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1756Backing Out Omitted Covariance From a Slope DropA desk regresses slippage Y on inventory pressure X. Without an urgency control, the OLS slope on X is 0.90. After adding a perfect measure of urgency U, the slope falls to 0.60. Suppose the structural model is Y = beta X + 0.5 U + noise and Var(X)=1. What is Cov(X,U)?统计简单derivation未尝试免费1757Recovering Signal-to-Noise From Two Attenuated SlopesA latent factor X* is measured twice: X1 = X* + e1 and X2 = X* + e2, where e1 and e2 are independent classical errors with the same variance and are independent of X*. Regressing Y on X1 alone gives slope 0.80. Regressing Y on the average (X1+X2)/2 gives slope 1.00. Under the same true structural slope beta, what is Var(X*)/Var(e)?统计简单derivation未尝试免费1758Selection-on-Survivors Bias in Strategy EvaluationA desk only records post-launch performance for strategies that first clear an internal backtest hurdle. Why does regressing realized performance on backtest score inside the launched set generally fail to recover the unconditional relationship?统计中等derivation未尝试面试订阅1759Averaging Two Noisy MeasurementsAgain the structural model is Y=2X+u with E[u\mid X]=0, but now you observe two noisy proxies: W 1=X+\eta 1, \qquad W 2=X+\eta 2, where \eta 1,\eta 2 are independent of each other and of X,u. Suppose Var (X)=4 and each noise term has variance 1. If you regress Y on the average proxy W=(W 1+W 2)/2, what is the probability limit of the slope?统计中等derivation未尝试面试订阅1760Wald Ratio With an Explicit Sign ConventionAn exchange latency shock Z in 0,1 moves the fraction of aggressive orders from 0.30 when Z=0 to 0.18 when Z=1, and average slippage from 4.2 bps when Z=0 to 5.4 bps when Z=1. Using the consistent orientation Delta Y / Delta X = (E[Y|Z=1]-E[Y|Z=0]) / (E[X|Z=1]-E[X|Z=0]), what Wald IV estimate do you get for slippage per one-unit increase in aggressive-order fraction?统计中等derivation未尝试面试订阅1761Direct and Total Effect After a Routing Channel SplitA routing signal X increases child-order fragmentation M by 2 units on average. The outcome obeys Y = 1.3 X + 0.4 M + noise, with X otherwise exogenous. What coefficient on X would you expect in a regression that controls for M, and what total effect of a one-unit increase in X would appear in a regression of Y on X alone?统计简单derivation未尝试免费1762A Tiny First Stage Is a Weak-Instrument WarningTwo candidate rollouts have the same reduced-form impact on PnL: E[Y\mid Z=1]-E[Y\mid Z=0]=0.02. For rollout A, the first stage is 0.20; for rollout B, the first stage is 0.01. Which rollout creates the weaker IV design, and why?统计中等multi part未尝试面试订阅1763Which Proposed Instrument Is More Plausible?You want to estimate the causal effect of quote-update intensity X on realized spread capture Y. Candidate instrument A is a randomized gateway assignment decided by the exchange. Candidate instrument B is same-day order-flow imbalance, which also directly moves spread capture. Which candidate is more plausible as an instrument, and what IV condition fails for the other one?统计中等multi part未尝试面试订阅1764Selection Bias from Looking Only at Filled OrdersA desk studies how aggressiveness X affects trade profitability Y, but Y is observed only for orders that actually fill. Fill probability is higher when latent market demand D is strong, and stronger demand also tends to improve profitability. Why can regressing observed Y on X using only filled orders be biased?统计中等multi part未尝试面试订阅1765Fixed Effects Still Miss a Moving Stress ChannelPanel fixed effects remove each desk's persistent skill, but an omitted intraday stress variable still changes day by day. Suppose higher stress raises both inventory pressure X and slippage Y within the same desk. After adding desk fixed effects, what confounding channel remains, and in what direction does it bias the within-desk slope on X?统计中等essay未尝试面试订阅1766What Remains After Controlling the Hedge ChannelA signal X changes the hedge ratio H immediately, and both X and H affect desk PnL Y. The direct effect of X on Y is +1.2 bps, while the channel through H contributes another +0.8 bps. If H is measured perfectly and you regress Y on X and H, what effect does the coefficient on X identify, and what number should it equal?统计简单derivation未尝试免费1767Weak-IV Risk From the First Stage AloneA proposed instrument shifts treatment by 0.02 with standard error 0.015 in the first stage. Even if the exclusion story sounds plausible, what first-stage F-statistic do you get, and what is the main identification concern?统计中等derivation未尝试面试订阅1768A Lagged Variable Is Not Automatically a Valid InstrumentSomeone proposes using yesterday's order-flow imbalance as an instrument for today's imbalance in a return-impact regression. Why is this not automatically a valid instrument in financial data?统计中等multi part未尝试面试订阅1770Why Selection on Implemented Trades Distorts Treatment EffectsWhy can studying only executed trades bias the estimated effect of an execution rule, even if the rule assignment itself was randomized upstream?统计简单essay未尝试免费