Derive correlation bounds and omitted-variable bias
Core Statistics Prompt
This is the core statistics round of a multi-stage Data Scientist interview, consisting of two independent questions.
Constraints & Assumptions
- All random variables are real-valued with finite second moments.
- In Part A, each variable has unit variance, so the covariance matrix and the correlation matrix coincide.
- In Part B, the "true model" assumes data are generated by a full linear model with classical OLS error assumptions (E[ε∣X1,X2] = 0), unless otherwise argued.
- Standard linear algebra and probability tools (eigen-decomposition, Cholesky, OLS normal equations) are permitted; simulation is not required.
Part A — Equal Pairwise Correlation
Let X, Y, Z be random variables with unit variance and equal pairwise correlation: Corr(X,Y) = Corr(Y,Z) = Corr(X,Z) = p.
- What values of p are feasible?
- Provide a method to construct (X,Y,Z) that achieves any feasible p.
- Generalize: for n variables with the same pairwise correlation p, what is the feasible range of p, and how would you construct them?
Part B — Omitted Variable Bias
Consider the true linear regression model y = X1β1 + X2β2 + ε, but you mistakenly fit the reduced model y = X1β̃1 + error, omitting X2.
- What is the impact on the estimated coefficient β̃1?
- Prove the result using matrix notation (OLS).
What a Strong Answer Covers
- Reduces "what values are possible / what is the effect" questions to precise linear algebra conditions (PSD-ness, normal equations) rather than informal reasoning.
- States assumptions explicitly (unit variance/PSD validity; E[ε∣X] = 0, full column rank) and flags where they matter.
- Distinguishes finite-sample (e.g., X1ᵀX2 = 0) from population (true correlation) statements where relevant.
Similar LeetCode problem: Correlation Coefficient
AI interview prep
Core dilemma
Balancing theoretical statistical constraints (PSD of correlation matrix, OLS properties) with practical construction and interpretation of results, especially when dealing with multiple correlated variables or model misspecification.
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Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Derive correlation bounds and omitted-variable bias
Core Statistics Prompt
This is the core statistics round of a multi-stage Data Scientist interview, consisting of two independent questions.
Constraints & Assumptions
- All random variables are real-valued with finite second moments.
- In Part A, each variable has unit variance, so the covariance matrix and the correlation matrix coincide.
- In Part B, the "true model" assumes data are generated by a full linear model with classical OLS error assumptions (E[ε∣X1,X2] = 0), unless otherwise argued.
- Standard linear algebra and probability tools (eigen-decomposition, Cholesky, OLS normal equations) are permitted; simulation is not required.
Part A — Equal Pairwise Correlation
Let X, Y, Z be random variables with unit variance and equal pairwise correlation: Corr(X,Y) = Corr(Y,Z) = Corr(X,Z) = p.
- What values of p are feasible?
- Provide a method to construct (X,Y,Z) that achieves any feasible p.
- Generalize: for n variables with the same pairwise correlation p, what is the feasible range of p, and how would you construct them?
Part B — Omitted Variable Bias
Consider the true linear regression model y = X1β1 + X2β2 + ε, but you mistakenly fit the reduced model y = X1β̃1 + error, omitting X2.
- What is the impact on the estimated coefficient β̃1?
- Prove the result using matrix notation (OLS).
What a Strong Answer Covers
- Reduces "what values are possible / what is the effect" questions to precise linear algebra conditions (PSD-ness, normal equations) rather than informal reasoning.
- States assumptions explicitly (unit variance/PSD validity; E[ε∣X] = 0, full column rank) and flags where they matter.
- Distinguishes finite-sample (e.g., X1ᵀX2 = 0) from population (true correlation) statements where relevant.
Similar LeetCode problem: Correlation Coefficient