Derive correlation bounds and omitted-variable bias

## 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**: \[ \mathrm{Corr}(X,Y)=\mathrm{Corr}(Y,Z)=\mathrm{Corr}(X,Z)=p. \] 1. What values of \(p\) are feasible? 2. Give a method to **construct** \((X,Y,Z)\) that achieves any feasible \(p\). 3. 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 \(X_2\). 1. What is the impact on the estimated coefficient \(\tilde\beta_1\)? 2. Prove the result using matrix notation (OLS).