Answer: -0.18

## Explanation This question deals with the omitted variable bias in regression analysis. When a relevant variable (X₂) is omitted from a regression model that should include it, the estimated coefficient for the included variable (X₁) becomes biased. ### Step 1: Understanding the formula Given large sample sizes, the OLS estimator $\hat{\beta}_1$ in the reduced model converges to: $$ \beta_1 + \beta_2 \delta $$ Where: $$ \delta = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_1)} $$ ### Step 2: Calculating δ We are given: - $\rho_{X_1 X_2} = 0.7$ (correlation between X₁ and X₂) - $\sigma^2_{X_1} = 25$ (variance of X₁) - $\sigma^2_{X_2} = 36$ (variance of X₂) First, calculate standard deviations: $$ \sigma_{X_1} = \sqrt{25} = 5 $$ $$ \sigma_{X_2} = \sqrt{36} = 6 $$ Using the correlation formula: $$ \rho_{X_1 X_2} = \frac{\text{Cov}(X_1, X_2)}{\sigma_{X_1} \sigma_{X_2}} $$ $$ 0.7 = \frac{\text{Cov}(X_1, X_2)}{5 \times 6} $$ $$ 0.7 = \frac{\text{Cov}(X_1, X_2)}{30} $$ $$ \text{Cov}(X_1, X_2) = 0.7 \times 30 = 21 $$ Now calculate δ: $$ \delta = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_1)} = \frac{21}{25} = 0.84 $$ ### Step 3: Understanding the problem context From the original text, we need to find $\hat{\beta}_1$ in the reduced model. The question implies that in the true model, we have: $$ Y_i = \alpha + \beta_1 X_{1i} + \beta_2 X_{2i} + \epsilon_i $$ But we're estimating: $$ Y_i = \alpha + \hat{\beta}_1 X_{1i} + e_i $$ From the omitted variable bias formula: $$ \hat{\beta}_1 \rightarrow \beta_1 + \beta_2 \delta $$ Looking at the answer choices (-0.45, 0.67, -0.18, 0.23), and given δ = 0.84, we need to find which combination of β₁ and β₂ gives one of these values. ### Step 4: Determining the correct answer From the answer choices and the provided correct answer (C: -0.18), we can deduce that: $$ \beta_1 + \beta_2 \times 0.84 = -0.18 $$ Without additional information about β₁ and β₂, we must rely on the given correct answer. The calculation shows that δ = 0.84, and the correct answer is -0.18, which matches option C. ### Key Insight This question tests understanding of: 1. Omitted variable bias formula 2. Relationship between correlation and covariance 3. How omitted variables affect coefficient estimates 4. The bias introduced when relevant variables are excluded from a regression model

Author: Nikitesh Somanthe