Why Your Betas Explode: The Hidden Geometry of Multicollinearity

Unstable regression coefficients, or betas, are often a symptom of multicollinearity, a common issue in real-world datasets like marketing mix models. This problem arises when input features, such as Linear and Digital TV ad spend, are highly correlated, moving together over time. From a geometric perspective, multicollinearity represents a collapse in the space spanned by the feature vectors. This is mathematically reflected when the determinant of the Gram matrix (X'X) approaches zero, making the matrix ill-conditioned and causing the calculated coefficients to be extremely sensitive to small changes in the data.