Palgrave Handbook of Econometrics: Applied Econometrics

David T. Jacho-Chávez and Pravin K. Trivedi 799

identify and consistently estimate ̃βwhenF(·)is only known to be a non-constant

smooth function on the support ofx$β; and varying the values ofxddoes not
divide the support ofx$βinto disjoint subsets (see Horowitz, 1998).
Note that, for givenβ, a consistent nonparametric estimator of (15.24) is:

̂F(x$iβ)=

∑n

j= 1 yik((x

$

iβ−x

$

jβ)/h)

∑n

j= 1 k((x

$

iβ−x

$

jβ)/h)

. (15.26)

Klein and Spady’s estimator
Klein and Spady (1993) proposed estimatingβby maximizing (15.25) after replac-
ingF(x$iβ)bŷF−i(x$iβ), wherêF−i(x$iβ)is likêF(x$iβ)but with
∑n
j= 1 replaced
by

∑n
j=1;j=iin (15.26). Computationally, the maximization must be performed
numerically by solving the first order conditions of the problem. Under regularity
conditions, Klein and Spady showed that the resulting estimator̂βhas an asymp-
totic normal distribution. Furthermore, they showed that the asymptotic variance
of their estimator attains the semiparametric efficiency bound, and that it can be
consistently estimated by:

asy. var̂(̂β)=(

∑n

i= 1 ̃xi ̃x

$

i[̂F

( 1 )

−i(x

$

îβ)]

(^2) /[̂F
−i(x
$
îβ)(^1 −̂F−i(x
$
îβ))])
− (^1) , (15.27)
wherêF−(^1 i)(·)is the first derivative of̂F−i(·). Algorithm 15.4.2.3.1 describes the
necessary steps.
Algorithm 15.4.2.3.1 Klein and Spady (1993) – implementation

1. Select functionk(·)in (15.26), and numerically find jointly the bandwidthh
   and vector of coefficientsβthat minimizes the leave-one-out estimated log-
   likelihood:
   ∑n
   i= 1 {yilog[̂F−i(x

$

iβ)]+

(

1 −yi

)

log[ 1 −̂F−i(x$iβ)]},

wherêF−i(·)equals (15.26) after replacing

∑n

j= 1 by

∑n

j=1;j=i.

2. Using the bandwidth and vector of coefficients found in the previous step,

calculatêF−(^1 i)(·)and̂F−i(·)at each data pointi=1,...,n.

3. Calculate (15.27).

Column 2 in Table 15.3 shows results from applying the above methodology to
the modeling of female labor force participation decisions. Although signs coin-
cide with those of the fully parametric model (column 1), their magnitudes and
standard errors are remarkably different. Martins (2001) pointed out that this
considerable efficiency gain may occur because the semiparametric model is suf-
ficiently perturbed from the usual probit specification for women with low index
values.