Financial Risk Manager Part 1
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A candidate wishes to regress the earnings of individuals. He uses a dummy variable "Men" which takes on the value "1" for men and is "0" otherwise and a second dummy variable "Women" which similarly takes on the value "1" for women and is "0" otherwise. If it's known that men typically earn more than women, we would expect:
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NNikitesh
A
A statistically significant difference in means
B
None of the OLS estimators to exist as men's and women's earnings would exhibit perfect multicollinearity
C
The slope coefficients of men to be equal to that of women
D
The coefficient of men to be positive and that of women to be negative
Explanation:
Explanation
Perfect multicollinearity occurs when there is an exact linear relationship between independent variables in a regression model. In this case:
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The candidate creates two dummy variables:
- Men = 1 if male, 0 otherwise
- Women = 1 if female, 0 otherwise
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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)
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This creates a perfect linear relationship: Men + Women = 1 for every observation
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In matrix algebra terms, this means the design matrix X has linearly dependent columns, making X'X singular (non-invertible).
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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).
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