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.

23