Editing Regression analysis (section)

===General linear model===
{{Hatnote|For a derivation, see [[linear least squares]]}}
{{Hatnote|For a numerical example, see [[linear regression]]}}
In the more general multiple regression model, there are <math>p</math> independent variables:

: <math> y_i = \beta_1 x_{i1} + \beta_2 x_{i2} + \cdots + \beta_p x_{ip} + \varepsilon_i, \, </math>

where <math>x_{ij}</math> is the <math>i</math>-th observation on the <math>j</math>-th independent variable.
If the first independent variable takes the value 1 for all <math>i</math>, <math>x_{i1} = 1</math>, then <math>\beta_1</math> is called the [[regression intercept]].

The least squares parameter estimates are obtained from <math>p</math> normal equations. The residual can be written as

:<math>\varepsilon_i=y_i -  \hat\beta_1 x_{i1} - \cdots - \hat\beta_p x_{ip}.</math>

The '''normal equations''' are

:<math>\sum_{i=1}^n \sum_{k=1}^p x_{ij}x_{ik}\hat \beta_k=\sum_{i=1}^n x_{ij}y_i,\  j=1,\dots,p.\,</math>

In matrix notation, the normal equations are written as

:<math>\mathbf{(X^\top X )\hat{\boldsymbol{\beta}}= {}X^\top Y},\,</math>

where the <math>ij</math> element of <math>\mathbf X</math> is <math>x_{ij}</math>, the <math>i</math> element of the column vector <math>Y</math> is <math>y_i</math>, and the <math>j</math> element of <math>\hat \boldsymbol \beta</math> is <math>\hat \beta_j</math>. Thus <math>\mathbf X</math> is <math>n \times p</math>, <math>Y</math> is <math>n \times 1</math>, and <math>\hat \boldsymbol \beta</math> is <math>p \times 1</math>. The solution is

:<math>\mathbf{\hat{\boldsymbol{\beta}}= (X^\top X )^{-1}X^\top Y}.\,</math>