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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.

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  • $\begingroup$ Saturated, not satiated? $\endgroup$
    – Russ Lenth
    Commented Mar 13, 2020 at 12:18
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    $\begingroup$ A satiated model has had all s/he desires. $\endgroup$
    – whuber ♦
    Commented Mar 13, 2020 at 13:06

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I believe this should be "saturated," rather than "satiated."

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

This is a kind of a non-parametric model, which is non-linear since it allows the means to be very flexible.

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  • $\begingroup$ Thank you for your answer :) $\endgroup$
    – econnewb_sad
    Commented Mar 14, 2020 at 5:53

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