Questions / Quant research, Regression & linear algebra
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Two columns of a regression design matrix are identical. What happens to the unregularised coefficient identification?
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Answer: B. Their separate coefficients are not uniquely identified.
- If both columns equal x, their fitted contribution is x(β₁ + β₂).
- Adding any amount to β₁ and subtracting it from β₂ leaves predictions and residual loss unchanged. The sum can be identified while the individual values are not.
Interview shortcut. Perfect collinearity: only the sum β₁ + β₂ is identified.
Takeaway. Identify which coefficient combinations affect the fitted values.
Why the other choices fail
- Their separate coefficients become uniquely equal to one another. Equal coefficients may be a solver convention or a regularized solution, but unregularized data fit alone does not require equality.
- They may be identified if the columns are scaled. Scaling does not separate identical columns.
- Their separate coefficients are both forced to zero. Duplicate columns make different coefficient pairs observationally equivalent. They do not require either coefficient to be zero.
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