Palgrave Handbook of Econometrics: Applied Econometrics

794 Computational Considerations in Microeconometrics

9.5 10.0 11.0 12.0

–3

–2

–1

0

LNNLINC

23456

–3

–2

–1

0

1

AGE

510 1520

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

EDUC

0123456

–0.5

0.0

0.5

1.0

NC

Figure 15.3 Generalized additive model: labor force participation

interest then becomes a finite dimensional parameter, whose estimator is numer-
ically calculated in multiple stages with fully nonparametric calculations in the
first stages. This is known as semiparametric estimation, and its use is becoming
increasingly popular with the advent of faster and cheaper computing. Although
we only discuss a few of these estimators, a more complete exposition can be found
in, e.g., Horowitz (1998), Pagan and Ullah (1999), and Li and Racine (2007).

15.4.2.1 Example: efficient estimation with heteroskedasticity of unknown form

The heteroskedastic linear model specifiesE[y|x]=x$βand var(y|x)=σ^2 (x),

where the variance functionσ^2 (·)is left unspecified. Robinson (1987) proposed
a semiparametric feasible generalized least squares (GLS) estimator̂βthat requires

the nonparametric estimation ofσi^2 ≡σ^2

(

xi

)

in:

̂β=(∑ni= 1 xix$îσ−^2

i )

− (^1) (∑n
i= 1 xiyîσ
− 2
i ), (15.15)
byk-nearest neighbor methods. In particular, letRxs≡Rn(xs)denote the Euclidean
distance betweenxsand itsks-th nearest neighbor amongxs;i, fori=1,...,nand