budget surveys. Nonetheless, it is useful to check that the strong performance of the household-level model holds when estimated in simulation settings, as is commonly done in the literature.<sup>36</sup>

# a. Model-based simulation

We begin by conducting a Monte Carlo simulation study, following the basic structure outlined in Das and Chambers (2017). We perform 100 simulations, drawing a new simulated population each time, in 80 areas. Each of these areas contains between 15 and 29 subareas, with the exact number chosen randomly along a uniform distribution. Each subarea contains 15 and 29 households, also chosen randomly from a uniform distribution. Each household is assumed to contain one member.

At the sub-area level, we draw two X variables, X1sa and X2sa, from a joint normal distribution. All households in the same sub-area are assigned the same random values of X1sa and X2sa. X1sa is distributed with mean 0 and variance 2. The mean of X2sa is set equal to 0.05 times the area index, which varies from 1 to 80, with a variance of 2. The covariance of X1sa and X2sa is set to 0.5. We draw area effects (ηa), subarea effects ( sa), and an idiosyncratic error term for each household (εi). These error terms are drawn from normal distributions with mean zero and variances of 1 for the area effect, uu 0.5 for the sub-area effect, and 2 for the household idiosyncratic error. We then generate the population welfare measure as (6) The variance structure of the two X variables and variance components was selected to achieve an yyii = 6 + 0.5 ∗XX1ssaa −0.6 ∗XX2ssaa + ηηaa + uussaa + εεii The variance structure of the two X variables and variance components was selected to achieve an approximate R<sup>2</sup> of 0.25 when regressing household welfare on X1sa and X2sa.

From this population, we set the poverty line at the 25<sup>th</sup> percentile of the welfare distribution and calculate population poverty rates for each area. We then draw a two-stage sample of 3 subareas per area in the first stage and 10 households per sampled subarea in the second. We construct sample weights as the inverse of the probability of being sampled, and take the weighted 25<sup>th</sup> percentile of the sample distribution of welfare as the poverty line. Finally, we estimate a small area estimation model in the sample using the modified EMDI package, utilizing the population values of X1sa and X2sa to simulate welfare for all households in the population. The resulting simulated welfare aggregate is then compared with the poverty line derived from the sample, and averaged over households for each area to obtain area headcount poverty rates and estimated mean squared errors.<sup>37</sup> This entire process, starting with generating the simulated population, is then repeated 100 times. Therefore, the final output from the procedure is a set of 8,000 population poverty rates and small area estimates, derived from 100 simulations of 80 areas.

Table 16 displays the results. The small area estimates reduce the mean squared error by a factor of 10 and average CV by a factor of roughly 3 compared with the traditional EA-clustered direct estimates, while maintaining a high coverage rate close to 95 percent. The mean poverty is nearly exactly 25 percent, and the correlation compared with poverty rates in the population increases from 0.84 for the direct estimates to 0.98 for the small area estimates. The root mean squared error of the predictions

> 36 Evaluations of small area estimation methods using model and/or design-based simulations include Das and Chambers (2017), Demombynes et al (2007), Elbers, Lanjouw and Leite (2008), Tarozzi and Deaton (2009), Tarozzi (2011) and Tzavidis et al (2018), among many others.

> 37 The results are not benchmarked to super-areas, because super-areas are not defined in this simulation.

36