Table 2: _Monte Carlo simulation setting 2 - dated census and recent survey, some variation in the explanatory variable between clusters, explanatory variable changes over time_

||True value|N<br>Bias|ew estima<br>RMSE|tor<br>Coverage|E<br>Bias|LL estim<br>RMSE|ator<br>Coverage|Survey est.<br>RMSE|
|---|---|---|---|---|---|---|---|---|
|_W_0(_._25)|0.2500|-0.0035|0.0128|0.9533|0.1186|0.1190|0.0000|0.0155|
|_W_0(_._50)|0.5000|0.0040|0.0144|0.9833|0.1374|0.1377|0.0000|0.0166|
|_W_0(_._75)|0.7500|-0.0011|0.0112|0.9867|0.0925|0.0927|0.0000|0.0154|
|_W_1(_._25)|0.0089|-0.0001|0.0007|0.9367|0.0065|0.0066|0.0000|0.0008|
|_W_1(_._50)|0.0234|-0.0002|0.0011|0.9767|0.0115|0.0116|0.0000|0.0012|
|_W_1(_._75)|0.0456|-0.0001|0.0014|0.9833|0.0154|0.0155|0.0000|0.0016|

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.

# **5 Application to census data from Brazil**

In order to test the proposed method in a real-world example, we use data extracts from the 2000 and 2010 Brazilian censuses provided by the Integrated Public Use Micro Sample (IPUMS, Minnesota Population Center, 2017), the preferred basis of welfare measurement in developing countries. Both censuses include information about monthly income at the level of the individual. In addition, the data sets provide information that is potentially useful in explaining incomes, including the location in which the household resides (urban / rural), the number of household members, ownership of specific assets, and employment status. This allows us to generate artificial surveys from the more recent census and predict income by dated census data. The poverty measures derived from the predicted income values can then be compared to the true ones based on the entire recent census.

The data sets are extracts from the respective censuses. Roughly ten million individuals are included in each data set, corresponding to 6 and 5 percent of the population in 2000 and 2010, respectively. The country is divided into 25 states and 1,980 municipalities. These municipalities constitute the smallest geographical unit which can be matched between 2000 and 2010. Accordingly, we consider them as clusters in the terminology used in the previous sections. Thus, we use averages over municipalities for the 2000 census to predict household incomes in 2010. Household incomes are calculated as the sum of individual incomes of all household members, adjusted for the household size according to the OECD-modified scale.<sup>9</sup> The poverty line is set to $5.5 in 2011 PPP per person and day.<sup>10</sup> For the sake of illustration, we focus on one single Brazilian state, Minas Gerais. In comparison to other states, it features a large number of municipalities (282) which we can match over the two censuses. The data sets comprise 303,134 and 359,051 observed households in 2000 and 2010, respectively, with full information on the used variables. Maintaining the ratio of number of households, we sample randomly about 18,188

> 9 `http://www.oecd.org/eco/growth/OECD-Note-EquivalenceScales.pdf` .

> 10The World Bank calculates poverty rates at three poverty lines for Brazil, see `http://databank.worldbank. org/data/download/poverty/B2A3A7F5-706A-4522-AF99-5B1800FA3357/9FE8B43A-5EAE-4F36-8838-E9F58200CF49/ 60C691C8-EAD0-47BE-9C8A-B56D672A29F7/Global_POV_SP_CPB_BRA.pdf` . We chose the highest one since otherwise there are very few households below the other two poverty lines in both years. Our main aim is to illustrate the method’s applicability even in settings in which the time span between the data sets is large and relevant changes in the welfare status have occurred over time.

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