Palgrave Handbook of Econometrics: Applied Econometrics

David T. Jacho-Chávez and Pravin K. Trivedi 795

s=1,...,q 1 , then:

̂σi^2 =

∑n

j= 1 ̂u

2

jK(x

c

i,x

c

j;Rxc)L(x

d

i,x

d

j;λ)

∑n

j= 1 K(x

c

i,x

c

j;Rxc)L(x

d

i,x

d

j;λ)

, (15.16)

wherêu^2 jis the residual from first-stage ordinary least squares (OLS) regression of
yionxi. Robinson (1987) showed that (15.15) is adaptive whenk 1 = ··· =kq 1 ,

k(u)=( 1 / 2 )×I(|u|≤ 1 ), and(xdi,xdj)is empty in (15.11), (15.12), and (15.16). Its
asymptotic variance can be estimated as:

asy. var̂(̂β)=(

∑n

i= 1 xix

$

îσ

− 2

i )

− (^1). (15.17)
For illustration we will fit a linear hedonic pricing model of house attributes using
data collected by Ho (1995) of 92 detached homes in the Ottawa area that were sold
in 1987. The data file contains continuous variables such as sale price (SALEPRIX),
average neighborhood income (AVGINC), distance to highway (DISTHWY), lot size
(LOTAREA), square footage of usable space (USESPACE), location coordinate in the
south (SOUTH), and west (WEST). It also contains discrete variables such as indi-
cators for the presence of fire place (FIREPLAC), garage (GARAGE), luxurious bath
(LUXBATH), and the number of bedrooms (NRBED). Using SALEPRIX asyand the
remaining variables as regressorsx, algorithm 15.4.2.1.1 provides implementation
details.
Algorithm 15.4.2.1.1 Semiparametric feasible GLS – implementation

1. Regressyonxby simple OLS and save the fitted squared residuals,{̂u^2 i}ni= 1.


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

CV[k,λ]=

∑n

i= 1

[

̂u^2 i−̂σ−^2 i

] 2

,

wherêσ−^2 iequals (15.16) after replacing

∑n

j= 1 by

∑n

j=1;j=i.

3. Usingkandλfound in the previous step, calculatêσi^2 in (15.16) for eachi=
   1,...,n.


4. Calculate (15.15) and (15.17).

The results are presented in Table 15.2. The first column shows the results
from running simple OLS with corrected standard errors. The second column
presents results using algorithm 15.4.2.1.1. Although there is not much differ-
ence between parameter point estimates, the estimated standard errors have been
reduced dramatically.