# 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.