Causal ML - Chapter 4

Chapter 4: Statistical Inference on Predictive and Causal Effects in High-Dimensional Linear Regression Models

PDF link to Chapter 4

Core concept: builds on Chapter 3 to create statistically robust confidence intervals for tools including lasso. In authors' viewpoint, this moves from predictive inference (i.e., what happens to the outcome) to statistical inference (i.e., is there a robust understanding of how the outcome depends on changes to input predictor).

Notes

• Chapters 1 and 2 covered estimation of a predictive effect + confidence interval for regression; Chapters 3 and 4 extend the same respective concepts to high-dimensional models.
• This chapter starts with double lasso, which is not the same as double/debiased ML. Double lasso is fairly simple:
  1. Run two lasso regressions: \(Y\) on \(W\) and \(D\) on \(W\) and get residuals \(\tilde{Y}\) and \(\tilde{D}\).
  2. Run least-squares regression of \(\tilde{Y}\) on \(\tilde{D}\) to get the estimator \(\hat{\alpha}\). \(\hat{\alpha}\) is the partialled-out effect of D on Y.
• There is a theoretical approach to calculating the standard error for \(\hat{\alpha}\), which is mildly complex but depends only on estimates of the matrices for the target regressor, the noise, and the number of samples.
• Again, authors argue that cross-validation using standard approaches is not suitable for a small sample due to overfitting. I appreciate the caution and the difference is clear in terms of the results in the notebook, but the examples do not show why the cross-validated result is wrong (other than an appeal to a priori values of the estimator). Relying on the outcome of the black-box package (hdmpy/HDM in R) doesn't really teach me anything or help me evaluate when to apply cross-validation vs theoretical penalties. I don't like this section as a technique; it has a valuable larger point, but the provided approach is insufficient.

Miscellaneous learnings from going through this chapter

• Installing the required R packages for the notebook on country-level growth convergence took 5 minutes on Colab, as opposed to 23 seconds for Python. Not impressed with speed of R here.
• Datasets in R are apparently just available without explicit calling? Makes me uncomfortable that importing a library appears that it could just populate an arbitrarily large namespace without knowing where/what.
• R notebook also defaults to a ludicrous number of significant figures in standard data presentation.
• R can choose to explicitly drop columns from selection if given a negative ordinal number, which is useful.
• R distinguishes between in, which is used only in context of for loops (eg, for (x in seq(10))), and the infix operator %in%, which checks if the input is contained in the output. In Python, these two have the same syntax and are distinguished through use.

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