# 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.

Words like "model" and "population" are often used very differently by different authors. Here is a description of a pretty standard usage:

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 first question: Yes, this is presumed, otherwise the standard algorithms don't work anymore. Note that the true model would still be the true model if it didn't satisfy those properties, we just hope that it does for convenience.

Your second question: Sometimes, the word model only refers to the functional form and includes all possible parameter settings. But sometimes authors describe the same functional form but with different parameters as different models. This has to be understood from the context.

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.

In a real situation we cannot know what the "true" model is and I'd even say there is none. ("All models are wrong but some are useful" said George Box.)

A true model is always a model that is assumed to be true (in some kind of artificial model world) in order to investigate mathematically what happens in case this is true.

In standard statistical thinking, there exists a "true" model, and statistical methods attempt to fit or estimate or select it (in the simplest and most standard case it is assumed that the model that we attempt to fit is the true model). This, however, is a "useful fiction". It makes sense to distinguish the unknown real truth from an estimator or fit of it, however a "true model" only exists in the world of mathematics.

In reality all you can do is to reject a certain model if you observe something that should hardly ever happen were the model indeed true. Models can often be motivated from knowledge of the situation, but that doesn't necessarily make them true.

When doing statistical theory and research, which model is assumed as true can be decided by the researcher and depends on what the researcher is interested in.

For example, in your situation with models

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

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

you could be interested what happens if the first model is true and you fit any of the two models (or both and compare them), or analogously what happens if the second model is true. You may have a model selection method and may be interested in whether it selects the true model with high probability, so you can assume either model as true and ask whether this will likely be found (in much theory such situations can be assessed in the same go, so they'd give you an indication what happens if any of the models of interest is true, but this does not always work).

Rather standard are results where you fit a certain model, you assume this model to be true, and then, by theory or simulation, you find out about how good your fit is expected to be.

You may also be interested (as people in robust statistics are) what happens if you try to fit a certain model, and in fact a different model is true.

So any model can be assumed as true, if you are interested in analysing what happens to a certain statistical method in case the model is true. It is the job of the author to explain clearly which model is assumed to be true where, so as a reader you should know, but in principle any model goes, there are no general criteria that a model assumed as true has to fulfill.