Answer: $\hat{\alpha} = \hat{Y}_i - 2.4476X_{1i} + 2.7312X_{2i}$

This is a tricky question that needs application of the omitted variables formula. Recall that if the regression model is stated as: $$Y_i = \alpha + \beta_1 X_{1i} + \beta_2 X_{2i} + e_i$$ If we omit $X_2$ from the estimated model, then the model is given by: $$Y_i = \alpha + \beta_1 X_{1i} + e_i$$ Now, in large sample sizes, the OLS estimator $\hat{\beta}_1$ converges to: $$\beta_1 + \beta_2 \delta_1$$ Where: $$\delta_1 = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_1)}$$ Maintaining this line of thought, the first regression equation suggests that: $$\beta_1 + \beta_2 \delta_1 = 0.5633$$ And the second equation suggests that: $$\beta_2 + \beta_1 \delta_2 = -0.7633$$ Where $\delta_2 = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_2)}$ Given: - $\text{Cov}(X_1, X_2) = 0.603$ - $\text{Var}(X_1) = 0.874$ - $\text{Var}(X_2) = 0.75$ We can calculate: $$\delta_1 = \frac{0.603}{0.874} = 0.6899$$ $$\delta_2 = \frac{0.603}{0.75} = 0.804$$ Now we have two equations: 1. $\beta_1 + 0.6899\beta_2 = 0.5633$ 2. $0.804\beta_1 + \beta_2 = -0.7633$ Solving these simultaneous equations: From equation 1: $\beta_1 = 0.5633 - 0.6899\beta_2$ Substitute into equation 2: $0.804(0.5633 - 0.6899\beta_2) + \beta_2 = -0.7633$ $0.4529 - 0.5547\beta_2 + \beta_2 = -0.7633$ $0.4529 + 0.4453\beta_2 = -0.7633$ $0.4453\beta_2 = -1.2162$ $\beta_2 = -2.7312$ Then $\beta_1 = 0.5633 - 0.6899(-2.7312) = 0.5633 + 1.884 = 2.4473$ Now for the intercept, we need to consider that the intercept from the first regression (0.5767) is actually $\alpha + \beta_2 \times \text{mean}(X_2)$ when $X_1$ is omitted, but since we don't have means, we can use the fact that the intercept should satisfy: From the first regression: $\hat{Y} = 0.5767 + 0.5633X_1$ From the second regression: $\hat{Y} = 0.6767 - 0.7633X_2$ When we have both variables, the intercept $\hat{\alpha}$ should be such that: $\hat{Y} = \hat{\alpha} + \beta_1 X_1 + \beta_2 X_2$ Given the calculated coefficients $\beta_1 = 2.4473$ and $\beta_2 = -2.7312$, and matching the intercept from either equation, we get the expression in option D: $\hat{\alpha} = \hat{Y}_i - 2.4476X_{1i} + 2.7312X_{2i}$ (note the sign change for $\beta_2$ since it's negative).

Author: Nikitesh Somanthe