Derive Coefficient and Covariance in Regression Analysis
Quick Overview
Evaluates statistics fundamentals across equicorrelation constraints, reverse regression slopes, covariance of uniform order statistics, and change of variables. Strong answers use positive semidefinite matrices, simple-regression identities, order-statistic expectations, and Jacobian transformations.
Derive Coefficient and Covariance in Regression Analysis
Company: Citadel
Role: Data Scientist
Category: Statistics & Math
Difficulty: medium
Interview Round: Technical Screen
##### Scenario
Assessing knowledge of correlation structure, regression relationships and covariance calculations.
##### Question
1) For three random variables X, Y, Z with identical pairwise correlations ρ, what is the smallest possible value of ρ?
2) In simple linear regression of Y on X, you know R² and the slope coefficient β(y|x). Derive the slope β(x|y) from regressing X on Y.
3) Let X and Y be i.i.d. Uniform(0, 1). Compute Cov(max(X, Y), min(X, Y)).
4) Given a monotone function Y = g(X), derive the pdf of X from the pdf of Y (inverse-function distribution).
##### Hints
Use positive-definite covariance matrices, β relations with R², Cov identities, and change-of-variables theorem.
Quick Answer: Evaluates statistics fundamentals across equicorrelation constraints, reverse regression slopes, covariance of uniform order statistics, and change of variables. Strong answers use positive semidefinite matrices, simple-regression identities, order-statistic expectations, and Jacobian transformations.
Derive Coefficient and Covariance in Regression Analysis
Citadel
Jul 12, 2025, 6:59 PM
mediumData ScientistTechnical ScreenStatistics & Math
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Derive Coefficient and Covariance in Regression Analysis
This statistics prompt tests correlation constraints, regression slope relationships, covariance of order statistics, and change-of-variables reasoning.
Constraints & Assumptions
Assume finite second moments for correlation and regression questions.
Assume simple linear regression with an intercept where slopes are discussed.
For the uniform order-statistic question, let
X
and
Y
be independent
Uniform(0,1)
.
State any monotonicity and differentiability assumptions for the change-of-variables result.
Clarifying Questions to Ask Guidance
Are
X
,
Y
, and
Z
standardized, or are we discussing correlations only?
Is
R^2
from simple linear regression with one predictor?
Should the final answers be formulas, derivations, or numerical values?
Part 1 - Equicorrelation Constraint
For three random variables X, Y, and Z with identical pairwise correlations rho, what is the smallest possible value of rho?
What This Part Should Cover Guidance
Equicorrelation matrix.
Positive semidefinite constraint.
Minimum
rho = -1/2
.
Part 2 - Reverse Regression Slope
In simple linear regression of Y on X, you know R^2 and the slope coefficient. Derive the slope coefficient from regressing X on Y.
What This Part Should Cover Guidance
Relationship between slopes, correlation, and standard deviations.
Product of the two simple-regression slopes equals
R^2
.
Edge case when the slope and
R^2
are zero.
Part 3 - Covariance of Maximum and Minimum
Let X and Y be i.i.d. Uniform(0,1). Compute the covariance between max(X,Y) and min(X,Y).
What This Part Should Cover Guidance
Expectations of minimum and maximum.
Identity
min(X,Y) * max(X,Y) = XY
.
Final covariance
1/36
.
Part 4 - Change of Variables
Suppose Y = g(X), where g is monotone. What is the density of Y in terms of the density of X?
What This Part Should Cover Guidance
Inverse transformation.
Absolute derivative/Jacobian term.
Correct handling of increasing versus decreasing transformations.
What a Strong Answer Covers Guidance
A strong answer gives clean derivations, states assumptions, and recognizes the matrix, regression, order-statistic, and transformation tools needed for each part.
Follow-up Questions Guidance
How does the minimum equicorrelation generalize to
n
variables?
What if
R^2
is known but the slope sign is not?
How would the covariance change for more than two uniform variables?