(PPML) version of the lasso and show how to choose the tuning parameter of this estimator using either cross-validation or the “plug-in”(or “theory-driven”) approach of Belloni, Chernozhukov, Hansen, and Kozbur (2016), which accounts for heteroskedasticity and clustered errors.<sup>1</sup> We apply our PPML-lasso estimators to a comprehensive data set on PTA provisions recently made available by the World Bank (Mattoo, Rocha and Ruta, 2020). Importantly, this database is very detailed, to the point where the number of provision variables we consider is larger than the number of PTAs we observe in our data. In addition, due to template effects and possible synergies between groups of provisions, the 305 provision variables in our data can be highly correlated with one another. We find that the number of provisions selected when using the PPML-lasso with the tuning parameter chosen by cross-validation is too large for the model to have a meaningful interpretation and that, in contrast, the number of provisions identified when using the plug-in penalty is too small to allow us to be confident that it includes the majority of relevant provisions.<sup>2</sup> To address these issues, we introduce two additional methods that seek to identify potentially important variables that may have been missed in an initial lasso step based on the plug-in penalty. One of the methods, that we call “iceberg lasso”, involves regressing each of the provisions selected by the plug-in lasso on all other provisions, with the purpose of identifying relevant variables that were initially missed due to their collinearity with the provisions selected in the initial step. The other method, termed “bootstrap lasso”, augments the set of variables selected by the plug-in lasso with the variables selected when the plug-in lasso is bootstrapped. As we show using simulations, these new methods present a favorable balance between the parsimony of the plug-in lasso and the lenience of cross-validation methods in small-to-moderate data sets where the true causal variables may be highly correlated with an unknown number of other variables.

To provide some headline results, the PPML-lasso based on cross-validation selects 133 provisions as being relevant, whereas using the plug-in penalty we find that only 8 provisions are associated with enhancing the trade-increasing effect of trade agreements. In turn, the iceberg lasso procedure identifies a set of 42 provisions and, depending on the cutoff used, the bootstrap lasso identifies between 30 and 74 provisions that may be impacting trade. Therefore, our iceberg lasso and bootstrap lasso methods select sets of provisions that are small enough to be interpretable and large enough to give us some confidence that they include the more relevant provisions, something that is confirmed by the simulation evidence we provide. Reassuringly, both the iceberg lasso and bootstrap lasso select similar sets of provisions, mainly related to technical barriers to trade, antidumping, trade facilitation, subsidies, and competition policy. Having identified the set of provisions that are more likely to have an impact on trade, we also discuss how our

> 1An R package (penppml) implementing penalized PPML regressions with high-dimensional fixed effects is available from CRAN and can be installed with install.packages("penppml"). For more details see https://github.com/tomzylkin/penppml.

> 2Our simulation results in Section 4 suggest that the lasso with a penalty parameter chosen by the plug-in method often fails to select the relevant regressors. A similar result, in a different context, is reported by Wüthrich and Zhu (2021).

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