Palgrave Handbook of Econometrics: Applied Econometrics

792 Computational Considerations in Microeconometrics

cost

2

4

6

8

time

20

40

60

0.0010.000

0.002

0.003

0.004

0.005

f^(cost, time|car)

1 cost

2

3

time

10

20

30

40

(^6050)
0.005
0.010
0.015
f^(cost, time|carpool)
1.5 cost
2.0
2.5
time
20
40
60
0.00
0.01
0.02
0.03
f^(cost, time|bus)
cost
1.5
2.0
2.5
3.0
time
20
40
60
0.005
0.010
0.015
0.020
0.025
^f (cost, time|rail)
Figure 15.2 Multivariate conditional p.d.f.
15.4.1.3 Example: additive models
Kernels, as well as many other nonparametric methods, do not perform well whenq
is large. The sparseness of data in this setting inflates the variance of the estimates,
and the numerical accuracy of estimates rapidly decreases with the number of
regressors. This problem is sometimes referred to as the “curse of dimensionality.”
Computationally, the curse of dimensionality means that for kernel methods, asq
becomes large relative to a fixed sample sizen, division by 0 becomes more frequent
in the calculation of̂m(x).
To overcome these difficulties, Stone (1985) proposed additive models, i.e.:
m(x)=m 1
(
xc 1
)
+···+mq 1
(
xcq 1
)
,
where thexcls are all univariate continuous variables, and theml(·)are unknown
functions forl = 1,...,q 1. A benefit of an additive model is that estimates
of the individual terms explain how the dependent variable changes with the
corresponding independent variables. Suppose q 1 = 4, then the following
identities:
E[y−m 2 (xc 2 )−m 3 (xc 3 )−m 4 (xc 4 )|xc 1 ]=m 1 (xc 1 ),
E[y−m 1 (xc 1 )−m 3 (xc 3 )−m 4 (xc 4 )|xc 2 ]=m 2 (xc 2 ),