True model

In statistics , the underlying true model is the actual model in the population that relates the response variable to the relevant independent variables. This relationship is superimposed by an additive disturbance variable , which is assumed to have an expected value of zero. The basic assumption of the model is that it is linear in parameters.

Multiple linear regression

The following multiple linear regression model is given :

${\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}+\varepsilon _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}+\varepsilon _{i},\quad \operatorname {E} (\varepsilon _{i})=0}$		(1)

Here is the number of unknown ( true ) parameters to be estimated . The regression parameters are unknown, constant parameters of interest (they should be estimated) and is an unobservable random variable called a disturbance or error term. Even if one knew the true population regression function , the observed value of the outcome would still differ from the predicted value by some degree, which corresponds to the confounding variable. ${\displaystyle k+1=:p}$ ${\displaystyle \beta _{0},\beta _{1},\beta _{2},\dotsc ,\beta _{k}}$ ${\displaystyle \beta _{0},\beta _{1},\beta _{2},\dotsc ,\beta _{k}}$ ${\displaystyle \varepsilon _{i}}$ ${\displaystyle y_{i}}$ ${\displaystyle {\hat {y}}_{i}}$

Formally, the above equation is the model in the population or the population model . This is sometimes called a true model because assuming a true model ensures that one estimates a model that differs from (1).

For example, you could add redundant independent variables. However, the inclusion of redundant independent variables does not always have to represent a specification error (one speaks of a specification error if the assumption that the expected value of the disturbance variable is zero is violated). For example, the underlying true model could be given by . The selected (specified) model (with the irrelevant independent variables ) could be the following model: . That the variable is assumed to be irrelevant means that the true value is equal to zero ( ). For this reason, the following applies: . In this case the KQ estimators are still fair to expectations for the true values ​​and there is no specification error. ${\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\varepsilon _{i}}$ ${\displaystyle x_{i3}}$ ${\displaystyle y_{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+x_{i3}\beta _{3}+\varepsilon _{i}^{*}}$ ${\displaystyle \beta _{3}}$ ${\displaystyle \beta _{3}=0}$ ${\displaystyle \operatorname {E} (\varepsilon _{i}^{*})=\operatorname {E} (\varepsilon _{i}-x_{i3}\beta _{3})=0}$

Individual evidence

1. ↑ Jeffrey Marc Wooldridge : Introductory econometrics: A modern approach. 5th edition. Nelson Education, 2015, p. 859.
2. ↑ Jeffrey Marc Wooldridge: Introductory econometrics: A modern approach. 5th edition. Nelson Education, 2015, p. 83.