Table 1: _Monte Carlo simulation setting 1 - simultaneous census and survey collection, some variation in the explanatory variable between clusters_

||True value|N<br>Bias|ew estima<br>RMSE|tor<br>Coverage|E<br>Bias|LL estima<br>RMSE|tor<br>Coverage|Survey est.<br>RMSE|
|---|---|---|---|---|---|---|---|---|
|_W_0(_._25)|0.2500|0.0025|0.0121|0.9800|0.0017|0.0081|0.9833|0.0137|
|_W_0(_._50)|0.5000|0.0062|0.0146|0.9767|0.0058|0.0102|0.9633|0.0159|
|_W_0(_._75)|0.7500|0.0028|0.0113|0.9800|0.0036|0.0084|0.9600|0.0144|
|_W_1(_._25)|0.0094|0.0000|0.0007|0.9500|-0.0000|0.0005|0.9700|0.0007|
|_W_1(_._50)|0.0240|0.0002|0.0012|0.9833|0.0001|0.0008|0.9800|0.0012|
|_W_1(_._75)|0.0473|0.0003|0.0015|0.9800|0.0002|0.0010|0.9900|0.0015|

The RMSEs is the root of the mean squared deviations of the estimates from the true value over 300 replications. Coverage rates are calculated for 95% bootstrap percentile intervals.

All results are based on 300 Monte Carlo replications with 500 bootstrap census data sets generated in each replication for the two methods which use census data. The bootstrap procedure to sample the error components applies a simple nonparametric version, i.e., both cluster effects and household errors are independently sampled with replacement from their sample analogs from the first-stage regression. See Section 2.3 for details.

In the first setting, the root mean squared error is, as expected, smallest for the ELL method, followed by our estimator and an estimator solely based on the survey (Table 1). Although the _R_<sup>2</sup> from the first-stage regression for the ELL method is seven times as large as for our new method, the root mean squared errors only differ by a factor of about 1.5 or two-thirds, respectively. The coverage rates of the two methods are close to the nominal one of 95% and the bias is negligible.

In the second and more interesting setting, the ELL method naturally is the worst in terms of prediction and generates invalid confidence intervals (Table 2). The upward bias originates from the data generating process above: as the expected values of _xch_ and thus _ych_ are larger in the recent survey and its underlying population than in the dated census, using the dated census data to predict current poverty statistics necessarily underestimates the current values of _ych_ and hence overestimates the magnitude of poverty. In contrast, the new method yields valid confidence intervals. It also results in a lower mean squared error in comparison to the purely survey-based estimate since additional census information is exploited. The last result typically holds on average if the model assumptions are fulfilled (as it is the case in this simulation setting) and census and survey size differ distinctly. The latter is often true in practice.<sup>8</sup>

> 8Note that under the stated conditions, our estimator performs better only in predicting the true value on average. In a single sample, the pure survey mean is superior to our approach if the sample mean is by chance equal or very close to the census mean. An extreme example includes the limiting case in which the recent survey is equal to the underlying census. Then, the survey mean is trivially the census mean, that is, there is no error at all. But our new method is still prone to idiosyncratic and (small) simulation error, even under correct model specification.

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