Answer: None of the OLS estimators to exist as men's and women's earnings would exhibit perfect multicollinearity

## Explanation Perfect multicollinearity occurs when there is an exact linear relationship between independent variables in a regression model. In this case: 1. The candidate creates two dummy variables: - Men = 1 if male, 0 otherwise - Women = 1 if female, 0 otherwise 2. For every observation in the dataset: - If Men = 1, then Women must = 0 - If Women = 1, then Men must = 0 - If Men = 0, then Women must = 1 (assuming all individuals are either male or female) 3. This creates a perfect linear relationship: **Men + Women = 1** for every observation 4. In matrix algebra terms, this means the design matrix X has linearly dependent columns, making X'X singular (non-invertible). 5. Ordinary Least Squares (OLS) requires the inversion of X'X to estimate coefficients. When perfect multicollinearity exists, this matrix cannot be inverted, making OLS estimators impossible to compute. **Why other options are incorrect:** - **A**: While there may be a statistically significant difference in means, this is not the primary statistical issue with the model specification. - **C**: The coefficients cannot be estimated at all due to multicollinearity, so we cannot compare them. - **D**: Again, coefficients cannot be estimated, and even if they could, this pattern doesn't logically follow from the given information. **Proper approach**: To avoid perfect multicollinearity when using dummy variables for categorical data, one should use k-1 dummy variables for k categories (e.g., include only a "Men" dummy variable, with "Women" as the reference category).

Author: Nikitesh Somanthe