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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