structural:summarization, structural:scholar_edition
Formulation

Given a data set { y i, x i 1, …, x i p } i = 1 n {\displaystyle \{y_{i},\,x_{i1},\ldots,x_{ip}\}_{i=1}^{n}} of n statistical units, a linear regression model assumes that the relationship between the dependent variable y and the vector of regressors x is linear. This relationship is modeled through a disturbance term or error variable ε —an unobserved random variable that adds "noise" to the linear relationship between the dependent variable and regressors. Thus the model takes the form y i = β 0 + β 1 x i 1 + ⋯ + β p x i p + ε i = x i T β + ε i, i = 1, …, n, {\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i1}+\cdots +\beta _{p}x_{ip}+\varepsilon _{i}=\mathbf {x} _{i}^{\mathsf {T}}{\boldsymbol {\beta }}+\varepsilon _{i},\qquad i=1,\ldots,n,} where denotes the transpose, so that x β is the inner product between vectors x and β.