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Multiple Linear Regression

Multiple regression
Y=β0+β1X1+β2X2+⋯+βkXk+εY = \beta_0 + \beta_1X_1 + \beta_2X_2 + \cdots + \beta_kX_k + \varepsilon

Each coefficient is the effect of its own predictor with the others held fixed, which is the entire point of adding them.

Multiple regression is the backbone of factor modelling: regress a return on market, size, value and momentum factors, and the coefficients are the exposures.

Coefficients are partial effects

βj\beta_j is the expected change in YY per unit of XjX_j holding the other predictors fixed. This differs from the slope you would get regressing YY on XjX_j alone, and the difference can be dramatic.

The clean way to see it: βj\beta_j is the effect of the part of XjX_j that is uncorrelated with the other predictors. Regressing a stock on the market alone gives its total market sensitivity; adding a sector factor gives its sensitivity to the market beyond what the sector explains.

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