800 Computational Considerations in Microeconometrics
Ichimura’s estimator
The semiparametric least squares (SLS) estimator of Ichimura (1993) numerically
minimizes:
∑n
i= 1 [yi−̂F−i(x
$
iβ)](^2) , (15.28)
with respect to β, where, as before, ̂F−i(·) is the kernel estimator for the
unknown link functionF(·). The resulting estimator,̂β, is asymptotically normally
distributed, and its asymptotic variance can be consistently estimated by:
asy. var̂(̂β)=n−^1 ̂−^1 ̂̂−^1 , (15.29)
where:
̂=n−^1 ∑ni= 1 ̃xi ̃x$i[̂F−(^1 i)(x$îβ)]^2 , and (15.30)
̂=n−^1 ∑ni= 1 ̃xi ̃x$i[̂F(^1 )
−i(x
$
îβ)]
(^2) [y
i−̂F−i(x
$
iβ)]
(^2). (15.31)
Column 3 in Table 15.3 presents the results of applying algorithm 15.4.2.3.2 to our
example.
Algorithm 15.4.2.3.2 Ichimura (1993) – implementation
- Select functionk(·)in (15.26), and numerically find jointly the bandwidthh
and vector of coefficientsβthat minimizes the semiparametric least squares
objective function (15.28). - Using the bandwidth and vector of coefficients found in the previous step,
calculatêF−(^1 i)(·)and̂F−i(·)at each data pointi=1,...,n. - Calculate (15.29), (15.30) and (15.31).
The results differ mainly in the magnitude of the effects of each regressor as
well as in their precision. Both semiparametric estimators provide estimates of the
participation probabilities (Part. prob.) closer to the actual 1,400 women in the
sample that are observed to work for wages. The parametric probit specification
overestimates this quantity.
It should be emphasized that, unlike Klein and Spady’s estimator, Ichimura’s is
applicable to a wider range of problems where the responseyiis not only binary,
but takes on different discrete or continuous values.
15.4.2.4 Further considerations
The numerical stability of likelihood cross-validated bandwidths in step 1 in
algorithms 15.4.1.1.1 and 15.4.1.2.1 might be affected by outliers. However,
the suggested likelihood cross-validated method may work well for thin-tailed
distributions.
In the semiparametric models discussed above, the first-order asymptotic dis-
tribution of the normalized and centered estimators does not depend on the