Derive correlation bounds and omitted-variable bias

Q: Derive correlation bounds and omitted-variable bias

This question evaluates understanding of multivariate correlation structure and linear regression properties, focusing on feasible ranges and constructions for equal pairwise correlations and the derivation of omitted-variable bias in ordinary least squares.

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Core Statistics Prompt

Answer the following related statistics questions.

Part A — Pairwise correlation constraints

Let $X, Y, Z$ be random variables with unit variance and equal pairwise correlation:

$\mathrm{Corr}(X,Y)=\mathrm{Corr}(Y,Z)=\mathrm{Corr}(X,Z)=p.$

What values of $p$ are feasible?
Give 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 $p$ ? How would you construct them?

Part B — Omitted variable bias

Consider the true linear regression model:

$\mathbf{y}=X_1\beta_1 + X_2\beta_2 + \varepsilon,$

but you mistakenly fit the reduced model $\mathbf{y}=X_1\tilde\beta_1+\text{error}$ , omitting $X_2$ .

What is the impact on the estimated coefficient $\tilde\beta_1$ ?
Prove the result using matrix notation (OLS).

Derive correlation bounds and omitted-variable bias

Overview

Core Statistics Prompt

Part A — Pairwise correlation constraints

Part B — Omitted variable bias

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