Palgrave Handbook of Econometrics: Applied Econometrics

798 Computational Considerations in Microeconometrics

follows that:

Pr[y= 1 |x]=E[y|x]=F(x$β), (15.24)

wherex$βis known as asingle index. In parametric probit and logit models, the
unknown parameterβin (15.23) is obtained by numerically maximizing the log-
likelihood function:
∑n
i= 1 {yilog[F(x

$

iβ)]+

(

1 −yi

)

log[ 1 −F(x$iβ)]}, (15.25)

whereF(·)is assumed to be a normal or logistic distribution function respectively.
Column 1 in Table 15.3 shows results from fitting a probit model to female labor
participation decisions in Portugal, using 2,339 observations taken from Martins
(2001). The regressors include variables such as the number of children younger
than 18 living in the family (CHILD), the number of children younger than three
years of age (YCHILD), the number of years of formal schooling (EDU), the log
of the husband’s monthly wages (LNHW), women’s age divided by 10 and 100
respectively (AGE and AGE2). The model does not include a constant and the
coefficient multiplying AGE has been normalized to be equal to one for comparison
purposes.
Notice that all regressors, except the number of years of formal schooling, have
a negative effect on the probability that a woman works for wages.
IfF(·)is unknown, thenF(x$β)is observationally equivalent toF

(

x$β−a

)

,

and ̃F

(

x$(β/c)

)

, withF(u)=F(u+a)and ̃F(u)=F(c·u)respectively∀a,c∈R.

Under these circumstances, restrictions are needed to identify the unknown vector
of parametersβ. For example, the so called location-scale normalization sets any
intercept inx$βequal to 0, and the coefficient of a continuous regressor (say the
first regressorx 1 ) equal to 1.

Let us define ̃xc =[xc 2 ,...,xcq 1 ], ̃x=[ ̃xc,xd],x=[x 1 , ̃x], ̃β =[β 2 ,...,βq 1 ,

βq 1 + 1 ,...,βq 2 ]$, andβ≡(1, ̃β$). Semiparametric methods have been shown to

Table 15.3 Female labor participation: estimated models

(1) Probit (2) Klein–Spady (1993) (3) Ichimura (1993)

Coef. Std. error Coef. Std. error Coef. Std. error

CHILD −0.079 0.071 −0.012 0.029 −0.145 0.123

YCHILD −0.129 0.026 −0.083 0.012 −0.096 0.076

EDU 0.144 0.009 0.076 0.003 0.134 0.013

LNHW −0.181 0.009 −0.043 0.004 −0.138 0.017

AGE2 −0.148 0.004 −0.145 0.002 −0.158 0.014

AGE 1 – 1 – 1 –

Part. prob. 1408.00 1400.11 1399.17

Comp. time 0.421 seconds 491.072 seconds 2462.740 seconds

Note: Part. prob. = participation probability.