# Rescaling model coefficients

I have the following model $$y \sim b_0 + Z \theta_0 + X \beta + \sum((X_j \circ Z) \theta_j)$$

Where...

• $$Z$$ is (n, k)
• $$X$$ is (n, p)
• $$\beta$$ is (p,)
• $$\theta$$ is (p, k)
• $$\theta_0$$ is (k,)
• $$\circ$$ is the element wise multiplication
• n is the number of samples
• p is the number of predictors
• k is the number of other predictors
• $$j$$ is the jth predictor of predictors p

y is centered by subtracting the mean of y and $$X$$ and $$Z$$ are scaled so they have 0 mean and 1 std. My question is how do I transform $$\beta$$ and $$\theta$$ so that their values are the values they would have had if X, Z and y were not modified.

PS: I can guess for simpler models like $$y = Xm + b$$ the variables $$m$$ and $$b$$ can be modified by $$m_{scaled} = m / \sigma_X$$ and $$b = \mu_y - (b + m \cdot X_{offset})$$ . But I'm not sure if that universally correct.