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1746Slope From Predictor Variance and Fitted VarianceIn a simple OLS regression with intercept, x has sample variance 9 and the fitted values have sample variance 4. If the slope is positive, what is the slope coefficient?统计简单derivation未尝试免费1747Why Duplicate Regressors Destabilize Coefficients but Not the SpanWhy does adding a regressor that is almost a copy of an existing regressor make the coefficient vector unstable even though the fitted values may barely change?统计中等essay未尝试面试订阅1748Fitted Variance From R-Squared and Response VarianceA regression with intercept has response variance 9 and R 2 = 4/9. What is the variance of the fitted values?统计简单derivation未尝试免费1749Intercept From Center of the CloudIn a simple OLS regression with intercept, x bar = 3, y bar = 5, and the slope estimate is -0.4. What is the intercept?统计中等derivation未尝试面试订阅1750Why High Leverage Is Not the Same Thing as a Big ResidualWhy can a data point have very high leverage without having a large residual, and why does that still make it influential?统计简单essay未尝试免费1751Slope and Intercept From Two Fitted BenchmarksA desk summarizes its fitted line by saying y hat = 2.4 when x = 1 and y hat = 0.9 when x = 6. What are the slope and intercept of the line?统计简单derivation未尝试免费1752Why OLS Hedging Is a Projection ProblemWhy is the OLS hedge ratio often described as projecting a cash-book return stream onto the span of hedge instruments?统计中等essay未尝试面试订阅1753Average Leverage and Average Residual-Maker DiagonalA regression uses n = 25 observations and three estimated parameters including the intercept. What are (i) the average leverage and (ii) the average diagonal entry of the residual-maker matrix I - H?统计中等derivation未尝试面试订阅1754Why Multicollinearity Can Leave Training Fit Looking FineWhy can a regression with severe multicollinearity still show a strong in-sample fit and a high R 2?统计简单essay未尝试免费1755Why the Prediction Problem Can Be Easier Than the Coefficient ProblemWhy can two very different coefficient vectors produce nearly the same predictions on the observed design points?统计简单essay未尝试免费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