Consider the model y = x′β0 + f*(z) + ε, where ε [d over =] N(0, σ02). We calculate the smallest asymptotic variance that n1/2 consistent regular (n1/2CR) estimators of β0 can have when the only information we possess about f* is that it has a certain shape. We focus on three particular cases: (i) when f* is homogeneous of degree r, (ii) when f* is concave, (iii) when f* is decreasing. Our results show that in the class of all n1/2CR estimators of β0, homogeneity of f* may lead to substantial asymptotic efficiency gains in estimating β0. In contrast, at least asymptotically, concavity and monotonicity of f* do not help in estimating β0 more efficiently, at least for n1/2CR estimators of β0.
[1]
G. Chamberlain.
Asymptotic efficiency in semi-parametric models with censoring
,
1986
.
[2]
Werner Krabs,et al.
Optimization and approximation
,
1979
.
[3]
Rosa L. Matzkin.
Restrictions of economic theory in nonparametric methods
,
1994
.
[4]
Gary Chamberlain,et al.
Efficiency Bounds for Semiparametric Regression
,
1992
.
[5]
W. Wong,et al.
Profile Likelihood and Conditionally Parametric Models
,
1992
.
[6]
C. Stein.
Efficient Nonparametric Testing and Estimation
,
1956
.
[7]
W. Newey,et al.
Semiparametric Efficiency Bounds
,
1990
.