Palgrave Handbook of Econometrics: Applied Econometrics

David T. Jacho-Chávez and Pravin K. Trivedi 797

wherê# = n−^1

∑n

i= 1 (zi−̂E[zi|xi])(zi−̂E[zi|xi])

$, and̂ = n− 1 ∑n

i= 1 ̂u

2
i(zi−
̂E[zi|xi])(zi−̂E[zi|xi])$, witĥui=yi−z$îβ−̂g(xi), and:

̂g

(

xi

)

=

∑n

j= 1 (yi−z

$

îβ)K(x

c

i,x

c

j;h)L(x

d

i,x

d

j;λ)

∑n

j= 1 L(x

d

i,x

d

j;λ)

. (15.22)

Algorithm 15.4.2.2.1 makes precise the necessary steps for the above calculations.

Algorithm 15.4.2.2.1 Partially linear model – implementation

1. For eachl= 1,...,p, select functionsk(·)andl(·)in (15.11) and (15.12),
   and choose vectors of smoothing parametershandλby leave-one-out cross-
   validation, i.e., select[h 1 ,...,hq 1 ,λ 1 ,...,λq 2 ]to minimize:

CV[h,λ]=

∑n

i= 1

[

yi−̂E−i[yi|xi]

] 2

CV[h,λ]=

∑n

i= 1

[

zl;i−̂E−i[zl;i|xi]

] 2

,

wherêE−iequals (15.19) and (15.20) after replacing

∑n

j= 1 by

∑n

j=1;j=i.

2. Using the bandwidths found in the previous step, calculate (15.19) and (15.20)
   for each data pointi=1,...,n.


3. Calculate (15.18) and (15.21).


4. Select functionsk(·)andl(·)in (15.11) and (15.12), and choose vectors of
   smoothing parametershandλby leave-one-out cross-validation, i.e., select
   [h 1 ,...,hq 1 ,λ 1 ,...,λq 2 ]to minimize:

CV[h,λ]=

∑n

i= 1

[

yi−z$iβ̂−̂E−i[yi−z$îβ|xi]

] 2

,

wherêE−iequals (15.19) after replacingyibyyi−z$îβand

∑n

j= 1 by

∑n

j=1;j=i.

5. Calculate (15.22) using the bandwidths found in the previous step for each
   i=1,...,n.

Since house location (WEST and SOUTH) has no natural parametric effect in

housing prices, we proceed to include a two-dimensional nonparametric effect,

g(·), in the hedonic pricing model of housing attributes in section 15.4.2.2. The

results are presented in column 3 of Table 15.2.

15.4.2.3 Example: binary choice model

The binary choice model specifies the following relationship:

y=I(x$β+u> 0 ), (15.23)

whereIis the indicator function, andurepresents unobserved characteristics with

continuous symmetric distributionF(·)which is assumed independent ofx. It then