Note that since inequalities �7� and �8� use the same poverty line, we can use Proposition 1 to test the comparability of poverty rates derived from them. Indeed, inequalities �7� and �8� satisfy the conditions of Proposition 1. First, the NISR (2016) approach is based on nominal household expenditure, which is constructed the same way in EICV3 and EICV4. Second, the COLI used for deflating consumption ������� expenditure in EICV3 (π����� ) is fully consistent with the index used for deflating consumption ������� expenditure in EICV4 (π����� ), i.e., the base month, the source price data, and the budget weights for the two indices are the same. Therefore, poverty rates estimated from inequalities �7� and �8� satisfy the necessary conditions of comparability, corroborating that the official poverty trend, where headcount poverty declined from 46 percent to 39 percent between EICV3 and EICV4, is credible.

# III.3. Establishing robustness of the declining poverty trend

To substantiate the credibility of the poverty trend based on the NISR (2016) approach, they conducted further analysis using a survey-to-survey imputation method (S2S). When consumption data from two rounds of a household survey are deemed non-comparable, the survey-to-survey imputation approach, as in Christiaensen et al. (2012), is often used to restore the comparability of consumption data and poverty rates. The methodology creates consumption models in a household survey where both consumption and non-consumption data are reliable, and then imputes household consumption expenditures using the models in a different survey where only non-consumption data are reliable. The consumption models use household and individual characteristics such as educational attainment and employment status of household members, household composition and the like. This S2S approach is useful if price adjustments or poverty lines are not comparable over time, but non-consumption data are comparable. Given some key assumptions are satisfied, Christiansen et al. (2012) and Ahmed et al. (2013) corroborate the reliability of this approach using data from various countries.

NISR (2016) estimates a poverty trend between 2010/11 and 2013/14 using the S2S approach and compares it with the official poverty trend between 2010/11 and 2013/14 in NISR (2015). Following Tarozzi (2002), NISR (2016) estimates a logistic regression model to predict the probability of being poor from nonconsumption data using EICV4, predicts a poverty rate from a model using EICV4, and applies it to nonconsumption data in EICV3. Using this approach, they find that the poverty rate of EICV3 is 44.6 percent, which is almost the same as the official poverty estimate for EICV3, i.e., 44.9 percent. NISR (2016) also estimates the poverty rate of EICV4 using this approach (39.0 percent), which is almost identical to the official estimate of EICV4 (39.1 percent). The S2S analysis, therefore, also confirms that the official trend is reliable.

In Rwanda, the reduction in food budget shares over time also corroborates the poverty decline. Typically, increases in income are strongly associated with a declining share of the budget spent on food (Engel’s law). We examine if this is true in the case of Rwanda. We estimate real food budget shares, expressed in January 2014 prices, for 2010/11 and 2013/14 and find that the real food share declines from 64 percent in EICV3 to 61 percent in EICV4. We also find that these differences are statistically significant.<sup>11</sup>

> 11 The real food budget share is estimated by deflating food expenditures (numerator) with an inflation rate calculated from the food CPI and total expenditures (denominator) with an inflation rate calculated from the overall CPI.

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