# 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 $$\beta_1$$:

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

2. "$$\beta_1$$ 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?

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 $$X$$.

• +1 That last paragraph is an important insight.
– Dave
Commented Jul 31, 2022 at 10:56
• Consider making a clearer distinction between causal and noncausal interpretations. The OPs formulation is essentially causal and is only correct when the regression can be interpreted causally. Commented Jul 31, 2022 at 12:36
• @RichardHardy I updated the answer. Commented Jul 31, 2022 at 14:51