random draws from the estimated distributions of the residuals, ηc and εh. For each subsample, these simulations are repeated 100 times. The mean across the 100 simulations of a poverty statistic provides a point estimate of the statistic for the subsample. To respond to apparent differences in consumption patterns between urban and rural areas, separate consumption models for urban and rural areas are estimated.

# **Estimation of Poverty Headcount Rates and Their Standard Errors: Subsample Imputation and Subsample Direct Estimation**

With either the Subsample Imputation or Subsample Direct Estimation, subsample selection can have a significant impact on poverty estimates. To see the statistical performance, we randomly drew subsamples and repeatedly estimated poverty rates with the subsamples. For the Subsample Direct Estimation approach, we estimated poverty numbers from each subsample and used the mean of poverty rates as the estimate of poverty rate and the standard deviation as the SE.

For the Subsample Imputation approach, we estimated consumption models from each subsample and estimated poverty rates using household expenditure imputed from the models. For the first 30 subsamples, we estimated the consumption models using PovMap2 software, and so were able to follow ELL’s method exactly. However, although most results become stable after the exercise is repeated 30 times, there remains a concern on the robustness of the findings.

To see the robustness of the results from the 30 observations, we simplified the imputation method slightly and repeated it 1,000 times. Like ELL, we assumed the error structure consists of the cluster (ηc) and household errors (εch). But, although ELL and PovMap2 allow for a flexible error structure, we assumed both cluster and household level errors are normally distributed with zero means.

The variances of the errors were estimated as follows: first, residuals from the ordinary least squares (OLS) estimation, ech, were calculated. Secondly, the empirical cluster effects, ec, were calculated using the residuals within each cluster. The SE of ηc is equal to the standard deviations of the empirical cluster effects (ec). Third, the empirical household effect was defined by the difference between the residuals and the empirical cluster errors, such that εch = ech −ec. Using the obtained two SEs of ηc and εch, the two error terms were randomly produced from two normal distributions and the imputed consumption was calculated using this formula: Such simplifications will increase the SEs of coefficients and model errors.

Finally, although the first 30 simulations trimmed extreme values, the following 1,000 simulations did not control them. Such trimming likely has a small effect on the estimation of poverty headcount rates, although it can have a significant impact on inequality measures, such as Gini coefficients. Because this paper focuses on poverty headcount rates, ignoring trimming is of less concern.

All in all, the 1,000 simulations following the first 30 simulations likely increased the SEs of poverty statistics estimated by the Subsample Imputation approach, but can be used to check whether the results of the first 30 simulations are vulnerable to the small sample errors.

SEs estimated by ELL’s method need to be modified to incorporate sampling errors because the ELL method was originally used to impute consumption for all households in census data. As a result, estimated SEs consist only of modeling errors and do not include sampling errors. However, in both Subsample Imputation and Full Sample Imputation, we imputed consumption for all households in a household survey, and this means that the estimated SEs should include sampling errors that are not originally incorporated into the ELL method. To incorporate sampling errors, we used the SEs from the Full Sample Direct Estimation approach, which included only sampling errors, and added them to the SEs obtained from the ELL method. We assumed zero correlation between the modeling and sampling errors since we could not predict the sign or size of the correlation.

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