An Analysis on Danish Micro Data - School of Economics and ...

where φ ( z)
is the standard normal density
φ
( z)
≡ Φ(
z)
≡ ( v)
z
∫ ∞
−
G φ
dv
− 1
2
( z) = ( 2π
) 2 exp(
− z 2)
Assuming that the errors, U, have a standard normal distribution, this model is called the probit
model. 63
A probit model is estimated with maximum likelihood methods. The density of Yi given Xi looks as
follows
f
( Y X ; ) G(
X β )
i
Y
1−Y [ ] [ 1−
G(
X ) ] ,
Consequently, the log likelihood function is given by
β = β
Y = 0,
1
i
( ) = Y log [ G(
X β ) ] + ( 1−
Y ) log[
1 G(
X β ) ]
l −
i β i
i
i
i
and is maximized with respect to β. There exists a unique optimum since the probit model has a
negative definit Hessian matrix. The probit estimator, ∧
β , is consistent and asymptotically normal.
As mentioned above from equation [5.3] it is apparent that a probit estimation provides the sign of
the marginal effects, but not the magnitude. In order to make conclusions about the variable of
interest, that also include the size of the coefficient, marginal effects can be calculated after running
the probit estimation. Marginal effects evaluated at the mean, X
( Xβ
) j
g β
are computed by evaluating the marginal effect at the mean value of the explanatory variables. The
marginal effects of dummy variables are not evaluated at the mean value, since there is no person in
the dataset with the average value of a dummy that only can take on the values 0 and 1. In this case
the marginal effect is calculated by changing the dummy variable from 0 to 1 and evaluating the
effect this has on the dependent variable.
5.2 Testing for exogeneity
As mentioned throughout the thesis, there is a risk that some of the explanatory variables are
endogenous. Particularly since the aim is to estimate the effect of antidepressants, but there is no
63 <strong>An</strong>other special case is the logit model, which ariseses when U has a standard logistic distribution. In practice probit
and logit models are not very different from each other.
i
46