answering the recall module. The latter module was specifically designed to inform the national statistics agency on the transition from diary to recall, which is planned for 2021.

Our methodological innovation is in the use of diary and recall measurements for the _same_ household. Compared to prior work, we derive conditions – building on Hu and Schennach (2008) – to identify nonparametrically any functional of the underlying distribution of usual consumption allowing for errors in both survey measurements. Our strategy also yields identification of survey error distributions for recall and diary measurements without assuming that errors are mutually independent or independent of usual consumption.<sup>3</sup> Moreover, our estimation framework leaves unrestricted the functional forms of usual consumption and survey error distributions. The availability of repeated diary-recall measurements of food spending or acquisition is not unique to our setting, the Consumer Expenditure Surveys in the United States and the Canadian Food Expenditure Survey being notable examples. This substantiates the external validity of our approach in a number of contexts beyond the specific case study.

Among the conditions needed for identification, three play a fundamental role (as shown in Figure 3). First, we assume that household acquisition in the diary week is a random variable centered on household usual consumption. We make the theoretical case for this assumption using a model of consumer demand (as in Meghir and Robin, 1992). This model also sets the reasons for expecting good estimates of consumption _averages_ from diary data, but not of inequality or poverty rates in general.<sup>4</sup> Second, we seek variability in survey measurements induced by a change in usual consumption. Specifically, we build on the IHSES sampling design and assume that household usual consumption covaries with the average acquisition of same-income households located in other census enumeration areas and interviewed the same week. Our key restriction is that the average acquisition of these peers correlates with household diary and recall measurements only through the household’s own consumption. Third, while we allow survey errors in recalled consumption and diary acquisitions to be dependent, we channel their dependence through household usual consumption. This assumption is standard in nonparametric identification with measurement error (Schennach, 2013), and its interpretation along with possible threats to its validity are discussed using the consumer demand model above.

Our first key insight is that diary and recall reports are not rank preserving, meaning that they do _not_ order the same household identically in the data. We rule out simple explanations for this result. For example, we show that answering the additional recall module does not affect the accuracy of diaries: households randomized to, and households excluded from, the recall module have the same distribution of acquisitions in diaries. Besides, important departures from rank invariance are found across the support of diary and recall measurements and of the income distribution, suggesting that survey effects are unlikely to arise only due to households with specific demographics and income. These results imply that the same household could end up above or below the poverty line depending on the survey mode employed, which has distribution-wide implications for welfare ranking relevant to policies using proxy means tests derived from consumption data. The departure from rank invariance is also worrisome because it is among the

> 3These assumptions are rejected in our data and would be likely violated in most empirical settings (as conjectured by Aguiar and Bils, 2015).

> 4When the between-group component of inequality is of interest, however, the same argument can be used to claim that data from diaries dominate recall measurements.

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