Sample regression function

In statistics , a designated sample regression function , even empirical regression function ( english function sample regression , in short: SRF ), the estimated version of the regression function of the population . The sample regression function is fixed, but unknown in the population. If the regression function is a straight line, then we also speak of a sample regression line or an empirical regression line . The sample regression line is obtained as a least-squares regression line (short: KQ regression line) from observation pairs that represent data points. According to the least squares criterion, it represents the best possible fit to the data.

Simple linear regression

If the least squares estimator for the slope and the least squares estimator for the absolute term are determined using the least squares estimation , the following KQ regression line is obtained ${\displaystyle {\hat {\beta }}_{1}=SP_{xy}/SQ_{x}}$ ${\displaystyle {\hat {\beta }}_{0}={\overline {y}}-{\hat {\beta }}_{1}{\overline {x}}}$

${\displaystyle {\hat {y}}={\widehat {\operatorname {E} (y\mid X=x)}}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}x\quad }$.

This is also called the sample regression function because it is an estimated variant of the (theoretical) regression function of the population

${\displaystyle \operatorname {E} (y\mid X=x)=\beta _{0}+\beta _{1}x\quad }$

is. The parameters and are also called empirical regression coefficients . Since the sample regression function is obtained from a given sample, a new sample yields a new slope and a new absolute term . In most cases, the least squares estimate for the slope can be represented as ${\displaystyle {\hat {\beta }}_{0}}$ ${\displaystyle {\hat {\beta }}_{1}}$ ${\displaystyle {\hat {\beta }}_{1}}$ ${\displaystyle {\hat {\beta }}_{0}}$

${\displaystyle {\hat {\beta }}_{1}=\Delta {\hat {y}}/\Delta x}$

This representation shows that the least-squares estimator for the slope shows how much the target variable changes when the influencing variable increases by one unit. ${\displaystyle y}$ ${\displaystyle x}$

Multiple linear regression

Given a typical multiple linear regression model , with the vector of the unknown regression parameters , the experiment plan matrix , the vector of the dependent variables and the vector of the disturbance variables . Then the KQ sample regression function or sample regression hyperplane is given by ${\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}}$ ${\displaystyle {\boldsymbol {\beta }}}$ ${\displaystyle p\times 1}$ ${\displaystyle n\times p}$ ${\displaystyle \mathbf {X} }$ ${\displaystyle n\times 1}$ ${\displaystyle \mathbf {y} }$ ${\displaystyle n\times 1}$ ${\displaystyle {\boldsymbol {\varepsilon }}}$ ${\displaystyle {\hat {\mathbf {y} }}}$

${\displaystyle {\hat {\mathbf {y} }}=\mathbf {X} {\hat {\boldsymbol {\beta }}}=\underbrace {\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }} _{=\mathbf {P} }\mathbf {y} =\mathbf {P} \mathbf {y} }$,

where represents the prediction matrix. ${\displaystyle \mathbf {P} }$

Web links

• Springer Gabler Verlag, Gabler Wirtschaftslexikon, keyword: sample regression line

Individual evidence

1. ↑ Jeffrey Marc Wooldridge : Introductory econometrics: A modern approach. 5th edition. Nelson Education, 2013, p. 31.
2. ^ Otfried Beyer, Horst Hackel: Probability calculation and mathematical statistics. 1976, p. 185.
3. ↑ Jeffrey Marc Wooldridge: Introductory econometrics: A modern approach. 5th edition. Nelson Education, 2013, p. 31.