Goal

You need to separate market-driven fluctuations from a product-caused decline in executed_trades per active user around a known release on 2025-07-10.

Setup and Assumptions

• Metric: executed_trades per active user (daily).
• Pre window: 2025-06-01–2025-07-09.
• Post window: 2025-07-10–2025-08-21.
• You may have staggered exposure (e.g., client versions, geographies) and/or subsets of assets affected.
• Market factors (e.g., VIX, SPX returns) can strongly influence trading; control for them.

Tasks

1. Design a difference-in-differences (DiD) around the 2025-07-10 release. Define treatment and control groups (e.g., cohorts exposed vs not yet exposed; assets affected vs unaffected; geographies rolling out later). Specify the model equation, fixed effects, and clustered standard errors. State identifying assumptions (parallel trends, no spillovers/no anticipation), and how you will test them. If pre-trends fail, describe a remedy (e.g., synthetic control, matching + staggered DiD, or event study with leads/lags) and why it would be valid.
2. Detect structural breaks and quantify effect size with at least two approaches: one of CUSUM or Bai–Perron multiple change points, and Bayesian Structural Time Series (BSTS). Explain how you will reconcile effect sizes and uncertainty when they disagree.
3. Compute the minimum detectable effect for a 10% decrease in executed_trades per active user with daily aggregation, α = 0.05, power = 0.80, mean active users/day = 200,000, baseline mean = 1.0 trades, SD = 1.5 trades. Use the pooled-variance two-sample t-test formula and state the resulting sample size or window length needed.
4. Propose robustness checks: placebo dates, symbol-level randomization inference, wild bootstrap standard errors, and sensitivity of results to volatility controls (e.g., VIX, SPX return) and holiday dummies. Define pass/fail criteria that would change your decision to ship a fix.