and near-zero coeffi cient on PTA in column (4) as an encouraging indication that the selected provisions completely explain the average PTA effect reported in column (1).

Next, column (5) returns to our original simple model from column (1) but adds a dummy variable for the EU.<sup>24</sup> Our reasons for treating the EU separately from other agreements are three-fold. First, we suspect that not all of the EU’s efforts to promote trade are captured in how their provisions variables are coded in our data. There could also be unobserved effects that are channeled through the EU’s secondary law process, in which the EU’s governing institutions are empowered to pass new regulations and directives on an ongoing basis. Second, our provisions data set does not include agreements that are no longer in effect. For the most part, the agreements that cannot be included are EU pre-accession agreements, which obviously are subsumed by the EU agreement once each new member joins the EU. As discussed in Section 2, we deal with this data issue in practice by dropping all observations associated with obsolete agreements. Nonetheless, this could lead to biased estimates of the EU agreement and the provisions associated with it. Third, the latest EU agreement has in place six of the eight provisions selected in column 2 (all except AD14 and TBT7); thus, we want to make sure we are not simply picking up an “EU effect”in the provisions that are selected.

As the PPML results in column (5) show, the estimated EU effect is large, several times that of non-EU PTAs in fact. However, when we treat the EU as a possible predictor in the lasso, we find that is not selected and consequently the set of provision variables selected is identical to that in column (2), which is our preferred set to work with in the subsequent iceberg lasso and bootstrap lasso analyses.

# 5.2 Cross-Validation Lasso Results

As discussed before, the plug-in approach to choosing penalty parameter tends to choose a relatively small set of regressors and may fail to pick the “correct” regressors. For comparison, we now discuss the choice of regressors when we use the cross-validation approach.<sup>25</sup> Figure 2 shows how the out-of-sample mean square error (MSE) varies with the log of the tuning parameter, which is scaled by<sup>�</sup> ijt<sup>yijtsothattheresultsdonotdepend</sup> on the scale of the data. At the optimal value of the tuning parameter, λ/<sup>�</sup> ijt<sup>yijt=</sup> 0.00025, the cross-validation approach selects 128 provisions to have non-zero effects. Additionally, some of the selected provisions are perfectly collinear with variables that are not selected; if we take this into account, the effective number of provisions selected is 133, which is many more than what we found using the plug-in approach.

For more illustration, Figures 3 and 4 show the corresponding regularization paths for selected provisions.<sup>26</sup> That is, the figures show how the value of the estimated (postlasso) coeffi cient on the selected provisions changes as we vary λ. As expected, fewer provisions are selected as we increase λ and, for values of λ/<sup>�</sup> ijt<sup>yijtaround 0.01, which</sup>

> 24 We use EU as shorthand for the EU and EC agreements.

> 25As explained before and in the Appendix, the cross validation is performed clustering by agreement.

> 26In each panel of the figures, the fourth set of estimates from the right corresponds to the variables selected by the cross-validation method.

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