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What do we mean by a true population model when talking about linear regression? Say I want to study the effects of years of schooling $S$ on wages. I posit the following two models:

$ log(wage) = \beta_0 + \beta_1 S + \varepsilon $

$ log(wage) = v_0 + v_1 S + v_2 A + v $

where $A$ is ability. Which is the "true" population model and what exactly are the requirements for a model to be a "true" model? I know that in this case $v_1=\beta_1$ because both measure partial effects of $S$ on $log(wages)$.

  1. Must the "true" population model satisfy the classical regression model assumptions (i.e. zero conditional mean $E[\varepsilon|S,A] = 0$ etc?
  2. Must the true model simply posit the "correct" functional form relationship between the variables of interest? (i.e. that the true population model must be linear or non-linear)
  3. If $ log(wage) = \beta_0 + \beta_1 S + \varepsilon $ is the "true" population model, if I split $\beta_1 = l_1 + g_1$ and rewrite the "true" population model as follows: $log(wage)=\beta_0+l_1S+e$ where $e=\varepsilon+g_1S$. Is this new model also a "true" population model?

I ask because many econometrics textbooks (especially Wooldridge) introduce a regression model and say that it is a "true" model but never clearly define what exactly constitutes a "true" population regression model.

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