Econometrica最新文献

Two opposed interested parties (IPs) compete to influence citizens with heterogeneous priors which receive news items produced by a variety of sources. The IPs fight to capture the coverage conveyed in these items. We characterize the equilibrium level of capture of item as well as the equilibrium level of information transmission. Capture increases the prevalence of the ex ante most informative messages and can explain the empirical distribution of slant at the news-item level. Opposite capturing efforts do not cancel each other and instead undermine social learning as rational citizens discount informative messages. Citizen skepticism makes efforts to capture the news strategic substitutes. Because of strategic substitution, competition for influence is compatible with horizontal differentiation between successful media. In equilibrium, rational citizens choose to consume messages from aligned sources despite knowledge of the bias in a manner consistent with recent empirical evidence.

We examine the split-sample robust inference (SSRI) methodology introduced by Chernozhukov, Demirer, Duflo, and Fernandez-Val for quantifying uncertainty in heterogeneous treatment effect estimates produced by machine learning (ML) models. Although SSRI properly accounts for the additional variability due to sample splitting, its computational cost becomes prohibitive with complex ML models. We propose an alternative approach based on randomization inference (RI) that preserves the broad applicability of SSRI while eliminating the need for repeated sample splitting. Leveraging cross-fitting and design-based inference, the RI procedure yields valid confidence intervals with substantially reduced computational burden. Simulation studies demonstrate that the RI method preserves the statistical efficiency of SSRI while scaling to much larger applications and more complex settings.

We warmly thank Kosuke Imai, Michael Lingzhi Li, and Stefan Wager for their gracious and insightful comments. We are particularly encouraged that both pieces recognize the importance of the research agenda the lecture laid out, which we see as critical for applied researchers. It is also great to see that both underscore the potential of the basic approach we propose—targeting summary features of the CATE after proxy estimation with sample splitting.

We are also happy that both papers push us (and the reader) to continue thinking about the inference problem associated with sample splitting. We recognize that our current paper is only scratching the surface of this interesting agenda. Our proposal is certainly not the only option, and it is exciting that both papers provide and assess alternatives. Hopefully, this will generate even more work in this area.

One potential concern with our approach is that it is demanding in terms of data, since it relies on repeated splitting of data into two parts: one used for CATE signal extraction and another used for post-processing. To examine potential improvements, Wager's discussion focuses on the special problem of testing the null effect—that is, whether the CATE function is zero. It is a specific setting, as typical machine learning algorithms are in fact able to learn the zero function consistently even in high-dimensional settings.¹ Nonetheless, the problem of testing the null of a zero-CATE remains very important.

Fixing a single split of data into K folds, Wager (2024) investigates relative gains in power generated by the sequential inference approach of Luedtke and Van Der Laan (2016). This approach uses progressively more data to estimate the “signal” and then generates a sequence of statistics to test if the “signal” is zero. The statistics can be aggregated to form a “single-split” p-value using the martingale properties of the construction. Wager shows in Monte Carlo experiments (reproduced below) that this improves power over a method of taking the median p-value over K equal-sized folds (which is not the method we propose, but is a sensible benchmark). It also outperforms the “naïve” approach that relies on cross-fitting à la the debiased machine learning (DML) approach, which is asymptotically valid in this special setting but suffers from size distortions.²

We believe that Wager's proposal is potentially a fruitful complement to what we propose, since we can use the sequential estimation within our “multiple-split” approach. We now show that this combination generates further size and power improvements.

In what follows, we report results from a numerical simulation using the same experiment and implementation details as in Wager (2024).³ As in Wager, we consider the following “single-split” approaches: