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I am having some trouble understanding exactly what is meant by the "true" Regression coefficients. Let's say it is stated that "the true regression coefficients are given as $y=a+bx+e$ where the error term fulfills all the OLS assumptions. Does this mean

a) Given knowledge of the value x takes one would describe their belief about the value y takes with the probability distribution on the right hand side.

b)a one unit change of x in a certain instance would cause the expected value of y to change by b.

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a) not sure what exactly you mean here by probability distribution but if you mean the distribution of potential Y given the distribution of estimated coefficient b and a it would be correct, although usually we look at expectation to get single number than on distribution in which case it would be $E(Y|X)$

b) is correct

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  • $\begingroup$ but aren't these two interpretations mutually exclusive? $\endgroup$ – Jemlin95 Jan 5 at 1:03
  • $\begingroup$ @Jemlin95 why do you think they are? $\endgroup$ – 1muflon1 Jan 5 at 1:07
  • $\begingroup$ Let's say the "true" model for health expenditure is given as h=a+bi+e where i is income. According to the second interpretation if I were to give a family an extra 1 unit a month I would expect their health expenditures to rise by b units. According to the first interpretation this may not be true since perhaps people who earn more spend a higher proprtion of their income on health expenditures. $\endgroup$ – Jemlin95 Jan 5 at 1:24
  • $\begingroup$ @Jemlin95 I am not sure if I understood your A interpretation correctly (b is clear and it’s correct) I understood your A interpretation in a way that at any point there is a distribution of Y for example if b=2 and standard error of b=0.1 then if x increases by 1 the mean expected increase of Y will be 2 but there will be an distribution around this expected increase t(2,0.1) so for example with 95% confidence the true increase will be somewhere between approximately 1.8 and 2.2. Maybe I misunderstood your A so let me know if you mean something else by distribution $\endgroup$ – 1muflon1 Jan 5 at 1:29
  • $\begingroup$ No, I mean that the expected value of y conditional on x is equal to a+bx $\endgroup$ – Jemlin95 Jan 5 at 1:32
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It might be too long for a comment so I write it as an answer. Apologies if I missed any ongoing discussions. Let me say the general background first and then my thoughts about OP's question.

The model $y=a+bx+e$ as such tells us nothing about what the true $b$ is. We need more restrictions to identify $b$. There can be many ways of making identifying restrictions. Some are:

  1. $E(e|x)=0$,

  2. $median(e|x)=0$,

  3. $e|x \sim N(0, \sigma^2)$,

  4. the distribution of $e$ conditional on $x$ is independent of $x$,

  5. $E(e|z_1, z_2)=0$ for some other given $z_1$ and $z_2$,

or many possible others. We have 3 => 1, 4 => 1, 3 => 2, and 4 => 2. Each identifying restriction is associated with the "true" $b$ parameter. The true $b$ identified by 1 is not necessary equal to that by 2. We know that the true $b$ by 1 (the probability limit of OLS) may be different from the true $b$ by 5 (the probability limit of 2SLS).

Identification by condition 1 gives OP's "interpretation b". OP's "interpretation a" seems related with option 3 or 4 above, but it is unclear since OP does not give enough information about the population distribution. The distribution of $y$ (or equivalently of $e$) given $x$ should be specified somehow to achieve the identification of the true $b$.

How to interpret $b$ is actually the same as how to define the true $b$, and defining $b$ requires restrictions. Like I said at the beginning, just writing $y=a+bx+e$ is uninformative about what the true $b$ is.

Now to the question, OP says, "the error term fulfills all the OLS assumptions", which I understand as $E(e|x)=0$. Then OP's interpretation "b" is correct. Note that "b" is correct because OP assumed so ("the OLS assumptions fulfilled") not because of any other fancy reasons.

The true $b$ associated with OP's "a" (if the probability distribution was specified) may or may not be equal to the true $b$ associated with $E(e|x)=0$. But if the distribution of $e$ conditional on $x$ is independent of the $x$ value, then its conditional mean should also be independent of $x$. So "a" is also correct if "a" is rephrased to something like: (c) "Given knowledge of the value $x$ takes one would describe their belief about the value $y$ takes with the probability distribution describe by the right hand side under the assumption that the distribution of $e$ conditional on $x$ is independent of $x$". That's because 4 implies 1. Here correct means the true parameter defined by "c" is the same as the true parameter defined by "b".

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