What is the correct interpretation of the $\beta_1$ coefficient in a linear regression model? Given a linear regression model:
$$
Y_i = \beta_0+\beta_1X_{i1}+\epsilon_i
$$
I have seen two interpretations of $\beta_1$:

*

*"$\beta_1$ is the amount $Y_i$ increases by when $X_{i1}$ increases by 1 unit"


*"$\beta_1$ is the amount $E(Y_i\mid X_{i1})$ increases by when $X_{i1}$ increases by 1 unit"
Which of the two is technically the correct interpretation?
 A: Here is what $E[Y|X]$ changes when $X$ changes by one unit (using that the expectation operator is linear and that $\epsilon$ is independent of $X$ with $E[\epsilon] = 0$):
$$
\begin{align}
E[Y|X = x_0+1] - E[Y|X=x_0] &= E[\beta_0 + \beta_1X + \epsilon|X=x_0+1] - E[\beta_0 + \beta_1X + \epsilon|X=x_0]\\
    &= \beta_0 + \beta_1 E[X|X=x_0+1] + E[\epsilon|X=x_0+1] - (\beta_0 + \beta_1 E[X|X=x_0] + E[\epsilon|X=x_0])\\
    &= \beta_0 + \beta_1(x_0+1)+ 0 - (\beta_0 + \beta_1x_0 + 0)\\
    &= \beta_1,
\end{align}
$$
so the second interpretation is correct.
However, when people use the first version, they mean the second, it is just a bit of a lax way of expressing themselves.

In the comments, @RichardHardy makes the point that one should distinguish between a purely probabilistic and causal interpretation of those two statements.
In the answer above, I presumed those statements to be meant probabilistically, noncausally. Of course, there is also a causal interpretation along the lines: "If $X$ is intervened upon and set to $x_0$ and to $x_0+1$, how would $Y$ change as a result? I.e., what is the average causal effect (ACE) of $X$ on $Y$?" Of course, there are many situations where $\beta_1$ cannot be interpreted as the ACE, e.g. if $Y$ would be causing $X$.
