information when the auxiliary data are aggregated to the subarea level.<sup>32</sup> The estimates produced by the Stata SAE with the traditional bootstrap are less accurate than the baseline EMDI estimates in Sri Lanka, but more accurate in Tanzania. The coverage rate of the traditional bootstrap estimates is 93 percent, which is very close to the target 95 percent rate. This high coverage rate results from two countervailing biases in the variance estimate; the traditional bootstrap overstates the variance by failing to hold the composition of the sample constant across replications, which counteracts the downward bias due to the failure to account for the positive correlation among the sample residuals. Overall, the traditional bootstrap performs quite well in these two settings, in terms of accuracy and coverage, but poorly in terms of efficiency.

The estimates produced by the latest version of Stata SAE with the parametric bootstrap are slightly less accurate, in terms of its correlation with the full census, than those produced by the EMDI package.<sup>33</sup> The Stata SAE estimates are also less efficient than the baseline EMDI estimate in Sri Lanka, as the mean MSE rises from 4.6 to 6.8. In Tanzania, in contrast, the Stata SAE package with the parametric bootstrap yields estimates that are slightly more efficient than the baseline estimates, but because the estimates are less accurate, the coverage rate falls significantly from 74.8 to 64.2 percent.

The fifth row of the table presents results without the heteroscedasticity correction. Disabling the correction brings the mean poverty rate closer to the true 4 percent rate in Sri Lanka, but further from the assumed poverty rate of 20 percent in Tanzania. In Sri Lanka, disabling the correction increases the accuracy of the estimates to slightly above the baseline estimates. In Tanzania, disabling the heteroscedasticity correction has minor effects on accuracy and slightly decreases precision, improving the coverage rate. The limited effects of the heteroscedasticity correction likely results from the negligible explanatory power of the “alpha model” that predicts the variance of the residuals. The R<sup>2</sup> of the alpha model, which like the “beta” prediction model was selected using the plugin Lasso method to avoid overfitting, is only 0.004 in Sri Lanka and 0.003 in Tanzania.

The bottom four rows of the table present a variety of robustness checks using the EMDI package. The first of these uses the publicly available version of the package that does not allow for household weights when aggregating the simulated household poverty outcomes into an area poverty rate. The estimator without weights is slightly less accurate than those with weights in each country, with a corresponding mild decline in both efficiency and coverage. The second EMDI robustness check uses a more standard Box-Cox transformation of the welfare variable instead of the ordered quantile normalization transformation. This leads to considerable downward bias in the estimates in both countries, leading us to prefer the ordered quantile normalization, although the correlation and coverage rate changes little when using the Box-Cox transformation. The third EMDI robustness check reports the results of a model selected using stepwise regression, with a probability 32 The penalty for failing to implement EB in a traditional bootstrap setting is smaller in Tanzania than Sri Lanka, although the estimates without EB are very imprecise in both countries. This is likely because the traditional bootstrap with EB performs gives less precise estimates in Tanzania than Sri Lanka, due to the smaller number of areas in Tanzania.

33 This may result from the failure of the Stata SAE package to append the sample to the census when simulating poverty. The sample constitutes about 0.5 percent of Sri Lankan households while the Tanzanian sample only contains about 0.1 percent of households, which is consistent with the greater fall in accuracy in Sri Lanka.

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