Linear model

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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:


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:


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:


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 .
  • The derivatives of the function are linear functions of that make it easy to find the parameter estimates .
  • The parameter estimates are linear functions of the random variables .


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