Subvector Inference in Partially Identified Models with Many Moment Inequalities

Subvector Inference in PI Models with Many Moment Inequalities

Introduction

This paper contributes to the growing literature on inference in partially identified econometric models defined by a large number of moment inequalities. As discussed by tamer2010partial, canay/shaikh:2017, and ho/rosen:2017, partially identified moment inequality models arise naturally in a large variety of economic problems and have been increasingly used in the empirical literature. As argued in chernozhukov2013testing, the number of moment inequalities implied by many econometric applications is frequently very large relative to the sample size. Examples of this include bajari2007estimating, ciliberto2009market, pakes2013moment, beresteanu2011sharp, galichon2011set, chesher2013instrumental, and chesher2017generalized.

There is a substantial literature on inference of the entire parameter vector in a partially identified moment inequality model. An earlier strand of this literature considered partially identified models defined by a finite number of unconditional moment inequalities\footnote{See chernozhukov2007estimation, andrews/berry/jiabarwick:2004, romano2008inference, rosen2008confidence, andrews2009validity, andrews2010inference, canay2010inference, bugni:2010, bugni2016comparison, andrews2012inference, romano2014practical, menzel2014consistent, bugni2015specification, and pakes/porter/ho/ishii:2015, among many others.} and conditional moment inequalities\footnote{See kim:2008, ponomareva:2010, andrews2013inference,chernozhukov2013intersection,lee2013testing,andrews2014nonparametric,armstrong2014weighted, armstrong2015asymptotically, armstrong2016multiscale, chetverikov2018adaptive, and armstrong2018choice, among many others.}. More recently, chernozhukov2013testing study the problem of the entire parameter vector in partially identified models with {\it many moment inequalities}. According to this asymptotic framework, the number of moment inequalities is allowed to grow with the sample size, possibly at exponential rates. Furthermore, the moment inequalities in this framework are allowed to be {\it unstructured}, in the sense that no restrictions are imposed on their correlation structure.\footnote{This feature distinguishes their framework from a model with conditional moment inequalities. While conditional moment conditions can generate an uncountable set of unconditional moment inequalities, their covariance structure is restricted by the conditioning structure.} As chernozhukov2013testing explain, the many moment inequalities framework substantially expands the scope of applications by allowing the number of unstructured moment inequalities to be much larger than the sample size.

The main contribution of this paper is to propose inference for a function of a parameter vector in a partially identified models with {\it many moment inequalities}. Our main motivating application is {\it subvector inference}, i.e., inference on a single component of the partially identified parameter vector that is associated with a treatment effect or a policy variable of interest. Our inference method is based on the {\it MinMax} test statistic, where the minimum is computed over the parameter values that satisfy the null hypothesis (i.e.\ profiling) and the maximum is computed over an index of the moment inequalities. We propose comparing the MinMax test statistic against critical values that can be based on bootstrap approximations or analytical bounds. We show that the resulting inference method with either of these critical values controls asymptotic size uniformly over a large class of data generating processes.

There is a recent literature on the problem of inference for a function of a partially identified parameter in a moment inequality model. This literature focuses on moment inequality models with finitely many moment conditions, which can be restrictive in certain applications. We now summarize this literature. andrews2009validity and andrews2010inference propose conducting projection-based inference, i.e., intersecting the confidence set for the entire parameter space with the subset of the parameter space that satisfies the null hypothesis. bugni2017inference show that projection-based inference can result in power losses relative to profiling-based methods. romano2008inference consider profiling-based inference with critical values constructed using subsampling. In turn, bugni2017inference and kaido2016confidence consider profiling-based methods with critical values constructed using the bootstrap. gafarovinference considers subvector inference in affine moment inequality models. As already mentioned, these references restrict attention to partially identified models with a finite number of moment conditions. This restriction becomes essential in proving the asymptotic validity of the proposed inference. For example, it implies that profiling-based test statistics, such as the MinMax statistic, converge in distribution to a function of a Gaussian process. In contrast, these statistics may not even converge in distribution in the many moment inequality setting considered in this paper.

More recently, chernozhukov2015constrained consider inference for subvector on partially identified conditional moment restriction models with a conditioning covariate of restricted dimension. This restriction imposes enough structure on the problem to allow them to use Koltichinskii's Hungarian couplings and approximate the relevant empirical processes by the corresponding Gaussian processes. By contrast, our approach is designed to be valid in a framework with many moment inequalities that does not impose the structure produced by having a small number continuous conditional moment inequalities, enabling a much broader set of applications.

Our inference method compares the MinMax test statistic with critical values based on bootstrap approximations or analytical bounds. Both methods have their relative merits. The first set of main results pertains to bootstrap approximations that exploit correlation structures to increase power. These procedures were inspired by the ideas in bugni2017inference which considered a finite number of inequalities but for a broader class of test statistics. By considering many moment inequalities to cover a wide range of applications, we develop new theoretical arguments to establish the validity of our bootstrap procedures. In particular, non-asymptotic bounds and new high-dimensional central limit theorems for the MinMax statistics were developed.\footnote{This is in sharp contrast to bugni2017inference, who use Donsker's functional central limit theorem to derive a limiting distribution.} Our second set of main results is on the construction of analytical bounds based on self-normalized moderate deviation theory which are computationally easier but do not exploit the underlying correlation structure. We build upon arguments in chernozhukov2013testing which considered the vector inference problem but also allowed for many moment inequalities.

As it is usual in this literature, our bootstrap approximation requires a threshold sequence $(\kappa_n)_{n=1}^{\infty}$ that determines whether each moment inequality is considered to be sufficiently close to binding or not. In models with finitely many moment inequalities, the threshold sequence is required to satisfy $\kappa_n \to \infty$ and $\kappa_n/\sqrt{n} \to 0$. We show that the many moment inequality setting imposes additional requirements on the threshold sequence. In order to facilitate this choice for practitioners, we propose a data-driven method to select this threshold sequence in an asymptotically valid way.

In deriving our primary results, we also obtain several auxiliary findings that might be of independent interest. First, we derive a new non-asymptotic coupling result for the MinMax of an empirical process (see Theorem (ref)). A key ingredient to obtain such coupling is a novel use of a smooth approximation of the MinMax functional. Second, the data-driven procedure proposed to estimate the anti-concentration of the MinMax statistic seems to be widely applicable to other contexts, as it allows one to bypass the need to development of anti-concentration bounds that are only available to a limited class of statistics, such as the Max statistic. Both results are new and could be of independent interest beyond our contribution.

The paper is organized as follows. Section (ref) formally describes the setting and problem. We provide an overview of some of our main results in Section (ref) where we discuss insights and key issues. Section (ref) derives our main theoretical results for different procedures. We discuss in detail the anti-concentration properties associated with the MinMax statistics in Section (ref). Section (ref) provides new coupling results for the MinMax of non-centered empirical processes and for the sum of independent random matrices which are of independent interest.

Setup and Test Statistics

Let $\left( S,\mathcal{S}\right) $ be a measurable space, and let $ (W_{i})_{i=1}^{n}$ denote an i.i.d.\ sequence of random variables taking values on $\left( S,\mathcal{S}\right) $ with common distribution $P\in \mathcal{P}$, and let $\theta \in \Theta \subseteq \mathbb{R}^{d_{\theta }}$ denote the parameter of the model. The econometric model predicts that true parameter value, $\theta ^{\ast }\in \Theta $, satisfies the following collection of $p_{I}$ moment inequalities and $p_{E}$ moment equalities:

If we let $m_{j}(W,\theta )\equiv m_{I,j}(W,\theta )$ for $j=1,\dots ,p_{I}$ and $m_{p_{I}+j}(W,\theta )\equiv m_{E,j}(W,\theta )$ for $j=1,\dots ,p_{E}$ , and $m_{p_{I}+p_{E}+j}(W,\theta )\equiv -m_{E,j}(W,\theta )$ for $ j=1,\dots ,p_{E}$, (ref) can be equivalently re-expressed as moment inequality model with $p\equiv p_{I}+2p_{E}$ moment inequalities:

We allow the econometric model to be partially identified, i.e., the moment inequalities in (ref) do not necessarily restrict $\theta^{*}$ to a single value, but rather constrain it the identified set, given by:

We are implicitly allowing the distribution ${\rm P}$ of the data to change with the sample size. In particular, the dimensionality of $\Theta$, denoted by $d_\theta$, and the number of moment inequalities $p$ to depend on $n$. In particular, we are primarily interested in the case in which $p=p_{n}\to \infty$, but the subscripts are omitted to keep the notation simple. In particular, $p$ can be much larger than the sample size $n$.

Let $h:\Theta \rightarrow \mathbb{R}^{d_{h}}$ be a known function, let $ h[\Theta ]$ denote the image of $h$, and let $\bar h \in h[\Theta ]$ be an arbitrary parameter value. Our goal is to conduct the following hypothesis test:

The main application of our test is the subvector inference problem, i.e., testing whether a subset of the partially identified parameter vector is equal to a particular value or not. For example, we could test whether, say, the $s$'th coordinate of $\theta ^{\ast }$, $\theta _{s}^{\ast }$, is equal to a particular value $\bar h \in \mathbb{R}$ or not, i.e.,

(ref) is a particular example of (ref) with $h(\theta )=\theta _{s}$.

To test (ref), we propose the \textquotedblleft MinMax\textquotedblright\ test statistic, defined as follows:

where $\Theta (\bar h )$ is the \textquotedblleft null set\textquotedblright\ associated to (ref), given by:

and

Finally, if $\hat{\sigma}_{\theta,j}=0$ in (ref), we define $\bar{m}_{\theta,j}/\hat{\sigma}_{\theta,j}\equiv \infty \times {\rm sign}(\bar{m}_{\theta,j})$.

Given the test statistic in (ref) and nominal size $\alpha \in (0,1)$, our goal is to present critical values $c_{n}(\bar h ,\alpha)$ that can be associated with $ T_{n}(\bar h)$ to produce asymptotically valid inference for (ref). Specifically, we propose to reject $H_{0}$ in (ref) if and only if

By exploiting the duality between hypothesis tests and confidence sets, we construct a confidence set for $h(\theta ^{\ast })$ by collecting all parameter values $\bar h $ for which we do not reject, i.e.,

Our formal results will have the following structure. Under $H_{0}$, we show that

where $c$ and $C$ are constants that depend only on other constants of the problem (i.e.\ they do not depend on $\bar h$, ${\rm P}$, or $n$). As a corollary of this, the convergence in (ref) occurs uniformly over a suitable class of probability distributions.

The main contribution of this paper is to propose methods to approximate the critical value $c_{n}(\bar h ,\alpha )$ that satisfies (ref) in the many inequalities setting. To this end, we develop an approximation to the asymptotic distribution of the MinMax test statistic $T_{n}(\bar h)$ based on suitably constructed bootstrap procedures and based on self-normalized moderate deviation theory.

Overview of Main Results

In this section we provide a simplified discussion of the issues and proposed methods to construct critical values that are asymptotically valid in the sense of (ref). This section intends to motivate the key issues we face and more general results and other procedures are discussed in Section (ref).

An important feature of the MinMax statistic ((ref)) is that it is a functional of non-centered sums. Therefore it is convenient to highlight the centering and rewrite the MinMax statistic as $$T_{n}(\bar h) = \inf_{\theta \in \Theta (\bar h) }\max_{j\in[p]} \left\{\hat v_{\theta,j} +\sqrt{n}{\rm E}[ m_{j}\left(W, \theta \right)] /\hat {\sigma}_{\theta,j}\right\},$$ where $\hat v_{\theta,j}$ is the centered empirical process indexed by $\theta \in \Theta$ and $j\in [p]$, i.e., $$\hat v_{\theta,j}~\equiv~ n^{-1/2}\sum_{i=1}^n \{ m_j(W_i,\theta)- {\rm E}[ m_{j}\left(W, \theta \right)] \}/\hat \sigma_{\theta,j}.$$

There are two main insights in the derivation of critical values for the MinMax statistics. Under $H_0$, $T_n(\bar h) \leq \max_{j\in[p]} \sqrt{n}\bar m_j(\theta^*)/\hat \sigma_{\theta^*,j} \leq \max_{j\in [p]} \hat v_{\theta^*,j} $. However, also under $H_0$, $h(\theta^*)$ is known and yet the parameter vector $\theta^*$ may not be known. Therefore, in addition to considering which inequalities can be binding, it is important to consider which values of the parameter $\theta$ are candidates to be $\theta^*$ based on the moment inequalities and $h(\theta^*)=\bar h$. Note however that these two “selection” problems have very different consequences in the analysis of the size of the test. That is, overselecting inequalities leads to potentially conservative but valid asymptotic size control, while over selecting values of $\theta$, because of the minimum, can lead to a procedure that fails to control asymptotic size. In particular, using a critical value based on bootstrapping $\min_{\theta \in \Theta(\bar h)}\max_{j\in [p]} \hat v_{\theta,j}$ will fail to control asymptotic size even if $p=2$ for simple data generating processes.\footnote{See Example (ref) in the Appendix. For the minimum of the sum of violations similar issues have been shown in bugni2017inference.}

We begin with a bootstrap-based procedure for the construction of a critical value. Consider the critical value $c_{n}(\bar h ,\alpha )$ based on the penalized bootstrap procedure

where the random variable $R_n^*$, conditional on the data, is defined as

where $\kappa_n$ is a tuning parameter, and $(g_i)_{i=1}^n$ are i.i.d.\ standard Gaussian random variables. It has been noted in the literature that the centering term cannot be consistently estimated in a uniformly fashion. Therefore the use of $\kappa_n=1$ would not make ((ref)) a valid choice. In ((ref)) we set $\kappa_n$ to dominate the effective noise in the centering, namely $$\bar w_n \geq \sup_{\theta\in \Theta(\bar h)}\max_{j\in[p]}|\sqrt{n}\hat\sigma^{-1}_{\theta,j}\{\bar m_{\theta,j}-{\rm E}[m_j(W,\theta)]\}|$$ with high probability. With that in mind we keep $\kappa_n$ as small as possible not to remove the true centering $\sqrt{n}\hat\sigma^{-1}_{\theta,j}{\rm E}[m_j(W,\theta)]$ completely (because the minimum over $\theta$ would lead to a too small critical value). We note that because of the many inequalities setting, we typically have $\bar w_n \to \infty$. \footnote{It follows we will be able to approximate $\bar w$ via a separate bootstrap procedure providing a data driven way to compute $\bar w$ that is theoretically valid.}

Moreover, the approximation errors of the new coupling results should not affect the size. Letting $\delta_n$ denote such coupling approximation error, we have that

The validity of the bootstrap approximation is guaranteed by ${\rm P}( |R_n^{*} - c_{n}(\bar h ,\alpha )|\leq \delta_n )\to 0$. In other words, the distribution of $R_n^*$ should not concentrate too much mass around $c_{n}(\bar h ,\alpha )$ as $n$ diverges. It is not hard to show that this concentration is bounded by $\delta_n$ multiplied by the maximum value of the density function of $R_n^{*}$. In moment inequality models with fixed number of moment conditions, this does not pose any problems as the limit of the corresponding density function is typically bounded. However, in moment inequality models with many moment inequalities, these statistics can have very different behavior. For example, chernozhukov2013testing show that the distribution of the Maximum statistic can concentrate but not too fast. Indeed it is shown that maximum density of the Maximum statistic is bounded by a logarithmic factor of the number of moment conditions $p$. In this paper, we are interested in the concentration properties associated to the MinMax statistic. Based on the previous derivation, we define unconditional and conditional anti-concentration parameters associated with $R_n^*$ as follows:

Importantly, the anti-concentration property that is needed pertains to the bootstrap-based statistic $R_n^{*}$, and not the original statistics $T_n(\bar h)$. Because of this feature, we can investigate $\mathcal{A}_{n}$ via bootstrap of the quantity $\mathcal{A}_{n}(W)$ conditionally on the data. This approximation is described in detail in Section (ref).

The corollary below shows that the choice of critical value ((ref)) effectively controls the size, in the sense of ((ref)), uniformly over a set of data generating processes. The conditions are simple to allow easier interpretability but allows for the many inequality setting with potentially $p\gg n$. (Our results hold much more generally as stated in Theorems (ref) and (ref).)

Corollary (ref) provides low-level conditions that apply to many cases of practical interest. For example, it includes moment conditions that are Lipschitz continuous and it allows the dimension of the parameter space $d_\theta$ and the number of moment inequalities to grow with the sample size. In particular, it allows for $p\gg n$.

Corollary (ref) requires the choice of $\kappa_n$ to be appropriate. Such requirements generalize the requirements in bugni2017inference where it was required $\kappa_n \to \infty$ and $\sqrt{n}/\kappa_n\to \infty$. Theorem (ref) characterizes the finite sample impact of $\kappa_n$ for a non-Donsker class of functions. Theorem (ref) also motivates data driven choices of $\kappa_n$ that accounts for the anti-concentration of the process $R_n^*$ which might be non-trivial since $\mathcal{A}_{n}$ could grow.

The bootstrap procedure discussed above and the others studied in Section (ref) adapt to the correlation structure of the process to improve power. However, that is achieved by levering conditions on the number of inequalities $p$, sample size $n$, effective noise $\bar w_n$, and anti-concentration $\mathcal{A}_{n}$.

An alternative approach is to rely on union bounds that are more conservative but are valid under weaker conditions. Moreover, since the bounds are based on the marginal distribution, there is no need to handle the anti-concentration. With that in mind, similarly to chernozhukov2013testing, ideas from self-normalized moderate deviation theory can be applied to control size as we describe next. By construction of the test statistics $T_n(\bar h)$ satisfies

Moreover, $H_0$ implies that ${\rm E}[m_j(W_i,\theta^*)]\leq 0$, and when combined with the union bound we have

where the last sum is approximately self-normalized. Such self-normalization has been exploited in the moderate deviation literature to establish central limit theorems that are valid in the tails of the distribution under mild moment conditions, see e.g. delapena. In turn this leads to pivotal choices for the critical value to control size of the test by setting $$c_{n}(\bar h ,\alpha ) \approx \Phi^{-1}(1-\alpha/p).$$ Since $c_{n}(\bar h ,\alpha )$ is of the order of $\sqrt{\log(p/\alpha)}$, it grows in the many moment inequality setting where $p\to \infty$. Therefore the Gaussian approximation should hold sufficiently deep in the tails to include $c_{n}(\bar h ,\alpha )$. This leads to the restriction $\log^3 (p/\alpha) =o(n)$ under suitable moment requirements. The following corollary of Theorem (ref) provides a result.

We also consider two-step versions of the self-normalization procedure that can improve power relative to the self-normalization procedure described above. However, because of the MinMax structure, substantial departure from chernozhukov2013testing is needed to establish the validity of a proposed critical value. These described in detail in Section (ref).

Critical Values and Theoretical Guarantees

In this section, we propose several methods to construct critical values $c_{n}(\bar h ,\alpha )$ in the hypothesis test procedure in ((ref)). We consider critical values based on the bootstrap approximations and based on self-normalized moderate deviation approximations. While the resulting hypothesis tests are shown to control asymptotic size, they have different power properties and are asymptotically valid under different conditions.

Bootstrap-based critical values

In the following subsections we propose and analyze bootstrap-based method for the construction of critical values that appropriately control size despite the high-dimensionality of the process being considered. For each $\theta \in \Theta$ and $j\in[p]$, we will denote the bootstrap process

where $(\xi_i)_{i=1}^n$ are i.i.d.\ $N(0,1)$ random variables independent of the data $(W_i)_{i=1}^n$. We will make the following assumption to analyse the bootstrap based critical values. \\ {\bf Condition MB.} {\it The set $\Theta(\bar h)$ is convex and the following conditions hold: (i) $\|\theta\|_\infty \leq \sqrt{n}$ and $\max_{ j \in [p]} \|\nabla_\theta{\rm E}[m_j(W,\theta)]/\sigma_{\theta,j}\| \leq L_G$ for every $\theta \in \Theta(\bar h)$. The set of functions $\{m_j(\cdot,\theta)/\sigma_{\theta,j} : \theta \in \Theta(\bar h), j\in [p]\}$ is VC type with measurable envelope $F$ and constants $\bar A$ and $v\geq 1$.\footnote{See Section (ref) for a definition.} Moreover, for constants $b\geq \sigma >0$ we have $\sup_{\theta\in\Theta(\bar h)}\max_{j\in[p]}{\rm E}[|m_j(W,\theta)/\sigma_{\theta,j}|^k] \leq \sigma^2 b^{k-2}$, $k=2,3,4$, and ${\rm E}[F^q]^{1/q} \leq b$ for $q>6$. (ii) $\max_{ j \in [p]} {\rm E}[\{m_j(W,\theta)/\sigma_{\theta,j}-m_j(W,\tilde\theta)/\sigma_{\tilde\theta,j}\}^2] \leq L_C\|\theta - \tilde\theta\|^{\chi}$ for every $\theta, \tilde \theta \in \Theta(\bar h)$ for some $\chi \geq 1$. (iii) For every $\theta \in \Theta(\bar h)\setminus \Theta_I$ we have $$\max_{j\in[p]} {\rm E}\left[ m_{j}\left( W,\theta\right) \right] /\sigma _{\theta _{n},j} \geq \vartheta_n \min \{ \delta ,\inf_{\tilde{\theta}\in \Theta (\bar h) \cap \Theta_I } \| \theta-\tilde{\theta} \| \}.$$ }

These conditions impose the existence of fourth moments of $m_j(W,\theta)/\sigma_{\theta,j}$ for each $j \in [p]$ and $\theta \in \Theta$. It also states that the functions $m_j(\cdot,\theta)/\sigma_{\theta,j}$ are well behaved in the sense of being a VC type, which covers a many applications of interest. Condition MB(iii) is a polynomial minorant condition as we move away from the identified set similar to the conditions imposed in chernozhukov2007estimation and bugni2017inference. However, we also allow for the parameter $\vartheta_n \to 0$ in our analysis.

For a sequence $\gamma_n = o(1)$, we define

Throughout the paper we will typically consider $\gamma_n = n^{-c}$ for a suitably small constant $c>0$. In words, $\bar w_n$ is an effective measure of the noise in the problem, $t_n^\sigma$ is a uniform rate of convergence for the estimation of the standard deviation on the moment conditions, and $K_n$ controls the entropy associated with the class of functions induced by the moment conditions. Importantly, we will be able to approximate $\bar w_n$ via bootstrap simulations of the Gaussian process $\{\hat v^*_{\theta,j}: \theta \in \Theta, j\in [p]\}$ conditional on the data. Under mild conditions, we have that $\bar w_n \lesssim \sqrt{d_\theta \log(pn/\gamma_n)}$ and $t_n^\sigma \bar w_n = o(1)$.

Discard Resampling or DR Bootstrap

The first strategy to construct critical values is based on a bootstrap procedure that discards parameter values and moment inequalities that are problematic for an asymptotic approximation. The definition of this bootstrap statistic requires certain sample objects. For some sequence $M_n \geq \bar w_n$, let:

By definition, $\widehat \Theta_n(\bar h)$ is an arbitrary non-empty subset of the minimizers that define the MinMax statistic and, for each $\theta \in \Theta$, $\widehat \Psi_\theta$ denotes the subset of the moment inequalities that are sufficiently close to being binding. In principle, we can choose $\widehat \Theta_n(\bar h)$ to be any of the minimizers that define the MinMax statistic.

The DR bootstrap statistic is defined as follows:

where $\hat v_{\theta,j}^{\ast }$ is the Gaussian multiplier bootstrap defined in ((ref)). For $\alpha \in (0,1)$, we define the corresponding conditional critical value as follows:

The following result establishes the relationship between the original MinMax statistic and the DR bootstrap statistic, and it exploits this result to show the asymptotic validity of the DR bootstrap approximation.

Theorem (ref) shows how the tail probability of the statistic of interest can be bounded by the tail probability of the bootstrap statistic (up to an approximation error). The result allows for both $p$ and $d_\theta$ to increase with the sample size. In particular, $p\gg n$ is allowed. However, our proof of the validity of this procedure requires the choice of $M_n$ to be such that binding inequalities are not missed. In particular, it would suffice to choose $M_n$ such that $M_n/\bar w_n \to \infty$ in cases where $\vartheta_n \geq c$ and $L_G \leq C$, which are common in applications. Importantly, we can rigorously approximate $\bar w_n$ as the $1-n^{-c}$ quantile of $\sup_{\theta\in \Theta(\bar h), j\in [p]}|\hat v_{\theta,j}^*|$ which provides a data-driven approach to set $M_n$. Note that this represents an additional restriction relative to $M_n\to \infty$ required by bugni2017inference in the context of finite number of moment inequalities.\footnote{Note that $\bar w_n$ would be uniformly bounded if $p$ and $d_\theta$ are fixed.} Instead, our requirement that $M_n/\bar w_n \to \infty$ indicates that $M_n$ is adaptive to the setting under consideration.

To guarantee that the DR bootstrap approximation provides asymptotic size control we require that $\gamma_n+\mathcal{A}^{DR}_{n}\delta_n^{DR} \approx 0$, i.e., the distribution DR bootstrap statistic cannot concentrate excessively at the quantile of interest relative to the approximation error $\delta_n^{DR}$. (See Section (ref) for further discussion.) The following corollary of Theorem (ref) provides simple sufficient conditions that covers many data generating process of interest.

Penalized Resampling or PR Bootstrap

The second strategy to construct critical values is based on a bootstrap procedure that takes into account the non-centered feature of the empirical process via penalization. The Penalized Resampling or PR bootstrap statistic is defined as follows

where $\kappa_n \geq 1$ is a user-specified tuning parameter and the $\hat v^{\ast}_{\theta,j}$ is the bootstrap process defined in ((ref)). For $\alpha \in (0,1)$, we define the corresponding conditional critical value as follows:

Similarly to Theorem (ref), the following result establishes the relationship between the original MinMax statistic and the PR bootstrap statistic, and it exploits this result to show the asymptotic validity of the PR bootstrap approximation.

Theorem (ref) bounds the probability distribution function of $T_n(\bar h)$ based on approximation errors and the probability distribution of $R^{PR*}_n$. The core of the proof constructs intermediary processes for which we can apply Theorems (ref) and (ref). Theorem (ref) also provides guideline on how to choose $\kappa_n$. Indeed we need to ensure that $\kappa_n/\bar w_n \to \infty$ and $n^{-1/2}\kappa_n (L_G\sigma^2K_n/\vartheta_n^2)\to 0$. These highlights the role of considering many moment inequalities on the threshold sequence. In fact, the moment inequality literature with a fixed number of moment conditions typically just requires that $\kappa_n\to \infty$ and $n^{-1/2}\kappa_n \to 0$.

Provided that $\gamma_n$ such as $\gamma_n + \delta_{n}^{PR} \mathcal{A}^{PR}_{n} \leq C_2n^{-c_2}$, Theorem (ref) implies that the critical value based on the PR bootstrap approximation provides asymptotic size control. However, note that the anti-concentration condition impacts the choice of threshold sequence $\kappa_n$ as it impact $\delta_{n}^{PR}$. (See Section (ref).) Corollary (ref) stated earlier provided conditions under which we obtain ((ref)).

Minimum Resampling or MR Bootstrap

We can combine the two previous bootstrap approximations by taking the minimum of their bootstrap statistics. Formally, the Minimum Resampling or MR bootstrap statistic is defined as follows: $$ R^{MR\ast}_n = \min\{ R^{DR\ast}_n, R^{PR\ast}_n\}. $$ For $\alpha \in (0,1)$, we define the corresponding critical values as follows $$c_n^{MR*}(\bar h,\alpha) = \mbox{conditional} \ \ (1-\alpha)\mbox{-quantile of} \ \ R_n^{MR\ast} \ \ \mbox{given} \ (W_i)_{i=1}^n.$$

The following result builds on Theorems (ref) and (ref). It establishes a relationship between the original MinMax statistic $T_n(\bar h)$ and the MR bootstrap statistic $R_n^{PR\ast}$, and it shows the asymptotic validity of the MR bootstrap approximation.

The proof of Theorem (ref) shows how the finite sample analysis used here lands itself for the combination of valid procedures. The approach allows us to avoid considering the limit distribution and the issues associated with its existence. Indeed, we do not require the existence of a distributional limit but instead control approximation errors for each $n$.

Self-Normalization based Critical Values

In this section, we discuss inference using critical values based on self-normalized moderate deviation theory. Although the resulting inference is potentially more conservative than the one based on the bootstrap, the method is easier to compute and asymptotically valid for a wider class of data generating processes.

The self-normalized inference was originally proposed by chernozhukov2013testing in the context of inference on the parameter vector $\theta^*$ in the partially identified many moment inequality model. The ideas proposed in this section follow closely the arguments in chernozhukov2013testing.

For $\alpha \in (0,1)$, we define the self-normalized or SN critical value as follows:

where $\Phi$ is the cumulative distribution function of the standard normal distribution.

The proof of Theorem (ref) follows from the proof of Theorem 4.1 in chernozhukov2013testing. Note that the assumptions of Theorem (ref) are only required to hold at the true parameter value $\theta^*$, rather than every $\theta \in \Theta(\bar h)$.

Next, we discuss a two-step version of the SN procedure. Similar to analogous procedures defined in chernozhukov2013testing, our two-step SN procedure takes advantage of restricting attention to relevant parameter values and moment inequalities. However, unlike the two-step procedure defined in chernozhukov2013testing, our two-step SN procedure takes into account the fact that $H_0$ does not completely specify the true parameter vector $\theta^*$.

For every $\theta \in \Theta(\bar h)$, and $\gamma_n \in (0,\alpha/4)$ define the following objects:

For every candidate parameter $\theta \in \Theta(\bar h)$, $\hat J_n^{SN}(\theta)$ is the subset of moment inequalities that are sufficiently close to binding and $c_{n}^{SN,2S}(\theta,\alpha)$ is the two-step SN critical value associated with the parameter value $\theta$. Finally, $\widehat \Theta_n^{SN}$ is a subset of $\Theta(\bar h)$ that is sufficiently close to the identified set.

Theorem (ref) shows that the two-step approach provides asymptotic size control under mild conditions compared to the bootstrap-based methods. For example, we are not making polynomial minorant assumption as in Condition MB(iii). The need for taking the maximum over $\widehat \Theta_n^{SN}$ arises from the fact that $H_0$ does not completely determine the true parameter value. As in Theorem (ref), note that the moment assumptions of Theorem (ref) are only required to hold at the true parameter value $\theta^*$. The proof builds upon Theorem 4.2 in chernozhukov2013testing but the additional selection over $\widehat \Theta_n^{SN}$ requires controlling additional approximation errors.

Anti-concentration and Data-driven estimates

In the high-dimensional settings the statistics of interest might have no limiting distribution and the use of simulation procedures (e.g. bootstrap) are useful to approximate the distribution of the statistics of interest, except for some special pivotal statistics (e.g. BC-SparseQR for quantile regression). However, even if the approximation has a vanishing error, such error can distort coverage. The anti-concentration property ensures that the random variable we aim to simulate does not concentrate too much at relevant points of the distribution. Unless the approximation errors introduced by the coupling procedures are of a smaller order of the rate in which the random variable concentrate, we cannot reliably use the simulated quantiles to approximate the quantile of the original statistics. The anti-concentration properties for the maximum of (correlated and non-centered) Gaussian random variables have been understood recently, see chernozhukov2012comparison,chernozhukov2015noncenteredprocesses. This is a prime example which covers several applications of interest including the construction of simultaneous confidence bands in high-dimensional regression settings, e.g., BCK-LAD,belloni2015uniformly.

In this section we are concerned with the anti-concentration property associated with the MinMax statistic that we propose for the problem of subvector inference in partially identified models with many moment inequalities. In contrast to the Max operator, the MinMax operation is not a convex function of its arguments, and this has substantial implications for our derivations. The anti-concentration bounds based on the maximum density function for the MinMax are qualitatively different than the case of the max discussed in Remark (ref). The following lemma shows this in the case of the statistic $Z = \min_{k\in [N]}\max_{j\in [p]} W_{kj}$ where $W \in \mathbb{R}^{N\times p}$ is a random matrix with i.i.d.\ $N(0,1)$ entries.

Proposition (ref) shows that the concentration properties of the MinMax case is different than the concentration properties of the Max case even for i.i.d. standard normal random variables. In particular, Proposition (ref)(ii) shows that in the high-dimensional case with $p=N$, the probability density function has values of the order $\log^{3/2} p$ which contrasts with the max operator. Indeed, the random variable $\max_{k,j\in[p]} W_{kj}$ has the maximum of its probability density function to be at most of the order of $\log^{1/2} p$. Nonetheless, Proposition (ref)(i) suggests it is possible for the anti-concentration parameter to increase logarithmically with the number of moment inequalities $p$.

We propose a data-driven procedure to estimate the anti-concentration property associated with the statistic of interest. This is applicable not only to the MinMax functional but to other functionals. This is particularly relevant in settings where it is hard to derive analytical bounds or when such bounds might be conservative. Indeed in the case of standardized by non-centered processes it is likely that the statistics of interest (and their anti-concentration property) will depend on a subset of the process associated with points that are more likely to determine the statistics. In that case, even if theoretical bounds for how fast some statistics concentrate are available, they can be conservative. Therefore, the use of an adaptive tool can be desired.

We now note several features of the anti-concentration parameter $\mathcal{A}_{n}$ associated to a bootstrap statistic $R_n^*$, as defined in ((ref)). First, note that the anti-concentration quantity defined in ((ref)) trivially satisfies $$\mathcal{A}_n \leq \max_{t'\in \mathbb{R}} f_{R^*_n}(t')$$ but it could be much smaller. Second, it allows for the distribution of $R^*_n$ to have a point mass and still be nicely bounded as we restrict the size of the smallest set.\footnote{This could be of interest to consider other bootstrap procedures other than the Gaussian multiplier bootstrap considered in this paper.} Third, we can construct upper bounds for $\mathcal{A}_{n}$ and $\mathcal{A}_{n}(W)$ by using a lower bound on $\delta_n$. In most cases of interest, $n^{-1/2}$ constitutes such a lower bound which makes upper bounding $\mathcal{A}_{n}(W)$ feasible.

A sufficient condition for the concentration of the statistics not to impact the size is $\mathcal{A}_{n} \delta_n \to 0$. This ensures that the process $R_n^*$ has suitable anti-concentration not to introduce additional size distortions because of the coupling approximation errors. The next proposition connects the control of distortions via the quantity $\mathcal{A}_{n}(W)$.

Theorem (ref) accounts for the fact that the selection of inequalities and centering are data-driven and therefore it does not follow from a direct application of Markov inequality and Theorems (ref) and (ref).

The next result shows how coupling results can be used to provide upper bounds on the unconditional anti-concentration quantity $\mathcal{A}_n$ based on the conditional anti-concentration quantity $\mathcal{A}_n(W)$ with high probability.

Although this procedure provides a way to conduct inference, theoretical bounds for the anti-concentration play an important role to inform of the requirements on the sample size that are needed for the overall validity of the procedure. Nonetheless, it is conceivable that in more complex applications the anti-concentration can vary substantially with the data generating process. To provide such additional adaptivity is precisely the goal of the proposed data-driven procedure.

Coupling for MinMax Functional

In this section we state results for coupling the MinMax of the sum of independent empirical processes with the MinMax of Gaussian processes. Such results are crucial in our analysis and may be of independent interest. They build upon and complement recent results on the literature for the maximum of processes due to chernozhukov2012Gaussian,chernozhukov2014clt,chernozhukov2014honestbands,chernozhukov2012comparison,chernozhukov2015noncenteredprocesses.

MinMax of Empirical Processes

Let $B:\mathcal{F}\to \mathbb{R}$ be a given functional. For $\eta >0$, let $N_B(\eta)$ denote the cardinality of a minimum $\eta$-cover of $\mathcal{F}$, i.e., $N_B(\eta)$ is the minimal integer $N$ such that there exist $f_1,\ldots,f_N \in \mathcal{F}$ with the property that for every $f\in\mathcal{F}$, $|B(f) - B(f_j)|< \eta$ for some $j\leq N$. In what follows, we let $(X_i)_{i=1}^{n}$ be i.i.d.\ random processes mapping $\mathcal{F}\to \mathbb{R}$, and we define ${\mathbb{G}}_n(f) \equiv n^{-1/2}\sum_{i=1}^n \{X_i(f)-{\rm E}[X_i(f)]\}$. Also, for i.i.d.\ random variables $\xi=(\xi_i)_{i=1}^n$ independent from $(X_i)_{i=1}^{n}$, we let ${\mathbb{G}}_n^\xi(f) \equiv n^{-1/2}\sum_{i=1}^n \xi_i\{X_i(f)-{\mathbb{E}_n}[X_i(f)]\}$. We say a class of functions is VC type with envelope $F$ and constants $\bar A$ and $v$ if for each $\epsilon>0$, its covering number satisfies $N(\mathcal{F},\|\cdot\|_Q,\epsilon\|F\|_Q)\leq (\bar A/\epsilon)^v$, see Dudley99 for definitions.

\\

{\bf Condition A.} {\it (i) There exists a countable subset $\mathcal{G}$ of $\mathcal{F}$ such that for every $f\in \mathcal{F}$ there is a sequence $g_m\in\mathcal{G}$ with $g_m\to f$ pointwise and $B(g_m) \to B(f)$. (ii) The class of functions $\mathcal{F}$ is VC type with measurable envelope $F$ and constants $\bar A\geq e$ and $v\geq 1$. (iii) There exists constants $b\geq \sigma>0$ and $q\in [4,\infty)$ such that $\sup_{f \in \mathcal{F}} {\rm E}[ |f|^k ] \leq \sigma^2 b^{k-2}$, $k=2,3,4$, and $\|F\|_{P,q}\leq b$. }

\\

Conditions A(i)-(iii) correspond to the conditions in chernozhukov2015noncenteredprocesses. These assumptions imply the existence of a centered Gaussian process $G_P$ indexed by $\mathcal{F}$ with uniformly continuous sample paths with respect to the $\mathcal{L}^2(P)$-seminorm $d(f,g)=\{{\rm E}[(f-g)^2]\}^{1/2}$ and covariance operator given ${\rm E}[G_P(f)G_P(g)]={\rm cov}(f(X),g(X))$.

Next we turn to conditions that are specific to the setting we consider.

\\ {\bf Condition B.} {\it (i) The class of functions $\mathcal{F}$ can be written as $\mathcal{F} = \{ f \in \mathcal{F}_\theta, \ \mbox{for some} \ \ \theta \in \Lambda \}$ where $\Lambda \subset \mathbb{R}^{d_\theta}$ satisfies ${\rm diam}(\Lambda) \leq n$. (ii) There is some ${\bar \chi} > 0$ such that for $\gamma_n \in (0,1)$, there are constants $C_{\mathcal{F},\gamma_n}$ and $\eta_{\mathcal{F},\gamma_n}$ such that for any $\epsilon >0$ we have { $$P\left( \sup_{\theta,\tilde\theta\in \Lambda, \|\theta - \tilde \theta\|\leq \epsilon} \left| \sup_{f\in \mathcal{F}_\theta} B(f)+ \mbox{\sf G}(f) - \sup_{\tilde f\in \mathcal{F}_{\tilde \theta}} B(\tilde f)+ \mbox{\sf G}(\tilde f) \right| > C_{\mathcal{F},\gamma_n}\epsilon^{\bar \chi} + \eta_{\mathcal{F},\gamma_n} \right) \leq \gamma_n, $$} for $\mbox{\sf G} = G_P, {\mathbb{G}}_n, {\mathbb{G}}_n^\xi$, and with $\xi=(\xi_i)_{i=1}^n$ are i.i.d.\ $N(0,1)$.}

\\

Condition B(i) postulates the function class is parameterized by two indices. Condition B(ii) is a high level condition that allows us to construct a net for the functionals $\{ \sup_{f\in \mathcal{F}_\theta} B(f)+ {\mathbb{G}}_n(f) : \theta \in \Lambda\}$ and $\{ \sup_{f\in \mathcal{F}_\theta} B(f)+ G_P(f) : \theta \in \Lambda\}$. It is implied by typical equicontinuity assumptions imposed on the whole process $\{ B(f)+ {\mathbb{G}}_n(f) : \theta \in \Lambda, f \in \mathcal{F}_\theta\}$ (see vdV-W for general definitions and for moment inequalities models see andrews2014nonparametric,andrews2013inference,bugni2017inference). However, Condition B(ii) allows for non-Donsker classes as the constant $C_{\mathcal{F},\gamma_n}$ can increase with the sequence of the data generating process. The case ${\bar \chi}=1$ and $\eta_{\mathcal{F},\gamma_n}=0$ covers the important case in which the functions in $\mathcal{F}_\theta$ and the operator $B$ are Lipschitz functions of $\theta$ whose constant can increase with the dimension $d_\theta$. The case ${\bar \chi}=1$ and $\eta_{\mathcal{F},\gamma_n}=o(1)$ allows for discontinuous functions. In Section (ref) below we provide simple conditions that imply Condition B(ii).

Recently new central limit theorems for high-dimensional vectors have been derived in a sequence of papers by chernozhukov2012Gaussian, chernozhukov2014clt, chernozhukov2014honestbands, chernozhukov2012comparison, chernozhukov2015noncenteredprocesses. They have been used to construct Gaussian approximations for the distribution of the maximum of the components as in the case of many moment studied in chernozhukov2013testing. In this work, we build upon these tools and ideas to develop new Gaussian approximation for the MinMax statistic.

In what follows it will be convenient to define

which is an entropy measure associated with the class of function $\mathcal{F}$. The following theorem provides a key result for our analysis.

Theorem (ref) above establishes that we can approximate the value of the MinMax statistic with the MinMax of Gaussian processes. A crucial step in the proof of Theorem (ref) was the development of a new smooth approximation function for the MinMax operator. Finally, in most applications the parameters $\eta_{\mathcal{F},\gamma_n}$ and $\eta$ are dominated by the other terms in $\delta_{n,\eta,\gamma_n}$ (e.g., see Section (ref)).

Our next result shows that the distribution of the MinMax of the Gaussian process is close to the distribution of the MinMax of our bootstrap approximation.

The combination of Theorems (ref) and (ref) provide a constructive way to approximate the distribution of $T$ by simulating $S$ which is associated with the process induced by the multiplier bootstrap procedure.

MinMax of High-dimensional Matrices

In this section we collect key results for coupling the MinMax of the components of high-dimensional matrices. They are of independent interest and apply for non-i.i.d.\ matrices. We first state a coupling result between (arbitrary) random matrices and Gaussian random matrices for the MinMax operator. In what follows let $\mathbb{V}\left(X_i\right)={\rm E}[(X_i-{\rm E}[X_i])(X_i-{\rm E}[X_i])']$.

The following theorem establishing a coupling result regarding two Gaussian random matrices for the MinMax operator. This result is important to establish the validity of the Gaussian multiplier bootstrap which relies on an estimate of the covariance matrix of the original process.

Theorems (ref) and (ref) parallels the results for non-central random vectors for the max operator obtained in chernozhukov2015noncenteredprocesses. Their proofs rely on the use of a novel smooth approximation of the MinMax that is a suitable composition of the logarithm of the sum of the exponentials.