effects models.<sup>12</sup> Below, as a robustness check, we show that giving equal weight to each municipality worsens the predictions.

Finally, we ignore heteroscedasticity in the dependent variable, which is the average AGEB labor market outcome taken from the sample. A sample of 40 individuals was taken from each AGEB, except for the few cases where AGEBs contain fewer than 40 working aged persons, which implies that heteroscedasticity is not a first order concern when estimating labor force participation rates. Different rates of labor force participation across sample AGEBs will affect the number of labor force participants used to calculate unemployment rates from the sample, but we do not adjust for this source of heteroscedasticity.<sup>13</sup> Population counts for each AGEB taken from the 2020 census were used as population weights when aggregating AGEBs to municipal estimates. We used a modified version 2.0 of the EMDI package in R, modified to incorporate sample and population weights, to estimate the model (Kreutzmann et al., 2018).<sup>14</sup>

# **e. Evaluating the estimates**

To evaluate the estimates, we compare each estimate to the population values from the 2020 census. We calculate the following statistics to compare these four separate estimation methods:

- Estimated mean squared error. For the direct estimates, this is assumed to be equal to the estimated variance of the mean, estimated using the Horwitz-Thompson approximation.<sup>15</sup> For the small area estimates, MSE estimates were generated using the parametric bootstrap procedure.

- The median estimated relative standard error across municipalities, where the relative standard error for each municipality is defined as the square root of the estimated mean squared error divided by the point estimate.

- Coverage rate: This is the share of municipalities for which the actual census value falls within the 95% confidence interval generated by the estimate.

> 12 See Rabe-Hesketh and Skondral (2006). In conditional random effect models, unlike in standard random effect models, multiplying the weights by a constant affects the parameter estimates.

> 13 This is partly because the high number of municipalities in the sample with zero unemployment is a more serious issue than heteroscedasticity when estimating unemployment rates.

> 14 The number of simulations and bootstraps, L and B, were each set to equal 100. The weighting strategy relies on passing through the specified weights to the lme function in the R nlme package that estimates the conditional random effect model (Pinheiro et al., 2020). In addition, we take a weighted mean instead of a simple mean when aggregating point estimates for AGEBs to generates estimates for municipalities. In the latter case, the aggregation weights are the population of the AGEBs.

15 We use the Horvitz-Thompson approximation of the variance described in Molina and Marhuenda (2015), in which the variance of the sample mean for municipality m is estimated as: �= ∑ 2 NNss Where m is a particular municipality, i is an individual within municipality m, N ss ii=1 ssii ssii ssii m is the number of relevant VV<sup>�</sup> �YY<sup>��</sup> ww (1 −ww )yy individuals in the municipality, w is the inverse probability sample weight, and ymi is the binary outcome variable (participation or unemployment) for individual i.

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