observable characteristics, **x**<sup>(</sup> _chi_<sup>_k_),throughthefollowinganthropometricmodel.</sup> where _β_<sup>(</sup><sup>_k_)</sup> is a _d_<sup>(</sup><sup>_k_)</sup> -vector of parameters and _u_<sup>(</sup> _chi_<sup>_k_)isadisturbanceterm.</sup> _u_<sup>(</sup> _chi_<sup>_k_)satisfies</sup> _E_ [ _uchi_<sup>(</sup><sup>_k_)</sup><sup>_|_</sup><sup>**x**</sup> _chi_<sup>(</sup><sup>_k_)]=0forall</sup><sup>_c_,</sup><sup>_h_,</sup><sup>_i_and</sup><sup>_k_.LetC</sup><sup>_≡_#(</sup><sup>_C_),H</sup><sup>_c≡_#(</sup><sup>_Hc_),andI</sup><sup>_ch≡_#(</sup><sup>_Ich_),where</sup> #( _·_ ) is the counting measure. The total number of observations is _N ≡_<sup>�</sup> _c∈C_ � _h∈Hc_<sup>I</sup><sup>_ch_</sup> and each cluster has a weight _wc_ which is normalized so that<sup>�</sup> _c_<sup>_wc_=1.Recallherethat</sup> the number of children under five in one household is usually small and can be 1 for some households.

As with the ELL study, we allow for location, or cluster-specific, effect and heteroskedastic household-specific effect. In addition, we have multiple indicators and the individual effect correlated across the indicators. Hence, _u_<sup>(</sup> _chi_<sup>_k_)=</sup><sup>_η_</sup> _c_<sup>(</sup><sup>_k_)</sup> + _ε_<sup>(</sup> _ch_<sup>_k_)+</sup><sup>_δ_</sup> _chi_<sup>(</sup><sup>_k_),where</sup><sup>_η_</sup> _c_<sup>(</sup><sup>_k_),</sup><sup>_ε_(</sup> _ch_<sup>_k_),and</sup> _δ_<sup>(</sup><sup>_k_)respectivelythelocation,householdandindividualeffect.Inprinciple,eachofthe</sup> _chi_<sup>are</sup> three components of _u_<sup>(</sup> _chi_<sup>_k_)couldbeheteroskedasticandcorrelatedacrosstheindicators.The</sup> particular choice we make is driven by the insights from previous studies and also by the limitation imposed by the data. For example, given that **I** _ch_ = 1 for many households in the CDHS data, it is extremely difficult to distinguish household effects from individual effects if both are heteroskedastic. Cluster-level heteroskedasticity is also difficult to estimate because of the limited number of clusters. We allow for flexibility in the correlational structure of the disturbance terms where such flexibility seems most crucial.

We shall hereafter denote ( _u_<sup>(1)</sup> _, · · · , u_<sup>(</sup><sup>_K_)</sup> )<sup>_T_</sup> by _⃗u_ , and use similar notation for other variables. We assume that _⃗ηc,⃗εch_ and _⃗δchi_ are uncorrelated and satisfy _E_ [ _⃗ηc_ ] = _E_ [ _⃗εch_ ] = _E_ [ _⃗δchi_ ] = _OK_ , where the last term is the _K_ -vector of zeros. For _l, k ∈{_ 1 _,_ 2 _, . . . , K}_ with _l_ = _k_ , we assume _E_ [ _ηc_<sup>(</sup><sup>_k_)</sup><sup>_η_</sup> _c_<sup>(</sup><sup>_l_)]=</sup><sup>_E_[</sup><sup>_ε_(</sup> _ch_<sup>_k_)</sup><sup>_ε_</sup> _ch_<sup>(</sup><sup>_l_)]=0.Wedenotethevarianceofeachcomponentofthedis-</sup> turbance term as ( _ση_<sup>(</sup><sup>_k_))2</sup><sup>_≡E_[</sup><sup>_η_</sup> _c_<sup>(</sup><sup>_k_)</sup> _· ηc_<sup>(</sup><sup>_k_)],(</sup><sup>_σ_</sup> _ε,ch_<sup>(</sup><sup>_k_))2</sup><sup>_≡E_[</sup><sup>_ε_(</sup> _ch_<sup>_k_)</sup><sup>_· ε_(</sup> _ch_<sup>_k_)],and(</sup><sup>_σ_</sup> _δ_<sup>(</sup><sup>_k_))2</sup><sup>_≡E_[</sup><sup>_δ_</sup> _chi_<sup>(</sup><sup>_k_)</sup><sup>_· δ_</sup> _chi_<sup>(</sup><sup>_k_)].</sup> Note that we need subscripts _ch_ to express the heteroskedasticity of the household effect.

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