Environmental Engineering Reference
In-Depth Information
The Heckman model, also known as the selectivity model or tobit type 2 model,
begins by formulating the following regression model to represent the consumption
equation:
y
i
¼
x
i
b þ e
i
ð
7
Þ
where, the sub-index
i
refers to each individual and the dependent variable
y
i
represents any variable that cannot be observed in the whole population.
3
The
explanatory variables of the model are represented by
x
i
and
b
. The random term is
represented by
e
i
. The Heckman model also requires a selection variable
z
such
that
y
i
is observed only if
z
exceeds a set threshold, e.g. zero. This variable is
explained by the following selection equation:
z
¼
w
i
c þ u
i
;
ð
8
Þ
where, the vector
w
i
contains the explanatory variables that affect the selection, the
vector
c
of unknown coef
cients, and
u
i
is the random error term.
Assuming the errors
e
i
and
u
i
have a normal bivariate distribution with mean
zero and correlation
q
, one can write:
E
½
y
i
jy
i
observed
¼
E
½
y
i
jz
[
0
¼
E
½
x
i
b þ e
i
jw
i
c þ u
i
[
0
¼
x
i
b þ E
½
e
i
jw
i
c þ u
i
[
0
¼
x
i
b þ E
½
e
i
ju
i
[
w
i
c
ð
9
Þ
2
3
/
w
i
c
r
u
4
5
;
¼
x
i
b þ qr
e
w
i
c
r
u
U
where, the standard deviations of the errors are, from equations (
7
) and (
8
),
r
e
and
r
u
. The expressions
/
(
) represent the density and probability functions,
respectively, of the standard normal distribution. In most instances, it is supposed
that
·
) and
(
·
U
˃
u
is equal to unity as the variable
z
cannot be observed. Therefore, Eq. (
9
)
corresponds to Eq. (
7
) with an additional explanatory variable called the inverse
Mills ratio evaluated at
w
i
c
:
Note that
c
is replaced with Probit estimates from the
rst stage (
8
).
From Eq. (
9
) it can be seen that if
e
i
and
u
i
are independent (
q
¼
0), the second
term on the right hand side would be zero, so the consumption equation could be
estimated directly by ordinary least squares (OLS). On the other hand, if
q 6
¼
0,
Eq. (
9
) could be estimated by the two-stage estimation procedure in Heckman [
9
],
or by maximising the following likelihood function [
6
]:
3
In this paper
y
i
is residential natural gas consumption.
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