Econometric Analysis of Cross Section and Panel Data - Free

Conditional Expectations and Related Concepts in Econometrics 21
Now suppose equation (2.23) contains an interaction in x 1 and z:
Eðy j x 1 ; x 2 ; zÞ ¼b 0 þ b 1 x 1 þ b 2 x 2 þ b 3 z þ b 4 x 1 z
ð2:26Þ
Then, again by the LIE,
Eðy j x 1 ; x 2 Þ¼b 0 þ b 1 x 1 þ b 2 x 2 þ b 3 Eðz j x 1 ; x 2 Þþb 4 x 1 Eðz j x 1 ; x 2 Þ
If Eðz j x 1 ; x 2 Þ is again given in equation (2.25), you can show that Eðy j x 1 ; x 2 Þ has
terms linear in x 1 and x 2 and, in addition, contains x 2 1 and x 1x 2 . The usefulness of
such derivations will become apparent in later chapters.
The general form of the LIE has other useful implications. Suppose that for some
(vector) function fðxÞ and a real-valued function gðÞ, Eðy j xÞ ¼g½fðxÞŠ. Then
E½y j fðxÞŠ ¼ Eðy j xÞ ¼g½fðxÞŠ
ð2:27Þ
There is another way to state this relationship: If we define z 1 fðxÞ, then Eðy j zÞ ¼
gðzÞ. The vector z can have smaller or greater dimension than x. This fact is illustrated
with the following example.
Example 2.3:
If a wage equation is
Eðwage j educ; experÞ ¼b 0 þ b 1 educ þ b 2 exper þ b 3 exper 2 þ b 4 educexper
then
Eðwage j educ; exper; exper 2 ; educexperÞ
¼ b 0 þ b 1 educ þ b 2 exper þ b 3 exper 2 þ b 4 educexper:
In other words, once educ and exper have been conditioned on, it is redundant to
condition on exper 2 and educexper.
The conclusion in this example is much more general, and it is helpful for analyzing
models of conditional expectations that are linear in parameters. Assume that, for
some functions g 1 ðxÞ; g 2 ðxÞ; ...; g M ðxÞ,
Eðy j xÞ ¼b 0 þ b 1 g 1 ðxÞþb 2 g 2 ðxÞþþb M g M ðxÞ
ð2:28Þ
This model allows substantial flexibility, as the explanatory variables can appear in
all kinds of nonlinear ways; the key restriction is that the model is linear in the b j .If
we define z 1 1 g 1 ðxÞ; ...; z M 1 g M ðxÞ, then equation (2.27) implies that
Eðy j z 1 ; z 2 ; ...; z M Þ¼b 0 þ b 1 z 1 þ b 2 z 2 þþb M z M
ð2:29Þ