Reduce residual confounding in time series

Hi @kbattocchi ,

This paper presents a method to partially correct for residual confounding in time series, including in the model a negative control exposure (the value of the exposure the day after the outcome event (Xt+1). This is the DAG used in the paper:

The authors say "Our main result has been that the effect estimator from the “extended” model (with the negative control exposure) tends to be less biased than that from the “final” model (without the negative control). That is, adding a negative control exposure to the model is expected to reduce residual confounding, given our assumptions".
My question is: How should I enter the negative control exposure Xt+1, for estimating the effect in econml? i.e.:

Y=Yt, T=Xt, X="other covariate"
and
W=[Ct, Xt+1]??? (Xt+1 is not a confounder)

Or is there another correct form to introduce the negative control exposure Xt+1?

I appreciate your cooperation.

No file, test, or entry point is identified. Start by reviewing the linked paper alongside EconML's estimator documentation and determine whether negative-control exposures are supported; done would be a documented, agreed-upon integration path or a separately scoped implementation issue.