Multivariate Linear Regression

Multivariate linear regression, as the term is commonly used in finance, is linear regression in which one variable is explained by two or more others at the same time. Statisticians call this multiple linear regression and reserve the word multivariate for models with several dependent variables, but the looser usage is widespread. The model is:

Here $y$ is the variable to be explained, $x_1$ to $x_k$ are the explanatory variables, $\beta_0$ is the intercept and $\varepsilon$ is the error. Each coefficient measures the effect of its variable with the others held fixed. The main use in markets is the factor model: a fund's returns are regressed on the market, size, value and other factors, and each slope is the fund's exposure, or beta, to that factor. Two cautions apply. Explanatory variables that move together make individual coefficients unstable, a problem called multicollinearity. And every added variable raises the coefficient of determination whether or not it belongs, which encourages overfitting.