# Linear model

In statistics , the term linear model ( LM for short ) is used in different ways and in different contexts. The term occurs most frequently in regression analysis and is mostly used synonymously with the term linear regression model . However, the term is also used in time series analysis , where it has a different meaning. In any case, the attribution “linear” is used to refer to a specific class of models.

## Linear regression models

In the case of linear regression, a linear model is defined as follows: Let the random sample be given with the realizations . The relationship between the dependent variables and the independent variables is formulated as follows: ${\ displaystyle (Y_ {i}; X_ {i1}, \ ldots, X_ {ip}), \, i = 1, \ ldots, n}$ ${\ displaystyle X_ {1} = x_ {1}, \ ldots, X_ {n} = x_ {n}}$ ${\ displaystyle Y}$ ${\ displaystyle x_ {1}, \ ldots x_ {n}}$

${\ displaystyle Y_ {i} = \ beta _ {0} + \ beta _ {1} \ phi _ {1} (x_ {i1}) + \ ldots + \ beta _ {p} \ phi _ {p} ( x_ {ip}) + \ varepsilon _ {i} \ quad, i = 1, \ ldots, n}$,

where can represent non-linear functions. In the above regression equation, the disturbance terms represent random variables. The epithet results from the requirement that the regression equation is linear in the regression parameters . For example would not be allowed. As an alternative to the above equation, one can also say that the predicted values ​​of the dependent variables are given by the following equation: ${\ displaystyle \ phi _ {1}, \ ldots, \ phi _ {p}}$ ${\ displaystyle \ varepsilon _ {i}}$ ${\ displaystyle \ beta _ {j}}$${\ displaystyle \ beta _ {j} ^ {2}}$

${\ displaystyle {\ widehat {Y}} _ {i} = b_ {0} + b_ {1} \ phi _ {1} (x_ {i1}) + \ ldots + b_ {p} \ phi _ {p} (x_ {ip}) \ quad, i = 1, \ ldots, n}$.

Assuming that the estimation of the regression parameters and the error variance is performed using the least squares method , the result is the following least squares minimization criterion:

${\ displaystyle Q = \ sum _ {i = 1} ^ {n} \ left (Y_ {i} - \ beta _ {0} - \ beta _ {1} \ phi _ {1} (x_ {i1}) - \ ldots - \ beta _ {p} \ phi _ {p} (x_ {ip}) \ right) ^ {2} \ rightarrow \ mathrm {Min!}}$.

From this one can easily see that the "linear" aspect of the model means:

• The function to be minimized is a quadratic function of the regression coefficients .${\ displaystyle \ beta _ {j}}$
• The derivatives of the function are linear functions of that make it easy to find the parameter estimates .${\ displaystyle \ beta _ {j}}$
• The parameter estimates are linear functions of the random variables .${\ displaystyle b_ {0}, \ ldots, b_ {n}}$${\ displaystyle Y_ {i}}$

## literature

• Ludwig Fahrmeir, Thomas Kneib, Stefan Lang: Regression: Models, Methods and Applications. 2nd Edition. Springer Verlag, 2009, ISBN 978-3-642-01836-7 .