A Unifying Framework for Testing Shape Restrictions

A Unifying Framework for Testing Shape Restrictions

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Introduction

Shape restrictions are ubiquitous in economics, often arising as characterizations or implications of economic theory; see Matzkin1994Handbook, MWG1995, and ChetverikovSantosAzeem2018Shape for surveys and textbook treatment, as well as CoupriePelusoTrannoy2010Power, Ellison2011Strategic, CardMasMorettiSaez2012Inequality, Gicheva2013Working, ChandraHeadTappata2014Border, BoffaPiolattoPonzetto2016Political, ScheuerWerning2017Superstar, and ElliottKudrinWuthrich2021Detecting for some recent concrete examples. Contrary to their extensive roles in economics and despite the sizable literature, the formal statistical analysis of shape restrictions appears to be relatively scant in empirical work. One possible explanation is that many existing inferential procedures are either problem-specific or lack computational tractability without sacrificing statistical power FangSeo2019Shape.

In this paper, we develop a unifying framework for testing shape restrictions concerning a parameter of interest $\theta_0$, where the hypotheses are formulated as:

for $\phi$ some known map taking nonnegative values. In the spirit of the Wald test, we base our framework on the plug-in statistic $\phi(\hat\theta_n)$ for an unconstrained estimator $\hat\theta_n$ of $\theta_0$. Statistical properties of the resulting test then depend on the choice of the map $\phi$, which we shall call the Wald functional in what follows. As an example, for $\Lambda$ the family of all elements in the parameter space satisfying the shape restriction (hereafter), FangSeo2019Shape opt for $\phi(\theta)=\|\theta-\Pi_\Lambda(\theta)\|_{\mathbf H}$, where $\Pi_\Lambda(\theta)$ is the closest element in $\Lambda$ to $\theta$ (i.e., the projection of $\theta$ onto $\Lambda$), and $\|\cdot\|_{\mathbf H}$ is some $L^2$-norm. While the fruitful analytic properties of the projection operator enable FangSeo2019Shape to develop a powerful test for a class of shape restrictions, there are regrettably other prominent choices of $\phi$ that fall beyond the scope of their framework.

First, there are other shape enforcing operators that have received increasing attention recently, notably the rearrangement ChernozhukovFernandezGalichon2009Improving,ChernozhukovFernandezGalichon2010Crossing,ChernozhukovFernandezValLuo2018Sorted,ChenChernozhukovFernandezKostyshakLuo2021Shape and the greatest convex minorization (GCM) or the least concave majorization (LCM) CarolanTebbs2005LR,DelgadoEscanciano2012SM,BeareMoon2015DR,Seo2018SM,ChenChernozhukovFernandezKostyshakLuo2021Shape. The use of these two operators have been, however, confined to the construction of point and interval estimates or testing problems where the parameter $\theta_0$ is $\sqrt n$-estimable. Empirical settings of shape restrictions (e.g., nonparametric regression and density estimation), on the other hand, are often concerned with $\theta_0$ that is only estimable at slower than the $\sqrt n$-rate. Second, one may be interested in projection defined by non-$L^2$-norms such as the sup norm. Analogous to the comparison of the Cram\'{e}r-von Mises ($L^2$-type) test vs.\ the Kolmogorov-Smirnov (sup-type) test, different norms lead to tests that are powerful against different classes of alternatives, and there are settings where it may be desirable to employ a particular test---see Chapter 14 in TSH2005, AndrewsShi2013CMI, Armstrong2018Choice, and references therein for related discussions. Unfortunately, the projection operator in general may not be well-defined/behaved with respect to non-$L^2$-norms, rendering the framework of FangSeo2019Shape not directly applicable.

As the first contribution of this paper, we show that the attractive statistical properties of FangSeo2019Shape's test are not unique to the $L^2$-projection operator. Indeed, as far as forming a suitable Wald functional $\phi$ is concerned, it suffices to require convexity, positive homogeneity, and Lipschitz continuity. The importance of convexity has been noted in FangSantos2018HDD, while positive homogeneity and Lipschitz continuity are mild and automatically fulfilled for the class of Wald functionals we consider. To accommodate non-$\sqrt n$ estimable parameters $\theta_0$, we appeal to strong approximations as in FangSeo2019Shape. We show that our test has asymptotic uniform size control and is uniformly consistent (under regularity conditions).

In turn, with the testing framework in hand, we examine some concrete choices of the Wald functional. First, for a shape enforcing operator $\Upsilon$, we consider

where $\|\cdot\|$ is a norm that may not be $L^2$. We show that this strategy works for GCM and LCM, but not for the rearrangement as the resulting $\phi$ is not convex. Second, since projection may not be well-defined, one may instead employ

The aforementioned analytic properties are then met whenever $\Lambda$ is a nonempty closed convex cone. Devising a powerful test, however, further demands that the restriction under the null be incorporated into the construction of critical values, as well understood in the literature. This may be accomplished by using $\Upsilon(\hat\theta_n)$ for (ref), but using $\Pi_\Lambda(\hat\theta_n)$ for (ref) as in FangSeo2019Shape is problematic as $\Pi_\Lambda$ may not be well-defined/behaved if $\|\cdot\|$ is not a $L^2$-norm. To circumvent this challenge, we make use of $\Upsilon(\hat\theta_n)$ for (ref) as well, provided $\Upsilon$ is a well-defined shape enforcing operator. In particular, the rearrangement is well suited to this end, despite that it is not suitable in forming (ref). Together, these results broaden the use of GCM/LCM and rearrangement to a wide range of nonparametric settings, and extend the $L^2$-projection test in FangSeo2019Shape to a distance-based test (defined by a general norm).

We conduct Monte Carlo simulations to evaluate the performance of our test. The simulation results show that our framework yields tests that are competitive (in terms of size and power) compared to the prominent sup tests of ChernozhukovLeeRosen2013Intersection and Chetverikov2018Monotonicity. To showcase the empirical relevance of our framework, we test the shape restrictions on the relationship between weakly working hours and the annual wage growth in the high-end labor market. Overall, our empirical findings support the theoretical predictions of Gicheva2013Working for both men and women. The application also highlights the importance of conducting formal statistical tests when it comes to shape restrictions, rather than just relying on eyeball inspection.

The literature on shape restrictions dates back to Hildreth1954Concave, AyerBrunkEwingReidSilverman1955Incomplete, Brunk1955MLEmonotone, Eeden1956MLEorder, and Grenander1956II. We contribute to the problem of testing shape restrictions. Our paper extends FangSeo2019Shape who employ (ref) defined by a $L^2$-norm. We also build upon some of the analytic results and computational algorithms in ChenChernozhukovFernandezKostyshakLuo2021Shape. These authors study the use of shape enforcing operators for point and interval estimation, which is different from our focus on testing. We note that recent tests of concavity based on LCM are limited to {\it univariate} settings where $\theta_0$ is $\sqrt n$-estimable---see aforementioned references as well as ,DelgadoEscanciano2013ConditionalSD,DelgadoEscanciano2016ConditionalMI, BeareSchmidt2016Pricing, BeareShi2019DRO, and Fang2019KW. Other recent work include ChernozhukovNeweySantos2019CCMM and Zhu2020Shape who study moment restriction models with partial identification, FreybergerReeves2017Shape and ChiangKatoSasakiUra2021LP who construct shape-constrained confidence bands, BreunigChen2020Adaptive who develop adaptive and rate-optimal tests in nonparametric instrumental variable models, KomarovaHidalgo2020Shape who devise a pivotal test based on the Khmaladze's transformation for univariate nonparametric regression models, and KostyshakLuo2021PMP who propose the partial monotonicity parameter as a generalization of regression monotonicity. For brevity, we refer the reader to FangSeo2019Shape for additional references.

We now introduce some notation and concepts. Set $\mathbf R_+=\{a\in\mathbf R: a\ge 0\}$. For a vector $a\in\mathbf R^k$, we let $a^{(j)}$ be its $j$th entry and set $\|a\|_p$ to equal $\{\sum_{j=1}^{k}|a^{(j)}|^p\}^{1/p}$ if $p\in[1,\infty)$ and $\max_{j=1}^k|a^{(j)}|$ if $p=\infty$. We denote by $\mathbf M^{m\times k}$ the space of $m\times k$ matrices. For a function $f:\mathcal Z\to\mathbf R$ with $\mathcal Z\subset\mathbf R^{d_z}$, we set its $L^p$ norm $\|f\|_p$ to be $\{\int_{\mathcal Z}|f(z)|^p\,\mathrm dz\}^{1/p}$ if $p\in[1,\infty)$ and $\sup_{z\in\mathcal Z}|f(z)|$ if $p=\infty$. In turn, we define $L^p(\mathcal Z)\equiv\{f:\mathcal Z\to\mathbf R: \|f\|_p<\infty\}$ for $p\in[1,\infty)$ and $\ell^\infty(\mathcal Z)\equiv\{f:\mathcal Z\to\mathbf R: \|f\|_\infty<\infty\}$. For a vector space $\mathbf B$, we recall that a map $\phi:\mathbf B\to\mathbf R$ is said to be positively homogeneous if $\phi(a\theta)=a\phi(\theta)$ for all $a\ge 0$ and $\theta\in\mathbf B$. For generic families of distributions $\mathbf P_n$, a sequence $\{a_n\}$ of positive scalars, and a sequence $\{\mathbb X_n\}$ of random elements in a normed space $\mathbf B$ with norm $\|\cdot\|_{\mathbf B}$, write $\mathbb X_n=o_p(a_n)$ uniformly in $P\in\mathbf P_n$ if $\lim_{n\to\infty}\sup_{P\in\mathbf P_n}P(\|\mathbb X_n\|_{\mathbf B}>a_n\epsilon)= 0$ for any $\epsilon>0$, and $\mathbb X_n=O_p(a_n)$ uniformly in $P\in\mathbf P_n$ if $\lim_{M\to\infty}\limsup_{n\to\infty}\sup_{P\in\mathbf P_n}P(\|\mathbb X_n\|_{\mathbf B}>Ma_n)= 0$.

The remainder of the paper is structured as follows. In Section (ref), we develop the testing framework, investigates a number of possible Wald functionals, and provide some implementation guidance. Section (ref) conducts Monte Carlo simulation studies, while Section (ref) tests some shape restrictions concerning labor supply. Section (ref) concludes. All proofs are relegated to the appendix.

The Testing Framework

As well understood in the literature ImbensManski2004PID,Mikusheva2007Uniform,AndrewsGuggenberger2009Validity, it is imperative to ensure that testing procedures in nonstandard settings (such as the present one) be uniformly valid. To this end, we shall make explicit the dependence of the parameter interest $\theta_0$ on the underlying distribution $P$ by instead writing $\theta_P$ whenever appropriate. Moreover, we denote by $\mathbf P_0$ the model under the null and by $\mathbf P_1$ the model under the alternative. That is, $\mathbf P_0=\{P\in\mathbf P: \phi(\theta_P)=0\}$ and $\mathbf P_1=\mathbf P\backslash\mathbf P_0$, where $\mathbf P$ is the posited family of distributions possibly generating the data. We allow $\mathbf P_0$, $\mathbf P_1$ and $\mathbf P$ to depend on the sample size $n$, but such dependence is suppressed for notational simplicity.

Size and Power

In order to present a unifying treatment, we assume throughout that the parameter of interest $\theta_0$ lives in an abstract Banach space $\mathbf B$ (i.e., a complete normed space) with a known norm $\|\cdot\|_{\mathbf B}$. The following assumption formalizes our main restrictions on the Wald functional $\phi: \mathbf B\to\mathbf R_+$.

Assumptions (ref)(i)(ii) effectively demand that the null parameter space $\Lambda\equiv\{\theta\in\mathbf B: \phi(\theta)=0\}$ be a nonempty convex cone, which must be closed under Assumption (ref)(iii). As illustrated in FangSeo2019Shape, this is satisfied by a number of common shape restrictions such as nonnegativity, monotonicity, convexity/concavity, Slutsky restriction, supermodularity and any intersections of these restrictions. Nonetheless, some other prominent restrictions such as quasi-convexity/concavity are excluded. We note that, given Assumption (ref)(i), Assumption (ref)(ii) is equivalent to $\phi$ being subadditive, i.e., $\phi(\theta_1+\theta_2)\le\phi(\theta_1)+\phi(\theta_2)$ for all $\theta_1,\theta_2\in\mathbf B$. Assumption (ref)(iii) is a convenient mild condition for our distributional and bootstrap approximations. As shall be discussed in Section (ref), Assumption (ref) is satisfied for the distance function generated by a closed convex cone and the map $\phi$ generated by GCM/LCM, but is violated for the map $\phi$ generated by the rearrangement operator.

Assumption (ref) implies that the map $a\mapsto\phi(h+a\theta_0)$ is weakly decreasing on $[0,\infty)$ for any $\theta_0\in\Lambda$ and $h\in\mathbf B$, a property shared by the $L^2$-distance map in FangSeo2019Shape. This in turn yields the following key relation that is fundamental in the development of our test: for an estimator $\hat\theta_n$ of $\theta_0$ and $0\le \kappa_n\le r_n$,

The display (ref) reveals both challenges and opportunities in devising a test based on $r_n\phi(\hat\theta_n)$. On one hand, while the law of $r_n\{\hat\theta_n-\theta_P\}$ may be consistently bootstrapped, the generic impossibility of consistently estimating $r_n\theta_P$ poses challenges for estimating the distribution of $r_n\phi(\hat\theta_n)$, an issue prevalent in other nonstandard settings AndrewsSoares2010,ChernozhukovNeweySantos2019CCMM. On the other hand, consistently estimating the law of the upper bound in (ref) is possible if $\kappa_n$ is suitably small, in view of the identity $\kappa_n\hat\theta-\kappa_n\theta_P=\kappa_n/r_n\cdot r_n\{\hat\theta_n-\theta_P\}$. Taken together, the inequality in (ref) thus suggests that a test with $r_n\phi(\hat\theta_n)$ as the test statistic and critical values from the upper bound may control size in large samples.

To formalize the above idea, we introduce our second main assumption.

Assumption (ref)(i) requires a uniform (in $P\in\mathbf P$) distributional approximation of $r_n\{\hat\theta_n-\theta_P\}$ by a {\it sequence} of coupling variables $\mathbb Z_{n,P}$, in lieu of an {\it asymptotic} distribution. While one may work with the latter if $r_n\{\hat\theta_n-\theta_P\}$ does converge in distribution, this becomes problematic in general nonparametric settings in which $r_n\{\hat\theta_n-\theta_P\}$ fails to converge as a process. Nonetheless, ChernozhukovLeeRosen2013Intersection and subsequently, BelloniChernozhukovChetverikovKato2015New, ChernozhukovNeweySantos2019CCMM, ChenChristensen2018SupNormOptimal, BelloniChernozhukovChetverikovFernandez2019QR, CattaneoFarrellFeng2018Partition, and LiLiao2020Uniform show that approximations as in Assumption (ref)(i) can be established under regularity conditions, thereby greatly facilitating inference on global constraints (e.g., shape) of nonparametric functions. Assumption (ref)(ii) simply says that the law of $\bar{\mathbb Z}_{n,P}$ (or equivalently the law of $\mathbb Z_{n,P}$) can be consistently bootstrapped by $\hat{\mathbb G}_{n}$, which may be verified by the same machineries developed in the aforementioned work. We stress that Assumption (ref) places no restrictions on the particular schemes underlying the estimator $\hat\theta_n$ or the bootstrap $\hat{\mathbb G}_n$. Thus, one may resort to sieve or kernel estimation in constructing $\hat\theta_n$ and various bootstrap/simulation methods for $\hat{\mathbb G}_n$. Finally, Assumption (ref) implicitly entails certain smoothness on $\theta_P$, as typically required for nonparametric estimation.

Given the estimator $\hat\theta_n$ and the bootstrap $\hat{\mathbb G}_n$, we may employ $\phi(\hat{\mathbb G}_n+\kappa_n \hat\theta_n)$ as a bootstrap estimator for the upper bound in (ref). As well understood in analogous nonstandard settings AndrewsSoares2010,RomanoShaikhWolf2014TwoStep, however, this may cause loss of power in finite samples because $\hat\theta_n$ (in the term $\kappa_n \hat\theta_n$) does not reflect the restriction $\theta_P\in\Lambda$ under the null. This motivates us to use the restricted estimator $\Gamma(\hat\theta_n)$, provided a suitable shape enforcing operator $\Gamma:\mathbf B\to\Lambda$ is available, an issue we shall revisit in Section (ref). In turn, for a given significance level $\alpha\in(0,1)$, we may then obtain the critical value $\hat c_{n,1-\alpha}$ as:

By construction, $\hat c_{n,1-\alpha}$ is an estimator of the $1-\alpha$ quantile, denoted $c_{n,P}(1-\alpha)$, of $\phi(\mathbb Z_{n,P}+\kappa_n\theta_P)$ (a distributional approximation of the upper bound in (ref)). To justify the construction of $\hat c_{n,1-\alpha}$, we further impose:

Assumption (ref) demands mild requirements on $\Gamma$, which are satisfied by, e.g., the rearrangement for monotonicity, GCM for convexity, and a suitable composition of these two for monotonicity jointly with convexity---see Section (ref). Assumption (ref)(i) is a mild technical restriction, while Assumption (ref)(iii)(iv) imply that the cdfs of $\psi_{\kappa_n,P}(\mathbb Z_{n,P})$ are suitably continuous around $c_{n,P}(1-\alpha)$. Assumption (ref)(ii) differs from FangSeo2019Shape who assume uniform boundedness of $E[\|\mathbb Z_{n,P}\|_{\mathbf B}]$, because $E[\|\mathbb Z_{n,P}\|_{\mathbf B}]$ may grow with $n$ in our setting (e.g., when $\mathbf B=\ell^\infty([0,1])$).

Assumptions (ref), (ref), (ref), and (ref) together deliver our first main result.

Theorem (ref) formally establishes the asymptotic size control of our test, under a proper choice of $\kappa_n$. While one may ignore $\zeta_n$ from $\kappa_n\zeta_n/r_n=o(c_n)$ if $\mathbf B$ is a Hilbert space FangSeo2019Shape, it should be taken into account when $\mathbf B$ is endowed with, e.g., a uniform norm. In the nonparametric regression settings of ChernozhukovLeeRosen2013Intersection, one may take $\zeta_n=\sqrt{\log n}$.\footnote{Thus, the rate $r_n$ may not necessarily be the convergence rate of $\hat\theta_n$ as $\mathbb Z_{n,P}$ may diverge.} Theorem (ref) further show that our test uniformly attains the nominal rejection rates in the limit over a class of null distributions that may heuristically be thought of as the “boundary”. We also note that the classes of alternatives against which our test is uniformly consistent depends on the Wald functional $\phi$. In general, there does not exist a choice that leads to the most powerful test, and a different $\phi$ “distributes” power over a different region of the parameter space.

A key element in implementing our test is the choice of the tuning parameter $\kappa_n$. Intuitively, while $\kappa_n$ should be small as dictated in Theorem (ref), it should not be “too small” in the sense of causing the upper bound in (ref) overly crude. Following FangSeo2019Shape, we provide a data-driven choice of $\kappa_n$ as follows. For some small $\gamma_n\in(0,1)$, set $\hat\kappa_n\equiv r_nc_n/\hat\tau_{n,1-\gamma_n}$ where

In order to validate the use of $\hat\kappa_n$, we need to introduce our final assumption where

for $\mathbf B^*$ the space of continuous linear functions $b^*:\mathbf B\to\mathbf R$ endowed with the norm $\|b^*\|_{\mathbf B^*}\equiv\sup_{b\in\mathbf B: \|b\|_{\mathbf B}\le 1}|\langle b^*,b\rangle|$, and $\langle b^*,b\rangle\equiv b^*(b)$.

Assumption (ref) is in line with Assumption (ref)(ii) which allows $\{\|\mathbb Z_{n,P}\|_{\mathbf B}\}$ to diverge. Intuitively, Assumption (ref) requires that there be enough variations in $\{\mathbb Z_{n,P}\}$ even if the sequence $\{E[\|\mathbb Z_{n,P}\|_{\mathbf B}]\}$ of expected norms diverges.

The condition $\gamma_n\to 0$ ensures that our data driven $\hat\kappa_n$ satisfies the rate condition in Theorem (ref) in probability, while $(r_nc_n)^{-2}\zeta_n^2\log\gamma_n\to 0$ formalizes the precise sense in which $\gamma_n$ and $\hat\kappa_n$ should not be “too small.” In line with prior findings in the literature FangSantos2018HDD,ChernozhukovNeweySantos2019CCMM,FangSeo2019Shape, our Monte Carlo simulations show that the testing results are quite insensitive to the choice of $\gamma_n$. We recommend $\gamma_n=0.01/\log n$ or $1/n$ for practical implementations.

The Wald Functional

We next introduce a number of concrete Wald functionals, and investigate their suitability in applying the previous framework.

Rearrangement

Rearrangment is an operation that enforces a specific shape, namely monotonicity---throughout monotonicity means “weakly increasing.” For $\theta: \mathcal Z\to\mathbf R$ a potentially non-monotonic function with some bounded set $\mathcal Z\subset\mathbf R$, then the rearrangement operator $\Upsilon$ monotonizes $\theta$ as follows: for any $z\in\mathcal Z$,

If $\theta$ is multivariate, then one may monotonize $\theta$ by applying (ref) along each argument of $\theta$---see Section (ref) for more details. While rerrangement has proven particularly convenient in obtaining restricted point and interval estimates Fougeres1997Density,DetteNeumeyerPilz2006Simple,ChernozhukovFernandezGalichon2009Improving,ChernozhukovFernandezGalichon2010Crossing,ChernozhukovFernandezValLuo2018Sorted,ChenChernozhukovFernandezKostyshakLuo2021Shape, it does not generate a convex $\phi$ as in (ref), a property that is crucial for implementing our test. This can be easily seen when $\theta_0\in\mathbf B$ is finite dimensional. For example, if $\mathbf B=\mathbf R^4$ is endowed with the max norm and $\phi(\theta)=\|\theta-\Upsilon(\theta)\|_\infty$ with $\Upsilon$ the rearrangement operator, then, for $\theta_1=[40,54,42,69]^\text{\scalebox{0.8}{$\intercal$}}$ and $\theta_2=[21,88,3,68]^\text{\scalebox{0.8}{$\intercal$}}$, simple calculations yield

implying that $\phi$ is not convex. The lack of convexity remains true for alternative (e.g., $L^1$ or $L^2$) norms. Nonetheless, rearrangement satisfies Assumption (ref) LiebLoss2001Analysis,ChernozhukovFernandezGalichon2009Improving,ChenChernozhukovFernandezKostyshakLuo2021Shape, and may thus be utilized to construct constrained estimators for the purpose of power improvement.

Greatest Convex Minorization

GCM is also specific to a particular shape restriction, namely, convexity. The greatest convex minorant (also abbreviated GCM) of a function $\theta$ is the pointwise supremum of convex functions lying below $\theta$. Analogously, the least concave majorant (LCM) of a function $\theta$ is the pointwise infimum of concave functions lying above $\theta$. In what follows, we shall focus on GCM, as the analysis of LCM is similar. Following ChenChernozhukovFernandezKostyshakLuo2021Shape, we define GCM directly through the Legendre-Fenchel transform. Concretely, let $\mathcal Z\subset\mathbf R^{d_z}$ be a nonempty convex set, and $\theta:\mathcal Z\to\mathbf R$. Then the conjugate $\mathcal Z^\star$ of $\mathcal Z$ is $\mathcal Z^\star\equiv\{y\in\mathbf R^{d_z}: \sup_{z\in\mathcal Z}\{\langle y,z\rangle-\theta(z)\}<\infty\}$ which is a nonempty convex set, and the convex conjugate of $\theta$ is a map $\theta^\star:\mathcal Z^\star\to\mathbf R$ defined by

for all $y\in\mathcal Z^\star$. In turn, the biconjugate $\theta^{\star\star}\equiv (\theta^\star)^\star: \mathcal Z\to\mathbf R$ of $\theta$ is

Thus, the shape enforcing operator $\Upsilon$ in this case assigns each $\theta$ with its biconjugate $\theta^{\star\star}$. While well understood in mathematics (see, e.g., Zeidler1990III), the connection of GCM to the Legendre-Fenchel transform appears to be largely unnoticed in econometrics and statistics until the recent work by ChenChernozhukovFernandezKostyshakLuo2021Shape. This connection enables ChenChernozhukovFernandezKostyshakLuo2021Shape to propose a linear programming algorithm for the computation of $\theta^{\star\star}$---see Section (ref) for more details.

Following the literature, notably BeareMoon2015DR, we may construct the Wald functional based on the GCM operator $\Upsilon$ as: for all $\theta\in\ell^\infty(\mathcal Z)$ and $p\in[1,\infty]$,

Our next theorem establishes the analytic properties of $\phi$.

Positive homogeneity and Lipschitz continuity are well understood BeareFang2016Grenander,ChenChernozhukovFernandezKostyshakLuo2021Shape, though proving convexity of $\phi$ is nontrivial (to us). Theorem (ref) remains true for concavity if we set $\Upsilon(\theta)=-(-\theta)^{\star\star}$ in the construction (ref). We note that the boundedness of $\mathcal Z$ may be dispensed with at the cost of introducing a suitable weighting function in the definition of the $L^p$ norm. Theorems (ref) and (ref) together extend the use of GCM/LCM to nonparametric settings where $\theta_0$ may be convex/concave with respect to multiple variables and/or may not be $\sqrt n$-estimable. As GCM/LCM may be obtained through linear programming, the test based on (ref) may be more desirable than one based on $L^2$-projection (which requires quadratic programming), if computation cost is a binding constraint.

Distance and Projection

If $\Lambda$ is the class of all elements in $\mathbf B$ satisfying the shape restriction in question, then it is natural to form $\phi$ as the distance function defined in (ref) with $\|\cdot\|$ being $\|\cdot\|_{\mathbf B}$. By construction, statistical properties of the resulting test heavily depends on the shape of $\Lambda$ and the norm $\|\cdot\|_{\mathbf B}$. If $\Lambda$ is a nonempty closed convex set and $\mathbf B$ is a Hilbert space (a complete inner product space), then every $\theta\in\mathbf B$ admits a unique element in $\Lambda$, denoted $\Pi_{\Lambda}(\theta)$ and called the projection of $\theta$ onto $\Lambda$, which is closest to $\theta$. Thus, the map $\phi$ in (ref) reduces to a particular instance of (ref): for any $\theta\in\mathbf B$,

This is pursued in FangSantos2018HDD, and further in FangSeo2019Shape who additionally exploit $\Lambda$ being a cone. Projection onto closed convex sets/cones in Hilbert spaces enjoy elegant analytic properties that these authors extensively utilize to establish the statistical properties of their tests.

In particular settings, however, practitioners may wish to work with a different norm, such as the uniform norm if he/she wants to guard against alternatives uniformly deviating from the null. Such an extension is nontrivial for two reasons. First, the projection operator $\Pi_\Lambda$ in non-Hilbert spaces is in general not well-defined/behaved in the sense that it may be empty-valued, set-valued, discontinuous, or continuous but not uniformly so---see BarbuPrecupanu2012Convexity, DontchevZolezzi1993WellPosed, and Alber1996Projection. Second, employing the constrained estimator $\Pi_\Lambda(\hat\theta_n)$ as in FangSeo2019Shape (to improve power) is also problematic in view of the erratic behaviors of $\Pi_\Lambda$. The first challenge prompts us to take a step back and work directly with the distance function (ref), which remains well-defined. Importantly, $\phi$ enjoys attractive analytic properties, as summarized in the following well known lemma.

To circumvent the second challenge, we employ shape enforcing operators designed for specific restrictions, such as the rearrangement for monotonicity and GCM/LCM for convexity/concavity. For the joint restriction of monotonicity and convexity/concavity, if $\Upsilon_1$ denotes rearrangement and $\Upsilon_2$ GCM, then the composition $\Upsilon_2\circ\Upsilon_1$ satisfies Assumption (ref) but not $\Upsilon_1\circ\Upsilon_2$---see Remark 17 in ChenChernozhukovFernandezKostyshakLuo2021Shape. Together with Theorem (ref), Lemma (ref) thus extends FangSeo2019Shape to a distance-based test under alternative norms. In particular, the sup-norm yields a competing test to existing sup-type tests such as ChernozhukovLeeRosen2013Intersection and Chetverikov2018Monotonicity.

Implementation

For the convenience of practitioners, we now provide an implementation guide.

\sc Step 1: Compute the test statistic $r_n\phi(\hat\theta_n)$.

There are two aspects involved: obtain the estimator $\hat\theta_n$ and its rate $r_n$, and compute $r_n\phi(\hat\theta_n)$ for a given $\hat\theta_n$. The former may be obtained by standard procedures such as kernel or sieve estimation---see FangSeo2019Shape for more details. Given $\hat\theta_n$ and $r_n$, the key remains to compute $\phi(\hat\theta_n)$ which we demonstrate through two examples. Let $\mathbf B=\ell^\infty([0,1])$ and set $\vartheta=[\hat\theta_n(z_0),\ldots,\hat\theta_n(z_{N-1})]^\text{\scalebox{0.8}{$\intercal$}}$ for a large enough $N$ and $z_j\equiv j/(N-1)$. Consider first the set $\Lambda$ of weakly increasing functions in $\ell^\infty([0,1])$, and $\phi$ being the distance function. Then we may approximate $\phi(\hat\theta_n)$ by solving

where $D_N\in\mathbf M^{(N-1)\times N}$ is the matrix such that $D_Nh=[h^{(2)}-h^{(1)},\ldots, h^{(N)}-h^{(N-1)}]^\text{\scalebox{0.8}{$\intercal$}}$. Note that (ref) is a linear programming problem BoydVandenberghe2004Convex. Consider now the set $\Lambda$ consisting of convex functions in $\ell^\infty([0,1])$, and the Wald functional $\theta\mapsto\phi(\theta)=\|\theta-\theta^{\star\star}\|_\infty$. Then we may approximate $\phi(\hat\theta_n)$ by $\|\vartheta-\vartheta^{\star\star}\|_\infty$ where $l$th entry of $\vartheta^{\star\star}$ is given by (see, e.g., Carolan2002LCM):

where the objective in (ref) becomes $\hat\theta_n(z_{l-1})$ if $j=k=l-1$.

Both examples may be extended to multivariate settings. Concretely, let $\mathbf B=\ell^\infty([0,1]^{d_z})$, $\{z_j\}_{j=1}^N$ be a class of grid points over $[0,1]^{d_z}$, and $\vartheta=[\hat\theta_n(z_1),\ldots,\hat\theta_n(z_N)]^\text{\scalebox{0.8}{$\intercal$}}$. Then one may approximate $\phi(\hat\theta_n)$ in the first example by solving (ref) but now subject to $Ah\ge 0$ for some matrix $A$---see Example B.1 in FangSeo2019Shape. For the second example, following ChenChernozhukovFernandezKostyshakLuo2021Shape, we approximate $\phi(\hat\theta_n)$ by $\|\vartheta-\vartheta^{\star\star}\|_\infty$ where $l$th entry of $\vartheta^{\star\star}$ is given by solving the linear programming problem:

\sc Step 2: Construct the critical value $\hat c_{n,1-\alpha}$ with $\alpha\in(0,1)$.

In this step, we presume that the practitioner is capable of computing/approximating $\phi(\theta)$ for any given $\theta\in\mathbf B$ (as described above).

\sc Step 3: Reject $\mathrm H_0$ if and only if $r_n\phi(\hat\theta_n)>\hat c_{n,1-\alpha}$.

Simulation Studies

We next conduct Monte Carlo simulations to examine the finite sample performance of our test, based on both univariate and bivariate designs of nonparametric regression models. Throughout, the significance level is 5%, the number of Monte Carlo replications is 3000, and the number of bootstrap samples for each replication is 200. The tuning parameter $\kappa_n$ will be selected as in Proposition (ref), which entails a choice of $\gamma_n$. Since prior studies have repeatedly shown that testing results are quite insensitive to $\gamma_n$, we choose three values for $\gamma_n$: $0.01$, $0.01/\log n$, and $1/n$.

Testing Monotonicity

In this section, we are concerned with monotonicity. For the univariate designs, the regression function $\theta_0:[-1,1]\to\mathbf R$ under the null is of the form:

where $\varphi$ is the standard normal pdf, and $(\mathsf a,\mathsf b,\mathsf c)$ equals $(0,0,0)$, $(0.1,0.5,0.5)$ or $(0.5,2,1)$, labeled D1, D2 and D3 respectively. We then consider two sets of alternatives, in order to showcase the relative advantages of our test. The first set of alternatives consists of functions as in (ref) but with $\{(\mathsf a,\mathsf b,\mathsf c): \mathsf a =\mathsf c=-\Delta\delta,\mathsf b = 0.2\delta,\delta=1,\ldots,10\}$ for $\Delta=0.05$. The second set of alternatives are defined by: for $\mathsf b= 0.5\delta$ with $\delta=1,2,\ldots,10$,

The specification in (ref) is inspired by AndrewsShi2013CMI and ChernozhukovLeeRosen2013Intersection, but modified so that the empirical power is close to one for $\delta$ close to $10$. Figure (ref) depicts the curves of $\theta_P$ based on these designs. Importantly, $\theta_0$ in Figure (ref)-(b) becomes sharp V-shaped as $\delta$ increases, while $\theta_0$ in Figure (ref)-(c) has a visually “flat” bottom for each $\delta$. Finally, we draw i.i.d.\ samples $\{Z_i^*,u_i\}_{i=1}^n$ with $n\in\{500,750,1000\}$ from the standard normal distribution in $\mathbf R^2$, and set $Z_i=-1+2\Phi(Z_i^*)\in[-1,1]$ and $Y_i=\theta_0(Z_i)+u_i$, with $\Phi$ the standard normal cdf.

{ \newlength\figurewidth \setlength\figurewidth{0.35\textwidth}

}

For the bivariate designs, the regression function $\theta_0:[0,1]^2\to\mathbf R$ is specified as in FangSeo2019Shape: for some $(\mathsf a,\mathsf b,\mathsf c)\in\mathbf R^3$,

where the first term on the right-hand side of (ref) equals $\mathsf a\sqrt{z_1z_2}$ when $\mathsf b=0$. We consider three choices of $(\mathsf a,\mathsf b,\mathsf c)$ under $\mathrm H_0$, namely $(0,0,0)$, $(0.2,1,0)$ and $(0.5,0,0.5)$ (labeled D1, D2, and D3, respectively), and, for the alternative, the collection $\{(\mathsf a,\mathsf b,\mathsf c): \mathsf a =\mathsf c=-\Delta\delta,\mathsf b = 0.2\delta,\delta=1,\ldots,10\}$ for $\Delta=0.05$. We draw i.i.d.\ samples $\{Z_{1i}^*,Z_{2i}^*,u_i\}_{i=1}^n$ with $n\in\{500,750,1000\}$ from the standard normal distribution in $\mathbf R^3$, and set $Y_i=\theta_0(Z_i)+u_i$ where $Z_i\equiv(Z_{1i},Z_{2i})$ with $Z_{ji}=\Phi(Z_{ji}^*)\in[0,1]$ for all $i$ and $j=1,2$.

The implementation of our test is based on series least squares estimation using B-splines. Specifically, we employ cubic B-splines with 3, 5, or 7 interior knots for the univariate designs, and quadratic as well as cubic B-splines each with one or zero knots for the bivariate designs. The knots in both cases are placed at the equispaced empirical quantiles of the regressors. We choose the test statistics based on the supremum distance as described in Section (ref). In turn, $\hat{\mathbb G}_{n,b}$ is obtained by score bootstrap with i.i.d.\ weights from the standard normal distribution, and the coupling rate $c_n=1/\log n$. For ease of reference, we label our test with quadratic B-splines and $j$ knots as F-Q$j$; similarly, F-C$j$ is the implementation with cubic B-splines and $j$ knots.

For the sake of fair comparisons, we implement the sup-test of ChernozhukovLeeRosen2013Intersection and the one-step test of Chetverikov2018Monotonicity (labelled as C-OS) which is also of sup-type. For the former, we closely follow the steps articulated in Section 6.1 of ChernozhukovLeeRosen2013Intersection, and label the resulting test as CLR-C$j$ if the estimation is based on cubic B-splines with $j$ interior knots---CLR-Q$j$ is similarly defined. The C-OS test is implemented as in FangSeo2019Shape---see also Chetverikov2018Monotonicity and its working paper version for more details.

{ {6.5pt}

}

{ {3pt}

}

Tables (ref) and (ref) present the empirical sizes. In the univariate designs, all tests seem to control size reasonably well, though our tests and the CLR tests slightly over-reject. In the bivariate designs, while the rejections rates of our tests and the C-OS test are close to the nominal level (at least in large samples), the CLR tests are notably over-sized even in large samples. One possible explanation is that they are derivative-based tests and so the slow rate of convergence Stone1982Global is exacerbated in the bivariate designs. Figure (ref) in turn depicts the power curves, where we only present our test with $\gamma_n=0.01/\log n$ as other choices lead to very similar results (here and below). Overall, these curves demonstrate that our test is competitive to the tests of ChernozhukovLeeRosen2013Intersection and Chetverikov2018Monotonicity in terms of power as well. We note that, while the C-OS test enjoys some adaptivity and optimality properties in univariate settings, its bivariate version only captures part of the discordance between the outcome and the regressors so that its power in our bivariate designs is relatively low.

\pgfplotstableread{ delta alpha MonFiveKn3 MonFiveKn5 MonFiveKn7 MonSevenKn3 MonSevenKn5 MonSevenKn7 MonTenKn3 MonTenKn5 MonTenKn7 0 0.05 0.0587 0.0643 0.0647 0.0613 0.0607 0.0630 0.0647 0.0690 0.0587 1 0.05 0.0713 0.0727 0.0667 0.0847 0.0777 0.0737 0.0860 0.0810 0.0753 2 0.05 0.1177 0.1017 0.0923 0.1527 0.1350 0.1120 0.1747 0.1577 0.1380 3 0.05 0.1947 0.1667 0.1453 0.2823 0.2290 0.2010 0.3730 0.3143 0.2590 4 0.05 0.3227 0.2727 0.2293 0.4827 0.4023 0.3533 0.6190 0.5437 0.4580 5 0.05 0.4820 0.4290 0.3557 0.6917 0.6213 0.5437 0.8363 0.7690 0.6733 6 0.05 0.6650 0.5900 0.5033 0.8567 0.8033 0.7263 0.9397 0.9173 0.8600 7 0.05 0.8070 0.7460 0.6587 0.9513 0.9217 0.8697 0.9907 0.9780 0.9537 8 0.05 0.9080 0.8637 0.8027 0.9883 0.9783 0.9533 0.9987 0.9963 0.9863 9 0.05 0.9630 0.9420 0.8950 0.9970 0.9973 0.9860 1.0000 0.9997 0.9983 10 0.05 0.9890 0.9810 0.9560 0.9997 0.9990 0.9983 1.0000 1.0000 0.9997 }\UniMona

\pgfplotstableread{ delta alpha MonFiveKn3 MonFiveKn5 MonFiveKn7 MonSevenKn3 MonSevenKn5 MonSevenKn7 MonTenKn3 MonTenKn5 MonTenKn7 0 0.05 0.0587 0.0643 0.0647 0.0613 0.0607 0.0630 0.0647 0.0690 0.0587 1 0.05 0.0663 0.0713 0.0720 0.0787 0.0667 0.0700 0.0793 0.0777 0.0717 2 0.05 0.1033 0.0900 0.0917 0.1310 0.0967 0.0923 0.1357 0.1157 0.1080 3 0.05 0.1587 0.1307 0.1210 0.2170 0.1593 0.1453 0.2543 0.2000 0.1727 4 0.05 0.2433 0.1847 0.1690 0.3473 0.2570 0.2177 0.4357 0.3213 0.2727 5 0.05 0.3543 0.2703 0.2247 0.5260 0.3900 0.3213 0.6427 0.4940 0.4177 6 0.05 0.4883 0.3757 0.3147 0.6877 0.5433 0.4473 0.8013 0.6683 0.5677 7 0.05 0.6213 0.4950 0.4143 0.8317 0.6867 0.5883 0.9167 0.8130 0.7100 8 0.05 0.7470 0.6110 0.5217 0.9260 0.8177 0.7157 0.9767 0.9153 0.8260 9 0.05 0.8500 0.7323 0.6303 0.9747 0.9117 0.8270 0.9943 0.9680 0.9140 10 0.05 0.9237 0.8203 0.7260 0.9923 0.9623 0.9020 0.9990 0.9933 0.9703 }\UniMonb

\pgfplotstableread{ delta alpha MonFiveQKn0 MonFiveQKn1 MonSevenQKn0 MonSevenQKn1 MonTenQKn0 MonTenQKn1 0 0.05 0.0590 0.0720 0.0603 0.0633 0.0563 0.0550 1 0.05 0.0803 0.0967 0.0877 0.0987 0.0847 0.0873 2 0.05 0.1113 0.1207 0.1307 0.1360 0.1337 0.1340 3 0.05 0.1560 0.1557 0.1977 0.1883 0.2117 0.2100 4 0.05 0.2107 0.2030 0.2723 0.2520 0.3120 0.2913 5 0.05 0.2773 0.2573 0.3783 0.3370 0.4463 0.4190 6 0.05 0.3650 0.3280 0.4923 0.4463 0.5893 0.5457 7 0.05 0.4650 0.4047 0.6070 0.5597 0.7283 0.6883 8 0.05 0.5700 0.5003 0.7190 0.6760 0.8410 0.8067 9 0.05 0.6633 0.5993 0.8223 0.7787 0.9257 0.8920 10 0.05 0.7407 0.6817 0.8957 0.8537 0.9703 0.9460 }\BiMonQ

\pgfplotstableread{ delta alpha MonFiveCKn0 MonFiveCKn1 MonSevenCKn0 MonSevenCKn1 MonTenCKn0 MonTenCKn1 0 0.05 0.0747 0.0627 0.0670 0.0640 0.0567 0.0600 1 0.05 0.1000 0.0830 0.1020 0.0863 0.0903 0.0823 2 0.05 0.1283 0.0997 0.1437 0.1110 0.1370 0.1140 3 0.05 0.1637 0.1293 0.1953 0.1497 0.2167 0.1640 4 0.05 0.2203 0.1627 0.2670 0.2007 0.3083 0.2333 5 0.05 0.2733 0.2077 0.3583 0.2700 0.4303 0.3103 6 0.05 0.3440 0.2577 0.4740 0.3437 0.5670 0.4197 7 0.05 0.4273 0.3237 0.5853 0.4453 0.7067 0.5370 8 0.05 0.5317 0.3983 0.7000 0.5473 0.8230 0.6567 9 0.05 0.6270 0.4710 0.8020 0.6523 0.9037 0.7600 10 0.05 0.7093 0.5497 0.8723 0.7390 0.9527 0.8533 }\BiMonC

\pgfplotstableread{ delta alpha MonFiveKn3 MonFiveKn5 MonFiveKn7 MonSevenKn3 MonSevenKn5 MonSevenKn7 MonTenKn3 MonTenKn5 MonTenKn7 0 0.05 0.0603 0.0650 0.0703 0.0623 0.0630 0.0677 0.0590 0.0630 0.0607 1 0.05 0.0737 0.0737 0.0700 0.0817 0.0703 0.0717 0.0737 0.0817 0.0663 2 0.05 0.1167 0.1007 0.0823 0.1447 0.1113 0.0937 0.1687 0.1400 0.1033 3 0.05 0.1997 0.1530 0.1107 0.2863 0.1953 0.1350 0.3623 0.2520 0.1660 4 0.05 0.3307 0.2320 0.1560 0.4723 0.3317 0.2173 0.6157 0.4490 0.2753 5 0.05 0.4883 0.3507 0.2327 0.6740 0.5187 0.3357 0.8210 0.6710 0.4493 6 0.05 0.6617 0.5020 0.3247 0.8537 0.7080 0.4847 0.9370 0.8370 0.6377 7 0.05 0.8057 0.6510 0.4440 0.9477 0.8473 0.6480 0.9870 0.9397 0.7913 8 0.05 0.9053 0.7817 0.5787 0.9840 0.9343 0.7873 0.9977 0.9813 0.8993 9 0.05 0.9593 0.8850 0.6967 0.9960 0.9807 0.8910 0.9997 0.9973 0.9580 10 0.05 0.9850 0.9450 0.8070 1.0000 0.9953 0.9523 1.0000 1.0000 0.9907 }\UniMonCLRa

\pgfplotstableread{ delta alpha MonFiveKn3 MonFiveKn5 MonFiveKn7 MonSevenKn3 MonSevenKn5 MonSevenKn7 MonTenKn3 MonTenKn5 MonTenKn7 0 0.05 0.0603 0.0650 0.0703 0.0623 0.0630 0.0677 0.0590 0.0630 0.0607 1 0.05 0.0663 0.0710 0.0707 0.0690 0.0663 0.0677 0.0653 0.0677 0.0640 2 0.05 0.0790 0.0790 0.0723 0.0913 0.0770 0.0773 0.0933 0.0827 0.0700 3 0.05 0.1043 0.0853 0.0753 0.1223 0.0940 0.0867 0.1417 0.1007 0.0850 4 0.05 0.1397 0.1060 0.0840 0.1770 0.1220 0.0983 0.2237 0.1397 0.1050 5 0.05 0.1927 0.1313 0.0960 0.2633 0.1583 0.1217 0.3357 0.1797 0.1253 6 0.05 0.2523 0.1607 0.1157 0.3590 0.2113 0.1490 0.4623 0.2457 0.1700 7 0.05 0.3320 0.2033 0.1350 0.4810 0.2663 0.1837 0.6093 0.3370 0.2130 8 0.05 0.4230 0.2520 0.1633 0.6123 0.3550 0.2290 0.7400 0.4347 0.2747 9 0.05 0.5177 0.3077 0.1943 0.7337 0.4287 0.2740 0.8483 0.5487 0.3377 10 0.05 0.6287 0.3720 0.2233 0.8240 0.5267 0.3227 0.9287 0.6570 0.4057 }\UniMonCLRb

\pgfplotstableread{ delta alpha MonFiveQKn0 MonFiveQKn1 MonSevenQKn0 MonSevenQKn1 MonTenQKn0 MonTenQKn1 0 0.05 0.0720 0.1027 0.0713 0.0897 0.0630 0.0833 1 0.05 0.0933 0.1193 0.0947 0.1107 0.0937 0.1000 2 0.05 0.1277 0.1353 0.1330 0.1323 0.1397 0.1237 3 0.05 0.1717 0.1587 0.1800 0.1573 0.1983 0.1547 4 0.05 0.2143 0.1850 0.2550 0.1900 0.2870 0.1870 5 0.05 0.2733 0.2140 0.3437 0.2247 0.3923 0.2350 6 0.05 0.3533 0.2470 0.4457 0.2653 0.5123 0.2893 7 0.05 0.4287 0.2817 0.5530 0.3267 0.6513 0.3517 8 0.05 0.5110 0.3263 0.6563 0.3767 0.7727 0.4260 9 0.05 0.5873 0.3763 0.7587 0.4370 0.8660 0.5033 10 0.05 0.6767 0.4150 0.8420 0.5020 0.9283 0.5813 }\BiMonQCLR

\pgfplotstableread{ delta alpha MonFiveCKn0 MonFiveCKn1 MonSevenCKn0 MonSevenCKn1 MonTenCKn0 MonTenCKn1 0 0.05 0.1003 0.1337 0.0923 0.1023 0.0797 0.0930 1 0.05 0.1193 0.1440 0.1110 0.1143 0.0997 0.1053 2 0.05 0.1377 0.1530 0.1363 0.1337 0.1190 0.1147 3 0.05 0.1577 0.1717 0.1570 0.1443 0.1453 0.1340 4 0.05 0.1797 0.1840 0.1850 0.1617 0.1770 0.1543 5 0.05 0.2047 0.2013 0.2160 0.1773 0.2177 0.1697 6 0.05 0.2370 0.2277 0.2473 0.2057 0.2670 0.1967 7 0.05 0.2657 0.2403 0.2927 0.2273 0.3147 0.2243 8 0.05 0.3003 0.2567 0.3420 0.2487 0.3790 0.2627 9 0.05 0.3307 0.2850 0.3877 0.2813 0.4453 0.2990 10 0.05 0.3747 0.3043 0.4403 0.3100 0.5157 0.3287 }\BiMonCCLR

\pgfplotstableread{ delta alpha FivePI FiveOS FiveSD SevenPI SevenOS SevenSD TenPI TenOS TenSD 0 0.05 0.0550 0.0550 0.0550 0.0543 0.0543 0.0543 0.0563 0.0563 0.0563 1 0.05 0.0630 0.0630 0.0630 0.0630 0.0630 0.0630 0.0613 0.0613 0.0613 2 0.05 0.0800 0.0800 0.0800 0.0937 0.0937 0.0937 0.1150 0.1150 0.1150 3 0.05 0.1177 0.1177 0.1177 0.1743 0.1743 0.1743 0.2350 0.2350 0.2350 4 0.05 0.2050 0.2050 0.2050 0.3240 0.3240 0.3240 0.4357 0.4357 0.4357 5 0.05 0.3290 0.3290 0.3290 0.5253 0.5253 0.5253 0.6920 0.6920 0.6920 6 0.05 0.4830 0.4830 0.4830 0.7183 0.7183 0.7183 0.8627 0.8627 0.8627 7 0.05 0.6457 0.6457 0.6457 0.8737 0.8737 0.8737 0.9527 0.9527 0.9527 8 0.05 0.7807 0.7807 0.7807 0.9503 0.9503 0.9503 0.9873 0.9873 0.9873 9 0.05 0.8860 0.8860 0.8860 0.9873 0.9873 0.9873 0.9990 0.9990 0.9990 10 0.05 0.9480 0.9480 0.9480 0.9963 0.9963 0.9963 1.0000 1.0000 1.0000 }\UniMonCheta

\pgfplotstableread{ delta alpha FivePI FiveOS FiveSD SevenPI SevenOS SevenSD TenPI TenOS TenSD 0 0.05 0.0550 0.0550 0.0550 0.0543 0.0543 0.0543 0.0563 0.0563 0.0563 1 0.05 0.0523 0.0523 0.0523 0.0547 0.0547 0.0547 0.0643 0.0643 0.0643 2 0.05 0.0647 0.0647 0.0647 0.0697 0.0697 0.0697 0.0917 0.0917 0.0917 3 0.05 0.0877 0.0877 0.0877 0.1083 0.1083 0.1083 0.1540 0.1540 0.1540 4 0.05 0.1220 0.1220 0.1220 0.1930 0.1930 0.1930 0.2603 0.2603 0.2603 5 0.05 0.1900 0.1900 0.1900 0.3207 0.3207 0.3207 0.4310 0.4310 0.4310 6 0.05 0.2910 0.2910 0.2910 0.4757 0.4757 0.4757 0.6323 0.6323 0.6323 7 0.05 0.4053 0.4053 0.4053 0.6427 0.6427 0.6427 0.7970 0.7970 0.7970 8 0.05 0.5387 0.5387 0.5387 0.7940 0.7940 0.7940 0.9140 0.9140 0.9140 9 0.05 0.6750 0.6750 0.6750 0.8993 0.8993 0.8993 0.9703 0.9703 0.9703 10 0.05 0.7820 0.7820 0.7820 0.9623 0.9623 0.9623 0.9927 0.9927 0.9927 }\UniMonChetb

\pgfplotstableread{ delta alpha FivePI FiveOS FiveSD SevenPI SevenOS SevenSD TenPI TenOS TenSD 0 0.05 0.0593 0.0593 0.0593 0.0560 0.0560 0.0560 0.0587 0.0587 0.0587 1 0.05 0.0657 0.0657 0.0657 0.0660 0.0660 0.0660 0.0697 0.0697 0.0697 2 0.05 0.0703 0.0703 0.0703 0.0733 0.0733 0.0733 0.0833 0.0833 0.0833 3 0.05 0.0793 0.0793 0.0793 0.0877 0.0877 0.0877 0.0960 0.0960 0.0960 4 0.05 0.0853 0.0853 0.0853 0.1050 0.1050 0.1050 0.1187 0.1187 0.1187 5 0.05 0.0973 0.0973 0.0973 0.1270 0.1270 0.1270 0.1427 0.1427 0.1427 6 0.05 0.1127 0.1127 0.1127 0.1523 0.1523 0.1523 0.1823 0.1823 0.1823 7 0.05 0.1357 0.1357 0.1357 0.1820 0.1820 0.1820 0.2300 0.2300 0.2300 8 0.05 0.1543 0.1543 0.1543 0.2317 0.2317 0.2317 0.2913 0.2913 0.2913 9 0.05 0.1853 0.1853 0.1853 0.2900 0.2900 0.2900 0.3727 0.3727 0.3727 10 0.05 0.2147 0.2147 0.2147 0.3520 0.3520 0.3520 0.4643 0.4643 0.4643 }\BiMonChet

Testing Convexity/Concavity

We adopt the same univariate designs as in Section (ref), with the only change being that the specification in (ref) is replaced by

The null hypothesis in this case is that $\theta_0: [-1,1]\to\mathbf R$ is convex. In bivariate designs, we instead test the concavity of $\theta_0$ by employing designs that are slight variations of (ref) (so that the power curves get close to one as $\delta$ increases):

where $(\mathsf a,\mathsf b,\mathsf c)$ is chosen to be the same as for (ref) except $\Delta=0.2$. We implement our test based on the GCM/LCM operators (i.e., based on (ref) with $p=\infty$), and compare it to the sup-test of ChernozhukovLeeRosen2013Intersection. The estimation and bootstrap steps are the same as those in Section (ref).

{ {6.5pt}

}

Tables (ref) and (ref) summarize the empirical sizes. In the univariate designs, both our tests and the CLR tests have reasonable size control, while in the bivariate designs the CLR tests once again exhibit notable over-rejections across the sample sizes and the sieve spaces. Note that both D1 and D2 are at the “boundaries” of the parameter spaces, and thus the rejections rates are expected to be close to $\alpha=5\%$. Figure (ref) shows that our test remains competitive to the sup-test of ChernozhukovLeeRosen2013Intersection as far as power is concerned. This is particularly the case for the designs based on (ref) in which $\theta_0$ has a flat region around $z=0$ for each $\delta$, in line with the discussions in ChernozhukovLeeRosen2013Intersection. One possible explanation is that these are settings where the set estimation step in implementing the CLR tests is statistically challenging.

{ {3pt}

}

\pgfplotstableread{ delta alpha ConFiveKn3 ConFiveKn5 ConFiveKn7 ConSevenKn3 ConSevenKn5 ConSevenKn7 ConTenKn3 ConTenKn5 ConTenKn7 0 0.05 0.0557 0.0523 0.0547 0.0560 0.0553 0.0607 0.0620 0.0627 0.0613 1 0.05 0.0867 0.0713 0.0683 0.1043 0.0853 0.0823 0.1127 0.1047 0.0920 2 0.05 0.1613 0.1273 0.1103 0.2277 0.1740 0.1420 0.2703 0.2253 0.1923 3 0.05 0.3027 0.2373 0.1903 0.4180 0.3317 0.2693 0.5497 0.4443 0.3587 4 0.05 0.4770 0.3883 0.3157 0.6530 0.5580 0.4727 0.7900 0.7007 0.6107 5 0.05 0.6713 0.5687 0.4810 0.8530 0.7683 0.6890 0.9423 0.8927 0.8220 6 0.05 0.8287 0.7430 0.6570 0.9550 0.9130 0.8473 0.9857 0.9713 0.9440 7 0.05 0.9247 0.8763 0.8083 0.9907 0.9793 0.9497 0.9983 0.9960 0.9893 8 0.05 0.9747 0.9527 0.9070 0.9980 0.9980 0.9910 0.9993 0.9990 0.9987 9 0.05 0.9930 0.9840 0.9673 1.0000 1.0000 0.9990 1.0000 0.9997 0.9997 10 0.05 0.9987 0.9960 0.9900 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 }\UniCona

\pgfplotstableread{ delta alpha ConFiveKn3 ConFiveKn5 ConFiveKn7 ConSevenKn3 ConSevenKn5 ConSevenKn7 ConTenKn3 ConTenKn5 ConTenKn7 0 0.05 0.0557 0.0523 0.0547 0.0560 0.0553 0.0607 0.0620 0.0627 0.0613 1 0.05 0.0947 0.0780 0.0750 0.1063 0.0887 0.0840 0.1163 0.1073 0.0917 2 0.05 0.1543 0.1207 0.1037 0.1890 0.1503 0.1303 0.2257 0.1957 0.1540 3 0.05 0.2623 0.1823 0.1517 0.3387 0.2433 0.2083 0.4180 0.3170 0.2593 4 0.05 0.3877 0.2720 0.2210 0.5277 0.3917 0.3163 0.6613 0.4987 0.4117 5 0.05 0.5330 0.3937 0.3120 0.7183 0.5583 0.4633 0.8480 0.6983 0.5997 6 0.05 0.6920 0.5273 0.4250 0.8740 0.7290 0.6130 0.9557 0.8663 0.7640 7 0.05 0.8320 0.6670 0.5430 0.9510 0.8683 0.7627 0.9910 0.9620 0.8897 8 0.05 0.9170 0.8087 0.6753 0.9863 0.9450 0.8787 0.9993 0.9900 0.9663 9 0.05 0.9690 0.9013 0.7953 0.9953 0.9800 0.9427 1.0000 0.9980 0.9893 10 0.05 0.9907 0.9527 0.8827 0.9997 0.9947 0.9793 1.0000 1.0000 0.9980 }\UniConb

\pgfplotstableread{ delta alpha ConFiveQKn0 ConFiveQKn1 ConSevenQKn0 ConSevenQKn1 ConTenQKn0 ConTenQKn1 0 0.05 0.0573 0.0643 0.0603 0.0683 0.0563 0.0627 1 0.05 0.0820 0.0827 0.0893 0.0937 0.0883 0.0923 2 0.05 0.1127 0.1043 0.1327 0.1200 0.1440 0.1353 3 0.05 0.1553 0.1357 0.1880 0.1673 0.2157 0.1937 4 0.05 0.2143 0.1827 0.2623 0.2327 0.3157 0.2743 5 0.05 0.2860 0.2377 0.3710 0.3123 0.4393 0.3820 6 0.05 0.3673 0.3007 0.4690 0.3953 0.5740 0.4967 7 0.05 0.4497 0.3823 0.5913 0.4877 0.7037 0.6163 8 0.05 0.5553 0.4617 0.7070 0.5973 0.8220 0.7340 9 0.05 0.6467 0.5483 0.8100 0.7073 0.9027 0.8337 10 0.05 0.7283 0.6247 0.8830 0.7953 0.9527 0.9063 }\BiConQ

\pgfplotstableread{ delta alpha ConFiveCKn0 ConFiveCKn1 ConSevenCKn0 ConSevenCKn1 ConTenCKn0 ConTenCKn1 0 0.05 0.0650 0.0643 0.0730 0.0700 0.0637 0.0603 1 0.05 0.0853 0.0767 0.0980 0.0917 0.0940 0.0767 2 0.05 0.1153 0.0913 0.1340 0.1077 0.1440 0.0967 3 0.05 0.1537 0.1073 0.1820 0.1347 0.2047 0.1313 4 0.05 0.2050 0.1327 0.2500 0.1627 0.2970 0.1720 5 0.05 0.2587 0.1523 0.3363 0.2050 0.4063 0.2337 6 0.05 0.3347 0.1897 0.4243 0.2530 0.5260 0.2963 7 0.05 0.4170 0.2290 0.5303 0.3127 0.6560 0.3683 8 0.05 0.5007 0.2753 0.6407 0.3810 0.7683 0.4640 9 0.05 0.5863 0.3130 0.7343 0.4557 0.8643 0.5513 10 0.05 0.6680 0.3847 0.8303 0.5303 0.9290 0.6537 }\BiConC

\pgfplotstableread{ delta alpha ConFiveKn3 ConFiveKn5 ConFiveKn7 ConSevenKn3 ConSevenKn5 ConSevenKn7 ConTenKn3 ConTenKn5 ConTenKn7 0 0.05 0.0543 0.0560 0.0643 0.0610 0.0630 0.0650 0.0577 0.0650 0.0613 1 0.05 0.0707 0.0637 0.0667 0.0823 0.0667 0.0670 0.0833 0.0780 0.0707 2 0.05 0.1077 0.0883 0.0737 0.1477 0.1040 0.0843 0.1693 0.1323 0.0933 3 0.05 0.1907 0.1297 0.0957 0.2727 0.1727 0.1090 0.3643 0.2237 0.1353 4 0.05 0.3150 0.1980 0.1317 0.4663 0.2837 0.1647 0.5900 0.3847 0.2080 5 0.05 0.4700 0.3107 0.1790 0.6590 0.4443 0.2493 0.7940 0.5803 0.3197 6 0.05 0.6430 0.4330 0.2573 0.8253 0.6267 0.3550 0.9233 0.7630 0.4777 7 0.05 0.7817 0.5870 0.3453 0.9373 0.7780 0.5043 0.9800 0.8873 0.6350 8 0.05 0.8893 0.7227 0.4563 0.9797 0.8923 0.6413 0.9947 0.9590 0.7820 9 0.05 0.9470 0.8343 0.5757 0.9940 0.9577 0.7830 0.9997 0.9873 0.8917 10 0.05 0.9763 0.9113 0.6837 0.9993 0.9863 0.8787 1.0000 0.9990 0.9537 }\UniConCLRa

\pgfplotstableread{ delta alpha ConFiveKn3 ConFiveKn5 ConFiveKn7 ConSevenKn3 ConSevenKn5 ConSevenKn7 ConTenKn3 ConTenKn5 ConTenKn7 0 0.05 0.0543 0.0560 0.0643 0.0610 0.0630 0.0650 0.0577 0.0650 0.0613 1 0.05 0.0663 0.0610 0.0663 0.0727 0.0700 0.0707 0.0720 0.0743 0.0660 2 0.05 0.0813 0.0703 0.0733 0.0913 0.0807 0.0723 0.1007 0.0817 0.0713 3 0.05 0.1020 0.0750 0.0730 0.1170 0.0883 0.0807 0.1230 0.0953 0.0743 4 0.05 0.1183 0.0853 0.0797 0.1490 0.0997 0.0827 0.1643 0.1123 0.0867 5 0.05 0.1373 0.0937 0.0837 0.1820 0.1123 0.0927 0.2043 0.1257 0.0937 6 0.05 0.1687 0.1063 0.0857 0.2213 0.1303 0.1007 0.2607 0.1440 0.1007 7 0.05 0.1977 0.1223 0.0943 0.2717 0.1493 0.1050 0.3257 0.1693 0.1063 8 0.05 0.2457 0.1297 0.0973 0.3240 0.1630 0.1120 0.3967 0.1863 0.1183 9 0.05 0.2853 0.1473 0.1067 0.3833 0.1847 0.1157 0.4800 0.2173 0.1277 10 0.05 0.3357 0.1543 0.1140 0.4593 0.2047 0.1273 0.5717 0.2497 0.1410 }\UniConCLRb

\pgfplotstableread{ delta alpha ConFiveQKn0 ConFiveQKn1 ConSevenQKn0 ConSevenQKn1 ConTenQKn0 ConTenQKn1 0 0.05 0.0713 0.0873 0.0627 0.0800 0.0653 0.0800 1 0.05 0.0977 0.0990 0.0897 0.0907 0.0863 0.0847 2 0.05 0.1200 0.1093 0.1257 0.1057 0.1223 0.1013 3 0.05 0.1590 0.1220 0.1690 0.1207 0.1697 0.1173 4 0.05 0.2003 0.1360 0.2273 0.1350 0.2357 0.1343 5 0.05 0.2387 0.1480 0.2877 0.1517 0.3120 0.1637 6 0.05 0.2933 0.1670 0.3623 0.1727 0.4187 0.1847 7 0.05 0.3523 0.1923 0.4420 0.1977 0.5320 0.2223 8 0.05 0.4217 0.2073 0.5293 0.2233 0.6350 0.2503 9 0.05 0.5067 0.2387 0.6213 0.2567 0.7330 0.2900 10 0.05 0.5730 0.2593 0.7073 0.2837 0.8157 0.3277 }\BiConQCLR

\pgfplotstableread{ delta alpha ConFiveCKn0 ConFiveCKn1 ConSevenCKn0 ConSevenCKn1 ConTenCKn0 ConTenCKn1 0 0.05 0.1010 0.1360 0.0863 0.1030 0.0763 0.0897 1 0.05 0.1143 0.1480 0.1047 0.1143 0.0927 0.0973 2 0.05 0.1367 0.1587 0.1193 0.1177 0.1137 0.1127 3 0.05 0.1593 0.1723 0.1463 0.1367 0.1470 0.1180 4 0.05 0.1847 0.1833 0.1833 0.1490 0.1823 0.1413 5 0.05 0.2157 0.1950 0.2190 0.1600 0.2333 0.1507 6 0.05 0.2463 0.2067 0.2650 0.1793 0.2957 0.1710 7 0.05 0.2893 0.2257 0.3190 0.1967 0.3683 0.1990 8 0.05 0.3307 0.2433 0.3803 0.2230 0.4537 0.2287 9 0.05 0.3787 0.2660 0.4477 0.2450 0.5373 0.2587 10 0.05 0.4347 0.2820 0.5250 0.2787 0.6230 0.2933 }\BiConCCLR

Testing Joint Restrictions

Finally, we test the joint restrictions of monotonicity and convexity/concavity, based on the same designs as those in Section (ref). We implement our test based on the supremum distance, coupled with the $\Gamma$ operator (see Assumption (ref)) obtained by taking the composition $\Gamma=\Upsilon_2\circ\Upsilon_1$, where $\Upsilon_1$ is the rearrangement operator and $\Upsilon_2$ is the GCM/LCM operator. That is, we apply rearrangement first and then the GCM/LCM operation. The remaining steps of our tests as well as the CLR tests are the same as before, beyond the need of incorporating the joint restrictions.

{ {6.5pt}

}

Tables (ref) and (ref) summarize the empirical sizes, while Figure (ref) presents the power curves. Consistent with our previous findings, our tests control size reasonably well in both univariate and bivariate designs, across sample sizes and sieve spaces, while the CLR tests tend to over-reject in bivariate designs (but otherwise perform well in terms of size). Moreover, our tests enjoy competitive power compared to the CLR tests, especially when $\theta_0$ has a flat region.

To conclude our simulations, we stress that our intention is not to show that our test is uniformly better than existing tests. As well understood in the literature, uniformly most powerful tests typically do not exist in nonparametric (and also many parametric) settings, and thus no single test would dominate all others in terms of both size and power. Indeed, our numerical results show that, depending on the sample size, the functional form of $\theta_0$, and the sieve space, the CLR tests and the C-OS test may perform better than our tests (in terms of size or power). Instead, we hope to convey the message that our tests may serve as competitive alternatives, and may perform better in particular settings. In addition, compared to tests such as Chetverikov2018Monotonicity that are designed for specific shapes, our framework readily accommodates additional/joint restrictions. Our numerical exercises also confirm the applicability and usefulness of shape enforcing operators in either forming the Wald functinal or enforcing the null restriction for the purpose of power improvement.

{ {3pt}

}

\pgfplotstableread{ delta alpha MConFiveKn3 MConFiveKn5 MConFiveKn7 MConSevenKn3 MConSevenKn5 MConSevenKn7 MConTenKn3 MConTenKn5 MConTenKn7 0 0.05 0.0553 0.0517 0.0533 0.0547 0.0547 0.0583 0.0603 0.0600 0.0600 1 0.05 0.0840 0.0700 0.0667 0.1007 0.0820 0.0810 0.1103 0.1030 0.0907 2 0.05 0.1590 0.1247 0.1070 0.2233 0.1727 0.1393 0.2663 0.2237 0.1897 3 0.05 0.2980 0.2333 0.1870 0.4150 0.3263 0.2660 0.5480 0.4420 0.3563 4 0.05 0.4720 0.3873 0.3117 0.6497 0.5557 0.4693 0.7880 0.6993 0.6060 5 0.05 0.6677 0.5657 0.4767 0.8517 0.7660 0.6873 0.9420 0.8900 0.8200 6 0.05 0.8263 0.7420 0.6520 0.9547 0.9120 0.8470 0.9857 0.9710 0.9430 7 0.05 0.9243 0.8740 0.8083 0.9907 0.9787 0.9493 0.9983 0.9960 0.9893 8 0.05 0.9747 0.9510 0.9063 0.9980 0.9980 0.9910 0.9993 0.9990 0.9990 9 0.05 0.9933 0.9833 0.9680 1.0000 1.0000 0.9987 1.0000 0.9997 0.9997 10 0.05 0.9987 0.9960 0.9893 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 }\UniMCona

\pgfplotstableread{ delta alpha MConFiveKn3 MConFiveKn5 MConFiveKn7 MConSevenKn3 MConSevenKn5 MConSevenKn7 MConTenKn3 MConTenKn5 MConTenKn7 0 0.05 0.0553 0.0517 0.0533 0.0547 0.0547 0.0583 0.0603 0.0600 0.0600 1 0.05 0.0927 0.0753 0.0727 0.1053 0.0873 0.0840 0.1153 0.1063 0.0913 2 0.05 0.1527 0.1203 0.1023 0.1880 0.1483 0.1297 0.2257 0.1920 0.1527 3 0.05 0.2610 0.1810 0.1500 0.3380 0.2423 0.2073 0.4170 0.3167 0.2563 4 0.05 0.3880 0.2700 0.2193 0.5263 0.3923 0.3143 0.6620 0.4993 0.4093 5 0.05 0.5327 0.3930 0.3120 0.7177 0.5577 0.4627 0.8480 0.6987 0.5990 6 0.05 0.6920 0.5277 0.4243 0.8737 0.7283 0.6113 0.9557 0.8660 0.7637 7 0.05 0.8323 0.6667 0.5420 0.9507 0.8683 0.7607 0.9910 0.9620 0.8893 8 0.05 0.9170 0.8077 0.6737 0.9863 0.9450 0.8797 0.9993 0.9900 0.9663 9 0.05 0.9693 0.9010 0.7953 0.9953 0.9800 0.9433 1.0000 0.9980 0.9893 10 0.05 0.9907 0.9520 0.8823 0.9997 0.9947 0.9790 1.0000 1.0000 0.9980 }\UniMConb

\pgfplotstableread{ delta alpha MConFiveQKn0 MConFiveQKn1 MConSevenQKn0 MConSevenQKn1 MConTenQKn0 MConTenQKn1 0 0.05 0.0597 0.0693 0.0597 0.0660 0.0590 0.0547 1 0.05 0.2857 0.2637 0.3840 0.3407 0.4533 0.4307 2 0.05 0.7570 0.6973 0.9040 0.8647 0.9727 0.9493 3 0.05 0.9823 0.9647 0.9993 0.9970 1.0000 1.0000 4 0.05 0.9997 0.9983 1.0000 1.0000 1.0000 1.0000 5 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 6 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 7 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 8 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 }\BiMConQ

\pgfplotstableread{ delta alpha MConFiveCKn0 MConFiveCKn1 MConSevenCKn0 MConSevenCKn1 MConTenCKn0 MConTenCKn1 0 0.05 0.0760 0.0673 0.0700 0.0677 0.0570 0.0587 1 0.05 0.2807 0.2060 0.3610 0.2683 0.4447 0.3210 2 0.05 0.7183 0.5583 0.8830 0.7593 0.9613 0.8640 3 0.05 0.9720 0.9063 0.9983 0.9883 1.0000 0.9993 4 0.05 0.9987 0.9940 1.0000 1.0000 1.0000 1.0000 5 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 6 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 7 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 8 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 }\BiMConC

\pgfplotstableread{ delta alpha MConFiveKn3 MConFiveKn5 MConFiveKn7 MConSevenKn3 MConSevenKn5 MConSevenKn7 MConTenKn3 MConTenKn5 MConTenKn7 0 0.05 0.0590 0.0640 0.0723 0.0653 0.0670 0.0667 0.0583 0.0663 0.0663 1 0.05 0.0760 0.0703 0.0713 0.0850 0.0763 0.0717 0.0787 0.0803 0.0763 2 0.05 0.1210 0.0997 0.0837 0.1543 0.1127 0.0967 0.1717 0.1413 0.1003 3 0.05 0.2107 0.1493 0.1163 0.2957 0.2077 0.1297 0.3733 0.2597 0.1647 4 0.05 0.3380 0.2240 0.1620 0.4897 0.3350 0.2203 0.6273 0.4577 0.2683 5 0.05 0.4963 0.3547 0.2303 0.6910 0.5223 0.3343 0.8290 0.6757 0.4380 6 0.05 0.6733 0.5067 0.3260 0.8623 0.7187 0.4907 0.9413 0.8420 0.6380 7 0.05 0.8107 0.6557 0.4447 0.9537 0.8497 0.6490 0.9893 0.9423 0.8017 8 0.05 0.9090 0.7970 0.5800 0.9860 0.9387 0.7970 0.9977 0.9847 0.9053 9 0.05 0.9623 0.8943 0.6987 0.9977 0.9817 0.8967 0.9997 0.9970 0.9610 10 0.05 0.9847 0.9533 0.8147 0.9993 0.9950 0.9560 1.0000 1.0000 0.9897 }\UniMConCLRa

\pgfplotstableread{ delta alpha MConFiveKn3 MConFiveKn5 MConFiveKn7 MConSevenKn3 MConSevenKn5 MConSevenKn7 MConTenKn3 MConTenKn5 MConTenKn7 0 0.05 0.0590 0.0640 0.0723 0.0653 0.0670 0.0667 0.0583 0.0663 0.0663 1 0.05 0.0633 0.0667 0.0673 0.0753 0.0750 0.0680 0.0690 0.0743 0.0687 2 0.05 0.0773 0.0740 0.0697 0.0980 0.0840 0.0770 0.0977 0.0863 0.0770 3 0.05 0.1003 0.0830 0.0793 0.1343 0.1037 0.0870 0.1547 0.1107 0.0863 4 0.05 0.1407 0.0983 0.0847 0.1880 0.1273 0.1047 0.2323 0.1497 0.1073 5 0.05 0.1897 0.1197 0.1003 0.2723 0.1653 0.1267 0.3373 0.1930 0.1287 6 0.05 0.2557 0.1520 0.1103 0.3677 0.2170 0.1470 0.4727 0.2523 0.1637 7 0.05 0.3407 0.1860 0.1277 0.4813 0.2647 0.1857 0.6213 0.3307 0.2010 8 0.05 0.4283 0.2290 0.1530 0.6150 0.3387 0.2207 0.7630 0.4347 0.2593 9 0.05 0.5233 0.2887 0.1803 0.7387 0.4230 0.2577 0.8610 0.5410 0.3160 10 0.05 0.6247 0.3527 0.2137 0.8347 0.5153 0.3030 0.9397 0.6440 0.3867 }\UniMConCLRb

\pgfplotstableread{ delta alpha MConFiveQKn0 MConFiveQKn1 MConSevenQKn0 MConSevenQKn1 MConTenQKn0 MConTenQKn1 0 0.05 0.0703 0.1023 0.0697 0.0907 0.0653 0.0830 1 0.05 0.2210 0.1733 0.2613 0.1680 0.3053 0.1760 2 0.05 0.5817 0.3103 0.7700 0.3613 0.8750 0.4287 3 0.05 0.9277 0.5380 0.9887 0.6757 0.9983 0.7973 4 0.05 0.9987 0.7923 1.0000 0.9190 1.0000 0.9820 5 0.05 1.0000 0.9447 1.0000 0.9957 1.0000 1.0000 6 0.05 1.0000 0.9927 1.0000 1.0000 1.0000 1.0000 7 0.05 1.0000 0.9997 1.0000 1.0000 1.0000 1.0000 8 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 0.05 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 }\BiMConQCLR

\pgfplotstableread{ delta alpha MConFiveCKn0 MConFiveCKn1 MConSevenCKn0 MConSevenCKn1 MConTenCKn0 MConTenCKn1 0 0.05 0.1113 0.1523 0.0913 0.1127 0.0793 0.0973 1 0.05 0.1647 0.1787 0.1620 0.1413 0.1517 0.1347 2 0.05 0.2720 0.2407 0.3157 0.2153 0.3620 0.2250 3 0.05 0.4603 0.3270 0.5763 0.3350 0.7047 0.3810 4 0.05 0.6907 0.4473 0.8490 0.5153 0.9513 0.6107 5 0.05 0.8930 0.5927 0.9747 0.7080 0.9980 0.8273 6 0.05 0.9790 0.7433 0.9990 0.8737 1.0000 0.9630 7 0.05 0.9983 0.8663 1.0000 0.9617 1.0000 0.9967 8 0.05 1.0000 0.9517 1.0000 0.9957 1.0000 1.0000 9 0.05 1.0000 0.9877 1.0000 1.0000 1.0000 1.0000 10 0.05 1.0000 0.9960 1.0000 1.0000 1.0000 1.0000 }\BiMConCCLR

Empirical Application

A central aspect of labor supply is the decision about working hours and its implications for earnings EhrenbergSmith2016Labor. By incorporating the worker's labor supply decision and inherent ability into the dynamic complete information framework of GibbonsWaldman1999Wage, Gicheva2013Working builds up a model that predicts a positive and convex intertemporal relationship between working hours and wage growth. In turn, Gicheva2013Working validates these shape predictions by analyzing the data set from a panel survey of registrants for the Graduate Management Admission Test (GMAT). Specifically, Gicheva2013Working employs a partially linear model to estimate the relationship between working hours and wage growth, and then verifies the theoretical predictions through eyeball inspection. Below we complement her results by carrying out formal statistical tests of these restrictions based on the same data.

The GMAT sample involves individuals who registered to take the GMAT between June 1990 and March 1991, and were living in the United States at the time of registration. The survey was conducted in four waves of interviews: shortly after the registration, 15 months after registration, 3.5-4 years after registration, and 7 years registration. The sampled workers are concentrated in the high-end labor market: they are college educated, tend to work long hours, and have high earnings. The final cleaned sample consists of $1,911$ respondents of the survey; see Section III in Gicheva2013Working for more details. Following Gicheva2013Working, we base our analysis on the model:

where $Y$ is the annual wage growth rate between the second and fourth interviews, $Z$ indicates weekly working hours, $W$ denotes a vector of demographic control variables (e.g., experience, education, gender, age, race, and family characteristics), and $E[u|Z,W]=0$. We then respectively test monotonicity, convexity, and monotonicity jointly with convexity of $\theta_0$, based on the full sample as in Gicheva2013Working as well as each gender group ---there are $808$ women and $1,103$ men in the sample.

We employ the sup-distance test for monotonicity and the joint restriction, and the sup test based on GCM for convexity. For all three tests, the number of bootstrap samples is $5,000$, while the grid points for $Z$ is $3,...,90$ (based on the realized weakly working hours). The estimation and bootstrap steps are otherwise the same as the univariate designs in Section (ref). Figure (ref) depicts the curves of $\hat\theta_n$ obtained based on the full sample, women and men respectively, with the sieve dimension $k_n=9$ (i.e., cubic B-splines with 5 interior knots). While the aforementioned shape restrictions are “overall” present in the data, there are nonetheless local violations. It is therefore crucial to conduct hypothesis test in order to formally quantify the sampling uncertainty.

\pgfplotstableread{ Grid HatTheta 3 0.2764 4 0.2803 5 0.2838 6 0.2869 7 0.2895 8 0.2918 9 0.2937 10 0.2952 11 0.2965 12 0.2974 13 0.2980 14 0.2984 15 0.2985 16 0.2984 17 0.2980 18 0.2975 19 0.2969 20 0.2960 21 0.2951 22 0.2940 23 0.2929 24 0.2917 25 0.2905 26 0.2892 27 0.2880 28 0.2867 29 0.2855 30 0.2844 31 0.2833 32 0.2823 33 0.2815 34 0.2808 35 0.2802 36 0.2798 37 0.2797 38 0.2797 39 0.2803 40 0.2832 41 0.2862 42 0.2868 43 0.2857 44 0.2840 45 0.2825 46 0.2821 47 0.2826 48 0.2837 49 0.2853 50 0.2871 51 0.2887 52 0.2903 53 0.2918 54 0.2932 55 0.2946 56 0.2959 57 0.2971 58 0.2984 59 0.2996 60 0.3008 61 0.3021 62 0.3034 63 0.3047 64 0.3060 65 0.3074 66 0.3089 67 0.3105 68 0.3122 69 0.3140 70 0.3159 71 0.3179 72 0.3201 73 0.3225 74 0.3250 75 0.3277 76 0.3306 77 0.3337 78 0.3371 79 0.3406 80 0.3445 81 0.3485 82 0.3529 83 0.3575 84 0.3624 85 0.3676 86 0.3732 87 0.3791 88 0.3853 89 0.3919 90 0.3988 }\Full

\pgfplotstableread{ Grid HatTheta 6 0.1661 7 0.1740 8 0.1810 9 0.1871 10 0.1923 11 0.1967 12 0.2004 13 0.2033 14 0.2055 15 0.2071 16 0.2081 17 0.2086 18 0.2086 19 0.2081 20 0.2072 21 0.2059 22 0.2043 23 0.2024 24 0.2003 25 0.1981 26 0.1956 27 0.1931 28 0.1905 29 0.1880 30 0.1854 31 0.1829 32 0.1806 33 0.1785 34 0.1765 35 0.1748 36 0.1736 37 0.1737 38 0.1759 39 0.1814 40 0.1910 41 0.1980 42 0.1972 43 0.1924 44 0.1871 45 0.1843 46 0.1839 47 0.1853 48 0.1879 49 0.1908 50 0.1935 51 0.1953 52 0.1963 53 0.1965 54 0.1960 55 0.1949 56 0.1933 57 0.1913 58 0.1889 59 0.1862 60 0.1834 61 0.1805 62 0.1776 63 0.1748 64 0.1722 65 0.1698 66 0.1679 67 0.1663 68 0.1653 69 0.1649 70 0.1652 71 0.1664 72 0.1684 73 0.1713 74 0.1754 75 0.1805 76 0.1869 77 0.1947 78 0.2038 79 0.2145 80 0.2267 81 0.2406 82 0.2562 83 0.2737 84 0.2932 85 0.3146 86 0.3382 87 0.3640 88 0.3921 89 0.4225 90 0.4554 }\Woman

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{ \newlength\newfigurewidth \setlength\newfigurewidth{0.35\textwidth}

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Table (ref) reports the $p$-values of our tests based on various choices of $k_n$ (the sieve dimension) and $\gamma_n$ (the tuning parameter that determines $\hat\kappa_n$). Consistent with Gicheva2013Working, the shape restrictions are significant in the full sample and in men, at all three conventional significance levels. The evidence is, however, less strong among women. Inspecting Figure (ref)-(b), we see that there are two segments of the working hours, namely $[18,36]$ and $[53,69]$, over which the wage growth rate declines. Following Gicheva2013Working, we also re-compute the $p$-values by restricting to observations with working hours $Z\in[35,65]$. This eliminates 231 observations in total, 128 women, and 103 men. The presence of the shape restrictions remains strong, but more so for women. Overall, our findings are in line with the theoretical predictions in Gicheva2013Working.

{ {3.5pt}

}

Conclusion

This paper develops a unifying framework for testing shape restrictions, and investigates the suitability of some Wald functionals in applying the framework. In particular, while the influential rearrangement operator has proven particularly useful in obtaining restricted point and interval estimates, it is inapplicable to our framework due to a lack of convexity. In contrast, the greatest convex minorization (resp.\ the least concave majorization) is shown to enjoy attractive analytic properties, and thus may be employed to test convexity (resp.\ concavity) in nonparametric settings that have not been explored previously. Finally, despite that the projection operator may not be well-defined/behaved in general Banach spaces, we show that one may nonetheless devise a powerful distance-based test by applying our framework.