A multiple linear regression model can be written as
where is the explained variable, are the explanatory variables, is the error, and are unknown coefficients to be estimated. Given observations , we have a system of linear equations that can be expressed in matrix notation.[3]
or
where and are each a vector of dimension , is the design matrix of order , and is a vector of dimension . Under the Gauss–Markov assumptions, the best linear unbiased estimator of is the linear least squares estimator , involving the two moment matrices and defined as
and
where is a square normal matrix of dimension , and is a vector of dimension .