Boosted Regression Forests, and Cubist regression. Unlike Krennmair et al., the machine learning models do not include a conditional random effect. Despite that, the results indicate that all three machine learning methods generate substantially more accurate estimates than the linear EBP model, particularly out of sample. Of the machine learning methods, boosted regression forests and extreme gradient boosting perform equally well except in Sri Lanka, where extreme gradient boosting produces slightly more accurate estimates. The random effect block residual bootstrap approach proposed by Krennmair et al (2022) also works well when applied to gradient boosting, providing coverage rates ranging from 94 to 97 percent.

Finally, Lee and Braithewaite (2022) propose an innovative iterative process that combines tree-based machine learning and deep learning to generate wealth estimates. This paper first trains a model using extreme gradient boosting to predict the probability of being in different wealth classes using interpretable geospatial features, and then uses the predicted values from this procedure to train a convolutional neural network on satellite imagery. The predicted probabilities of being in different wealth classes from the convolutional neural network are subsequently fed back into the boosting model. This process is repeated iteratively until there is convergence. This improves upon the methodology in Jean et al. (2016) by avoiding the use of night-time lights to train models. The authors report that the average out-of-sample R<sup>2</sup> from withheld countries is 0.90.

In general, several studies suggest that small area estimates generated by combining survey and geospatial data are more accurate than those based solely on survey data, sometimes by significant margins. This is notable because small area estimates based on geospatial data are subject to model bias; for example, a model that uses night-time lights as a predictor may underestimate poverty in a poor area that happens to contain a highway, if the high level of night-time lights associated with highways makes the area look less poor from the sky than it actually is. However, at least when predicting poverty rates at higher levels such as subdistricts, the evidence so far indicates that modelbased estimates based on geospatial indicators are more accurate than direct survey estimates. This implies that the benefits of reducing sampling error by supplementing survey data with model-based predictions derived from geospatial indicators outweighs the introduction of model bias.

c. Geospatial data are a second-best option when recent census data are unavailable Although geospatial data are strongly correlated with welfare across space, recent census data remain the gold standard for auxiliary data for small area estimation. Unfortunately, in many cases census data are old or unavailable, which creates two problems. First, because survey and census data cannot typically be linked at the household level to preserve confidentiality, it is standard when estimating a unit-level model to assume that the predictors follow the same distribution in the survey and the census data. This assumption becomes less tenable as the temporal gap between the census and survey increases. If the census and survey distributions are sufficiently different, estimating a linked model with primary-sampling-unit-level (PSU) aggregates from the census is preferable to assuming a common distribution (Lange, Pape, and Putz, 2018). A second and more important problem is that old census data, even if they are linked directly to the survey data, cannot reflect any changes that affect wealth or poverty that occurred since the census. Current geospatial data may be more likely to reflect current conditions than old census data, especially when including geospatial indicators such as precipitation and vegetation that better predict short-run shocks.

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