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I want to make sure my understanding is correct. Particularly in econometrics, when authors write down a model:

$Y_i = \beta_0 + \beta_1 X_i + \epsilon$

Can I think of this as a 'structural model'- or a linear approximation to the true underlying data generating process that creates the observed joint distribution of x and y? Is this also a description of the conditional expectation function (CEF)? What is the relation between the CEF and this underlying 'structural model'?

So it is a model saying beta is the causal effect of x on y, regardless of whether $\epsilon$ is correlated with x. But then when we go to the regression model, we are estimating the linear projection/CEF, which will give us an observed relationship, and is it only here where the statistical relationship between $\epsilon$ and $x_i$ becomes important? so if the error term is correlated with $x_i$, then the linear projection model will not recover $\beta_1$? so is it about aligning the porjection model with the underlying 'structural model'?

sorry if this is too many question packed into one.

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I am going to try to answer the overall theme as well as I can, but may not get every individual point.

Without any more context, simply writing an equation $Y = \beta_0 + X\beta_1 + \epsilon$ is not enough to glean any real interpretation. Authors who are clear should distinguish whether they are treating a model as a ground truth or as a proposal for an approximation in some sense, but in intro texts these distinctions are not always as clear as they ought to be.

So I can't exactly answer your question, but I will try to give you some insight into a few of the possible perspectives one might have in economics when a linear regression model like the above is proposed.

I will talk broadly about 4 perspectives on linear regression – structural, non-structural, causal, and model robust (or skeptical). They are not exhaustive nor mutually exclusive. Crucially, however, the perspectives that we bring to bear on the linear regression will inform the way we interpret the resulting coefficients and fits.

Structural - My (likely limited) view-point on structural is that is used to describe situations where we use a family of distributions to encode economic ideas or theory to describe some economic process of note. See this https://stanford.io/3oFv6gr for a more thorough discussion on structural and non-structural modeling and economics. In the absence of such a process we are in some sense taking a non-structural view.

Causal – A causal perspective comes from viewing the linear regression equation as representing some notion of a counter-factual, i.e how the outcome distribution will change on average as we manipulate one (or potentially several) of the covariates (usually called a treatment or exposure). This requires fairly strict assumptions on the errors ($E[\epsilon|X] =0$ everywhere in the covariate space) which in observational data (as opposed to in an experiment) are likely to be best justified from a structural perspective. One of the important features in this lens is that while we might be interested in a particular parameter, we are thinking about that parameter as an average causal effect of some sort and not just a parameter to be recovered.

We pay the price of making stricter assumptions in structural and causal view-points, but for the gain of more interesting interpretations. It is then natural to ask, if we are skeptical about our knowledge of the underlying system, what can we say then. The first thing is that linear regression will always give us the best linear projection onto our covariates. For the purpose of some (especially predictive) tasks this might be more than enough. But what can we say about the actual coefficients themselves if we don’t entirely believe or trust the model.

The model robust framework championed by Hal White in the 1980s gives us one place to start. Here we think of the linear regression model as a working model or approximation to the true conditional mean function. This could theoretically be done from a structural perspective, but where for example you believe you have well specified the important variables underlying an economic process, but do not believe you know the true conditional expectation function. More common perhaps is the view best expressed by Box that the very notion of a model is to suggest simplification. Thus we are never going to get the model right in some sense and any model we propose is an convenient approximation to that truth and if we are lucky that model might still be meaningful.

A more recent series of papers building on Hal White's work in the 1980s by Buja et al (Part I https://whr.tn/2LNK2KM and Part II https://whr.tn/39pTRqG) argue that without believing our models we can think of the parameters as statistical functions of the underlying distribution. The "error term " (they prefer the term offsets) in this framework now consists of a random part (like $\epsilon$ in the normal framework) and an approximation error, $\eta(X) = E[Y|X] - E[\hat{\beta}|X]$, where $\hat{\beta} = \arg \min_{\beta} E[|Y - X\beta|^2|X]$.

One of the crucial differences in this framework is that the parameters will actually depend on the distribution of the regressors (if the true CEF is not linear), which is not something we are used to thinking about in most treatments of linear regression. The parts of the actual conditional expectation function that we approximate well will also depend on the distribution of the covariates. It is worth looking at the papers above and specifically looking at the examples they give of their intepretations. I don’t think I could possibly do as good of a job here.

So in short, yes interpretation depends on which assumptions you make and the extent to which you believe them. One of the challenging things when first getting into this space is to sort out the ways in which these lenses conflict, overlap, or enhance on another.

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