Generally speaking, there is no concept of potential outcomes in regression
because regression models estimate conditional expectations of a response
variable. However, under the key assumptions of *conditional exchangeability*
and *consistency*:
$$ E \\Biggl( \\overset \\sim Y_{g,t}\^{ \\left(\\left\\{ x_{g,t,i}\^{(\\ast)} \\right\\}\\right) } \\Big\| \\bigl\\{z_{g,t,i}\\bigr\\} \\Biggr) = E \\Biggl( \\overset \\sim Y_{g,t} \\Big\| \\bigl\\{z_{g,t,i}\\bigr\\}, \\big\\{x_{g,t,i}\^{(\\ast)}\\bigr\\} \\Biggr) $$

**Key assumptions**

- Conditional exchangeability:

  \\( \\overset \\sim Y_{g,t}\^{(\\{ x_{g,t,i}\^{(\\ast)} \\})} \\)
  is independent of the random variables
  \\(\\bigl\\{ X_{g,t,i}\^{(\\ast)} \\bigr\\}\\) for any counterfactual scenario
  \\(\\bigl\\{ x_{g,t,i}\^{(\\ast)} \\bigr\\}\\). So, the set of potential outcomes
  is conditionally independent of the advertiser's historical media execution
  decision.
- Consistency:

  \\( \\overset \\sim Y_{g,t} = \\overset \\sim Y_{g,t}\^{
  (\\{ x_{g,t,i}\^{(\\ast)} \\})
  } \\) when \\(\\bigl\\{ X_{g,t,i}\^{(\\ast)} \\bigr\\} =
  \\bigl\\{ x_{g,t,i}\^{(\\ast)} \\bigr\\}\\). So, the observed KPI is the
  realization of the potential outcome for the counterfactual scenario
  matching the advertiser's historical media execution.

Under these assumptions, you have the previously stated result:
$$ E \\Biggl( \\overset \\sim Y_{g,t}\^{ \\left(\\left\\{ x_{g,t,i}\^{\\ast} \\right\\}\\right) } \\Big\| \\bigl\\{ z_{g,t,i} \\bigr\\} \\Biggr) \\overset{\\text{exchangeability}}{=} E \\Biggl( \\overset \\sim Y_{g,t}\^{ \\left(\\left\\{ x_{g,t,i}\^{\\ast} \\right\\}\\right) } \\Big\| \\bigl\\{ z_{g,t,i} \\bigr\\},\\ \\bigl\\{ x_{g,t,i}\^{(\\ast)} \\bigr\\} \\Biggr) \\overset{\\text{consistency}}{=} E \\Biggl( \\overset \\sim Y_{g,t}\\ \\Big\| \\bigl\\{ z_{g,t,i} \\bigr\\},\\ \\bigl\\{ x_{g,t,i}\^{(\\ast)} \\bigr\\} \\Biggr) $$

The consistency assumption is fairly intuitive, and holds unless the
counterfactual is poorly defined or is not accurately represented in the data.
For more information, see [Hernan MA, Robins JM, (2020) Causal Inference: What
If](https://miguelhernan.org/whatifbook).

The conditional exchangeability assumption is a bit less intuitive. This
assumption holds if all confounding variables are measured and included in the
control array \\(\\{z_{g,t,i}\\}\\). *Confounding variables* are anything that has
a causal effect on both the observed treatment \\(\\{x_{g,t,i}\\}\\) and outcome
\\(\\{\\overset \\sim y_{g,t}\\}\\). A causal effect on treatment can mean an effect
of the advertiser's overall budget level, the allocation across channels, the
allocation across geos, or the allocation across time periods. In practice, it
is difficult to know whether all of the confounding variables are measured
because it is purely an assumption, and there is no statistical test to
determine this from your data. However, it can be helpful to know that the
conditional exchangeability assumption holds if you assume a causal graph that
meets a condition known as the *backdoor criterion* (Pearl, J., 2009). For more
information, see [Causal graph](https://developers.google.com/meridian/docs/causal-inference/causal-graph).