conditional means. To account for the deviations of the observed consumption values from these means, random disturbance terms have to be added by simulation. Assume that the aim is to estimate a poverty measure _W_ , where the indices from (1) are dropped for notational convenience.

A bootstrap procedure is applied to generate _R_ pseudo censuses and resultant poverty measures:

1. Draw all model coefficients from their respective sampling distribution estimated by the model in the first stage, including regression coefficients, random term variances and possible heteroscedasticity parameters. Multivariate normal distributions with first-stage estimates for the means and variance-covariance matrix are used to draw the regression coefficients and the heteroscedasticity parameters.<sup>5</sup>

2. Conditional on the parameters describing the error components’ distributions from the first step, cluster effects and household errors are drawn from their respective distributions. One option is to use a parametric bootstrap, i.e., to assume certain parametric distributions for which the estimates from the first stage regression might give some indication. However, a nonparametric bootstrap procedure is a valid alternative or supplement. In this case, a cluster effect can be estimated as the mean of the deviations between observed and predicted values in one cluster, i.e. _η_ � _ct_ = 1 _/Hc_ � _Hh c_<sup>(</sup><sup>_y_�</sup><sup>_cht −_</sup><sup>**x**</sup> _c_<sup>_′_</sup> **_._** _,t−_ 1<sup>**_β_**�),whilethehouseholdresidualsarecomputedasthosedeviations</sup> � � minus the cluster effects, i.e., _echt_ = ( _ycht −_ **x**<sup>_′_</sup> _c_ **_._** _,t−_ 1<sup>**_β_**�)</sup><sup>_−η_�</sup><sup>_ct_.Therearedifferentstrategiestodraw</sup> from these sampling distributions. One may draw with replacement from all estimated cluster effects and all household residuals. Alternatively, the household residuals may be drawn only from the location to which the cluster effect belongs. This strategy generally allows the estimated two error components to be related in a nonlinear way, even though they are by construction uncorrelated.

3. Calculate the predicted consumption values for all households and all individuals as well as the poverty measure _W_<sup>�(</sup><sup>_r_)</sup> derived from those values.

4. Repeat steps 1 to 3 _R_ times.

For the poverty measure _W_ , the (simulated) expected value is then given by

> 5One may also assume a distribution for the error components’ variances such as the gamma distribution, e.g., but in many cases it is reasonable to treat their estimates from the first stage as fixed, especially if the numbers of enumeration areas and households in the survey are large since then there is not much uncertainty in the variance estimators. The household error variance estimator is usually very precise as it is based on the (large) number of households in the survey. The amount of enumeration areas in the survey is smaller but the uncertainty in the variance estimator of the enumeration area effects is often still negligible. In practice, one may check whether the estimated variances of the error components’ variances are small enough in order to treat them as fixed in all bootstrap replications.

7