Nonparametric Estimation of Triangular Simultaneous Equations Models under Weak Identification Sukjin Han⇤ Department of Economics University of Texas at Austin
[email protected] February 21, 2017
Abstract This paper analyzes the problem of weak instruments on identification, estimation, and inference in a simple nonparametric model of a triangular system. The paper derives a necessary and sufficient rank condition for identification, based on which weak identification is established. Then, nonparametric weak instruments are defined as a sequence of reduced-form functions where the associated rank shrinks to zero. The problem of weak instruments is characterized as concurvity and to be similar to the ill-posed inverse problem, which motivates the introduction of a regularization scheme. The paper proposes a penalized series estimation method to alleviate the e↵ects of weak instruments and shows that it achieves desirable asymptotic properties. The findings of this paper provide useful implications for empirical work. To illustrate them, Monte Carlo results are presented and an empirical example is given in which the e↵ect of class size on test scores is estimated nonparametrically. Keywords: Triangular models, nonparametric identification, weak identification, weak instruments, series estimation, inverse problem, regularization, concurvity. JEL Classification Numbers: C13, C14, C36. ⇤ I am very grateful to my advisors, Donald Andrews and Edward Vytlacil, and committee members, Xiaohong Chen and Yuichi Kitamura for their inspiration, guidance and support. I am deeply indebted to Donald Andrews for his thoughtful advice throughout the project. The earlier version of this paper has benefited from discussions with Joseph Altonji, Ivan Canay, Philip Haile, Keisuke Hirano, Han Hong, Joel Horowitz, Seokbae Simon Lee, Oliver Linton, Whitney Newey, Byoung Park, Peter Phillips, Andres Santos, and Alex Torgovitsky. I gratefully acknowledge financial support from a Carl Arvid Anderson Prize from the Cowles Foundation. I also thank the seminar participants at Yale, UT Austin, Chicago Booth, Notre Dame, SUNY Albany, Duke, Sogang, SKKU, and Yonsei, as well as the participants at NASM and Cowles Summer Conference.
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1
Introduction
Instrumental variables (IVs) are widely used in empirical research to identify and estimate models with endogenous explanatory variables. In linear simultaneous equations models, it is well known that standard asymptotic approximations break down when instruments are weak in the sense that (partial) correlation between the instruments and endogenous variables is weak. The consequences of and solutions for weak instruments in linear settings have been extensively studied in the literature over the past two decades; see, e.g., Bound et al. (1995), Staiger and Stock (1997), Dufour (1997), Kleibergen (2002, 2005), Moreira (2003), Stock and Yogo (2005), and Andrews and Stock (2007), among others. Weak instruments in nonlinear parametric models have recently been studied in the literature in the context of weak identification by, e.g., Stock and Wright (2000), Kleibergen (2005), Andrews and Cheng (2012), Andrews and Mikusheva (2016b,a), Andrews and Guggenberger (2015), and Han and McCloskey (2015). One might expect that nonparametric models with endogenous explanatory variables will generally require stronger identification power than parametric models as there is an infinite number of unknown parameters to identify, and hence, stronger instruments may be required.1 Despite the problem’s importance and the growing popularity of nonparametric models, weak instruments in nonparametric settings have not received much attention.2 Furthermore, surprisingly little attention has been paid to the consequences of weak instruments in empirical research using nonparametric models; see below for references. Part of the neglect is due to the existing complications embedded in nonparametric models. In a simple nonparametric framework, this paper analyzes the problem of weak instruments on identification, estimation, and inference, and proposes an estimation strategy to mitigate the e↵ect. Identification results are obtained so that the concept of weak identification can subsequently be introduced via localization. The problem of weak instruments is characterized as concurvity and is shown to be similar to the ill-posed inverse problem. An estimation method is proposed through regularization and the resulting estimators are shown to have desirable asymptotic properties even when instruments are possibly weak. As a nonparametric framework, we consider a triangular simultaneous equations model. The specification of weak instruments is intuitive in the triangular model because it has an explicit reduced-form relationship. Additionally, clear interpretation of the e↵ect of weak instruments can be made through a specific structure produced by the control function approach. To make our analysis succinct, we specify additive errors in the model. This particular model is considered in Newey et al. (1999) (NPV) and Pinkse (2000) in a situation without weak instruments. Although relatively recent 1
This conjecture is shown to be true in the setting considered in this paper; see Theorem 5.1 and Corollary 5.2. Chesher (2003, 2007) mentions the issue of weak instruments in applying his key identification condition in the empirical example of Angrist and Keueger (1991). Blundell et al. (2007) determine whether weak instruments are present in the Engel curve dataset of their empirical section. They do this by applying the Stock and Yogo (2005) test developed in linear models to their reduced form, which is linearized by sieve approximation. Darolles et al. (2011) briefly discuss weak instruments that are indirectly characterized within their source condition. 2
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developments in nonparametric triangular models contribute to models with nonseparable errors (e.g., Imbens and Newey (2009), Kasy (2014)), such flexibility complicates the exposition of the main results of this paper.3 Also, having a form analogous to its popular parametric counterpart, the model with additive errors is broadly used in applied research such as Blundell and Duncan (1998), Yatchew and No (2001), Lyssiotou et al. (2004), Dustmann and Meghir (2005), Skinner et al. (2005), Blundell et al. (2008), Del Bono and Weber (2008), Frazer (2008), Mazzocco (2012), Coe et al. (2012), Breza (2012), Henderson et al. (2013), Chay and Munshi (2014), and Koster et al. (2014). One of the contributions of this paper is that it derives novel identification results in nonparametric triangular models that complement the existing results in the literature. With a mild support condition, we show that a particular rank condition is necessary and sufficient for the identification of the structural relationship. This rank condition is substantially weaker than what is established in NPV. The rank condition covers economically relevant situations such as outcomes resulting from corner solutions or kink points in certain economic models. More importantly, deriving such a rank condition is the key to establishing the notion of weak identification. Since the condition is minimal, a “slight violation” of it has a binding e↵ect on identification, hence resulting in weak identification. To characterize weak identification, we consider a drifting sequence of reduced-form functions that converges to a non-identification region, namely, a space of reduced-form functions that violate the rank condition for identification. A particular rate is designated relative to the sample size, which e↵ectively measures the strength of the instruments, so that it appears in asymptotic results for the estimator of the structural function. The concept of nonparametric weak instruments generalizes the concept of weak instruments in linear models such as in Staiger and Stock (1997). In the nonparametric control function framework, the problem of weak instruments becomes a nonparametric analogue of a multicollinearity problem known as concurvity (Hastie and Tibshirani (1986)). Once the endogeneity is controlled by a control function, the model can be rewritten as an additive nonparametric regression, where the endogenous variables and reduced-form errors comprise two regressors, and weak instruments result in the variation of the former regressor being mainly driven by the variation of the latter. This problem of concurvity is related to the illposed inverse problem inherent in other nonparametric models with endogeneity or, in general, to settings where smoothing operators are involved; see Carrasco et al. (2007) for a survey of inverse problems. Although the sources of ill-posedness in the two problems are di↵erent, there is sufficient similarity that the regularization methods used in the literature to solve the ill-posed inverse problem can be introduced to our problem. Due to the problems’ distinct features, however, among the regularization methods, only penalization (i.e., Tikhonov-type regularization) alleviates the e↵ect of weak instruments. 3 For instance, the control function employed in Imbens and Newey (2009) requires large variation in instruments, and hence discussing weak instruments (i.e., weak association between endogenous variables and instruments or little variation in instruments) in such a context requires more care.
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This paper proposes a penalized series estimator for the structural function and establishes its asymptotic properties. Our results on the rate of convergence of the estimator suggest that, without penalization, weak instruments characterized as concurvity slow down the overall convergence rate, exacerbating bias and variance “symmetrically.” We then show that a faster convergence rate is achieved with penalization. In controlling the penalty bias, we introduce a high level condition that restrict the weakness of instruments relative to the smoothness of the unknown function. This condition is related to source conditions (Engl et al. (1996)) in the literature on ill-posed inverse problems. We also derive consistency and asymptotic normality with mildly weak instruments. The problem of concurvity in additive nonparametric models is also recognized in the literature where di↵erent estimation methods are proposed to address the problem—e.g., the backfitting methods (Linton (1997), Nielsen and Sperlich (2005)) and the integration method (Jiang et al. (2010)); also see Sperlich et al. (1999). In particular, as closely related work to the asymptotic results of this paper, Jiang et al. (2010) establish pointwise asymptotic normality for local linear and integral estimators in an additive nonparametric model with highly correlated covariates. In the present paper, where an additive model results from a triangular model accompanied with the control function approach, the problem of concurvity is addressed in a more direct manner via penalization. In addition, although the main conclusions of this paper do not depend on the choice of nonparametric estimation method, using series estimation in our penalization procedure is also justified in the context of design density. In situations where the joint density of x and v becomes singular, such as in our case with weak instruments, it is known that series and local linear estimators are less sensitive than conventional kernel estimators; see, e.g., Hengartner and Linton (1996) and Imbens and Newey (2009) for related discussions. Another possible nonparametric framework in which to examine the problem of weak instruments is a nonparametric IV (NPIV) model (Newey and Powell (2003), Hall and Horowitz (2005) and Blundell et al. (2007), among others). Unlike in a triangular model, the absence of an explicit reduced-form relationship forces weak instruments in this setting to be characterized as a part of the ill-posed inverse problem. Therefore, in this model, the performance of the estimator can be severely deteriorated as the problem is “doubly ill-posed.”4 Further, it may also be hard to separate the e↵ects of the two in asymptotic theory. As a related recent work, Freyberger (2015) provides a framework by which the completeness condition can be tested in a NPIV model. Instead of using a drifting sequence of distributions, he indirectly defines weak instruments as a failure of a restricted version of the completeness condition. While he applies his framework to test weak instruments, our focus is on estimation and inference of the function of interest in a di↵erent nonparametric model with a more explicit definition of weak instruments. The findings of this paper provide useful implications for empirical work. First, when estimating a nonparametric structural function, the results of IV estimation and subsequent inference can be 4
In Section 8, we illustrate this point in an empirical application by comparing estimates calculated from the triangular and NPIV models.
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misleading even when the instruments are strong in terms of conventional criteria for linear models.5 Second, the symmetric e↵ect of weak instruments on bias and variance implies that the bias–variance trade-o↵ is the same across di↵erent strengths of instruments, and hence, weak instruments cannot be alleviated by exploiting the trade-o↵. Third, penalization on the other hand can alleviate weak instruments by significantly reducing variance and sometimes bias as well. Fourth, there is a tradeo↵ between the smoothness of the structural function (or the dimensionality of its argument) and the requirement of strong instruments. Fifth, if a triangular model along with its assumptions is considered to be reasonable, it makes the data to be informative about the relationship of interest more than a NPIV model does, which is an attractive feature especially in the presence of weak instruments. Sixth, although a linear first-stage reduced form is commonly used in applied research (e.g., in NPV, Blundell and Duncan (1998), Blundell et al. (1998), Dustmann and Meghir (2005), Coe et al. (2012), and Henderson et al. (2013)), the strength of instruments can be improved by having a nonparametric reduced form so that the nonlinear relationship between the endogenous variable and instruments can be fully exploited. The last point is related to the identification results of this paper. In Section 8, we apply the findings of this paper to an empirical example, where we nonparametrically estimate the e↵ect of class size on students’ test scores. The rest of the paper is organized as follows. Section 2 introduces the model and obtains new identification results. Section 3 discusses weak identification and Section 4 relates the weak instrument problem to the ill-posed inverse problem and defines our penalized series estimator. Sections 5 and 6 establish the rate of convergence and consistency of the penalized series estimator and the asymptotic normality of some functionals of it. Section 7 presents the Monte Carlo simulation results. Section 8 discusses the empirical application. Finally, Section 9 concludes.
2
Identification
We consider a nonparametric triangular simultaneous equations model y = g0 (x, z1 ) + ",
x = ⇧0 (z) + v,
E["|v, z] = E["|v] a.s.,
E[v|z] = 0 a.s.,
(2.1a) (2.1b)
where g0 (·, ·) is an unknown structural function of interest, ⇧0 (·) is an unknown reduced-form
function, x is a dx -vector of endogenous variables, z = (z1 , z2 ) is a (dz1 + dz2 )-vector of exogenous variables, and z2 is a vector of excluded instruments. The stochastic assumptions (2.1b) are more general than the assumption of full independence between (", v) and z and E[v] = 0. Following the 5
For instance, in Coe et al. (2012), the first-stage F -statistic value that is reported is (sometimes barely) in favor of strong instruments, but the judgement is based on the criterion for linear models. The majority of empirical works referenced above do not report first-stage results.
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control function approach, E[y|x, z] = g0 (x, z1 ) + E["|⇧0 (z) + v, z] = g0 (x, z1 ) + E["|v] = g0 (x, z1 ) + where
0 (v)
0 (v),
(2.2)
= E["|v] and the second equality is from the first part of (2.1b). In e↵ect, we capture
endogeneity (E["|x, z] 6= 0) by an unknown function
0 (v),
which serves as a control function. Once
v is controlled for, the only variation of x comes from the exogenous variation of z. Based on equation (2.2) we establish identification, weak identification, and estimation results. First, we obtain identification results that complement the results of NPV. For useful comparisons, we first restate the identification condition of NPV which is written in terms of ⇧0 (·). Given (2.2), the identification of g0 (x, z1 ) is achieved if one can separately vary (x, z1 ) and v in g(x, z1 ) + (v). Since x = ⇧0 (z) + v, a suitable condition on ⇧0 (·) will guarantee this via the separate variation of z and v. In light of this intuition, NPV propose the following identification condition. Proposition 2.1 (Theorem 2.3 in NPV) If g(x, z1 ), (v), and ⇧(z) are di↵erentiable, the boundary of the support of (z, v) has probability zero, and
Pr rank
✓
@⇧0 (z) @z20
◆
= dx = 1,
(2.3)
then g0 (x, z1 ) is identified up to an additive constant. The identification condition can be seen as a nonparametric generalization of the rank condition. One can readily show that the order condition (dz2
dx ) is incorporated in this rank condition.
Note that this condition is only a sufficient condition, which suggests that the model can possibly be identified with a relaxed rank condition. This observation motivates our identification analysis. We find a necessary and sufficient rank condition for identification by introducing a mild support condition. The identification analysis of this section is also important for our later purpose of defining the notion of weak identification. Henceforth, in order to keep our presentation succinct, we focus on the case where the included exogenous variable z1 is dropped from model (2.1) and z = z2 . With z1 included, all the results of this paper readily follow similar lines; e.g., the identification analysis follows conditional on z1 . We first state and discuss the assumptions that we impose. Assumption ID1 The functions g0 (x),
0 (v),
and ⇧0 (z) are continuously di↵erentiable in their
arguments. This condition is also assumed in Proposition 2.1 above. Before stating a key additional assumption for identification, we first define the supports that are associated with x and z. Let X ⇢ Rdx
and Z ⇢ Rdz be the marginal supports of x and z, respectively. Also, let Xz be the conditional 6
support of x given z 2 Z. We partition Z into two regions where the rank condition is satisfied, i.e., where z is relevant, and otherwise.
Definition 2.1 (Relevant set) Let Z r be the subset of Z defined by r
r
Z = Z (⇧0 (·)) =
⇢
z 2 Z : rank
✓
@⇧0 (z) @z 0
◆
= dx .
Let Z 0 = Z\Z r be the complement of the relevant set. Let X r be the subset of X defined by
X r = {x 2 Xz : z 2 Z r }. Given the definitions, we introduce an additional support condition.
Assumption ID2 The supports X and X r di↵er only on a set of probability zero, i.e., Pr [x 2 X \X r ] = 0.
Intuitively, when z is in the relevant set, x = ⇧0 (z) + v varies as z varies, and therefore, the support of x corresponding to the relevant set is large. Assumption ID2 assures that the corresponding support is large enough to almost surely cover the entire support of x. ID2 is not as strong as it may appear to be. Below, we show this by providing mild sufficient conditions for ID2. If we identify g0 (x) for any x 2 X r , then we achieve identification of g0 (x) by Assumption ID2.6
Now, in order to identify g0 (x) for x 2 X r , we need a rank condition, which will be minimal. The following is the identification result:
Theorem 2.2 In model (2.1), suppose Assumptions ID1 and ID2 hold. Then, g0 (x) is identified on X up to an additive constant if and only if
Pr rank
✓
@⇧0 (z) @z 0
◆
= dx > 0.
(2.4)
This and all subsequent proofs can be found in the Appendix. The rank condition (2.4) is necessary and sufficient. By Definition 2.1, it can alternatively be written as Pr [z 2 Z r ] > 0. The condition is substantially weaker than (2.3) in Proposition 2.1, which is Pr [z 2 Z r ] = 1 (with z = z2 ). That is, Theorem 2.2 extends the result of NPV in the sense that when Z r = Z, ID2 is trivially satisfied with X = X r . Theorem 2.2 shows that it is enough
for identification of g0 (x) to have any fixed positive probability with which the rank condition is satisfied.7 This condition can be seen as the local rank condition as in Chesher (2003), and we achieve global identification with a local rank condition. Although this gain comes from having the additional support condition, the trade-o↵ is appealing given the later purpose of building a weak 6 The support on which an unknown function is identified is usually left implicit in the literature. To make it more explicit, g0 (x) is identified if g0 (x) is identified on the support of x almost surely. 7 A similar condition appears in the identification analysis of Hoderlein (2009), where endogenous semiparametric binary choice models are considered in the presence of heteroskedasticity.
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identification notion. Even without Assumption ID2, maintaining the assumptions of Theorem 2.2, we still achieve identification of g0 (x), but on the set {x 2 X r }.
Lastly, in order to identify the level of g0 (x), we need to introduce some normalization as in NPV. Either E["] = 0 or 0 (¯ v ) = ¯ suffices to pin down g0 (x). With the latter normalization, it follows that g0 (x) = E[y|x, v = v¯] ¯ , which we apply in estimation as it is convenient to implement. The following is a set of sufficient conditions for Assumption ID2. Let Vz be the conditional
support of v given z 2 Z.
Assumption ID20 Either (a) or (b) holds. (a) (i) x is univariate and x and v are continuously distributed, (ii) Z is a cartesian product of connected intervals, and (iii) Vz = Vz˜ for all z, z˜ 2 Z 0 ; (b) Vz = Rdx for all z 2 Z.
Lemma 2.1 Under Assumption ID1, Assumption ID20 implies Assumption ID2. In Assumption ID20 , the continuity of the r.v. is closely related to the support condition in Proposition 2.1 that the boundary of support of (z, v) has probability zero. For example, when z or v is discrete their condition does not hold. Assumption ID20 (a)(i) assumes that the endogenous variable is univariate, which is most empirically relevant in nonparametric models. An additional condition is required with multivariate x, which is omitted in this paper. Even under ID20 (a)(i), however, the exogenous covariate z1 in g(x, z1 ), which is omitted in the discussion, can still be a vector. ID20 (a)(ii) and (iii) are rather mild. ID20 (a)(ii) assumes that z has a connected support, which in turn requires that the excluded instruments vary smoothly. The assumptions on the continuity of the r.v. and the connectedness of Z are also useful in deriving the asymptotic theory
of the series estimator considered in this paper; see Assumption B below. ID20 (a)(iii) means that the conditional support of v given z is invariant when z is in Z 0 . This support invariance condition is the key to obtaining a rank condition that is considerably weaker than that of NPV. Our support
invariance condition is di↵erent from the support invariance condition introduced in Imbens and Newey (2009). Using the notations of the present paper, Imbens and Newey (2009) require that the support of v conditional on x equals the marginal support of v, which inevitably requires a large support of z. On the other hand, ID20 (a)(iii) requires that the support of v conditional on z is invariant (for z 2 Z 0 ), and therefore imposes no restriction on the support of z. Also, the conditional support does not have to equal the marginal support of v here. ID20 (a)(iii), along with the control
function assumptions (2.1b), is a weaker orthogonality condition for z than the full independence condition z ? v. Note that Vz = {x
⇧0 (z) : x 2 Xz }. Therefore, ID20 (a)(iii) equivalently means
that Xz is invariant for those z satisfying E[x|z] = const. Moreover, one can introduce a condition that is weaker than ID20 (a)(iii): Xz ⇢ X r for those z satisfying E[x|z] = const.8 These conditions in terms of Xz can be checked from the data. 8
Therefore, some types of heteroskedasticity of v can be allowed under ID20 (a)(iii) or under this weaker condition.
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Figure 1: Identification under Assumption ID20 (a), univariate z and no z1 . Given ID20 (b) that v has a full conditional support, ID2 is trivially satisfied and no additional restriction is imposed on the joint support of z and v. ID20 (b) also does not require univariate x or the connectedness of Z. This assumption on Vz is satisfied with, for example, a normally distributed error term (conditional on regressors).
Figure 1 illustrates the intuition of the identification proof under ID20 (a) in a simple case where z is univariate. With ID20 (b), the analysis is even more straightforward; see the proof of Lemma 2.1 in the Appendix. In the figure, the local rank condition (2.4) ensures global identification of g0 (x). The intuition is as follows. First, by @E[y|v, z]/@z = (@g0 (x)/@x) · (@⇧0 (z)/@z) and the rank
condition, g0 (x) is locally identified on x corresponding to a point of z in the relevant set Z r . As such a point of z varies within Z r , the x corresponding to it also varies enough to cover almost the
entire support of x. At the same time, for any x corresponding to an irrelevant z (i.e., z outside of Z r ), one can always find z inside of Z r that gives the same value of such an x. The probability
Pr [z 2 Z r ] being small but bounded away from zero only a↵ects the efficiency of estimators in the estimation stage. This issue is related to the weak identification concept discussed later.
Note that the strength of identification of g0 (x) is di↵erent for di↵erent subsets of X . For
instance, identification must be strong in a subset of X corresponding to a subset of Z where
⇧0 (·) is steep. In addition, g0 (x) is over-identified on a subset of X that corresponds to multiple
subsets of Z where ⇧0 (·) has a nonzero slope, since each association of x and z contributes to identification. This discussion implies that the shape of ⇧0 (·) provides useful information on the
strength of identification in di↵erent parts of the domain of g0 (x). Lastly, it is worth mentioning that the separable structure of the reduced form along with ID20 (a)(iii) allows “global extrapolation” in a manner that is analogous to that in a linear model. The identification results of this section apply to economically relevant situations. Let x be an economic agent’s optimal decision induced by an economic model and z be a set of exogenous components in the model that a↵ects the decision x. One is interested in a nonlinear e↵ect of the 9
optimal choice on a certain outcome y in the model. We present two situations where the resulting Pr [z 2 Z r ] is strictly less than unity in this economic problem: (a) x is realized as a corner solution beyond a certain range of z. In a returns-to-schooling example, x can be the schooling decision of a potential worker, z the tuition cost or distance to school, and y the future earnings. When the tuition cost is too high or the distance to school is too far beyond a certain threshold, such an instrument may no longer a↵ect the decision to go to school. (b) The budget set has kink points. In a labor supply curve example, x is the before-tax income, which is determined by the labor supply decision, z the worker’s characteristics that shift her utility function, and y the wage. If an income tax schedule has kink points, then the x realized at such points will possibly be invariant at the shift of the utility. The identification results of this paper imply that even in these situations, the returns to schooling or the labor supply curve can be fully identified nonparametrically as long as Pr [z 2 Z r ] > 0.
3
Weak Identification
The previous section discusses the structure of the joint distribution of x and z that contributes to the identification of g 0 (·). Specifically, (2.4) imposes a minimal restriction on the shape of the conditional mean function E[x|z] = ⇧0 (z). This necessity result suggests that “slight violation” of (2.4) will result in weak identification of g 0 (·). Note that this approach will not be successful with (2.3) of NPV, since violating the condition, i.e., Pr [rank (@⇧0 (z)/@z 0 ) = dx ] < 1, can still result in strong identification. In this section, we formally construct the notion of weak identification via localization. We define nonparametric weak instruments as a drifting sequence of reduced-form functions that are localized around a function with no identification power. Such a sequence of models or drifting data-generating process (Davidson and MacKinnon (1993)) is introduced to define weak instruments relative to the sample size n. As a result, the strength of instruments is represented in terms of the rate of localization, and hence, it can eventually be reflected in the local asymptotics of the estimator of g0 (·). Let C(Z) be the class of conditional mean functions ⇧(·) on Z that are bounded, Lipschitz
and continuously di↵erentiable. Define a non-identification region C0 (Z) as a class of functions that satisfy the lack-of-identification condition motivated by (2.4)9 : C0 (Z) = {⇧(·) 2 C(Z) :
Pr [rank (@⇧(z)/@z 0 ) < dx ] = 1}. Define an identification region as C1 (Z) = C(Z)\C0 (Z). We consider a sequence of triangular models y = g0 (x) + " and x = ⇧n (z) + v with corresponding stochastic
assumptions. Although g(x) is identified with ⇧n (·) 2 C1 (Z) for any fixed n by Theorem 2.2, g(x) is ¯ in C0 (Z). Namely, the noise (i.e., v) cononly weakly identified as ⇧n (·) drifts toward a function ⇧(·) tributes more than the signal (i.e., ⇧n (z)) to the total variation of x 2 {⇧n (z) + v : z 2 Z, v 2 V} 9
The lack of identification condition is satisfied either when the order condition fails (dz < dx ), or when z are jointly irrelevant for one or more of x, almost everywhere in their support.
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as n ! 1. In order to facilitate a meaningful asymptotic theory in which the e↵ect of weak instruments is reflected, we further proceed by considering a specific sequence of ⇧n (·).
Assumption L (Localization) For some > 0, the true reduced-form function ⇧n (·) satisfies ˜ 2 C1 (Z) that does not depend on n and for z 2 Z the following. For some ⇧(·) @⇧n (z) =n @z 0
˜ @ ⇧(z) + op (n @z 0
).
˜ · ⇧(z) + c + op (n
)
·
Assumption L is equivalent to ⇧n (z) = n
(3.1)
for some constant vector c. This specification of a uniform convergent sequence over Z can be justified by our identification analysis. The “local nesting” device in (3.1) is also used in Stock and
Wright (2000) and Jun and Pinkse (2012) among others. In contrast to these papers, the value measures the strength of identification here and is not specified to be 1/2.10 Unlike a linear
of
reduced form, to characterize weak instruments in a more general nonparametric reduced form, we need to control the complete behavior of the reduced-form function, and the derivation of local asymptotic theory seems to be more demanding. Nevertheless, the particular sequence considered in Assumption L makes the weak instrument asymptotic theory straightforward while embracing the most interesting local alternatives against non-identification.11
4
Estimation
Once the endogeneity is controlled by the control function in (2.2), the problem becomes one of estimating the additive nonparametric regression function E[y|x, z] = g0 (x) +
0 (v).
In a weak
instrument environment, however, we face a nonstandard problem called concurvity: x = ⇧n (z) + v ! v a.s. as n ! 1 under the weak instrument specification (3.1) of Assumption L with c = 0 P as normalization. With a series representation g0 (x) + 0 (v) = 1 j=1 { 1j pj (x) + 2j pj (v)}, where the pj (·)’s are the approximating functions, it becomes a familiar problem of multicollinearity as
pj (x) ! pj (v) a.s. for all j. More precisely, pj (x) pj (v) = Oa.s. (n ) by mean value expansion ˜ ˜ pj (v) = pj (x n ⇧(z)) = pj (x) n ⇧(z)@p x)/@x with an intermediate value x ˜. Alternatively, j (˜ by plugging this expression of pj (v) back into the series, we can see that the variation of the regressor It would be interesting to have di↵erent rates across columns or rows of @⇧@zn0(·) . One can also consider di↵erent rates for di↵erent elements of the matrix. The analyses in these cases can analogously be done by slight modifications of the arguments. 11 In defining weak instruments in Assumption L, one can consider an intermediate case where @⇧@zn0(·) converges to a matrix with reduced-rank rather than that with zero rank. Extending the analysis in this case can follow analogously but omitted in the paper for succinctness. 10
11
shrinks as n ! 1. This feature is reminiscent of the ill-posed inverse problem that commonly occurs,
e.g., in estimating a standard NPIV model of y = g0 (x) + " with endogenous x and E["|z] = 0. By P P1 writing g0 (x) = 1 j=1 1j pj (x) it follows that E[y|z] = j=1 1j E [pj (x)|z]. Analogous to the weak ⇥ ⇤ instruments problem, the variation of the regressor E [pj (x)|z] shrinks since E E[pj (x)|z]2 ! 0 as j ! 1 (Kress (1999, p. 235)). Blundell and Powell (2003, p. 321) also acknowledge that the ill-posed inverse problem is a functional analogue to the multicollinearity problem.
Given the connections between the weak instruments, concurvity, and ill-posed inverse problems, the regularization methods used in the realm of research concerning inverse problems are suitable for use with weak instruments. There are two types of regularization methods used in the literature: the truncation method and the penalization method.12 In this paper, we introduce the penalization scheme. The nature of our problem is such that the truncation method does not work properly. Unlike in the ill-posed inverse problem, the estimators of
1j
and
2j
above can still be unstable
even after truncating the series since we still have pj (x) ! pj (v) a.s. for j J < 1 as n ! 1. On the other hand, the penalization directly controls the behavior of the
1j ’s
and
2j ’s,
and hence, it
successfully regularizes the weak instrument problem. We propose a penalized series estimation procedure for h0 (w) = g0 (x) +
0 (v)
where w = (x, v).
We choose to use series estimation rather than other nonparametric methods as it is more suitable in our particular framework. Because x ! v a.s. the joint density of w becomes concentrated along a lower dimensional manifold as n tends to infinity. As mentioned in the introduction, series estimators are less sensitive to this problem of singular density. See Assumption B below for related discussions. Furthermore with series estimation, it is easy to impose the additivity of h0 (·) and to characterize the problem of weak instruments as a multicollinearity problem. The estimation procedure takes two steps. In the first stage, we estimate the reduced form ⇧n (·) using a standard series estimation method and obtain the residual vˆ. In the second stage, we estimate the structural function h0 (·) using a penalized series estimation method with w ˆ = (x, vˆ) as the regressors. The theory that follows uses orthonormal polynomials and regression splines as approximating functions.13 Let {(yi , xi , zi )}ni=1 be the data with n observations, and let rL (zi ) =
(r1 (zi ), ..., rL (zi ))0 be a vector of approximating functions of order L for the first stage. Define ˆ a matrix R = (rL (z1 ), ..., rL (zn ))0 . Then, regressing xi on rL (zi ) gives ⇧(·) = rL (·)0 ˆ where ˆ =
n⇥L 0 (R R) 1 R0 (x
1 , ..., xn )
0,
and we obtain vˆi = xi
ˆ i ). Define a vector of approximating ⇧(z
functions of order K for the second stage as pK (w) = (p1 (w), ..., pK (w))0 . To reflect the additive structure of h0 (·), there are no interaction terms between the approximating functions for g0 (·) and those for
0 (·)
in this vector; see the Appendix for the explicit expression. Denote a matrix of approximating functions as Pˆ = (pK (w ˆ1 ), ..., pK (w ˆn ))0 where w ˆi = (xi , vˆi ). Note that L = L(n) n⇥K
and K = K(n) grow with n. 12
In Chen and Pouzo (2012) closely related concepts are used in di↵erent terminologies: minimizing a criterion over finite sieve space and minimizing a criterion over infinite sieve space with a Tikhonov-type penalty. 13 For detailed descriptions of power series and regression splines, see Newey (1997).
12
We define a penalized series estimator : ˆ ⌧ (w) = pK (w)0 ˆ⌧ , h
(4.1)
where the “interim” estimator ˆ⌧ optimizes a penalizing criterion function ˆ⌧ = arg min
˜2RK
where y = (y1 , ..., yn )0 and ⌧n
⇣
y
⌘0 ⇣ Pˆ ˜ y
⌘ Pˆ ˜ /n + ⌧n ˜0 ˜,
(4.2)
0 the penalization parameter. To control bias, ⌧n is assumed to
converge to zero. The optimization problem (4.2) yields a closed form solution: ˆ⌧ = (Pˆ 0 Pˆ + n⌧n IK )
1
Pˆ 0 y.
The multicollinearity feature discussed above is manifested here by the fact that the matrix Pˆ 0 Pˆ is nearly singular under Assumption L, since the two columns of Pˆ become nearly identical. In terms of ⇥ ⇤ the population second moment matrix Q = E pK (wi )pK (wi )0 , the challenge is that the minimum eigenvalue of Q is not bounded away from zero, which is manifested as in Lemma A.2 in the Appendix) where
max
max (Q
1)
= O(n2 ) (shown
denotes the maximum eigenvalue. The term n⌧n IK
mitigates such singularity, without which the performance of the estimator of h0 (·) would deteriorate severely.14 The relative e↵ects of weak instruments (n2 ) and penalization (⌧n ) will determine the ˆ ⌧ (·). Given h ˆ ⌧ (·), with the normalization that (¯ asymptotic performance of h v ) = ¯ , we have ˆ ⌧ (x, v¯) ¯ . gˆ⌧ (x) = h
5
Consistency and Rate of Convergence
First, we state the regularity conditions and key preliminary results under which we find the rate of convergence of the penalized series estimator introduced in the previous section. Let X = (x, z). Assumption A {(yi , xi , zi ) : i = 1, 2, ...} are i.i.d. and var(x|z) and var(y|X) are bounded functions of z and X, respectively.
Assumption B (z, v) is continuously distributed with density that is bounded away from zero on Z ⇥ V, and Z ⇥ V is a cartesian product of compact, connected intervals. Assumption B is useful to bound below and above the eigenvalues of the “transformed” second moment matrix of approximating functions. This condition is worthy of discussion in the context of 14
In linear settings, the introduction of a regularization method is less appealing as it creates the well-known biased estimator of ridge regression. In contrast, we do not directly interpret ˆ⌧ in the current nonparametric setting, since ˆ ⌧ (·). More importantly, the overall bias of h ˆ ⌧ (·) is unlikely to it is only an interim estimator calculated to obtain h be worsened in the sense that the additional bias introduced by penalization can be dominated by the existing series estimation bias.
13
identification and weak identification. Let fu and fw denote the density functions of u = (z, v) and w = (x, v), respectively. An identification condition like Assumption ID20 in Section 2 is embodied in Assumption B. To see this, note that fu being bounded away from zero means that there is no functional relationship between z and v, which in turn implies Assumption ID20 (a)(iii).15 On the other hand, an assumption written in terms of fw like Assumption 2 in NPV (p. 574) cannot be imposed here. Observe that w = (⇧n (z) + v, v) depends on the behavior of ⇧n (·), and hence fw is not bounded away from zero uniformly over n under Assumption L and approaches a singular density. Technically, making use of a transformation matrix (see the Appendix), an assumption is made in terms of fu , which is not a↵ected by weak instruments, and the e↵ect of weak instruments can be handled separately in the asymptotics proof. Note that the assumption for the Cartesian products of supports, namely Z ⇥ V and its compactness can be replaced by introducing a trimming
function as in NPV, that ensures bounded rectangular supports. Assumption B can be weakened to hold only for some component of the distribution of z; some components of z can be allowed to be discrete as long as they have finite supports. Next, Assumption C is a smoothness assumption on the structural and reduced-form functions. Let W be the support of w = (x, v). Let Fx , Fv and Fw be the distribution functions of x, v and w, respectively.
Assumption C g0 (x) and
0 (v)
are in L2 (Fx ) and L2 (Fv ), respectively, and are Lipschitz and con-
tinuously di↵erentiable of order s on W. ⇧n (z) is bounded, Lipschitz and continuously di↵erentiable of order s⇡ on Z.
This assumption ensures that the series approximation error shrinks as the number of approxP 2 imating functions increases. Let K (K) for a nonnegative function (·). Since j=1 |hh0 , pj i| h0 (w) is in L2 (Fw ) by Assumption C,
(K) = O(1).
p p L/n + L s⇡ /dz ) ! 0, K 4 ( L/n + L s⇡ /dz ) ! 0, p p K 3 ! 0, L2 /n ! 0; (b) for power series, n K 7/2 ( L/n+L s⇡ /dz ) ! 0, K 8 ( L/n+L s⇡ /dz ) !
Assumption D (i)(a) For splines, n K 2 ( n
0, n
K 11/2 ! 0, L3 /n ! 0; (ii) ⌧n ! 0, n ⌧n ! 0; (iii) n2
(K) ! 0.
Assumption D(i) restricts the rate of growth of the numbers K and L of the approximating functions. The conditions on K and L are more restrictive than the corresponding assumption for the power series in NPV (Assumption 4, p. 575) where weak instruments are not considered. D(ii) restricts the rate of convergence of ⌧n . D(iii) is reminiscent of the source condition in the literature on inverse problem; see e.g., Darolles et al. (2011) and Chen and Pouzo (2012). The source condition relates the degree of ill-posedness to the smoothness of the unknown function of interest. As discussed earlier, the weak instrument problem is related to the ill-posed inverse problem, and n2 15
(K) = O(1) controls the weakness of instruments relative to the smoothness
The definition of a functional relationship can be found, e.g., in NPV (p. 568).
14
of h0 . Now, we provide results for the rate of convergence in probability of the penalized series ˆ ⌧ (w) in terms of L2 and uniform distance. Let F (w) = Fw (w) for simplicity. estimator h Theorem 5.1 Suppose Assumptions A–D and L are satisfied. Let Rn = min{n , ⌧n ⇢ˆ h
ˆ ⌧ (w) h
h0 (w)
i2
1/2
dF (w)
⇣ p = Op Rn ( K/n + K
s/dx
+ ⌧n +
1/2
p L/n + L
}. Then
s⇡ /dz
⌘ ) .
Also, for r = 1/2 for splines and r = 1 for power series, ˆ ⌧ (w) sup h
w2W
⇣ p h0 (w) = Op Rn K r ( K/n + K
s/dx
+ ⌧n +
p L/n + L
s⇡ /dz
⌘ ) .
Suppose there is no penalization (⌧n = 0). Then with Rn = Op (n ), Theorem 5.1 provides the ˆ rates of convergence of the unpenalized series estimator h(·). For example, with k·kL2 denoting the L2 norm above,
ˆ h
h0
L2
⇣ p = Op n ( K/n + K
s/dx
+
p L/n + L
s⇡ /dz
⌘ ) .
(5.1)
Compared to the strong instrument case of NPV (Lemma 4.1, p. 575), the rate deteriorates by p the leading n rate, the weak instrument rate. Note that the terms K/n and K s/dx correspond p to the variance and bias of the second stage estimator, respectively,16 and L/n and L s⇡ /dz are
those of the first stage estimator. The latter rates appear here due to the fact that the residuals vˆi are generated regressors obtained from the first-stage nonparametric estimation. The way that n enters into the rate implies that the e↵ect of weak instruments (hence concurvity) not only exacerbates the variance but also the bias.17 Moreover, the symmetric e↵ect of weak instruments on bias and variance implies that the problem of weak instruments cannot be resolved by the choice of the number of terms in the series estimator. This is also related to the discussion in Section 4 that the truncation method does not work as a regularization method for weak instruments. More importantly, in the case where penalization is in operation (⌧n > 0), the way that Rn enters into the convergence rates implies that penalization can reduce both bias and variance by the same mechanism working in an opposite direction to the e↵ect of weak instruments. Explicitly, when Rn = min{n , ⌧n ˆ⌧ h
h0
1/2
} = ⌧n
L2
1/2
, penalization plays a role and yields the rate
⇣ p = Op ⌧n 1/2 ( K/n + K
s/dx
+ ⌧n +
Here, the overall rate is improved since the multiplying rate ⌧n 16
p L/n + L
1/2
s⇡ /dz
⌘ ) .
(5.2)
is of smaller order than the mul-
The dimension of w is reduced to the dimension of x as the additive structure of h0 (w) is exploited; see e.g., Andrews and Whang (1990). 17 This is di↵erent from a linear case where multicollinearity only results in imprecise estimates but does not introduce bias. This is also di↵erent from the ill-posed inverse problem where the degree of ill-posedness only a↵ects variance.
15
tiplying rate n of the previous case. There is the additional bias term introduced by penalization, namely, the penalty bias ⌧n . Even after multiplying the multiplier ⌧n
1/2
1/2
, ⌧n
! 0 by Assumption
D(ii). When the e↵ect of weak instruments prevails over the e↵ect of penalization, the penalty (with 1/2
the multiplier n ) becomes ⌧n n which converges to zero since ⌧n n ! 0 in this situation.
Next, we find the optimal L2 convergence rate. For a more concrete comparison between the
rates n and ⌧n
1/2
, let ⌧n = n
2
⌧
for some
⌧
> 0. For example, the larger
⌧
is, the faster the
penalization parameter converges to zero, and hence, the smaller the e↵ect of penalization is. Corollary 5.2 (Consistency) Suppose the Assumptions of Theorem 5.1 are satisfied. Let K = ˆ ⌧ h0 O(n1/(1+2s/dx ) ) and L = O(n1/(1+2s⇡ /dz ) ). Then h = Op (n q ) = op (1), where q = 2 L n o s⇡ s , dz +2s , 2 min{ , ⌧ }. min dx +2s ⌧ ⇡ For the penalization to play a role we want that converge too slowly (i.e., the overall
rate.18
⌧
⌧
< . But at the same time, ⌧n cannot
cannot be too small) lest q becomes too small, which will deteriorate
It can be readily shown that q > 0 from Assumption D(i) and (ii). Given the 3 1+2s/dx
choice of K and L in the corollary with splines, Assumption D(i)(a) implies that s⇡ dz +2s⇡
2 1+2s/dx .
<
<
For example, when the nonparametric functions are smooth (s = s⇡ = 1), it
should be that 0 <
< 1/2, requiring instruments to be mildly weak compared to the n
1/2
rate
typically introduced in parametric settings. Corollary 5.2 has several implications. Let dx = dz = d, s = s⇡ and so that q =
s d+2s
min{ ,
⌧ }.
s d+2s
<2
Consider a weak instruments-prevailing case of
⌧
for simplicity <
⌧.
When
the structural function is less smooth or has a high dimensional argument (i.e., small s or large d, and hence, small
s d+2s ),
instruments should not be too weak (i.e., small ) to achieve the same
optimal rate (i.e., holding q fixed). This implies a trade-o↵ between the smoothness of the structural function (or the dimensionality) and the required strength of instruments, which is relevant to the source condition (Assumption D(iii)). This, in turn, implies that the weak instrument problem can be mitigated with some smoothness restrictions, which is in fact one of our justifications for introducing the penalization method. Once the penalization e↵ect is prevailing ( and a faster rate is
⌧
< ), q increases
achieved.19
When implementing the penalized series estimator in practice, there remains the issue of choosing tuning parameters, namely, the penalization parameter ⌧n and the orders K and L of the series. In the simulations, we present results with a few chosen values of ⌧ , K, and L. A data-driven procedure such as the cross-validation method can also be used (Arlot and Celisse (2010)), and is done for choosing ⌧ in the empirical section below. There may be, however, no optimal method in the literature for choosing the tuning parameters; in the context of the NPIV, see a related 18
Relatedly, Assumption D(ii) implies that /2 < ⌧ . With weak instruments, optimal rates in the sense of Stone (1982) are not attainable. Also the uniform convergence rate does not attain Stone’s (1982) bound even without the weak instrument factor (Newey (1997, p. 151)) and hence is not discussed here. 19
16
discussion in Blundell et al. (2007, pp. 1636–1637)). It would be interesting to further investigate ˆ = Pˆ 0 Pˆ /n and recall that the sensitivity of the penalized estimator to the choice of ⌧ . Denote Q ˆ ⌧ (·) = pK (·)0 (Q ˆ ⌧ (·) to ⌧ , we ˆ + ⌧ I) 1 Pˆ 0 y/n is a function of ⌧ . To determine the sensitivity of h h consider the sharp bound on
ˆ
max ((Q+⌧ I)
1 ),
which is (
ˆ
min (Q)+⌧ )
1
by (A.29) in the Appendix.
As a measure of sensitivity, we calculate @(
ˆ + ⌧) @⌧
min (Q)
1
=(
ˆ + ⌧)
min (Q)
2
.
(5.3)
ˆ That is, as instruments become weaker Note that the sensitivity is a decreasing function of min (Q). ˆ ⌧ (·) becomes more sensitive to a change in ⌧ . ˆ becomes smaller), the performance of h (i.e., min (Q) This has certain implications in practice. Theorem 5.1 leads to the following theorem, which focuses on the rate of convergence of the structural estimator gˆ⌧ (·) after subtracting the constant term which is not identified. Theorem 5.3 Suppose Assumptions A–D and L are satisfied. Let Rn = min{n , ⌧n ˆ (x) = gˆ⌧ (x) g0 (x), (ˆ
ˆ (x)
ˆ
ˆ (x)dFw
ˆ ⌧ (x, v¯) Also, if gˆ⌧ (x) = h series, sup |ˆ g⌧ (x)
x2X
2
dFw
)1/2
¯ and ¯ =
⇣ p = Op Rn ( K/n + K v ), 0 (¯
s/dx
+ ⌧n +
p L/n + L
1/2
}. For
s⇡ /dz
⌘ ) .
then, for r = 1/2 for splines and r = 1 for power
⇣ p g0 (x)| = Op Rn K r ( K/n + K
s dx
+ ⌧n +
p L/n + L
s⇡ dz
⌘ ) .
The optimal rate results for gˆ⌧ (·) and the related analyses can be followed analogously, and we omit them here. The convergence rate is net of the constant term. We can further assume E["] = 0 to identify the constant. We discuss one of the practical implications of the identification and asymptotic results thus far. In applied research that uses nonparametric triangular models, a linear specification of the reduced form is largely prevalent; see, e.g., NPV, Blundell and Duncan (1998), Blundell et al. (1998), Yatchew and No (2001), Lyssiotou et al. (2004), Dustmann and Meghir (2005), and Del Bono and Weber (2008). While a linear reduced-form relationship is rarely justified by economic theory, linear specification is introduced to avoid the curse of dimensionality with many covariates, or for an ad hoc reason that it is easy to implement and that the nonparametric structural equation is of primary interest. When the reduced form is linearly specified, however, any true nonlinear relationship is “flattened out,” and the situation is more likely to have the problem of weak instruments, let alone the problem of misspecification. On the other hand, one can achieve a significant gain in the performance of the estimator by nonparametrically estimating the relationship of x and z. According 17
to (2.4), identification power can be enhanced by exploiting the entire nonlinear relationship between x and z. This phenomenon may be interpreted in terms of the “optimal instruments” in the GMM settings of Amemiya (1977); see also Newey (1990) and Jun and Pinkse (2012). The nonparametric first stage estimation is not likely to worsen the overall convergence rate of the estimator, since the nonparametric rate from the second stage is already present.
6
Asymptotic Distributions
ˆ ⌧ (·). We establish the asymptotic normality of the functionals of the penalized series estimator h We consider linear functionals of h0 (·) that include h0 (·) at a certain value (i.e., h0 (w)) ¯ and the ´ weighted average derivative of h0 (·) (i.e., #(w) [@h0 (w)/@x] dw). The linear functionals of h = h0 (·) ˆ ⌧ ) of ✓0 = a(h) is the natural “plug-in” estimator. are denoted as a(h). Then, the estimator ✓ˆ⌧ = a(h Let A = (a(p1K ), a(p2K ), ..., a(pKK )), where pjK (·) is an element of pK (·). Then, ˆ ⌧ ) = a(pK (x)0 ˆ⌧ ) = A ˆ⌧ . ✓ˆ⌧ = a(h Then the following variance estimator of ✓ˆ⌧ can naturally be defined: ⇣ ⌘ ˆ 1 ⌃ ˆ⌧ + H ˆ⌧ Q ˆ 1⌃ ˆ 1Q ˆ 1H ˆ0 Q ˆ 1 A0 , Vˆ⌧ = AQ ⌧ ⌧ ⌧ 1 1
ˆ⌧ = ⌃
n X
pK ( w ˆi )pK (w ˆi )0 [yi
i=1
ˆ⌧ = H
n X
pK ( w ˆi )
i=1
⇢h
ˆ ⌧ (w h ˆi )]2 /n,
ˆ1 = ⌃
n X
vˆi2 rL (zi )rL (zi )0 /n,
i=1
i0
ˆ ⌧ (w ˆ i ))/@⇡ rL (zi )0 /n, @h ˆi )/@w @!(Xi , ⇧(z
ˆ 1 = R0 R/n, Q
where X is a vector of variables that includes x and z and !(X, ⇡) is a vector of functions of X and ⇡ where ⇡ is a possible value of ⇧(z). The following are additional regularity conditions for the asymptotic normality of ✓ˆ⌧ . Let ⌘ = y h. Assumption E
2 (X)
h i ⇥ ⇤ = var(y|X) is bounded away from zero, E ⌘ 4 |X is bounded, and E kvk4 |X
is bounded. Also, h0 (w) is twice continuously di↵erentiable in v with bounded first and second derivatives. This assumption strengthens the boundedness of conditional second moments in Assumption A. For the next assumption, let |h|r = max|µ|r supw2W |@ µ h(w)|. Also let p˜K (w) be a generic vector of approximating functions.
Assumption F a(h) is a scalar, |a(h)| |h|r˜ for some h r˜ 0, andithere exists 2 K0 K ! 1, a(p p˜K (w)0 K ! 0. K ) is bounded away from zero while E 18
K
such that as
This assumption includes the case of h at a certain value. The next condition restricts the rate of growth of Kand L and the rate of convergence of ⌧n . 1/2
Assumption G For ⌧ > 0, Rn = ⌧n = n ⌧ , and the following converge to zero as n ! 1: p p p p nK s/dx , nL s⇡ /dz , n⌧n Rn (K); also for spline, n4 ⌧ 1 (K 2 +L2 ), n ⌧ 1/2 (K 3/2 L3/2 +K 2 L), p p n3 ⌧ 1/2 (K 7/2 L/ n + K 5/2 L1/2 + K log(K)), n K 7/2 , and for polynomials, n4 ⌧ 1 (K 3 + L3 ), p p p L3/2 log(L)/ n, n ⌧ 1/2 (K 3 L3/2 + K 1/2 L3/2 + K 9/2 L1/2 + K 4 L), n3 ⌧ 1/2 (K 13/2 L/ n+ K 4 L1/2 + p K 3/2 log(K)), n K 6 . By Rn = ⌧n
1/2
, Assumption G supposes that the penalization is in operation, while imposing
more restrictions on the behavior of ⌧n (and of K and L) than Assumption D. The conditions p p nK s/dx ! 0 and nL s⇡ /dz ! 0 introduce overfitting in that the bias (K s/dx ) shrinks faster p than 1/ n, the usual rate of standard deviation of the estimator. The same feature is found in the corresponding assumption in NPV (Assumption 8, p. 582), while the overall conditions p p on K and L in Assumption G are stronger than that in NPV. The condition n⌧n Rn (K) = p +1/2 ⌧ n (K) ! 0 essentially implies that the unknown function should be smooth when penalization has a mediocre e↵ect in the presence of weak instruments.
Theorem 6.1 If Assumptions A–G and L are satisfied, then ✓ˆ⌧ = ✓0 + Op (n2 p
nVˆ⌧
1/2
(✓ˆ⌧
⌧
1 2
⇣r˜v (K)) and
✓0 ) !d N (0, 1).
In addition to asymptotic normality, the results also provide the bound on the convergence rate ˆ of ✓⌧ . The bound on the rate achieved here is slower than that of NPV due to the penalization p introduced for weak instruments. Under the following assumption, n-consistency is achieved. Let p˜⇤K (z, v) be a “transformation” of p˜K (w) purged of the weak instruments e↵ect (see the Appendix). h i Assumption H There exists ⌫(w) and ↵K such that E k⌫(w)k2 < 1, a(h) = E [⌫(w)h0 (w)], h i 2 a(pj ) = E [⌫(w)pj (w)], and E ⌫(⇧n (z) + v, v) p˜⇤K (z, v)0 ↵K ! 0 as K ! 1. This assumption includes the case of the weighted average derivative of h, in which case ⌫(w) = fw (w)
1 @#(w)/@w.
Let ⇢(z) = E [⌫(w)@h0 (w)/@v 0 |z]. Under this assumption, the asymptotic
variance of ✓ˆ in this case can be expressed as
⇥ ⇤ ⇥ ⇤ V¯ = E ⌫(w)⌫(w)0 var(y|X) + E ⇢(z)var(x|z)⇢(z)0 .
Corollary 6.2 If Assumptions A–E, G, H and L are satisfied, then p
n(✓ˆ⌧
✓0 ) !d N (0, V¯ ),
19
Vˆ⌧ !p V¯ .
In this case, instruments can be regarded as nearly strong in that
p
n-consistency is achieved as
if instruments are strong, as in NPV. There still remain issues when the results of Theorem 6.1 and Corollary 6.2 are used for inference, e.g., for constructing pointwise asymptotic confidence intervals. As long as nuisance parameters are present, such an inferential procedure may depend on the strength of instruments or on the choice of the penalization parameter. Developing a robust procedure against weak instruments in nonparametric models is beyond the scope of our paper, and we leave it to future research.
7
Monte Carlo Simulations
In this section, we document the problems of weak instruments in nonparametric estimation and investigate the finite sample performance of the penalized estimator. We are particularly interested in the finite sample gain in terms of the bias, variance, and mean squared errors (MSE) of the penalized series estimators defined in Section 4 (“penalized IV (PIV) estimators”) relative to those of the unpenalized series estimators (“IV estimators”) for a wide range of strength of instruments. We consider the following data generating process: y=
✓
x
µx x
◆
+ ",
x = ⇡1 + z⇡ + v,
2 2 0 where " y, x, and # z are univariate, z ⇠ N (µz , z ) with µz = 0 and z = 1, and (", v) ⇠ N (0, ⌃) with 1 ⇢ ⌃= . Note that |⇢| measures the degree of endogeneity, and we consider ⇢ 2 {0.2, 0.5, 0.95}. ⇢ 1 The sample {zi , "i , vi } is i.i.d. with size n = 1000. The number of simulation repetitions is s 2
{500, 1000}. We consider di↵erent strengths of the instrument by considering di↵erent values of ⇡. Let the intercept ⇡1 = µx ⇡. Note that
2 x
=
⇡ 2 z2
⇡µz with µx = 2 so that E[x] = µx does not depend on the choice of
+ 1 still depends on ⇡, which is reasonable since the signal contributed
to the total variation of x is a function of ⇡. More specifically, to measure the strength of the P instrument, we define the concentration parameter (Stock and Yogo (2005)): µ2 = ⇡ 2 ni=1 zi2 / v2 . Note that since the dimension of z is one, the concentration parameter value and the first-stage F -
statistic are similar to each other. For example, in Staiger and Stock (1997), for F = 30.53 (strong instrument), a 97.5% confidence interval for µ2 is [17.3, 45.8], and for F = 4.747 (weak instrument), a confidence interval for µ2 is [2.26, 5.64]. The candidate values of µ2 are {4, 8, 16, 32, 64, 128, 256}, which range from a weak to a strong instrument in the conventional sense.20 Also, with ⇡ = n under Assumption L and
2 z
= 1, the concentration parameter is related to
Suppose ⇡ ˜ = 1 then the range of
that corresponds to the chosen range of
by µ2
µ2
⇡
n1 2
⇡ ˜
⇡ ˜.
is approximately
{0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1}. As for the penalization parameter ⌧ , we consider candidate values 20 The simulation results seem to be unstable when µ2 = 4 (presumably because instruments in this range are severely weak in nonparametric settings), and hence need caution when interpreting them.
20
of {0.001, 0.005, 0.01, 0.05., 0.1}.
The approximating functions used for g0 (x) and
0 (v)
are polynomials with di↵erent choices of
(K1 , K2 ), where K1 is the number of terms for g0 (·), K2 for the normalization g0 (x) = h0 (x, 1)
0 (1)
0 (·),
and K = K1 + K2 . We introduce
= ⇢, where ⇢ is chosen because of the joint normality of (", v). Then,
⇢, where h(x, v) = g(x) + (v).
In the first part of the simulation, we calculate gˆ⌧ (·) and gˆ0 (·), the penalized and unpenalized IV estimates, respectively, and compare their performances. For di↵erent strengths of the instrument, we compute estimates with di↵erent values of the penalization parameter. We choose K1 = K2 = 6 and ⇢ = 0.5.21 As one might expect, the choice of orders of the series is not significant as long as we are only interested in comparing gˆ⌧ (·) and gˆ(·). Figures 2 and 3 present some representative results. Results with di↵erent values of µ2 and ⌧ are similar and hence are omitted to save space. In Figure 2, we plot the mean of gˆ⌧ (·) and gˆ(·) with concentration parameter µ2 = 16 and penalization parameter ⌧ = 0.001. In Figure 2(a), the plot for the unpenalized estimate indicates that with the given strength of the instrument, the variance is very large, which implies that functions with any trends can fit within the 0.025–0.975 quantile ranges; it indicates that the bias is also large. The graph for the penalized estimate shows that the penalization significantly reduces the variance so that the quantile range implies the upward trend of the true g0 (·). Note that the bias of gˆ⌧ (·) is no larger than that of gˆ(·). Although µ2 = 16 is considered to be strong according to the conventional criterion, this range of the concentration parameter value can be seen as the case where the instrument is “nonparametrically” weak in the sense that the penalization induces a significant di↵erence between gˆ⌧ (·) and gˆ(·). Figure 2(b) is drawn with µ2 = 256, while all else remains the same. In this case, the penalization induces no significant di↵erence between gˆ⌧ (·) and gˆ(·). This can be seen as the case where the instrument is “nonparametrically” strong. It is noteworthy that the bias of the penalized estimate is no larger than the unpenalized one even in this case. Figure 3 presents similar plots but with penalization parameter ⌧ = 0.005. Figure 3(a) shows that with a larger value of ⌧ than the previous case, the variance is significantly reduced, while the biases of the two estimates are comparable to each other. The change in the patterns of the graphs from Figure 3(a) to 3(b) is similar to those in the previous case. Furthermore, the comparison between Figure 2 and Figure 3 shows that the results are more sensitive to the change of ⌧ in the weak instrument case than in the strong instrument case. This provides evidence for the theoretical discussion on sensitivity; see (5.3) in Section 5. The fact that the penalized and unpenalized estimates di↵er significantly when the instrument is weak has a practical implication: Practitioners can be informed about whether the instrument they are using is worryingly weak by comparing penalized series estimates with unpenalized estimates. Because of the bivariate normal assumption for (", v)0 , we implicitly impose linearity in the function E["|v] = (v). Although K2 being smaller than K1 would better reflect the fact that 0 (·) is smoother than g0 (·), we assume that we are agnostic about such knowledge. 21
21
A similar approach can be found in the linear weak instruments literature; for example, the biased TSLS estimates and the approximately median-unbiased LIML estimates of Staiger and Stock (1997) can be compared to detect weak instruments. Tables 1 reports the integrated squared bias, integrated variance, and integrated MSE of the penalized and unpenalized IV estimators and least squares (LS) estimators of g0 (·). The LS estimates are calculated by series estimation of the outcome equation (with order K1 ), ignoring the endogeneity. We also calculate the relative integrated MSE for comparisons. We use K1 = K2 = 6 and ⇢ = 0.5 as before. Results with di↵erent choices of orders K1 and K2 between 3 and 10 and a di↵erent degree of endogeneity ⇢ in {0.2, 0.95} show similar patterns. Note that the usual bias and variance trade-o↵s are present as the order of the series changes. In the table, as the instrument
becomes weaker, the bias and variance of the unpenalized IV (⌧ = 0) increase with a greater proportion in variance. The integrated MSE ratios between the IV and LS estimators (M SEIV /M SELS ) indicate the relative performance of the IV estimator compared to the LS estimator. A ratio larger than unity implies that IV performs worse than LS. In the table, the IV estimator does poorly in terms of MSE even when µ2 = 16, which is in the range of conventionally strong instruments; therefore, this can be considered as the case where the instrument is nonparametrically weak. The rest of the results in Table 1 are for the penalized IV (PIV) estimator gˆ⌧ (·). Overall the variance is reduced significantly compared to that of IV without sacrificing much bias. In the case of ⌧ = 0.001, the variance is reduced for the entire range of instrument strength (compared to the unpenalized estimator). Remarkably, the bias is no larger even though penalization is in operation and is reduced when the instrument is weak. This provides evidence for the theoretical discussion in Section 5 that the penalty bias can be dominated by the existing series estimation bias. This feature diminishes as the increased value of ⌧ introduces more bias. The integrated MSE ratios between PIV and IV (M SEP IV /M SEIV ) in Table 1 suggest that PIV outperforms IV in terms of MSE for all the values of ⌧ considered here. For example, when µ2 = 8, the MSE of PIV with ⌧ = 0.001 is only about 12% of that of IV, while the bias (squared) of PIV is only about 39% of that of IV. These results imply that PIV performs substantially better than LS unlike the previous case of IV. The simulation results can be summarized as follows. Even with a strong instrument in a conventional sense, unpenalized IV estimators do poorly in terms of mean squared errors compared to LS estimators. Variance seems to be a bigger problem, but bias is also worrisome. Penalization alleviates much of the variance problem induced by the weak instrument, and it also works well in terms of bias for relatively weak instruments and for some values of the penalization parameter.
8
Application: E↵ect of Class Size
To illustrate our approach and apply the theoretical findings, we nonparametrically estimate the e↵ect of class size on students’ test scores. Estimating the e↵ect of class size has been an interesting 22
topic in the schooling literature, since among school inputs that a↵ect students’ performance, class size is thought to be easier to manipulate. Angrist and Lavy (1999) analyze the e↵ect of class size on students’ reading and math scores in Israeli primary schools. With linear models, they find that the estimated e↵ect is negative in most of the specifications they consider. This specific empirical application is chosen for the following reasons: (i) Although Angrist and Lavy (1999) use an instrument that is considered to be strong for their parametric model, it may not be sufficiently strong when applied in a nonparametric specification of the relationship; see below for details. (ii) The instrument is continuous in this example and presents a nonlinear relationship with the endogenous variable; see Figure 1 in Angrist and Lavy (1999). (iii) We also compare estimates calculated from our triangular model and the NPIV model in Horowitz (2011), where the same example is considered. In this section, we investigate whether the results of Angrist and Lavy (1999) are driven by their parametric assumptions. It is also more reasonable to allow a nonlinear e↵ect of class size, since it is unlikely that the marginal e↵ect is constant across class-size levels. We nonparametrically extend their linear model by considering scoresc = g(classizesc , disadvsc ) + ↵s + "sc for school s and class c, where scoresc is the average test score within class, classizesc the class size, disadvsc the fraction of disadvantaged students, and ↵s an unobserved school-specific e↵ect. Note that this model allows for di↵erent patterns for di↵erent subgroups of school/class characteristics (here, disadvsc ). Class size is endogenous because it results from choices made by parents, schooling providers or legislatures, and hence is correlated with other determinants of student achievement. Angrist and Lavy (1999) use Maimonides’ rule on maximum class size in Israeli schools to construct an IV. According to the rule, class size increases one-for-one with enrollment until 40 students are enrolled, but when 41 students are enrolled, the class size is dropped to an average of 20.5 students. Similarly, classes are split when enrollment reaches 80, 120, 160, and so on, so that each class does not exceed 40. With es being the beginning-of-the-year enrollment count, this rule can be expressed as fsc = es /{int((es
1)/40) + 1}, which produces the IV. This rule generates discontinuity in the
enrollment/class-size relationship, which serves as exogenous variation. Note that with the sample around the discontinuity points, IV exogeneity is more credible in addressing the endogeneity issue. The dataset we use is the 1991 Israel Central Bureau of Statistics survey of Israeli public schools from Angrist and Lavy (1999). We only consider fourth graders. The sample size is n = 2019 for the full sample and 650 for the discontinuity sample. Given a linear reduced form, first stage tests have F = 191.66 with the discontinuity sample (±7 students around the discontinuity points) and F = 2150.4 with the full sample. Lessons from the theoretical analyses of the present paper suggest that an instrument that is strong in a conventional sense (F = 191.66) can still be weak in
23
nonparametric estimation of the class-size e↵ect, and a nonparametric reduced form can enhance identification power. We consider the following nonparametric reduced form classizesc = ⇧(fsc , disadvsc ) + vsc . The sample is clustered, an aspect which is reflected in ↵s of the outcome equation. Hence, we use the block bootstrap when computing standard errors and take schools as bootstrap sampling units to preserve within-cluster (school) correlation. This produces cluster-robust standard errors. We use b = 100 bootstrap repetitions. With the same example and dataset (only the full sample), Horowitz (2011, Section 5.2) uses the model and assumptions of the NPIV approach to nonparametrically estimate the e↵ect of class size. To address the ill-posed inverse problem, he conducts regularization by replacing the operator with a finite-dimensional approximation. First, we compare the NPIV estimate of Horowitz (2011) with the IV estimate obtained by the control function approach of this paper. Figure 3 in Horowitz (2011) is the NPIV estimate of the function of class size (g(·, ·)) for disadv = 1.5(%) with the full
sample. The solid line is the estimate of g and the dots show the cluster-robust 95% confidence band. As noted in his paper (p. 374), “the result suggests that the data and the instrumental variable assumption, by themselves, are uninformative about the form of any dependence of test scores on class size.” Using the same scales in the axes for comparison, Figure 4 in the present paper depicts the (unpenalized) IV estimate calculated with the full sample using the triangular model (2.1) and the control function approach. Although not entirely flexible, the nonparametric reduced form above is justified for use in the comparison with the NPIV estimate, since the NPIV approach does not specify any reduced-form relationship. The sample, the orders of the series, and the value of disadv are identical to those for the NPIV estimate. The dashed lines in the figure indicate the cluster-robust 95% confidence band. The result clearly presents a nonlinear shape of the e↵ect of class size and suggests that the marginal e↵ect diminishes as class size increases. The overall trend seems to be negative, which is consistent with the results of Angrist and Lavy (1999). It is important to note that the control function and NPIV approaches maintain di↵erent sets of assumptions. For example in terms of orthogonality conditions for IV, assumptions (2.1b) are not stronger or weaker than E["|z] = 0, the orthogonality condition introduced in the NPIV model; only if v ? z is assumed, then E["|v, z] = E["|v] with E["] = 0 implies E["|z] = 0. Therefore,
this comparison does not imply that one estimate performs better than the other. It does, however, imply that if the triangular model and control function assumptions are considered to be reasonable, they make the data to be informative about the relationship of interest. Moreover, since the NPIV approach su↵ers from the ill-posed inverse problem even without the problem of weak instruments, the control function approach may be a more appealing framework than the NPIV approach in the possible presence of weak instruments. We proceed by calculating the penalized IV estimates from the proposed estimation method of
24
this paper. For all cases below, we find estimates for disadv = 1.5(%) as before. To better justify the usage of our method in this part, we use the discontinuity sample and a linear reduced-form where the instrument is possibly weak in this nonparametric setting. For the penalization parameter ⌧ , we use the 5-fold cross-validation (CV) to choose a value from among {0.005, 0.01, 0.015, 0.02, 0.05}. Table 2 suggests that ⌧ = 0.02 is the MSE-minimizing value. Figure 5 depicts the penalized and
unpenalized IV estimates. There is a certain di↵erence in the estimates, but the amount is small. It is possible that either the instrument is not very weak in this example or that CV chooses a smaller value of ⌧ . Similar to before, the results suggest a nonlinear e↵ect of class size with the overall negative trend.
9
Conclusions
This paper analyzes identification, estimation, and inference in a nonparametric triangular model in the presence of weak instruments and proposes an estimation strategy to mitigate the e↵ect. The findings and implications of this paper can be adapted to other nonparametric models, such as nonparametric limited dependent variable models of Das et al. (2003) and Blundell and Powell (2004) or IV quantile regression models of Chernozhukov and Hansen (2005) and Lee (2007). The results can also directly be applicable in semiparametric versions of the model of this paper. As more structure is imposed on the model, the identification condition of Section 2 and the regularity conditions of Sections 5 and 6 can be weakened. Subsequent research can consider two specification tests: a test for the relevance of the instruments and a test for endogeneity. These tests can be conducted by adapting the existing literature on specification tests where the test statistics can be constructed using the series estimators of this paper; see, e.g., Hong and White (1995). Testing whether instruments are relevant can ˆ be conducted with the nonparametric reduced-form estimate ⇧(·). A possible null hypothesis is H0 : Pr {⇧0 (z) = const.} = 1, which is motivated by our rank condition for identification. Testing whether the model su↵ers from endogeneity can be conducted with the control function estimate ˆ (v) = h(w) ˆ gˆ(x). A possible null hypothesis is H0 : Pr { 0 (v) = const.} = 1. When using existing results on the specification test, one needs to take into account the generated regressors vˆ. Constructing a test for instrument weakness would be more demanding than the above-mentioned tests. Developing inference procedures that are robust to identification of arbitrary strength is also an important research question.
25
A
Appendix
A.1
Proofs in Identification Analysis (Section 2)
Throughout the Appendices, we suppress the subscript “0” for the true functions and, when no confusion arises, the subscript “n” of the true reduced-form function in Assumption L. In this section, we prove Lemma 2.1 and Theorem 2.2. For Lemma 2.1, we first introduce a preliminary lemma. For nonempty sets A and B, define the following set A + B = {a + b : (a, b) 2 A ⇥ B} .
(A.1)
Then, for nonempty sets A, B, and C, A + B = B + A (commutative)
(Rule 1)
A + (B [ C) = (A + B) [ (A + C) (distributive 1)
(Rule 2)
A + (B \ C) = (A + B) \ (A + C) (distributive 2)
(Rule 3)
c
c
(A + B) ⇢ A + B ,
(Rule 4)
where the last rule is less obvious than the others but can be shown to hold. Let Lebesgue measure on
Rd x ,
Leb
denote a
and @V and int(V) denote the boundary and interior of V, respectively.
Lemma A.1 Suppose Assumptions ID1 and ID20 (a)(i) and (ii) hold. Suppose Z r 6= Then, (a) ⇧(z) + v : z 2 Z 0 , v 2 int(V) ⇢ X r , and (b)
Leb (⇧(Z
0 ))
and Z 0 6= .
= 0 and @V is countable.
We prove the main lemma first. Proof of Lemma 2.1: When Z r =
or Z r = Z we trivially have X r = X . Suppose Z r 6=
and
dx
Z 0 6= . First, under Assumption ID20 (b) that V = R , we have the conclusion by
n o n o X r = ⇧(z) + v : z 2 Z r , v 2 Rdx = Rdx = ⇧(z) + v : x 2 Z, v 2 Rdx = X .
Now suppose Assumption ID20 (a) holds. By Assumption ID20 (a)(iii), for z 2 Z 0 , the joint support of (z, v) is Z 0 ⇥ V. Hence
⇧(z) + v : z 2 Z 0 , v 2 int(V) = ⇧(z) + v : (z, v) 2 Z 0 ⇥ int(V) = ⇧(Z 0 ) + int(V). c
But by Lemma A.1(a), ⇧(Z 0 ) + int(V) ⇢ X r or contrapositively, X \X r ⇢ ⇧(Z 0 ) + int(V) . Also, by (Rule 4), ⇧(Z 0 ) + int(V)
c
⇢ ⇧(Z 0 ) + @V. Therefore,
X \X r ⇢ ⇧(Z 0 ) + @V.
26
(A.2)
Let @V = {⌫1 , ⌫2 , ..., ⌫k , ...} = [1 k=1 {⌫k } by Lemma A.1(b). Then, Leb
⇧(Z 0 ) + @V =
Leb 1 X
⇧(Z 0 ) + ([1 k=1 {⌫k }) = Leb (⇧(Z
k=1
0
) + {⌫k }) =
1 X
Leb
0 [1 k=1 (⇧(Z ) + {⌫k })
Leb (⇧(Z
0
)) = 0,
k=1
where the second equality is from (Rule 2) and the third equality by the property of Lebesgue measure. The last equality is by Lemma A.1(b) that Leb (⇧(Z 0 )) = 0. Since x is continuously ⇥ ⇤ distributed, by (A.2), Pr [x 2 X \X r ] Pr x 2 ⇧(Z 0 ) + @V = 0. ⇤
In the following proofs, we explicitly distinguish the r.v.’s with their realization. Let ⇠, ⇣, and
⌫ denote the realizations of x, z, and v, respectively. We now prove Lemma A.1. Proof of Lemma A.1(a): First, we claim that for any ⇡ 2 ⇧(Z 0 ) there exists [1 n=1 {⇡n } ⇢ ⇧(Z r ) such that limn!1 ⇡n = ⇡. By Proposition 4.21(a) of Lee (2011, p. 92), for any space S,
the path components of S form a partition of S. Note that a path component of S is a maximal
nonempty path connected subset of S. Then for Z 0 ⇢ Rdz , we have Z 0 = [◆2I Z◆0 where partitions Z◆0 are path components. Note that, since Z◆0 is path connected, for any ⇣ and ⇣˜ in Z◆0 , there exists a path in Z◆0 , namely, a piecewise continuously di↵erentiable function : [0, 1] ! Z◆0 such ˜ Note that { (t) : t 2 [0, 1]} ⇢ Z 0 . Consider a composite function that (0) = ⇣ and (1) = ⇣. ◆ ⇧
: [0, 1] ! ⇧(Z◆0 ) ⇢ Rdx . Then, ⇧( (·)) is di↵erentiable, and by the mean value theorem, there
exists t⇤ 2 [0, 1] such that
⇧( (1))
⇧( (0)) =
@⇧( (t⇤ )) (1 @t
0) =
@⇧( (t⇤ )) @ (t⇤ ) . @⇣ 0 @t
Note that @⇧( (t⇤ ))/@⇣ 0 = 0dx ⇥dx since (t⇤ ) 2 Z◆0 ⇢ Z 0 and dx = 1. This implies that ⇧( (1)) = ˜ Therefore for any ⇣ 2 Z 0 , ⇧(⇣) = c◆ for some constant c◆ . ⇧( (0)), or ⇧(⇣) = ⇧(⇣). ◆
⇢ Z such that lim ⇣n = ⇣¯ 2 Z. Then for each n, @⇧(⇣n )/@⇣ 0 = 0 = @⇧(lim 0 0 ¯ (0, 0, ..., 0) = 0 by the definition of Z 0 , and @⇧(⇣)/@⇣ n!1 ⇣n )/@⇣ = limn!1 @⇧(⇣n )/@⇣ = Consider any
[1 n=1 {⇣n }
⇢
Z0
0 where the second equality is by continuity of @⇧(·)/@⇣ 0 . Therefore, ⇣¯ 2 Z 0 , and hence Z 0 is closed,
which implies that Z◆0 is closed for each ◆. That is, Z 0 is partitioned to a closed disjoint union of Z◆0 ’s. But Assumption ID20 (a)(ii) says Z is a connected set in Euclidean space (i.e., Rdz ). There-
fore, for each ◆ 2 I, Z◆0 must contain accumulation points of Z r (Taylor (1965, p. 76)). Now, for
any ⇡ = ⇧(⇣) 2 ⇧(Z 0 ), it satisfies that ⇣ 2 Z◆0 for some ◆ 2 I. Let ⇣c 2 Z◆0 be an accumulation r point of Z r , that is, there exists [1 n=1 {⇣n } ⇢ Z such that limn!1 ⇣n = ⇣c . Then, it follows that
⇡ = ⇧(⇣) = c◆ = ⇧(⇣c ) = ⇧(limn!1 ⇣n ) = limn!1 ⇧(⇣n ), where the second and third equalities are from ⇧(⇣) = c◆ for ⇣ 2 Z◆0 and the fourth by continuity of ⇧(·). Let ⇡n = ⇧(⇣n ), then ⇡n 2 ⇧(Z r ) for every n
r 1. Therefore, we can conclude that for any ⇡ 2 ⇧(Z 0 ), there exists [1 n=1 {⇡n } ⇢ ⇧(Z )
such that limn!1 ⇡n = ⇡.
27
Next, we prove that if ⇠ 2 ⇧(z) + v : z 2 Z 0 , v 2 int(V) then ⇠ 2 X r . Suppose ⇠ 2 {⇧(z) + v :
z 2 Z 0 , v 2 int(V) , i.e., ⇠ = ⇡ + ⌫ for ⇡ 2 ⇧(Z 0 ) and ⌫ 2 int(V). Then, by the result above,
r there exists [1 n=1 {⇡n } ⇢ ⇧(Z ) such that limn!1 ⇡n = ⇡. Define a sequence ⌫n = ⇠
n
1. Notice that ⌫n is not necessarily in V. But, ⌫n = (⇡ + ⌫)
⇡n = ⌫ + (⇡
⇡n for
⇡n ), hence
limn!1 ⌫n = ⌫. Since ⌫ 2 int(V), there exists an open neighborhood B" (⌫) of ⌫ for some " such that B" (⌫) ⇢ int(V). Also, by the fact that limn ⌫n = ⌫, there exists N" such that for all n
N" ,
⌫n 2 B" (⌫). Therefore, by conveniently taking n = N" , ⇠ satisfies that ⇠ = ⇡N" + ⌫N" where ⇡N" 2 ⇧(Z r ) and ⌫N" 2 B" (⌫) 2 int(V) ⇢ V. That is, ⇠ 2 X r . ⇤
Proof of Lemma A.1(b): Recall dx = 1. Note that V ⇢ R can be expressed by a union of
disjoint intervals. Since we are able to choose a rational number in each interval, the union is a countable union. Since each interval has at most two end points which are the boundary of it, @V is countable. To prove that
Leb (⇧(Z
0 ))
= 0, note that Z 0 is the support where @⇧(z)/@zk = 0 for
k dz . Therefore, its bilateral (directional) derivative D↵ ⇧(z) in the direction ↵ = (↵1 , ↵2 , ..., ↵dz )0 Pz satisfies D↵ ⇧(z) = dk=1 ↵k · @⇧(z)/@zk = 0. Since the bilateral derivative is zero, each unilateral
derivative is also zero; see, e.g., Giorgi et al. (2004, p. 94) for the definitions of various derivatives. Then, by Corollary 6.1.3 in Garg (1998),
Leb (⇧(Z
0 ))
= 0. ⇤
Proof of Theorem 2.2: Consider equation (2.2) with z = z2 , E[y|x, z] = E[y|v, z] = g(⇧(z) + v) + (v),
(2.2)
where the conditional expectations and ⇧(·) are consistently estimable, and v can also be estimated. By di↵erentiating both sides of (2.2) with respect to z, we have @E[y|v, z] @g(x) @⇧(z) = · . 0 @z @x0 @z 0
(A.3)
Now, suppose Pr [z 2 Z r ] > 0. For any fixed value z¯ such that z¯ 2 Z r , we have rank (@⇧(¯ z )/@z 0 ) =
dx by definition, hence the system of equations (A.3) has a unique solution @g(x)/@x0 for x in the conditional support Xz¯. That is, @g(x)/@x0 is locally identified for x 2 Xz¯. Now, since the above argument is true for any z 2 Z r , we have that @g(x)/@x0 is identified on x 2 X r . Now by
Assumption ID2, the di↵erence between X r and X has probability zero, thus @g(x)/@x0 is identified. Once @g(x)/@x0 is identified, we can identify @ (v)/@v 0 by di↵erentiating (2.2) with respect to v: @E[y|v, z] @g(x) @ (v) = + . @v 0 @x0 @v Next, we prove the necessity part of the theorem. Suppose Pr[z 2 Z r ] = 0. This implies
Pr[z 2 Z 0 ] = 1, but since Z 0 is closed Z 0 = Z. Therefore, for any z 2 Z, the system of equations (A.3) either has multiple solutions or no solution, and hence g(⇧(z) + v) is not identified. ⇤ 28
A.2
Key Technical Steps for Asymptotic Theory (Section 5)
For asymptotic theory, we require a key preliminary step to separate out the weak instrument factor from the second moment matrices of interest. Define a vector of approximating functions of orders K = K1 + K2 + 1 for the second stage,
0 . . p (w) = (1, p1 (x), ..., pK1 (x), p1 (v), ..., pK2 (v)) = 1 .. pK1 (x)0 .. pK2 (v)0 , 0
K
where pK1 (x) and pK2 (v) are vectors of approximating functions for g0 (·) and
0 (·)
of orders K1
and K2 , respectively. Note that this rewrites pK (w) = (p1 (w), ..., pK (w))0 of the main body for expositional convenience. Since g0 (·) and
0 (·)
can only be separately identified up to a constant,
when estimating h0 (·), we include only one constant function. Define a K ⇥ K sample second moment matrix
ˆ0 ˆ ˆ=PP = Q n
ˆ + ⌧n I K ) Then, ˆ⌧ = (Q
1P ˆ 0 y/n.
Pn
i=1 p
K (w ˆ
i )p
K (w ˆ i )0
n
.
(A.4)
For the rest of this section, we consider univariate x for simplicity. This corresponds to Assump0
tion ID2 (a). The analysis can also be generalized to the case of a vector x by using multivariate mean value expansion, but omitted for succinctness. Note that z is still a vector. Under Assump˜ after applying a normalization c = 0 and suppressing op (n ) for simplicity tion L, ⇧n (·) = n ⇧(·) in (3.1). Omitting op (n
) does not a↵ect the asymptotic results developed in the paper. For
r 2 {1, 2} define its r-th derivative as @ r pj (x) = dr pj (x)/dxr . By mean value expanding each element of pK1 (xi ) around vi , we have, for j K1 , pj (xi ) = pj (n
˜ i ) + vi ) = pj (vi ) + n ⇧(z
˜ i )@pj (˜ ⇧(z vi ),
(A.5)
where v˜i is a value between xi and vi . Define @ r pK1 (x) = [@ r p1 (x), @ r p3 (x), ..., @ r pK1 (x)]0 . Then, by (A.5) the vector of regressors pK (wi ) for estimating h(·) can be written as .. K1 . 0 .. K2 0 p (wi ) = 1 . p (xi ) . p (vi ) = 1 .. pK1 (vi )0 + n K
0
Let = K1 = K2 = (K
. ˜ i )@pK1 (˜ ⇧(z vi )0 .. pK2 (vi )0 .
(A.6)
1)/2. Again, K1 , K2 , L, K, and all depends on n. Note that
K ⇣ K1 ⇣ K2 ⇣ , where an ⇣ bn denote that an /bn is bounded below and above by constants that are independent of n. This setting can be justified by g0 (·) and
0 (·)
with the same smoothness,
which is imposed in Assumption C. Extending the analysis to a general case of K1 6= K2 can follow by a slight modification of the argument with = min {K1 , K2 }, which we omit for succinctness.
29
Now we choose a transformation matrix Tn to be 2
1
6 Tn = 4 0⇥1 0⇥1
01⇥ n I n I
01⇥
3
7 0⇥ 5 . I
After multiplying Tn on both sides of (A.6), the weak instrument factor is separated from pK (wi )0 : with ui = (zi , vi ),
where
p⇤K (ui )0
0
K
p (wi ) Tn = 1 = 1
.. . p (vi )0 + n
. ˜ i )@p (˜ ⇧(z vi )0 .. p (vi )0 · Tn
.. ˜ . . ⇧(zi )@p (˜ vi )0 .. p (vi )0 = p⇤K (ui )0 + mK0 i ,
.. ˜ .. . ˜ 0 0 K0 v )0 = 1 . ⇧(zi )@p (vi ) . p (vi ) and mi = 0 .. ⇧(z i ) (@p (˜ i
(A.7)
. @p (vi )0 ) .. (0⇥1 )0 .
To illustrate the role of this linear transformation, rewrite the original vector of regressors in (A.6) as pK (wi )0 = pK (wi )0 Tn Tn 1 = p⇤K (ui ) + mK i
0
Tn 1 .
(A.8)
Ignoring the remainder vector mK i which is shown to be asymptotically negligible below, the original vector pK (wi ) is separated into p⇤K (ui ) and Tn 1 . Note that p⇤K (ui ) is not a↵ected by the weak instruments and can be seen as a new set of regressors.22 Now, consider
⇥ ⇤ Q = E pK (wi )pK (wi )0 .
(A.9)
By equations (A.9) and (A.7), it follows
⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇤K K0 , Tn0 QTn = Q⇤ + E mK (ui )0 + E p⇤K (ui )mK0 + E mK i p i i mi
(A.10)
⇥ ⇤ where the newly defined Q⇤ = E p⇤K (ui )p⇤K (ui )0 is the population second moment matrix with ˜ the new regressors. Furthermore, since ⇧(·) 2 C1 (Z) can have nonempty Z0 as a subset of its ⇥ ⇤K ⇤ ⇥ ⇤ r⇤ ⇤K 0 domain, we define Q = E p (ui )p (ui ) |zi 2 Z r and Q0⇤ = E p⇤K (ui )p⇤K (ui )0 |zi 2 Z 0 . Also ⇥ ⇤ define the second moment matrix for the first-stage estimation as Q1 = E rL (zi )rL (zi )0 . Assumptions B, C, D, and L of the main text serve as sufficient conditions for high-level as-
sumptions stated here. Appendix A.4.1 proves that the latter are implied by the former. For a symmetric matrix B, let
min (B)
and
max (B)
denote the minimum and maximum eigenvalues of
B, respectively, and det(B) the determinant of B. Assumption B† (i)
min (Q
r⇤ )
is bounded away from zero for all K(n) and
min (Q1 )
is bounded
For justification that p⇤K (ui ) can be regarded as regressors, see Assumption B in Section 5 and Assumption B† below. 22
30
away from zero for all L(n); (ii)
max (Q)
is bounded by a fixed constant, for all K(n), and
max (Q1 )
bounded by a fixed constant, for all L(n).
Assumption C† There exist ˜ CK
s/dx
=(
1 , ...,
˜) K
˜ ! 1 and supz2Z ⇧0 (z) as K
and
= ( 1 , ...,
˜ pL (z)0
˜ CL
˜) L
s⇡ /dz
such that supw2W h0 (w)
˜
pK (w)0
˜ ! 1. as L
For the next assumption let ⇣rv () and ⇠rv (L) satisfy max sup k@ µ p (v)k ⇣rv (),
max sup @ µ rL (z) ⇠r (L),
|µ|r v2V
|µ|r z2Z
which impose nonstochastic uniform bounds on the vectors of approximating functions. Let p L/n + L s⇡ /dz . Assumption D† (i) n 1/2 ⇣1v () ⇠0
(L)2 L/n
! 0; also, (v) ⌧n ! 0,
⇡ 2 n
! 0; (ii) n (K) ! 0.
1/2 ⇣2v () ! 0; (iii) ⇣1v ()⇣2v ()
⇡
⇡
=
! 0; (iv)
Under these assumptions, Lemma A.2 below obtains the order of magnitudes of eigenvalues of the second moment matrices in term of the weak instrument. In proving this lemma, we frequently apply two useful mathematical lemmas (Lemmas A.4 and A.5) that are stated and proved in Appendix p A.4.2. For any matrix A, let the matrix norm be the Euclidean norm kAk = tr(A0 A). Lemma A.2 Suppose Assumptions ID, A, B† , D† , and L are satisfied. Then, (a) ˆ 1 ) = Op (n2 ). O(n2 ) and (b) max (Q
max (Q
1)
=
In all proofs, let C denote a generic positive constant that may be di↵erent in di↵erent use. TR, CS, MK are triangular inequality, Cauchy-Schwartz inequality and Markov inequality, respectively. Proof of Lemma A.2: Consider (a) first. Let p⇤i = p⇤K (ui ) and mi = mK i for brevity. Recall ⇤ 0 0 (A.10) that Tn0 QTn = Q⇤ + E [mi p⇤0 i ] + E [pi mi ] + E [mi mi ]. Then,
Tn0 QTn
⇣ ⌘1/2 ⇣ ⌘1/2 Q⇤ 2E kmi k kp⇤i k + E kmi k2 2 E kmi k2 E kp⇤i k2 + E kmi k2
.. ˜ v )0 by CS. But mi = mK i i = [0 . ⇧(zi ) @p (˜
. @p (vi )0 .. (0⇥1 )0 ]0 where v˜ is the intermediate value
between x and v. Then, by mean value expanding @p (˜ vi ) around vi and |˜ vi have
˜ i )@ 2 p (¯ kmi k2 = ⇧(z vi ) (˜ vi =n
2
vi )
4
2
˜ i ) ⇣2v ()2 Cn ⇧(z 31
2
˜ i ) ⇣2v ()2 |xi ⇧(z 2
⇣2v ()2 ,
vi | |xi
vi |, we
vi |2 (A.11)
˜ i ) < 1. where v¯ is the intermediate value between v and v˜, and by Assumption L that supz ⇧(z Therefore
E kmi k2 Cn Then,
⇥ ⇤ ⇤ ⇤ E p⇤0 i pi = tr(Q ) tr(IK )
⇤)
2
⇣2v ()2 .
max (Q
⇤
(A.12)
) C · K = Op (),
(A.13)
C is by the fact that the polynomials or splines are defined on bounded sets and ˜ 2 C1 (Z). Therefore, by combining (A.12) and (A.13) it follows by Assumption L that ⇧(·) where
max (Q
Tn0 QTn
Q⇤ O(1/2 n
⇣2v ()) + n
2
⇣2v ()2 ) = o(1)
(A.14)
by Assumption D† (ii), which shows that all the remainder terms are negligible. Now, by Lemma A.4, we have 0 min (Tn QTn )
min (Q
Combine the results (A.14) and (A.15) to have with simpler notations p1 = Pr [z 2
Zr]
and p0 =
by a variant of Lemma A.4 (with the fact that it follows that
min
(Q⇤ )
p1 ·
Since p1 > 0, it holds that Therefore,
(Qr⇤ )
min ⇤ min (Q )
1 = 0 min (Tn QTn )
Let T0n =
Then, by solving
"
n n
˜
n
n max (I ) = 1, it follows
0 1
0 1
˜
#
follows
0 max (Tn Tn )
) Tn0 QTn
0 min (Tn QTn ) = ⇥ ⇤ Pr z 2 Z 0 , we 1(
B) =
(Q0⇤ )
+ p0 ·
min (Qr⇤ )
p1 ·
min
min
(Q⇤ )
⌦ I ,
+ o(1)
so that
Q⇤ min (Q
have
k (B)
= p1 ·
1
)=
⇤)
Q⇤
+ o(1). But note that,
= p1 Qr⇤ + p0 Q0⇤ . Then,
for any symmetric matrix B),
min (Q
r⇤ ),
because
min (Q
1 = O(1). c + o(1)
Tn =
"
0⇤ )
= 0.
(A.16)
1
01⇥2
02⇥1
T0n
#
.
= 0, we have ˜ = n or 1 for eigenvalues of T0n , and since
=
max (T0n )
=n .
(A.17)
n2 by Lemma A.5. Since (A.16) implies
max (Q
(A.15)
c > 0 for all K(n) by Assumption B† (i).
1
max (Tn )
Note that
⇤
0 1 0 max (Tn (Tn QTn ) Tn )
O(1)
0 1 max ((Tn QTn ) )
0 max (Tn Tn )
by applying Lemma A.5 again. The proof of part (b) proceeds similarly as above. Using (A.6),
32
pK ( w ˆ i )0
= O(1), it
= O(n2 )
. . = 1 .. pK1 (xi )0 .. pK2 (ˆ vi )0 =
. 1 .. pK1 (vi )0 + n 0
K
. ˜ i )@pK1 (˜ ⇧(z vi )0 .. pK2 (ˆ vi )0 and
p (w ˆ i ) Tn = 1 = 1
.. . p (vi )0 + n
. ˜ i )@p (˜ ⇧(z vi )0 .. p (ˆ v i ) 0 · Tn
.. . n (p (vi )0
. ˜ i )@p (˜ p (ˆ vi )0 ) + ⇧(z vi )0 .. p (ˆ vi )0 = p⇤K (ˆ ui )0 + mK0 ˆi0 , i +r
. ˜ . . (v )0 .. p (ˆ 0 with u 0 = 0 .. n (p (v )0 where p⇤K (ˆ ui )0 = 1 .. ⇧(z )@p v ) ˆ = (z , v , v ˆ ) and r ˆ i i i i i i i i i
p (ˆ vi )0 )
P .. . (0⇥1 )0 . Let pˆ⇤i = p⇤K (ˆ ui ) and recall mi = mK i . For a random matrix Xi , denote i Xi /n as
En Xi for simplicity. Then by (A.8),
ˆ n=Q ˆ ⇤ + En [mi pˆ⇤0 Tn0 QT p⇤i m0i ] + En [mi m0i ] i ] + En [ˆ + En [ˆ ri pˆ⇤0 p⇤i rˆi0 ] + En [ˆ ri m0i ] + En [mi rˆi0 ] + En [ˆ ri rˆi0 ] i ] + En [ˆ and thus ˆ n Tn0 QT
ˆ ⇤ = 2En kmi k kˆ Q p⇤i k + En kmi k2 + 2En kˆ ri k kˆ p⇤i k + 2En kˆ ri k kmi k + En kˆ ri k2 ⇣ ⌘1/2 ⇣ ⌘1/2 ⇣ ⌘1/2 ⇣ ⌘1/2 2 En kmi k2 En kˆ p⇤i k2 + 2 En kˆ r i k2 En kˆ p⇤i k2 ⇣ ⌘1/2 ⇣ ⌘1/2 + 2 En kˆ ri k2 En kmi k2 + En kmi k2 + En kˆ r i k2 .
Similar as (A.13), the bound on En kˆ p⇤i k2 can be derived as ⇥ ⇤ ⇥ ⇤ En pˆ⇤0 ˆ⇤i = tr(En pˆ⇤i pˆ⇤0 i p i ) tr(IK )
max (En
p⇤i pˆ⇤0 max (En [ˆ i ])
⇥
⇤ pˆ⇤i pˆ⇤0 i ) Op (K) = Op (),
(A.18)
Op (1) is by the fact that the polynomials or splines are defined on bounded ˜ 2 C1 (Z). For En kˆ sets and by Assumption L that ⇧(·) ri k2 , note that where
kˆ ri k2 = n (p (vi )0
p (ˆ vi )0 )
and therefore En kˆ ri k2 = Op (n2 ⇣1v ()2 ˆ n Tn0 QT
ˆ ⇤ Op (n Q
2 ). ⇡
2
n2 ⇣1v ()2 |vi
vˆi |2 = Op (n2 ⇣1v ()2
2 ⇡)
Combining this with (A.18) and (A.12) yields
1/2 ⇣2v () + n 1/2 ⇣1v ()
⇡
+ ⇣1v ()⇣2v ()
⇡
+n
2
⇣2v ()2 + n2 ⇣1v ()2
2 ⇡)
= op (1) by Assumption D† (i)–(iii). Also, by Lemma A.4, we have
33
0 ˆ min (Tn QTn )
min (Q
⇤)
ˆ n Tn0 QT
Q⇤ .
0 ˆ min (Tn QTn )
Combining the two results yields 1 0 ˆ min (Tn QTn )
ˆ
Therefore, we have
A.3
max (Q
1)
=
min (Q
+ op (1). Similar as before
1 1 = Op (1). ⇤ ) + o (1) (Q c + o min p p (1)
=
0 ˆ 1 0 max (Tn (Tn QTn ) Tn )
=
⇤)
Op (1)
0 max (Tn Tn )
(A.19)
= Op (n2 ). ⇤
Proofs of Rate of Convergence (Section 5)
Given the results of the lemmas above, we derive the rates of convergence. We first derive the ˆ ˆ rate of convergence of the unpenalized series estimator h(·) defined as h(w) = pK (w)0 ˆ where ˆ = (Pˆ 0 Pˆ ) 1 Pˆ 0 y. Then, we prove the main theorem with the penalized estimator h ˆ ⌧ (·) defined in Section 4. Lemma A.3 Suppose Assumptions A–D, and L are satisfied. Then, ˆ h
h0
L2
⇣ p = Op n ( K/n + K
Proof of Lemma A.3: Let ˆ h
h0
L2
= =
⇢ˆ h ⇢ˆ h ⇢⇣
ˆ
C ˆ
=(
ˆ h(w)
1 , ...,
h0 (w)
s⇡ /dz
⌘ ) .
1/2
K
K
0
+ O(K
s/dx
)
p L/n + L
dF (w) dF (w)
p (w) ( ˆ ⌘0
i2
+
By TR of L2 norm (first inequality),
i2
0
K
K ).
s/dx
1/2
Ep (w)p (w) )
⇣
+ ⌘
ˆ
⇢ˆ
⇥
1/2
pK (w)0
+ O(K
by Assumption B† (ii) and using Lemma A.5 (last eq.). As ˆ
h0 (w) s/dx
⇤2
1/2
dF (w)
)
= (Pˆ 0 Pˆ )
1P ˆ 0 (y
Pˆ ), it follows
that ˆ
2
= (y
Pˆ )0 Pˆ (Pˆ 0 Pˆ )
= (y
ˆ Pˆ )0 Pˆ Q
Op (n2 )(y
1/2
1
ˆ Q
(Pˆ 0 Pˆ ) 1
ˆ Q
1/2
⇣ ⌘ Pˆ )0 Pˆ Pˆ 0 Pˆ
Pˆ 0 (y
Pˆ )
Pˆ 0 (y
Pˆ )/n2
1
1
Pˆ 0 (y
Pˆ )/n
by Lemma A.5 and Lemma A.2(b) (last ineq.). ˜ = (h(w Let h = (h(w1 ), ..., h(wn ))0 and h ˆ1 ), ..., h(w ˆn ))0 . Also let ⌘i = yi (⌘1 , ..., ⌘n
)0 .
Let W = (w1 , .., wn
)0 ,
h0 (wi ) and ⌘ =
then E [yi |W ] = h0 (wi ) which implies E [⌘i |W ] = 0. Also 34
⇥ ⇤ similar to the proof of Lemma A1 in NPV (p. 594), by Assumption A, we have E ⌘i2 |W being bounded and E [⌘i ⌘j |W ] = 0 for i 6= j, where the expectation is taken for y. Then, given that ˜ + (h ˜ Pˆ ), we have, by TR, y Pˆ = (y h) + (h h) ˆ
ˆ 1/2 Pˆ 0 (y Pˆ )/n = Op (n ) Q n ˆ 1/2 Pˆ 0 ⌘/n + Q ˆ Op (n ) Q
1/2
˜ ˆ h)/n + Q
Pˆ 0 (h
1/2
˜ Pˆ 0 (h
Pˆ )/n
o
.
(A.20)
For the first term of equation (A.20), consider E
h
2
P ⇤ )0 ⌘/n
(P Tn
i h W = E M 0 ⌘/n = Op (n
2
2
i 1 X W C 2 kmi k2 n i
1 v ⇣2 ()2 )
= op (1)
by (A.11) and op (1) is implied by Assumption D† (ii). Therefore, by MK, (P Tn
P ⇤ )0 ⌘/n = op (1).
(A.21)
Also, E
"
⇣
Pˆ Tn
P Tn
⌘0
2
⌘/n
W
#
C
1 X (ˆ pi n2
2
pi ) 0 T n
i
1 O(n2 )Op (⇣1v ()2 n
2 ⇡)
C
1 X n2
max (Tn )
2
i
= Op (n2 ⇣1v ()2
2 ⇡ /n)
kˆ pi
p i k2 (A.22)
by (A.17) and kˆ pi
pi k2 = kp (xi )
p (xi )k2 + k@p (¯ vi ) (ˆ vi vi )k2 1X C⇣1v ()2 |ˆ vi vi |2 Op (⇣1v ()2 2⇡ ). n
(A.23)
i
Therefore,
by Assumption D† (i) and MK. Also E P ⇤0 ⌘/n
2
⇣
Pˆ Tn
h
P Tn
= E E[ P ⇤0 ⌘/n
2
⌘0
⌘/n = op (1)
i
|W ] = E
"
X i
⇤ 2 2 p⇤0 i pi E[⌘i |W ]/n
1 X ⇥ ⇤0 ⇤ ⇤ C 2 E pi pi = Ctr(Q⇤ )/n = O(/n) n i
35
(A.24)
#
by Assumptions A (first ineq.) and equation (A.13) (last eq.). By MK, this implies P ⇤0 ⌘/n Op ( Hence by TR with (A.21), (A.24), and (A.25), ⇣
Tn0 Pˆ 0 ⌘/n
Pˆ Tn
P Tn
⌘0
p /n).
(A.25)
P ⇤ )0 ⌘/n + P ⇤0 ⌘/n Op (
⌘/n + (P Tn
p /n).
Therefore, the first term of (A.20) becomes ˆ Q
1/2
Pˆ 0 ⌘/n
2
⌘ 0 Pˆ Tn n
=
!
ˆ n) (Tn0 QT
1
Tn0 Pˆ 0 ⌘ n
!
Op (1) Tn0 Pˆ 0 ⌘/n
2
= Op (/n)
(A.26)
by Lemma A.5 and (A.19). ⇣ ⌘ 1 Because I Pˆ Pˆ 0 Pˆ Pˆ 0 is a projection matrix, hence is p.s.d, the second term of (A.20) becomes
ˆ Q
1/2
˜ h)/n
Pˆ 0 (h
2
⇣ ⌘ 1 ˜ 0 Pˆ Pˆ 0 Pˆ ˜ = (h h) Pˆ 0 (h h)/n (h X X = (h(wi ) h(w ˆi ))2 /n = ( (vi ) i
C
X i
|vi
2
vˆi | /n =
X
˜ 0 (h h)
˜ h)/n
(ˆ vi ))2 /n
i
2
ˆ i ) /n = Op ( ⇧(z
⇧n (zi )
2 ⇡)
(A.27)
i
by Assumption C (Lipschitz continuity of (v)) (last ineq.). Similarly, the last term is ˆ Q
1/2
˜ Pˆ 0 (h
Pˆ )/n
2
˜ = (h
⇣ ⌘ Pˆ )0 Pˆ Pˆ 0 Pˆ
1
˜ Pˆ )0 (h ˜ Pˆ )/n (h X = h(w ˆ i ) pK ( w ˆ i )0
˜ Pˆ 0 (h
2
Pˆ )/n
/n = Op (K
2s/dx
)
(A.28)
i
by Assumption C† . Therefore, by combining (A.26), (A.27), and (A.28) ˆ Consequently, since ⇣ K, ˆ h
h0
L2
h p Op (n ) Op ( /n) + Op (
h p Op (n ) Op ( K/n) + O(K
and we have the conclusion of the lemma. ⇤
s/dx
⇡)
+ O(K
) + Op (
s/dx
i ) .
i ) + O(K ⇡
s/dx
)
ˆ ⌧ = (Pˆ 0 Pˆ +n⌧n IK )/n = Q+⌧ ˆ n IK . Define Pˆ# = Pˆ +n⌧n Pˆ (Pˆ 0 Pˆ ) Proof of Theorem 5.1: Recall Q 36
1.
Note that the penalty bias emerges as Pˆ# 6= Pˆ . Consider 2
ˆ⌧
= (y
Pˆ# )0 Pˆ (Pˆ 0 Pˆ + n⌧n IK )
= (y
ˆ ⌧ 1/2 Q ˆ ⌧ 1Q ˆ ⌧ 1/2 Pˆ 0 (y Pˆ# )0 Pˆ Q ˆ
max (Q⌧
1
ˆ 1/2 Pˆ 0 (y ) Q ⌧
1
(Pˆ 0 Pˆ + n⌧n IK )
1
Pˆ 0 (y
Pˆ# )
Pˆ# )/n2
Pˆ# )/n
2
.
First note ˆ
max (Q⌧
1
1 1 1 = ˆ ˆ ˆ min (Q + ⌧n I) min (Q) + min (⌧n I) min (Q) + ⌧n ( ) n o 1 1 min , = min Op (n2 ), ⌧n 1 , ˆ min (Q) ⌧n
)=
ˆ ⌧ 1/2 Pˆ 0 (y Op (Rn ) Q
where the last eq. is by Lemma A.2(b). Then ˆ⌧ Rn = min{n , ⌧n
1/2
is p.s.d. Therefore
ˆ ⌧ 1 Pˆ c c0 Pˆ 0 Q ˆ }. Also, note that c0 Pˆ 0 Q
ˆ 1/2 Pˆ 0 (y Q ⌧
ˆ Pˆ# )/n Q
1/2
Pˆ 0 (y
1P ˆc
(A.29) (A.30)
Pˆ# )/n by letting
ˆ for any vector c, since (Q
1
ˆ ⌧ 1) Q
Pˆ# )/n
ˆ Q
1/2
Pˆ 0 (y
ˆ h)/n + Q
1/2
Pˆ 0 (h
Pˆ
n⌧n Pˆ (Pˆ 0 Pˆ )
ˆ Q
1/2
Pˆ 0 (y
ˆ h)/n + Q
1/2
Pˆ 0 (h
ˆ Pˆ )/n + ⌧n Q
1
)/n
1/2
.
The third term (squared) is ˆ ⌧n Q
1/2
2
ˆ = ⌧n2 0 Q
where the ineq. is by Lemma A.2 and
PK
j=1
1
2 j
⌧n2 n2
(K) = Op (⌧n2 ),
(K), and the last eq. by the source condition
(D(iii)). Consequently, by combining (A.26), (A.27), and (A.28) in Lemma A.3, ˆ⌧ h
h0
L2
⇣ p = Op Rn ( K/n + K
s/dx
+ ⌧n +
p L/n + L
s⇡ /dz
⌘ ) .
This proves the first part of the theorem. The conclusion of the second part follows from ˆ h(w)
h0 (w)
1
pK (w)0
h0 (w)
1
+ pK (w)0 ( ˆ⌧
)
1
O(K
s/dx
) + ⇣0v (K) ˆ⌧
.
⇤ Proof of Theorem 5.3: The proof follows directly from the proofs of Theorems 4.2 and 4.3 of NPV (p. 602). As for notations, we use v instead of u of NPV and the other notations are identical.
37
⇤
A.4
Technical Proofs
A.4.1
Proofs of Sufficiency of Assumptions B, C, D, and L h i ˜ = 1 and we prove after Proof that B and L imply B† : For simplicity, assume Pr z 2 Z r (⇧) ˜ is piecewise one-to-one. Here, we prove replacing Qr⇤ with Q⇤ in Assumption B† (i). Note that ⇧(·) ˜ ˜ the caseh where ⇧(·) i is one-to-one. The general cases where ⇧(·) is piecewise one-to-one or where ˜ < 1 can be followed by conditioning on z in appropriate subset of Z. 0 < Pr z 2 Z r (⇧) ˜ Consider the change of ˜ = (˜ z , v˜) where z˜ = ⇧(z) and v˜ = v. Then, variables of u = (z, v) 0into u . . it follows that p⇤K (ui ) = 1 .. z˜i @p (vi )0 .. p (vi )0 = pK (˜ ui ) where pK (˜ ui ) is one particular form of a vector of approximating functions as specified in NPV (pp. 572–573). Moreover, the joint density of u ˜ is ˜ fu˜ (˜ z , v˜) = fu (⇧ ˜ Since @ ⇧
1
˜ @⇧
(˜ z ), v˜) ·
1 (˜ z)
@ z˜
0
0
1
˜ = fu ( ⇧
1
(˜ z ), v˜) ·
˜ 1 (˜ @⇧ z) . @ z˜
˜ 2 C1 (Z) (bounded derivative) and fu is bounded away from zero and the 6= 0 by ⇧
1 (˜ z )/@ z˜
support of u is compact by Assumption B, the support of u ˜ is also compact and fu˜ is also bounded away from zero. Then, by the proof of Theorem 4 in Newey (1997, p. 167), is bounded away from zero. Therefore
min (Q
⇤)
K u )pK (˜ ui ) 0 ) min (Ep (˜ i
is bounded away from zero for all .
As for Q1 that does not depend on the e↵ect of weak instruments, the density of z being bounded away from zero implies that The maximum eigenvalues
min (Q1 ) is bounded of Q⇤ , Q and Q1 are
away from zero for all L by Newey (1997, p. 167).
bounded by fixed constants by the fact that the ˜ 2 C1 (Z). ⇤ polynomials or splines are defined on bounded sets and by Assumption L that ⇧(·) Proof that C implies C† : The results follow by Theorem 8 of Lorentz (1986, p. 90) for power series and by Theorem 12.8 of Schumaker (1981) for splines. ⇤ Proof that D implies D† : It follows from Newey (1997, p. 157) that with power series, ⇣rv (K) = 1
K 1+2r and with splines, ⇣rv (K) = K 2 +r , and similarly for the restriction on L. The same results holds for ⇠r (L). ⇤ A.4.2
Matrix Algebra
The following are mathematical lemmas that are useful in proving Lemma A.2 and other results. Lemma A.4 For symmetric k ⇥ k matrices A and B, let values such that
1
2
···
| j (A)
k.
j (A)
and
j (B)
denote their jth eigen-
Then, the following inequality holds: For 1 j k, j (B)|
|
1 (A
38
B)| kA
Bk .
Lemma A.5 If K(n) ⇥ K(n) symmetric random sequence of matrices An satisfies
max (An )
=
Op (an ), then kBn An k kBn k Op (an ) for a given sequence of matrices Bn .
Proof of Lemma A.4: First, by Weyl (1912), for symmetric k ⇥ k matrices C and D i+j 1 (C
where i + j
+ D)
i (C)
+
j (D),
(A.31)
1 k. Also, for any k ⇥ 1 vector a such that kak = 1, (a0 Da)2 = tr(a0 Daa0 Da) =
tr(DDaa0 ) tr(DD)tr(aa0 ) = tr(DD). Since
j (D)
= a0 Da for some a with kak = 1, we have
| j (D)| kDk ,
(A.32)
for 1 j k. Now, in (A.31) and (A.32), take j = 1, C = B, and D = A
B, and we have the
desired results. ⇤
tr Bn U D2 U
A.5
1B0 n
tr Bn U U
1B0 n
·
max (An )
2
Then, kBn An k2 =
1.
Proof of Lemma A.5: Let An have eigenvalue decomposition An = U DU = kBn k2 Op (n )2 . ⇤
Proof of Asymptotic Normality (Section 6)
˜ = P 0 P/n where Assumption G in Section 6 implies the following technical assumption. Define Q P = (pK (w1 ), ..., pK (wn ))0 and define
n⇥K
ˆ Q Q1 H
p v = ⇣1v ()2 2⇡ + 1/2 ⇣1v () ⇡ , = ⇣ () log()/n, ˜ Q = ˆ + 0 Q Q p p p = ⇠(L) log(L)/n, L/n + L s⇡ /dz , ⇡ = h = Rn ( K/n + K p = L1/2 ⇣1v () ⇡ + K 1/2 ⇠(L)/ n.
Assumption G† The following quantities converge to zero as n ! 1: p p p n⌧n Rn (K), Rn4 (⇣0v (K)2 K+⇠(L)2 L)/n, n⇣0v (K) 2⇡ , L1/2 Q1 , Rn L1/2 L1/2 ⇣0v (K)⇣1v (K)
h,
and n
⇣2v (K)K. Also Rn = ⌧n
1/2
p
nK
H,
˜ Q s/dx
+
s/dx ,
Rn3 K 1/2
⇡)
p
nL
Q,
s⇡ /dz ,
Rn ⇣1v (K)2
= n ⌧.
p p Proof that Assumption G implies Assumption G† : First by nK s/dx ! 0 and nL s⇡ /dz ! p p p p p 0, we have ⇡ = L/n(1 + L 1/2 nL s⇡ /dz ) = O( L/n) and h = O(Rn ( K/n + L/n)).
39
2, ⇡
Therefore p
n⇣0v (K) L1/2
2 ⇡ Q1
Rn L1/2
H
Rn3 K 1/2
Q
Rn ⇣1v (K)2
2 ⇡
L1/2 ⇣0v (K)⇣1v (K)
h
p = ⇣0v (K)L/ n p = L1/2 ⇠(L) log(L)/n
p = Rn (⇣1v (K)L3/2 + K 1/2 L1/2 ⇠(L))/ n p p p = Rn3 (⇣1v (K)2 K 1/2 L/ n + ⇣1v (K)K L + ⇣0v (K)K 1/2 log(K)) = Rn ⇣1v (K)2 L/n
p = Rn ⇣0v (K)⇣1v (K)(K 1/2 L1/2 + L)/ n 1
1
Then by plugging in ⇣rv (K) = K 2 +r and ⇠(L) = L 2 with splines and ⇣rv (K) = K 1+2r and ⇠(L) = L with polynomials, it is readily seen that Assumption G† is followed by Assumption G. ⇤ Proof of Theorem 6.1: The proofs of Theorem 6.1 and Corollary 6.2 are a modification of the proof of Theorem 5.1 in NPV (with their trimming function being an identity function) in the setting of weak instruments and penalization. We use the components established in the proof of the convergence rate, which are distinct from NPV. The rest of the notations are the same as those of NPV. Let “MVE” abbreviate mean value expansion. Under Assumptions A, B† (ii), and E and given our choice of basis functions, Theorem 4.6 of ˜ Q = Op ( ˜ ) and Q ˆ1 I = Belloni et al. (2015) (which improves over Newey (1997)) yields Q Q ˆ Op ( Q1 ) by letting Q1 = I. Also similar to the proof of Lemma A1 of NPV, but using Q 1 Pn ˆ Q ˜ = Op ( ˆ ). Also, pi pˆ0i pi p0i ) instead of their (A.5), we have Q i=1 (ˆ n Q ˆ⌧ Q
ˆ Q⌧ Q
˜ = Q
ˆ Q + ⌧n I Q
˜ + Q ˜ Q + k⌧n Ik Q p Op ( Qˆ ) + Op ( Q˜ ) + O(⌧n K) = Op ( Q⌧ ).
Let X is a vector of variables that includes x and z, and !(X, ⇡) a vector of functions of X and ⇡ and, trivially, !(X, ⇡) = (x, x
⇧n (z)). Let ri = rL (zi ). Define Q⌧ = Q + ⌧n I and
V⌧ = AQ⌧ 1 ⌃ + HQ1 1 ⌃1 Q1 1 H 0 Q⌧ 1 A0 , ⇥ ⇤ ⇥ ⇤ ⌃ = E pi p0i var(yi |Xi ) , H = E pi [@h(wi )/@w]0 @!(Xi , ⇧n (zi ))/@⇡ ri0 .
Note that H is a channel through which the first-stage estimation error a↵ects into the variance of the estimator of h(·). We first prove p
nV ⌧ 1/2 (✓ˆ⌧
For notational simplicity, let F = V ⌧
1/2
✓0 ) !d N (0, 1).
˜ = (h(w . Let h = (h(w1 ), ..., h(wn ))0 and h ˆ1 ), ..., h(w ˆn ))0 .
40
h0 (wi ) and ⌘ = (⌘1 , ..., ⌘n )0 .
Also let ⌘i = yi
Let ⇧n = (⇧n (z1 ), ..., ⇧n (zn ))0 , vi = xi
⇧n (zih), and U = (v1 , i..., vn )0 . Similar to NPV, once we prove that (i) F AQ⌧ 1 = O(Rn ), (ii) p p ˜ Pˆ# ) = op (1), (iv) F AQ ˆ 1 Pˆ 0 (h nF a(pK0 ˜) a(h0 ) = op (1), (iii) nF A(Pˆ 0 Pˆ +n⌧n I) 1 Pˆ 0 (h ⌧ p p p p 1 0 1 0 1 0 ˜ ˆ ⌧ Pˆ ⌘/ n = F AQ⌧ P ⌘/ n + op (1) below, then h)/ n = F AQ⌧ HR U/ n + op (1), and (v) F AQ we will have, p
⇣ nV ⌧ 1/2 ✓ˆ⌧
Then, for any vector
⌘ p ⇣ ⌘ ✓0 = nF a(pK0 ˆ⌧ ) a(pK0 ˜) + a(pK0 ˜) a(h0 ) p p = nF A ˆ⌧ nF A ˜ + op (1) p p = nF A(Pˆ 0 Pˆ + n⌧n I) 1 Pˆ 0 (h + ⌘) nF A(Pˆ 0 Pˆ + n⌧n I) p ˜ Pˆ# ˜) + op (1) + nF A(Pˆ 0 Pˆ + n⌧n I) 1 Pˆ 0 (h p p ˜ ˆ 1 Pˆ 0 ⌘/ n F AQ ˆ 1 Pˆ 0 (h h)/ = F AQ n + op (1) ⌧ ⌧ p p = F AQ⌧ 1 (P 0 ⌘/ n + HR0 U/ n) + op (1). with k k = 1, let Zin =
0 F AQ 1 [p ⌘ i i ⌧
1
˜ Pˆ 0 h
(A.33)
p + Hri ui ] / n. Note Zin is i.i.d. for
each n. Also EZin = 0, var(Zin ) = 1/n. Furthermore, F AQ⌧ 1 = O(Rn ) and F AQ C F AQ
1
= O(n ) by CI
1H
HH 0 being p.s.d, so that, for any " > 0,
⇥ ⇤ ⇥ ⇤ ⇥ ⇤ 2 nE 1 {|Zin | > "} Zin = n"2 E 1 {|Zin /"| > 1} (Zin /")2 n"2 E (Zin /")4 h i n"2 4 2 4 k k4 F AQ⌧ 1 {kpi k2 E kpi k2 E[⌘i4 |Xi ] n " h i + kri k2 E kri k2 E[u4i |zi ] } n o CO(Rn4 ) ⇣0v (K)2 E kpi k2 + ⇠(L)2 E kri k2 /n CO(Rn4 ) ⇣0v (K)2 tr(Q) + ⇠(L)2 tr(Q1 ) /n O(Rn4 ⇣0v (K)2 K + ⇠(L)2 L /n) = o(1) by G† . Then,
p
nF (✓ˆ⌧
✓0 ) !d N (0, 1) by Lindbergh-Feller theorem and (A.33).
Now, we proceed with detailed proofs of (i)–(v). For simplicity as before, the remainder of the proof will be given for the scalar ⇧(z) case. Under Assumption F and by CS, for hK (w) = pK (w)0 |a(hK )| = |A
K|
kAk k
away from zero, Q⌧
1
Kk
= kAk EhK
min (Q⌧
1 )I
1/2 (w)2
= CI. Also since
zero by Assumption E and ⌧n ! 0, we have ⌃ V⌧
so kAk ! 1. But since
AQ⌧ 1 ⌃Q⌧ 1 A0
2 (X)
2
1)
is bounded
= var(y|X) is bounded away from
CQ⌧ . Hence
CAQ⌧ 1 Q⌧ Q⌧ 1 A0
1 0 ˜ CAQ ⌧ A
C˜˜ kAk2 ,
Therefore, F is bounded. Now, by (A.34) F AQ⌧ 1/2
min (Q⌧
K,
= tr(F AQ⌧ 1 A0 F ) tr(CF V ⌧ F ) = C. 41
(A.34)
Also
max (Q⌧
1)
= O(Rn2 ), which can readily be shown analogous to (A.30). Using these results,
F AQ⌧ 1 = F AQ⌧ 1/2 Q⌧ 1/2
1 1/2
max (Q⌧
F AQ⌧ 1/2 = O(Rn ).
)
Then, combining with (A.30), (i) follows by ⇣ ˆ 1 F AQ 1 + F AQ 1 Q ˆ⌧ F A0 Q ⌧ ⌧ ⌧
ˆ⌧ O(Rn ) + O(Rn )Op (Rn2 ) Q = O(Rn ) + Op (Rn3
where Rn3
Q
Q)
⌘ ˆ 1 Q⌧ Q ⌧ Q⌧
= Op (Rn ),
= op (1) is by Assumption G† . Also
ˆ ⌧ 1/2 F AQ
2
F AQ⌧ 1/2
2
⇣ ˆ⌧ + tr(F AQ⌧ 1 Q ˆ⌧ Q
C + F A0 Q⌧ 1
⌘ ˆ ⌧ 1 A0 F ) Q⌧ Q
ˆ 1 O(Rn )Op ( F A0 Q ⌧
Q⌧
Q )O(Rn )
= op (1)
by G† . To prove (ii), by C† and G† , h nF a(pK0 ˜)
a(h0 )
˜ ˆ 1 Pˆ 0 (h F AQ ⌧
p p p ˆ 1 Pˆ 0 / n Pˆ# ˜)/ n F AQ n sup pK (w)0 ˜ ⌧
p
i
=
p
h nF a(pK0 ˜
p C nK
s/dx
h0 )
i
p
n |F | sup pK (w)0 ˜ w
h0 (w)
= op (1).
For (iii),
w
h0 (w)
p ˆ 1 ˜/ n + n⌧n F AQ ⌧ p p ˆ ⌧ 1/2 ˆ⌧ 1 ˜ F AQ nO(K s/dx ) + n⌧n F AQ p p p op (1)O( nK s/dx ) + Op ( n⌧n Rn (K)) = op (1)
ˆ 1 ˜ = Op (Rn ) can be shown analogous to F AQ ˆ 1 = Op (Rn ). To prove by G† , where F AQ ⌧ ⌧ ¯ = P pˆi d(Xi )r0 /n. (iv), let = ( 1 , ..., L )0 , di = d(Xi ) = [@h(wi )/@w]0 @!(Xi , ⇧0 (zi ))/@⇡ and H i
By a second order MVE of each h(w ˆi ) around wi ˆ ⌧ 1 Pˆ 0 (h F AQ
h X p ˜ ˆ i) ˆ⌧ 1 h)/ n = F AQ pˆi di ⇧(z i
i p ⇧n (zi ) / n + ⇢ˆ
p ˆ H ¯Q ˆ 1 R0 U/ n + F AQ ˆ 1H ¯Q ˆ 1 R0 (⇧n = F AQ ⌧ ⌧ 1 1 X ⇥ 0 ⇤ p 1 ˆ + F AQ pˆi di ri ⇧n (zi ) / n + ⇢ˆ. ⌧ 1
i
42
p R0 )/ n (A.35)
2 p ˆ ⌧ 1/2 ⇣ v (K) P ⇧(z ˆ i ) ⇧n (zi ) /n = op (1)Op (pn⇣ v (K) 2⇡ ) = op (1). But kˆ ⇢k C n F AQ 0 0 i 1 ¯0 ¯ ˆ Also, by di being bounded and nH Q1 H being equal to the matrix sum of squares from the ¯Q ˆ 1H ¯ 0 P pˆi pˆ0 d2 /n C Q ˆ CQ ˆ ⌧ . Therefore, the second multivariate regression of pˆi di on ri , H i i 1 i
term in (A.35) becomes
p p p ˆ ⌧ 1H ¯Q ˆ 1 R0 / n R0 )/ n F AQ n sup ⇧n (z) 1
ˆ ⌧ 1H ¯Q ˆ 1 R0 (⇧n F AQ 1
rL (z)0
Z
n ⇣ ⌘o1/2 p ˆ ⌧ 1H ¯Q ˆ 1Q ˆ 1Q ˆ 1H ¯ 0Q ˆ ⌧ 1 A0 F 0 tr F AQ O( nL s⇡ /dz ) 1 1 n ⇣ ⌘o1/2 p ˆ 1Q ˆ⌧ Q ˆ 1 A0 F 0 C tr F AQ O( nL s⇡ /dz ) ⌧ ⌧ p p ˆ ⌧ 1/2 O( nL s⇡ /dz ) = op (1)Op ( nL s⇡ /dz ) = op (1) C F AQ
by G† . Similarly, the third term is ˆ 1 F AQ ⌧
X
p p ˆ 1/2 O( nL ⇧n (zi )]/ n C F AQ ⌧
pˆi di [ri0
i
s⇡ /dz
p ) = op (1)Op ( nL
s⇡ /dz
) = op (1).
ˆ ⌧ 1H ¯Q ˆ 1 R0 U/pn in (A.35). Note that E kR0 U/pnk2 = Next, we consider the first term F AQ 1 p tr(⌃1 ) Ctr(IL ) L by E[u2 |z] bounded, so by MR, kR0 U/ nk = Op (L1/2 ). Also, we have ˆ ⌧ 1H ¯Q ˆ 1 Op (1) F AQ ˆ ⌧ 1/2 = op (1). F AQ 1 Therefore ˆ ⌧ 1 H( ¯ Q ˆ 1 F AQ 1
p ˆ 1H ¯Q ˆ 1 I)R0 U/ n F AQ ⌧ 1 = op (1)Op (
ˆ1 Q
Q1 )Op (L
I
1/2
p R0 U/ n
) = op (1)
by G† . Similarly, ˆ ⌧ 1 (H ¯ F AQ
p ˆ⌧ 1 H)R0 U/ n F AQ = Op (Rn )Op (
¯ where H
H = Op (
H)
¯ H
p R0 U/ n
H
H )Op (L
1/2
) = op (1),
instead of (A.12) in NPV, and the last eq. by G† . Note that HH 0 is
the population matrix mean-square of the regression of pi di on ri so that HH 0 C, it follows that p 2 p E kHR0 U/ nk = tr(H⌃1 H 0 ) CK and therefore, kHR0 U/ nk = Op (K 1/2 ). Then, ˆ⌧ 1 F A(Q
p Q⌧ 1 )HR0 U/ n
ˆ
max (Q⌧
1
) F AQ⌧ 1
= O(Rn2 )O(Rn )Op (
43
Q⌧
Q )Op (K
ˆ⌧ Q 1/2
p HR0 U/ n
) = op (1).
ˆ ⌧ 1H ¯Q ˆ 1 R0 U/pn = F AQ Combining the results above and by TR, F AQ 1
1 HR0 U/pn
+ op (1), and
thus we have the result of (iv). Lastly, to prove (v), similar to (A.22), p Pˆ )0 ⌘/ n = Op (Rn ⇣1v (K)2
ˆ 1/2 (P Q ⌧
2 ⇡)
= op (1)
by G† (and by (A.6) of NPV), which implies p ˆ 1/2 P )0 ⌘/ n F AQ ⌧
ˆ ⌧ 1 (Pˆ F AQ
ˆ ⌧ 1/2 (P Q
p Pˆ )0 ⌘/ n = op (1)op (1) = op (1).
Also, by E[⌘|X] = 0,
p 2 ˆ ⌧ 1 Q⌧ 1 )P 0 ⌘/ n |Xn F A(Q ⇣ hX i ⌘ ˆ 1 Q 1) ˆ 1 Q 1 )A0 F tr F A(Q pi p0i var(yi |Xi )/n (Q ⌧ ⌧ ⌧ ⌧ ⇣ ⌘ ⇣ ˆ ⌧ 1 Q⌧ 1 ) Q ˆ ⌧ (Q ˆ ⌧ 1 Q 1 ⌧ )A0 F = Ctr F AQ⌧ 1 (Q ˆ⌧ Ctr F A(Q
E
Op (Rn2 ) F AQ⌧ 1
2
ˆ⌧ Q
Q⌧
2
Op (Rn2
Q)
2
ˆ ⌧ 1 (Q ˆ⌧ Q⌧ ) Q
Q⌧ )Q⌧ 1 A0 F
= op (1)
⌘
by G† . Combining all of the previous results and by TR, we have the result of (v). Next, note CI CQ⌧ . Therefore,
H⌃1 H 0 is p.s.d and since var(y|X) is bounded by Assumption E, ⌃ CQ
V ⌧ = AQ⌧ 1 ⌃ + HQ1 1 ⌃1 Q1 1 H 0 Q⌧ 1 A0 = AQ⌧ 1 ⌃ + H⌃1 H 0 Q⌧ 1 A0 AQ⌧ 1 (CQ⌧ + CI) Q⌧ 1 A0 = C AQ⌧ 1/2
2
+ C AQ⌧ 1
2
.
The first-stage estimation error is not cancelled out with Q⌧ 1 , since the first-stage does not su↵er multicollinearity. Since a(pK0 A0 ) = AA0 , kAk2 a(pK0 A0 ) ApK
so
kAk ⇣r˜(K). Hence
=
O(Rn4 ⇣r˜(K)2 ).
Thus, V ✓ˆ⌧
⇣r˜(K) kAk by CS, r˜ 2 AQ⌧ = AQ⌧ 1 A0 C kAk = O(Rn2 ⇣r˜(K)2 ). Likewise, AQ⌧ 1 2 2 4 2 4 2 ⌧ = O(Rn ⇣r˜(K) ) + O(Rn ⇣r˜(K) ) = O(Rn ⇣r˜(K) ). Therefore, 1/2 2
2
p p p 2 2 v ✓0 = Op (V 1/2 ⌧ / n) = Op (Rn ⇣r˜(K)/ n) Op (Rn ⇣r˜ (K)/ n),
since max|µ|r supw2W @ µ pK (w)
2
supx2X k@ r p (x)k2 + supv2V k@ r p (v)k2 ⇣rv ()2 + ⇣rv ()2 =
2⇣rv ()2 and thus ⇣r (K) C⇣rv (K). This result for scalar a(h) covers the case of Assumption F. Now we can prove
by showing F Vˆ⌧ F
p
nVˆ⌧
1/2
(✓ˆ⌧
✓0 ) !d N (0, 1)
1 !p 0. Then, V ⌧ 1 Vˆ⌧ !p 1, so that
44
p ˆ nV ⌧
1/2
(✓ˆ⌧
✓0 ) =
p
nV ⌧
1/2
(✓ˆ
✓0 )/(V ⌧ 1 Vˆ⌧ )1/2 !d N (0, 1). The rest part of the proof can analogously be followed by the relevant part of the proof of NPV (pp. 600–601), using Q 6= I because of weak instruments and F = V⌧
1/2
.
Therefore the following replace the corresponding parts in the proof: For any matrix B, we have kB⌃k C kBQ⌧ k by ⌃ CQ⌧ . Therefore, ˆ 1 ⌃Q ˆ 1 F A(Q ⌧ ⌧ ˆ 1 F A(Q ⌧
Q⌧ 1 ⌃Q⌧ 1 )A0 F 0
ˆ 1 A0 F 0 + F AQ 1 ⌃(Q ˆ 1 Q⌧ 1 )⌃Q ⌧ ⌧ ⌧
ˆ ⌧ 1 (Q⌧ F AQ ˆ⌧ 1 F AQ
2
ˆ⌧ 1 C F AQ Op (Rn2 )Op (
Q⌧ 1 )A0 F 0
ˆ ⌧ )Q⌧ 1 ⌃Q ˆ ⌧ 1 A0 F 0 + F AQ⌧ 1 ⌃Q⌧ 1 (Q⌧ Q ˆ ⌧ )Q⌧ 1 ⌃ + F AQ⌧ 1 Q
(Q⌧ 2
(Q⌧
Q)
⌃Q⌧ 1 (Q⌧
ˆ ⌧ )Q⌧ 1 Q⌧ + C F AQ⌧ 1 Q
+ Op (Rn2 )Op (
Q)
ˆ ⌧ )Q ˆ ⌧ 1 A0 F 0 Q ˆ⌧ ) Q
ˆ⌧ 1 F AQ ˆ⌧ ) Q
Q⌧ Q⌧ 1 (Q⌧
ˆ⌧ 1 F AQ
= op (1)
by Assumption G† . Also note that in our proof, Q⌧ is introduced by penalization but the treatment is the same as above. Also, recall ⇣r (K) ⇣rv (K) and is, by
⇣0v (K)
2 h , Rn
Q
+
⇣0v (K)2 K/n
, and
Q are redefined in this paper. h and v 1/2 v ⇣0 (K)L ⇣1 (K) h converging to zero, we can
That prove
the following: ˆ 1 (⌃ ˆ⌧ F AQ ⌧ ˆ ⌧ 1 (⌃ ˜ F AQ
ˆ ⌧,i ˜ Q ˆ 1 A0 F 0 Ctr(D) ˆ max h ⌃) ⌧ in
ˆ ⌧ 1 A0 F 0 F AQ ˆ⌧ 1 ⌃)Q
ˆ⌧ H
¯ C H
n X i=1
2
˜ ⌃
hi Op (1)Op (⇣0v (K)
⌃ Op (Rn2 )Op (
kˆ pi k2 kri k2 /n
!1/2
= Op (⇣0v (K)L1/2 )Op (⇣1v (K)
n X
dˆ⌧,i
i=1
h)
Q
h)
= op (1),
+ ⇣0v (K)2 K/n) = op (1),
2
di /n
!1/2
= op (1).
h i0 ˆ ⌧ (wi )/@w @w(Xi , ⇧0 (zi ))/@⇡. The rest of the proof thus follows. ⇤ where dˆ⌧,i = @ h
Proof of Corollary 6.2: Now instead, suppose Assumption H holds. For the functional c0 a(g), this assumption is satisfied with ⌫(w) replaced by c0 ⌫(w). Therefore, it suffices to prove the result ⇥ ⇤ for scalar ⌫(w). Then, A = a(pK ) = E ⌫(wi )pK (wi )0 . Let ⌫K (w) = AQ 1 pK (w), which is 1⇥K
(transpose of) mean square projection of ⌫(·) on approximating functions pK (w). Note that Q 1 ⇥ ⇤ is singular with weak instruments. Therefore, define A⇤ = a(p⇤K ) = E ⌫(wi )p⇤K (ui )0 . Also, let ⇤ (u) = A⇤ Q⇤ ⌫K
1 p⇤K (u),
which is (transpose of) mean square projection of ⌫(·) on approximating
45
⇤ k2 E ⌫ ⌫K
functions p⇤K (u). Then, E k⌫
⌫K = E[⌫(wi )pK (wi )0 ]Q
⇤ E k⌫K
2
! 0 by Assumption H. But,
p (w) = E[⌫(wi )pK (wi )0 ]Tn (Tn0 QTn )
1 K
= E[⌫(wi )p⇤K (ui )0 ](Tn0 QTn ) Let R⇤ = Tn0 QTn
0 p⇤K ↵K
1 ⇤K
p
1
Tn0 pK (w)
(u).
⇤ 0 0 Q⇤ = E [mi p⇤0 i ] + E [pi mi ] + E [mi mi ]. Then, we have
⌫K k2 = E A⇤ [Q⇤
(Tn0 QTn )
1
1
]p⇤K (ui )
C kA⇤ k2 kR⇤ k2 E p⇤K (ui )
2
2
= E A⇤ Q⇤ 1 R⇤ (Tn0 QTn ) 1 p⇤K (ui ) ⇣ ⌘2 C˜ kA⇤ k2 2E kmi k kp⇤i k + E kmi k2 .
2
But by CS and Assumptions B and H, a(p⇤j ) = E[⌫(wi )p⇤j (ui )] E[v(wi )2 ]E[pj (u⇤i )2 ] < 1, and therefore kA⇤ k2 = a(p⇤K0 ) and by Assumption D (n n
2
⇣2v ()2 K 1/2 = n
2
K6
2
= O(K). Then, by using previous results of (A.12) and (A.13),
! 0) which implies that n
⇣2v ()1/2 K 1/2 = n
⌫K k2 ! 0. Lastly,
⇤ ⇣2v (K)2 K 1/2 converge to zero, we conclude that E k⌫K
let ⌫⌧,K (w) = AQ⌧ 1 pK (w). Then E k⌫K
⌫⌧,K k2 = E AQ⌧ 1 (Q⌧
1 K
Q)Q
p (wi )
0 K = O(⌧n Rn2 )E ↵ ˜K p (wi )
by Assumption F. Given
q
E k⌫
⌫⌧,K k2
2
q E ⌫
2
= O(⌧n Rn2 )E AQ
⇣2v (K)K and
1 K
p (wi )
2
= o(1) ⇤ ⌫K
2
+
q
⇤ E ⌫K
⌫K
2
+
q
E k⌫K
we have the desired result that E k⌫ ⌫⌧,K k2 ! 0. ⇥ ⇤ ⇥ ⇤ Let b⌧,KL (z) = E di ⌫⌧,K (wi )rL (zi )0 rL (z) and bL (z) = E di ⌫(wi )rL (zi )0 rL (z). Then, h E kb⌧,KL (zi )
i h bL (zi )k2 E d2i k⌫⌧,K (wi )
i h ⌫(wi )k2 CE k⌫⌧,K (wi )
h as K ! 1. Furthermore by Assumption E, E kbL (zi )
⌫⌧,K k2 ,
i ⌫(wi )k2 ! 0
i ⇢(zi )k2 ! 0 as L ! 1, where ⇢(z) is
a matrix of projections of elements of ⌫(w)d(X) on L which is the set of limit points of rL (z)0
L.
Therefore (as in (A.10) of NPV), by Assumption E ⇥ V⌧ = E ⌫⌧,K (wi )⌫⌧,K (wi )0
2
⇤ ⇥ ⇤ (Xi ) + E b⌧,KL (zi )var(xi |zi )b⌧,KL (zi )0 ! V¯ ,
⇥ ⇤ where V¯ = E ⌫(wi )⌫(wi )0 2 (Xi ) + E [⇢(zi )var(xi |zi )⇢(zi )0 ]. This shows F is bounded. p Then, given nF (✓ˆ⌧ ✓0 ) !d N (0, 1) from the proof of Theorem 6.1, the conclusion follows p from F 1 ! V¯ 1/2 so that F 1 nF (✓ˆ⌧ ✓0 ) !d N (0, V¯ ). ⇤
46
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51
µ2 4
8
16
32
64
128
256
Bias2
0.0377
0.0335
0.0054
0.0008
0.0000
0.0003
0.0000
V ar
99.0147
3.8019
0.9395
0.1419
0.0711
0.0310
0.0186
M SE
99.0524
3.8354
0.9449
0.1426
0.0711
0.0313
0.0186
M SEIV /M SELS
374.7291
15.9165
3.3232
0.5875
0.2790
0.1472
0.0901
Bias2
0.0328
0.0131
0.0030
0.0010
0.0002
0.0000
0.0000
V ar
0.3727
0.2557
0.1497
0.0829
0.0427
0.0349
0.0174
M SE
0.4055
0.2688
0.1527
0.0839
0.0429
0.0349
0.0174
M SEP IV /M SEIV
0.0035
0.1203
0.5365
0.6888
0.8297
0.9074
0.9452
Bias2
0.1145
0.0682
0.0305
0.0150
0.0042
0.0017
0.0010
V ar
0.4727
0.1332
0.0732
0.0894
0.0345
0.0248
0.0354
M SE
0.5872
0.2014
0.1037
0.1045
0.0387
0.0265
0.0364
M SEP IV /M SEIV
0.0024
0.0894
0.3501
0.5594
0.6462
0.7795
0.7464
Bias2
0.1566
0.1068
0.0685
0.0346
0.0158
0.0047
0.0022
V ar
0.2117
0.1981
0.2965
0.0318
0.0265
0.0183
0.0132
M SE
0.3684
0.3049
0.3649
0.0664
0.0423
0.0230
0.0154
M SEP IV /M SEIV
0.0037
0.0795
0.3862
0.4655
0.5942
0.7345
0.8238
⌧ =0
⌧ = 0.001
⌧ = 0.005
⌧ = 0.01
Table 1: Integrated squared bias, integrated variance, and integrated MSE of the penalized and unpenalized IV estimators gˆ⌧ (·) and gˆ(·).
⌧
CV Value
0.005
37.5315
0.01
37.5133
0.015
37.5065
0.02
37.5035
0.05
37.5070
Table 2: Cross-validation values for the choice of ⌧ .
52
(a) with a weak instrument
(b) with a strong instrument
Figure 2: Penalized versus unpenalized estimators (ˆ g⌧ (·) vs. gˆ(·)), ⌧ = 0.001. The (blue) dotted-dash line is the true g0 (·). The (black) solid line is the (simulated) mean of gˆ(·) with the dotted band representing the 0.025-0.975 quantile ranges. Note that the di↵erence between g0 (·) and the mean of gˆ(·) is the (simulated) bias. The (red) solid line is the mean of gˆ⌧ (·) with the dashed 0.025-0.975 quantile ranges.
(a) with a weak instrument
(b) with a strong instrument
Figure 3: Penalized versus unpenalized estimators (ˆ g⌧ (·) vs. gˆ(·)), ⌧ = 0.005.
53
Figure 4: Unpenalized IV estimates with nonparametric first-stage equations, full sample (n = 2019), 95% confidence band
Figure 5: Penalized IV estimates with the discontinuity sample (n = 650, F = 191.66).
54
