Empty model

In the statistic a is empty model , also empty model , or null model called ( English intercept-only model ), a model in which only the intercept ( English intercept is taken into account), and in all other regression parameters are next to the intercept equal to zero. It represents the simplest specification of a linear model and contains one as the only trivial regressor. In the null model, all regression parameters except for the intercept are zero and the “ best ” estimate of the dependent variable is provided by the arithmetic mean . The null model is usually the reference model to evaluate the goodness of fit of a regression . Successive regression is included in the model in order to assess how the goodness of fit has improved compared to the null model. In the so-called forward selection, the explanatory variables are included one after the other for the zero model, which have a significant influence on the target variable , thus making a significant contribution to improving the model, i.e. H. the increase in the multiple coefficient of determination .

The model

The null model, i.e. the model which only consists of the axis intercept is

${\displaystyle Y_{i}=\beta _{0}+\varepsilon _{i}\quad ,i=1,\ldots ,n}$.

Here is the axis intercept and represents a stochastic error term . ${\displaystyle \beta _{0}}$ ${\displaystyle \varepsilon _{i}}$

Estimation of the model

If you start with the general definition of the KQ estimator , the KQ estimator for the axis intercept is: ${\displaystyle \mathbf {b} =(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} }$

${\displaystyle \mathbf {b} =(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} =\left({\begin{pmatrix}1&\cdots &1\end{pmatrix}}{\begin{pmatrix}1\\\vdots \\1\end{pmatrix}}\right)^{-1}\mathbf {X} ^{\top }\mathbf {y} =n^{-1}\sum \limits _{i=1}^{n}y_{i}={\overline {y}}={\hat {\beta }}_{0}\quad }$, With ${\displaystyle \quad \mathbf {X} =1\!\!1_{n}}$

that is, the average of the target size . This estimate is unbiased because . ${\displaystyle {\overline {y}}}$ ${\displaystyle \operatorname {E} ({\hat {\beta }}_{0})=\operatorname {E} ({\overline {Y}})=n^{-1}\sum \limits _{i=1}^{n}\operatorname {E} (\beta _{0}+\varepsilon _{i})=\beta _{0}}$

Individual evidence

1. ↑ Lothar Sachs , Jürgen Hedderich: Applied Statistics: Collection of Methods with R. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2018, ISBN 978-3-662-56657-2 , p. 840
2. ↑ Lothar Sachs, Jürgen Hedderich: Applied Statistics: Collection of Methods with R. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2018, ISBN 978-3-662-56657-2 , p. 835