Required assumptions

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 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.

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.