Causal Impact of Optional Device Ownership on Engagement (Observational Design)

Scenario

A new, optional smart speaker is released. Some users purchase and link the speaker to their account; others do not. Users self-select into purchasing, and you cannot run a forced A/B test.

Assume you have: (a) user-level panel data (daily/weekly) on engagement (e.g., streaming hours, sessions), (b) a time-stamped indicator of speaker activation/ownership, (c) rich covariates (prior engagement levels and trends, demographics/geo, devices, marketing exposure), and (d) staggered adoption timing across users.

Task

How would you measure the causal impact of owning the speaker on engagement? Describe your preferred design and why.

In your answer, specify:

1. Identification strategy and main assumptions.
2. How you would construct the control group and handle staggered adoption.
3. Model/estimator choice and diagnostics (e.g., pre-trend checks).
4. Sensitivity analyses and alternatives if assumptions fail.

You may discuss quasi-experimental approaches such as difference-in-differences (event study), propensity-score matching/weighting, instrumental variables, and pre-post checks.