The Influence Function of Semiparametric Estimators∗
Hidehiko Ichimura
University of Tokyo

Whitney K. Newey
MIT

July 2016

Abstract
Often semiparametric estimators are asymptotically equivalent to a sample average. The
object being averaged is referred to as the influence function. The influence function is useful in formulating primitive regularity conditions for asymptotic normality, in efficiency
comparions, for bias reduction, and for analyzing robustness. We show that the influencefunction of a semiparametric estimator can be calculated as the limit of the Gateaux derivative of a parameter with respect to a smooth deviation as the deviation approaches a pointmass. We also consider high level and primitive regularity conditions for validity of theinfluence function calculation. The conditions involve Frechet differentiability, nonparametric convergence rates, stochastic equicontinuity, and small bias conditions. We apply theseresults to examples.

JEL Classification: C14, C24, H31, H34, J22
Keywords: Influence function, semiparametric estimation, bias correction.

∗

The NSF and JSPS provided partial financial support. We are grateful for comments by V. Chernozhukov,
K. Kato, U. Mueller, J. Porter and participants at seminars at UC Berkeley, NYU, University of Kansas, and
Yale.

1

Introduction

There are many economic parameters that depend on nonparametric first steps. Examples
include games, dynamic discrete choice, average consumer surplus, and treatment effects. Often
those estimators are asymptotically equivalent to a sample average. The thing being averaged isreferred to as the influence function. The influence function is useful for a number of purposes.
It can be used to construct estimators with improved properties, Chernozhukov, Escanciano,
Ichimura, and Newey (2016). Its variance is the asymptotic variance of the estimator and so it
can be used for asymptotic variance estimation and asymptotic efficiency comparisons. Also,
the form of remainder terms follow from the form of the influence function so knowing theinfluence function is a good starting point in formulating regularity conditions. Furthermore,
the influence function approximately gives the influence of a single observation on the estimator,and so can be used for robustness comparisons. Indeed this interpretation is where the influencefunction gets its name in the robust estimation literature, see Hampel (1974).
BETTER We show how the influence function can be calculated as the limit of a derivativeof the object the estimator converges to. The derivative is taken with respect to a weight on a
general alternative to the true CDF. The limit is taken as the alternative approaches the CDF
of constant. We impose restrictions on the alternative so that the object being differentiated iswell defined. This calculation is similar to that of FIX Von Mises (1947) and Hampel (1974),
of a derivative with respect to a CDF of a constant, but we specify the alternative so that
it is smooth and also allow it to satisfy other restrictions. As a result the calculation given
here applies generally to semiparametric estimator including those where the first step is aconditional expectation or density.
We also consider regularity conditions for validity of the influence function calculation. The
conditions involve Frechet differentiability as well as convergence rates for nonparametric estimators. They also involve stochastic equicontinuity and small bias conditions. When estimators
depend on nonparametric objects like conditional expectations and pdf’s, the Frechet differentiability condition is generally satisfied for intuitive norms, e.g. as is well known from Goldsteinand Messer (1992). The situation is different for functionals of the empirical distribution where
Frechet differentiability is only known to hold under special norms, Dudley (1994). The asymptotic theory here also differs from functionals of the empirical distribution in other ways as will
be discussed below.
Newey (1994) previously showed that the influence function of a semiparametric estimatorcan be obtained by solving a functional equation involving pathwise derivatives and scoresof parametric models. That approach has proven useful in many settings but does require the[1]solution to a functional equation. The approach of this paper is an explicit calculation that does
not require finding the solution to a functional equation. Here we simply calculate a derivativeand find its limit. This calculation is accomplished by specifying a parametric model, i.e. apath, in the right way to obtain the influence function.
Regularity conditions for functionals of nonparametric estimators involving Frechet differentiability have previously been formulated by Ait-Sahalia (1991), Goldstein and Messer (1992),
Newey and McFadden (1994), Newey (1994), Chen and Shen (??), Chen, Linton, and Keilegom(2003), and Ichimura and Lee (2010), among others. Newey (1994) gave stochastic equicontinuity and small bias conditions for functionals of series estimators. In this paper we update
those using Belloni, Chernozhukov, Chetverikov, and Kato (2015). Bickel and Ritov (2003)
formulated similar conditions for kernel estimators. Andrews (2004) gave stochastic equicontinuity conditions for the more general setting of GMM estimators that depend on nonparametricestimators.
In Section 2 we describe the estimators we consider. Section 3 presents the method for
calculating the influence function. In Section 4 we outline some conditions for validity of theinfluence function calculation. Section 5 gives primitive conditions for linear functionals of
kernel density and series regression estimators. Section 6 outlines additional conditions forsemiparametric GMM estimators. Section 7 concludes.

2

Semiparametric Estimators

This paper is about estimators where parameters of interest depend on a first step nonparametric estimator. We refer to these estimators as semiparametric. We could also refer to themestimators where nonparametric first step estimators are “plugged in.” This terminology seems
awkward though, so we simply refer to them as semiparametric estimators. We denote suchan estimator by β̂, which is a function of the data z1 , ..., zn where n is the number of observations. Throughout the paper we will assume that the data observations zi are i.i.d. We denotethe object that β̂ estimates as β 0 , the subscript referring to the parameter value under the
distribution that generated the data.
We adopt a general framework where the estimator of the parameter of interest is a generalized method of moments estimator depending on a nonparametric first step. To describe thetype of estimator we consider let m(z, β, γ) denote a vector of functions of the data observationz, parameters of interest β ∈ Rq , and a function γ that may be vector valued. Here γ represents some possible value of a nonparametric estimator. A GMM estimator can be based on a[2]moment condition where β 0 is the unique parameter vector satisfying
E[m(zi , β 0 , γ 0 )] = 0,

(2.1)

and γ 0 is the true γ. Here it is assumed that this moment condition identifies β. Let γ̂ denote
some nonparametric estimator of γ 0 . Plugging in γ̂ to obtain m(zi , β, γ̂) and averaging over zi
Pgives the estimated sample moments m̂(β) = ni=1 m(zi , β, γ̂)/n. For Ŵ a positive semi-definite
weighting matrix a semiparametric GMM estimator isβ̂ = arg min m̂(β)T Ŵ m̂(β).
β∈B

We note that this class of estimators includes an explicit functional µ(F ) of the distribution Fof a single observations, where m(z, β, γ) = µ(F ) − β and F = γ. Many other estimators arealso included as special cases.
Examples can help illustrate the results. We consider one example here and more below.
The first example is an estimator of a bound on average surplus of a price change when there
are bounds on income effects, as in Hausman and Newey (2016a,b). Let y denote quantity
consumed of some good, x = (x1 , x2 )0 where x1 is price, x2 is income, w2 (x2 ) be a weightfunction for income (such as an indicator for some interval), and γ 1 (x) a possible conditional
expectation function E[yi |xi = x]. We assume that B is a uniform bound on the derivative of
demand with respect to income, i.e. the income effect. The object of interest is a bound on theweighted average over income of equivalent variation for a price change from x̆1 to x̄1 , given byZ x̄1Z
β 0 = E[w2 (x2i )γ 10 (u, x2i )e−B(u−x̆1 ) du] = E[w2 (x2i ) w1 (u)γ 10 (u, x2i )du],x̆1

where w1 (x1 ) = 1(x̆1 ≤ x1 ≤ x̄1 )e−B(x1 −x̆1 ) . If B is an upper (lower) bound on income effects
then β 0 is a lower (upper) bound on average equivalent variation over income and individual
heterogenetiy of a price change from x̆1 to x̄1 . This object is identified from the semiparametricmoment function.
Zm(z, β, γ 1 ) = w2 (x2 )w1 (u)γ 1 (u, x2 )du − β.

We will consider additional examples below.
The results of this paper apply generally to asymptotically linear estimators. An asymptotically linear estimator is one satisfying√n(β̂ − β 0 ) =nX√ψ(zi )/ n + op (1), E[ψ(zi )] = 0, E[ψ(zi )T ψ(zi )] < ∞.

(2.2)

i=1

The function ψ(z) is referred to as the influence function, following terminology of Hampel(1974). It gives the influence of a single observation in the leading term of the expansion in[3]

equation (2.2). It also quantifies the effect of a small change in the distribution on the limit ofβ̂ as we further explain below.
A semiparametric GMM estimator will be asymptotically linear under regularity conditions
that are summarized below. Let
M=∂E[m(zi , β, γ 0 )], W = plim(Ŵ ).
∂β
β=β 0

A standard expansion argument along with well understood properties of semiparametric estimators leads to an influence function of the form
ψ(z) = −(M T W M )−1 M T W [m(z, β 0 , γ 0 ) + φ(z)],(2.3)

where φ(z) is an adjustment term for the estimator γ̂ of γ 0 , as discussed in Newey (1994).
Here m(z, β 0 , γ 0 ) + φ(z) will be the influence function of m̂(β 0 ). This formula for the influencefunction is valid under weak regularity conditions, that allow for m(z, β, γ 0 ) to not be smoothin β, e.g. as in Chen, Linton, et al. (??).

3

Calculating the Influence Function

In this Section we provide a method for calculating the influence function. The key object onwhich the influence function depends is the limit of the estimator when zi has a CDF F thatis unrestricted except for regularity conditions. We denote this object by β(F ). One can thinkof β(F ) as the object that is estimated by β̂ when misspecification is allowed. The idea is thatevery estimator converges to something under some regularity conditions. The function β(F )is that something. It describes how the limit of the estimator varies as the distribution of adata observation varies. Formally, it is mapping from a set F of CDF’s into real vectors,β(·) : F −→ <.
In the surplus bound exampleZβ(F ) =

w2 (x̃2 )w1 (x̃1 )EF [yi |xi = x̃]dx̃1 F2 (dx̃2 ),(3.4)

where EF [yi |xi ] denotes the conditional expectation under distribution F and F2 (x2 ) is themarginal CDF of x2i .
How β(F ) varies as F varies near the true distribution F0 can be used to calculate theinfluence function. An important feature of β(F ) is that it may only be well defined when F
is restricted in some way. In the average surplus example β(F ) will only be well defined when
the uniform distribution on (x̆1 , x̄1 ) is absolutely continuous with respect to the distribution of[4]

x1i . In formal terms this feature means that the domain F of β(·) is restricted. To allow for a
restricted domain we consider only variations in F that are contained in F. The specific kind ofvariation we consider is a convex combination Fτ = (1 − τ )F0 + τ Gjz of the true distribution F0
with some other distribution Gjz where Fτ ∈ F. The superscript j and subscript z designate Gjz
as a member of sequence of CDF’s approaching the CDF of the constant z. Under conditions
given below the influence function can be calculated asdjψ(z) = limβ((1 − τ ) · F0 + τ · Gz ) ,j−→∞ dτ(3.5)

where all derivatives with respect to τ are right derivatives at τ = 0. The derivative in thisexpression is the Gateaux derivative of the functional β(F ) with respect to a deviation τ [Gjz −F0 (z)] from the true distribution F0 . This formula says that the influence function is the limitof this Gateaux derivative as Gjz approaches the CDF of the constant z.
Equation (3.5) can be thought of as a generalization of the influence function calculation of
Von Mises (1947) and Hampel (1974). That calculation is based on Gjz = λz where λz is the
CDF of the constant z. If (1 − τ ) · F0 + τ · λz is in F then the influence function is given by the
Gateaux derivative
ψ(z) =dβ((1 − τ ) · F0 + τ · λz )dτThe problem with this formula is that Fτ = (1 − τ ) · F0 + τ · λz will not be in the domain F formany semiparametric estimators. In many cases F ∈ F (i.e. β(F ) being well defined) reqiures
that certain marginal distributions of F are continuous. The CDF (1 − τ ) · F0 + τ · λz does
not satisfy that restriction. Equation (3.5) circumvents this problem by restricting Fτ to be in
F. The influence function is then obtained as the limit of a Gateaux derivative as Gjz −→ λz
rather than the Gateaux derivative with respect to the CDF of a point. This generalization
applies to most semiparametric estimators.
We can relate equation (3.5) to the pathwise derivative characterization of the influencefunction in Newey (1994). Denote one of a class of parametric models as Fθ , where θ denotes
a vector of parameters, with Fθ ∈ F equal to the true distribution F0 at θ = 0. Restrict eachparametric model in the class to be regular in the sense used in the semiparametric efficiency
bounds literature, so that Fθ has a score S(z) (derivative of the log-likelihood in many cases,
e.g. see Van der Vaart, 1998, p. 362) at θ = 0 and possibly other conditions are satisfied.
We assume that the set of scores over all regular parametric families has mean square closurethat includes all functions with mean zero and finite variance. This assumption is the precise
meaning of the statment that we are not restricting F except for regularity conditions. As
shown by Newey (1994) the influence function ψ(z) is then the unique solution to the derivative[5]

equation of Van der Vaart (1991),∂β(Fθ )= E[ψ(zi )S(zi )], E[ψ(zi )] = 0,
∂θ

(3.6)

as the score S(z) varies over those for regular parametric models.
The advantage of equation (3.5) is that it is a direct calculation while the outer productformula (3.6) is a functional equation that must be solved to find the influence function. It
is true that Newey (1994), Hahn (1998), Hirano, Imbens, and Ridder (2003), Bajari, Hong,
Krainer, and Nekipelov (2010), Bajari, Chernozhukov, Hong, and Nekipelov (2009), Hahn
and Ridder (2013, 2016), and Ackerberg, Chen, Hahn, and Liao (2015) have solved equation(3.6) for important models. Generally though, these results have required the solution to afunctional equation, such as a Riesz representation in Propositions 4 and 5 of Newey (1994)
and in Ackerberg et. al. (2015). No such solution is required to apply equation (3.5). Instead,all that is required is an expression for ∂β(Fτ )/∂τ and for its limit. This advantage of equation(3.5) is highlighted in examples to follow. MOVE.
To use the derivative formula to calculate the influence function we need to specify Gjz .
Various kinds of restrictions on Gjz may be needed to insure that Fτ ∈ F. In the surplus
bound example the Lebesgue measure on [x̌1 , x̄1 ] must be absolutely continuous with respect
to the distribution of the price variable x1i for β(F ) to be well defined. In other examples anidentification condition may need to be satisfied. We are free to choose Gjz in whatever way isconvenient for imposing these restrictions and ensuring that equation (3.5) holds. Here we use
Gjz (z̃) = E[1(zi ≤ z̃)δ(zi )],(3.7)

where δ(zi ) is a bounded nonnegative function with E[δ(zi )] = 1. The variable z̃ represents
a possible value of the random variable zi , and we supress a j superscript and z subscripton δ(zi ) for notational convenience. We will assume that z ∈ Rr and that F0 is absolutely
continuous with respect to a product measure µ on Rr . This assumption allows for componentsof zi to be continuously or discretely distributed, or some mixture of the two. The distributionFτ = (1 − τ )F0 + τ Gjz will have a pdf with respect to µ given byfτ (z̃) = f0 (z̃)[1 − τ + τ δ(z̃)] = f0 (z̃)[1 + τ S(z̃)], S(z̃) = δ(z̃) − 1.
Also, for any measureable function yi and components xi of zi the marginal pdf fτ (x̃) of xi ,conditional expectation Eτ [yi |xi ] of yi given xi , and its derivative arefτ (x̃) = f0 (x̃){1 + τ E[S(zi )|xi = x̃]},E[yi |xi ] + τ E[yi S(zi )|xi ] ∂Eτ [yi |xi ]Eτ [yi |xi ] =,= E[{yi − E[yi |xi ]}S(zi )],1 + τ E[S(zi )|xi ]
∂τ
[6]

(3.8)

as shown in Lemma A1 of the Appendix. These formulae will be useful for calculating theinfluence function in many cases.
The characterization of the influence function in equation (3.6) justifies the influence function formula (3.5) for this choice of Gjz . Consider Fτ = (1 − τ ) · F0 + τ · Gjz as a model withparameter τ passing through the truth at τ = 0. This model is regular, as shown in the proofof Theorem 1 below. Also, the score is given by ∂ ln fτ (z̃)/∂τ = S(z̃). Then equation (3.6) andE[S(zi )] = 0 givesZ∂β(Fτ )= E[ψ(zi )δ(zi )] = ψ(z̃)δ(z̃)f0 (z̃)dµ.
∂τAt any point z where ψ(z̃) is continuous, equation (3.5) will follow by taking the limit asδ(z̃)f0 (z̃) approaches a spike at z, which corresponds to Gjz approaching the CDF of a constantat z.
A particular choice of δ(z) will be useful in the calculations. Let K(u) be a pdf that issymmetric around zero, has bounded support, and is continuously differentiable of all orders
with bounded derivatives. The smoothness of K is useful for some of the examples. Also letRµ̄j` = j −1 K((z` − z̃` )/j)dµ` (z̃` ), and −1K((z` − z̃` )/j).
(3.9)g(z̃) = Πr`=1 κj` (z̃` ), κj` (z̃` ) = j µ̄j`
We take δ(z̃) to be the function
δ(z̃) = g(z̃)1(f0 (z̃) ≥ 1/j)f0 (z̃)−1 .

(3.10)For this choice of δ(z̃) equation (3.5) will hold when ψ(z̃) is continuous at z and f0 (z̃) is bounded
away from zero on a set of z̃ that has full µ measure locally to z, as shown and further discussed
below.
We can calculate the influence function for the surplus bound using this δ(z). We assume
that the joint pdf f0 (ỹ, x̃) of (yi , xi ) is bounded away from zero on a neighborhood of z so that
δ(z) = g(z)/f0 (z̃) where g(z) is positive for large enough j. Now assume j is large enough soRthat holds. Then for any a(y, x) we have E[a(yi , xi )S(zi )|xi = x̃] = a(ỹ, x̃)g(z̃)dỹ/f0 (x̃). LetZ∆ =w2 (x̃2 )w1 (x̃1 )E[{yi − γ 10 (xi )} S(zi )|xi = x̃]dx̃1 f2 (x̃2 )dµx2ZZ={w2 (x̃2 )w1 (x̃1 )[ỹ − γ 10 (x̃)]f2 (x̃2 )/f0 (x̃)}g(z̃)dµ = α(x̃)[ỹ − γ 1 (x̃)]g(z̃)dµ,α(x̃) = w2 (x̃2 )w1 (x̃1 )f2 (x̃2 )/f0 (x̃),
where µx2 is the dominating measure for x2i and f2 (x̃2 ) the pdf of x2i with respect to thismeasure. Then plugging Fτ in equation (3.4) and applying the chain rule givesZZdβ(Fτ ) = E[w2 (x2i )w1 (x̃1 )γ 10 (x̃1 , x2i )dx̃1 S(zi )] + ∆ = m(z̃, β 0 , γ 10 )g(z̃)dµ + ∆,dτ[7]where the second equality follows by iterated expecations, the definition of m(z, β, γ 1 ) above,and by E[S(zi )] = 0. Assume that γ 10 (x̃), w1 (x̃1 ), w2 (x̃2 ), and f0 (x̃1 |x̃2 ) = f20 (x̃2 )/f0 (x̃) arecontinuous at x, so that α(x̃)[ỹ − γ 10 (x̃)] is continuous at z. Also assume that γ(x̃1 , x̃2 ) isR
continuous at (x̃1 , x2 ) for all x̃1 ∈ (x̌1 , x̄1 ), so that w1 (x̃1 )γ 10 (x̃1 , x̃2 )dx̃1 is continuous at x2 .
R
Then m(z̃, γ 10 , β 0 ) is continuous at z. By the construction of g (z̃) we have a(z̃)g(z̃)dµ −→a(z) for any a(z̃) function that is continuous at z. Therefore as j −→ ∞ we havedβ(Fτ ) −→ m(z, β 0 , γ 10 ) + α(x)[y − γ 10 (x)].
dτ(3.11)

We can also characterize the influence function in this example using Proposition 4 of Newey(1994). To do so we need to find the solution ot the Riesz representation in equation (4.4) of
Newey (1994). Multiplying and dividing by f0 (x̃1 |x̃2 ) givesZ
E[w2 (x2i ){ w1 (x̃1 )γ 1 (x̃1 , x2i )dx̃1 }] = E[α(xi )γ 1 (xi )].
Here we see that α(xi ) is the Riesz representor in Proposition 4 of Newey (1994), so theconclusion of that result implies that the influence function of the surplus bound is the expression
on the right of equation (3.11).
This example shows the advantage of the derivative formula in equation (3.5) over solving thefunctional equation (3.6). In the example the influence function was derived by a straighforward
calculation of a limit. At no point in that calculation did we need to solve for the function
α(x). Instead the expression for α(x) emerged from the derivative calculation. Thus this new
example demonstrates how the influence function can be obtained from a derivative without
solving a functional equation. We will show this approach is similarly useful in an even morechallenging and original example in the next Section.
We give a precise theoretical justification for the formula in equation (3.5) by assuming thatan estimator is asymptotically linear and then showing that equation (3.5) is satisfied under afew mild regularity conditions. One of the regularity conditions we use is local regularity of β̂√
along the path Fτ . This property is that for any τ n = O(1/ n), when z1 , ..., zn are i.i.d. with
distribution Fτ n ,√dn[β̂ − β(Fτ n )] −→ N (0, V ), V = E[ψ(zi )ψ(zi )T ].

That is, under a sequence of local alternatives, when β̂ is centered at β(Fτ ), then β̂ has the same
limit in distribution as for F0 . This is a very mild regularity condition. Many semiparametricestimators could be shown to be uniformly asymptotically normal for τ in a neighborhood of
0, which would imply this condition. Furthermore, it turns out that asymptotic linearity of β̂
and Gateaux differentiability of β(Fτ ) at τ = 0 with the correct derivative are sufficient for[8]local regularity. For these reasons we view local regularity as a mild condition for the influencefunction calculation.
We will prove that (3.5) is valid for Gjz as specified in equation (3.7). It would be straightforward to extend this validity result to more general classes of Gjz but the result we give should
suffice for most cases.
Theorem 1: Suppose that β̂ is asymptotically linear with influence function ψ(z) and there
is an open set N containing z such that a) there is N̄ ⊆ N such that µ(N ) = µ(N̄ ) and ψ(z̃)
is continuous at z for z̃ ∈ N̄ ; b) there is ε > 0 such thatµ(N ∩{z : f0 (z ) ≥ ε}) =µ(N ).
If β̂ is locally regular for the parametric model (1 − τ )F0 + τ Gjz for τ in a neighborhood ofzero then dβ(Fτ )/∂τ exists and satisfies equation (3.5).
PROOF
This result shows that if an estimator is asymptotically linear and locally regular then theinfluence function satisfies equation (3.5), justifying that calculation. This result is like Van
der Vaart (1991) in having differentiability of β(Fτ ) as a conclusion. It differs in restricting
the paths to have the form (1 − τ )F0 + τ Gjz . Such a restriction on the paths actually weakens
the local regularity hypothesis because β̂ only has to be locally regular for a particular kind ofpath rather than the general class of paths in Van der Vaart (1991). We view locally regularity
for such paths as a very weak condition because the deviations are bounded smooth densities.
We expect that these deviations are regular enough so that Fτ will generally satisfy whateever
regularity conditions are needed for asymptotic linearity uniformly in τ near zero, so β̂ will
be locally regular. The conditions of Theorem 1 are stronger than Van der Vaart (1991) inassuming that the influence function is continuous at z and that the pdf of zi is bounded awayfrom zero on a neighborhood of z. We view this as a weak restriction that will be satisfied
almost everywhere with respect to the dominating measure µ in many cases. We also notethat this result allows for distributions to have a discrete component because the dominating
measure µ may have atoms.
The weak nature of the local regularity condition highlights the strength of the asymptotic
linearity as hypothesis. Primitive conditions for asymptotic linearity can be quite strong andcomplicated. For example, it is known that asymptotic linearity of estimators with a nonparametric first step generally requires some degree of smoothness in the functions being estimated,
see Bickel and Ritov (1988). Our purpose here is to bypass those conditions in order to calculate
the influence function, which result can then be used in all the important ways outlined in the[9]introduction, including as a starting point for formulating regularity conditions for asymptotic
linearity.
It is interesting to note that the scores for the parametric families (1 − τ )F0 + τ Gjz with Gjz
as given in equation (3.7) all satisfy S(z) ≥ −1. Thus to calculate the influence function we do
not require a family of parametric models where the set of scores can approximate any randomvariable with zero mean and finite variance, as is required in Newey (1994). Also, apparently
this restriction means that Theorem 1 is not a special case of the results of Van der Vaart(1998), where it is assumed that the set of scores is a cone. The proof of Theorem 1 does use
some of Van der Vaart’s (1991) reasoning to show differentiability of β(Fτ ) but otherwise isvery straightforward.
We want to emphasize that the purpose of Theorem 1 is quite different than Van der Vaart(1991, 1998) and other important contributions to the semiparametric efficiency literature.
Here β (F ) is not a parameter of interest for some semiparametric model. Instead β(F ) is
associated with an estimator β̂, being the limit of that estimator when F is a distribution thatis unrestricted except for regularity conditions, as formulated in Newey (1994). Our goal is touse β(F ) to calculate the influence function of β̂ under the assumption that β̂ is asymptoticallylinear. The purpose of Theorem 1 is to justify this calculation via equation (3.5). In contrast,
the goal of the semiparametric efficiency literature is to find the efficient influence function for
a parameter of interest when F belongs to a family of distributions.

4

Nonparametric Instrumental Variables

In this Section we derive the influence function for a semiparametric GMM estimator where thefirst step γ 0 is the nonparametric two stage least squares estimator (NP2SLS) of Newey andPowell (1989, 2003) and Newey (1991), abbreivated NP henceforth. As with consumer surplus,
the form of the influence function emerges from the calculation of the derivative. Also, thelimit of the NP2SLS estimator exists and is unique under deviations similar to those specified
above as is essential for calculation of the influence function. The uniqueness and existence
result is made possible by the specification Fτ = (1 − τ )F0 + τ Gjz of deviations from the true
distribution. In this way the approach of this paper provides the key intermediate step ofexistence and uniqueness of the limit of the NP2SLS estimator.
The first step will be based on a linear, nonparametric, instrumental variables model in NP
where
yi = γ 0 (wi ) + εi , E[εi |xi ] = 0,

(4.12)

where wi are right hand side variables that may be correlated with the disturbance εi and xi are[10]

instrumental variables. We begin with the case where wi and xi have the same dimension andare continuously distributed with rectangular supports. The identification condition for γ 0 (wi )in this model is completeness of the conditional expectation given xi , meaning E[∆(wi )|xi ] = 0
implies ∆(wi ) = 0. Under our conditions on wi and xi completeness holds generically, as shownby Andrews (2011) and Chen, Chernozhukov, Lee, and Newey (2014). Genericity justifies ourassumption of completness, although as with other important generic conditions (e.g. existenceof moments), completeness cannot be be tested (see Canay, Santos, Shaik, 2013).
P
The NP2SLS estimator minimizes the objective function Q̂(γ) = i {yi − Ê[γ(·)|xi ]}2 /nover γ ∈ Γn where Ê[·|xi ] is a conditional expectation estimator and Γn imposes restrictions on
the function, such as it being a linear combination of known functions. We first consider thecase where Γn leaves γ unrestricted in the limit, except for having finite second moment. For
fixed γ the limit of the objective function will be Q(γ, F ) = EF [{yi − EF [γ(wi )|xi ]}2 ] where F
denotes the distribution of a single observation. The objective function will also be the limit
of other regularized objective functions such as Darolles, Fan, Florens, and Renault (2011),
so we expect that the corresponding estimators converge to the same object. As usual foran extremum estimator the limit of the minimizer will be the minimizer of the limit underappropriate regularity conditions. Therefore the limit γ(F ) of the NP2SLS estimator will beγ(F ) = arg min Q(γ, F ) = arg min EF [{yi − EF [γ(wi )|xi ]}2 ].
γγ

A problem with this calculation is that γ(F ) need not exist nor be unique. This problem
occurs because γ appears inside a conditional expectation. Our framework helps. Under ourconditions γ(Fτ ) does exist and is unique for Fτ = (1 − τ )F0 + τ Gjz when τ is small enough.
The use of deviations from the truth of the form we have considered allows us to specify Gjz insuch a way that γ(Fτ ) exists and is unique.
We modify slightly our choice of δ(z) in order to allow for weak conditions on the marginal
pdf f0 (x̃) of xi as outlined below. Let gy (y), gw (w), and gx (x) be as specified in equation (3.9)
except that for gx (x) (and gy (y), gw (w)) the product is only taken over components of x (y, w).
R
For example we let µ̄jx` = j −1 K((x` − x̃` )/j)dµ̃` , −1gx (x̃) = Πr`=1 κj` (x̃` ), κj` (x̃` ) = j µ̄j`
K((x` − x̃` )/j).

(4.13)We then choose δ (z) to be−1Zδ(z) = gy (y)gw (w)gx (x) f0 (y, w|x) f0 (x̃)gx (x̃)dµx̃1(f (y, w|x) ≥ 1/j).
With this δ(z) we can make a key assumption that we use to show existence and uniqueness ofγ(Fτ ).
[11]

Assumption 1: a) wi and xi have the same dimension, b) E[∆(wi )|xi ] is complete, and c)R
for each j there is ∆j (wi ) such that E ∆j (wi )|xi = gx (xi )/ gx (x̃)f0 (x̃)dx̃.
Recall that K(u) is continuously differentiable of all orders with bounded support so thatgx (x̃) is also continuously differentiable with bounded derivatives of all orders. Assumption 1c) is then satisfied if E [∆(wi )|xi ] is a compact operator with singular values that do not decline
too fast and other technical conditions hold, as discussed in the Appendix. If we had usedthe δ(z) from earlier we would need to replace Assumption 1 c) with existence of ∆j (wi ) withE ∆j (wi )|xi = gx (xi )/fx (xi ), which would only hold if fx (x̃) were very smooth near x. The
choice of δ(z) allows us to avoid these smoothness conditions for f0 (x̃).
The following result shows that γ(Fτ ) exists for small enough τ and gives a useful formula
Lemma 2: If Assumption 1 is satisfied then for each j and τ small enough, γ(Fτ ) =arg minγ Q(γ, Fτ ) exists and is unique and there is c(τ ) with c(0) = 0 andZ∂cj (τ )j= [ỹ − γ 0 (w̃)]gy (ỹ)gw (w̃)dµỹ dµw̃γ(wi , Fτ ) = γ 0 (wi ) + cj (τ )∆ (wi ),∂τWith this result in place we can derive the influence function for a variety of different
estimators with NP2SLS first step. We begin with a plug in estimator of the form
β̂ =

n
X

v(wi )γ̂(wi )/n,(4.14)

i=1

where v(w) is a known function. This β̂ is an estimator of β 0 = E[v(wi )γ 0 (wi )]. The limit ofβ̂ will beβ(F ) = EF [v(wi )γ(wi , F )].
As shown by Severini and Tripathi (2012), a necessary condition for root-n consistent estimability here is that β 0 there exists α(xi ) such thatv(wi ) = E[α(xi )|wi ].

(4.15)

We will assume that α(x) is unique, which is equivalent to completeness of E[a(xi )|wi ], as holds
generically like discussed above.
To calculate the influence β̂ note that by equation (4.15) and Assumption 1,
E[v(wi )∆j (wi )] = E[E[α(xi )|wi ]∆j (wi )] = E[α(xi )∆j (wi )] = E[α(xi )E[∆j (wi )|xi ]]Z= E[α(xi )g̃x (xi )], g̃x (xi ) = gx (xi ) / gx (x̃) f0 (x̃)dx̃.

[12]By the chain rule and Lemma 2∂β(Fτ )
∂τ=∂EFτ [v(wi )γ(wi , Fτ )]∂τ

(4.16)∂cj (τ )E[v(wi )∆j (wi )]∂τ= E[{v(wi )γ 0 (wi ) − β 0 }δ(zi )]Z+ [ỹ − γ 0 (w̃)]gy (ỹ)gw (w̃)dµỹ dµw̃ E[α(xi )g̃x (xi )]= E[v(wi )γ 0 (wi )S(zi )] += E[ψ(zi )δ(zi )], ψ(z) = v(w)γ 0 (w) − β 0 + α(x)[y − γ 0 (w)].
R
As j −→ ∞ we will have [ỹ − γ 0 (w̃)]gy (ỹ)gw (w̃)dµỹ dµw̃ −→ y − γ 0 (w) for γ 0 (w̃) continuous atw and E[α(xi )g̃x (xi )] −→ α(x)f0 (x)/f0 (x) = α(x) for α(x̃) and f0 (x̃) continuous at x, so we
have
Theorem 3: If Assumption 1 is satisfied, there exists a unique solution α(xi ) to v(wi ) =E[α(xi )|wi ], each of α̃(x), f0 (x̃), v(w̃), and γ 0 (w̃) are continuous at (w, x), the conditional pdf
f (ỹ, w̃|x̃) is bounded away from zero in a neighborhood of (y, w, x), and f0 (x) > 0,then for
NP2SLSlimj−→∞∂β(Fτ )= ψ(z) = v(w)γ 0 (w) − β 0 + α(x)[y − γ 0 (w)].
∂τHere we find that the influence function of β̂ of equation (4.14) is ψ(z) of Theorem 3.
Like the consumer surplus example a nonparametric residual y − γ 0 (w) residual emerges in thecalculation of ψ(z). Unlike the surplus example the residual is from the structural equation(4.12) rather than a nonparametric regression. The function α(x) of the instrumental variables
is also a key component of the influence function. Here α(x) is defined implicitly rather thanhaving an explicit form. This implicit form seems inherent to the NP2SLS first step, withexistence of α(x) solving equation (4.15) being required for root-n consistency of β̂.
Note that α(x) is the solution of a ”reverse” structural equation involving an expectation
conditional on the endogenous variable wi rather than the instrument xi . An analogous ”reverse” structural equation also appears in a linear instrumental variables (IV) setting. Let dˆ =PP( ni=1 Xi WiT )−1 ni=1 Xi yi be the linear IV estimator having limit d0 = (E[Xi WiT ])−1 E[Xi yi ].
A linear IV analog of the structural function γ 0 (w) is wT d0 and of parameter β 0 isb0 = E[v(Wi )(wiT d0 )].
A corresponding estimator of b0 is b̂ =PnT ˆ
i=1 v(Wi )Wi d/n. It is straightforward to show thatthe influence function of b̂ is−1v(w)(wT d0 ) − b0 + a(x)[y − wT d0 ], a(x) = xT E[wi xTi ]E[wi v(wi )].
[13]

Here a(x) is the function obtained from ”reverse” IV where x is the right hand side variable
and w the instrumental variable. The functional α(x) is a nonparametric analog of a(x) where
linear IV is replaced by the solution to a conditional expectation equation.
A solution α(x) to equation (4.15) will only exist when v(w) satisfies certain conditions.
When E[v(wi )2 ] < ∞ a function α(x) can only exist when v(wi ) has Fourier coefficients, withrespect to the singular value basis corresponding to the (assumed to be compact) operator
E[·|wi ], that decline fast enough relative to the inverse of the singular values. This condition
requires some ”smoothness” of v(wi ) and will rule out some functions, such as v(w) that thathave jumps (e.g. indicator functions).
The influence function of Theorem 3 is consistent with the semiparametric efficiency boundgiven in Severini and Tripathi (2012). WHAT ABOUT AI AND CHEN 2007.
There is a different way of estimating β 0 that is analogous to Santos (2014). By equation(4.15), the conditional moment restriction (4.12), and iterated expectations,
β 0 = E[E[α(xi )|wi ]γ 0 (wi )] = E[α(xi )γ 0 (wi )] = E[α(xi )yi ].
Based on the last equality an estimator for β 0 could be constructed as β̃ =(4.17)Pn

i=1 α̂(xi )yi /nwhere α̂ is an estimator of the solution of equation (4.15). The influence function for thisestimator is the same as in Theorem 3. This equality of influence functions occurs because
equation (4.17) is satisfied for any Fτ where equation (4.15) holds and equation (4.15) will holdfor the Fτ we are considering using arguments like those of Lemma 2. Thus β̃ will have thesame limit as β̂ for a general distribution and thus the same influence function.
This influence function calculation can be extended beyond the estimator of (4.14) to othersemiparametric GMM estimators. This extension requires a corresponding extension of equation(4.15). The following condition provides such an extension:
Assumption 2: There exists α(xi ) with E[α(xi )2 ] < ∞ such that for all Fτ
∂E[m(zi , β 0 , γ(Fτ ))]∂E[γ(wi , Fτ )|xi ]= E[α(xi )]∂τ∂τ(4.18)For m(z, β, γ) = v(w)γ(w) − β Assumption 2 is equivalent to equation (4.15). If equation(4.15) is satisfied then∂E[m(zi , β 0 , γ(Fτ ))]∂γ(wi , τ )∂E[γ(wi , Fτ )|xi ]= E[α(xi )] = E[α(xi )
],
∂τ
∂τ
∂τ
where the last equality follows by iterated expectations and interchanging the order of differentiation and integration. Also, if Assumption 2 is satisfied for all ∆τ (wi ) = ∂γ(wi , Fτ )/∂τ wehave
E[v(wi )∆τ (wi )] = E[α(xi )∆τ (wi )] = E[E[α(xi )|wi ]∆τ (wi )],[14]

by iterated expectations. The only way this equation can hold over all ∆τ (wi ) is if equation(4.15) is satisfied.
The existence of α(xi ) satisfying Assumption 2 will follow from other conditions. If the∂E[m(zi , β 0 , γ(Fτ ))]/∂τ is a continuous linear functional of ∂E[γ(wi , Fτ )|xi ]/∂τ which is contained in a closed linear set L and that functional can be extended to be continuous on all of
L then existence of α(x) satisfying Assumtpion 2 follows by the Riesz represenation theorem.
Similar uses of the Riesz representation theorem are given in Newey (1994), Ai and Chen (2007),and Ackerberg, Chen, Hahn, and Liao (2014).
We can use Assumption 2 and Lemma 2 to calculate the influence function of a semiparametric GMM estimator with a NP2SLS first step. Recall from the discussion of equation (2.3)
that the influence function of semiparametric GMM is determined by the correction term φ(z)for the first step and that φ(z)is the influence function of E[m(zi , β, γ(F ))]. When Assumption
2 is satisfied it follows exactly as in equation (4.16) that
∂E[m(zi , β 0 , γ(Fτ ))]∂τ

∂cj (τ )E[∆j (wi )|xi ]]∂τ= E[φ(zi )δ(zi )], φ(z) = α(x)[y − γ 0 (w)],= E[α(xi )giving the following result:
Theorem 4: If Assumptions 1 and 2 are satisfied then for the NP2SLS first step∂E[m(zi , β 0 , γ(Fτ ))]= φ(z) = α(x)[y − γ 0 (w)].
j−→∞
∂τ
lim

An interesting example of this result is the average derivative estimator of Ai and Chen
(2007), where m(z, β, γ) = v̄(w)∂γ(w)/∂w−β for some known v̄(w). Let v(w) = −f0 (w)−1 ∂[v̄(w)f0 (w)]/∂w.
Assume that equation (4.15) is satisfied for this v(w), so that there exist α(xi ) with−f0 (wi )−1 ∂[v̄(wi )f0 (wi )]/∂w = E[α(xi )|wi ].

(4.19)

Then integration by parts and iterated expectations gives∂E[m(zi , β 0 , γ(Fτ ))]∂τ==∂E[v̄(wi )∂γ(wi , Fτ )/∂w]∂E[v(wi )γ(wi , Fτ )]=∂τ∂τ∂E[α(xi )E[γ(wi , Fτ )|xi ]]∂E[γ(wi , Fτ )|xi ]= E[α(xi )
],
∂τ
∂τ

so Assumption 2 is satisfied. It then follows by m(z, β, γ) = v̄(w)∂γ(w)/∂w − β and Theorem
4 that the influence function for the weighted average derivative isψ(z) = v̄(w)∂γ 0 (w)− β 0 + α(x)[y − γ 0 (w)].
∂w[15]Comparison with Ai and Chen ??.
ESTIMATOR; NP2SLS estimator of ??.
We can also calculate the influence function for NP2SLS for a nonlinear, possibly misspecified residual ρ(z, γ) where the NP2SLS estimatori is based on orthogonality of ρ(z, γ 0 ) with aset A of functions of instrumental variables x. We will assume that there is a fixed countablebasis (a1 (x), a2 (x), ...) that spans A in mean square for each Fτ for τ small enough. NP2SLS
is here based on the orthogonality condition
E[aj (xi )ρ(zi , γ 0 )] = 0 for all j.
For example, if A is all functions of xt with finite second moment this restriction is equivalent
to E[ρ(zi , γ 0 )|xi ] = 0. Here aj (x) could be power series in a bounded one-to-one function of x
or could be regression splines. More generally aj (x) could be functions of only a subvector of x
or could be restricted to be additive in subvectors of x.
The NP2SLS estimator is like that given in NP. It minimizes over γ ∈ Γn the sample second
moment of the predicted values from the ordinary least squares regression of the residual ρ(zi , γ)
on pK (xi ) = (p1K (xi ), ..., pKK (xi ))T , where each function pkK (x) is one of the basis functions
aj (x) for some j. We assume that as K −→ ∞ any element of A may be approximated in meansquare by a linear combination of pK (xi ). Let b(zi ) denote any random variable with finite
variance and π τ (ρ(γ), xi ) and π τ (b, xi ) denote the population orthogonal projection of ρ(zi , γ)and b(zi ) on A when the true distribution is Fτ . Also let Eτ [·] denote the expectation underFτ . The limit of the 2SLS objective function will beQτ (γ) = Eτ [π τ (ρ(γ), xi )2 ].
We will assume that γ(w) is restricted to belong to a linear set Γ and that Γn ⊂ Γ for all n. For
example, γ(w) might be restricted to be additive in subvectors of w. It then follows as usualfor extemum estimators that the limit of the NP2SLS estimator will beγ τ = arg min Qτ (γ).
Γ

Here we will just assume that γ τ exists and is unique. We do this, rather than prove existence and uniqueness, because it is difficult to show uniqueness of a minimum when ρ(z, γ)
is nonlinear in γ and when there are more instrumental variables than endogenous variables.
Given the previous results of this Section and Chen, Chernozhukov, Lee, and Newey (2014) we
conjecture that conditions for local existence and uniquenss could be formulated, but we leavethat formulation to future work.

[16]

To find the form of the adjustment term in this setting we need an extended version ofAssumption 2. The following condition provides that extension.
Assumption 3: There exists α(xi ) ∈ A with E[α(xi )2 ] < ∞ such that for all Fτ
∂E[m(zi , β 0 , γ(Fτ ))]∂π 0 (ρ(γ τ ), xi )= −E[α(xi )]∂τ∂τ(4.20)

To derive the adjustment term for NP2SLS it is helpful to consider first order conditionsfor γ τ . Let ζ denote a scalar and ∆(w) some function of the endogenous variables w such that∆(wi ) ∈ Γ. Then we have γ τ (wi ) + ζ∆(wi ) ∈ Γ for any ζ by Γ linear. We will impose thefollowing condition:
Assumption 4: For all τ small enough π τ (ρ(γ τ + ζ∆), xi ) is differentiable in at ζ = 0 andthere is dτ (zi ) such that∂π τ (ρ(γ τ + ζ∆), xi )= π τ (dτ ∆, xi ).
∂ζζ=0
To illustrate consider the endogenous quantile model of Chernozhukov and Hansen (2004)
and Chernozhukov, Imbens, and Newey (2007) where ρ(z, γ) = 1(y < γ(w)) − η for a scalar ηwith 0 < η < 1. Suppose that for τ small enough the distribution of yi conditional on xi andwi is continuous in a neighborhood of γ τ (wi ) with conditional pdf fτ (y|w, x). Let derivatives
with respect to ζ be evaluated at ζ = 0. Then∂Eτ [ρ(zi , γ τ + ζ∆)|wi , xi ]= −fτ (γ τ (wi )|wi , xi )∆(wi ) = −dτ (wi , xi )∆(wi ),∂ζdτ (wi , xi ) = fτ (γ τ (wi )|wi , xi ).

(4.21)

Assuming that the order of differentiation and projection can be interchanged, it follows by
iterated projections that∂π τ (ρ(γ τ + ζ∆), xi )
= πτ
∂ζ



∂E[ρ(zi , γ τ + ζ∆)|wi , xi ]
, xi
∂ζ


= π τ (dτ ∆, xi ),so that Assumption 3 is satisfied. More generally Assumption 3 will hold if ρ(z, γ) = ρ(z, γ(w))
with dτ (w, x) = ∂E[ρ(zi , γ τ (w) + ζ)|wi , xi ]/∂ζ.
By calculus of variations and Assumption 3 the first order conditions for γ τ are that for any∆(wi ) ∈ Γ,
0 = Eτ [π τ (ρ(γ τ ), xi )∂π τ (ρ(γ τ + ζ∆), xi )] = Eτ [π τ (ρ(γ τ ), xi )π τ (dτ ∆, xi )].
∂ζ
[17]This is an identity in τ . Differentiating this identity in τ at τ = 0 and applying the chain rulegives
∂π 0 (ρ(γ τ ), xi )] + E[π 0 (ρ(γ 0 ), xi )π 0 (d0 ∆, xi )S(zi )]∂τ∂π τ (ρ(γ 0 ), xi )∂π τ (d0 ∆, xi )+E[π 0 (d0 ∆, xi )] + E[π 0 (ρ(γ 0 ), xi )]∂τ∂τ
∂π 0 (dτ ∆, xi )+E[π 0 (ρ(γ 0 ), xi )]∂τ

0 = E[π 0 (d0 ∆, xi )(4.22)

The following result helps us evaluate the third and fourth terms in this equation.
Lemma 5: For any a(xi ) ∈ A and b(zi ) with finite variance∂E[a(xi )π τ (b|xi )]= E[a(xi ){b(zi ) − π 0 (b|xi )}S(zi )].
∂τProof of Lemma 5: Note that for each j the definition of the projection implies thatEτ [aj (xi )b(zi )] = Eτ [aj (xi )π τ (b|xi )]identically in τ . Differentiating both sides at τ = 0 and applying the chain rule and ?? gives
E[aj (xi )b(zi )S(zi )] = E[aj (xi )π 0 (b|xi )S(zi )] +∂E[aj (xi )π τ (b|xi )].
∂τSolving it follows that for each j,∂E[a(xi )π τ (b|xi )]= E[aj (xi ){b(zi ) − π 0 (b|xi )}S(zi )].
∂τConsider λ = (λ1 , ..., λJ )T such that (a1 (xi ), ..., aJ (xi ))λ −→ a(xi ) in mean square. The conclusion then follows by S(z) bounded. Q.E.D..
We can apply Lemma 5 to the third and fourth terms of equation (4.22) and solve for the
first term to obtain
E[π 0 (d0 ∆, xi )∂π 0 (ρ(γ τ ), xi )∂π 0 (dτ ∆, xi )] = −E[φ∆ (zi )S(zi )] − −E[π 0 (ρ(γ 0 ), xi )
],
∂τ
∂τ

φ∆ (z) = π 0 (ρ(γ 0 ), x)[d0 (w, x)∆(w) − π 0 (d∆, x)] + π 0 (d∆, x)[ρ(z, γ 0 ) − π 0 (ρ(γ 0 ), x)]+π 0 (ρ(γ 0 ), x)π 0 (d∆, x) − E[π 0 (ρ(γ 0 ), xi )π 0 (d∆, xi )].
This result can be combined with Assumption 3 to obtain the adjustment term when the firststep has is the NP2SLS estimator. We state this result as a Proposition, similarly to Newey[18]

(1994), because its derivation uses formal caclulations without specifying a sufficient set of
regularity conditions.
Proposition 6: If the model is correctly specified, so π 0 (ρ(γ 0 ), xi ) = 0, and there is asequence ∆j (w) such that π 0 (d0 ∆j , xi ) −→ α(xi ) in mean square then the adjustment term is
φ(z) = α(x)ρ(z, γ 0 ).
If the model is misspecified with π 0 (ρ(γ 0 ), xi ) 6= 0, ∂π 0 (dτ ∆, xi )/∂τ = 0, and there exists ∆(w)
such that α(xi ) = π 0 (d∆, xi ) then the adjustment term is φ(z) = φ∆ (z).
Note here that the result with misspecification assumes that ∂π 0 (dτ ∆, xi )/∂τ = 0. We do
not know if an influence function exists when this condition does not hold. The problem isthat dτ may be a nonparametric object evaluated at a point and hence the derivative of the
projection of dτ ∆ on A may not have a representation as an expected product with the score.
For example, for quantile IV dτ (wi , xi ) = fτ (γ τ (wi )|wi , xi ) which is nonparametric conditional
density evaluated at a point. In such cases it may be the case that the influence function does
not exist.
The existence of ∆ (w) with α(xi ) = π 0 (d∆, xi ) in the misspecified case is restrictive. Thiscondition requires α(x) be smooth in a way similar to v(w) being smooth because of v(wi ) =E[α(xi )|wi ]. Because these conditions are similar that the one for v(w) is necessary for rootn consistent estimability under correct specification it may be that existence of ∆ (w) withα(xi ) = π 0 (d∆, xi ) is necessary for root-n consistent estimability under misspecification. The
condition under correct specification, that there is a sequence ∆j (w) such that π 0 (d0 ∆j , xi ) −→α(xi ), is a much weaker condition. For example, for a model where d0 is constant, A is the setof all functions of xi with finite variance, and the dimension of wi is equal to the dimension ofzi this condition automatically holds.
To illustrate we can apply Proposition 6 to obtain the adjustment term when the first stepis quantile IV, so that ρ(z, γ) = 1(y < γ(w)) − τ , and the model is correctly specified. Similar
to the above discussion of equation (4.21) assume that
∂π 0 (ρ(γ τ ), xi )∂γ (·)= −π 0 (d0 τ , xi ).
∂τ∂τSuppose also that there is some v(wi ) such that
∂E[m(zi , β 0 , γ(Fτ ))]∂γ (wi )= E[v(wi ) τ].
∂τ∂τThen by α(xi ) ∈ A and E[α(xi )π 0 (b, xi )] = E[α(xi )b(zi )] for any b(zi ) with finite variance
Assumption 3 becomesE[v(wi )∂γ τ (wi )∂γ (wi )] = E[α(xi )d0 (wi , xi ) τ
].
∂τ
∂τ
[19]

This equation will hold ifv(wi ) = E[α(xi )d0 (wi , xi )|wi ].
Applying Proposition 6 then gives
Proposition 7: If ρ(z, γ) = 1(y < γ(w)) − η, π 0 (ρ(γ 0 ), xi ) = 0, and i) there there exists
α(xi ) such that v(wi ) = E[d0 (wi , xi )α(xi )|wi ] and ii) there exists a sequence ∆j (w)π 0 (d0 ∆j , xi ) −→α(xi ) in mean square then the adjustment term is
φ(z) = α(x)ρ(z, γ 0 ).
For example consider the average derivative for quantile IV where β 0 = E[v̄(wi )∂γ 0 (wi )/∂w]
for known v̄(w). Here condition i) of Proposition 7 is existence of α(x) such thatv(wi ) = −f0 (wi )−1 ∂[v̄(wi )f0 (wi )]/∂w = E[α(xi )d0 (wi , xi )|wi ].
This is a weighted (by d0 (wi , xi )) modification of equation (4.19) that will only hold when thefunction on the left satisfies certain restrictions, similar to the above discussion. Condition ii)
may place some additional restrictions on v(wi ). These restrictions will be weaker the richer isthe instrumental variable set A. Consider the case where A is the set of all functions of xi so that
π 0 (d0 ∆j , xi ) = E[d0 (wi , xi )∆j (wi )|xi ]. If the operator E[d0 (wi , xi )∆j (wi )|xi ] is compact thencondition ii) will hold by standard arguments as in ??. More generally if A were restricted thatwould impose corresponding restrictions on α(x), and hence on v(w). When both conditions i)
and ii) are satisfied the conclusion of Proposition 7 implies that the influence function of theweighted average derivative for quantile IV isψ(z) = v̄(w)5∂γ 0 (w)− β 0 + α (x) [1(y < γ 0 (w)) − η].
∂w

Sufficient Conditions for Asymptotic Linearity

One of the important uses of the influence function is to help specify regularity conditions forasymptotic linearity. The idea is that an formula for ψ(z) determines the remainder terms that
can then be analyzed in order to formulate primitive regularity conditions. In this Section we
formulate such regularity conditions using a functional expansion approach that applies quite
broadly. It may be possible to formulate regularity conditions for particular estimators thatare weaker than we consider.
In this section we consider estimators that are functionals of a nonparametric estimator
taking the formβ̂ = β(F̂ ),[20]

where F̂ is some nonparametric estimator of the distribution of zi . Both the integrated squared
density and the average consumer surplus estimators have this form, as discussed below. We
consider a more general class of estimators in Section 7.
R
Since β 0 = β(F0 ), adding and subtracting the term ψ(z)F̂ (dz) gives√n(β̂ − β 0 ) −

n
X

√
√
√
nR̂1 (F̂ ) + nR2 (F̂ ),ψ(zi )/ n =(5.23)

i=1Z
ψ(z)F (dz) −R̂1 (F ) =nXZ
R2 (F ) = β(F ) − β(F0 ) −ψ(zi )/n,ψ(z)F (dz).

i=1

If

√

nR̂1 (F̂ ) and√nR2 (F̂ ) both converge in probability to zero then β̂ will be asymptoticallylinear. To the best of our knowledge little is gained in terms of clarity or relaxing conditions
by considering R̂1 (F ) + R2 (F ) rather than R̂1 (F ) and R2 (F ) separately, so we focus on theindividual remainders.
The form of the remainders R̂1 (F ) and R2 (F ) are motivated by ψ(z) being a derivative ofβ(F ) with respect to F . The derivative interpretation of ψ(z) suggests a linear approximationof the formZβ(F ) ≈ β(F0 ) +Z
ψ(z)(F − F0 )(dz) = β(F0 ) +ψ(z)F (dz),where the equality follows by E[ψ(zi )] = 0. Plugging in F̂ in this approximation givesRψ(z)F̂ (dz)as a linear approximation to β̂ − β 0 . The term R2 (F̂ ) is then the remainder from linearizingβ̂ = β(F̂ ) around F0 . The term R̂1 (F̂ ) is the difference between the linear approximation
Rψ(z)F (dz) evaluated at the nonparametric estimator F̂ and at the empirical distribution F̃ ,R
Pwith ψ(z)F̃ (dz) = ni=1 ψ(zi )/n.
It is easy to fit the kernel estimator of the integrated squared density into this framework. We
R
let F̂ be the CDF corresponding to a kernel density estimator fˆ(z). Then for β(F ) = f (z)2 dz,
the fact that fˆ2 − f 2 = (fˆ − f )2 + 2f (fˆ − f ) gives an expansion as in equation (5.23) withZR̂1 (F̂ ) =ψ(z)fˆ(z)dz −nXZψ(zi )/n, R2 (F̂ ) =[fˆ(z) − f0 (z)]2 dz.

i=1

Applying this framework to a series regression estimator requires formulating that as an estimator of a distribution F . One way to do that is to specify a conditional expectation operator
conditional on x and a marginal distribution for x, since a conditional expectation operator
implies a conditional distribution. For a series estimator we can take F̂ to have a conditional
expectation operator such thatnEF̂ [a(q, x)|x] =1Xa(qi , x)pK (xi )T Σ̂−1 pK (x).
n
i=1

[21]

Then it will be the case such thatZZ
ˆβ(F̂ ) = W (x)EF̂ [q|x]dx = W (x)d(x)dx= β̂,
which only depends on the conditional expectation operator, leaving us free to specify any
marginal distribution for x that is convenient. Taking F̂ to have a marginal distribution whichis the true distribution of the data we see thatZ
Z
Z
β(F̂ ) − β 0 = EF̂ [W (x){q − d0 (x)}|x]dx = EF̂ [ψ(z)|x]f0 (x)dx = ψ(z)F̂ (dz).
In this case R2 (F ) = 0 andnZR̂1 (F̂ ) =EF̂ [ψ(z)|x]f0 (x)dx −1X
ψ(zi ).
ni=1

Next we consider conditions for both of the remainder terms R̂1 (F̂ ) and R2 (F̂ ) to be smallRenough so that β̂ is asymptotically linear. The remainder term R̂1 (F̂ ) = ψ(z)(F̂ − F̃ )(dz) isthe difference between a linear functional of the nonparametric estimator F̂ and the same linear
functional of the empirical distribution F̃ . It will shrink with the sample size due to F̂ andF̃ being nonparametric estimators of the distribution of zi , meaning that they both converge
to F0 as the sample size grows. This remainder will be the only one when β(F ) is a linear
functional of F̂ .
This remainder often has an important expectation component that is related to the bias ofβ̂. Often F̂ can be thought of as a result of some smoothing operation applied to the empirical
distribution. The F̂ corresponding to a kernel density estimator is of course an example ofthis. An expectation of R̂1 (F̂ ) can then be thought of as a smoothing bias for β̂, or more
precisely a smoothing bias in the linear approximation term for β̂. Consequently, requiring that√√pnR̂1 (F̂ ) −→ 0 will include a requirement that n times this smoothing bias in β̂ goes to zero.
√
Also n times the deviation of R̂1 (F̂ ) from an expectation will need to go zero in order for√√pnR̂1 (F̂ ) −→ 0. Subtracting an expectation from nR̂1 (F ) will generally result in a stochastic
equicontinuity remainder, which is bounded in probability for fixed F and converges to zero as
F approaches the empirical distribution. In the examples the resulting remainder goes to zerounder quite weak conditions.
To formulate a high level condition we will consider an expectation conditional on somesigma algebra χn that can depend on all of the observations. This set up gives flexibility in thespecification of the stochastic equicontinuity condition.
Assumption 1: E[R̂1 (F̂ )|χn ] = op (n−1/2 ) and R̂1 (F̂ ) − E[R̂1 (F̂ )|χn ] = op (n−1/2 ).

[22]

We illustrate this condition with the examples. For the integrated square density let χn be aconstant so that the conditional expectation in Assumption 1 is the unconditional expectation.
R
Let ψ(z, h) = ψ(z + hu)K(u)du and note that by a change of variables u = (z − zi )/h we haveR
P RP
ψ(z)fˆ(z)dz = n−1 h−r nψ(z)K((z − zi )/h)dz = n ψ(zi , h)/n. Then
i=1

i=1

Z Z
E[R̂1 (F̂ )] = E[ψ(zi , h)] = [ ψ(z + hu)f0 (z)dz]K(u)du,(5.24)nR̂1 (F̂ ) − E[R̂1 (F̂ )] =1X
{ψ(zi , h) − E[ψ(zi , h)] − ψ(zi )} .
ni=1Here E[R̂1 (F̂ )] is the kernel bias for the convolution ρ(t) =function and the true pdf.

R

ψ(z + t)f0 (z)dz of the influence

It will be o(n−1/2 ) under smoothness, kernel, and bandwidth con-ditions that are further discussed below. The term R̂1 (F̂ ) − E[R̂1 (F̂ )] is evidently a stochastic
equicontinuity term that is op (n−1/2 ) as long as limh−→0 E[{ψ(zi , h) − ψ(zi )}2 ] = 0.
R
For the series estimator for consumer surplus let δ̂(x) = [ W (x)pK (x)dx]T Σ̂−1 pK (x) andP
note that β̂ = ni=1 δ̂(xi )qi /n. Here we take χn = {x1 , ..., xn }. Then we havenE[R̂1 (F̂ )|χn ] =R̂1 (F̂ ) − E[R̂1 (F̂ )|χn ] =1Xδ̂(xi )d0 (xi ) − β 0 ,
n
1
n

i=1
n
X

(5.25)[δ̂(xi ) − δ(xi )][qi − d0 (xi )].

i=1

Here E[R̂1 (F̂ )|χn ] is a series bias term that will be op (n−1/2 ) under conditions discussed below.
The term R̂1 (F̂ ) − E[R̂1 (F̂ )|χn ] is a stochastic equicontinuity term that will be op (n−1/2 ) as
δ̂(x) gets close to δ(x). In particular, since δ̂(x) depends only on x1 , ..., xn , the expected squarePof this term conditional on χn will be n−2 ni=1 [δ̂(xi )−δ(xi )]2 V ar(qi |xi ), which is op (n−1 ) whenP
V ar(qi |xi ) is bounded and n−1 ni=1 [δ̂(xi ) − δ(xi )]2 = op (1).
Turning now to the other remainder R2 (F ), we note that this remainder results from linearizing around F0 . The size of this remainder is related to the smoothness properties of β(F ).
We previously used Gateaux differentiability of β(F ) along certain directions to calculate the influence function. We need a stronger smoothness condition to make the remainder R2 (F̂ ) small.
Frechet differentiability is one helpful condition. If the functional β(F ) is Frechet differentiable
at F0 then we will have
R2 (F ) = o(kF − F0 k),
for some norm k·k . Unfortunately Frechet differentiability is generally not enough for R2 (F̂ ) =op (n−1/2 ). This problem occurs because β(F ) and hence kF − F0 k may depend on features of√F which cannot be estimated at a rate of 1/ n. For the integrated squared error kF − F0 k =

[23]

√
is the root integrated squared error. Consequently n F̂ − F0 is not√bounded in probability and so nR2 (F̂ ) does not converge in probability to zero.
R[f (z) − f0 (z)]2 dz

1/2This problem can be addressed by specifying that F̂ − F0converges at some rate andthat β(F ) satisfies a stronger condition than Frechet differentiability. One condition that iscommonly used is that R2 (F ) = O(kF − F0 k2 ). This condition will be satisfied if β(F ) is
twice continuously differentiable at F0 or if the first Frechet derivative is Lipschitz. If it is also
assumed that F̂ converges faster than n−1/4 then Assumption A1 will be satisfied. A moregeneral condition that allows for larger R2 (F ) is given in the following hypothesis.
Assumption 2: For some 1 < ζ ≤ 2, R2 (F ) = O(kF − F0 kζ ) and F̂ − F0 = op (n−1/2ζ ).
This condition separates nicely into two parts, one about the properties of the functional andanother about a convergence rate for F̂ . For the case ζ = 2 Assumption 2 has been previously
been used to prove asymptotic linearity, e.g. by Ait-Sahalia (1991), Andrews (1994), Newey(1994), Newey and McFadden (1994), Chen and Shen (1997), Chen, Linton, and Keilegom(2003), and Ichimura and Lee (2010) among others.
R
In the example of the integrated squared density R2 (F ) = [f (z)−f0 (z)]2 dz = O(kF − F0 k2 )R
for kF − F0 k = { [f (z) − f0 (z)]2 dz}1/2 . Thus Assumption 2 will be satisfied with ζ = 2 whenfˆ converges to f0 faster than n−1/4 in the integrated squared error norm.
The following result formalizes the observation that Assumption 1 and 2 are sufficient forasymptotic linearity of β̂.
Theorem 2: If Assumptions 1 and 2 are satisfied then β̂ is asymptotically linear with
influence function ψ(z).
√n(β̂ − β 0 ) was given by Ait√
Sahalia (1991). Instead of using Assumption 1 Ait-Sahalia used the condition that n(F̂ − F0 )An alternative set of conditions for asymptotic normality of

converged weakly as a stochastic process to the same limit as the empirical process. Asymp√ R
totic normality of n ψ(z)F̂ (dz) then follows immediately by the functional delta method.
This approach is a more direct way to obtain asymptotic normality of the linear term in the√
expansion. However weak convergence of n(F̂ − F0 ) requires stronger conditions on the nonparametric bias than does the approach adopted here. Also, Ait-Sahalia’s (1991) approach does
not deliver asymptotic linearity, though it does give asymptotic normality.
These conditions for asymptotic linearity of semiparametric estimators are more complicatedthan the functional delta method outlined in Reeds (1976), Gill (1989), and Van der Vaart
and Wellner (1996). The functional delta method gives asymptotic normality of a functionalof the empirical distribution or other root-n consistent distribution estimator under just two[24]

conditions, Hadamard differentiability of the functional and weak convergence of the empirical
process. That approach is based on a nice separation of conditions into smoothness conditions
on the functional and statistical conditions on the estimated distribution. It does not appear
to be possible to have such simple conditions for semiparametric estimators. One reason is√that they are only differentiable in norms where n F̂ − F0 is not bounded in probability. In
addition the smoothing inherent in F̂ introduces a bias that depends on the functional and so
the weakest conditions are only attainable by accounting for interactions between the functional
and the form of F̂ . In the next Section we discuss this bias issue.

6

Linear Functionals

In this Section we consider primitive conditions for Assumption 1 to be satisfied for kernel
density and series estimators. We focus on Assumption 1 because it is substantially more
complicated than Assumption 2. Assumption 2 will generally be satisfied when β(F ) is sufficiently smooth and F̂ converges at a fast enough rate in a norm. Such conditions are quite wellunderstood. Assumption 1 is more complicated because it involves both bias and stochastic
equicontinuity terms. The behavior of these terms seems to be less well understood than the
behavior of the nonlinear terms.
Assumption 1 being satisfied is equivalent to the linear functional β(F ) =Rψ(z)F (dz) being

an asymptotically linear estimator. Thus conditions for linear functionals to be asymptoticallylinear are also conditions for Assumption 1. For that reason it suffices to confine attention toRlinear functionals in this Section. Also, for any linear functional of the form β(F ) = ζ(z)F (dz)R
we can renormalize so that β(F ) − β 0 = ψ(z)F (dz) for ψ(z) = ζ(z) − E[ζ(zi )]. Then without
Rloss of generality we can restrict attention to functionals β(F ) = ψ(z)F (dz) with E[ψ(zi )] = 0.

6.1

Kernel Density Estimators

Conditions for a linear functional of a kernel density estimator to be asymptotically linear
were stated though (apparently) not proven in Bickel and Ritov (2003). Here we give a brief
exposition of those conditions and a result. Let z be an r × 1 vector and F̂ have pdf fˆ(z) =RPn−1 h−r i K((z − zi )/h). As previously noted, for ψ(z, h) = ψ(z + hu)K(u)du we havePβ̂ = n−1 ni=1 ψ(zi , h). To make sure that the stochastic equicontinuity condition holds weassume:Assumption 3: K(u) is bounded with bounded support,

R

K(u)du = 1, ψ(z) is continuous

almost everywhere, and for some ε > 0, E[sup|t|≤ε ψ(zi + t)2 ] < ∞.

[25]

From Bickel and Ritov (2003, pp. 1035-1037) we know that the kernel bias for linear
functionals is that of a convolution. From equation (5.24) we see thatZ
Z
Z
E[β̂] − β 0 = ρ(hu)K(u)du, ρ(t) = ψ(z + t)f0 (z)dz = ψ(z̃)f0 (z̃ − t)dz̃.
Since ρ(0) = 0 the bias in β̂ is the kernel bias for the convolution ρ(t). A convolution is smootherthan the individual functions involved. Under quite general conditions the number of derivativesof ρ(t) that exist will equal the sum of the number of derivatives sf of f0 (z) that exist andthe number of derivatives sψ of ψ(z) that exist. The idea is that we can differentiate the firstexpression for ρ(t) with respect to t up to sψ times, do a change of variables z̃ = z + t, andthen differentiate sf more times with respect to t to see that ρ(t) is sψ + sf times differentiable.
Consequently, the kernel smoothing bias for β̂ behaves like the kernel bias for a function thatis sψ + sf times differentiable. If a kernel of order sf + sψ is used the bias of β̂ will be oforder hsψ +sf that is smaller than the bias order hsf for the density. Intuitively, the integration
inherent in a linear function is a smoothing operation and so leads to bias that is smaller orderthan in estimation of the density.
Some papers have used asymptotics for kernel based semiparametric estimators based on
the supposition that the bias of the semiparametric estimator is the same order as the bias of
the nonparametric estimator. Instead the order of the bias of β̂ is the product of the orderof kernel bias for f0 (z) and ψ(z) when the kernel is high enough order. This observations ismade in Bickel and Ritov (2003). Newey, Hsieh, and Robins (2004) also showed this result fora twicing kernel, but a twicing kernel is not needed, just any kernel of appropriate order.
As discussed in Bickel and Ritov (2003) a bandwidth that is optimal for estimation of f0
may also give asymptotic linearity. To see this note that the optimal bandwidth for estimationof f0 is n−1/(r+2sf ) . Plugging in this bandwidth to a bias order of hsψ +sf gives a bias in β̂that goes to zero like n−(sψ +sf )/(r+2sf ) . This bias will be smaller than n−1/2 for sψ > r/2.
Thus, root-n consistency of β̂ is possible with optimal bandwidth for fˆ when the number ofderivatives of ψ(z) is more than half the dimension of z. Such a bandwidth will require use ofa sψ + sf order kernel, which is higher order than is needed for optimal estimation of f0 . Bickel
and Ritov (2003) refer to nonparametric estimators that both converge at optimal rates andfor which linear functionals are root-n consistent as plug in estimators, and stated sψ > r/2 asa condition for existence of a kernel based plug in estimator.
We now give a precise smoothness condition appropriate for kernel estimators. Let λ =P(λ1 , ..., λr )T denote a vector of nonnegative integers and |λ| = rj=1 λj . Let ∂ λ f (z) = ∂ |λ| f (z)/∂z1λ1 ·· · ∂zrλr denote the λth partial derivative of f (z) with respect to the components of z.
Assumption 4: f0 (z) is continuously differentiable of order sf , ψ(z) is continuously dif[26]ferentiable of order sψ , K(u) is a kernel of order sf + sψ ,√nhsf +sψ −→ 0, and there is ε > 0such that for all λ, λ0 , λ00 with |λ| ≤ sψ , λ0 = sψ , and λ0 ≤ sf
Z
Z
0
00
λ
sup ∂ ψ(z + t) f0 (z)dz < ∞,∂ λ ψ(z) sup ∂ λ f (z + t) dz < ∞
|t|≤ε

|t|≤ε

Here is a result on asymptotic linearity of kernel estimators of linear functionals.
Theorem 3: If Assumptions 3 and 4 are satisfied thenRψ(z)F̂ (dz) =Pn

i=1 ψ(zi )/n +op (n−1/2 ).
There are many previous results on asymptotic linearity of linear functionals of kernel density
estimators. Newey and McFadden (1994) survey some of these. Theorem 3 differs from many ofthese previous results in Assumption 4 and the way the convolution form of the bias is handled.
We follow Bickel and Ritov (2003) in this.

6.2

Series Regression Estimators

Conditions for a linear functional of series regression estimator to be asymptotically linear were
given in Newey (1994). It was shown there that the bias of a linear functional of a series
estimator is of smaller order than the bias of the series estimator. Here we provide an updateto those previous conditions using Belloni, Chernozhukov, Chetverikov, and Kato (2015) on
asymptotic properties of series estimators. We give conditions for asymptotic linearity of alinear functional of a series regression estimator of the formZˆβ̂ = W (x)d(x)dx.
We give primitive conditions for the stochastic equicontinuity and bias terms from equation(5.25) to be small.
R
Let δ̂(x) = [ W (x)pK (x)dx]T Σ̂−1 pK (x) = E[δ(x)pK (x)T ]Σ̂−1 pK (x) and δ(x) = f0 (x)−1 W (x)Ppas described earlier. The stochastic equicontinuity term will be small if ni=1 [δ̂(xi )−δ(xi )]2 /n −→0. Let Σ = E[pK (xi )pK (xi )T ] and γ = Σ−1 E[pK (xi )d0 (xi )] be the coefficients of the population
regression of d0 (xi ) on pK (xi ). Then the bias term from equation (5.25) satisfies
n

n

X
1X
δ̂(xi )d0 (xi ) = ΓT Σ̂−1
pK (xi )[d0 (xi ) − pK (xi )T γ]/n + E[δ(xi ){pK (xi )T γ − d0 (xi )}],
n
i=1
i=1
(6.26)
The first term following the equality is a stochastic bias term that will be op (n−1/2 ) under relatively mild conditions from Belloni et. al. (2015). For the coefficients γ δ = Σ−1 E[pK (xi )δ(xi )]of the population projection of δ(xi ) on pK (xi ) the second term satisfies
E[δ(xi ){pK (xi )T γ − d0 (xi )}] = −E[{δ(xi ) − γ Tδ pK (xi )}{d0 (xi ) − pK (xi )T γ}][27]where the equality holds by d0 (xi ) − pK (xi )T γ being orthogonal to pK (xi ) in the population.
As pointed out in Newey (1994), the size of this bias term is determined by the product of series
approximation errors to δ(xi ) and to d0 (xi ). Thus, the bias of a series semiparametric estimator
will generally be smaller than the nonparametric bias for a series estimate of d0 (x). For example,
for power series if d0 (x) and δ(x) are continuously differentiable of order sd and sδ respectively,
x is r-dimensional, and the support of x is compact then by standard approximation theory ,E[{δ(x) − γ Tδ pK (x)}{d0 (x) − pK (x)T γ}] ≤ CK −(sd +sδ )/r
As discussed in Newey (1994) it may be possible to use a K that is optimal for estimationof d0 and also results in asymptotic linearity. If sδ > r/2 and K is chosen to be optimal for√
estimation of d0 then nK −(sd +sδ )/r −→ 0. Thus, root-n consistency of β̂ is possible with
optimal number of terms for d0 when the number of derivatives of δ(x) is more than half thedimension of z.
Turning now to the regularity conditions for asymptotic linearity, we follow Belloni et. al.
(2015) and impose the following assumption that takes care of the stochastic equicontinuity
condition and the random bias term.:
Assumption 5: var(qi |xi ) is bounded, E[δ(xi )2 ] < ∞, the eigenvalues of Σ = E[pK (xi )pK (xi )T ]are bounded and bounded away from zero uniformly in K, there is a set χ with Pr(xi ∈ χ) = 1
pand cK and `K such that E[{d0 (xi ) − pK (xi )T γ}2 ] ≤ cK , supx∈χ |d0 (x) − pK (x)T γ| ≤ `K cK ,q√
and for ξ K = supx∈χ pK (x) , we have K/n + ξ 2K (ln K) /n(1 + K`K cK ) + `K cK −→ 0.
The next condition takes care of the nonrandom bias term.
Assumption 6:p√
E[{δ(xi ) − pK (xi )T γ δ }2 ] ≤ cδK , cδK −→ 0, and ncδK cK −→ 0.

Belloni et. al. (2015) give an extensive discussion of the size of cK , `K , and ξ K for various
kinds of series approximations and distributions for xi . For power series Assumptions 5 and 6
are satisfied with cK = CK −sd /r , cδK = CK −sδ /r , `K = K, ξ K = K, andp√
K 2 (ln K) /n(1 + K 3/2 K −sd /r ) + K 1−(sd /r) −→ 0, nK −(sd +sδ )/r −→ 0.
For tensor product splines of order o, Assumptions 5 and 6 are satisfied with cK = CK − min{sd ,o}/r ,√cδK = CK − min{sδ ,o}/r , `K = C, ξ K = K, andp√√
K (ln K) /n(1 + KK − min{sd ,o}/r ) −→ 0, nK −(min{sd ,o}+min{sδ ,o})/r −→ 0.

R

Theorem 4: If Assumptions 5 and 6 are satisfied then for ψ(z) = δ(x)[q − d0 (x)] we have
ˆ = Pn ψ(zi )/n + op (n−1/2 ).
W (x)d(x)
i=1

[28]

Turning now to the consumer surplus bound example, note that in this case W (x) is noteven continuous so that δ(x) is not continuous. This generally means that one cannot assume
a rate at which cδK goes to zero. As long as pK (x) can provide arbitrarily good mean-squareapproximation to any square integrable function, then cδK −→ 0 as K grows. Then Assumption√
6 will require that ncK is bounded. Therefore for power series it suffices for asymptotic
linearity of the series estimator of the bound thatp√
K 2 (ln K) /n(1 + K 3/2 K −sd /2 ) + K 1−(sd /2) −→ 0, nK −sd /2 ≤ C.
For this condition to hold it suffices that d0 (x) is three times differentiable, K 2 ln(K)/n −→ 0,
and K 3 /n is bounded away from zero. For regression splines it suffices thatp

K (ln K) /n(1 +√√KK − min{sd ,o}/2 ) −→ 0, nK − min{sd ,o}/2 ≤ C.

For this condition to hold it suffices that the splines are of order at least 2, d0 (x) is twice
differentiable, K ln(K)/n −→ 0 and K 2 /n is bounded away from zero. Here we find weakersufficient conditions for a spline based estimator to be asymptotically linear than for a power
series estimator.

7

Semiparametric GMM Estimators

A more general class of semiparametric estimators that has many applications is the class ofgeneralized method of moment (GMM) estimators that depend on nonparametric estimators.
Let m(z, β, F ) denote a vector of functions of the data observation z, parameters of interest β,
and a distribution F . A GMM estimator can be based on a moment condition where β 0 is theunique parameter vector satisfying
E[m(zi , β, F0 )] = 0.
That is we assume that this moment condition identifies β.
Semiparametric single index estimation provides examples. For the conditional mean restriction, the model assumes the conditional mean function to only depend on the index, sothat E(y|x) = φ(xT θ0 ). With normalization imposed, first regressor coefficient is 1 so thatθ0 = (1, β T0 )T . Let θ = (1, β T )T . Ichimura (1993) showed that under some regularity conditions,
min E{[y − E(y|xT θ)]2 }βidentifies β 0 . Thus in this case, z = (x, y) andm(z, β, F ) =∂{[y − EF (y|xT θ)]2 }
.
∂β
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For the conditional median restriction, the model assumes the conditional median function
M (y|x) to only depend on the index, so that M (y|x) = φ(xT θ0 ). Ichimura and Lee (2010)
showed that under some regularity conditions,
min E{|y − M (y|xT θ)|}β

identifies β 0 . Thus in this case,m(z, β, F ) =∂{|y − MF (y|xT θ)|}.
∂β

Let x = (x1 , x̃T )T . Note that at β = β 0 , the derivative of E(y|xT θ) with respect to β equals
φ0 (xT θ0 )[x̃ − E(x̃|xT θ0 )].
Thus the target parameter β 0 satisfies the first order condition
0 = E{φ0 (xT θ0 )[x̃ − E(x̃|xT θ0 )][y − E(y|xT θ0 )]}.
Analogously, at β = β 0 , the derivative of M (Y |X T θ) with respect to β equals
φ0 (xT θ0 )[x̃ − E(x̃|xT β)]/fy|x (M (y|xT θ0 )|x).
Thus the target parameter β 0 satisfies the first order condition
0 = E{φ0 (xT θ0 )[x̃ − E(x̃|xT θ0 )][2 · 1{y < M (y|xT θ0 )} − 1]/fy|x (M (y|xT θ0 )|x)}.
Estimators of β 0 can often be viewed as choosing β̂ to minimize a quadratic form in sample
Pn
moments evaluated at some estimator F̂ of F0 . For m̂(β) =i=1 m(zi , β, F̂ )/n and Ŵ apositive semi-definite weighting matrix the GMM estimator is given byβ̂ = arg min m̂(β)T Ŵ m̂(β).
β∈B

In this Section we discuss conditions for asymptotic linearity of this estimator.
For this type of nonlinear estimator showing consistency generally precedes showing asymptotic linearity. Conditions for consistency are well understood. For differentiable m̂(β) asymptotic linearity of β̂ will follow from an expansion of m̂(β̂) around β 0 in the first order conditions.
This gives√√n(β̂ − β 0 ) = −(M̂ T Ŵ M̄ )−1 M̂ T Ŵ nm̂(β 0 ),with probability approaching one, where M̂ = ∂ m̂(β̂)/∂β, M̄ = ∂ m̂(β̄)/∂β, and β̄ is a meanpvalue that actually differs from row to row of M̄ . Assuming that Ŵ −→ W for positive semippdefinite W , and that M̂ −→ M = E[∂m(zi , β 0 , F0 )/∂β] and M̄ −→ M, it will follow that
[30]

p

(M̂ T Ŵ M̄ )−1 M̂ T Ŵ −→ (M T W M )−1 M T W. Then asymptotic linearity of β̂ will follow fromasymptotic linearity of m̂(β 0 ).
With an additional stochastic equicontinuity condition like that of Andrews (1994), asymptotic linearity of m̂(β 0 ) will follow from asymptotic linearity of functionals of F̂ . For F ∈ F letµ(F ) = E[m(zi , β 0 , F )] andnR̂3 (F ) =1X{m(zi , β 0 , F ) − m(zi , β 0 , F0 ) − µ(F )}ni=1Note that√nR̂3 (F ) is the difference of two objects that are bounded in probability (byE[m(zi , β 0 , F0 )] = 0) and differ only when F is different than F0 . Assuming that m(zi , β 0 , F )√is continuous in F in an appropriate sense we would expect that nR̂3 (F ) should be close tozero when F is close to F0 . As long as F̂ is close to F0 in large samples in that sense, i.e. isconsistent in the right way, then we expect that the following condition holds.
√
Assumption 7:

pnR̂3 (F̂ ) −→ 0.

This condition will generally be satisfied when the nonparametrically estimated functions are
sufficiently smooth with enough derivatives that are uniformly bounded, see Andrews (1994) and
Van der Vaart and Wellner (1996). Under Assumption 7 asymptotic linearity of µ(F̂ ) will suffice√
for asymptotic linearity of nm̂(β 0 ). To see this suppose that µ(F̂ ) is asymptotically linearwith influence function ϕ(z). Then under Assumption 7 and by µ(F0 ) = E[m(zi , β 0 , F0 )] = 0,
√

n

n

i=1

i=1

√
1 X
1 X
nm̂(β 0 ) = √m(zi , β 0 , F0 ) + nµ(F̂ ) + op (1) = √[m(zi , β 0 , F0 ) + ϕ(zi )] + op (1).
nnThus Assumption 7 and asymptotic linearity of µ(F̂ ) suffice for asymptotic linearity of m̂(β 0 )with influence function m(z, β 0 , F0 ) + ϕ(z). In turn these conditions and others will imply thatβ̂ is asymptotically linear with influence function
ψ(z) = −(M T W M )−1 M T W [m(z, β 0 , F0 ) + ϕ(z)].
The influence function ϕ(z) of µ(F ) = E[m(zi , β 0 , F )] can be viewed as a correction term
for estimation of F0 . It can be calculated from equation (3.5) applied to the functional µ(F ).
Assumptions 1 and 2 can be applied with β(F ) = µ(F ) for regularity conditions for asymptotic
linearity of µ(F̂ ). Here is a result doing sopp

Theorem 5: If β̂ −→ β 0 , Ŵ −→ W , m̂(β) is continuously differentiable in a neighborhoodppof β 0 with probability approaching 1, for any β̄ −→ β 0 we have ∂ m̂(β̄)/∂β −→ M, M T W M
is nonsingular, Assumptions 1 and 2 are satisfied for β(F ) = E[m(zi , β 0 , F )] and ψ(z) =[31]

ϕ(z), and Assumption 7 is satisfied then β̂ is asymptotically linear with influence function−(M T W M )−1 M T W [m(z, β 0 , F0 ) + ϕ(z)].
For brevity we do not give a full set of primitive regularity conditions for the general GMM
setting. They can be formulated using the results above for linear functionals as well as Frechet
differentiability, convergence rates, and primitive conditions for Assumption 7.

8

Conclusion

In this paper we have given a method for calculating the influence function of a semiparametricestimator. We have also considered ways to use that calculation to formulate regularity conditions for asymptotic linearity. Consideration of other uses of the influence function are outside
the scope of this paper.

9

Appendix A: Proofs

We first give the formulas for the marginal pdf fτ (ã) of a measureable function a(zi ) conditional
expectation Eτ [b(zi )|a(zi )] when the expectation is Eτ [b(zi )] = E[b(zi ){1 + τ S(zi )}].
Lemma A1: For fτ (z̃) = f0 (z̃)[1 − τ + τ δ(z)] and S(z) = δ(z) − 1 the marginal pdf of any
measureable function a(zi ) is fτ (ã) = f0 (ã){1 + τ E[S(zi )|a(zi ) = ã]} and for any b(zi ) with
E[|b(zi )|] < ∞,Eτ [b(zi )|a(zi )] =E[b(zi )|a(zi )] + τ E[b(zi )S(zi )|a(zi )].
1 + τ E[S(zi )|a(zi )]Proof: Let 1i = 1(a(zi ) ∈ A) for any measureable set A. By iterated expectations,τZ
1(ã ∈ A)fτ (a)dµ = E[1i ] + τ E[1i E[S(zi )|ai ]] = E[1i ] + τ E[1i S(zi )] = Eτ [1i ],E[b(zi )|a(zi )] + τ E[b(zi )S(zi )|a(zi )]]1 + τ E[S(zi )|a(zi )]= E[1i {E[b(zi )|a(zi )] + τ E[b(zi )S(zi )|a(zi )]}] = E[1i b(zi )] + τ E[1i b(zi )S(zi )]Eτ [1i •= E[1i b(zi ){1 + τ S(zi )}] = E[1i b(zi )].Q.E.D.
1/2
Proof of Theorem 1: Note that in a neighborhood of τ = 0, (1 − τ )f0 (z̃) + τ gzh (z̃)
is continuously differentiable and we havesτ (z̃) =i1/2 1∂ hgzh (z̃) − f0 (z̃)
gzh (z̃) + f0 (z̃)≤
C
(1 − τ )f0 (z̃) + τ gzh (z̃)=.
∂τ
2 [τ g h (z̃) + (1 − τ )f0 (z̃)]1/2
f0 (z̃)1/2
z

By f0 (z̃) bounded away from zero on a neighborhood of z and the support of gzh (z̃) shrinking
to zero as h −→ 0 it follows that there is a bounded set B with gzh (z̃)/f0 (z̃)1/2 ≤ C1(z̃ ∈ B) for[32]h small enough. Therefore, it follows thatZ hZ
gz (z̃) + f0 (z̃)
dµ ≤ C 1(z̃ ∈ B)dz̃ + 1 < ∞.
f0 (z̃)1/2
1/2
Then by the dominated convergence theorem (1 − τ )f0 (z̃) + τ gzh (z̃)
is mean-square differR
entiable and I(τ ) = sτ (z̃)2 dz̃ is continuous in τ on a neighborhood of zero for all h smallenough. Also, by gzh (z̃) −→ 0 for all z̃ 6= z and f0 (z̃) > 0 on a neighborhood of it follows
that gzh (z̃) 6= f0 (z̃) for all τ and h small enough and hence I(τ ) > 0. Then by Theorem 7.2√
and Example 6.5 of Van der Vaart (1998) it follows that for any τ n = O(1/ n) a vector of nobservations (z1 , ..., zn ) that is i.i.d. with pdf fτ n (z̃) = (1 − τ n )f0 (z̃) + τ n gzh (z̃) is contiguous toa vector of n observations with pdf f0 (z̃). Therefore,
√

n

1 X
n(β̂ − β 0 ) = √ψ(zi ) + op (1)ni=1holds when (z1 , ..., zn ) are i.i.d. with pdf fτ n (z̃).
Next by ψ(z̃) continuous at z, ψ(z̃) is bounded on a neighborhood of z. Therefore for small
RR
R
enough h, kψ(z̃)k2 gzh (z̃)dz̃ < ∞, and hence kψ(z̃)k2 fτ (z̃)dz̃ = (1 − τ ) kψ(z̃)k2 fτ (z̃)dz̃ +RRτ kψ(z̃)k2 gzh (z̃)dz̃ is continuous in τ in a neighborhood of τ = 0. Also, for µhz = ψ(z̃)gzh (z̃)dz̃
R
note that ψ(z̃)fτ (z̃)dz̃ = τ µhz .
Suppose (z1 , ..., zn ) are i.i.d. with pdf fτ n (z̃). Let β(τ ) = β((1−τ )F0 +τ Ghz ) and β n = β(τ n ).
Adding and subtracting terms,
n

√ 
√
√
√
1 X
n β̂ − β n=n(β̂ − β 0 ) − n(β n − β 0 ) = √ψ(zi ) + op (1) − n(β n − β 0 )
n
i=1

=

n
√
√
1 X
√
ψ̆ n (zi ) + op (1) + nτ n µhz − n(β n − β 0 ), ψ̆ n (zi ) = ψ(zi ) − τ n µhz .
n
i=1

R

Note that ψ̆ n (z̃)fτ n (z̃)dz̃ = 0. Also, for large enough n,
Z
Z
2
lim
1( ψ̆ n (z̃) ≥ M ) ψ̆ n (z̃) fτ n (z̃)dz̃ ≤ lim C 1(kψ(z̃)k ≥ M/2)(kψ(z̃)k2 +C)f0 (z̃)dz̃ −→ 0,
M −→∞

M −→∞

so the Lindbergh-Feller condition for a central limit theorem is satisfied. Furthermore, it follows
Rby similar calculations that ψ̆ n (z̃)ψ̆ n (z̃)T fτ n (z̃)dz̃ −→ V. Therefore, by the Lindbergh-FellerP√ddcentral limit theorem, ni=1 ψ̆ n (zi ) −→ N (0, V ). Therefore we have n(β̂ − β n ) −→ N (0, V ) if
and only if√nτ n µhz −√n(β n − β 0 ) −→ 0.

Suppose that β(τ ) is differentiable at τ = 0 with derivative µhz . Then√n(β n − β 0 ) −√nτ n µhz =√no(τ n ) =[33]√nτ n o(1) −→ 0

(9.27)

by√nτ n bounded. Next, we follow the proof of Theorem 2.1 of Van der Vaart (1991), and√suppose that eq. (9.27) holds for all τ n = O(1/ n). Consider any sequence rm −→ 0. Let nm
be the subsequence such that
−1/2
(1 + nm )−1/2 < rm ≤ nm.

√
Let τ n = rm for n = nm and τ n = n−1/2 for n ∈/ {n1 , n2 , ...}. By construction, τ n = O(1/ n),so that eq (9.27) holds. Therefore it also holds along the subsequence nm , so that


√
√
√
β(rm ) − β 0= nm rm µhz − nm [β(rm ) − β 0 ] −→ 0.
nm rm µhz −rm
By construction√

nm rm is bounded away from zero, so that µhz − [β(rm ) − β 0 ] /rm −→ 0.

Since rm is any sequence converging to zero it follows that β(τ ) is differentiable at τ = 0 with
derivative µhz .

√We have now shown that eq. (9.27) holds for all sequences τ n = O(1/ n) if and only ifβ(τ ) is differentiable at τ = 0 with derivative µhz . Furthermore, as shown above eq. (9.27)
holds if and only if β̂ is regular. Thus we have shown that β̂ is regular if and only if β(τ ) isdifferentiable at τ = 0 with derivative µhz .
Finally note that as h −→ 0 it follows from continuity of ψ(z̃) at z, K(u) bounded withbounded support, and the dominated convergence theorem that
Z
Z
Z
h
h
−r
µz = ψ(z̃)gz (z̃)dz̃ = h
ψ(z̃)K((z̃ − z)/h)dz̃ = ψ(z + hu)K(u)du.Q.E.D.
Proof of Theorem 2: We will first prove that Eτ [∆(wi )|xi ] is complete as a function of∆(wi ) for all τ small enough. Consider τ ∈ [0, τ̄ ) for τ̄ = min{1/|1 − Eκ [∆∗ (w)]|, 1}. Consider
any ∆(wi ) 6= 0. Note that for large enough j by f (y, w|x) bounded away from zero,ZE[δ(zi )∆(wi )|xi ] = Eκ [∆(wi )]δ x (xi ), Eκ [∆(w̃)] = ∆(wi )κw (w̃)dµw̃ .
From equation (3.8), Eτ [∆(wi )|xi ] = 0 only if
0 = (1 − τ )E [∆(wi )|xi ] + τ Eκ [∆(wi )] δ x (xi ).
If Eκ [∆(wi )] = 0 note that (1 − τ )E[∆(wi )|xi ] 6= 0 by hypothesis i), so that Eτ [∆(wi )|xi ] 6= 0.
If Eκ [∆(wi )] 6= 0 then E[∆(wi )|xi ] = Cδ x (xi ) for C = τ Eκ [∆(wi )]/(1 − τ ) 6= 0. Note that
E[C∆∗ (wi )|xi ] = Cδ x (xi ) by hypothesis ii) so by hypothesis i), ∆(wi ) = C∆∗ (wi ). Substituting
this back in the above equation it follows that0 = {(1 − τ )C + τ CEκ [∆∗ (wi )]}δ x (xi ).

[34]

By C 6= 0 and δ x (xi ) positive with positive probability this equation implies 0 = (1 − τ ) +τ Eκ [∆∗ (wi )] , which does not hold by τ ∈ [0, τ ). Therefore Eτ [∆(wi )|xi ] 6= 0 for all τ smallenough.
Next consider γ(wi , C) = γ 0 (wi ) + C∆∗ (wi ) for a constant C. Note that Eτ [γ(wi , C)|xi ] =Eτ [yi |xi ] if and only if
E[γ(wi , C)|xi ] + τ E[δ(zi )γ(wi , C)|xi ] = E[yi |xi ] + τ E[δ(zi )yi |xi ].
Noting that E[γ 0 (wi )|xi ] = E[yi |xi ] and E[∆∗ (wi )|xi ] = δ x (xi ), this equation holds if any only
ifCδ x (xi ) + τ (Eκ [γ 0 (wi )] + CEκ [∆∗ (w̃)])δ x (xi ) = τ Eκ [ỹ]δ x (xi ).
Since δ x (xi ) is positive with positive probablity this equation holds if and only if
C + τ (Eκ [γ 0 (w̃)] + CEκ [∆∗ (w̃)]) = τ Eκ [ỹ].
Solve for C = c(τ ) to obtainc(τ ) =

τ Eκ [ỹ − γ 0 (x̃)],
1 + τ Eκ [∆∗ (wi )]for all τ small enough. Then by construction and by differentiating c(τ ) at τ = 0 the conclusion
holds. Q.E.D.

Proof of Theorem 2: This follows as outlined in the text from Assumptions 1 and 2 andeq. (5.23) and the fact that if several random variables converge in probability to zero then sodoes their sum. Q.E.D.
Proof of Theorem 3: By the first dominance condition of Assumption 4,R

ψ(z + t)f (z)dz

is continuously differentiable with respect t up to order sζ in a neighborhood of zero and for all
λ with |λ| ≤ sζ ,∂λZZψ(z + t)f0 (z)dz =∂ λ ψ(z + t)f0 (z)dz.

For any λ with |λ| = sζ it follows by a change of variables z̃ = z + t and the second dominancecondition thatZ

∂ λ ψ(z + t)f0 (z)dz =Z∂ λ ψ(z̃)f0 (z̃ − t)dz̃

is continuously differentiable in t up to order sf in a neighborhood of zero and that for any λ0
with λ0 ≤ sf
∂

λ0

Z

λ

Z

∂ ψ(z̃)f0 (z̃ − t)dz̃ =

[35]

0

∂ λ ψ(z̃)∂ λ f0 (z̃ − t)dz̃.

Therefore ρ(t) =Rψ(z + t)f0 (z)dz is continuously differentiable of order sζ + sf in a neighbor-hood of zero. Since ρ(0) = 0 and K(u) has bounded support and is order sζ + sf the usual
expansion for kernel bias givesZE[β̂] − β 0 =ρ(hu)K(u)du = O(hsζ +sf ).

√Therefore, E[ nR̂1 (F̂ )] −→ 0.
Next, by continuity almost everywhere of ψ(z) in Assumption 3 it follows that ψ(zi +hu) −→ψ(zi ) as h −→ 0 with probability one (w.p.1). Also, by Assumption 3 sup|t|≤ε |ψ(zi + t)| is finite
w.p.1, so that by K(u) having bounded support and the dominated convergence theorem, w.p.1,
Z
ψ(zi , h) = ψ(zi + hu)K(u)du −→ ψ(zi ).
Furthermore, for h small enoughψ(zi , h)2 ≤ C sup ψ(zi + t)2 ,|t|≤εso it follows by the dominated convergence theorem that E[{ψ(zi , h)−ψ(zi )}2 ] −→ 0 as h −→ 0.
Therefore,√

V ar( nR̂1 (F̂ )) = V ar(n

−1/2

n
X

{ψ(zi , h) − ψ(zi )}) ≤ E[{ψ(zi , h) − ψ(zi )}2 ] −→ 0.

i=1

Since the expectation and variance of√nR̂1 (F̂ ) converges to zero it follows that Assumption

1 is satisfied. Assumption 2 is satisfied because β(F ) is a linear functional, so the conclusion
follows by Theorem 2. Q.E.D.
Proof of Theorem 4: Since everything in the remainders is invariant to nonsingular linear transformations of pK (x) it can be assumed without loss of generality that Σ =E[pK (xi )pK (xi )T ] = I. Let δ̃(xi ) = ΓT pK (xi ) = γ 0δ pK (xi ) so that by Assumption 6, E[{δ̃(xi ) −δ(xi )}2 ] −→ 0. Note that by V ar(qi |xi ) bounded and the Markov inequality,nX{δ̂(xi ) − δ(xi )}2 V ar(qi |xi )/n ≤ C

i=1

n
X
{δ̂(xi ) − δ(xi )}2 /n
i=1

n
n
X
X
≤ C
{δ̃(xi ) − δ(xi )}2 /n + C{ΓT (Σ̂−1 − I)pK (xi )}2 /n
i=1

i=1
T

≤ op (1) + Γ (Σ̂−1− I)Σ̂(Σ̂−1− I)Γ = op (1),where the last equality follows as in Step 1 of the proof of Lemma 4.1 of Belloni et. al. (2015).
We also haveΓT Γ = E[δ(x)pK (xi )T ]Σ−1 E[δ(x)pK (xi )] = E[{γ Tδ pK (xi )}2 ].
[36]

By cK −→ 0 it follows that E[{γ Tδ pK (xi )}2 ] −→ E[δ(xi )2 ] > 0, so that Γ 6= 0. Let Γ̄ =
Γ/(ΓT Γ)1/2 , so that Γ̄T Γ̄ = 1. Note that
T

Γ̄ Σ̂

−1

n
X

K

K

T

T

−1

p (xi )[d0 (xi ) − p (xi ) γ]/n = Γ̄ (γ̃ − γ), γ̃ = Σ̂

i=1

n
X

pK (xi )d0 (xi )/n

i=1

Let R1n (Γ) and R2n (Γ) be defined by the equations√nΓ̄T (γ̃ − γ) = Γ̄TnX√
pK (xi )[d0 (xi ) − pK (xi )T γ]/ n + R1n (Γ̄) = R1n (Γ) + R2n (Γ̄).

i=1

By eqs. (4.12) and (4.14) of Lemma 4.1 of Belloni et. al. (2015) and by Assumption 5 we haveq√pp
R1n (Γ̄) = Op ( ξ 2K (ln K) /n(1 + K`K cK )) −→ 0, R2n (Γ̄) = Op (`K cK ) −→ 0.
Noting that ΓT Γ ≤ E[δ(xi )2 ] = O(1), we have
T

−1

Γ Σ̂

n
X

p

pK (xi )[d0 (xi ) − pK (xi )T γ]/n = (ΓT Γ)1/2 Γ̄T (γ − γ) = O(1)op (1) −→ 0.

i=1

Also, note that E[pK (xi ){d0 (xi ) − pK (xi )T γ}] = 0, so that by the Cauchy-Schwarz inequality,√n E[δ(xi ){d0 (xi ) − pK (xi )T γ}] =√n E[{δ(xi ) − pK (xi )T γ δ }{d0 (xi ) − pK (xi )T γ}] ≤√ncδK cK −→ 0.

Then the conclusion follows by the triangle inequality and eq. (6.26). Q.E.D.
Proof of Theorem 5: As discussed in the text it suffices to prove that m̂(β 0 ) is asymptotically linear with influence function m(z, β 0 , F0 ) + α(z). By Assumption 7 it follows thatnm̂(β 0 ) =1Xm(zi , β 0 , F0 ) + µ(F̂ ) + op (n−1/2 ).
ni=1Also, by the conclusion of Theorem 1 and µ(F0 ) = 0 we havenµ(F̂ ) =1X
ϕ(zi ) + op (n−1/2 ).
ni=1

By the triangle inequality it follows thatnm̂(β 0 ) =1X[m(zi , β 0 , F0 ) + ϕ(zi )] + op (n−1/2 ).Q.E.D.
n
i=1

[37]

10

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