<p>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 <span class="math-container">$E(Y|X)$</span></p>

<p>b) is correct </p>

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2020-01-05T01:03:35.490Z
but aren't these two interpretations mutually exclusive?
Jemlin95
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2020-01-05T01:07:14.927Z
@Jemlin95 why do you think they are?
1muflon1
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2020-01-05T01:24:53.610Z
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
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2020-01-05T01:29:40.047Z
@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
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2020-01-05T01:32:59.670Z
No, I mean that the expected value of y conditional on x is equal to a+bx
Jemlin95
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2020-01-05T02:09:39.447Z
@Jemlin95 well for your regression E(Y|X) = a + bX. I did not put the other side of equality there because I already described it in text. Or am I still misunderstanding something?
1muflon1
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2020-02-18T10:52:16.353Z
ANSWER To (b)  part is incorrect.
Subhash C. Davar
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2020-01-05T01:00:51.393Z
2020-01-05T01:00:51.393Z
1muflon1
<p>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.</p>

<p>The model <span class="math-container">$y=a+bx+e$</span> as such tells us nothing about what the true <span class="math-container">$b$</span> is. We need more restrictions to identify <span class="math-container">$b$</span>. There can be many ways of making identifying restrictions. Some are:</p>

<ol>
<li><p><span class="math-container">$E(e|x)=0$</span>,</p></li>
<li><p><span class="math-container">$median(e|x)=0$</span>,</p></li>
<li><p><span class="math-container">$e|x \sim N(0, \sigma^2)$</span>,</p></li>
<li><p>the distribution of <span class="math-container">$e$</span> conditional on <span class="math-container">$x$</span> is independent of <span class="math-container">$x$</span>,</p></li>
<li><p><span class="math-container">$E(e|z_1, z_2)=0$</span> for some other given <span class="math-container">$z_1$</span> and <span class="math-container">$z_2$</span>,</p></li>
</ol>

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

<p>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 <span class="math-container">$y$</span> (or equivalently of <span class="math-container">$e$</span>) given <span class="math-container">$x$</span> should be specified somehow to achieve the identification of the true <span class="math-container">$b$</span>.</p>

<p>How to <em>interpret</em> <span class="math-container">$b$</span> is actually the same as how to <em>define</em> the true <span class="math-container">$b$</span>, and defining <span class="math-container">$b$</span> requires restrictions. Like I said at the beginning, just writing <span class="math-container">$y=a+bx+e$</span> is uninformative about what the true <span class="math-container">$b$</span> is.</p>

<p>Now to the question, OP says, "the error term fulfills all the OLS assumptions", which I understand as <span class="math-container">$E(e|x)=0$</span>. 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.</p>

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

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2020-01-05T04:32:22.877Z
2020-01-05T04:32:22.877Z
chan1142
<p>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 <span class="math-container">$y=a+bx+e$</span> where the error term fulfills all the OLS assumptions. Does this mean</p>

<p>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.</p>

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

CC BY-SA 4.0
2020-01-23T19:20:35.730Z
Chen & Pearl ["Regression and Causation: A Critical Examination of Six Econometrics Textbooks"](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2338705) (2013) could be relevant. They show how $b$ is commonly ill defined (conflating probabilistic and causal languages) in popular textbooks and explain the background a bit.
Richard Hardy
<econometrics><regression>
Meaning of regression coefficients
Jemlin95
# Meaning of regression coefficients

(Asked by: username_0 on 2020-01-05)

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.

**Tags:** econometrics, regression
> **Comments:**
> **username_1**: Chen & Pearl ["Regression and Causation: A Critical Examination of Six Econometrics Textbooks"](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2338705) (2013) could be relevant. They show how $b$ is commonly ill defined (conflating probabilistic and causal languages) in popular textbooks and explain the background a bit.  

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## 2 answers

### Answer by username_2 on 2020-01-05. Score: 1

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

> **Comments:**
> **username_0**: but aren't these two interpretations mutually exclusive?  
> **username_2**: @Jemlin95 why do you think they are?  
> **username_0**: 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.  
> **username_2**: @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  
> **username_0**: No, I mean that the expected value of y conditional on x is equal to a+bx  
> **username_2**: @Jemlin95 well for your regression E(Y|X) = a + bX. I did not put the other side of equality there because I already described it in text. Or am I still misunderstanding something?  
> **username_3**: ANSWER To (b)  part is incorrect.  

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### Answer by username_4 on 2020-01-05. Score: 0

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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