Does anyone know exactly what mechanisms are causing/why coefficients for variables change when they are run alone in a univariate regression, vs. when they are run together with other explanatory variables, in a multivariate regression?
The same vector does not necessarily fit an n-dimensional space in the same way as it fits a 2-dimensional space.
One common reason is related to multicollinearity. Correlations between predictors make the parameter estimates unstable. Small changes to the model can result in large changes to the coefficients.
Another is Simpson's Paradox. Different models, for good/appropriate reasons, can produce very different parameter estimates.
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