# F. Model selection using the Lasso

Model selection in small area estimation remains an unsettled issue and has traditionally been treated as both art and science. Practice can vary widely. Tzavidis et al (2018), for example, select only 6 covariates present in Mexican census and survey data, based in part on the Akaike Information Criterion (AIC) from a standard OLS model. Most small area applications using ELL, however, use a considerably larger set of variables. Zhao and Lanjouw (2008), for example, note that many successful applications include less than 20 household level variables. However, they also advocate including cluster-level means, which can boost the number of variables significantly past 20 or 30.

In recent years, the least absolute shrinkage and selection operator (LASSO) has become an increasingly popular tool for selecting prediction models. Small area estimation offers a natural application of the “post-lasso” procedure, in which the variables selected via the lasso are used as predictors into the mixed model used to estimate the parameters of the small area estimation model.<sup>19</sup> The post-lasso procedure offers the benefit of a convenient and data-driven approach to model selection that, in its most popular variant, maximizes out-of-sample predictive accuracy, while roughly equalizing in and out of sample R<sup>2</sup> to prevent overfitting the model to the sample. The LASSO framework formalizes and extends the advice in Zhao and Lanjouw (2008) to use out-of-sample cross-validation to ensure that the model is stable and predicts well out of sample. Several varieties of LASSO have been implemented in popular statistical software. We use the plugin lasso estimator, also known as the rigorous lasso, implemented in Stata version 16 (StataCorp, 2019). This variant of lasso is computationally faster and more parsimonious than others that use cross-validation to minimize mean out-of-sample prediction error, and the resulting models maintain strong predictive power.<sup>20</sup>

# G.  Benchmarking

Benchmarking refers to the procedure of forcing model-dependent estimators to agree with the direct sample estimates for super-areas, defined as the lowest geographic level for which the sample is considered representative. Benchmarking can be desirable to ensure that the population-weighted average of small area poverty estimates, when aggregated to super-areas, match official published statistics derived from the survey. Simulations indicate that in the cross-sectional context, simple “ratio” benchmarking performs equally well as other methods and there is no significant loss of efficiency from benchmarking in general (Pfefferman et al, 2014, Wang et al, 2008). We therefore apply a simple ratio benchmarking procedure by multiplying the estimated headcount rate in each area by a scaling factor unique to each super-area, for both the Fay-Herriot and household level model estimates. The scaling 19 See Hastie et al (2015) for a review of the lasso and related estimators, and Belloni and Chernozhukov (2013) for more details on the post-lasso procedure.

20 We use the variant of the Stata plugin estimator that allows for heteroscedastic errors. The plugin method uses an iterative formula to select the lambda shrinkage parameter in the lasso instead of a grid search. See StataCorp (2019), Belloni and Chernozhukov (2011), and appendix A of Belloni, Chernozhukov, and Hansen (2014) for technical details regarding the implementation of the plugin lasso estimator.

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