Meaning of regression coefficients

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.

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

• but aren't these two interpretations mutually exclusive? – Jemlin95 Jan 5 at 1:03
• @Jemlin95 why do you think they are? – 1muflon1 Jan 5 at 1:07
• 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. – Jemlin95 Jan 5 at 1:24
• @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 – 1muflon1 Jan 5 at 1:29
• No, I mean that the expected value of y conditional on x is equal to a+bx – Jemlin95 Jan 5 at 1:32

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".