Instrumental Variables from an Encouragement Design

You want the effect of enabling push notifications on 30-day sessions, but users choose whether to enable them. You randomise an in-app prompt: half of new users see it, half do not. Results:

• Prompt shown: 45% enable notifications; mean 30-day sessions 12.4
• Prompt not shown: 15% enable notifications; mean 30-day sessions 11.5

1. Compute the intention-to-treat effect and the IV (Wald) estimate of the effect of enabling notifications.
2. State the three IV assumptions in this context and give one way the exclusion restriction could fail here.
3. The estimated effect is much larger than the naive adopters-vs-non-adopters gap of +0.5 sessions. Explain why the two numbers describe different populations, and what "LATE" means here.