# What is a satiated model, and how is it non-linear?

I am just beginning to learn econometrics, and am a little confused by my lecture notes.

They say that the conditional mean function has a known functional form, and is linear in parameter, e.g.,:

$$m(x_1,x_2) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1^2$$.

Okay - so looking above, whilst we have some non-linear regressors, the $$\beta$$'s are all linear, so this is what it means "linear in parameter"?

I think I understand so far.

Next, the notes say:

"There are cases in which linearity is not a binding constraint. This is sometimes referred to as a satiated model. For example, let $$x_1$$ and $$x_2$$ be binary variables, both taking values 0 and 1." In this case:

$$m(x_1,x_2) = m(0,0) + m(1,0)x_1(1-x_2)+m(0,1)(1-x_1)x_2 + m(1,1)x_1x_2$$

I am really confused by what this sentence is trying to say or convey.

• Saturated, not satiated? Commented Mar 13, 2020 at 12:18
• A satiated model has had all s/he desires.
– whuber
Commented Mar 13, 2020 at 13:06

This just means you use regression to calculate four means for all the possible combinations of $$x_1$$ and $$x_2$$: $$(0,0),(1,0),(0,1),(1,1)$$