Adaptive, Rate-Optimal Hypothesis Testing in Nonparametric IV Models

Adaptive, Rate-Optimal Hypothesis Testing in Nonparametric IV Models

Keywords: {Sieve two-stage least squares}, {shape restrictions}, {Hilbert projection onto closed convex sets}, {composite hypothesis}, {nonparametric alternatives}, {minimax rate of testing}, {adaptive hypothesis testing}, {power}, {random exponential scan}, {sieve U-statistics}

Introduction

In this paper, we propose computationally simple, optimal hypothesis testing in a nonparametric instrumental variables (NPIV) model. The maintained assumption is that there is a nonparametric structural function $h$ satisfying the NPIV model

where $X$ is a $d_{x}$-dimensional vector of possibly endogenous regressors, $W$ is a $d_{w}$-dimensional vector of conditional (instrumental) variables (with $d_{w}\ge d_{x}$), and the joint distribution of $(Y,X,W)$ is unspecified beyond ((ref)). With the danger of abusing terminology, we call a function $h$ satisfying model ((ref)) a NPIV function. We are interested in testing a (composite) null hypothesis that a NPIV function $h$ satisfies some simplifying economic restrictions, such as parametric or semiparametric equality restrictions or inequality restrictions (e.g., nonnegativity, monotonicity, convexity, supermodularity, quasi-concavity). Our new test builds on a simple data-driven choice of tuning parameter that ensures asymptotic size control and non-trivial power uniformly against a large class of nonparametric alternatives.

Let $L^{2}(X)$ denote the space of square integrable function of $X$. Our new test is designed to test a composite null hypothesis $\mathcal{H}_{0}$ that is a closed, convex strict subset of $L^{2}(X)$ satisfying the NPIV model ((ref)). Before presenting the theoretical properties of our new test, we derive the minimax rate of testing $r_{n}$ in $L^{2}$, which is the fastest rate of separation in root-mean squared distance between the null hypothesis $\mathcal{H}_{0}$ and the class of nonparametric alternative NPIV functions $\mathcal{H}_{1} (\delta r_{n})$ that enables consistent testing uniformly over the latter, with the rate $r_{n}$ shrinking to zero as the sample size $n$ goes to infinity and $\delta >0$ being a finite constant independent of $n$. We establish the minimax result in two steps: First, we derive, uniformly over all possible tests, a lower bound for the sum of the supremum of type I error over $\mathcal{H}_{0}$ and the supremum of type II error over $\mathcal{H}_{1} (\delta r_{n})$ separated from the null hypothesis by a rate $r_{n}$. Thus, there exists no other test that provides a better performance with respect to the sum of those errors. Second, we propose a test whose sum of the type I and the type II errors is bounded from above (by the nominal level) at the same separation rate $r_{n}$. This test is based on a modified leave-one-out sample analog of a quadratic distance between the restricted and unrestricted sieve NPIV (i.e., sieve two-stage least squares) estimators of $h$. The test is shown to attain the minimax rate of testing $r_{n}$ when the sieve dimension is chosen optimally according to the smoothness of the nonparametric alternative functions and the degree of the ill-posedness of the NPIV model (that depends on the smoothness of the conditional density of $X$ given $W$). This test is called minimax rate-optimal (with known model regularities).

In practice, the smoothness of the nonparametric alternative functions and the degree of the ill-posedness of the NPIV model are both unknown. Our new test is a data-driven version of the minimax rate-optimal test that adapts to the unknown smoothness of the nonparametric alternative NPIV functions in the presence of the unknown degree of the ill-posedness. Our test rejects the null hypothesis as soon as there is a sieve dimension (say the smallest sieve dimension) in an estimated index set such that the corresponding normalized leave-one-out quadratic distance estimator exceeds 1; and fails to reject the null otherwise. The normalization builds on Bonferroni corrected chi-squared critical values. The simple Bonferroni correction is computed using the cardinality of the estimated index set, which is in turn determined by a random exponential scan (RES) procedure that automatically takes into account the unknown degree of ill-posedness.

We show that our new test attains the minimax rate of testing in $L^{2}$ for severely ill-posed NPIV models, and is within a $\sqrt{\log \log (n)}$ multiplicative factor of the minimax rate of testing for mildly ill-posed NPIV models. This extra $\sqrt{\log \log (n)}$ term is the necessary price to pay for adaptivity to unknown smoothness of nonparametric alternative functions.\footnote{This is needed even for adaptive minimax hypothesis testing in nonparametric regressions (without endogeneity); see spokoiny1996, horowitz2001, and guerre2005.} A key technical part to establish our adaptive minimax rate of testing in $L^{2}$ is to derive a sharp upper bound on the convergence rate of a leave-one-out sieve estimator of a quadratic functional of a NPIV function, which is proved using an exponential inequality for U-statistics with increasing dimensions. We show that our adaptive test has asymptotic size control under a composite null by deriving a tight, slowly divergent lower bound for Bonferroni corrected chi-squared critical value. By inverting our adaptive tests, we obtain $L^{2}$ confidence sets on restricted NPIV functions. These confidence sets are free of additional choices of tuning parameters. The adaptive minimax rate of testing determines the $L^{2}$ radius of the confidence sets.

Our adaptive minimax $L^{2}$ rate of testing decreases to zero strictly faster than the optimal $L^{2}$ rate of estimation (with known smoothness) for mildly ill-posed NPIV models, and coincides with the optimal $L^{2}$ rate of estimation for severely ill-posed NPIV models. In the existing literature on testing for parametric, semiparametric, or shape NPIV restrictions against nonparametric alternatives, all of the non-adaptive tests achieve their asymptotic size controls by choosing some deterministic tuning parameters such that the $L^{2}$ estimation bias for $h$ is of a smaller order than the $L^{2}$ standard deviation (aka, under-smoothing), which leads to a $L^{2}$ separation rate of testing shrinking to zero strictly slower than the optimal $L^{2}$ rate of estimation, and hence strictly slower than our adaptive minimax $L^{2}$ rate of testing for both mildly and severely ill-posed NPIV models. In particular, among all of the existing NPIV tests that have asymptotic size controls, our new adaptive test is asymptotically more powerful, uniformly over a larger class of nonparametric alternatives.

In Monte Carlo simulations, we analyze the finite-sample properties of our adaptive test for the null of monotonicity or a parametric hypothesis using various simulation designs from others' work. The simulations reveal the following patterns of our adaptive test in comparison to recent non-adaptive tests: First, while the competing tests can be over-sized at the boundary of the null hypothesis, our test delivers adequate size control under different composite null hypotheses, across different sample sizes, and for varying strengths of instruments. Second, our test is as powerful as the competing tests when alternative functions are relatively simple, and is more powerful when alternatives are more nonlinear/complex. The great power gains of our adaptive test are present even for relatively weak strength of instruments or small sample sizes. These findings highlight the importance of our data-driven choice of the sieve dimension to simultaneously ensure size control and powerful performance uniformly against a larger class of nonparametric alternative NPIV functions. Finally, unlike many NPIV tests using bootstrapped critical values, our powerful adaptive test uses simple Bonferroni corrected chi-squared critical values and hence is fast to compute.

We present two empirical applications of our adaptive test. The first is to test the connected substitutes shape restrictions in demand for differential products using market level data (e.g., berry2014). The second is to test for monotonicity, convexity, and parametric forms in Engel curves (e.g., BCK07econometrica).

There is a growing number of papers on testing equality and inequality (shape) restrictions in NPIV type models. See, for example, Horowitz2006, Santos12, Breunig2015, chen2013, chernozhukov2015, zhu2020, fang2019 and references therein.\footnote{There are also papers on NPIV estimation by directly imposing shape restrictions; see, for example, horowitzlee2012, blundell2017, CW2017, and Freyberger2019. See chetverikov2018 for a review on shape restrictions and chetverikov2019testing for adaptive kernel testing for monotonicity of a regression without endogeneity.} Most of these papers assume that some non-random sequences of key tuning (regularization) parameters satisfy some theoretical rate conditions with known smoothness of NPIV functions. None of the published work achieves the adaptive minimax $L^{2}$ rate of testing for NPIV models. Our paper makes an important contribution by providing the first data-driven choice of a key tuning parameter that leads to a new minimax rate-adaptive and powerful test for equality and inequality (shape) restrictions in NPIV models. Our paper also complements a concurrent work by chen2021, which constructs honest and near-adaptive uniform confidence bands for a NPIV function and its partial derivatives using a bootstrapped Lepski's procedure (in sup-norm).

The rest of the paper is as follows. Section (ref) describes our new hypothesis test. Section (ref) establishes the oracle minimax optimal rate of testing. Section (ref) shows that this minimax optimal rate is attained (within a $\sqrt{\log \log (n)}$ term) by our new test. Section (ref) presents simulation studies and Section (ref) provides empirical illustrations. Appendices (ref) and (ref) present proofs of Theorems (ref), (ref), (ref), and (ref). The Supplemental Material includes Appendix C for additional simulation results, Appendix D for proofs of Corollaries (ref) and (ref), and Appendix E for additional lemmas and their proofs.

\paragraph{Basic Notation} For a random variable $X$, we let $L^{2}(X)$ denote the Hilbert space of real-valued measurable functions $\phi $ of $X$ with finite second moment, with the norm $\|\phi \|_{L^{2}(X)}:=\sqrt{\operatorname{\mathrm{E}} [\phi ^{2}(X)]}$ and the inner product $\langle \cdot ,\cdot \rangle _{X}$. Let $\|\phi \|_{\infty}:=\sup_{x} |\phi (x)|$ be the sup-norm and $L^{\infty}=\{\phi :\|\phi \|_{\infty}<\infty \}$. For a matrix $M$, let $M'$ be its transpose and $M^{-}$ be its generalized inverse. For a $J\times J$ matrix $M=(M_{jl})_{1\leq j,l\leq J}$, we define its Frobenius norm as $\|M\|_{F}=\sqrt{\sum_{j,l=1}^{J}M_{jl}^{2}}$. Let $\|\cdot \|$ be the Euclidean norm when applied to a vector and the operator norm induced by the Euclidean norm when applied to a matrix. For sequences of positive real numbers $\{a_{n}\}$ and $\{b_{n}\}$, we use the notation $a_{n} \lesssim b_{n}$ if $\limsup_{n\to \infty}a_{n}/b_{n}<\infty $, and $a_{n}\sim b_{n}$ if $a_{n}\lesssim b_{n}$ and $b_{n}\lesssim a_{n}$.

Preview of the Adaptive Hypothesis Testing

We first introduce the null and the alternative hypotheses as well as the concept of minimax rate of testing in Section (ref). We then describe our new rate-adaptive test for NPIV type models in Section (ref).

Null Hypotheses and Nonparametric Alternatives

Let $\mathcal H$ denote a closed subset of $L^{2}(X)$ that captures some unknown degree of smoothness. Let $\{(Y_{i},X_{i},W_{i})\}_{i=1}^{n}$ denote a random sample from the distribution $\mathrm{P}_{h}$ of $(Y,X,W)$ satisfying the NPIV model ((ref)):

Here, $\operatorname{\mathrm{E}} _{h}$ denotes the (conditional) expectation under $\mathrm{P}_{h}$. In this paper, we assume that the joint distribution of $(X,W)$ does not depend on $h\in \mathcal H$ and that the conditional density of $X$ given $W$ is continuous on its support. The conditional expectation operator $T: L^{2}(X) \mapsto L^{2}(W)$ given by $T h(w) := \operatorname{\mathrm{E}} [h(X)|W= w]$ is uniquely defined by the conditional density of $X$ given $W$ and hence does not depend on $h$. We can then equivalently express the NPIV model ((ref)) as $\operatorname{\mathrm{E}} _{h}[Y|W]=(T h)(W)$ for $h\in \mathcal H$. For ease of presentation, we mainly consider a nonparametric class of functions as the maintained hypothesis $\mathcal H$. Nevertheless, our theoretical results allow for semiparametric structures $\mathcal H$ as well (see Section (ref)).

Let $\mathcal H_{0}$ denote the null class of functions in $\mathcal H$ that satisfies a conjectured restriction in ((ref)). We assume that $\mathcal H_{0}$ is a nonempty, closed and convex, strict subset of $\mathcal H$. For any $h\in \mathcal H$, there is a unique element $\Pi _{\mathcal H_0}h \in \mathcal H_{0}$ such that $\inf_{\phi \in \mathcal H_{0}}\|h-\phi \|_{L^{2}(X)}=\|h-\Pi _{ \mathcal H_0}h\|_{L^{2}(X)}$ (by the Hilbert projection theorem). In addition to a simple null $\mathcal H_{0}=\{h_{0}\}$ (with a known function $h_{0}\in \mathcal H$), we allow for general parametric, semi/nonparametric equality and inequality composite null restrictions. We present two examples of composite null restrictions below (see Section (ref) for additional examples).

To analyze the power of any test of the null class $\mathcal H_{0}$ against nonparametric alternatives, we require some separation in $\|\cdot \|_{L^{2}(X)}$- distance between the null and the class of nonparametric alternatives for all $h\in \mathcal H$. Below, we use the notation $\|h-\mathcal H_{0}\|_{L^{2}(X)} := \inf_{\phi \in \mathcal H_{0}}\|h- \phi \|_{L^{2}(X)}=\|h-\Pi _{\mathcal H_0}h\|_{L^{2}(X)}$. We consider the class of nonparametric alternatives

for some constant $\delta >0$ and a separation rate of testing $r_{n}>0$ that decreases to zero as the sample size $n$ goes to infinity. We say that a test statistic $\mathtt {T}_{n}$ with values in $\{0, 1\}$ is consistent uniformly over $\mathcal H_{1}(\delta r_{n})$ if $\sup_{h\in \mathcal H_{1}(\delta r_{n})} \mathrm{P}_{h}(\mathtt {T}_{n}=0)=o(1)$.

We refer to Part (i) as the lower bound and Part (ii) as the upper bound, and the test statistic $\mathtt {T}_{n}:=\mathtt {T}_{n}(\alpha )$ in Part (ii) attaining the matching lower and upper bound as an optimal test. We use $r_{n}^{*}$ to denote the minimax (separation) rate of testing as the matching lower and upper bound.

In Section (ref), we first establish a minimax rate of testing $r_{n}^{*}$ assuming the knowledge of the smoothness of alternative NPIV functions $h\in \mathcal H$ and the inversion property of the conditional expectation operator $T: L^{2}(X) \mapsto L^{2}(W)$. Both are unknown in practice. The minimax rate $r_{n}^{*}$ is attained by a sieve test statistic using an optimal choice of sieve dimension (a tuning parameter) that depends on these unknown objects, and hence is infeasible. In Section (ref), we provide a data-driven modification of the optimal sieve test, that is, a feasible testing procedure that adapts to the unknown smoothness of the unrestricted NPIV function $h\in \mathcal H$ in the presence of unknown smoothing properties of the inverse of the operator $T$. Precisely, we propose a feasible test statistic $\widehat{\mathtt {T}}_{n}$ with data-driven tuning parameters in Section (ref). We show that $\widehat{\mathtt {T}}_{n}$ attains the minimax rate of testing $r_{n}^{*}$ within a $\sqrt{\log \log (n)}$ multiplicative factor, has asymptotic size control over the composite null, and is consistent uniformly over the class of nonparametric alternatives in Theorem (ref). We call our test $\widehat{\mathtt {T}}_{n}$ adaptive and rate-optimal (or sometimes simply adaptive).

Our Adaptive Test

Our test is based on a consistent estimate of the quadratic distance, $\|h-\Pi _{\mathcal H_0}h\|^{2}_{L^{2}(X)}=\|h-\mathcal H_{0}\|^{2}_{L^{2}(X)}$, between the NPIV function $h\in \mathcal H$ and its projection $\Pi _{\mathcal H_0}h$ onto $\mathcal H_{0}$ under the $\|\cdot \|_{L^{2}(X)}$. We first introduce some notation. Let $\{\psi _{j}\}_{j=1}^{\infty}$ and $\{b_{k}\}_{k=1}^{\infty}$ be complete basis functions for the Hilbert spaces $L^{2}(X)$ and $L^{2}(W)$, respectively. Let $\psi ^{J}(\cdot )$ and $b^{K}(\cdot )$ be vectors of basis functions of dimensions $J$ and $K=K(J)\geq J$, respectively. These can be cosine, power series, spline, or wavelet basis functions. Let $G=\operatorname{\mathrm{E}} [\psi ^{J}(X)\psi ^{J}(X)']$, $G_{b}=\operatorname{\mathrm{E}} [b^{K(J)}(W)b^{K(J)}(W)']$, and $S=\operatorname{\mathrm{E}} [b^{K(J)}(W)\psi ^{J}(X)']$. We assume that $G$, $G_{b}$, and $S'G_{b}^{-1}S$ have full ranks. Then the $J\times K(J)$ matrix $ A= G^{1/2} [S'G_{b}^{-1}S]^{-1}S'G_{b}^{-1}$ is well defined. Let $\Psi _{J}$ denote the closed linear subspace of $L^{2}(X)$ spanned by $\{\psi _{1},\dots ,\psi _{J}\}$. We define a population 2SLS projection of $h\in L^{2}(X)$ onto the sieve space $\Psi _{J}$ as

For any NPIV function $h\in \mathcal H$ in ((ref)), we have $Q_{J} h(\cdot )=\psi ^{J}(\cdot )'G^{-1/2}A\operatorname{\mathrm{E}} _{h}[b^{K}(W)Y]$, and

which approximates $\|h-\Pi _{\mathcal H_{0}}h\|^{2}_{L^{2}(X)}$ well as $J$ grows large (see Lemma (ref)).

For each sieve dimension $J$, we construct a test based on an estimated quadratic distance $\|Q_{J} (h-\Pi _{\mathcal H_{0}}h)\|^{2}_{L^{2}(X)}$ between the unrestricted and restricted NPIV estimators of a function $h$ satisfying ((ref)). Let $\Psi = (\psi ^{J}(X_{1}),\dots ,\psi ^{J}(X_{n}))'$, $B = (b^{K}(W_{1}),\dots ,b^{K}(W_{n}))'$, $P_{B}= B(B'B)^{-} B'$, and $\widehat A= \sqrt n(\Psi ' \Psi )^{1/2}[\Psi 'P_{B}\Psi ]^{-} \Psi ' B(B'B)^{-}$. Let $\mathrm Y=(Y_{1},\ldots , Y_{n})'$. Our unrestricted sieve NPIV estimator solves a sample 2SLS problem (BCK07econometrica):

Let $\mathcal H_{0,J}$ denote a nonempty, closed and convex, finite-dimensional subset of $\mathcal H_{0}$. A restricted NPIV estimator for $\Pi _{\mathcal H_{0}}h \in \mathcal H_{0}$ is given by

The choice of $\mathcal H_{0,J}$ is allowed to depend on the structure of the null class of NPIV functions $\mathcal H_{0}$. For a general nonparametric or a semi-nonparametric composite null hypothesis, $\mathcal H_{0,J}$ depends on sieve dimension $J$ and grows dense in $\mathcal H_{0}$ as the sample size increases. For instance, we let $\mathcal H_{0,J}=\Psi _{J}\cap \mathcal H_{0}$ under a nonparametric composite null whenever $\Psi _{J}\cap \mathcal H_{0}\neq \emptyset $ (which holds for the nonparametric inequality restrictions in Example (ref)). We can also let $\mathcal H_{0,J}=\mathcal H_{0}$ under a simple null ($\mathcal H_{0}=\{h_{0}\}$ for a known function $h_{0}$), or under a parametric composite null ($\mathcal H_{0}=\{F( \cdot ;\theta ), \theta \in \Theta \}$ for some known mapping $F$).

For each sieve dimension $J$, we compute a $J$- dependent test statistic $n \widehat{D}_{J} /\widehat{V}_{J}$, which is a standardized, centered (or leave-one-out) version of the sample analog of ((ref)):

where $\widehat{ V}_{J}$ estimates the population normalization factor

which is the variance of $\frac{2}{n(n-1)}\sum_{1\leq i< i'\leq n} (Y_{i}- h(X_{i}) )b^{K(J)}(W_{i})' A' Ab^{K(J)}(W_{i'}) (Y_{i'}- h(X_{i'}) )$.

We compute our adaptive test for the null hypothesis $\mathcal H_{0}$ against nonparametric alternatives in three simple steps. \paragraph{Step 1.} Compute a random exponential scan (RES) index set:

where $\underline J:=\lfloor \sqrt{\log \log n}\rfloor $, $j_{\max}:=\lceil \log _{2}(n^{1/3}/\underline J)\rceil $, and the empirical upper bound

where $\widehat s_{J}$ is the minimal singular value of $(B'B)^{-1/2}B'\Psi (\Psi ' \Psi )^{-1/2}$, and $\zeta (J)=\sqrt{J}$ for spline, wavelet, or trigonometric sieve basis, and $\zeta (J)=J$ for power series.

\paragraph{Step 2.} Let $\#(\widehat{\mathcal I}_{n})$ be the cardinality of the RES index set. For a nominal level $\alpha \in (0,1)$, we compute a Bonferroni corrected chi-squared critical value as

where $q (a, J)$ is the $100(1-a)\%$-quantile of the standard chi-squared distribution with $J$ degrees of freedom.

\paragraph{Step 3.} Let $\widehat {\mathcal{W}}_{J}(\alpha ) := n \widehat{D}_{J}/(\widehat \eta _{J}(\alpha ) \widehat{V}_{J})$ for all $J\in \widehat{\mathcal I}_{n}$. Compute the test

where $\mathbbm{1}\{\cdot \}$ is the indicator function. Under the nominal level $\alpha \in (0,1)$, $\widehat {\mathtt{T}}_{n}=1$ indicates rejection of the null $\mathcal H_{0}$ and $\widehat {\mathtt{T}}_{n}=0$ indicates a failure to reject the null.

The Minimax Rate of Testing

This section derives the minimax separation rate of hypothesis testing in NPIV models, when $\mathcal H$ is a relative compact subset of $L^{2}(X)$. For simplicity, we assume in this paper that $\mathcal H$ is a standard Sobolev ellipsoid of smoothness $p>0$, which can be expressed as

where $\{\widetilde \psi _{j}\}_{j=1}^{\infty}$ is the orthonormal basis for $L^{2}(X)$ that is constructed from the basis $\{\psi _{j}\}_{j=1}^{\infty}$ (using the Gram--Schmidt procedure). Assuming the smoothness $p$ is known, we first establish a lower bound for the $L^{2}$ rate of testing in Section (ref), and then show that the lower bound can be achieved by a sieve test if the sieve dimension $J$ can be chosen optimally in Section (ref).

The Lower Bound

Before we state the lower bound for the rate of testing, we introduce the main assumptions.

Assumptions (ref)(i), (ii), (iii) are basic regularity conditions imposed in the paper. Assumption (ref)(iv) specifies the smoothing property of the conditional expectation operator $T$ relative to the basis $\{\widetilde \psi _{j}\}$. The smoother $T$ is (i.e., the smoother the conditional density of $X$ given $W$ is), the faster the sequence ${\nu _{j}}$ in Assumption (ref)(iv) decreases to zero, and the harder it is to detect properties of the NPIV function in the $L^{2}(X)$ metric.

In this paper, we call a decreasing sequence $\{\nu _{j}\}$ regularly varying if $\nu _{J}^{-4}J \lesssim \sum_{j=1}^{J} \nu _{j}^{-4}$. The regularly varying sequence $\{\nu _{j}\}$ allows for very broad decreasing patterns, and includes two leading special cases: (1) mildly ill-posed case where $\nu _{j}=j^{-a/d_{x}}$ for some $a>0$; and (2) severely ill-posed case where $\nu _{j}=\exp (-j^{a/d_{x}}/2)$ for some $a>0$.

According to Theorem (ref), the lower bound of the $L^{2}$ rate of testing is $n^{-2p/(4(p+a)+d_{x})}$ in the mildly ill-posed case, which goes to zero faster than the lower bound $n^{-p/(2(p+a)+d_{x})}$ of the $L^{2}$ rate of estimation (HallHorowitz2005 and ChenReiss2011). For the severely ill-posed NPIV models, the lower bound of the $L^{2}$ rate of testing is $(\log n)^{-p/a}$, which coincides with the lower bound of estimation in both the $L^{2}$ norm (ChenReiss2011) and the sup-norm (ChenChristensen2017).

In the literature on linear ill-posed inverse problem with a compact operator $T$, an “exact link condition” is commonly used to describe the smoothing (or compact embedding) property of $T$, which can be stated as follows:

for some finite constants $C\geq c>0$ and a positive decreasing function $\nu $ with $\nu _{j}:=\nu (j)$. The RHS inequality of (ref) (i.e., Assumption (ref)(iv)) is used for the lower bound calculation, and the LHS inequality of (ref) is imposed for the upper bound calculation. However, to have matching lower and upper bound, that is, to establish the rate is minimax optimal, the exact link condition (ref) or something similar is typically imposed even with a known $T$; see, for example, ChenReiss2011. We note that any compact operator $T$ has a unique singular value decomposition. If the basis $\{\widetilde \psi _{j}\}$ is an eigenfunction basis associated with the operator $T$, then (ref) is automatically satisfied with $C=c=1$ and $\{\nu _{j}\}_{j=1}^{\infty}$ being its singular values in decreasing order. More generally, (ref) is also satisfied when $\{\widetilde \psi _{j}\}$ is a Riesz basis (see BCK07econometrica). Since the conditional expectation operator $T$ is compact under very mild conditions (such as when the conditional density of $X$ given $W$ is continuous), it typically satisfies (ref), which is an alternative way to express the smoothing property of the operator $T$.

In our proof of Theorem (ref), we reduce the lower bound calculation for the NPIV model to that for a model with a known operator $T$. Consequently, Assumption (ref)(iv) is sufficient to establish the lower bound. However, for the upper bound calculation of the NPIV model, we need to estimate the unknown operator $T$. Therefore, in addition to the LHS inequality of (ref), some extra sufficient conditions will be used to address the error of estimating $T$ nonparametrically. See the next subsection for details.

An Upper Bound Under a Simple Null Hypothesis

For a simple null $\mathcal H_{0}=\{h_{0}\}$, we redefine $\widehat{D}_{J}$ in (ref) with $\widehat h_{J}^{\text{\textsc r}} = h_{0}$ as

We also redefine our test statistic $\widehat{\mathtt {T}}_{n}$ with a singleton RES index set $\{J\}$ as

The test $ \mathtt{T}_{n,J}$ with optimally chosen $J$ serves as a benchmark of our adaptive testing procedure (given in (ref)) for the simple null hypothesis.

We define the projections $\Pi _{J} h(\cdot )=\psi ^{J}(\cdot )'G^{-1} \langle \psi ^{J},h \rangle _{L^{2}(X)}$ for $h\in L^{2}(X)$ and $\Pi _{K} m(\cdot )=b^{K}(\cdot )'G_{b}^{-1} \operatorname{\mathrm{E}} [b^{K}(W)m(W)]$ for $m\in L^{2}(W)$. Further, let $s_{J}=\inf_{h\in \Psi _{J}}\|\Pi _{K} T h \|_{L^{2}(W)} /\|h\|_{L^{2}(X)}$, that is, $s_{J}$ coincides with the minimal singular value of $G_{b}^{-1/2}SG^{-1/2}$. Let $\zeta _{J}=\max (\zeta _{\psi ,J}, \zeta _{b,K})$, $\zeta _{\psi ,J}=\sup_{x}\|G^{-1/2}\psi ^{J}(x)\|$, and $\zeta _{b,K}=\sup_{w}\|G_{b}^{-1/2}b^{K}(w)\|$. We assume throughout the paper that $\zeta _{J}=O(\sqrt{J})$ (which holds for polynomial spline, wavelet, and cosine bases), or $\zeta _{J}=O(J)$ (which holds for orthogonal polynomial bases).

Let $\Psi _{J,1}:=\{h\in \Psi _{J}: \|h\|_{L^{2}(X)}=1\}$. Then $\tau _{J}:= [\inf_{h\in \Psi _{J,1}}\|T h \|_{L^{2}(W)} ]^{-1}$ is the sieve measure of ill-posedness that has been used in sieve estimation of NPIV models (see, e.g., BCK07econometrica). We have $s_{J} \leq \tau _{J}^{-1}$ by definition.

Assumption (ref)(i) is an extra condition on the data-generating process (DGP) since it imposes upper bounds on conditional second moment and finiteness of unconditional fourth moment. We note that the DGP displayed in our proof of Theorem (ref) already satisfies this assumption; it has no effect on our lower bound result. Assumptions (ref)(ii), (iii), (iv) are imposed since our test statistic involves linear sieve estimated operator $T$ to achieve the separation rate. Assumptions (ref)(ii), (iii) impose restrictions on the sieve dimension $J$, which are satisfied by $J_{*}$ given in (ref) of Theorem (ref). Assumption (ref)(iv) imposes an upper bound on the smoothing properties of the conditional expectation operator $T$. It is akin to the $L^{2}$ stability condition used in sieve NPIV estimation and is satisfied by Riesz bases (see BCK07econometrica). Assumption (ref)(i) is a mild condition on the approximation properties of the basis used for the instrument space (see ChenChristensen2017). It implies that $s_{J}$ and $\tau _{J}^{-1}$ are asymptotically equivalent:

while Assumption (ref)(ii) implies $\tau _{J}^{-1}=\inf_{h\in \Psi _{J,1}}\|T h \|_{L^{2}(W)} \geq \sqrt c \nu _{J}$ for all $J$. Assumption (ref) thus implies

Further, $s_{J} \sim \tau _{J}^{-1} \leq \|T \widetilde\psi _{J} \|_{L^{2}(W)} \leq \sqrt C \nu _{J}$ under Assumption (ref)(iv) and $\{\widetilde\psi _{j}\}$ being an orthonormal basis in $L^{2}(X)$, and Assumption (ref)(iv) is satisfied under Assumptions (ref)(iv) and (ref). Therefore, Assumptions (ref) and (ref) have no effect on the lower bound calculation in Theorem (ref).

The next theorem provides an upper bound on the separation rate of testing in $L^{2}$ under a simple null using the test statistic $\mathtt {T}_{n,J}$.

Theorem (ref) shows that, under Assumptions (ref)(i)--(iii) and (ref), the test statistic $\mathtt{T}_{n,J}$ given in (ref) attains the $L^{2}$ separation rate of testing $r_{n,J}$ in (ref). Given a sieve dimension $J$, this rate consists of a standard deviation term ($n^{-1/2}s_{J}^{-1}J^{1/4}$) and a bias term ($J^{-p/d_{x}}$). A central step to achieve this rate result is to establish a rate of convergence of the quadratic distance estimator $\widehat D_{J}(h_{0})$ (see Theorem (ref)), which we show is sufficient for the consistency of $\mathtt {T}_{n,J}$ uniformly over $\mathcal H_{1}(\delta ^{\circ }r_{n,J})$. In addition, under Assumption (ref), Theorem (ref) implies that the sieve test $\mathtt{T}_{n,J_{*0}}$ achieves the $L^{2}$ minimax rate of testing for a simple null, with known smoothness $p$ of the nonparametric alternatives and known degree of ill-posedness.

Given a sieve dimension $J$, the $L^{2}$ rate of sieve estimation for any NPIV function $h\in \mathcal H$ is $\max \{n^{-1/2}s_{J}^{-1}J^{1/2}, J^{-p/d_{x}}\}$ (see, e.g., ChenReiss2011). Comparing the $L^{2}$ rate of estimation and of testing via the sieve NPIV procedures, while both have the same bias term $J^{-p/d_{x}}$, the $L^{2}$ rate of testing has a smaller “standard deviation” term $n^{-1/2}s_{J}^{-1}J^{1/4}$. Intuitively, we may obtain a higher precision in testing as the $L^{2}$ rate of testing is determined by estimating a quadratic norm of the unrestricted NPIV function $h \in \mathcal H$. Interestingly, although this leads to a faster optimal $L^{2}$ rate of sieve testing $r_{n,J_{*0}}\sim n^{-2p/(4(p+a)+d_{x})}$ than the optimal $L^{2}$ rate of estimation $n^{-p/(2(p+a)+d_{x})}$ in the mildly ill-posed case, the optimal $L^{2}$ rate of sieve testing $r_{n,J_{*0}}\sim (\log n)^{-p/a}$ in the severely ill-posed case is the same as the optimal rate of sieve estimation in both the $L^{2}$ norm (ChenReiss2011) and the sup-norm (ChenChristensen2017). This is because, in the severely ill-posed case, the bias term dominates the standard deviation term for the optimally chosen sieve dimension in both sieve testing and estimation.

Adaptive Inference

This section establishes theoretical properties of our test $\widehat {\mathtt{T}}_{n}$ defined in (ref). We show that it adapts to the unknown smoothness $p>0$ of the functions in $\mathcal H$. Section (ref) establishes the rate optimality of our test for simple null hypotheses. Section (ref) extends this result to testing for composite null problems. Section (ref) proposes $L^{2}$ confidence sets by inverting the adaptive test under imposed restrictions on the NPIV function.

Adaptive Testing Under a Simple Null Hypothesis

Under the simple null hypothesis $\mathcal H_{0}=\{h_{0}\}$ with a known function $h_{0}$ satisfying (ref), our test $\widehat{\mathtt{T}}_{n}$ given in (ref) simplifies to

where $\widehat{D}_{J}(h_{0})$ is defined in ((ref)), and $\widehat{\mathcal I}_{n}$, $\widehat{V}_{J}$, $\widehat \eta _{J}(\alpha )$ are given in Section (ref).

Recall that the RES index set $\widehat{\mathcal I}_{n}$, given in (ref), depends on an upper bound $\widehat J_{\max}$ given in (ref). To establish our asymptotic results below, we introduce a non-random index set $\mathcal I_{n}$ with a deterministic upper bound $\overline J$ as follows:

with $\overline J=\sup \{J: \zeta _{J}^{2} \sqrt{(\log J)/n}\leq \overline c s_{J}\}$ for some sufficiently large constant $\overline c>0$. We show in Lemma (ref)(i) that $\widehat J_{\max}\leq \overline J$ (and thus $\widehat{\mathcal I}_{n}\subset \mathcal I_{n}$) holds with probability approaching 1 uniformly over all functions $h\in \mathcal H$. Thus, $\overline J$ serves as a deterministic upper bound for the RES index set $\widehat{\mathcal I}_{n}$.

Assumptions (ref)(i), (iii) strengthen Assumptions (ref)(ii), (iii), (iv) to hold uniformly over the deterministic index set $\mathcal I_{n}$. They are used to establish Lemma (ref). Assumption (ref)(i) restricts the growth of the deterministic upper bound $\overline J$ of the RES index set $\widehat{\mathcal I}_{n}$. Assumption (ref)(ii) is satisfied if $\{s_{j}\}$ is regularly varying, which is implied by Assumptions (ref)(iv) and (ref) with $\{\nu _{j}\}$ regularly varying. We note that Assumptions (ref)(ii), (ref)(i), and (ref)(i) together imply that $ V_{J} \sim s_{J}^{-2}\sqrt{J}$ uniformly for $h\in \mathcal H$ and $J\in \mathcal I_{n}$ (see Lemmas (ref) and (ref)).

Theorem (ref) establishes an upper bound for the testing rate of the adaptive test $\widehat {\mathtt{T}}_{n}$ under a simple null hypothesis. The proof of Theorem (ref) relies on a novel exponential bound for degenerate U-statistics based on sieve estimators (see Lemma (ref)). In particular, we control the type I error using tight lower bounds for adjusted chi-squared critical values (see Lemma (ref)) and establish the consistency of $\widehat{\mathtt{T}}_{n}$ uniformly over $\mathcal H_{1}(\delta ^{\circ }\text{\textsf r}_{n})$.

From Theorem (ref), we see that the adaptive test attains the oracle minimax rate of testing within a $\sqrt{\log \log (n)}$ term in the mildly ill-posed case. For the adaptive testing in regression models without endogeneity (i.e., when $X=W$), it is well known that the extra $\sqrt{\log \log (n)}$ term is required (see spokoiny1996). In the severely ill-posed case, our adaptive test attains the exact minimax rate of testing and hence, there is no price to pay for adaptation. This is because, in the severely ill-posed case, the bias term dominates the standard deviation term when the sieve dimension coincides with $J^{\circ}$, irrespective of the $\sqrt{\log \log (n)}$ term.

Adaptive Testing Under Composite Null Hypotheses

We extend the results from Section (ref) to adaptive testing for a general composite null hypothesis $\mathcal H_{0}$, which is a nonempty, closed and convex strict subset of $\mathcal H$. Without loss of generality, we assume $0 \in \mathcal H_{0}$. This is satisfied for the inequality restrictions in Example (ref) and the semiparametric equality restrictions considered in Example (ref) if, for instance, $F(\cdot ;\theta ,g)= 0$ for some $\theta \in \Theta $ and $g\in \mathcal G$.

Below, we impose some conditions on the complexity of the closed and convex null class of functions $\mathcal H_{0}$. Let $\mathcal S^{K}=\{\mathsf e\in \mathbb R^{K}: \mathsf e_{1}^{2}+ \cdots +\mathsf e_{K}^{2}=1\}$ denote the $(K-1)$-dimensional unit sphere. Let $K^\circ =K(J^\circ )$, $\widetilde b^{K}(\cdot )=G_{b}^{-1/2}b^{K}(\cdot )$, and $Z:=(X',W')'$. For any $h\in \mathcal H_{1}(\delta ^{\circ}\text{\textsf r}_{n})$, we consider the following class of functions:

with its envelope function denoted by $F_{h,\mathsf e}$. Let $N_{[]}(\epsilon , \mathcal F, L^{2}(Z))$ be the $L^{2}(Z)$ covering number with bracketing for $\mathcal F$, which is the minimal number of $\epsilon $-brackets, in $L^{2}(Z)$ sense, needed to cover $\mathcal F$. We let $\mathcal C_{h} :=\max_{\mathsf e\in \mathcal S^{K^\circ }}\int _{0}^{1} (1+\log N_{[]} (\epsilon \|F_{h,\mathsf e}\|_{L^{2}(Z)}, \mathcal F_{h,\mathsf e}, L^{2}(Z) ) )^{1/2}\,d\epsilon $.

Assumption (ref) restricts the complexity of the composite null hypothesis $\mathcal H_{0}$. Assumption (ref)(i) implies that $\widehat {\mathtt{T}}_{n}$ has size control uniformly over the composite null $\mathcal H_{0}$. Assumption (ref)(ii) ensures the consistency of $\widehat {\mathtt{T}}_{n}$ uniformly over $\mathcal H_{1}(\delta ^{\circ}\text{\textsf r}_{n})$. Note that Assumption (ref) imposes estimation rate conditions on $\widehat h_{J}^{\text{\textsc r}}$ under the composite null and the nonparametric alternatives, which can be viewed as NPIV extensions of the parametric estimation rate conditions imposed in horowitz2001 for testing for a parametric regression against nonparametric regressions.

The next result establishes an upper bound for the rate of testing under a composite null hypothesis using the test statistic $\widehat {\mathtt{T}}_{n}$ given in (ref).

Theorem (ref) states that $\widehat {\mathtt{T}}_{n}$ attains the same adaptive rate of testing $\text{\textsf r}_{n}$ for a composite null as that for a simple null. Moreover, (ref) shows that $\widehat{\mathtt {T}}_{n}$ simultaneously has asymptotic size control over the composite null, and is consistent uniformly over the largest class of nonparametric alternatives $\mathcal H_{1}(\delta ^{\circ }\text{\textsf r}_{n})$. The asymptotic size control is established by controlling the sieve approximation error uniformly over the index set $\widehat{\mathcal I}_{n}$ under the null, due to a projection property built in the construction of our test $\widehat {\mathtt{T}}_{n}$. See Lemma (ref), in which we utilize the convergence of von Neumann's alternating projection algorithm. Theorem (ref) is applicable to any composite null hypothesis $\mathcal H_{0}$ that is a closed convex strict subset of $\mathcal H$, including closed convex cone null restrictions as special cases.

Theorem (ref) shows that our adaptive test has asymptotic size control and non-trivial power against a large class of nonparametric NPIV alternatives without using under-smoothed choice of sieve dimensions in testing. This is different from the existing non-adaptive tests for semiparametric or shape NPIV restrictions, which achieve asymptotic size controls via under-smoothed choice of tuning parameters in $L^{2}$ estimation. For instance, in their bootstrap test for convex cone restrictions of a NPIV function, fang2019 estimated the unrestricted NPIV function by a sieve 2SLS estimator assuming known smoothness, and chose the sieve dimension $J$ deterministically such that the estimation bias $J^{-p/d_{x}}$ is of a smaller order than the standard deviation $n^{-1/2}s_{J}^{-1}J^{1/2}$ in $L^{2}$ estimation, which leads to a non-adaptive rate of testing $n^{-1/2}s_{J}^{-1}J^{1/2}$ that is suboptimal for $L^{2}$ testing of NPIV models.

Adaptive Testing in Semiparametric Models

Partially parametric models are often used in empirical work and can be easily incorporated in our framework either as restricted models or as the maintained models. Let $\Theta \oplus \mathcal G =\{h(x_{1},x_{2})= x_{1}'\theta +g(x_{2}): \theta \in \Theta , g\in \mathcal G\}$, where $\Theta $ denotes a finite-dimensional parameter space, and $\mathcal G$ denotes a class of nonparametric functions.

Let the NPIV model (ref) be the maintained hypothesis. We can test inequality restrictions as in Example (ref) and a semiparametric structure simultaneously. For example, we can test for a partial linear structure with a nondecreasing function $g$ by setting $\mathcal H_{0}=\{h\in \Theta \oplus \mathcal G: \partial _{x_{2}} g \geq 0\}$. The class of alternative functions can then be written as $\mathcal H_{1}(r_{n}): = \{ g\in \mathcal G: \|g-\mathcal G_{0} \|_{L^{2}(X_{2})}\geq r_{n} \}$, where $\mathcal G_{0}=\{g\in \mathcal G: \partial _{x_{2}} g\geq 0\}$ and the rate of testing $r_{n}$ does not depend on the dimensionality of $X_{1}$. We can also test for the nonnegativity of the coefficient $\theta $ and a partial linear restriction by setting $\mathcal H_{0}=\{h\in \Theta \oplus \mathcal G: \partial _{x_{1}} h \geq 0\}$. As in Example (ref), we can test semiparametric equality restriction by taking $\mathcal H_{0}=\Theta \oplus \mathcal G$.

Let the partial linear IV model be the maintained hypothesis in model (ref) with $\mathcal H=\Theta \oplus \mathcal G$. The maintained partial linear structure can be easily enforced in the sieve space used to estimate the unconstrained NPIV function. For instance, we impose a partial linear structure $\mathcal H$ in our empirical illustration on demand for differential products in Section (ref). Monotonicity in all arguments of $h$ can be imposed by $\mathcal H_{0}=\{h\in \Theta \oplus \mathcal G: \theta \geq 0, \partial _{x_{2}} g\geq 0\}$. We also allow for second or higher order derivatives in the hypotheses considered above.

Confidence Sets in $L^{2}$

One can construct $L^{2}$ confidence sets for a NPIV function by inverting our adaptive test. For any small $\alpha >0$, the $(1-\alpha )$ confidence set for a NPIV function $h$ belonging to a restricted nonparametric class $\mathcal{H}_{0}$ is given by

This confidence set does not depend on additional tuning parameters. The following corollary exploits our previous results to characterize the asymptotic size and power properties of our procedure.

Corollary (ref) result ((ref)) shows that the $L^{2}$ confidence set $\mathcal C_{n}(\alpha )$ controls size uniformly over the class of functions $\mathcal H_{0}$. Moreover, result ((ref)) establishes power uniformly over the class $\mathcal H_{1}(\delta ^{\circ }\text{\textsf r}_{n})$. We immediately see from Corollary (ref) that the diameter of the $L^{2}$ confidence ball, $\textsf{diam}(\mathcal C_{n}(\alpha ))=\sup \{\|h_{1}-h_{2}\|_{L^{2}(X)}: h_{1},h_{2}\in \mathcal C_{n}(\alpha ) \}$, depends on the degree of ill-posedness and the unknown smoothness $p$ of $\mathcal H$.

Corollary (ref) yields a confidence set whose diameter shrinks to zero at the adaptive optimal testing rate (of the order $ (J^\circ )^{-p/d_{x}}$) and whose implementation does not require specifying the values of any unknown regularity parameters. Our confidence set $\mathcal C_{n}(\alpha )$ thus adapts to the unknown smoothness $p$ of $\mathcal H$ (the class of unrestricted NPIV functions).

Monte Carlo Studies

This section presents Monte Carlo performance of our adaptive test for monotonicity and parametric form of an NPIV function using simulation designs based on chernozhukov2015. See Supplemental Appendix C for additional simulation results using other designs. All the simulation results reported here are based on $5000$ Monte Carlo replications for each experimental design and at $\alpha =0.05$ nominal level. The simulation results clearly indicate that our simple adaptive test has size-control and finite-sample non-trivial power uniformly against a large class of NPIV alternatives, even for models with relatively weak instruments. In addition, simulation and real data application results reported in BC2020, but not here due to the lack of space, have demonstrated that our adaptive test and its bootstrapped version perform similarly well in both finite-sample size and power.

For all the designs in this section, $Y$ is generated according to the NPIV model (ref) for scalar-valued random variables $X$ and $W$. We let $X_{i} =\Phi (X_{i}^{*})$ and $ W_{i} =\Phi (W_{i}^{*})$, where $\Phi $ denotes the standard normal distribution function, and generate the random vector $(X_{i}^{*},W_{i}^{*}, U_{i})$ according to

The parameter $\xi $ captures the strength of instruments and varies in the experiments below. As $\xi $ increases, the instrument becomes stronger (or the ill-posedness gets weaker). While chernozhukov2015 fixed $\xi = 0.5$ in their design, we let $\xi \in \{0.3, 0.5, 0.7\}$ in our simulation studies. The functional form of $h$ varies in different Monte Carlo designs below.

Adaptive Testing for Monotonicity

We generate $Y$ using (ref) and (ref) with $h$ from the chernozhukov2015 design:

This function $h(x)$ is decreasing in $x$, where $c_{0}$ captures the degree of monotonicity. We note that $c_{0}=0$ corresponds to $h(x)\equiv 0$ (the boundary case); $h(x) \approx 0$ for $c_{0}$ close to zero and $h(x)\approx \phi (0)(1-2x)$ for $c_{0}$ close to 1, where $\phi $ denotes the standard normal probability density function. The null hypothesis is that the NPIV function $h$ is weakly decreasing on the support of $X$.

We implement our adaptive test statistic $\widehat {\mathtt{T}}_{n}$ given in (ref) using quadratic B-spline basis functions with varying number of knots for $h$. Due to piecewise linear derivatives, monotonicity constraints are easily imposed on the restricted function at the derivative at $J-1$ points. For the instrument sieve $b^{K(J)}(W)$, we also use quadratic B-spline functions with a larger number of knots with $K(J)\in \{2 J, 4J, 8J\}$. Implementation of the restricted sieve NPIV estimator $\widehat h_{J}^{\text{\textsc r}}$ is straightforward using the R package coneproj. We compare our adaptive test to the nonadaptive test of fang2019, which involves approximately computing $[n^{-1/2}s_{J}^{-1}J^{1/2}]^{-1}\min_{h\in \mathcal H_{0}}\| \widehat h_{J}- h\|_{L^{2}(X)}$ for a deterministic choice of sieve dimensions $J$ and $K\geq J$ in their B-spline 2SLS estimate $\widehat h_{J}$. Their 2019 arXiv preprint presents a simulation study with $J=3$, $K\in \{3, 4, 5\}$, and other tuning parameter choices $c_{n} = (\log n)^{-1}$ and $\gamma _{n} = 0.01 / \log n$, such that their test achieves approximately empirical size control with a sample size $n=500$. Below, we use FS to denote their test with $J=3$ and $K=5$ (as $K=5$ yields the best empirical power in their simulation), which is computed using R language translation of their Matlab program code. To study the sensitivity to the choice of $K$, we also implement their test with $K = 12, 24$. In our simulations, we implement their test using 200 bootstrap iterations.

\paragraph{Size.} Table (ref) presents the average data-driven choice of tuning parameter $J$, denoted by $\widehat J$. Specifically, $\widehat J$ is the average choice of $J$ that maximizes $\widehat{\mathcal{W}}_{J} (\alpha )$ over the RES index set $\widehat{\mathcal I}_{n}$ when the null is not rejected; and is the smallest $J\in \widehat{\mathcal I}_{n}$ such that $\widehat{\mathcal{W}}_{J} (\alpha )>1$ when the null is rejected. This data-driven choice of $J$ corresponds to early stopping when the null is rejected. Table (ref) shows that, for the same sample size $n$, the average data-driven choice $\widehat J$ increases as the instrument strength (captured by the parameter $\xi $) increases; while for the same instrument strength $\xi $, $\widehat J$ weakly increases as the sample size $n$ increases. Table (ref) also reports empirical rejection probabilities under the null hypothesis using our adaptive test $\widehat {\mathtt{T}}_{n}$ and the FS test. Our adaptive test is slightly under-sized across different sample sizes $n\in \{500, 1000, 5000\}$, different instrument strength $\xi \in \{0.3,0.5,0.7\}$, different degrees of monotonicity $c_{0}\in \{0,0.01,0.1\}$ with $c_{0}=0$ being the “boundary” case. Table (ref) shows that our adaptive test has empirical size control for all $K(J) \in \{2J, 4J, 8J\}$, which is in line with our theoretical results establishing asymptotic size control for any deterministic relation of $K\sim cJ$ for some fixed constant $c\geq 1$. The difference between the empirical size (of our adaptive test) for different choice of $K(J)$ is small when $n$ is large or $\xi = 0.7$. While the FS test with $K=5$ has empirical size control, the FS test with $K=12$ can be slightly over-sized, and with $K=24$ can be heavily over-sized for all $\xi $ and $n=500$, especially so for functions at or close to the boundary.\footnote{In our previous version (arxiv:2006.09587v4), we implemented what we called a nonadaptive bootstrap test $T^{B}_{n,3}$ of fang2019, which is essentially their test, but uses empirical root-mean squared metric instead of their trapezoid rule approximated $\|\widehat h_{J}- h\|_{L^{2}(X)}$, and a cone projection onto a $J$-dimensional sieve space instead of their optimization over grid points (for $x$). The “nonadaptive bootstrap test” $T^{B}_{n,3}$ has an empirical size closer to that of our adaptive test.}

\paragraph{Power.} We next examine the rejection probabilities of our adaptive test when the data are generated according to (ref) and (ref) using the NPIV function

where $c_{A} \in [0, 2]$ and $c_{B}\in \{0, 0.5, 1\}$. The null hypothesis is that the NPIV function $h(\cdot )$ is weakly decreasing over the support of $X$. When $c_{B}=0$, the null is satisfied only if $c_{A}\leq 0.1$. When $c_{B}=0.5$, the null hypothesis is satisfied only if $c_{A}\leq 0.1/(1+\pi /2)\approx 0.04$. When $c_{B}=1$, the null is satisfied only if $c_{A}\leq 0.1/(1+\pi )\approx 0.02$.

Figure (ref) depicts the empirical power function of our adaptive test $\widehat {\mathtt{T}}_{n}$ (dashed plus lines for $K(J)=4J$ and solid circle lines for $K(J)=8J$), and of the FS test (dotted square lines, $J=3$, $K=5$), under the 5% nominal level for different instrument strengths $\xi \in \{0.3, 0.5,0.7\}$, and sample size $n=500$.\footnote{The finite-sample power of our adaptive test with $K(J)=2J$ is slightly smaller than that with $K(J)=4J$ when $n=500$, but the power difference disappears when $n$ becomes larger.} Figure (ref) shows these power curves for a larger sample size $n=5000$. From both figures, we see that our adaptive test becomes more powerful for $c_{A}>0.1$ as the instrument strength $\xi $ and the sample size $n$ increase. For weak instrument strength $\xi =0.3$ and a small sample size (i.e., $n=500$), our adaptive test with a larger $K(J)=8J$ is more powerful.

Figures (ref) and (ref) highlight the importance of adaptation for the power of nonparametric monotonicity tests. When the alternative is of a simple quadratic form (i.e., $c_{B} =0$), there is little difference between our adaptive test $\widehat {\mathtt{T}}_{n}$ and the FS test. But, as the alternative becomes more nonlinear when $c_{B}>0$ increases, the FS test becomes much less powerful than our adaptive test. This shows that a test with a tuning parameter $J$ that is a deterministic nondecreasing function of $n$ can be powerful in a certain direction but not for other nonlinear deviations.

In Supplemental Appendix C, we present another simulation design, which is based on an NPIV monotonicity design of CW2017. Simulation results using that design reveal that the empirical size and power of our adaptive test have patterns very similar to the ones reported in this subsection.

Testing for Parametric Restrictions

We now test for a parametric specification. We assume that the data are generated according to the design (ref) and (ref) with the NPIV function $h$ given by (ref) with $c_{A} \in [0,4]$ and $c_{B}\in \{0, 0.5\}$. The null hypothesis is $h$ being linear (i.e., $c_{A}=c_{B}=0$).

We implement our adaptive test $\widehat {\mathtt{T}}_{n}$ given in (ref) using quadratic B-spline basis functions with varying number of knots and where the constrained function coincides with the parametric 2SLS estimator. The number of knots varies within the RES index set $\widehat{\mathcal I}_{n}$ as implemented in the last subsection, with $K(J)\in \{2J, 4J, 8J\}$. We compare our adaptive test to the asymptotic $t$-test and the test by Horowitz2006 (denoted by JH).\footnote{Horowitz2006 already demonstrated in his simulation studies, with a sample size $n=500$ and $1000$ Monte Carlo replications, that his test is more powerful than several existing tests including bierens1990's.} To compute the JH test that involves kernel density estimation, we follow Horowitz2006 to estimate the joint density $f_{XW}$ using the kernel $K(v)=(15/16)(1 - v^{2})^{2}\mathbbm{1}\{|v|\leq 1\}$, with the kernel bandwidth chosen via cross-validation minimizing mean squared error of estimating $f_{XW}$.

\paragraph{Size.} Table (ref) reports empirical rejection probabilities of several tests under the null hypothesis of linearity of $h$. Results are presented under different sample sizes $n\in \{500, 1000, 5000\}$ and instrument strength $\xi \in \{0.3, 0.5,0.7\}$. It also reports our adaptive test with different $K(J)$ and $\widehat J$ (which is defined the same way as that in Table (ref)). We note that $\widehat J$ is again weakly increasing with sample size and with instrument strength. While the JH test can be slightly over-sized, our adaptive test $\widehat {\mathtt{T}}_{n}$ provides adequate size control across different sample size $n$, different instrument strength $\xi $, and different $K(J)$. The difference in empirical size of our adaptive test with different $K(J)$ is again small for large $n$, which is consistent with our theory.

\paragraph{Power.} Figure (ref) provides empirical power curves for the 5% level tests with sample sizes $n\in \{500,5000\}$. From this figure, we see that our adaptive test $\widehat {\mathtt{T}}_{n}$ (dashed plus lines with $K(J)=4J$ and solid circle lines with $K(J)=8J$) has power similar to the asymptotic $t$-test (dotted triangle lines) and the JH test (dotted square lines) for a simple quadratic alternative with $c_{B}=0$. When the alternative function in (ref) becomes more nonlinear/complex with $c_{B}= 0.5$, our adaptive test becomes more powerful than the JH test. This is theoretically sensible since the Horowitz2006 test is designed to have power against $n^{-1/2}$ smooth alternative only. Since our adaptive test is slightly under-sized for small sample sizes or for weak instrument strength, the size-adjusted empirical power of our test is even better (see our arxiv:2006.09587v3 version). To sum up, our adaptive minimax test not only controls size, but also has very good finite-sample power uniformly against a large class of nonparametric alternatives.

Finally, in Supplemental Appendix C, we present additional simulation comparisons of our adaptive test against our adaptive version of bierens1990's (bierens1990) type test when the dimension of conditional instrument $W$ is larger than the dimension of the endogenous variables $X$. We observe that our adaptive test $\widehat {\mathtt{T}}_{n}$ again has size control and even better finite-sample power when $d_{w} > d_{x}$.

Empirical Applications

We present two empirical applications of our adaptive test for NPIV models. The first one tests for connected substitutes restrictions in differentiated products demand using market level data. The second one tests for monotonicity, convexity, or parametric specification of Engel curves for non-durable good consumption using household level data. The applications demonstrate that our simple adaptive test is powerful to detect economic shape restrictions.

In both empirical applications, we implement our adaptive test $\widehat {\mathtt{T}}_{n}$ given in (ref) with $K(J)=4J$. The null hypothesis is rejected at the nominal level $\alpha =0.05$ whenever $\widehat {\mathcal{W}}_{J} (\alpha )>1$ for some $J\in \widehat{\mathcal I}_{n}$ (the RES index set). Let $\widehat{\mathcal J}$ be $\{J\in \widehat{\mathcal I}_{n}: \widehat {\mathcal{W}}_{J}(\alpha )>1 \}$ when our test rejects the null, and be $\arg \max_{J\in \widehat{\mathcal I}_{n}} \widehat {\mathcal{W}}_{J}( \alpha )$ when our test fails to reject the null. Let $\widehat J$ be the minimal integer of $\widehat{\mathcal J} \subset \widehat{\mathcal I}_{n}$. Tables in this section report $\widehat{\mathcal J} $ and $\widehat {\mathcal{W}}_{\widehat J}$. We also report the corresponding $p$ value, which should, by Bonferroni correction, be compared to the nominal level $\alpha =0.05$ divided by the cardinality of $\widehat{\mathcal I}_{n}$. Finally, since our test is based on a leave-one-out version, the value of $\widehat {\mathcal{W}}_{\widehat J}$ could be negative.

Adaptive Testing for Connected Substitutes in Demand for Differential Products

berry2014 provided conditions under which a nonparametric demand system for differentiated products can be inverted to NPIV equations using market level data. A key restriction is what they called “connected substitutes.” compiani2018 applied their nonparametric identification results and estimated the system of inverse demand by directly imposing the connected substitutes restrictions in his implementation of sieve NPIV estimator, and obtained informative results as an alternative to BLP demand in simulation studies and a real data application.

We revisit compiani2018's empirical application using the 2014 Nielsen scanner data set that contains market (store/week) level data of consumers in California choosing from organic strawberries, non-organic strawberries, and an outside option. While compiani2018 directly imposed “connected substitutes” restriction in his sieve NPIV estimation of inverse demand, we want to test this restriction. Following compiani2018, we consider

where $h$ denotes the inverse of the demand for organic strawberries, $X_{o}$ denotes a measure of taste for organic products, $X_{no}$ denotes the availability of other fruit, $S_{o}$ and $S_{no}$ denote the endogenous shares of the organic and non-organic strawberries, respectively. $(X_{o}, X_{no})$ are the two included instruments for the two endogenous shares $(S_{o}, S_{no})$. $In$ denotes store level (zip code) income and $U$ unobserved shocks for organic produce. The vector $\textbf{P}=(P_{o},P_{no},P_{\mathrm{out}})$ denotes the endogenous prices of organic strawberries, non-organic strawberries, and non-strawberry fresh fruit, respectively. We follow compiani2018 and let $\textbf{W}_{p}=(W_{o},W_{no},W_{\mathrm{out}},W_{s1},W_{s2})$ be a five-dimensional vector of conditional instruments for the price vector $\textbf{P}$, including three Hausman-type instrumental variables $(W_{o},W_{no},W_{\mathrm{out}})$ and two shipping-point spot prices $(W_{s1},W_{s2})$ (as proxies for the wholesale prices faced by retailers).

As shown by compiani2018, the connected substitutes assumption of berry2014 implies the following shape restrictions on the function $h$: First, $h$ is weakly increasing in the organic product price $P_{o}$. Second, $h$ is weakly increasing in the organic product share $S_{o}$. Third, $h$ is weakly increasing in the non-organic product share $S_{no}$. Fourth, $\partial h/\partial s_{o}\geq \partial h/\partial s_{no}$ (the so-called diagonal dominance). Below, we test for these inequality restrictions.

We use the data set of compiani2018,\footnote{For details on the construction of the data and descriptive statistics, see compiani2018.} where income ranges from the first and to the third quartile of its distribution and prices for organic produces are restricted to be above its 1st and below its 99th percentile. The resulting sample has size $n= 11910$. We implement our adaptive test $\widehat {\mathtt{T}}_{n}$ by making use of a semiparametric specification of the function $h$: we consider the tensor product of quadratic B-splines $\psi ^{J_{1}}(P_{o})$ and the vector $(1,In, P_{no}, \psi ^{3}(S_{o}))$, where we use a cubic B-spline transformation of $S_{o}$ without knots and without intercept, hence $J=6J_{1}$. The variables $( P_{\mathrm{out}}, S_{no}, S_{no}P_{no}, S_{no}S_{o})$ are included additively and we set $K(J)=4J$. We obtain the RES index set $\widehat{\mathcal I}_{n}=\{22,28,34\}$.

According to Table (ref), at the nominal level $\alpha =0.05$, our adaptive test fails to reject that $h$ is weakly increasing in the own price (but rejects $\partial h/\partial p_{o}\leq 0$), and fails to reject that $h$ is weakly increasing in the own share (but rejects $\partial h/\partial s_{o}\leq 0$). Our test fails to reject that $h$ is weakly increasing or decreasing in the non-organic share (i.e., fails to reject a constant partial effect of $h$ with respect to the non-organic share). Our test also fails to reject the diagonal dominance (but rejects $\partial h/\partial s_{o}\leq \partial h/\partial s_{no}$). In summary, our adaptive test provides strong empirical evidence for the connected substitutes restriction.

Adaptive Testing for Engel Curves

The system of Engel curves plays a central role in the analysis of consumer demand for non-durable goods. It describes the $i$th household's budget share $Y_{\ell ,i}$ for non-durable goods $\ell $ as a function of its log-total expenditure $X_{i}$ and other exogenous characteristics such as family size and age of the head of the $i$th household. The most popular class of parametric demand systems is the almost ideal class, pioneered by deaton1980, where budget shares are assumed to be linear in log-total expenditure. banks1997 proposed a popular extension of this system of linear Engel curves to include a squared term in log-total expenditure, and their parametric Student $t$-test rejects linear form in favor of quadratic Engel curves.

BCK07econometrica estimated a system of nonparametric Engel curves as functions of endogenous log-total expenditure and family size, using log-gross earnings of the head of household as a conditional instrument $W$. We use a subset of their data from the 1995 British Family Expenditure Survey, with the head of household aged between 20 and 55 and in work, and household with one or two children. This leaves a sample of size $n = 1027$. As an illustration we consider Engel curves $h_{\ell} (X)$ for four non-durable goods $\ell $: “food in,” “fuel,” “travel,” and “leisure”: $\operatorname{\mathrm{E}} [Y_{\ell} - h_{\ell} (X) | W]=0$. We use the same quadratic B-spline basis with up to three knots to approximate all the Engel curves and set $K(J)=4J$. Hence, the RES index set $\widehat{\mathcal I}_{n}=\{3,4,5\}$ is the same for the different Engel curves.

Table (ref) reports our adaptive test for weak monotonicity of Engel curves. It shows that our test rejects increasing Engel curves for “food in,” “fuel,” and “travel” categories, and also rejects decreasing Engel curve for “leisure” at the $0.05$ nominal level. Previously, to decide whether the Engel curves are strictly monotonic, estimated derivatives of these functions together with their non-adaptive 95% uniform confidence bands were also provided in ChenChristensen2017. Those uniform confidence bands are constructed using sieve score bootstrapped critical values with non-data-driven choice of sieve dimension $J$, and contain zero almost over the whole support of household expenditure. It is interesting to see that our adaptive test is more informative about monotonicity in certain directions that are not obvious from their 95% uniform confidence bands. Table (ref) reports our adaptive test for convexity and concavity of these Engel curves. At the 5% nominal level, we reject convexity of travel goods and reject concavity of Engel curves for fuel consumption. These are in line with ChenChristensen2017, but again, statistically significant statements about the convexity/concavity of Engel curves are only possible using our adaptive testing procedure. Finally, Table (ref) presents our adaptive tests for linear or quadratic specifications (against nonparametric alternatives) of the Engel curves for the four goods. At the nominal level $\alpha =0.05$, this table shows that our adaptive test fails to reject a quadratic form for all the goods, while it rejects a linear Engel curve for fuel and travel goods. Our results are consistent with the conclusions obtained by banks1997 using Student $t$-test for linear against quadratic forms of Engel curves.

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\vskip 1cm This supplementary appendix contains materials to support our main paper. Appendix (ref) presents additional simulation results. Appendix (ref) provides proofs of our results on confidence sets in Section 4.3. Appendix (ref) presents additional technical lemmas and all the proofs.