metadata: authors: 0: author: Battistin,Erich
1: author: De Nadai,Michele
2: author: Krishnan,Nandini
count: World
docna: 0: docna: The Insights and Illusions of Consumption Measurements
docty: Policy Research Working Paper
subtopic: Inequality,Demographics,Educational Sciences,Crime and Society,Labor & Employment Law
teratopic: Poverty Reduction,Health, Nutrition and Population,Education,Social Development,Law and Development
abstracts: cdata!: Although household well-being is
            anchored in long-term average rates of consumption, welfare
            comparisons typically rely on shorter-duration survey
            measurements. This paper develops a new strategy to identify
            the distribution of these long-term rates by leveraging a
            large-scale randomization that elicited repeated
            short-duration measurements from diaries and recall
            questions. Identification stems from diary-recall
            differences in reports from the same household, does not
            require these reports to be error-free, and hinges on a
            research design with broad replicability. This strategy
            delivers cost-effective suggestions for designing survey
            modules to yield the most accurate measurements of
            consumption well-being, and offers new insights for
            interpreting and reconciling diary-recall differences in
            household expenditure surveys.
repnme: repnme: The Insights and Illusions of Consumption Measurements
display_title: The Insights and Illusions of
            Consumption Measurements
listing_relative_url: /research/2020/05/32057625/insights-illusions-consumption-measurements
topicv3: Poverty m1326687 695,Criminal Justice m1326993 1014,Demographic Change m1327040 1448,Education m1327048 470,Household Surveys m1504071 2761
publishtoextweb_dt: 2020-05-21T00:00:00Z
historic_topic: Poverty Reduction,Health, Nutrition and Population,Education,Social Development,Law and Development
origu: EFI-SAR-POV-Poverty and Equity (ESAPV)
no_of_pages: 40
author_keys: 0: author_key: 1717131
1: author_key: 806943
model_extractions: Policy Research Working Paper 9255 The Insights and Illusions of Consumption Measurements _Erich Battistin Michele De Nadai Nandini Krishnan_ Poverty and Equity Global Practice May 2020
Policy Research Working Paper 9255

# **Abstract**

Although household well-being is anchored in long-term average rates of consumption, welfare comparisons typically rely on shorter-duration survey measurements. This paper develops a new strategy to identify the distribution of these long-term rates by leveraging a large-scale randomization that elicited repeated short-duration measurements from diaries and recall questions. Identification stems from diary-recall differences in reports from the same household, does not require these reports to be error-free, and hinges on a research design with broad replicability. This strategy delivers cost-effective suggestions for designing survey modules to yield the most accurate measurements of consumption well-being, and offers new insights for interpreting and reconciling diary-recall differences in household expenditure surveys.

This paper is a product of the Poverty and Equity Global Practice. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at nkrishnan@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# The Insights and Illusions of Consumption Measurements<sup>_∗_</sup>

Erich Battistin<sup>_†_</sup> Michele De Nadai<sup>_‡_</sup> University of Maryland, CEPR, FBK-IRVAPP and IZA University of New South Wales Nandini Krishnan<sup>_§_</sup> World Bank JEL Classification: C81, D31, D63, E21, I32 Keywords: Household Consumption, Household Expenditure Surveys, Inequality, Poverty and Household Well-Being, Modes of Data Collection

> _∗_ This version Thursday 21st May, 2020. The manuscript benefited from comments by Kathleen Beegle, Leah Bevis, Joachim De Weerdt, Beatriz Godoy, Arthur Lewbel, Juan Munoz, Sergio Olivieri, Thomas Pave Sohnesen and Tara Vishwanath as well as seminar participants at the CEPR conference on New Consumption Data (September 2019), NEUDC (October 2019) and the IARIW-World Bank conference on “New Approaches to Defining and Measuring Poverty in a Growing World”(November 2019). The views expressed here are those of the authors alone.

> _†_ Department of Agricultural and Resource Economics, 7998 Regents Drive, Symons Hall, College Park, MD 20742, United States of America. Contact: ebattist@umd.edu.

> _‡_ UNSW Business School UNSW Sydney, NSW 2052, Australia. Contact: m.denadai@unsw.edu.au.

> _§_ Poverty and Equity Global Practice, World Bank, 1818 H Street, NW Washington, DC 20433, United States of America. Contact: nkrishnan@worldbank.org.
# **1 Introduction**

Household surveys are the dominant tool to measure and monitor welfare in low- and middle-income countries. In these contexts, measures of material well-being typically consider a combination of household consumption and expenditures, of which food comprises a large share. Indeed, global monitoring of extreme poverty relies on food-based estimates of welfare, such as the Cost of Basic Needs approach. Proxy means-tested targeting of interventions also derives from these estimates of welfare. Many programs worldwide target food consumption and nutrition, ranging from school meals to food subsidies like the Supplemental Nutrition Assistance Program (SNAP). Because of its centrality to policy and welfare measurement, a large body of the literature has questioned the accuracy of different modes of collecting consumption data to assess household well-being. We revisit this issue at a time when developing countries have increasingly moved to recall modes to collect food consumption in their national expenditure surveys.

Acquisition diaries have become the workhorse for benchmarking comparisons of household measurements from alternative collection modes. Beegle et al. (2012) randomize households in Tanzania to individual assisted diaries to define a benchmark against alternative survey designs that use recall questions. This is possibly the most important experiment on consumption measurement in low- and middle-income countries in the last decade (a more recent study for Niger is Backiny-Yetna et al., 2017). The accuracy of consumption measurements is also of concern in the developed world. For example, Brzozowski et al. (2017) compare recall questions on food spending from the Canadian Food Expenditure Survey to data from expenditure diaries. Battistin and Padula (2016), among others, do so using the Consumer Expenditure Surveys in the United States. In these instances, too, the accuracy of reports is assessed against diaries.

The assumption that diaries provide an error-free benchmark has little statistical justification. Diaries arguably minimize various types of reporting errors, including recall errors, telescoping, rule of thumb, and errors of omission (Lyberg and Kasprzyk, 2004). However, diaries are far from perfect. Most diary surveys are characterized by differences in reporting within the interview period, usually with a declining pattern over time (Silberstein and Scott, 2004). Possible explanations are not only declining cooperation due to fatigue but also the tendency of participants to deviate from the usual purchasing behavior as an effect of the diary or social pressure (Peterson Zwane et al., 2011). The proxy reporting of events experienced by persons other than the respondent yields figures different from those obtained from individual diaries (Beegle et al., 2012). The ratio between food totals computed from household surveys using diaries and national accounts have been found to be considerably low (Gieseman, 1987, and Bee et al., 2013). Moreover, diaries are expensive, as completion involves a reasonable standard of literacy and commitment, and regular visits from interviewers to ensure quality standards boost costs.

Faced with high costs of implementation, respondent fatigue, the need for enumerator effort, and the risks of enumerator shirking, many developing countries have moved to collecting recall data.<sup>1</sup> In recall interviews, households are typically asked to report on food _consumption_ during the specified period and

> 1The World Bank’s Living Standards Measurement Study (LSMS) survey finder includes a list of household consumption and expenditure surveys: of almost 90 surveys from more than 25 countries, more than 75 use recall. See also the guidelines on the use of recall interviews endorsed by the United Nations Statistical Commission (FAO and The World Bank, 2018).

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to value that consumption, irrespective of source, at market prices. Diaries instead collect information on food _acquired_ during the interview period regardless of when the food was consumed. There is little evidence on the best approach for managing the transition to recall interviews, with most countries accepting a break in the series on poverty, food security, and welfare aggregates, given that the survey mode affects indicators of poverty and inequality in the cross-section (Beegle et al., 2012) and over time (Battistin, 2003, Pudney, 2008, Attanasio and Pistaferri, 2016, and Coibion et al., 2020).

This paper studies how to obtain the most accurate measurements of population well-being in household expenditure surveys. Specifically, we define a household’s well-being as their _usual_ (or _average_ ) rate of consumption in a fixed period, which in our application is one week. Usual consumption is a longer-run average of weekly intakes and the quantity grounding welfare comparisons on the theoretical concept of consumption smoothing. The empirical challenge then is to use survey measurements collected as a snapshot, typically covering one week, to infer usual consumption. We develop a strategy to identify the distribution of usual consumption, which is _unobserved_ because of the short-term nature of interviews, without applying any corrections to raw data or assuming _ex ante_ which mode of collection – diaries or recall questions – yields the most accurate measurement.

We frame the identification of consumer welfare in a general setting, encompassing “survey errors” – defined here as differences between survey measurements and usual consumption – related to household preferences and misreporting behavior. Specifically, households with different consumption preferences in the interview week but planning for the same dollar amount of consumption over a longer period (e.g., a month) will have the same well-being. Such _irregularity of consumption_ has implications for welfare analysis even when survey measurements are not misreported (as discussed by Kay et al., 1984, and Deaton and Grosh, 2000, among others). In addition, our analysis allows for survey errors from any of two broad sources. The first source is _infrequency of acquisitions_ , which arises because cash outlays occurred during the interview week may present extreme values resulting from no acquisition or positive acquisition of infrequently purchased items.<sup>2</sup> The second source of survey errors arises because of _misreporting_ , which is the object of a voluminous literature spanning both developed and developing countries (see Deaton and Zaidi, 2002, and Carroll et al., 2015). Our investigation disentangles the relative contribution of any of the above channels in both diary and recall data.

We leverage a unique large-scale experiment on consumption measurement in Iraq designed for the Iraq Household and Socio-Economic Survey (IHSES) in 2012. The survey ran continuously for one year and was administered to approximately 25,000 households across the country. All households filled out a 7-day diary on their acquisitions with the assistance of enumerators during regular visits. One-third of households were randomized to an additional survey module, administered before the diary, that asked them to recall food consumption in the last 7 days. The size of the experimental sample, approximately 8 _,_ 000 households, is of a different order of magnitude compared to other studies on consumption measurement in developing countries. The household diary employed was of the standard acquisition type, with one respondent recording on behalf of the household. Diaries were filled out by the same person

> 2If stocks are observed, one can impute consumption from diaries by measuring stock inflows and outflows (as in Beegle et al., 2012). However, the measurement of stocks is costly, prone to additional measurement concerns (Sharp et al., 2019), and seldom available in empirical work.

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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.

3
weakest assumptions needed to address measurement error in empirical work.

Our research design yields a second important finding: the presumption that diaries should be the benchmark for measuring household well-being is an illusion. Although a consumer demand model implies that acquisition is an unbiased estimate of usual consumption, we demonstrate empirically that the difference between these two quantities can be substantial. For example, the likelihood of measuring less than half of the actual value of usual consumption with a diary is 15.1% and 14.1% for households in the first and fourth consumption quartiles, respectively. The likelihood of attributing a value for usual consumption at least twice the size of the actual one is 4.1% and 3.7% for these households, respectively. Importantly, we find that diary errors remain large for perishable components of food, which implies that our conclusions are not mechanically driven by consumption from stocks or large cash outlays. We conclude that, if enumerator visits are effective to reduce misreporting in diary entries, mistaken conclusions about welfare measurement from diary data must primarily depend on how households decide to smooth their consumption rather than on the frequency of acquisitions. This implies that even when error-free measurements of consumption in the interview week are attained, misleading conclusions about actual well-being could still ensue, as well-being is a longer-run average of weekly intakes.

Our investigation does not find evidence of zero-mean or classical errors in recall data, which is a conclusion often conjectured in the literature. Our research design allows us to prove this conjecture, and also provides guidance on how to sign the bias from recall errors in empirical work. Specifically, we show that – in our case study – overreporting of usual consumption is between three and four times more likely than underreporting. This finding is not only consistent with telescoping effects but also with more general cognitive errors by respondents when they compute the market value of their consumption. Moreover, we find that the quality of recall data improves with the actual value of usual consumption, which suggests that errors may be comparatively less important for households with higher permanent income and human capital.

Further, the above properties of survey errors have important consequences for the computation of aggregate welfare statistics in our study. Precisely, we find that diary errors yield poverty rates and inequality figures that are less reliable than those obtained from recall data, which yield uniformly larger probabilities of correct classification for households with low usual consumption. However, we also find that the upper tail of the distribution of usual consumption is closer to that from diary data, suggesting that deciding which survey measurement will work best ultimately depends on the policy question of interest.

What are the implications for the design of household surveys? First, there is no loss in accuracy from using a recall module despite the higher costs of using a diary: a recall survey is almost as close to the optimal design as a diary when inequality measurement is of concern. The choice of the most appropriate collection mode can be seen as the solution to a decision problem in which each household is assigned to a diary or a recall module to minimize the overall impact of survey errors. We therefore study which assignment would make the inequality and poverty measurements computed from raw data as close as possible to their true values. We show that the optimal solution to this problem is a mix of diaries and recall interviews, with relatively more well-off households assigned to diaries. Although a 4
survey employing only recall interviews is suboptimal in our case study, we find no evidence that such survey would compromise the accuracy of policy conclusions about poverty and inequality compared to a diary survey. We also show that recall distributions are closer to the underlying distribution of usual consumption than in diaries. Thus, our findings provide the theoretical and empirical rationale for the choice of switching to recall modules in expenditure surveys.

Second, our empirical investigation demonstrates the cost-effectiveness of eliciting _repeated measurements_ from the same household, which is in the spirit of Browning and Crossley (2009). More specifically, when diaries are the collection mode of choice – as in the Iraq IHSES – the inclusion of one additional module on recalled consumption for a subsample of interviewees will allow for learning about survey errors and disparities in well-being at relatively low cost. When a recall survey is chosen instead, resources should be allocated by statistical agencies to conduct follow-up diary interviews on a smaller sample of interviewees. Our methodology therefore offers actionable recommendations to design a protocol for harmonizing time series of poverty and inequality in those countries that consider a transition to new modes of data collection.

The remainder of the paper is organized as follows. The next section presents the general formulation of the problem. Section 3 describes the institutional background and presents the diary-recall experiment. Section 4 presents the conditions for nonparametric identification. Estimation is also discussed, along with possible threats to the validity in our research design. Section 5 presents our results on the nature and consequences of survey errors. Section 6 looks at the optimal combination of diary and recall interviews. Section 7 concludes the paper.

# **2 General Formulation of the Problem**

## **2.1 Quantity of Interest**

Suppose that the household has a target monetary amount of consumption over a period of _τ_ time units. _Consumption_ is equal to _acquisition_ over this period. Without loss of generality the time unit coincides with the interview period, which is one _week_ in our case study.

We define _Y_<sup>_∗_</sup> as the average household consumption _per time unit_ , although we will often use the terms _usual consumption_ or _consumption rate_ as shorthand. The quantity _Y_<sup>_∗_</sup> is _not observable_ , as it results from a utility maximization problem solved by the household. It is interpreted as the average weekly rate of consumption that one would measure only by observing the household over _τ_ weeks.<sup>5</sup> The empirical challenge is to learn about the distribution of _Y_<sup>_∗_</sup> , _FY ∗_ [ _y_ ], using diary ( _Y_<sup>_d_</sup> ) and recall ( _Y_<sup>_r_</sup> ) measurements for the interview period. These are possibly inaccurate indicators of _Y_<sup>_∗_</sup> , and this problem is inherently related to the short-term nature of interviews. Even perfectly measured values of consumption in a random week (obtained, for example, from stock inflows and outflows) will be different from _Y_<sup>_∗_</sup> in general.

The distribution _FY ∗_ [ _y_ ] is needed to retrieve key inequality functionals for policy analyses, which are

> 5For example, if the household targets $1000 of consumption over _τ_ = 4 weeks and the desired amount is consumed in this period, the usual rate of consumption in one week must be _Y_<sup>_∗_</sup> = $1000 _/_ 4 = $250. In this example, 4 weeks and $1000 are choices made by the household to maintain smooth living standards depending on preferences and material resources.

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the quantities of interest here. Household and area characteristics are also available, the conditioning on which is left implicit in what follows.

# **2.2 Diary Interviews**

Diaries are a self-administered form of data collection in which respondents must record their market purchases and the estimated market values of acquisitions from non-market sources. In our case study, the quantity _Y_<sup>_d_</sup> is defined as acquisitions captured over the one week diary.

The theoretical case for differences between consumption _Y_<sup>_∗_</sup> and acquisition _Y_<sup>_d_</sup> can be made with the aid of a model of frequency of purchase and consumer demand (see Meghir and Robin, 1992, and Coibion et al., 2020, for examples). Using the notation above, households decide simultaneously on the desired consumption over _τ_ weeks (e.g., one month) and a purchasing strategy. Assume, for simplicity, that acquisition is only conducted through cash outlays. Let _N_<sup>_∗_</sup> be the desired number of purchases in one week implied by utility maximization. An implication of this choice is that _Y_<sup>_∗_</sup> _/N_<sup>_∗_</sup> must be the expected amount of each purchase. If _N_ denotes the number of purchases recorded in the diary week, the assumption of purchases of a constant amount yields:<sup>6</sup> The main take-away message from this model is that household acquisition in the diary week is a random variable centered at the quantity of interest _Y_<sup>_∗_</sup> . Specifically, in a randomly chosen diary week, we have _E_ [ _N |N_<sup>_∗_</sup> = _n_<sup>_∗_</sup> _, Y_<sup>_∗_</sup> = _y_<sup>_∗_</sup> ] = _n_<sup>_∗_</sup> and:

The model also implies that distributional indicators obtained from diary acquisitions may be liable to provide incorrect policy conclusions. For example, calculations reported in the Online Appendix show that if acquisitions occur independently at the same rate _N_<sup>_∗_</sup> across households, we have: where _CV_ ( _Y_<sup>_∗_</sup> ) is the coefficient of variation of consumption. The last expression implies that for items usually acquired once a week ( _N_<sup>_∗_</sup> = 1), the variance of acquisition must be at least twice as large as the variance of consumption. When the coefficient of variation is equal to one, the variance of acquisition becomes three times the variance of consumption.

We conclude that acquisition diaries may yield a distorted idea of consumption _Y_<sup>_∗_</sup> even if all entries are recorded correctly. However, entries may not be exempt from errors. As the effects of fatigue on accuracy have long been known to researchers (Silberstein and Scott, 2004), it is common practice to assist households through frequent visits by enumerators during the recording process. It is generally presumed that this design yields the most accurate data (see Brzozowski et al., 2017, for discussion).

> 6The model can be easily generalized to allow for random spending around _Y ∗/N ∗_ .

6
Misreporting of diary entries affects equation (1) by introducing a disturbance term on the right-hand side.

# **2.3 Recall Interviews**

Many household surveys implemented in the developing world ask respondents to recall quantities in a preselected list of food items consumed over a fixed period and their associated market value. In our case study, the recall period is seven days, ending with the interview (Lyberg and Kasprzyk, 2004, Beegle et al., 2012, and Crossley and Winter, 2014 discuss the effects of changing this time window). Information is collected in two steps. The respondent is first asked if the items in the list were consumed during the past week, going vertically down in the survey form from the first item to the last. For the items for which there is a positive response, the quantity consumed and the unit of measurement are reported. The respondent then must backcast the monetary value of this quantity by reporting estimated expenditures (if market purchased) or market values (if self-produced, received as a gift, or received in barter). The monetary equivalent of recalled consumption (quantity consumed times implicit price per unit, or unit value) is the quantity we consider for the measurement _Y_<sup>_r_</sup> .

Why should one expect differences between _Y_<sup>_∗_</sup> and recalled consumption _Y_<sup>_r_</sup> ? As discussed above, consumption in the week before the interview need not coincide with usual consumption. Arguments can be made in favor of using a longer reference period for recall questions, or asking directly about “usual week” consumption. However, limited cognitive abilities and the difficulty of recalling the timing of events may challenge respondents’ ability to perform the computations. Estimating the monetary equivalent of the quantities consumed from non-market sources adds an additional layer of difficulty. Errors may arise because the respondent solves a prediction problem rather than reporting a noisy measurement. Households may form their answers using the information available, for example including other features of the survey, so that the value _Y_<sup>_r_</sup> is their best predictor of the unobserved value _Y_<sup>_∗_</sup> . Experimental evidence in psychology research supports this interpretation (see Comerford et al., 2009). Rounding errors in providing the quantities consumed or telescoping (the act of recalling consumption occurring over a longer period of time as within the reference period; Neter and Waksberg, 1964) were found to be important factors in the computation. These concerns may be particularly salient in a context such as Iraq, where many households have relatively low educational attainment.

The textbook assumption of classical errors is violated in general, with no obvious direction of bias. For example, if the best prediction is obtained by respondents in terms of quadratic loss, recall errors must be centered at zero and not correlated with _Y_<sup>_r_</sup> but correlated with consumption _Y_<sup>_∗_</sup> . If an absolute value loss is used instead, the household would report the median of _Y_<sup>_∗_</sup> given the information available (see Hyslop and Imbens, 2001, and Hoderlein and Winter, 2010).

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## **2.4 Aggregation over Items**

Our empirical analysis uses measurements computed by aggregating over a number of food items. The definitions above can be adjusted to allow for _I_ items: where _Yi_<sup>_∗_is usual consumption of item</sup><sup>_i_= 1</sup><sup>_, . . . , I_,</sup><sup>_Y∗≡_�</sup> _i_<sup>_Y_</sup> _i_<sup>_∗_is (total) food consumption, and</sup><sup>_b∗_</sup> _i_<sup>=</sup> _Y_<sup>_Y_</sup> _<u>i</u>_<sup>_<u>∗∗</u>_</sup> is the budget share allocated to the item.

Even after aggregation, acquisition from diaries remains an unbiased estimate of usual consumption _Y_<sup>_∗_</sup> . Specifically, a generalization of the frequency of purchase model to the case of a vector of items still yields equation (2) for total food acquisition _Y_<sup>_d_</sup> , as we show in the Online Appendix. We also show there that, despite the aggregation across items, severely misleading conclusions about inequality from diary data are still possible. For example, assuming that acquisitions are independent across goods and occur at the same rates _Ni_<sup>_∗_’sforallhouseholdsyields:</sup> The lower bound can be significantly larger than one depending on assumptions about the correlation of consumption across items. For example, if consumption is independent across items and goods are acquired once a week ( _Ni_<sup>_∗_= 1),thevarianceof</sup><sup>_Yd_mustbeatleasttwiceaslargeasthevarianceof</sup><sup>_Y∗_.</sup> Equation (2) may still be valid if survey participation affects the purchasing behavior of respondents, or with misreporting. For example, recorded cash outlays tend to be higher at the beginning of the diary period (in the first day in particular; see Lyberg and Kasprzyk, 2004, and Silberstein and Scott, 2004). If this pattern arises because of antedated expenditures that would normally take place later in the same week, equation (2) is still valid. Moreover, acquisition _Y_<sup>_d_</sup> is still an unbiased estimate of _Y_<sup>_∗_</sup> under more general forms of manipulation or misreporting (e.g., overspending on goods linked to social status or underreporting due to an embarrassment motive) that result in any zero-mean, possibly non-classical, disturbance term at the right hand side of _Y_<sup>_d_</sup> in (3).

# **3 Data, Descriptives and Graphical Analysis**

## **3.1 The Iraq Household and Socio-Economic Expenditure Survey**

Iraq is an upper middle income, resource-rich, fragile, and conflict-affected country facing development challenges in line with those of far poorer countries. The median education level in 2012 was primary schooling. While infant mortality rates remain below the norm for similar countries, a third of Iraqi children aged 0-5 are stunted, 58 percent of adult males of working age are employed, and only one in 8
five women of working age participate in the labor market. International standards for measuring poverty and socioeconomic indicators were adopted after the fall of Saddam Hussein.

The implementation of a new LSMS began in 2007, the IHSES. We use the second round of this survey, which was carried out in 2012 to obtain official statistics on poverty, food security, income, labor market outcomes, health, and education. The survey is the basis for information on the household consumption component in the national accounts and for consumer price indices. The IHSES collects a more detailed labor and income module than the norm (relative to, for instance, countries where the LSMS is complemented by a Labor Force Survey) and has special modules to fill critical data gaps (such as anthropometrics and time use).

IHSES 2012 had an intended sample size of 25,488 households and a final sample of 24,750 households. The population was stratified in 119 districts ( _qadahs_ ). Within each district, 24 census enumeration areas (EAs) were randomly selected, and 9 households were sampled within each EA. Teams of interviewers were responsible for fieldwork in two districts distributed over one year. In each month, interviews were conducted by each team in four randomly selected EAs, two from each of the assigned districts. It follows that, by design, households in one EA had an equal probability of being interviewed in any of the survey months.<sup>7</sup>

# **3.2 Food Measurements**

A 7-day diary of the standard acquisition type was used in all interviews, with one respondent (the head of household or the most informed person) recording acquisitions on behalf of the household. The completion of the diary was assisted by enumerators, who visited households on alternate days to check the quality of entries, clarify any questions, and enter recorded data into computers. Instructions emphasized that households should record daily acquisitions of food and non-food commodities and meals away from home starting from the day after the first visit. Qualitative analyses confirmed a high level of enumerator effort expended to ensure that diary entries were correctly and regularly recorded.

One-third of households within each EA were also administered a recall module on food consumption during the first visit. Specifically, _two_ (diary and recall) measurements are available for household 3, 6 and 9 in each EA, while _one_ (diary) measurement is available for all remaining households. The availability of repeated measurements and the scale of the experiment mark a departure from past research on consumption measurement in developing economies. The recall module asked the market value of the quantities consumed by the household in the 7 days prior to the enumerator’s visit. A list of expenditure groups was selected based on an assessment of their importance in IHSES 2007 diaries. Fieldwork and resource constraints meant that this list was consolidated to include 20 such groups of aggregated food items (we list these groups in Appendix Figure A.2).<sup>8</sup>

> 7One of the important rationales for year-long surveys is the ability to capture seasonality in consumption and livelihoods. Seasonality in consumption is experienced due to differences in the availability of food items and, importantly, due to the month-long observations of Ramadan, and two Eid holidays. To ease fieldwork, a team would visit the 2 EAs from one of the districts in wave 1 (days 1 to 14 of the month) and the 2 EAs from the other district in wave 2 (days 15 to 29) of the month. Moreover, within each wave, the teams alternated the EA visited each day.

> 8The scale of IHSES precluded the use of individual diaries, which is deemed the most reliable instrument in Beegle et al. (2012). Our experiment is interesting because the cheaper option is randomized (recall questions) while the expensive option (diaries) remains the baseline for all households surveyed. Our recall module bears the closest resemblance to the subset list

9
Randomization to different interview modes was adopted to guide an eventual transition to recall as a cheaper way of collecting IHSES data on food.<sup>9</sup> As non-food expenditures are already measured using different recall periods depending on the frequency of purchase, the experiment was limited to food measurements. Recall groups were defined and matched to their counterparts in the diary using the COICOP classification, yielding the quantities _Y_<sup>_d_</sup> and _Y_<sup>_r_</sup> defined above. Both quantities include household estimations of the market value of food items consumed that are not acquired from the market such as self-produced items and gifts. In practice, however, market purchases account for almost all entries in our case study. We used the square root of family size to equivalize survey measurements in our analysis. Acquisitions and valuations of recalled consumption, in Iraqi dinars, were transformed to US dollars throughout.

There are no incentives for households to report specific consumption levels, for instance, to meet thresholds for social programs. The most important social safety net, the Public Distribution System (PDS), is a universal in-kind food subsidy, and other much smaller public transfers were not meanstargeted at the time of the survey.<sup>10</sup> A separate IHSES module collected information about ration items received, consumed, bartered, sold or given away by the household during the last 30 days. The diary recorded market transactions to purchase ration items over and above the monthly allocation, but these transactions in our data are rare and small in magnitude. Because our focus is on comparing diary entries with recalled consumption, the consumption of rationed products is excluded from our analysis.

# **3.3 Covariate Balance and Raw Distributions**

Table 1 documents the demographic balance for households assigned to recall modules. This randomly selected group of roughly 8,000 households constitutes our “treatment” sample, while all remaining households form the “baseline” sample. This table shows treatment-control differences from regressions that control for strata (EAs) used in the randomization design. All variables are well balanced between groups, as can be seen from the small and nonsignificant coefficient estimates in the two panels.

Randomization to the recall module did not affect the reporting of diary entries, as shown in Table 2. Respondents might adjust their diary to reconcile aggregates with those in the recall module, as in bounded interviews (see Neter and Waksberg, 1964), but the results in column 2 rule out this possibility. Specifically, Panel A reports treatment-control differences in cumulative distributions of diary acquisitions _Y_<sup>_d_</sup> at selected percentiles, as well as differences in averages. Treatment-control differences in the dispersion of _Y_<sup>_d_</sup> are reported in Panel B. Results in column 2 imply that diaries collected in treatment and baseline samples yielded statistically indistinguishable distributions of household acquisitions.

Recall data yielded higher consumption figures than diary acquisition. This can be seen from column 3 in Panel A of Table 2, which presents coefficients from the same regressions in column 2 but using

> of the 17 most important food items considered in the Beegle et al. (2012) Tanzania experiment.

> 9As noted in the sampling and fieldwork documentation, the recall module “ _should be administered in the first visit to the household, before the recording of food consumption by diaries. Asking these questions afterwards (when both the respondent and the interviewer will know the diary records) would defeat the purpose of this module, which is to compare the results obtained from the two instruments, to assess the possibility of applying in future surveys the recall method instead of diaries_ ”.

> 10The PDS includes 13 rationed products, of which four, whole wheat flour, rice, vegetable oil/cooking oil and sugar represent almost 98% of total rationed expenditures. Because they are heavily subsidized, ration expenditures account for only 6% of average household spending.

10
Table 1: Summary statistics and covariate balance<sup>a</sup>

||**All Ho**|**useholds**|**Rural H**|**ouseholds**|**Urban H**|**ouseholds**|
|---|---|---|---|---|---|---|
||Baseline<br>mean|Treatment<br>diference|Baseline<br>mean|Treatment<br>diference|Baseline<br>mean|Treatment<br>diference|
||(1)|(2)|(3)|(4)|(5)|(6)|
|||**Panel **|**A. Househ**|**old demogr**|**aphics**||
|Log income|6.1908|-0.0042|5.9676|-0.0265|6.2743|0.0041|
||[_0.0079_]|(_0.0118_)|[_0.0114_]|(_0.0187_)|[_0.0097_]|(_0.0146_)|
|Log implicit rent|4.6725|-0.0091|3.8909|-0.0016|5.0159|-0.0126|
||[_0.0120_]|(_0.0100_)|[_0.0133_]|(_0.0137_)|[_0.0124_]|(_0.0133_)|
|Log price index|0.9921|-0.0027|0.9489|-0.0019|1.0082|-0.0030|
||[_0.0014_]|(_0.0018_)|[_0.0017_]|(_0.0023_)|[_0.0017_]|(_0.0024_)|
|Household size|6.7184|0.0843|7.7662|0.1452|6.3263|0.0614|
||[_0.0365_]|(_0.0654_)|[_0.0609_]|(_0.1168_)|[_0.0432_]|(_0.0786_)|
|N. of children|2.1690|-0.0369|2.8358|0.0190|1.9195|-0.0580|
||[_0.0228_]|(_0.0392_)|[_0.0385_]|(_0.0725_)|[_0.0269_]|(_0.0466_)|
|N. of adults|3.9653|0.0455|4.1670|0.0440|3.8899|0.0461|
||[_0.0250_]|(_0.0450_)|[_0.0409_]|(_0.0730_)|[_0.0306_]|(_0.0556_)|
|N. of occupied adults|1.5105|-0.0076|1.5842|-0.0377|1.4829|0.0037|
||[_0.0123_]|(_0.0212_)|[_0.0200_]|(_0.0354_)|[_0.0151_]|(_0.0259_)|
|||**Panel B. **|**Household**|**head dem**|**ographics**||
|Male|0.8841|0.0005|0.9175|-0.0083|0.8716|0.0039|
||[_0.0041_]|(_0.0069_)|[_0.0047_]|(_0.0090_)|[_0.0053_]|(_0.0089_)|
|Age|46.7432|0.1303|44.7398|-0.1067|47.4930|0.2194|
||[_0.1661_]|(_0.2786_)|[_0.2388_]|(_0.4218_)|[_0.2087_]|(_0.3489_)|
|Has primary education|0.6419|-0.0183*|0.5626|-0.0132|0.6715|-0.0202|
||[_0.0056_]|(_0.0099_)|[_0.0085_]|(_0.0153_)|[_0.0070_]|(_0.0124_)|
|Has secondary education|0.2527|-0.0093|0.1629|-0.0007|0.2863|-0.0125|
||[_0.0056_]|(_0.0093_)|[_0.0067_]|(_0.0118_)|[_0.0072_]|(_0.0120_)|
|Reads and writes|0.7780|-0.0013|0.7161|-0.0022|0.8012|-0.0010|
||[_0.0047_]|(_0.0081_)|[_0.0076_]|(_0.0122_)|[_0.0058_]|(_0.0101_)|
|Employed|0.7313|-0.0108|0.7514|-0.0164|0.7238|-0.0087|
||[_0.0054_]|(_0.0097_)|[_0.0075_]|(_0.0140_)|[_0.0069_]|(_0.0122_)|
|Number of districts|119|119|119|119|119|119|
|Number of EAs|2,828|2,828|2,828|2,828|2,828|2,828|
|Number of households|16,530|8,220|16,530|8,220|16,530|8,220|

> a This table reports tests for covariate balance using household-level demographics (Panel A) and demographics of the household head (Panel B). Household is the unit of analysis. Columns (1), (3) and (5) report statistics for the _baseline sample_ , which consists of households excluded from the recall-module experiment. Means and standard deviations (in square brackets) are reported for variables listed at left. The _treatment sample_ consists of households randomized to the recall-module experiment. For these households, both diary and recall measurements are available. Columns (2), (4) and (6) report coefficients from regressions of each variable on the treatment dummy and a full set of dummies for strata (enumeration areas, EAs) used in the randomization design, pooling data from the two samples. Standard errors for the coefficient on the treatment dummy (in round brackets) are clustered on EAs. * _p <_ 0 _._ 10, ** _p <_ 0 _._ 05, *** _p <_ 0 _._ 01.

11
Table 2: Between-sample differences in survey measurements<sup>a</sup>

||**Baseline**|**Treatment **|**Diference**|
|---|---|---|---|
||Diary|Diary-Diary|Recall-Diary|
||(1)|(2)|(3)|
||**Panel A.**|**Cumulative D**<br>**and Locatio**|**istribution**<br>**n**|
|5th percentile|2.5291|0.0059<br>_(0.0045)_|-0.0431***<br>_(0.0042)_|
|25th percentile|2.0622|0.0121|-0.1482***|
|||_(0.0085)_|_(0.0086)_|
|50th percentile|2.5291|0.0097|-0.1731***|
|||_(0.0099)_|_(0.0101)_|
|75th percentile|2.9788|0.0100<br>_(0.0085)_|-0.0944***<br>_(0.0093)_|
|95th percentile|3.6601|0.0032|-0.0032|
|||_(0.0037)_|_(0.0039)_|
|Mean|2.5184|-0.0158|0.2706***|
|||_(0.0099)_|_(0.0105)_|
||**P**|**anel B. Disper**|**sion**|
|Std. deviation|0.7560|0.0182|-0.0650***|
|||_(0.0157)_|_(0.0244)_|
|Range 10%-90%|1.8207|0.0044<br>_(0.0387)_|-0.3227***<br>_(0.0320)_|
|Range 25%-75%|0.9166|-0.0044<br>_(0.0236)_|-0.1286***<br>_(0.0207)_|
|Number of districts|119|119|119|
|Number of EAs|2,828|2,828|2,828|
|Number of households|16,530|24,750|24,750|

> a Household is the unit of analysis. Column (1) reports selected statistics of the log( _Y_<sup>_d_</sup> ) distribution in the _baseline sample_ . Panel A: Columns (2) and (3) report coefficients on the treatment dummy in regressions that control for a full set of strata effects (enumeration areas, EAs) pooling data for treatment and control samples. Standard errors (in brackets) are clustered on EAs. For cumulative distributions, the outcome in column (2) is an indicator for having log( _Y_<sup>_d_</sup> ) below the value in column (1); the outcome in column (3) is an indicator for having log( _Y_<sup>_d_</sup> ) below the value in column (1) for households in the baseline sample, and an indicator for having log( _Y_<sup>_r_</sup> ) below the value in column (1) for households in the treatment sample. For the mean, the outcome in column (2) is log( _Y_<sup>_d_</sup> ). The outcome in column (3) is log( _Y_<sup>_d_</sup> ) for households in the baseline sample, and log( _Y_<sup>_r_</sup> ) for households in the treatment sample. Panel B: The value in column (2) is the difference of statistics computed using log( _Y_<sup>_d_</sup> ); the value in column (3) is the difference of statistics computed using log( _Y_<sup>_d_</sup> ) for households in the baseline sample, and log( _Y_<sup>_r_</sup> ) for households in the treatment sample. Standard errors for these differences are computed via bootstrap using 1,000 replications. * _p <_ 0 _._ 10, ** _p <_ 0 _._ 05, *** _p <_ 0 _._ 01.

12
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income distribution are, on average, in the second quintile of the consumption distribution. Households in the top income quintile are, on average, in the third consumption quintile. This relationship does not change with the survey mode employed, suggesting that diary-recall differences cannot be explained by differential errors at different points of the income distribution.

# **4 Nonparametric Identification**

## **4.1 Assumptions**

We assume that usual consumption _Y_<sup>_∗_</sup> and its measurements ( _Y_<sup>_d_</sup> _, Y_<sup>_r_</sup> ) are continuously distributed with bounded density. Consumption is defined by aggregation over items. The distribution of consumption on each item need not be continuous, for example because of zero consumption on single items. The assumption here is that aggregation across items makes the distribution of household consumption smooth enough. A visual inspection of the data corroborates this idea. Moreover, the theoretical case for lognormally distributed consumption is made in Battistin et al. (2009).

We maintain the assumption that the difference between household consumption and acquisition is centered at zero. The theoretical case for this restriction stems from the consumer demand model leading to _Y_<sup>_d_</sup> in equation (3) – see also the Online Appendix. As we discussed above, the following assumption also allows for possibly nonclassical reporting errors in diary entries.

**Assumption 1** _(Unbiasedness of Diary Measurements) E_ [ _Y_<sup>_d_</sup> _− Y_<sup>_∗_</sup> _|Y_<sup>_∗_</sup> = _y_<sup>_∗_</sup> ] = 0 _._

Nonparametric identification requires additional restrictions imposing structure between observables and usual consumption that we discuss with the aid of a directed acyclic graph (DAG). In Figure 3 survey measurements _Y_<sup>_d_</sup> and _Y_<sup>_r_</sup> depend on consumption _Y_<sup>_∗_</sup> . We require the existence of an observable variable _Z_ which affects survey measurements only through consumption.

**Assumption 2** _(Exclusion Restriction) There exists a continuous variable Z such that_ ( _Y_<sup>_r_</sup> _, Y_<sup>_d_</sup> ) _⊥Z|Y_<sup>_∗_</sup> _._

This is a standard exclusion restriction in the literature which has been used and justified in a number of empirical settings, for example, in education, labor and earning dynamics (see Schennach, 2013, and Wilhelm, 2019). It can be interpreted through the lens of the (unfeasible) regressions of _Y_<sup>_r_</sup> and _Y_<sup>_d_</sup> on _Y_<sup>_∗_</sup> : the variable _Z_ must be an excluded instrument for _Y_<sup>_∗_</sup> in both equations. A more general interpretation follows from the requirement of conditional independence: knowledge of _Z_ must not yield any more information on the reporting of survey measurements than _Y_<sup>_∗_</sup> would otherwise provide.<sup>12</sup> The DAG in Figure 3 also imposes that _Y_<sup>_∗_</sup> is the only common cause for the correlation among survey reports, which is the meaning of Assumption 3 below.

**Assumption 3** _(Conditional Independence) Survey measurements are conditionally independent given usual consumption Y_<sup>_r_</sup> _⊥Y_<sup>_d_</sup> _|Y_<sup>_∗_</sup> _._

> 12This assumption is also known as _surrogacy_ , meaning that the surrogate variable _Y ∗_ must mediate the link between _Z_ and the survey reports (see, for example, Athey et al., 2019).

15
Figure 3: Directed acyclic graph of model assumptions <!-- Start of picture text --> Y d<br>Z Y ∗<br>Y r<br><!-- End of picture text -->

**Note** . A directed acyclic graph (DAG) is a collection of nodes representing random variables and directed edges among nodes representing causal effects which are not mediated by other variables. This figure is a visual representation of the key assumptions used to achieve non-parametric identification. Diary and recall measurements from the same household are denoted by _Y_<sup>_d_</sup> and _Y_<sup>_r_</sup> , respectively. These survey measurements are manifestations of usual consumption _Y_<sup>_∗_</sup> , which is unobserved. The theoretical case for having survey measurements differ from consumption _Y_<sup>_∗_</sup> is made in Section 2. Assumption 2 says that the _observable_ variable _Z_ affects _unobservable_ consumption _Y_<sup>_∗_</sup> and is correlated with survey measurements only through _Y_<sup>_∗_</sup> . Assumption 3 implies that the correlation between survey measurements _Y_<sup>_d_</sup> and _Y_<sup>_r_</sup> follows from the fact that they both measure _Y_<sup>_∗_</sup> . The model in Section 2 sheds light on the meaning of this assumption.

This assumption is sufficiently general to allow for correlation between errors in recalled consumption and errors in diary acquisitions. Assumption 3 also allows for correlation between the respondents’ ability to recall and their usual acquisition frequencies _Ni_<sup>_∗_,ascanbeseenfromequation(3).Inthisrespect,</sup> our analysis allows for forms of misreporting which are considerably more general than independent and classical survey errors. On the other hand, the assumption imposes that only _one_ unobservable drives the correlation between survey measurements. For example, the equations in (3) show that this restriction would be violated if, after conditioning on _Y_<sup>_∗_</sup> , recall errors were still correlated with household preferences over frequency of acquisition.

Two additional formal conditions are needed for identification, which are implicit in Figure 3 and unlikely to be a limitation in our setting. These conditions are _not_ testable because _Y_<sup>_∗_</sup> is unobservable. First, usual consumption must have a causal effect on the distribution of recalled consumption _Y_<sup>_r_</sup> _,_ as shown in Figure 3. The next condition is satisfied, for example, if _E_ [ _Y_<sup>_r_</sup> _|Y_<sup>_∗_</sup> ] increases monotonically in _Y_<sup>_∗_</sup> , but it is much weaker than that.

**Assumption 4** _(Informativeness of Recall) The relationship between Y_<sup>_r_</sup> _and Y_<sup>_∗_</sup> _satisfies FY r_ [ _y|Y_<sup>_∗_</sup> = _y_ 1<sup>_∗_] =</sup><sup>_FYr_[</sup><sup>_y|Y∗_=</sup><sup>_y_</sup> 2<sup>_∗_]</sup><sup>_,foranyvaluesy_</sup> 1<sup>_∗_=</sup><sup>_y_</sup> 2<sup>_∗._</sup>

We also require the following condition.

**Assumption 5** _(Completeness) The relationship between Y_<sup>_d_</sup> _, Y_<sup>_∗_</sup> _and Z satisfies:_

_1. Y_<sup>_d_</sup> _is complete for Y_<sup>_∗_</sup> _;_ 16
## _2. Y_<sup>_∗_</sup> _is complete for Z._

Completeness is common in the literature on nonparametric identification with instrumental variables (as in Newey and Powell, 2003), and is a weak regularity condition. Formally, completeness of _Y_<sup>_d_</sup> for _Y_<sup>_∗_</sup> is guaranteed if _E_ [ _h_ ( _Y_<sup>_d_</sup> ) _|Y_<sup>_∗_</sup> = _y_<sup>_∗_</sup> ] = 0 for all _y_<sup>_∗_</sup> only when _h_ ( _Y_<sup>_d_</sup> ) = 0. Intuitively, we require sufficient variability in the conditional distribution of _Y_<sup>_d_</sup> at different values of _Y_<sup>_∗_</sup> and, similarly, for the conditional distribution of _Y_<sup>_∗_</sup> given _Z_ . The latter requirement is a generalization of the rank condition for the instrument _Z_ to the non-parametric setting.

# **4.2 Empirical Specifications**

Assumptions 1-5 imply: and that there exists only one set of unobservable distributions on the right-hand side yielding the observable distribution on the left-hand side. In other words, under the assumptions stated, data on ( _Y_<sup>_d_</sup> _, Y_<sup>_r_</sup> _, Z_ ) are sufficient to retrieve the distribution of usual consumption: and the distributions of diary survey errors ( _fY d|Y ∗_ [ _y_<sup>_d_</sup> _|y_<sup>_∗_</sup> ]) and recall survey errors ( _fY r|Y ∗_ [ _y_<sup>_r_</sup> _|y_<sup>_∗_</sup> ]). This identification result does not hinge on any parametric assumption, and follows from adapting results in Hu and Schennach (2008) to the case of multiple (diary and recall) survey measurements.

We constructed _Z_ by matching each household to the _set_ of all other households in the _same_ income decile, filling out the diary in the _same_ survey wave and receiving visits from a _different_ team of interviewers. We defined _Z_ as average diary acquisition in this set, which is also an estimate of the average of _Y_<sup>_∗_</sup> in this set because of Assumption 1. The precision of this estimate depends on the sample size used to compute _Z_ for each household, which in our data is about 100 households. The idea then is to use the correlation between a household’s own consumption, _Y_<sup>_∗_</sup> , and the average consumption of similar peers, _Z_ , to learn about the properties of survey measurements. Considering peers in the same income decile and interviewed in the same wave strengthens the case for this correlation. The key exclusion restriction here is that variability in the average consumption of peers in different EAs (and thus visited by different interviewers) cannot explain a household’s diary and recall reports other than through her _Y_<sup>_∗_</sup> . Violations of this assumption due to interviewer effects are ruled out by defining _Z_ from averages of diary acquisitions supervised by non-overlapping survey teams.

We estimated the conditional densities on the right hand side of equation (4) using a flexible class of exponential families (Barron and Sheu, 1991). We let the density of log _Y_<sup>_r_</sup> given log _Y_<sup>_∗_</sup> depend on a vector of parameters **_β_**<sup>_r_</sup> as follows: 17
where the normalizing constant is: _Lk_ ( _x_ ) is the _k_ -th Legendre polynomial defined on the interval [ _−_ 1 _,_ 1], and: We set ∆ _r_ = 4 to ensure that the argument of _Lk_ ( _x_ ) lies in the interval ( _−_ 1 _,_ 1) for any reasonable choice of log _y_<sup>_r_</sup> and log _y_<sup>_∗_</sup> . The smoothing constant _Kr_ determines the degree of departure from normality of the distribution in (5). The value _Kr_ = 2 is equivalent to imposing a normal distribution, while a value _Kr >_ 2 allows for significant departures from normality. The quantities _δ_ 0<sup>_r_and</sup><sup>_δ_</sup> 1<sup>_r_were set to 3 and 7, respectively,</sup> to ensure that the argument of _Lj_ ( _x_ ) is in ( _−_ 1 _,_ 1). Our empirical investigation considers _Jr_ = 2, thus allowing each parameter _θk_<sup>_r_todependonlog</sup><sup>_y∗_viaaquadraticfunction.Thisspecificationimpliesthat</sup> the density in (5) depends on a vector of 3 _× Kr_ parameters **_β_**<sup>_r_</sup> = ( _β_ 10<sup>_r, β_</sup> 11<sup>_r, β_</sup> 12<sup>_r, . . . , β_</sup> _K_<sup>_r_</sup> _r_ 0<sup>_, β_</sup> _K_<sup>_r_</sup> _r_ 1<sup>_, β_</sup> _K_<sup>_r_</sup> _r_ 2<sup>)</sup><sup>_′._</sup> We used the same specification for the density of log _Y_<sup>_d_</sup> given log _Y_<sup>_∗_</sup> , which depends on a 3 _× Kd_ vector **_β_**<sup>_d_</sup> (the density-specific smoothing constant _Kd_ is defined by analogy with _Kr_ ). Finally, we let the density of log _Y_<sup>_∗_</sup> given log _Z_ : depend on a 3 _× Ky_ vector **_β_**<sup>_y_</sup> , we set: with ∆ _y_ = 8, _δ_ 0<sup>_y_= 0,</sup><sup>_δ_</sup> 1<sup>_y_= 3,and</sup><sup>_Jy_= 2,and</sup><sup>_M_(</sup><sup>**_θ_**</sup><sup>_y_)isthenormalizingconstant.</sup> The unknown parameters **_β_**<sup>_r_</sup> , **_β_**<sup>_d_</sup> and **_β_**<sup>_y_</sup> were estimated by sieve maximum likelihood (Grenander, 1981), maximizing the pseudo-likelihood obtained by substituting the unknown conditional densities in (4) with their approximations described above.<sup>13</sup> By allowing the smoothing constants _Kr_ , _Kd_ and _Ky_ to increase with sample size, this procedure yields non-parametric estimates of the conditional densities of interest. We ended up selecting _Kr_ = _Kd_ = 6 and _Ky_ = 3, as negligible improvements to the likelihood were found for larger values of these parameters.

Both diary and recall error distributions are therefore estimated away from normality. The conditional distribution of logged consumption is also not normal, although the degree of departure from this

> 13Log transformations were considered to ensure numerical stability. Distributions of variables in levels, used in the sections below, can be obtained straightforwardly from the distribution of their logs. The likelihood was constructed using all households in the sample. Under Assumptions 1-5 the contribution to the likelihood of households not answering the recall module is: _fY d|Y ∗_ [ _y_<sup>_d_</sup> _|y_<sup>_∗_</sup> ] _fY ∗|Z_ [ _y_<sup>_∗_</sup> _|z_ ] _dy_<sup>_∗_</sup> _._ �

18
Table 3: Survey measurements, usual consumption and survey errors<sup>a</sup>

|||||**Sur**|**vey Err**|**ors**|
|---|---|---|---|---|---|---|
||**Measur**|**ements**|**Consumption**|**Mean**|**M**|**SE**|
||Diary|Recall||Recall|Diary|Recall|
||(1)|(2)|(3)|(4)|(5)|(6)|
|1st decile|4.7270|7.4550|7.1882|1.408|0.288|0.620|
|2nd decile|6.8615|9.7729|8.8986|1.356|0.267|0.512|
|3rd decile|8.5786|11.9128|10.3768|1.320|0.257|0.446|
|4th decile|10.3563|14.1222|11.8958|1.290|0.250|0.396|
|5th decile|12.4364|16.2122|13.4062|1.265|0.246|0.358|
|6th decile|14.7070|18.9989|15.2380|1.239|0.243|0.321|
|7th decile|17.5878|21.9847|17.3200|1.214|0.242|0.288|
|8th decile|21.7675|26.0026|20.3705|1.185|0.242|0.253|
|9th decile|29.4806|32.9604|25.2176|1.149|0.246|0.214|
|Mean|15.2130|18.6712|15.2050||||
|Gini index|0.3765|0.3143|0.2656||||

> a This table compares distributions of survey reports and usual consumption. Columns (1) considers data on _Y_<sup>_d_</sup> for all households. Columns (2) considers data on _Y_<sup>_r_</sup> for the _treatment sample_ – see note to Table 1 for definitions. Column (3) reports the estimated distribution of _Y_<sup>_∗_</sup> . Column (4) reports the estimated mean of _Y_<sup>_r_</sup> _/Y_<sup>_∗_</sup> conditional on deciles of _Y_<sup>_∗_</sup> at left. The mean of _Y_<sup>_d_</sup> _/Y_<sup>_∗_</sup> is not reported, and this is always one because of Assumption 1. Column (5) reports the Mean Squared Error (MSE) of _Y_<sup>_d_</sup> _/Y_<sup>_∗_</sup> conditional on deciles of _Y_<sup>_∗_</sup> at left. Column (6) shows the MSE of _Y_<sup>_r_</sup> _/Y_<sup>_∗_</sup> conditional on deciles of _Y_<sup>_∗_</sup> at left.

benchmark is less pronounced.

# **5 Errors in Survey Reports**

## **5.1 Diary and Recall Measurements**

The presumption that diaries outperform recall data for the measurement of well-being finds little empirical support. This can be seen in Figure 1, where diary and recall histograms are compared with the distribution of usual consumption, _fY ∗_ [ _y_ ] (the continuous line). The lower tail of the recall distribution is closer to that of usual consumption, and mismeasurement from diaries is more substantial at the bottom end. As diary measurements are centered at _Y_<sup>_∗_</sup> (because of Assumption 1), the difference in the modes of _fY ∗_ [ _y_ ] and _fY d_ [ _y_ ] suggests that diary errors must have a thick lower tail. Figure 1 also illustrates how the upper tail of the distribution of recall data is substantially thicker than the distribution of usual consumption, which is better approximated using data from diaries. These conclusions are confirmed by the first three columns of Table 3, where different percentiles of empirical and estimated distributions are compared. The bottom three deciles of the recall distribution are closer to the deciles of _Y_<sup>_∗_</sup> than are those of the diaries. The opposite conclusion applies for the remaining deciles. Consistent with predictions from the model in Section 2, the Gini coefficient from diaries overstates the real value of this index; the distance from the real value is minimized using recall data.

These findings have important implications for the computation of aggregate poverty statistics. This 19
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column (4) of Table 3. Predictions from diaries are also subject to a large variance, which is the takeaway message from Panel A of Figure 5. It follows that when aggregate statistics such as poverty rates or inequality are of interest, recall data yield more reliable figures than diary interviews.

On the bright side, diary errors present important similarities with classical errors in measurement. This conclusion can be reached by noticing that the error densities in Panel B of Figure 5 are reasonably stable across households with different levels of consumption. Column (5) of Table 3 also shows that MSE varies little across consumption percentiles. By taking logs, we conclude that log _Y_<sup>_d_</sup> is centered below log _Y_<sup>_∗_</sup> and has variance approximately independent of log _Y_<sup>_∗_</sup> , which is a finding with important implications for applied research.<sup>15</sup>

# **5.2 Consumption Measurements and the Frequency of Purchases**

Large deviations from the mean in diary error distributions in Figure 5 can hardly be explained by the misreporting of diary entries, as respondents are assisted by frequent visits from enumerators. The most likely explanation is differences between acquisition in the diary week and the household’s usual consumption _Y_<sup>_∗_</sup> . As mentioned in the Introduction, one possible reason for these differences is acquisition patterns arising from stock inflows or consumption out of existing stocks. The practical question then arises of whether diary errors would be substantially lower if one could adjust for stock inflows and outflows, as in Beegle et al. (2012) or Sharp et al. (2019). Although theory implies a negative answer unless household consumption is smoothed exactly across weeks (that is, the case _τ_ = 1 in the consumer demand model), this ultimately remains an empirical matter.

As we cannot observe stock inflows and outflows during the diary week, we replicate the analysis in the last section considering only the more perishable items for which consumption is arguably close to acquisition. Panel A of Figure 7 shows that diary acquisitions of perishable components of food consumption are characterized by large errors. Specifically, items entering our definition of consumption in Panel A are eggs, lamb, beef, chicken and fish, and the budget share devoted to these items is approximately 50% of total acquisition. Panel B of Figure 7 shows the results for the remaining items, which we label non-perishable for brevity. Diary errors _Y_<sup>_d_</sup> _/Y_<sup>_∗_</sup> in Panel A present marginally lower dispersion at the top end of the consumption spectrum, as was the case in Figure 5 (note that _Y_<sup>_∗_</sup> in this panel now refers to usual consumption of perishable items). Errors can be large and lead to important under- or overstatements of the underlying consumption. For example, using a diary, the probability of attributing a value of consumption that is half ( _Y_<sup>_d_</sup> _/Y_<sup>_∗_</sup> _<_ 0 _._ 5) or more than double ( _Y_<sup>_d_</sup> _/Y_<sup>_∗_</sup> _>_ 2) the real value is 15% and 3.2%, respectively, for a household with median _Y_<sup>_∗_</sup> (the continuous line in Panel A).

The important implications of our approach for empirical work can be understood from Panel B of Figure 7, which shows how consumption from stocks or large acquisitions in the diary week affect the values of _Y_<sup>_d_</sup> _/Y_<sup>_∗_</sup> . Distributions in this panel are centered at one – because of Assumption 1 – and strikingly bimodal, with spikes at zero signaling no acquisition in the week identified in the interview. For this group of items, the probability of attributing a value of consumption that is half ( _Y_<sup>_d_</sup> _/Y_<sup>_∗_</sup> _<_ 0 _._ 5) or more than

> 15Specifically, our findings imply that log _Y ∗_ = log _Y r_ + log _εr_ , where log _εr_ has constant variance and average below one, is a good specification to consider for empirical work.

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distribution arising from this design is: the percent of survey participants filling out a diary being: For example, in a recall survey _p_ ( _y_<sup>_∗_</sup> ) = 0 at all values _y_<sup>_∗_</sup> . We are interested in the effects of the collection mode on functionals of the distribution of usual consumption _FY ∗_ [ _y_<sup>_∗_</sup> ]. Possible choices for these functionals are quantiles, share below the poverty line, and the Gini coefficient. The difference: represents the distance between the true statistic, _ν_ ( _FY ∗_ [ _y_<sup>_∗_</sup> ]), and the same statistic that would be computed under the assignment design described above, _ν_ ( _FY_ [ _y_ ]). In this section, we find the configuration of weights _p_ ( _y_<sup>_∗_</sup> ) that minimizes the distance in (8) at any given choice for the value of _p_ (the algorithm used to obtain the solution is described in the Appendix). In an effort to contain the computational burden, we use _FY d|Y ∗_ [ _y|y_<sup>_∗_</sup> ], _FY r|Y ∗_ [ _y|y_<sup>_∗_</sup> ] and _FY ∗_ [ _y_<sup>_∗_</sup> ] as if they were _known_ quantities and discard the additional variability arising from estimation errors.

Distributions retrieved from a recall survey are closer to the underlying distribution of usual consumption than in a diary survey. This can be seen in Panel A of Figure 9, where the statistic _ν_ represents the Kullback-Leibler distance (KLD) from the consumption distribution and _ν_ ( _FY ∗_ [ _y_<sup>_∗_</sup> ]) = 0. The continuous line here shows the value of (8) at different values of _p_ . The value corresponding to _p_ = 0 is the KLD from a survey with only recall questions. The panel conveys two important messages. First, as more resources are allocated to diaries, the KLD improves until it reaches the value _p_ = 0 _._ 42. In particular, Appendix Figure A.1 shows that weights _p_ ( _y_<sup>_∗_</sup> ) at the value _p_ = 0 _._ 42 are such that households with large values of _Y_<sup>_∗_</sup> should always be assigned to diaries. The second important message is that the KLD is larger at _p_ = 1 than at _p_ = 0. This implies that, in our case study, the data distribution obtained from a fully fledged recall survey has better properties than one obtained from diaries alone.

Distributions obtained from diary data yield worse measurements of poverty and inequality. This can be seen in the remaining panels of Figure 9, where the continuous lines represent alternative definitions for the statistic _ν_ ( _FY_ [ _y_ ]): the Gini coefficient (Panel B), the standard deviation of logged consumption (Panel C), and the share of consumption for households in the bottom decile (Panel D). The horizontal lines in panels represent the value of the true statistic _ν_ ( _FY ∗_ [ _y_<sup>_∗_</sup> ]). Diaries yield more disperse consumption distributions, as shown in Panel B and Panel C. Moreover, the Gini coefficient and the standard deviation computed from a recall survey are closer to their true values than one would obtain by using only diaries. It is also clear that allocating more resources to diaries would improve inequality measurement only until approximately the value _p_ = 0 _._ 10. This suggests that a fully fledged recall survey is almost as close to the optimal design when inequality measurement is of concern. Panel D yields very similar conclusions 26
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for poverty measurement.

We conclude our analysis by showing that a recall survey is the best option even in the worst case scenario when no information on _Y_<sup>_∗_</sup> is available.<sup>17</sup> This can be seen from the dashed lines in Figure 9, which are obtained by setting _p_ ( _y_<sup>_∗_</sup> ) = _p_ in (6) and allocating to diaries a _random_ share _p_ of households independently of their consumption. The lines in the four panels suggest that there is no gain from mixing diary and recall interviews without a selective allocation of households that depends on their consumption.

The take-away message from this section is that we do not find evidence of any loss in accuracy from using recall questions compared to a diary. Moreover, the above definition of optimal design does not take into account costs and budget constraints in the optimization. Our conclusions on the value of recall surveys would be strengthened given the much higher costs of running a diary survey.

# **7 Conclusion**

Using a large-scale randomization in Iraq, we found little empirical support for the idea that diaries yield better quality data for assessing household welfare. Diaries provide a more reliable measurement of usual consumption _averages_ , and we have shown that the cognitive errors arising from the process of recalling consumption lead to overstated actual consumption. However, when inequality and poverty measurements are of interest, the benefits of diaries are far less clear-cut. Diary measurements, despite being correct on average, have large variance. We have demonstrated that this is not the consequence of measurement errors but mostly a reflection of heterogeneous frequency of consumption across households. We have found that recall modules provide a more reliable mode of data collection for inequality and poverty. The use of surveys with both diary and recall interviews can yield improved measurements because these two collection modes work best for eliciting consumption in different segments of the distribution. Nevertheless, our calculations have shown that a fully fledged recall survey can yield inequality measurements that are not too different from those that would be obtained from the optimal mix of diary and recall interviews.

The first implication of our research is that the loss in accuracy from using recall questions is minimal compared to the higher costs of using diaries. This finding provides an empirical justification for considering a transition to recall modules in household surveys in developing countries. However, more research is needed to assess what makes a good recall module, given that the length of the recall list will affect the propensity of respondents to engage in the survey. The answer to this question could be given, for example, by randomizing households to recall modules of different length and using our research design. The decision between using diaries or recall modules is not confined to developing economies. For example, the problems associated with the Consumer Expenditure Surveys in the United States have intensified the call for a redesign of the survey, which is underway following the recommendations of a specially appointed panel (the Gemini project).

Diary surveys should collect information on the purchasing behavior of households to correct for the unwelcome effects of infrequent purchases on inequality measurement – which is the second implication

> 17Allocation based on proxies of _Y ∗_ , such as income or residence in areas with homogeneous compositions of household demographics, falls between this case and the one above.

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of this research. Respondents should be asked to estimate the number of purchases made over a fixed reference period, anchoring questions on consumption rather than on acquisitions (as in the 2016 Barbados Survey of Living Conditions and the 2017/18 Jordan Household Income and Expenditure Survey). If information on purchasing behavior is available, simple models can be used to estimate the effects of using diaries on inequality measurements.

A third implication of our findings is that surveys should be designed to elicit repeated measurements from the same respondents. In our case, for example, a recall module was administered to all households before starting the diary week. The same setting can be found in other important family expenditure surveys, such as in Canada and the United States. The methodology we have presented can be applied to any of these contexts to study how consumption inequality has evolved over time, raising the problem of how our approach can be extended to allow for longitudinal information. We hope to address this and some of the related problems in future research.

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