Econometric Analysis of Cross Section and Panel Data - Free

16
Chapter 2
Example 2.1 (continued):
In equation (2.2) we have
qEðy j xÞ
qx 1
¼ b 1 ;
qEðy j xÞ
qx 2
¼ b 2
As expected, the partial e¤ects in this model are constant. In equation (2.3),
qEðy j xÞ
qx 1
¼ b 1 ;
qEðy j xÞ
qx 2
¼ b 2 þ 2b 3 x 2
so that the partial e¤ect of x 1 is constant but the partial e¤ect of x 2 depends on the
level of x 2 . In equation (2.4),
qEðy j xÞ
qx 1
¼ b 1 þ b 3 x 2 ;
qEðy j xÞ
qx 2
¼ b 2 þ b 3 x 1
so that the partial e¤ect of x 1 depends on x 2 , and vice versa. In equation (2.5),
qEðy j xÞ
qx 1
¼ expðÞðb 1 =x 1 Þ;
qEðy j xÞ
qx 2
¼ expðÞb 2 ð2:7Þ
where expðÞ denotes the function Eðy j xÞ in equation (2.5). In this case, the partial
e¤ects of x 1 and x 2 both depend on x ¼ðx 1 ; x 2 Þ.
Sometimes we are interested in a particular function of a partial e¤ect, such as an
elasticity. In the determinstic case y ¼ f ðxÞ, we define the elasticity of y with respect
to x j as
qy
qx j
x j
y
qf ðxÞ
¼
qx j
x j
f ðxÞ
ð2:8Þ
again assuming that x j is continuous. The right-hand side of equation (2.8) shows
that the elasticity is a function of x. Wheny and x are random, it makes sense to use
the right-hand side of equation (2.8), but where f ðxÞ is the conditional mean, mðxÞ.
Therefore, the (partial) elasticity of Eðy j xÞ with respect to x j , holding x 1 ; ...; x j 1 ;
x jþ1 ; ...; x K constant, is
qEðy j xÞ
qx j
x j
x j
Eðy j xÞ ¼ qmðxÞ
qx j mðxÞ :
ð2:9Þ
If Eðy j xÞ > 0 and x j > 0 (as is often the case), equation (2.9) is the same as
q log½Eðy j xÞŠ
q logðx j Þ
ð2:10Þ