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