labor markets, for example). They also are, again, broadly in line with the findings obtained with iceberg lasso.

# 5.5 Predicting the effect of trade agreements

Having identified sets of provisions that are more likely to positively affect trade flows, it is natural to think of ways to use this information to evaluate the effects of different PTAs, and even to predict the impact of new ones. In the reminder of this section we discuss ways to perform these prediction exercises and the associated caveats.<sup>29</sup> The simulation results presented in Section 4 suggest that, in small to moderate samples, the most reliable predictions are the ones based on the (post-lasso) PPML estimates of a model whose regressors are the provisions selected by the plug-in lasso. This kind of prediction can easily be obtained using the results in column (3) of Table 5. For example, we have noted that the latest EU agreement includes all the provisions selected by the plug-in lasso, with the exception of AD14 and TBT7. Therefore, the effect of the latest EU agreement is estimated to be 87% (exp (0.118 + 0.184 + 0.123 + 0.113 + 0.089) − 1 = 0.87). This result is comparable to the effect estimated when the EU dummy is included in the model as in column (5) of Table 5, which is 86% (exp (0.618) − 1 = 0.86).<sup>30</sup> In results that are summarized in the third column of Table 9, we repeat this exercise for each of the PTAs in our data.<sup>31</sup> As in Baier, Yotov, and Zylkin (2019), we find a wide variety of effects, ranging from very large impacts in agreements such as the Eurasian Economic Union, which includes all of the selected provisions, to no effect at all in agreements that do not include any, such as ASEAN.<sup>32</sup> In comparison with column 1 of this Table, which describes results for PPML with the full set of provision variables, we see an immediate advantage of using the plug-in method to model PTA heterogeneity: it greatly cuts down on overfitting. The range spanned by the estimates obtained with the full set of provision reaches implausibly large positive and negative values at the extremes, and their standard deviation is thousands of times that of the estimates produced using plug-in lasso. As shown in column 2, overfitting may also be a problem for the predictions generated by cross-validation lasso, which also lead to some implausible estimates. These results resonate with what we found in the simulations reported in Section 4, where both the model with all regressors and the model with regressors selected by cross-validation performed poorly.

We next consider the performance of the two extensions of the plug-in lasso we have proposed, the iceberg lasso and the bootstrap lasso. The iceberg lasso has the advantage

> 29As in Section 4, in this section we compute penalized predictions when using cross-validation, and post-lasso unpenalized predictions for the plug-in, iceberg, and bootstrap lasso. For the bootstrap lasso, the predictions are obtained by averaging the post-lasso predictions in each of the bootstrap samples.

> 30Of course, using the delta method it is possible to obtain confidence intervals for these effects. However, such confidence intervals do not take into account model uncertainty, which is likely to be the main source of uncertainty in this context. We consider this issue below.

> 31 Note that the average estimated effect is 13.8%, which is very close to the estimated PTA effect of 14.0% corresponding to result in column 1 of Table 7.

> 32In contrast to Baier, Yotov, and Zylkin (2019), we are able to identify heterogeneity across different PTAs but not within PTAs.

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