You have obtained data on test scores and student-teacher ratios in region A and region B of your state. Region B, on average, has lower student-teacher ratios than region A. You decide to run the following regression
Yi = ?0+ ?1X1i + ?2X2i + ?3X3i+ui
where X1 is the class size in region A, X2 is the difference between the class size between region A and B, and X3 is the class size in region B. Your regression package shows a message indicating that it cannot estimate the above equation. What is the problem here and how can it be fixed? Explain the problem in terms of the rank of the X matrix.
What will be an ideal response?
Answer: There is perfect multicollinearity here, in that X2 = X1-X3, hence the X matrix (and the X'X) matrix does not have full rank (rank = 3 here, not 4). If the X'X is singular, you cannot invert it, since its determinant is zero. Dropping one of the three explanatory variables allows you to estimate the above equation.
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