We then find the empirical distributions of each disturbance component. We approximate the distributions of _ηc_ , _εch_ , and _δchi_ respectively by the distributions of _uc··_ , ( _uch· − uc··_ ), and ( _uchi − uch·_ ) standardized to have mean zero and a unit standard error.

**Simulation** We explicitly take into account the model and idiosyncratic errors using MonteCarlo simulation, where the model error is the error associated with the estimation of model parameters _α_ and _β_ , and the idiosyncratic error is the variations of the anthropometric indicators that remain unexplained by the model. Now, let _R_ be the number of simulations, which must be sufficiently large in order to make the computational errors small enough. In our study, we set _R_ = 100. In _r_ -th simulation where _r ∈{_ 1 _,_ 2 _, . . . , R}_ , we need the following parameters: _β_<sup> ̃</sup> (<sup>(</sup> _r_<sup>_k_</sup> )<sup>),</sup><sup>_α_ ̃</sup> (<sup>(</sup> _r_<sup>_k_</sup> )<sup>),( ̃</sup><sup>_στ,_(</sup><sup>_r_))2,( ̃</sup><sup>_σ_</sup> _δ,_<sup>(</sup><sup>_k_</sup> (<sup>)</sup> _r_ )<sup>)2,( ̃</sup><sup>_σ_</sup> _δ,_<sup>(</sup><sup>_k,l_</sup> ( _r_ )<sup>))2,( ̃</sup><sup>_σ_</sup> _η,_<sup>(</sup><sup>_k_</sup> (<sup>)</sup> _r_ )<sup>)2,</sup><sup>_A_(</sup> _∗_<sup>_k_</sup> _,_ (<sup>)</sup> _r_ )<sup>,</sup><sup>_B_</sup> _∗_<sup>(</sup><sup>_k_</sup> _,_ (<sup>)</sup> _r_ )<sup>for</sup><sup>_∀k_=</sup><sup>_l_.</sup> We randomly draw _α_  ̃( _r_ ) and _β_<sup> ̃</sup> ( _r_ ) from the normal distribution with mean _α_ ˆ and _β_<sup>ˆ</sup> , and variance-covariance matrix _V ar_<sup>�</sup> [ˆ _α_ ] and _V ar_<sup>�</sup> [ _β_<sup>ˆ</sup> ] respectively. For the rest of the parameters, we create a two-stage bootstrapping sample of _u_ ˆ in each round of simulation, and compute the parameters using the bootstrapping sample.<sup>6</sup> It is straightforward to calculate ( ̃ _σε,ch,_<sup>(</sup><sup>_k_)</sup> ( _r_ )<sup>)2</sup> from ( ̃ _στ,_ ( _r_ ))<sup>2</sup> , _A_<sup>(</sup> _∗_<sup>_k_</sup> _,_ (<sup>)</sup> _r_ )<sup>,</sup><sup>_B_</sup> _∗_<sup>(</sup><sup>_k_</sup> _,_ (<sup>)</sup> _r_ )<sup>fromEq(3).</sup>

For each census record, we draw the standardized cluster, household and individual effects. Each of these disturbance components is drawn jointly for multiple indicators to capture the intra-personal effect. Letting the standardized components of disturbance terms drawn in the _r_ -th simulation be _η_ ̃ _c,_<sup>(</sup><sup>_k_</sup> (<sup>)</sup> _r_ )<sup>,</sup><sup>_ε_ ̃(</sup> _ch,_<sup>_k_)</sup> ( _r_ )<sup>,and</sup><sup>_δ_ ̃</sup> _chi_<sup>(</sup><sup>_k_)respectively,the</sup><sup>_k_-thimputedanthropometric</sup> indicator for the individual in the census in _r_ -th simulation is<sup>7</sup> :

> 6Another possible implementation is to draw _α_  ̃ and _β_  ̃ from the bootstrapping sample.

7 To eliminate extreme values, we drop census observations for which the point estimate **x**<sup>(</sup> _chi_<sup>_k_)</sup><sup>_β_ˆ(</sup><sup>_k_)isnot</sup> in the range of _y_<sup>(</sup><sup>_k_)</sup> for at least one indicator. In each simulation, we censored _y_ ̃ _chi,_<sup>(</sup><sup>_k_)</sup> ( _r_ )<sup>attheminimumand</sup> maximum observed in the survey. We also censored _D_<sup>(</sup><sup>_k_)</sup> at the minimum and maximum.

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