_Note:_ Squares represent the 5 km by 5 km grids for which the hazard measures are available; the (starred) centroid of the square represents the household or cluster location; shaded squares represent the grids used to spatially aggregate the hazard measure for a 10 km radius (darker shade) and a 20 km radius (lighter shade) around the household’s location.

The results presented in this paper are based on matching a household’s geographic location with a spatially aggregated hazard measure within a 20 km radius. In the absence of any GPS data at the household or cluster level, the hazard measures are aggregated to the geographic or closest administrative level that is coded in the household survey. We chose the level of spatial aggregation so as to provide a roughly consistent area of spatial aggregation for surveys where matching was done using GPS data and administrative data.

Initial analysis suggested that this level of spatial aggregation was appropriate. We compared the sensitivity of the relationship between welfare and hazard measures aggregated using different radii in the case of Malawi, a country that has GPS coordinates for surveyed households and multiple years of data. The statistical significance of results increased for radii of up to 20 km and then started to decrease again with larger radii, suggesting that a radius of 20 km is able to reflect the local shock to a household’s crop yields and is also large enough to capture some localized impacts on prices and wages in the local market area. However, as shown in section 4.2, for the full data set a larger geographic area it is not clear that 20 km works better in every setting, and there is very little difference between the results across aggregating radii.

# 3.5. Using standardized anomalies for analysis

The hazard measures generated are in levels as well as standardized anomalies (z-scores).<sup>13</sup> The latter assume that differences in the level of the hazard from the historic average (i.e., level changes) matter not in an absolute sense but in proportion to an area’s usual variation. Welfare measures are only partly affected by the relation between hazards and crop yields and are the outcome of various ex ante and ex post risk-coping strategies developed by households over the years for the purpose of reducing income fluctuations and/or protecting their level of welfare (e.g., consumption). Hence the extent to which the experienced hazard deviates from what households expect (i.e., the historic mean value of the hazard) appears to be a better specification than the actual level of the hazard. Thus it is anticipated that the specification of shocks as standardized anomalies (in standard deviation units or z- scores) is likely to yield more sensible welfare impact estimates than those obtained using the level of the hazard (i.e., rainfall in millimeters). In the case of droughts, a positive anomaly means the hazard experienced was less severe than the historical mean, and a negative anomaly means it was more severe.

When analysis of the specification of the hazard level as a level includes enough years of welfare data, controlling for fixed effects at some spatial level (e.g., district level) transforms the hazard and all other control variables as deviations from the mean (Dell, Jones, and Olken 2014). In cases such as ours, where the sample of welfare data contains a limited number of observations over time, this is not the case, and it is preferable to use a full time series of hazard data prior to regression analysis to generate a standardized anomaly.<sup>14</sup> 13 Some hazard measures such as the WRSI cannot be expressed as a standardized anomaly because by construction they are top coded (for example, the top value of WRSI is 100 percent). However, WRSI can be expressed as an anomaly (i.e., the ratio of current value with the mean or median).

14 Consider, for example, a cross-sectional household data set with three survey rounds and an identifier for the same districts (or clusters, though that is less likely) in each round. Using fixed effects for districts (clusters) transforms all the variables as deviations from the sample district (cluster) mean over all the household observations across the years in the same district (cluster) within the sample. With three survey rounds over the 20-year period analyzed, the district-specific sample mean of any given hazard measured is likely to differ significantly from the district mean estimated over a 20-year period. As a consequence, estimates of the welfare impacts of shocks are likely to differ significantly depending on whether the analysis uses the specification of the 15