Linear regression is a estimation of conditional expectation? I am studying the topic of regression for the first time and some questions arise. First, linear regression is a estimation of conditional expectation? And also the conditional expectation estimate is the so-called $y ̂$ estimate? This is:
$$y=E(Y|X)+e$$ $$y=y ̂+e$$ $$y ̂=E(Y|X)$$ $$(|)=+$$
Second, the linearity of the parameters is an assumption of the linear regression to estimate the conditional expectation? $$ $$ Third, Hansen's book on econometrics says about this problem: "the linear CEF model is empirically unlikely to be accurate unless $x$ is discrete and low-dimensional so all interactions are included. Consequently in most cases it is more realistic to view the linear specification as an approximation". What interpretation can be given to this phrase?
 A: *

*Yes and yes. There is a subtle technical point here, though I hesitate to mention it until you’ve gotten used to the idea of regression predicting an expected value instead of just a number that “should” be the right answer.

(Don’t read this parenthetical part for a few months or years until you’re much more comfortable with regression. The subtle point is that we often don’t see the predictors as random variables, so there isn’t a multivariate distribution where we condition on many variables to examine $Y$. We think of $Y\vert X$ as a family of univariate distributions that are parameterized by the predictor variables. This is technically correct in many cases but not especially useful, particularly not to a beginner.)


*Right again!

For the first two, I think it makes sense when you start simulating regressions. I’ll let you think about how to do that and can come back and edit this answer with some R code. But I do think it’s a good exercise to think through it for a while.


*This gets into a George Box quote: “All models are wrong, but some are useful.” No, we probably don’t have real phenomena following perfectly linear patterns, much like real data don’t follow perfectly normal distributions. However, a linear model might provide a good enough model for us to do something useful.

