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Media mix modelling is concerned with estimating causal impact of marketing investments , a goal which have several challenges. In general, multiple regression models are deployed mapping up total sales(dependent variable) with marketing budgets(independent variables), control variables and a baseline/intercept to be able to measure incremental sales due to marketing.

While reading the paper Challenges and opportunities in media mix modelling

I came across the following issue of selection bias(page 8): enter image description here

I dont understand how this way of self-selection bias can be an issue. We are essentially interested in the causal effect of paid search ads on total sales. Total sales is being driven by an underlying demand that affect both our baseline sales(our intercept) as well as the effectiveness of paid search advertising. I assume that the bias is positive(meaning that the true effect is lower than what our OLS-estimates retrieve) since this is standard in other posts on this issue.

Question how can this actually becomes an issue when estimating the causal effect of paid search on sales; can someone elaborate on why my OLS-estimates would be biased given this type of selection bias as depicted in the report.?

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Let $y_a$ be the sales to "intersted users" that have been exposed to paid search ads, let $y_o$ be all other sales, so total sales is $y = y_a + y_o$.

If you regress total sales on a collection of regressors $X$ that includes intercept, controls and the spending on paid search ads, your OLS estimates will be

$$\hat \beta = (X'X)^{-1}X'y = (X'X)^{-1}X'y_a + (X'X)^{-1}X'y_o.$$

Since spending on targetd search ads relates only to $y_a$ you would want your estimator to be just $$\tilde \beta = (X'X)^{-1}X'y_a.$$

But we do not expect to have $X'y_o = 0$ (this is a vector), even if paid search ads (a column in the regressor matrix $X$) are orthogonal to "sales to other users" $y_o$.

Hence the bias.

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  • $\begingroup$ i just need clarification on a few points and i'll accept and upvote. "we do not expect to have $X'y_o = 0$ since we assume that the underlying demand affects both paid search ads aswell as $y_o$ ? thus demand spikes leads both our search ad spend to increase aswell as the $y_o$ creating a spuriious correlation making it impossible to separate the true effect of paid search ads spending? even if we intervene and randomize it we will still have other variables that correlate with $y_o$ hence "even if paid search ads are orthogonal to "sales to other users" $y_0$"? $\endgroup$
    – kurt eriksson
    Commented Sep 3, 2023 at 6:57
  • $\begingroup$ doesnt this essentially mean that if we have some sort of unobserved heterogeneity and 2 variables that solely intervene with different groups in this unobserved heterogeneity there will always be biasness? $\endgroup$
    – kurt eriksson
    Commented Sep 3, 2023 at 7:00
  • $\begingroup$ @kurteriksson The $X$ matrix contains regressors that are correlated with $y_o$, control variables and other marketing spending that reasonably are expected to affect $y_o$ too. Unobserved heterogeneity usually creates biases, indeed. $\endgroup$
    – Alecos Papadopoulos
    Commented Sep 3, 2023 at 9:30

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