Let's say we have the following regression model: $$z_i = \beta_0 + \beta_1 X_{1,i} + \beta_2 X_{2,1} + u_i$$ Where $z_i = \frac{y_i - \bar{y}}{\sigma_y}$ is the (standardized) dependent variable.

How do you interpret the effect that a marginal increase in $X_1$ has on the expected value of $y_i$ (not $z_i$)?

I found this question, but the dependent variable is standardized according to the group it belongs to.


1 Answer 1


I gave it some thought (sometimes posting here helps me structure my thought process) and this is what I came up with:

$\frac{\delta z_i}{\delta X_1} = \frac{\delta \frac{y_i - \bar{y}}{\sigma_y}}{\delta X_1} = \frac{\frac{\delta y_i}{\sigma_y}}{\delta X_1} = \beta_1$


$\frac{\delta y_i}{\delta X_1} = \beta_1 \sigma_1$

This means that for a given value of $X_{2i}$, a marginal increase in $X_{1i}$ will increase the expected value of $y_i$ in $\beta_1$ standard deviations ($\sigma_y$).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.