Palgrave Handbook of Econometrics: Applied Econometrics

796 Computational Considerations in Microeconometrics

Table 15.2 Hedonic prices of housing attributes: estimated models

(1) OLS (2) Semip. FGLS (3) Partially linear

Robust Efficient Robust

Coef. std. error Coef. std. error Coef. std. error

Intercept 73.978 18.367 77.821 5.821 – –

FIREPLAC 11.795 5.916 9.634 1.759 9.548 3.792

GARAGE 11.838 4.439 10.870 1.821 3.826 3.849

LUXBATH 60.736 10.351 58.821 4.205 44.329 10.820

AVGINC 0.477 0.199 0.502 0.059 −0.414 1.037

DISTHWY −15.277 5.596 −12.954 1.479 -23.079 32.358

LOTAREA 3.243 1.937 2.542 0.625 3.978 1.928

NRBED 6.586 4.484 4.276 1.439 1.668 3.222

USESPACE 21.128 12.070 27.125 4.043 17.767 7.059

WEST 3.206 1.884 2.931 0.702 – –

SOUTH −7.527 1.782 −7.254 0.527 – –

s 21.95 1.006 14.603

AdjustedR^2 0.569 0.947 0.847

Comp. time 0.64 seconds 196.62 seconds 127.17 seconds

Notes:srepresents the squared root of the estimated variance of the regression. FGLS = feasible

generalized least squares.

15.4.2.2 Example: partially linear model

The partially linear model specifiesE[y|z,x]=z$β+g(x), whereg(x)is left
unspecified, andz=[z 1 ,...,zp]∈Rp. In particular, assuming that, foru =

y−z$β−g(x),E[u|z,x]=0, and var(u|z,x)=σ^2 (z,x), and after observing that
E[y|x]=E[z|x]$β+g(x), Robinson (1988) proposed an estimator̂βthat requires
the nonparametric estimation ofE[yi|xi], andE[zi|xi],i=1,...,n, in:

̂β=(∑ni= 1 (zi−̂E[zi|xi])(zi−̂E[zi|xi])$)−^1 (∑ni= 1 (zi−̂E[zi|xi])(yi−̂E[yi|xi])),

(15.18)

by the Nadaraya–Watson estimator, i.e., for vectors of smoothing parametersh
andλ:

̂E[yi|xi]=

∑n

j= 1 yiK(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)

, (15.19)

̂E[zl;i|xi]=

∑n

j= 1 zl;iK(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)

, forl=1,...,p. (15.20)

Its asymptotic variance can be estimated consistently by:

asy. var̂(̂β)=n−^1 ̂#−^1 ̂#̂−^1 , (15.21)