# Interpret regression coefficients when dependent variable is standardized

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.

$$\frac{\delta z_i}{\delta X_1} = \frac{\delta \frac{y_i - \bar{y}}{\sigma_y}}{\delta X_1} = \frac{\frac{\delta y_i}{\sigma_y}}{\delta X_1} = \beta_1$$
$$\frac{\delta y_i}{\delta X_1} = \beta_1 \sigma_1$$
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 ($$\sigma_y$$).