Question about `ate_inference()`

Hello,

I have a question regarding the standard error computation in ate_inference() when X is not None.

For a linear CATE model, $\tau(x) = \psi(x)'\beta$, the ATE is $\theta = \Psi'\beta$, where $\Psi = E[\psi(X)]$.

The point estimate $\hat{\theta} = \hat{\Psi}'\hat{\beta}$ (where $\hat{\Psi} = \frac{1}{n}\sum_{i=1}^n \psi(X_i)$ and $\hat{\beta}$ are the final model coefficients) aligns with ate_inference().

However, it appears that ate_inference() calculates the standard error as $\sqrt{\hat{\Psi}' \hat{V} \hat{\Psi}}$, treating $\hat{\Psi}$ as a constant and disregarding its sampling variation.

Wouldn't it be preferable to use the Delta Method for standard error calculation in this case?

I'd appreciate your insights on this matter.

Thank you!

Start at the EconML implementation of ate_inference() for the X is not None path and trace how the linear CATE estimate and covariance are used. Compare the current standard-error calculation with the proposed Delta Method, then establish a documented decision and validation for the sampling variation of Ψ.