# Alternative interpretation of multiple regression coefficients?

If we have a linear regression of the form

$$Y = \beta_0 + \beta_1X_1 + \beta_2X_2$$

is it valid to interpret the coefficient $$\beta_1$$ as the associated change in $$Y$$ when $$X_1$$ increases by a unit of 1, when $$X_2=0$$?

• The fact that rate of change of (expected) $Y$ wrt $X_i$ is the same no matter what the values of the $X$ variables are is essentially our assumption that the relationship is linear. Sep 13, 2022 at 22:06

Just omit the last caveat and your interpretation is basically accurate, though I will add one small edit based on dipetkov's helpful comment:

$$\beta_1$$ is the associated change in the expected $$Y$$ when $$X_1$$ increases by a unit of 1.

This holds whatever the value of $$X_2$$ (conditional on the model being accurate, though this caveat is also covered by the addition of 'expected' to the definition). Variation in $$X_2$$ is irrelevant here because there is no interaction term in your model.

• I'd like to be explicit and say "the associated change in the expected Y" or "the expected change in Y associated with". Sep 13, 2022 at 5:35
• Thanks, is what I have a special case or is what I have incorrect as it stands? Sep 13, 2022 at 6:24
• @dipetkov Thanks, that is a good idea. I have edited to reflect that.
– mkt
Sep 13, 2022 at 6:49
• @Stef: it is a linear regression model - typically not all points fit the equation exactly, but there is assumed to be an error term with zero mean and independent of the $X_i$. Sep 13, 2022 at 13:17
• @Stef Regression is a model for the conditional mean E(Y | X = x). To understand this better, consider the Difference between confidence intervals and prediction intervals Also take a look at Section 10.2 in Regression and Other Stories. It's freely available online. Sep 13, 2022 at 13:27