Incorrect Code in Chapter 20 (and theoretical nitpicking)

[aliquod]

First of all, thank you for making this very accessible book!

In the section about continuous treatment in chapter 20, you defined

$$Y^*_i := (Y_i- \bar{Y})\dfrac{(T_i - M(T_i))}{(T_i - M(T_i))^2}$$

to be the pseudo-outcome¹ and then you threw away the denominator since you are interested in comparing treatment effects, not their absolute values. But doing so does not preserve order². Instead why don't we just simplify it to be

$$Y^*_i = \dfrac{Y_i- \bar{Y}}{T_i - M(T_i)}?$$

Now onto the actual issue: the code block that came after

$$Y^*_i = (Y_i- \bar{Y})(T_i - M(T_i))$$

is

y_star_cont = (train["price"] - train["price"].mean()
               *train["sales"] - train["sales"].mean())

but this is missing some parentheses, so it actually computes

$$Y^*_i \overset{???}{=} Y_i- (\bar{Y} \times T_i) - M(T_i).$$

Footnotes

1. The denominator I assume is an estimate of the conditional variance Var(T|X), but for most regression methods this residual is an underestimate. ↩

2. In the end we will average those values up to estimate the CATE. But unlike the randomized treatment case where every term is scaled by σ² and can be un-scaled without changing order, here each term has a different factor. ↩

[diepala]

Adding to this also. In the proof, we see that subtracting $\bar{Y}$ from $Y^*_i$ is not really necessary, but I think doing this reduces variance. I believe variance could be further reduced by substituting $\bar{Y}$ with a model for estimating $Y$ as a function of the confounders, $f(x) \approx \mathcal{E}[Y | X = x]$, then

$Y^*_i := (Y_i - f(x))\frac{T_i - M(T_i)}{(T_i - M(T_i))^2}$

The proof that this estimates the pseudo CATE would still be correct, but variance would be lower due to the (generally) smaller first term.