Explanation

When an important variable is omitted from a regression model, the assumption that $E(\epsilon_i | X_i) = 0$ is violated, which means:

• The OLS estimator becomes biased (Option A is correct)
• This occurs because the omitted variable might be correlated with the included variables
• The error term becomes correlated with the included variables
• This violates the key assumption needed for the unbiasedness of OLS estimators

Why other options are incorrect:

• Option B: The product of residuals and independent variables is still zero by construction in OLS
• Option C: The sum of residuals is still zero by construction in OLS
• Option D: The coefficient of determination (R²) is not necessarily zero - it could still be positive if other variables explain some variation

The violation of $E(\epsilon_i | X_i) = 0$ specifically leads to bias in the OLS estimator, making it inconsistent and unreliable for parameter estimation.