# **1 Introduction**

Collecting data to produce reliable and timely estimations of poverty requires elaborate and expensive surveys called Household Budget Surveys (HBSs). Only a few countries are able to collect data annually on income or expenditure to facilitate the estimation of poverty and inequality. Most National Statistical Offices (NSOs) worldwide collect these surveys only every four to five years. Therefore, producing reliable indices annually for monitoring poverty results remains quite challenging for many countries. To overcome this challenge, scholars have focused on developing methods to compare welfare indicators over time from surveys that are little comparable. These techniques, broadly known as Survey-to-Survey Imputation Techniques (SSITs), proved successful at predicting comparable poverty indicators, but as this paper argues, were less effective at predicting comparable inequality indicators.

SSITs are a derivative of the poverty map literature (Elbers et al., 2003; Tarozzi and Deaton, 2009, Christiaensen et al., 2012; Mathiassen, 2013). By imputing income onto censuses, poverty maps have been applied to many developing countries’ data to provide geographically disaggregated estimates of poverty. In addition to survey-to-census imputation, there has been a recent trend of survey-to-survey imputation, mapping from surveys with consumption data to those with other outcomes of interest, but lacking standard welfare aggregates (Elbers et al., 2004; Dabalen et al., 2014; Dang et al., 2019).

While this approach is well established for poverty estimates, a cursory overview of the literature shows that little attention is devoted to potential problems in obtaining accurate inequality measures with this method. Using actual data from a census of households in a set of rural Mexican communities, Demombynes and Hoogeveen (2007) show that correlations between the estimated and true welfare at the local level are highest for mean expenditure and poverty measures and lower for inequality measures. Douidich et al. (2016) obtain accurate estimates for quarterly poverty rates using a classic survey-to-survey imputation method that combines HBSs and Labor Force Surveys (LFSs). Using a log-linear regression to impute a completely missing variable (total expenditure) for certain years, they obtain reliable estimates for Head Count Ratio (HCR) with an exogenous poverty line. Krafft et al. (2019) impute consumption from HBSs onto LFSs and obtain information on poverty and inequality. Resulting measures of consumption, poverty, and inequality are similar across survey pairs, particularly for the data in Jordan and the Arab Republic of Egypt. These estimates are obtained based on the hypothesis that the model is time invariant, that is, the coefficients used to impute from one survey to another are stable over time (e.g., that _βt_ = _β_ ), and that the error term is homoscedastic and normally distributed. However, the concern is that incorrectly assuming normal errors or ignoring heteroscedasticity has the potential to introduce bias when estimating poverty or inequality. Specifically, in the prediction of inequality, the bias can be significant. Dang et al. (2019) provide an alternative method of simulating the final consumption imputation, which also seems extremely accurate when estimat2