# What exactly is a "true" population model in linear regression?

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:

$$log(wage)=β_0+β_1S+ε$$

$$log(wage)=v_0+v_1S+v_2A+v$$

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 $$v1=β1$$ because both measure the same partial effects of S on log(wages).

1. Must the "true" population model satisfy the classical regression model assumptions (i.e. zero conditional mean $$E[ε|S,A]=0$$ 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 $$log(wage)=β_0+β_1S+ε$$ is the "true" population model, if I split $$β_1=l_1+g_1$$ and rewrite the "true" population model as follows: $$log(wage)=β_0+l_1S+e$$ where $$e=ε+g_1S$$. 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.

We presume that there is a data generation process that creates all your data rows $$(wage, S, A)$$ and that those triplets satisfy: $$wage = f(S, A, \varepsilon),$$ where $$\varepsilon$$ is some error random variable. Then, this $$f$$ describes the "true" model. Usually, we don't know $$f$$, but we often think we can justify presuming that this $$f$$ can be approximated by a linear function such as yours: $$wage = v_0 + v_1 S + v_2 A + \varepsilon .$$ Then, once we have convinced ourselves that the true model is linear, we are then interested in approximating the "true" parameters $$v_i$$.
Your third question: Your second formula has an error term $$e$$ that is not anymore independent of the variable $$S$$ which then violates those standard rules that make the application of standard methods possible.