# Monotonic constraints in regression model with interaction

I'm trying to figure out what constraints I need to use to have monotonicity (both on $x$ and $z$) on the regression model with interaction. My model is:

$$\mathbb{E}[Y|x,z]= \beta_0+\beta_1x+\beta_2z+\beta_3xz,$$

thus, to get an increasing function in x and z, I need the derivatives wrt x and z to be positive: $$\beta_1+\beta_3z\ge0 \quad\text{ and } \quad\beta_2+\beta_3x\ge0$$

If x and z are both positive, I conclude: $\beta_i \ge 0, \quad i= 1,\ldots,3$.

Since I would estimate the model with standardised $x$ and $z$, I also can have negative values. How can I solve the system in that case?

• How do you derive this conclusion? If $x$ and $z$ can be arbitrarily large, then you cannot have $\beta_3$ be negative. Note, too, that this surface (qua function of $(x,z)$) is not a plane: it's a piece of a hyperboloid. – whuber Jan 27 '18 at 19:22