Abstract
Consider the model y ϭ x ' b 0 ϩ f *~z ! ϩ «, where « ϭ d N~0, s 0 2 !+ We calculate the smallest asymptotic variance that n 102 consistent regular~n 102 CR! estimators of b 0 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 n 102 CR estimators of b 0 , homogeneity of f * may lead to substantial asymptotic efficiency gains in estimating b 0 + In contrast, at least asymptotically, concavity and monotonicity of f * do not help in estimating b 0 more efficiently, at least for n 102 CR estimators of b 0 +
Abstract
The objective of this paper is to determine the minimum asymptotic variance, hereafter called the efficiency bound, for estimating finite dimensional parameters in an important class of econometric models+ These models, referred to as partially linear models~henceforth PLM's!, take the form y ϭ x ' b 0 ϩ f *~z ! ϩ «+ Here y is the response variable,~x, z! the covariates, b 0 a finite dimensional parameter of interest, and « an unobserved "error" term+ The functional form of f * , although unknown, is known to possess certain shape properties such as homogeneity, concavity, or monotonicity+ These properties are often the restrictions that economic theory imposes upon unknown functional forms+ We let F denote the set of certain shape restricted functions+ Semiparametric estimation of b 0 , when F is just a set of functions satisfying some smoothness properties, has been studied extensively+ However, when dealing with economic data, the functions in F often have to satisfy certain shape restrictions besides smoothness+ Although these restrictions have been extensively studied in economic theory, they have received only limited use in econometric practice notwithstanding their tremendous usefulness+ As pointed out in Matzkin~1994, p+ 2525!, I thank Joel Horowitz and two anonymous referees for comments that greatly improved this paper+ I am also indebted to Rosa Matzkin and Tom
INTRODUCTION
The objective of this paper is to determine the minimum asymptotic variance, hereafter called the efficiency bound, for estimating finite dimensional parameters in an important class of econometric models+ These models, referred to as partially linear models~henceforth PLM's!, take the form y ϭ x ' b 0 ϩ f *~z ! ϩ «+ Here y is the response variable,~x, z! the covariates, b 0 a finite dimensional parameter of interest, and « an unobserved "error" term+ The functional form of f * , although unknown, is known to possess certain shape properties such as homogeneity, concavity, or monotonicity+ These properties are often the restrictions that economic theory imposes upon unknown functional forms+ We let F denote the set of certain shape restricted functions+ Semiparametric estimation of b 0 , when F is just a set of functions satisfying some smoothness properties, has been studied extensively+ However, when dealing with economic data, the functions in F often have to satisfy certain shape restrictions besides smoothness+ Although these restrictions have been extensively studied in economic theory, they have received only limited use in econometric practice notwithstanding their tremendous usefulness+ As pointed out in Matzkin~1994, p+ 2525!, imposing shape restrictions may lead to a reduction in the variance of parameter estimators+ Omitting shape restrictions, when economic theory demands otherwise, would typically lead to inefficient estimation procedures, thus reducing the power of subsequent statistical analysis+ A very readable account of computing efficiency bounds in semiparametric models can be found in Newey~1990!+ However, as this paper illustrates, much of the research to date has concentrated upon developing efficiency bounds for distribution free models, i+e+, models in which the distribution of the error terms is unknown~Chamberlain, 1986; Cosslett, 1987!+ Where shape restrictions have been involved, they have been imposed on the error distribution~Newey, 1988!, rather than on the unknown function+ In fact, to the best of my knowledge, this paper is the first of its kind to develop efficiency bounds for models where the shape restrictions are imposed on the unknown functional form rather than on the distribution of the error term+ The bounds that we obtain are local in nature because we have specified the distribution of the error term to be Gaussian+ As Rilstone~1993, p+ 361! points out, this may not be as severe a restriction as one might think+ Furthermore, imposition of normality is a convenient assumption that allows us to focus on the main object of our investigation, namely, shape restrictions and their effect upon efficiency bounds+ The results in this paper should be of particular interest to practitioners in the field, because they provide some new insights into incorporating shape restrictions in estimation procedures+ The paper is organized as follows+ Section 2 lists the maintained assumptions+ Efficiency bounds for b 0 , when f * is homogeneous of degree r, are described in Section 3+ A simple example is also provided to illustrate possible efficiency gains due to homogeneity+ Section 4~resp+ Section 5! explores the case when f * is concave~resp+ decreasing!+ The form of the efficiency bounds obtained in these two cases is used as evidence to conclude that in the class of all n 102 consistent regular~henceforth n 102 CR! estimators of b 0 , concavity and monotonicity of f * do not help in estimating b 0 more efficiently in large samples, i+e+, asymptotically+ Section 6 concludes+ Section 7 provides the notation for the Appendixes and a useful definition+ A quick review of calculating efficiency bounds, along with all computational and technical details for the different cases, is relegated to the Appendixes+
Notation
Let S be a convex, compact subset of R q , C 2~S ! the set of all real valued twice continuously differentiable functions on S, and L 2~S ! the set of all square integrable~w+r+t+ the probability distribution on S! functions on S+ F h ʚ C 2~S ! is the set of all C 2~S ! functions that are also homogeneous of degree r, and F c ʚ C 2~S ! is the set of all C 2 functions that are concave on S+ In the onedimensional case q ϭ 1, S becomes a compact interval in R, C 1~S ! denotes the set of all continuously differentiable functions on S, and 
THE PARTIALLY LINEAR MODEL

Consider the regression
The following assumptions are maintained throughout the paper+ A+1! The data~x i , y i , z i ! are independent and identically distributed~i+i+d+! random vectors in
Consider the regression
2 Ͻ`, and z is drawn from a distribution with support S+ The joint probability density function~p+d+f+! of x and z exists and is denoted by
Consider the regression
As b 0 is ancillary to s 0 2 and g x, z , efficiency bounds for b 0 will remain unchanged whether s 0 2 and g x, z are known or not~Newey, 1990, p+ 109!+ Therefore, to remain focused upon the shape restrictions and to minimize algebraic details, assume w+l+o+g that s 0 2 and g x, z are known+ Assumption~A+5! ensures identification of b 0 , because b 0~w ithout any intercept term! is identified iff
EFFICIENCY BOUNDS WHEN f * IS HOMOGENEOUS
We now describe the efficiency bounds for estimating b 0 in the PLM when the only thing we know about f * is that it is homogeneous 1 of degree r, where r is known; i+e+, f * ʦ F h + Keep in mind that merely knowing the efficiency bounds is not enough+ To be useful, these bounds must be attainable; i+e+, we must be able to construct an estimator of b 0 with asymptotic variance equal to the efficiency bound+ To keep our presentation concise we have excluded the estimation part of the problem from this paper+ Following Severini and Wong~1992!, in Tripathi~1997! we construct such an estimator and also show that it attains the efficiency bounds+ Because "a nonparametric problem is at least as difficult as any of the parametric problems obtained by assuming we have enough knowledge of the unknown state of nature to restrict it to a finite dimensional set"~Stein, 1956, p+ 187!, efficiency bounds for b 0 may be obtained by finding the worst parametric submodel that is contained in the full model+ Following the details in Appendixes A and B, efficiency bounds for b 0 , when f * is homogeneous of degree r, are given by s 0 2~E SS ' ! Ϫ1 where S ϭ x Ϫ @z q r E~xz q r 6~z 1 0z q !, + + + ,~z qϪ1 0z q !!0 E~z q 2r 6~z 1 0z q !, + + + ,~z qϪ1 0z q !!#+ Note that apart from the projection-based ap-proach described in Appendix B, efficiency bounds for the homogeneity case can also be obtained by using the result of Chamberlain~1992!+ 2 As these bounds are obtained by an orthogonal projection onto a proper subspace of L 2~S !, not surprisingly, 3 there is an efficiency gain due to homogeneity of f * + But what about the magnitude of this gain? It is easy to construct examples where large gains in efficiency may be obtained by using the identifying power of homogeneity+ However, as the following example demonstrates, large gains in efficiency are possible under homogeneity, even when the parameter of interest is identified+
EFFICIENCY BOUNDS WHEN f * IS HOMOGENEOUS
,1!, and f * is linearly homogeneous+ Because f * is homogeneous of degree one, use q, r ϭ 1 in Lemma B+2~in Appendix B! to see that the efficiency bound for estimating b 0 is 6+1+ However, if homogeneity is not imposed upon f * the bound increases to 12+ Thus the loss in efficiency by not imposing homogeneity, when f * is truly homogeneous, is 96+7%+
EFFICIENCY BOUNDS WHEN f * IS CONCAVE
Following the details in Appendix C, efficiency bounds for estimating b 0 , when f * ʦ F c , are given by s 0 2~E @x Ϫ E~x6z!# @x ' Ϫ E~x ' 6z!# ! Ϫ1 + But these are the same bounds that would be obtained if the only thing we knew about f * was that it was twice continuously differentiable+ See, e+g+, Newey~1990, p+ 109!+ Hence, in the class of n 102 CR estimators of b 0 , computing efficiency bounds for b 0 when f * is concave is equivalent to computing efficiency bounds for b 0 when f * is just a C 2~S ! function+ Therefore, at least asymptotically, concavity of f * does not help us in estimating b 0 more efficiently+ Moreover, this conclusion does not change if "concavity" is replaced by "convexity+" Note, however, that as these bounds are asymptotic in nature, this result should not be interpreted as a prescription for disregarding concavity of f * when estimating b 0 + It may be quite worthwhile to impose concavity upon f * to obtain better finite sample results for estimating b 0 + Furthermore, even asymptotically, this result leaves open the possibility that we could do better in the presence of concavity if we looked at nonregular estimators of b 0 + As far as I know, these problems are still open and merit further research+
EFFICIENCY BOUNDS WHEN f * IS DECREASING
Next we look at the case when f * is decreasing+ As a result of the difficulty of working with monotonicity restrictions on f * in a multivariate setting, we confine ourselves to the one-dimensional case, i+e+, q ϭ 1+ So let f * ʦ F d + Using the details provided in Appendix D, efficiency bounds for estimating b 0 , when
EFFICIENCY BOUNDS WHEN f * IS DECREASING
+ Therefore, as in Section 4, we conclude that in the class of all n 102 CR estimators of b 0 the knowledge that f * is decreasing does not help us estimate b 0 more efficiently, at least asymptotically+ This result remains unchanged if the word decreasing is replaced by increasing+ As before, it does not mean that one should disregard monotonicity of f * while estimating b 0 + The possibility that monotonicity of f * could help in estimating b 0 more efficiently, even asymptotically, in some class of nonregular estimators remains to be investigated+
CONCLUSION
We have shown that in the class of n 102 consistent regular~n 102 CR! estimators of b 0 , concavity and monotonicity of f * , unlike homogeneity, do not help in estimating b 0 more efficiently in large samples+ Of course, these results are asymptotic in nature+ One would expect concavity, and monotonicity of f * to help in estimating b 0 more efficiently in finite samples+ To gain some intuition with regard to these asymptotic results one can visualize homogeneity as imposing equality restrictions on f * , whereas concavity and monotonicity impose inequality restrictions on the second and first derivatives, respectively, of f * + In the case of homogeneity the equality restrictions upon f * bind on the tangent space~for details, see Appendix B!, whereas in the case of concavity and monotonicity these inequality restrictions do not bind~see Appendixes C and D!+ Because lower bounds for the asymptotic variance of n 102 CR estimators of b 0 are obtained by an orthogonal projection onto the tangent space~see Appendix A!, homogeneity of f * reduces these bounds, whereas concavity and monotonicity of f * do not+ It is useful to note that regular estimators of b 0 exclude those estimators that impose concavity, or monotonicity, upon f * + The intuition behind this 4 is that such estimators would require inequality restrictions to be imposed upon estimators of the unknown functional form in the model and this would destroy their regularity+ This is best illustrated in parametric models with inequality restrictions on the nuisance parameters, where an estimator of the parameter of interest is obtained by truncating the estimator of the nuisance parameter into its parameter space+ 5 Although this procedure destroys regularity by introducing a discontinuity in the asymptotic distribution of the estimator sequence, it leads to improved estimators; i+e+, under inequality restrictions one can construct, at least in parametric models, truncated, but nonregular, estimator sequences that improve the performance of any regular estimator of the parameter of interest+ However, as van der Vaart~1989, p+ 1492! points out, this seems to be an unresolved problem for semiparametric models+ In light of these remarks, imposing regularity on estimators of b 0 , when f * is concave or monotone, may be overly restrictive+ On the other hand, it is not clear how the efficiency results of van der Vaart~1989! could be extended in the absence of a regularity requirement on estimator sequences+ Therefore, we leave for future research the issue of whether, and how, concavity and monotonicity of f * would help in asymptotic estimation of b 0 within a class of estimators distinct from the usual "n 102 consistent regular" estimators+
NOTATION FOR APPENDIXES AND A USEFUL DEFINITION
Let ¹ 2 f be the Hessian and 7 f 7 C 2 ϭ sup zʦS 6 f~z!6 ϩ sup zʦS,1ՅjՅq 6~]f~z!0]z j !6 ϩ sup zʦS,1Յi, jՅq 6~]
NOTATION FOR APPENDIXES AND A USEFUL DEFINITION
2 f~z!0]z i ]z j !6 be the C 2 norm of f+ When q ϭ 1, the C 1 norm of f is given by 7 f 7 C 1 ϭ sup zʦS 6 f~z!6 ϩ sup zʦS 6 f '~z !6+ The expressions F h and F c are closed~w+r+t+ 7{7 C 2 ! convex cones in C 2~S !, whereas F d is a closed~w+r+t+ 7{7 C 1 ! convex cone in C 1~S !+ If E is a subset of a linear space X, lin E denotes the linear subspace generated by E; i+e+, lin E is the set of all finite linear combinations of elements of E+ O E is the L 2 -closure of E+ DEFINITION 1+ Let M be a subset of a Banach space~X,7{7 X !+ A vector _x ʦ X is said to be tangent to M at the point x 0 ʦ M if there exist a t 0 Ͼ 0 and a mapping t ‫ۋ‬ r~t! of the interval~0, t 0 ! into X such that x 0 ϩ t _x ϩ r~t! ʦ M for all t ʦ~0, t 0 !, and 7r~t!7 X 0t r 0 as t f 0+ The set of vectors tangent to M at the point x 0 ʦ M is denoted by T~M, x 0 ! and is a closed~in the topology generated by 7{7 X ! nonempty cone in X+ By a "cone" we mean that if _x ʦ T~M, x 0 !, then l _x ʦ T~M, x 0 ! for all l Ͼ 0+ The expression T~M, x 0 !~resp+ lin T~M, x 0 !! is referred to as the tangent cone~resp+ tangent space! to M at x 0 + For more on this see Krabs~1979!+ NOTES 1+ A function f : S r R is said to be homogeneous of degree r ʦ R on S, iff f~lz! ϭ l r f~z! for all l Ͼ 0 such that lz ʦ S+ 2+ We thank a referee for pointing this out+ 3+ Because efficiency bounds for b 0 , when homogeneity is not imposed on f * , are obtained by an orthogonal projection onto the space L 2~S ! itself+ 4+ We thank one of the referees for this+ 5+ See, e+g+, Example 2+5+1 in Tripathi~1997!, which imposes monotonicity in a simple linear regression model+ 6+ Because we can always find a "one-dimensional subproblem" that has the same asymptotic variance as any "multidimensional subproblem"~Stein, 1956, p+ 188!, it suffices to look at "onedimensional subproblems+"
NOTATION FOR APPENDIXES AND A USEFUL DEFINITION
7+ To see this let $g n % be a sequence in , where S ϭ x Ϫ d * and the nonsingularity of E~SS ' ! follows from Assumption~A+5!+ Therefore, to calculate these bounds, all we need to do is to determine d * +
APPENDIX B: BOUNDS UNDER HOMOGENEITY-TECHNICAL DETAILS
For the homogeneity case we make an additional assumption about S, the support of z+ Besides being used in the proof of Lemma B+1, the following assumption also ensures that the components of d * obtained in Lemma B+2 are elements of L 2~S !+ Assumption B.1. S is such that for any z ʦ S, z q~t he last component of z! is bounded away from zero; i+e+, ∃m Ͼ 0 such that 6z q 6 Ն m+ To show the reverse direction, pick anyf ʦ T~F c , f * !+ Thenf ʦ C 2~S ! because F c ʚ C 2~S !, and so T~F c , f * ! ʚ C 2~S ! always holds+ However, we can strengthen this result slightly for the one-dimensional case q ϭ 1+ To see this, use Definition 1 to conclude that iff ʦ T~F c , f * !, then ∃t 0 Ͼ 0, and a map t ‫ۋ‬ r t of~0, t 0 ! into C 2~S !, such that f t~z ! ϭ f *~z ! ϩ tfz! ϩ r t~z ! ʦ F c for all t ʦ~0, t 0 !, and r t 0t
APPENDIX B: BOUNDS UNDER HOMOGENEITY-TECHNICAL DETAILS
C 2

{'paper_id': '40358905', 'title': 'LOCAL SEMIPARAMETRIC EFFICIENCY BOUNDS UNDER SHAPE RESTRICTIONS', 's2_url': 'https://api.semanticscholar.org/CorpusID:40358905', 'source': 'S2ORC'}