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.
I gave it some thought (sometimes posting here helps me structure my thought process) and this is what I came up with:
$\frac{\delta z_i}{\delta X_1} = \frac{\delta \frac{y_i - \bar{y}}{\sigma_y}}{\delta X_1} = \frac{\frac{\delta y_i}{\sigma_y}}{\delta X_1} = \beta_1$
Therefore
$\frac{\delta y_i}{\delta X_1} = \beta_1 \sigma_1$
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 ($\sigma_y$).