Selection Bias With Opt-in Discount Banner

Context

You observe that users can opt in to see a discount banner (treatment = 1 if they opted in and saw the banner). In observational data, treated users have higher conversion than untreated users.

Tasks

(a) Define selection bias and, for this opt-in scenario, list at least three plausible selection mechanisms. For each mechanism, state the expected direction of bias it induces on the naive difference in conversion between treated and untreated users.

(b) Propose two study designs or estimators that can yield unbiased (or less biased) causal estimates in this setting (for example, randomized encouragement with an IV, regression discontinuity with thresholded eligibility, front-door adjustment, IV with assignment instrument). For each, clearly state:

• The setup/design.
• The identifying assumptions.
• How you would diagnose or test those assumptions in practice.

(c) Provide a small numerical example where the naive treated-minus-untreated difference is biased, but a corrected method recovers an unbiased or less biased estimate.