ ) are normal-mixtures, then _p_ ( _ua|e_ ̄ _a_ ) is a normal mixture too. This is a powerful result as the integral in equation 3 will generally have to be computed by simulation, and sampling from normal mixtures is straightforward. Lemma 2 shows how the parameters that define _p_ ( _ua|e_ ̄ _a_ ) can be obtained as a function of the parameters of the normal-mixtures _p_ ( _ua_ ) and _p_ ( _εah_ ). Implementing EB estimation is thus as easy as sampling the area errors from the normal-mixture _p_ ( _ua|e_ ̄ _a_ ) whenever _e_ ̄ _a_ is observed in the survey sample.

7
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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.

9
point estimates and their corresponding standard errors are obtained by computing respectively the average and the standard deviation over these simulated values.

It should be noted that ELL draw the area error _u_ ̃ _a_ from the estimated unconditional distribution, which is estimated non-parametrically. (The distribution for _ε_ ̃ _ah_ too is estimated non-parametrically.) The advantage of this approach is that it is fully flexible in that it does not restrict the shape of the error distributions. A possible shortcoming is that it does not take full advantage of all the available data. Ideally one would want to draw _u_ ̃ _a_ from a distribution that is conditioned on all relevant data that has been sampled from area _a_ .

Molina and Rao (2010; henceforward MR) do exactly that, they closely follow ELL but then draw the area error from the conditional distribution. This is referred to as Empirical Bayes (EB) estimation. Importantly, MR also differ from ELL in that they restrict the errors to be normally distributed. As we have seen, under this assumption the conditional distribution of _u_ ̃ _a_ is normal too and its parameter can be easily determined.<sup>6</sup> When errors are non-normal, it is not obvious what form the conditional distribution for _u_ ̃ _a_ will take; it will generally be of a form than the unconditional distribution.

As is noted in section 2.3, getting the error distributions right is not merely a matter of efficiency. When the welfare function _W_ is a non-linear function of the error terms, using wrong error distributions will also introduce a bias in the welfare estimates. Whether the magnitude of this bias due to misspecification is important in practice is an empirical question.

The choice between non-normal errors combined with non-EB estimation (which is more flexible, but does not fully utilize all available data) or normal errors combined with EB estimation (which is less flexible, but fully utilizes the available data) may be determined/motivated by: (a) the degree of non-normality found in the data, and (b) how much information one stands to ignore/lose. The latter depends largely on: (i) how many areas have been sampled by the income survey (as for areas not represented in the survey EB and non-EB estimation are equivalent), and (ii) the size of the area error relative to the total error.

The approach developed in this paper aims to combine the best of both worlds; we adopt EB estimation while permitting non-normal distribution functions. The non-parametric estimator for the distribution functions used by ELL does not lend itself for EB estimation. We propose a non-parametric estimator that does.

**Remark 3** _Note that an added advantage of representing the non-normal error distributions by normal-mixtures is that E_ [ _w_ ( _yah_ )] _can be evaluated as the sum of expectations over normal errors, i.e. E_ [ _w_ ( _yah_ )] = _E_ [ _w_ ( _x_<sup>_T_</sup> _ah_<sup>_β_+</sup><sup>_ua_+</sup><sup>_εah_)] = �</sup> _ij_<sup>_κijE_[</sup><sup>_w_(</sup><sup>_xT_</sup> _ah_<sup>_β_+</sup><sup>_u_</sup> _a_<sup>(</sup><sup>_i_)+</sup><sup>_ε_(</sup> _ah_<sup>_j_))]</sup><sup>_,whereu_(</sup> _a_<sup>_i_)</sup> _and ε_<sup>(</sup> _ah_<sup>_j_)</sup><sup>_arenormaldistributedwithmeansμiandνj,andvariancesσ_</sup> _i_<sup>2</sup><sup>_andω_</sup> _j_<sup>2</sup><sup>_.Thismeansthat_</sup> _evaluating E_ [ _W_ ( _yah_ )] _analytically under normal-mixture errors is no more difficult than under normal errors._

> 6This is arguably why errors are generally assumed normal in the literature on EB estimation.

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# **3 Estimation of normal mixtures when errors are nested**

## **3.1 Estimation of unrestricted normal mixtures**

The objective is to estimate the parameters ( _πi, μi, σi_<sup>2)and(</sup><sup>_λj, νj, ω_</sup> _j_<sup>2),whichdeterminethe</sup> normal mixtures (NM) _Fu_ ( _u_ ) =<sup>�</sup><sup>_i_</sup> _i_<sup>=</sup> =1<sup>_mu_</sup> _πiFi_ ( _u_ ; _μi, σi_<sup>2)and</sup><sup>_Gε_(</sup><sup>_ε_)=�</sup><sup>_i_</sup> _j_<sup>=</sup> =1<sup>_mελjGj_(</sup><sup>_ε_;</sup><sup>_νj, ω_</sup> _j_<sup>2),re-</sup> spectively, where _mu_ and _mε_ denote the number of components.

Estimation of NM to observed data is a routine task, mostly using the EM algorithm, see e.g. Dempster et al. (1977) and McLachlan and Peel, (2000). Suppose that some data _y_ is drawn from a NM with _m_ components and parameter vector _θ_ . The algorithm introduces a latent random variable _zij_ that equals 1 if observation _yj_ has been drawn from component _i_ , and 0 otheriwse. Let the log-likelihood function that treats both _y_ and _z_ as data be denoted by _L_ ( _θ_ ; _y, z_ ), and its derivative with respect to _θ_ by _Lθ_ ( _θ_ ; _y, z_ ). The corresponding maximumlikelihood (ML) estimator solves the following moment condition: The EM algorithm essentially solves the same moment condition but iteratively: where _θ_<sup>ˆ(</sup><sup>_k_)</sup> denotes the iteration- _k_ estimate for _θ_ . Solving equation (15) yields _θ_<sup>ˆ(</sup><sup>_k_+1)</sup> . The procedure results in a non-decreasing sequence of likelihoods _L_ ( _y_ ; _θ_<sup>ˆ</sup><sup>_k_</sup> ). The advantage of the EM algorithm is that solving equation (15) is often considerably easier than solving equation (14). Note that the unobserved data _z_ is integrated out conditional on the observed data _y_ and the current estimate of _θ_ .

The nested error structure however poses a challenge that prevents a straightforward application of the EM algorithm. Conventionally, mixture distributions are estimated to data that are observed directly. We wish to estimate the distributions for _ua_ and _εah_ but we do not observe either of them. In some sense _ua_ and _εah_ are both observed with error (even if the total error _eah_ is known with certainty), which makes this a convolution problem.

We propose a modification of the EM algorithm that is able to deal with this convolution problem. In a nutshell, we treat the measurement error as a latent variable and integrate it out the same way as the latent variable _z_ is integrated out by the EM algorithm. When estimating _Fu_ (the NM for _ua_ ), the observable data includes _e_ ̄ _a_ = _ua_ + ̄ _εa_ , where _ε_ ̄ _a_ =<sup>�</sup> _h_<sup>_εah/na_will act as</sup> ̄ measurement error. The modified EM algorithm will then integrate out _εa_ along with _za_ . When estimating _Gε_ (the NM for _εah_ ), the observable data includes _eah_ = _ua_ + _εah_ . Now _ua_ takes on the role of measurement error and will hence be integrated out jointly with _za_ . Note that we would need some initial estimate of _Gε_ when estimating _Fu_ this way (as we have to integrate out _ε_ ̄). Similarly, we need some initial estimate of _Fu_ when estimating _Gε_ (to integrate out _u_ ). This by itself is very much in the spirit of the EM algorithm which requires an initial estimate of _θ_ to determine to distribution of _z_ conditional on the observed data.

11
~~[——} (——)~~ ( ~~—) [——)~~
that _p_ ( _ua|ea_ ) _≈ p_ ( _ua|e_ ̄ _a_ ). It follows that _ua|e_ ̄ _a_ too is normal-mixture distributed.

**Lemma 5** _The probability density function of ua conditional on_ ̄ _ea, which we denote by p_ ( _ua|e_ ̄ _a_ ) _, is a normal-mixture with known parameters:_

_where:_ ̃ _and where wijk_ = _wijk/_<sup>�</sup> _ijk_<sup>_w_ ̃</sup><sup>_ijkwith:_</sup> _where φ denotes the normal probability density function, and where_ ( _πi, μi, σi_<sup>2)</sup><sup>_and_(</sup><sup>_λj, νj, ω_</sup> _j_<sup>2)</sup> _denote the parameters associated with the normal-mixture distributions Fu and Gε, respectively._ It should be noted that by working with _p_ ( _ua|e_ ̄ _a_ ) instead of _p_ ( _ua|ea_ ) we will be solving a modified moment condition, namely:

This denotes a genuine departure from the original EM algorithm where one conditions on all the data that features in the log-likelihood function. We will refer to the resulting estimator as a pseudo-EM estimator. We expect the loss in precision to be minor, while the gain in practicality is substantial.

The resulting estimators, in the form of iterative equations, are presented below (see Annex 7.3 for a derivation). Given some initial estimate of _p_ ( ̄ _εa|e_ ̄ _a_ ), we may implement the estimator for _Fu_ . Subsequently, given this estimate for _Fu_ , we may implement the estimator ̄ for _Gε_ . The newly obtained estimates can in turn be used to update our estimates for _p_ ( ̄ _εa|ea_ ) and _p_ ( _ua|e_ ̄ _a_ ), after which we may obtain a new round of estimates for _Fu_ and _Gε_ . This is continued until convergence. (In practice, one iteration is found to be sufficient to obtain accurate estimates.)

# **Estimator for** _Fu_

The fixed-point solution to the following set of iterative equations yields the estimator (ˆ _πi,_ ˆ _μi,_ ˆ _σi_<sup>2)</sup> 13
»| <u>: |</u> 5 | ~~<u>) E</u> 5~~ | <u>(</u> ~~ST)~~

=| <u>: |</u> = <u>El ) |</u> = <u>| |</u> = :
In order to get the iterative scheme started, we would of course need suitable initial estimates for _p_ ( ̄ _εa|e_ ̄ _a_ ) and _p_ ( _ua|e_ ̄ _a_ ). Note that even without any prior knowledge about the distributions for _ua_ and _εah_ , we should be able to obtain a reasonable estimate of the distribution for _ε_ ̄ _a_ by appealing to the Central Limit Theorem (CLT). Under the assumption that _εah_ are independend across households, we have that the distribution of _ε_ ̄ _a_ tends to a normal distribution with mean zero and variance equal to _σε_<sup>2</sup><sup>_/na_.Inatypicalhouseholdincomesurvey,</sup><sup>_na_willbeintherange</sup> of 10 to 100, which is sufficiently large for the CLT to take effect. The corollaries below derive the initial estimates for _p_ ( ̄ _εa|e_ ̄ _a_ ) and _p_ ( _ua|e_ ̄ _a_ ) under the assumption of normal distributed _ε_ ̄ _a_ (with known variance).

**Corollary 6** _The probability density function of ε_ ̄ _a conditional on e_ ̄ _a, which we denote by p_ ( ̄ _εa|e_ ̄ _a_ ) _, is a normal-mixture with known parameters:_

̃ _where wi_ = _wi/_<sup>�</sup> _i_<sup>_w_ ̃</sup><sup>_iwith:_</sup> _where φ denotes the normal probability density function, and where πi, μi, and σi_<sup>2</sup><sup>_denotethe_</sup> _parameters associated with the normal-mixture distribution Fu._

**Corollary 7** _The probability density function of ua conditional on e_ ̄ _a, which we denote by p_ ( _ua|e_ ̄ _a_ ) _, is a normal-mixture with known parameters:_

_with γai_ = _σi_<sup>2</sup><sup>_/_(</sup><sup>_σ_</sup> _i_<sup>2+</sup><sup>_σ_</sup> _ε_<sup>2</sup><sup>_/na_)</sup><sup>_,andwhereαi_=</sup><sup>_α_ ̃</sup><sup>_i/_�</sup> _i_<sup>_α_ ̃</sup><sup>_iwith:_</sup> _where φ denotes the normal probability density function, and where πi, μi, and σi_<sup>2</sup><sup>_denotethe_</sup> _parameters associated with the normal-mixture distribution Fu._ The following propositions provide some properties of the estimators.

**Proposition 8** _At every iteration-k, the mean and variance for the probability density function g_ ˆ _ε_<sup>(</sup><sup>_k_)(</sup><sup>_ε_) = �</sup> _j_<sup>_λ_ˆ(</sup> _j_<sup>_k_)</sup><sup>_φ_(</sup><sup>_ε_; ˆ</sup><sup>_ν_</sup> _j_<sup>(</sup><sup>_k_)</sup> _,_ ˆ _ωj_<sup>2(</sup><sup>_k_)</sup> ) _solve:_

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**Proposition 9** _At every iteration-k, the mean and variance for the probability density function f_ ˆ _u_<sup>(</sup><sup>_k_)(</sup><sup>_u_) = �</sup> _i_<sup>_π_ˆ</sup> _i_<sup>(</sup><sup>_k_)</sup> _φ_ ( _u_ ; ˆ _μ_<sup>(</sup> _i_<sup>_k_)</sup> _,_ ˆ _σi_<sup>2(</sup><sup>_k_)</sup> ) _solve:_

# **3.2 Some practical parameter restrictions**

Let us also consider the case where the component distributions are assumed to be given, so that only the mixing probability vectors _π_ and _λ_ will have to be estimated. This is obviously a special case of the more general approach where the probabilities are jointly estimated with the parameters of the component distributions. Keeping the latter fixed markedly benefits the numerical convergence, indicating that the approach deserves to be considered a distinct option in and of itself. This is precisely the approach advocated by Cordy and Thomas (1997).

The estimators for _π_ and _λ_ are obtained as solutions to the following iterative equations: and: ̃ where _πi_ denotes the estimator for _πi_ . Note that Cordy and Thomas (1997) only consider the estimator for _Fu_ (i.e. the estimator for _πi_ for _i_ = 1 _, . . . , mu_ ), as they are not interested in the distribution function for _εah_ .

Since the parameters of the component distributions are not being estimated, the modeler will have to set the values for the means _μi_ and _νj_ , and variances _σi_<sup>2and</sup><sup>_ω_</sup> _j_<sup>2beforehand.Cordy</sup> and Thomas (1997) recommend to set a common variance (as is also done in kernel density estimators where the common variance is refered to as the bandwidth parameter). A natural choice is to set this common variance equal to: _σi_<sup>2=</sup> _mσ_ ˆ _<u>u</u>_<sup>2</sup> _u_<sup>forall</sup><sup>_i_(andhence</sup><sup>_ω_</sup> _j_<sup>2=</sup> _m_<sup>_σ_ˆ</sup> _<u>ε</u>_<sup>2</sup> _ε_<sup>).Note</sup> that for any given common variance _σ_ ̄<sup>2</sup> we have: _var_ [ _u_ ] = _σ_ ̄<sup>2</sup> +<sup>�</sup> _i_<sup>_πiμ_</sup> _i_<sup>2,with�</sup> _i_<sup>_piμ_</sup> _i_<sup>2</sup><sup>_≥_0.</sup> ̄ This means that at a minimum the common variance must be chosen so that: _var_ [ _u_ ] _− σ_<sup>2</sup> _≥_ 0. Obviously this is satisfied for _σ_ ̄<sup>2</sup> = _m_<sup>_σ_</sup> _<u>u</u>_<sup>2</sup> _u_<sup>,whichyields:�</sup> _i_<sup>_πiμ_</sup> _i_<sup>2= (</sup><sup>_<u>m</u>_</sup> _m_<sup>_<u>u−</u>_</sup> _u_<sup><u>1</u>)</sup><sup>_σ_</sup> _u_<sup>2.</sup> A convenient choice for the mean parameters is to take equally spaced _μi_ (and _νj_ ) with a range that respects the values attained by the observed data. Note that the estimated mixing probabilities are expected to satisfy: _E_ [ _u_ ] =<sup>�</sup> _i_<sup>_π_ˆ</sup><sup>_iμi_=�</sup> _a_<sup>_e_ ̄</sup><sup>_a/A_(matchingfirstmoment),as</sup> well as:<sup>�</sup> _i_<sup>_π_ˆ</sup><sup>_iμ_</sup> _i_<sup>2= (</sup><sup>_<u>m</u>_</sup> _m_<sup>_<u>u−</u>_</sup> _u_<sup><u>1</u>)ˆ</sup><sup>_σ_</sup> _u_<sup>2(matching second moment), although this is not guaranteed by the</sup> estimation approach. This is in contrast with the approach where the component distribution parameters are also being estimated, in which case the first and second moments are guaranteed.

16
**Remark 10** _One possible extension to the set of component distributions proposed above (with different μi but common σ_<sup>2</sup> _) can be obtained by adding components with common zero mean (i.e. μr_ = 0 _) but different variances σr_<sup>2</sup><sup>_._</sup>

# **4 A small simulation study**

This section presents a modest Monte-Carlo simulation experiment. We will focus our attention to the following two questions: (1) How effective is our approach in fitting non-normal error distributions (that are not necessarily normal-mixtures)?, and (2) What are the implications for the estimation of poverty and inequality? Do distinct deviations from normality have the potential to introduce a meaningful bias in estimates of poverty and inequality if this nonnormality is ignored?

We make the following assumptions. The simulated “country” consists of 500 domains (or target areas), which denotes the level at which measures of poverty and inequality will be estimated. Each domain is home to 3000 households. Household per capita incomes will be generated by means of the following model: where _yah_ denotes the log of household income, and where _xah_ represents a single covariate. For the model parameters we consider the values: _var_ [ _ua_ + _εah_ ] = _σu_<sup>2+</sup><sup>_σ_</sup> _ε_<sup>2= 0</sup><sup>_._3,</sup><sup>_β_= 1,</sup><sup>_E_[</sup><sup>_xah_] = 0,</sup> and where _var_ [ _xah_ ] is chosen so that _R_<sup>2</sup> = 0 _._ 4 (which represents a goodness-of-fit that is typical for empirical household income models).

It will be convenient to introduce a parameter that measures the size of the random area effect relative to the total error term. Let us denote this parameter by _ρ_ = _σu_<sup>2</sup><sup>_/_(</sup><sup>_σ_</sup> _u_<sup>2+</sup><sup>_σ_</sup> _ε_<sup>2).We</sup> will be considering both the case of small “area effect” ( _ρ_ = 0 _._ 05) and medium/large “area effect” ( _ρ_ = 0 _._ 25).

The non-normal errors _ua_ and _εah_ will be drawn from a Log-Dagum distribution with probability density function: where _p >_ 0 is the parameter that determines the shape (or skewness) of the distribution. Smaller values of _p_ will result in larger deviations from normality. The remaining two parameters _a_ and _b_ will be fixed by imposing zero mean and setting the variance to _σu_<sup>2or</sup><sup>_σ_</sup> _ε_<sup>2.Note that the</sup> Dagum distribution is not an uncommon choice when modeling income data (see e.g. Kleiber (2007), and the references therein). The covariate _xah_ will be drawn from a normal distribution. Finally, our artificial survey will sample 15 households from each domain.

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# **4.1 Estimation of** _Fu_ **and** _Gε_

The first test will be whether we can successfully uncover the probability density functions for _ua_ and _εah_ . We will keep the shape parameter for _Fu_ fixed at _pu_ = 0 _._ 5, but will consider three different values of the shape parameter for _Gε_ : _pε_ = (0 _._ 10 _,_ 0 _._ 25 _,_ 0 _._ 50). The benchmark density functions are obtained as a kernel density estimate applied to the actual realizations of the errors in the census. Our estimators for _Fu_ and _Gε_ are based on normal-mixtures with 3 and 2 component distributions, respectively.

Figure 1 presents our estimates of the probability density function for _ua_ . The estimates in the right panel show a remarkably good fit. These correspond to the case where the random area effect is large, with _ρ_ = 0 _._ 25, in which case the contribution of _ua_ matters. The imperfect fit shown in the left panel suggests that it is harder to estimate _Fu_ when the random area effect is small (in this case _ρ_ = 0 _._ 05). This is arguably due to the poor signal-to-noise ratio in this case; _ua_ will only make up a small fraction of the total residual which is what is observed. However, this lack of precision will also have little to no implication for estimates of poverty and inequality, precisely because of the smallness of _ua_ . In sum, in this example estimates of _Fu_ are precise when they matter, and less precise when they matter less.

Probability density function for _u_ <!-- Start of picture text --> −0.6 −0.4 −0.2 0.0 0.2 0.4 −1.5 −1.0 −0.5 0.0 0.5 1.0<br>x x<br>3.5 1.5<br>3.0<br>2.5<br>1.0<br>2.0<br>1.5<br>dens.uhat(x) dens.uhat(x)<br>0.5<br>1.0<br>0.5<br>0.0 0.0<br><!-- End of picture text --> Figure 1: Normal-mixture density (Black) versus benchmark density (Red): (a) _ρ_ = 0 _._ 05 (left panel); (b) _ρ_ = 0 _._ 25 (right panel) Our estimates for the probability density functions for _εah_ are presented in Figure 2. What is apparent is that, despite using no more than two components, our estimates provide remarkably accurate fits regardless of the size of the random area effect. When _ρ_ is small, we have that _εah_ is large relative to _ua_ , and so the signal-to-noise ratio is in our favour when estimating _Gε_ . It matters less in this case that we do not have a precise estimate for _Fu_ . On the other hand, when _ρ_ is large and hence _εah_ is of a more modest magnitude relative to _ua_ , our estimation of _Gε_ may be helped by having a precise estimate of _Fu_ . (Note that we first estimate _Fu_ , and then use this estimate to subsequently estimate _Gε_ .) 18
## Probability density function for _ε_

<!-- Start of picture text --> −4 −3 −2 −1 0 1 2 −3 −2 −1 0 1 2<br>x x<br>1.0<br>0.8<br>0.8<br>0.6<br>0.6<br>0.4<br>0.4<br>dens.epshat(x) dens.epshat(x)<br>0.2 0.2<br>0.0 0.0<br><!-- End of picture text --> Figure 2: Normal-mixture density (Black) versus benchmark density (Red): (a) _ρ_ = 0 _._ 05 (left panel); (b) _ρ_ = 0 _._ 25 (right panel)

# **4.2 Implications for estimates of poverty**

Next we investigate whether having more precise estimates of _Fu_ and _Gε_ will also give us more precise estimates of poverty, and whether any gain in precision is economically meaningful. For ease of exposition we will focus on the percentage of households with incomes below the poverty line as the measure of poverty which we wish to estimate. Tables 1 and 2 provide estimates of the bias for different values of the shape parameter for _Gε_ (the shape parameter for _Fu_ is fixed at _pu_ = 0 _._ 5), and for different values of the (log) poverty line. The poverty rates associated with the different (log) poverty lines are roughly 15, 20, 30, and 45 percent. It should be noted that the bias in this case is estimated as the difference between the estimated and the true poverty rate averaged over the 500 target areas for one given replication of the “census”.

At least two observations stand out. First, estimates obtained under the (incorrect) assumption of normal errors can be severly biased, with a bias of 3 to 4 percent on a poverty rate that ranges between 20 and 30 percent depending on the shape parameter for _Gε_ . Our approach provides far superior estimates of poverty in these cases with a bias of less than a percent. Second, the benefit of accommodating non-normal errors as opposed to assuming normal errors changes with the value of the poverty line. At the far left tail of the (log) income distribution (i.e. for particularly low values of the poverty line), it will be hard to empirically separate the two distributions. The difference will be more pronounced where the distributions have more “mass”, i.e. for intermediate to higher values of the poverty line.

# **4.3 Implications for estimates of inequality**

Table 5 compares the bias for estimates of income inequality for different values of the shape parameter and different values of the area location effect. We focus on the Mean-Log-Deviation (MLD) as the measure of inequality which we wish to estimate. Note that by definition we have that the realizations of the area error _ua_ will drop out of the true measure of MLD for that area, 19
|Poverty|_ρ_=|0_._05|||_ρ_= 0_._25||
|---|---|---|---|---|---|---|
|line/shape|0.50<br>0.|25|0.10<br>0|.50|0.25|0.10|
|-0.75|0.67 (1.61)<br>0.64|(1.69)|1.32 (2.06)<br>1.38|(2.51)|1.40 (2.70)|1.06 (2.55)|
|-0.50|0.76 (2.51)<br>0.82|(3.29)|1.35 (4.01)<br>1.39|(3.09)|1.55 (3.78)|1.32 (3.92)|
|-0.25|0.67 (2.73)<br>0.92|(4.16)|1.14 (5.15)<br>1.01|(2.82)|1.32 (3.92)|1.34 (4.37)|
|0|0.31 (1.96)<br>0.73|(3.60)|0.60 (4.51)<br>0.20|(1.53)|0.61 (2.73)|0.89 (3.32)|

Table 1: Bias for EB estimates of head-count poverty: (a) Normal-Mixture distribution (left number); (b) Normal distribution (right number)

|Poverty||_ρ_= 0_._05||_ρ_=|0_._25||
|---|---|---|---|---|---|---|
|line/shape|0.50|0.25<br>0.|10|0.50<br>0.|25|0.10|
|-0.75|0.19 (0.98)|0.12 (1.07)<br>0.91|(1.45)|0.23 (0.79)<br>0.22|(0.97)|-0.08 (0.85)|
|-0.50|0.25 (1.95)|0.22 (2.73)<br>0.82|(3.47)|0.36 (1.63)<br>0.41|(2.31)|0.16 (2.48)|
|-0.25|0.30 (2.40)|0.44 (3.83)<br>0.62|(4.82)|0.41 (2.03)<br>0.57|(3.13)|0.53 (3.60)|
|0|0.23 (19.7)|0.58 (3.60)<br>0.35|(4.51)|0.28 (1.66)<br>0.57|(2.86)|0.79 (3.47)|

Table 2: Bias for NON-EB estimates of head-count poverty: (a) Normal-Mixture distribution (left number); (b) Normal distribution (right number)

|Poverty||_ρ_= 0_._05|||_ρ_= 0_._25||
|---|---|---|---|---|---|---|
|line/shape|0.50|0.25|0.10|0.50|0.25|0.10|
|-0.75|0.03 (0.03)|0.03 (0.03)|0.03 (0.03)|0.04 (0.05)|0.04 (0.05)|0.04 (0.05)|
|-0.50|0.04 (0.05)|0.04 (0.05)|0.04 (0.06)|0.05 (0.06)|0.05 (0.06)|0.05 (0.06)|
|-0.25|0.05 (0.06)|0.05 (0.07)|0.05 (0.07)|0.06 (0.07)|0.06 (0.07)|0.06 (0.08)|
|0|0.06 (0.06)|0.06 (0.07)|0.06 (0.07)|0.07 (0.07)|0.07 (0.07)|0.07 (0.08)|

Table 3: RMSE for EB estimates of head-count poverty: (a) Normal-Mixture distribution (left number); (b) Normal distribution (right number)

|Poverty|_ρ_=|0_._05|||_ρ_= 0_._25||
|---|---|---|---|---|---|---|
|line/shape|0.50<br>0|.25|0.10|0.50|0.25|0.10|
|-0.75|0.04 (0.04)<br>0.04|(0.04)|0.04 (0.04)|0.10 (0.10)|0.09 (0.09)|0.09 (0.09)|
|-0.50|0.05 (0.06)<br>0.05|(0.06)|0.05 (0.06)|0.12 (0.12)|0.12 (0.12)|0.12 (0.12)|
|-0.25|0.07 (0.07)<br>0.07|(0.08)|0.07 (0.08)|0.14 (0.15)|0.14 (0.15)|0.14 (0.15)|
|0|0.07 (0.08)<br>0.07|(0.08)|0.08 (0.09)|0.15 (0.15)|0.16 (0.16)|0.16 (0.16)|

Table 4: RMSE for NON-EB estimates of head-count poverty: (a) Normal-Mixture distribution (left number); (b) Normal distribution (right number) which varies around 20 for our simulated data (we multiplied the numbers by a factor 100). Note however that the size of the “location effect” may still have a bearing on our estimates of inequality, to the extent that it impacts on our estimates of the distribution for _εah_ . Similarly to what we observed for poverty, the potential gain of accommodating non-normal errors as opposed to incorrectly assuming normal errors can be quite large. We observe biases of 2 to 3 with true inequality around 20 when assuming normal errors, compared to biases of roughly 0 _._ 5 when using our approach.

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||S|hape paramet|er|
|---|---|---|---|
|MLD|0.50|0.25|0.10|
|_ρ_= 0_._05|0.36 (1.67)|0.52 (2.59)|1.12 (3.71)|
|_ρ_= 0_._25|0.37 (1.26)|0.61 (2.18)|0.47 (2.25)|

Table 5: Bias for estimates of inequality: (a) Normal-Mixture distribution (left number); (b) Normal distribution (right number)

# **5 A small empirical study**

For this small empirical application we will treat the 2010 micro census from the United States, a 1 percent sample, as a “new” country (i.e. as if the data constitutes a full population census). The data is publically available at IPUMS USA. There are a total of 1 _._ 25 million households<sup>7</sup> residing in 422 counties, which is the level at which we will be estimating poverty and inequality (i.e. the “small area level”). The census includes individual income data as well as data on a wide range of individual characteristics. We use the income data to compute household income per capita. All other individual level variables are also aggregated at the household level as this denotes the level at which we will build the income model. We defined the head of the household as the member of the household with the highest level of individual income. (In many empirical applications much of the available data is at the household level.) Table 6 provides some descriptive statistics of the data.

|Variable|Description|Mean|Std. Dev.|
|---|---|---|---|
|lnycap|Log of household income per capita|9.939|0.998|
|metro<br>~~s~~uburb|In sub-urb of major city (dummy)|0.290|0.454|
|metro<br>~~n~~one|Away from major city (dummy)|0.201|0.401|
|hh<br>~~s~~ize<br>~~1~~|Household of size one (dummy)|0.300|0.458|
|hh<br>~~s~~ize<br>~~2~~|Household of size two (dummy)|0.342|0.474|
|hh<br>~~s~~ize<br>~~3~~|Household of size three (dummy)|0.147|0.354|
|share<br>~~c~~hild15|Share of children aged younger than 15|0.108|0.199|
|hh<br>~~h~~edu2|At least one member with a masters degree (dummy)|0.366|0.482|
|hh<br>~~e~~mployed|Number of employed household members|1.104|0.936|
|hd<br>~~f~~emale|Head of household is female (dummy)|0.417|0.493|
|hd<br>~~l~~ogage|Log of age of household head|3.889|0.371|
|hd<br>~~l~~edu|Household head has lower education only (dummy)|0.048|0.214|
|hd<br>~~h~~edu1|Household head has college degree (dummy)|0.226|0.418|
|hd<br>~~h~~edu2|Household head has a masters degree (dummy)|0.312|0.463|
|hd<br>~~e~~mployed|Household head is employed (dummy)|0.660|0.474|
|hd<br>~~h~~isp|Household head is Hispanic (dummy)|0.091|0.288|
|hd<br>~~b~~lack|Household head is African-American (dummy)|0.106|0.308|

Table 6: Descriptive statistics of variables used in empirical application (derived from 2010 IPUMS USA) We draw a balanced sample of 6330 households (15 households per county). The log of

> 7The exact number of households in the census is 1 _,_ 242 _,_ 967.

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household per capita income will be our dependent variable. Our independent variables include: urbanity, household size, age composition, gender, ethnicity, education, and employment. There are a total of 16 regressors (excluding the constant). The regression model is shown in Table 7. We obtained an adjusted R-squared of 0 _._ 432, which denotes a typical level for this type of income (or consumption) regression model. The location effect is estimated at 0 _._ 027, which is not unusual for developed nations but rather small for developing countries where one might find location effects in the range of 0 _._ 05 to 0 _._ 25.

||lnyc<br>|ap<br>|
|---|---|---|
|metro<br>suburb|0.171<sup>_∗∗∗_</sup>|(8.21)|
|metro<br>none|-0.0542<sup>_∗_</sup>|(-1.68)|
|hh<br>~~s~~ize<br>1|0.733<sup>_∗∗∗_</sup>|(16.32)|
|hh<br>~~s~~ize<br>2|0.620<sup>_∗∗∗_</sup>|(16.00)|
|hh<br>~~s~~ize<br>3|0.344<sup>_∗∗∗_</sup>|(9.73)|
|share<br>~~c~~hild15|-0.160<sup>_∗∗_</sup>|(-2.20)|
|hh<br>~~h~~edu2|0.277<sup>_∗∗∗_</sup>|(6.47)|
|hh<br>~~e~~mployed|0.257<sup>_∗∗∗_</sup>|(14.24)|
|hd<br>~~f~~emale|-0.212<sup>_∗∗∗_</sup>|(-10.80)|
|hd<br>~~l~~ogage|0.898<sup>_∗∗∗_</sup>|(30.46)|
|hd<br>~~l~~edu|-0.331<sup>_∗∗∗_</sup>|(-6.40)|
|hd<br>~~h~~edu1|0.175<sup>_∗∗∗_</sup>|(7.06)|
|hd<br>~~h~~edu2|0.450<sup>_∗∗∗_</sup>|(9.81)|
|hd<br>~~e~~mployed|0.484<sup>_∗∗∗_</sup>|(15.25)|
|hd<br>~~h~~isp|-0.159<sup>_∗∗∗_</sup>|(-4.36)|
|hd<br>~~b~~lack|-0.248<sup>_∗∗∗_</sup>|(-7.78)|
|~~c~~ons|5.201<sup>_∗∗∗_</sup>|(39.18)|
|_N_|6330||
|adj. _R_<sup>2</sup>|0.432||

_t_ statistics in parentheses _∗ ∗∗ ∗∗∗ p <_ 0 _._ 10, _p <_ 0 _._ 05, _p <_ 0 _._ 01 Table 7: Regression model with log household income per capita as dependent variable (sample from 2010 IPUMS USA) Figure 3 shows the estimates of the probability density function associated with the area error _ua_ and the household error _εah_ , respectively. The “red” line denotes the normal mixture density estimate, “blue” line denotes the normal density function that best fits the data. As we are dealing with empirical data, not simulated data, we do not know the distributions from which the errors are drawn, and hence are not able to include the “true” density function as a benchmark. The estimates suggest that the household errors are arguably drawn from a distribution that deviates visibly from a normal distribution. We do not see a similar deviation from the normal density for the area error. The smallness of the variance of the area error relative to the total error however, suggests that the signal to noise ratio is low making it difficult to identify the underlying density function. The small variance also means that the area error makes only a minor contribution to the total error and so its density function is less important. It is clearly more important that our estimate of the density associated with the 22
Probability density functions for _u_ and _ε_ <!-- Start of picture text --> −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 −4 −2 0 2<br>x x<br>3.0 0.6<br>2.5 0.5<br>2.0 0.4<br>1.5 0.3<br>dens.uhat(x) dens.epshat(x)<br>1.0 0.2<br>0.5 0.1<br>0.0 0.0<br><!-- End of picture text --> Figure 3: Normal-mixture density (Red) versus normal density (Blue) estimates for: (a) area error _ua_ (left panel); (b) household error _εah_ (right panel) household error is accurate.

The log poverty line is set at 9 _._ 5 which yields an aggregate poverty rate of about 28 _._ 3 percent. County level estimates of poverty and inequality are obtained using both EB and nonEB estimates under the assumption of normal errors and under the less restrictive assumption of normal mixture errors. For each of the specifications we obtain 422 estimates of poverty and inequality which we then compare to the true estimates derived from the full census. We will judge the accuracy of the estimates on the basis of two summary statistics: the aggregate bias and the RMSE (both obtained by aggregating the county level differences between our estimates and the true rates over the 422 counties). The results are presented in Table 8.

||Bias|RMSE|
|---|---|---|
|EB|0.029 (0.048)|0.045 (0.059)|
|NON-EB|0.024 (0.044)|0.042 (0.056)|

Table 8: Bias and RMSE for estimates of county-level poverty using 2010 data from the United States: (a) Normal-Mixture distribution (left number); (b) Normal distribution (right number) We observe a rather large bias when estimating county level poverty under the assumption of normal distributed errors; almost 5 percentage points given a national poverty rate of about 28 percent. By relaxing the normality assumption, and assuming normal mixture distributions instead, we are able to reduce this bias to just below 3 percent which denotes a non-trivial improvement. We see similar reductions in the RMSE. Working with EB or non-EB estimates makes little to no difference in this empirical application. This was to be expected given the smallness of the location effect. Note that the fact that some bias still remains suggests that a degree of model misspecification still persists; these could include mis-specifications in the structural model but also forms of heteroskedasticity that are currently being ignored.

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# **6 Concluding remarks**

We have developed an estimator for the error distribution functions associated with the nested errors from a linear mixed model. An attractive feature of the estimator is that it accommodates Empirical Bayes prediction. This is achieved without making any restrictive assumptions about the shape of the distrbution functions. Monte-Carlo simulations presented in this paper show that estimates of poverty and inequality can be severely biased when non-normality of the errors is ignored. The bias can be as high as 2 to 3 percent on a poverty rate of 20 to 30 percent. Most of this bias is resolved when using our approach.

# **7 Appendix**

## **7.1 A lemma and corollary that are used in the proofs of Lemmas 4 and 5**

**Lemma 11** _Let v_ 1 _and v_ 2 _be two independent normal random variates, with Evi_ = _μi,_ var _vi_ = _σi_<sup>2</sup><sup>_fori_=1</sup><sup>_,_2</sup><sup>_._</sup> _Let v_ 3 = _v_ 1 + _v_ 2 _. Denote the normal density function with expectation μ and variance σ_<sup>2</sup> _by φ_ ( _·_ ; _μ, σ_ ) _. Then for the joint density p_ ( _v_ 1 _, v_ 3) _and the conditional density p_ ( _v_ 1 _|v_ 3) _we have_

**Proof** This can be directly verified. Alternatively, note that both _v_ 3 and _v_ 1 _|v_ 3 are normally distributed. The distribution of _v_ 1 _|v_ 3 has expectation

and variance

**Corollary 12** _Let x and y be independent random variables distributed as normal mixtures, and let q_ = _x_ + _y. Then (i) q is also distributed as a normal mixture, and (ii) the conditional distributions p_ ( _x|q_ ) _and p_ ( _y|q_ ) _are also normal mixtures. Moreover, the mixing parameters of p_ ( _q_ ) _, p_ ( _x|q_ ) _and p_ ( _y|q_ ) _are closed expressions in terms of the distributional parameters of x and_

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# **7.3 A derivation of the EM-type estimator for** _Fu_

Below we will derive the EM-type estimator for the normal-mixture distribution _Fu_ . The estimator for _Gε_ is obtained in a very similar fashion.

Let Ψ denote the vector that contains the parameters of this normal-mixture, which includes the mixing probabilities _πi_ as well as the parameters _μi_ and _σi_<sup>2ofthecomponentdistributions.</sup> Assume for the moment that _ua_ denotes observed data. The EM algorithm would then proceed as follows. It introduces the random variable _zai_ that equals 1 if _ua_ comes from component _i_ and 0 otherwise, and defines _L_ (Ψ; _u, z_ ) as the log likelihood function that would apply if both _u_ and _z_ were observed. It then integrates out _z_ by taking expectations over _z_ conditional on an estimate of Ψ that is available at iteration _k_ , which we shall denote Ψ<sup>ˆ(</sup><sup>_k_)</sup> . The solution ˆΨ<sup>(</sup><sup>_k_+1)</sup> that maximizes the resulting log likelihood function _Q_ (Ψ; ˆΨ<sup>(</sup><sup>_k_)</sup> ) = _E_ [ _L_ (Ψ; _u, z_ ) _|_ ˆΨ<sup>(</sup><sup>_k_)</sup> ] can be obtained analytically, unlike the solution to the original log likelihood function. The EM estimator for Ψ in this case is obtained as the fixed point of this iterative equation.

Unfortunately, we do not observe _ua_ . According to the nested error model, _ua_ = _e_ ̄ _a − ε_ ̄ _a_ . We ̄ ̄ observe _ea_ , but not _εa_ . The EM algortihm however can still be applied. Instead of integrating ̄ out _z_ , we will be integrating out both _z_ and _εa_ . With slight abuse of notation, the resulting log likelihood function is now given by: _Q_ (Ψ; Ψ<sup>ˆ(</sup><sup>_k_)</sup> ) = _E_ [ _L_ (Ψ; ̄ _e,_ ̄ _ε, z_ ) _|_ Ψ<sup>ˆ(</sup><sup>_k_)</sup> ]. Note that this function _Q_ (Ψ; Ψ<sup>ˆ(</sup><sup>_k_)</sup> ) will be different from the function that is obtained when _u_ is treated as observed data and only _z_ is integrated out. (But it is obtained respecting the principles of the EM algorithm.)

It will be convenient to wait with taking the expectations over _ε_ ̄ until after we have evaluated the partial derivatives of _Q_ (Ψ; Ψ<sup>ˆ(</sup><sup>_k_)</sup> ) with respect to Ψ. It follows that: where expectations are taken over _ε_ ̄ conditional on _e_ ̄, and where: with: Evaluating the partial derivative of _Q_ (Ψ; Ψ<sup>ˆ(</sup><sup>_k_)</sup> ) with respect to Ψ = ( _π, μ, σ_<sup>2</sup> ) and bringing this inside the expectation operator yields: 29
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>>>
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