Palgrave Handbook of Econometrics: Applied Econometrics

David T. Jacho-Chávez and Pravin K. Trivedi 793

E[y−m 1 (xc 1 )−m 2 (xc 2 )−m 4 (xc 4 )|xc 3 ]=m 3 (xc 3 ), and

E[y−m 1 (xc 1 )−m 2 (xc 2 )−m 3 (xc 3 )|xc 4 ]=m 4 (xc 4 ),

imply the backfitting algorithm 15.4.1.3.1.

Algorithm 15.4.1.3.1 Standard backfitting algorithm – implementation

1. Select initial estimateŝm[ 10 ](xc 1 ),...,̂m[ 40 ](xc 4 ), saŷm[l^0 ]

(

xcl

)

=0 for alll=1,...,4.

2. For steps=1, 2,..., obtain:

m 1 [s](xc 1 )=̂E[yi−m̂[ 2 s−^1 ](x 2 c)−m̂[ 3 s−^1 ](xc 3 )−m̂[ 4 s−^1 ](xc 4 )|xc1;i=xc 1 ],

..

.=

..

.

m 4 [s](xc 4 )=̂E[yi−m̂[ 1 s−^1 ](x 1 c)−m̂[ 2 s−^1 ](xc 2 )−m̂[ 3 s−^1 ](xc 3 )|xc4;i=xc 4 ],

where, for a random variableai,̂E[ai|xcl;i=xcl]is the univariate kernel estimator

ofE[ai|xcl;i=xcl]using bandwidthhl.

3. Continue iterations in step 2. until a pre-specified convergence criterion is
   reached.

Suppose it takesditerations for the above algorithm to converge, then the rate

of convergence of

∑ 4

l= 1 m̂

[d]

l

(

xcl

)

is the same as if the regression model were a

function of a single continuous variable instead.
We use a cross-section of 872 observations from Switzerland (see Gerfin, 1996)
in order to illustrate the methodology. Whether or not a woman participates in
the labor market,E[y= 1 |x]≡Pr[x], is modeled as:

log

{

Pr[x]

1 −Pr[x]

}

=

∑^4

l= 1

ml

(

xl

)

,

wherex 1 is the log of non-labor income (LNNLINC),x 2 is age in years divided by
10 (AGE),x 3 is the number of years of formal education (EDUC), andx 4 is the
number of children (NC). Figure 15.3 illustrates the results. Each panel shows the
regressors’ individual effects on their entire observed support. Both LNNLINC and
AGE certainly have quadratic effects, while the estimated effects of EDUC and NC
might suffer from boundary effects.
Marginal integration is an alternative to backfitting (see, e.g., Linton and Nielsen,
1995; Linton and Härdle, 1996).

15.4.2 Semiparametric estimation

Another way to alleviate the curse of dimensionality is to finitely parameterize
certain aspects of the joint distribution ofyandx, e.g., its mean, while allowing
others to remain unknown, e.g., the conditional variance var(y|x). The object of