A Simple and Adaptive Confidence Interval when Nuisance Parameters Satisfy an Inequality

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abstractInequalities may appear in many models. They can be as simple as assuming a parameter is nonnegative, possibly a regression coefficient or a treatment effect. This paper focuses on the case that there is only one inequality and proposes a confidence interval that is particularly attractive, called the inequality-imposed confidence interval (IICI). The IICI is simple. It does not require simulations or tuning parameters. The IICI is adaptive. It reduces to the usual confidence interval (calculated by adding and subtracting the standard error times the $1-\alpha/2$ standard normal quantile) when the inequality is sufficiently slack. When the inequality is sufficiently violated, the IICI reduces to an equality-imposed confidence interval (the usual confidence interval for the submodel where the inequality holds with equality). Also, the IICI is uniformly valid and has (weakly) shorter length than the usual confidence interval; it is never longer. The first empirical application considers a linear regression when a coefficient is known to be nonpositive. A second empirical application considers an instrumental variables regression when the endogeneity of a regressor is known to be nonnegative.

{\bf Keywords:} Restricted Parameter Space, Inequality Constraint, Model Selection, Parameter on the Boundary, Uniformly Valid Inference.

Introduction

Inequalities may appear in many models. In a linear regression, an inequality could be the assumption that the coefficient on a control variable is nonnegative. In an instrumental variables (IV) regression, an inequality could be the assumption that the sign of the endogeneity of a regressor is nonnegative. Other models that commonly have inequalities include random coefficient models, models of conditional heteroskedasticity, and structural equation models.\footnote{For random coefficient models with inequalities, see Ketz2018, Ketz2019 and the references therein. For models of conditional heteroskedasticity with inequalities, see FrancqZakoian2007, FrancqZakoian2009, FrancqZakoian2019, Pedersen2017, FrancqThieu2018, and CavaliereNielsenRahbek2017, CNPR2022, CavalierePereraRahbek2024 and the references therein. For structural equation models with inequalities, see Stoeletal2006 and the references therein. Andrews1997, Andrews2002 consider other models with inequalities.} One can think of the inequalities as defining the boundary of the parameter space.

A substantial literature documents and addresses the complications caused by inequalities on nuisance parameters in otherwise regular models.\footnote{See below for references to this literature. By “otherwise regular,” we mean that a consistent and asymptotically normal estimator for the parameters exists. This rules out weakly or partially identified models. The only irregularity is the presence of the inequalities.} These complications arise when the true value of the parameters is at or near the boundary of the inequalities. Many applied researchers ignore the inequalities (or assume the true value of the parameters is in the interior of the parameter space) in order to avoid these complications. This paper focuses on the case that there is only one inequality and proposes a confidence interval (CI) that is particularly attractive. We leave to future research the generalization of the proposed CI to models with multiple inequalities.

A researcher that ignores the inequality calculates the usual CI (UCI) by adding and subtracting the standard error times the $1-\alpha/2$ standard normal quantile. The new CI, called the inequality-imposed CI (IICI), is equal to the UCI when the inequality is estimated to be sufficiently slack. This is an attractive feature of the IICI because it means the researcher does not need to report a different CI in order to be inequality-robust (or boundary-robust). Indeed, the inequality can be estimated after reporting the UCI as a robustness check to verify that the inequality is sufficiently slack.

If the inequality is assumed to hold with equality, then a researcher can calculate an equality-imposed CI (EICI) for the parameter of interest that uses the usual formula with a (weakly) smaller standard error. When the inequality is estimated to be sufficiently violated, then the IICI is equal to the EICI. This is an attractive feature of the IICI because it means that the researcher can ignore the possible slackness of the inequality and use the CI that assumes equality.

In between (when the inequality is estimated to be neither sufficiently slack nor sufficiently violated), the IICI simply interpolates between the UCI and the EICI. One of the endpoints of the IICI is equal to an endpoint of the UCI, while the other endpoint is equal to an endpoint of the EICI; see Figure (ref) for a picture. This is an attractive feature of the IICI because it does not depend on any tuning parameters or simulations. The endpoints of the IICI are a closed-form function of an asymptotically normal estimator and its asymptotic covariance matrix. This also means that the IICI is never longer than the UCI. The length of the IICI shrinks as the inequality becomes more violated until it reaches the length of the EICI. This is an attractive feature of the IICI because it means there is essentially no cost to imposing an inequality if one is available.

The final and most surprising feature of the IICI is that it is valid.\footnote{It is the personal opinion of the author that the IICI is so simple that it must be the case that no one has investigated whether it is valid; everyone has just assumed it is not.} Below, we show that the IICI is finite-sample valid in a normal model with known covariance matrix and asymptotically uniformly valid in an asymptotically normal model with consistently estimated asymptotic covariance matrix. The IICI also has exact coverage when the inequality holds with equality.

The first empirical application revisits a linear regression in BlattmanJamisonSheridan2017, hereafter BJS, that analyzes an experiment designed to evaluate the effects of cognitive behavioral therapy on crime and violence. BJS analyze the effects of two treatments together with their combination. The first treatment is the therapy and the second treatment is an unconditional cash grant. The IICI can be calculated for the coefficients in the linear regression assuming the treatment effect for the cash grant is nonpositive (not increasing crime/violence). While the UCI for the treatment effect for therapy easily includes zero, we find that the IICI for the treatment effect for therapy is shorter and barely includes zero. Also, the IICI for the treatment effect for both treatments combined is shorter and more negative than the UCI.

The second empirical application revisits an IV regression in Shapiro2021, hereafter S21, that analyzes an implicit subsidy given to industries with higher carbon emissions because they face lower tariffs and non-tariff barriers (NTBs). S21 uses an instrument to account for the possibility of measurement error in the measure of carbon emissions for an industry. The sign of the endogeneity of the measure of carbon emissions can be evaluated when the endogeneity is caused by measurement error. (This is a general result that is closely related to the attenuation bias of least squares when there is measurement error.) We find that for NTBs, the IICI is a subset of the UCI that excludes the smallest values of the carbon subsidy. This example demonstrates that the inequality need not be violated in order for the IICI to be shorter than the UCI.

Related Literature. It is well-known that estimators and standard test statistics have nonstandard limit theory when the true value of the parameters is at or near the boundary of the parameter space; see Chernoff1954, Geyer1994, Andrews1999, Andrews2001, MoonSchorfheide2009, and FanShi2023. There are a couple of distinct problems where the complications due to a parameter on the boundary arise.

(1) The first problem is inference on a parameter of interest with inequalities on the nuisance parameters. (This is the problem considered in this paper, albeit with only one inequality.) In this problem, one is not trying to test whether the inequalities hold. Instead, one assumes the inequalities hold and uses them as “overidentifying” restrictions on the parameter of interest. MoonSchorfheide2009 show how this problem covers the general setting of moment inequality models when moment equalities are sufficient for identification. Several papers in the literature cover this problem. Some papers modify the estimator/test statistic so the asymptotic distribution continues to be normal or chi-squared even when the parameter is at or near the boundary; see CalzolariFiorentiniSentana2004, Ketz2018, and FrazierRenault2020. This results in a CI that is asymptotically equivalent to the UCI. Some papers propose modifications to make tests inequality-robust; see MoonSchorfheide2009, Pedersen2017, PedersenRahbek2019, CNPR2022, CavalierePereraRahbek2024, and FanShi2023. Some papers use the inequalities as a source of information and propose hypothesis tests or CIs that have higher power or shorter length; see ElliottMullerWatson2015, Ketz2018, and KetzMcCloskey2023. AndrewsGuggenberger2010ET consider the one-inequality case and show using simulations that subsampling is not inequality-robust and that the two-sided CI with the inequality-imposed t-statistic is. In Section (ref), we compare the IICI to CIs from these papers.\footnote{Some of these papers recommend hypothesis tests instead of CIs. Throughout this paper, when we refer to a CI from a particular paper that focuses on hypothesis testing, it is understood to be the CI that is calculated by collecting the hypotheses that fail to reject.}

(2) A closely related but distinct problem is testing inequalities. This problem tests the hypothesis that the inequalities hold against the unrestricted alternative that at least one inequality does not. There is an established literature in statistics that covers this problem; see SilvapulleSen2005 and the references therein. Most of this literature is based on the (least-favorable) distribution of the likelihood-ratio statistic as a mixture of chi-squared distributions. SelfLiang1987 and KopylevSinha2011 point out that when there are nuisance parameters on the boundary, then the distribution of the likelihood-ratio statistic no longer takes this form. The literature on testing moment inequalities, including Rosen2008, AndrewsSoares2010, AndrewsBarwick2012, RomanoShaikhWolf2014, ChernozhukovChetverikovKato2019, CoxShi2023, AndrewsRothPakes2023, and FangSantosShaikhTorgovitsky2024, while focused on partially identified models, can also be applied to the testing inequalities problem.

(3) A third problem arises when only the parameter of interest is subject to inequalities. That is, there are no nuisance parameters or the nuisance parameters are not subject to the inequalities. Papers that cover this problem include Andrews2000, FrancqZakoian2009, AndrewsGuggenberger2010JoE, MullerNorets2016, CavaliereNielsenRahbek2017, FrancqThieu2018, and Li2024a, Li2024b. This problem is different because, under the null, the slackness of the inequalities is known.

Another related literature consists of papers that discuss inference after model selection and adaptation to submodels. Papers that point out the challenges of inference after model selection and propose solutions include LeebPotscher2005, AndrewsGuggenberger2009ECMA, McCloskey2017, McCloskey2020, and MuralidharanRomeroWuthrich2023. For a class of regression models, ArmstrongKolesar2018 state a general bound on adapting to a submodel that does not apply in this case because it requires the parameter space to be centro-symmetric.

Outline. Section (ref) presents the setup and examples. Section (ref) defines the IICI. Section (ref) states theorems for validity of the IICI. Section (ref) compares the IICI to other CIs in the literature by simulation. Section (ref) presents the empirical applications. Section (ref) concludes. An appendix contains the proofs. Code that implements the IICI in Stata and Matlab is available on the author's website.

Setting and Examples

We consider a general setting with one inequality on a vector of parameters. Let $\theta$ denote a $k$-dimensional vector of parameters. The inequality on $\theta$ is given by $g(\theta)\le 0$, where $g(\cdot)$ is a known function of $\theta$. Let $\widehat\theta$ be an estimator of $\theta$. Below, we assume $\widehat\theta$ is consistent and asymptotically normal at the $\sqrt{n}$ rate, where $n$ is the sample size. We can think of $\widehat\theta$ as an unrestricted estimator; $\widehat\theta$ is not required to satisfy the inequalities. The existence of such estimators is a standard result in many identified parametric and semiparametric models. Such estimators may be defined by linear or IV regression, generalized method of moments (GMM), (quasi-) maximum likelihood, minimum distance, or indirect inference.\footnote{Sometimes, an objective function cannot be defined when the inequality is violated. Even in that case, Ketz2018 shows how to define a consistent and asymptotically normal estimator.} Let $\widehat V$ be a symmetric estimator of the asymptotic variance of $\sqrt{n}(\widehat\theta-\theta)$. Suppose the parameter of interest is the first element of $\theta$, $e'_1\theta$, where $e_1$ denotes the first standard normal basis vector in $\mathbb{R}^k$.

We now give two examples of models that fit this setting.

example[Linear Regression] Consider a linear regression model: \begin{equation} Y_i=\beta'X_i+\epsilon_i, \end{equation} where $Y_i$ is the dependent variable, $\beta$ is $k$-vector of coefficients, $X_i$ is a $k$-vector of regressors, $\epsilon_i$ is an unobserved error, and $i\in\{1,...,n\}$. Suppose $\beta_1=e'_1\beta$ is a scalar parameter of interest. An inequality could be the restriction that one of the other coefficients is nonnegative or nonpositive. In that case, $g(\beta)=\pm e'_j\beta$, where $e_j$ denotes the $j^{\text{th}}$ standard normal basis vector in $\mathbb{R}^k$ for $j\in\{2,...,k\}$. The setting is satisfied if we take $\widehat\beta$ to be the least squares estimator and $\widehat V$ to be a heteroskedasticity-robust estimator of the asymptotic variance of $\widehat\beta$.\footnote{Note that heterogeneous treatment effects are covered. In general, implementing the IICI for $\beta_1$ does not change the interpretation of $\beta_1$ as an average of treatment effects.} The assumption that a coefficient is nonnegative/nonpositive is reasonable in many applications of linear regressions. Here, we describe an example that evaluates multiple treatments. Suppose $X_{1i}=e'_1X_i$ and $X_{2i}=e'_2 X_i$ indicate treatments from a randomized controlled trial with multiple treatments. Then, $\beta_1$ and $\beta_2$ are the average treatment effects. Many treatments can reasonably be assumed to have nonnegative average effects. Indeed, an ethics board may not approve a proposed treatment that has a negative average effect on the participants. If the second treatment has a nonnegative average effect, then $\beta_2\ge 0$ and the IICI can be implemented for $\beta_1$. In Section (ref), we implement the IICI in an empirical application with this setup. We can try to provide some intuition for why and how the IICI differs from the UCI in a simple case with multiple treatments. Suppose there are only two treatments with disjoint treatment groups, $T_1$ and $T_2$, and a control group, $C$. Let $\overline{Y}_S$ denote the average outcome over individuals in $S$ for $S\subset \{1,...,n\}$. Then, the least-squares estimators of $\beta_1$ and $\beta_2$ are $\widehat\beta_1=\overline{Y}_{T_1}-\overline{Y}_{C}$ and $\widehat\beta_2=\overline{Y}_{T_2}-\overline{Y}_C$. If $\beta_2$ is assumed to be nonnegative but $\widehat\beta_2<0$, then there is clearly some error in the estimate of $\beta_2$. $\widehat \beta_2=\overline{Y}_{T_2}-\overline{Y}_C$ must be under-estimating the true value of $\beta_2$. Either $\overline{Y}_{T_2}$ under-estimates the mean of $Y_i$ over $i\in T_2$ or $\overline{Y}_C$ over-estimates the mean of $Y_i$ over $i\in C$ (or both). This error can be partially corrected by imposing the inequality. Let $\ddot\beta_1$ and $\ddot\beta_2$ denote the least-squares estimators for $\beta_1$ and $\beta_2$ that impose the inequality. If $\widehat\beta_2<0$, then $\ddot\beta_2=0$ and $\ddot\beta_1=\overline{Y}_{T_1}-\overline{Y}_{C\cup T_2}$. This implicitly estimates the mean of $Y_i$ over $i\in C$ using the average value of $Y_i$ over $i\in C\cup T_2$. Essentially, the individuals in $T_2$ are counted as part of the control group. Note that $\overline{Y}_{C\cup T_2}<\overline{Y}_{C}$. This means that imposing the inequality increases the estimate of $\beta_1$. This is intuitive because the original estimator used $\overline{Y}_C$, which is likely to over-estimate the mean of $Y_i$ over $i\in C$ when $\widehat\beta_2<0$. This explains why the IICI is centered on a different value than the UCI. Also, $\overline{Y}_{C\cup T_2}$ is estimated with a larger sample, resulting in a smaller standard error. This explains why the IICI is shorter than the UCI.

example[Instrumental Variables Regression] Consider an IV regression: \begin{align} Y_i&=\beta'X_i+\delta'W_i+\epsilon_i\\ X_i&=\Pi_1Z_i+\Pi_2W_i+u_i, \end{align} where $Y_i$ is the dependent variable, $X_i$ is a $k_1$-vector of endogenous regressors, $W_i$ is a $k_2$-vector of exogenous regressors, $\beta$ and $\delta$ are vectors of coefficients, $Z_i$ is an $\ell$-vector of excluded exogenous instruments, $\Pi_1$ and $\Pi_2$ are matrices of coefficients, $\epsilon_i$ and $u_i$ are unobserved errors, and $i\in\{1,...,n\}$. The endogeneity of the endogenous regressors is determined by $\gamma=\mathbb{E}(X_i\epsilon_i)$.\footnote{The regressors in $X_i$ are called endogenous because we do not assume the endogeneity is zero. The true values of the components of $\gamma$ can be zero or nonzero.} Knowledge of the sign of a component of $\gamma$ can be exploited as an inequality on a nuisance parameter.\footnote{A known sign of one of the coefficients, $\beta$ or $\delta$, also counts as an inequality. In this case, the usual IV/GMM estimator and asymptotic variance can be used to calculate the IICI. Relatedly, a known sign of a first-stage coefficient in $\Pi_1$ or $\Pi_2$, as considered in AndrewsArmstrong2017, also counts, although the moments used to estimate $\Pi_1$ and $\Pi_2$ need to be added to the GMM specification. Also note that the model must be identified. This excludes the case of one endogenous regressor and one instrument with a known sign of the first-stage coefficient because the first-stage coefficient must be nonzero for strong identification.} It is common for empirical studies to suggest a sign of $\gamma$ when explaining why an instrument is needed. MoonSchorfheide2009 review empirical studies and report: “In almost all of the papers the authors explicitly stated their beliefs about the sign of the correlation between the endogenous regressor and the error term; yet none of the authors exploited the resulting inequality moment condition in their estimation.” Specifying the sign of $\gamma$ is the first step in signing the omitted variable bias of the least-squares estimator. A special case of this is when $X_i$ includes classical measurement error. Suppose $e'_j X_i$ is measured with error for $j\in\{1,...,k\}$. Then, $e'_j\gamma=-e'_j\beta\sigma^2$, where $\sigma^2$ denotes the variance of the measurement error. Therefore, the sign of $e'_j\gamma$ is the negative of the sign of $e'_j\beta$. In Section (ref), we implement the IICI in an empirical application with this setting. We can estimate $\beta$ and $\delta$ jointly with $\gamma$ by GMM. Suppose $\Pi_1$ has rank $k_1$ so the structural coefficients are strongly identified. Then, $\beta$ and $\delta$ are identified using the moments $\mathbb{E}Z_i(Y_i-\beta'X_i-\delta'W_i)=0$ and $\mathbb{E}W_i(Y_i-\beta'X_i-\delta'W_i)=0$. We identify $\gamma$ by including the moments $\mathbb{E}X_i(Y_i-\beta'X_i-\delta'W_i)-\gamma=0$. The GMM estimator is asymptotically normal with the usual asymptotic variance-covariance matrix. We point out that the GMM estimator for the structural coefficients (and their standard errors) is unchanged by the inclusion of the moments for $\gamma$. Code that demonstrates how to jointly estimate $\beta$, $\delta$, and $\gamma$ in Stata is available on the author's website.

An Inequality-Imposed Confidence Interval

The UCI is defined to be

equation[equation omitted — 105 chars of source]

where $\hat s=n^{-1/2}(e'_1\widehat Ve_1)^{1/2}$ is the standard error of $e'_1\widehat\theta$ and $z_{1-\alpha/2}$ denotes the $1-\alpha/2$ standard normal quantile. The UCI is valid with or without the inequality. The UCI is appealing because it does not require any modification to standard practice. The downside is that it does not use the inequality.

We next define the EICI. Suppose, for simplicity, $g(\theta)$ is linear in $\theta$ with $g(\theta)=a'\theta+b$, where $a$ and $b$ are constants. When the inequality is assumed to hold with equality, the resulting equality-imposed estimator (EIE) is given by

equation[equation omitted — 108 chars of source]

The standard error for the EIE is given by

equation[equation omitted — 141 chars of source]

This is (weakly) smaller than $\hat s$ with equality if $e'_1\widehat V a=0$. The (weakly) smaller standard error represents the fact that imposing the equality on the estimator may reduce the noise in estimating $e'_1\theta$. The EICI is then given by

equation[equation omitted — 115 chars of source]

When $g(\theta)$ is not linear, then the EICI can be defined using ((ref))-((ref)) with $a$ replaced by $\hat a=\frac{d}{d\theta}g(\widehat\theta)$.\footnote{Alternatively, $\ddot\theta$ can be defined to minimize $(\widehat\theta-\theta)'\widehat V^{-1}(\widehat\theta-\theta)$ subject to $g(\theta)=0$. This is asymptotically equivalent.}

The IICI is defined to transition between the UCI, when the inequality is estimated to be sufficiently slack, and the EICI, when the inequality is estimated to be sufficiently violated. Let

equation[equation omitted — 207 chars of source]

be a threshold that determines when to transition between the UCI and the EICI.\footnote{One can show that the formula for $\ddot c$ is continuous in the entries of $\widehat V$ and $a$, so that when $e'_1\widehat Va\rightarrow 0$, $\ddot c\rightarrow 0$; see ((ref)) in the appendix.} Let $IICI=[\ell, u]$, where

align[align omitted — 428 chars of source]

When $e'_1\widehat V a=0$, then $e'_1\widehat\theta=e'_1\ddot\theta$ and $\hat s=\ddot s$, so $IICI=UCI=EICI$.

figure[figure omitted — 3,217 chars of source]

Figure (ref) illustrates the UCI, EICI, and IICI in the case that $\theta$ is two-dimensional, $g(\theta)=e'_2\theta$ is assumed to be nonpositive, and $\widehat V=n\left[

smallmatrix1&0.7\\0.7&1

\right]$. (Scaling $\widehat V$ by $n$ removes the dependence of the CIs on $n$ and, in particular, prevents their length from shrinking to zero at the $n^{-1/2}$ rate.) Figure \ref{Comparison1} depicts the CIs for $e'_1\theta$ as a function of $e'_2\widehat\theta$, which is the estimated slackness/violation of the inequality. The value of $e'_1\widehat\theta$ is normalized to $0$. The CIs are the vertical intervals between the upper and lower bounds. We make several remarks on Figure (ref).

Remarks. (1) In Panel A, the endpoints of the UCI (EICI) are depicted by dashed blue (dash-dotted red) lines. The endpoints of the IICI are depicted by the solid purple lines. When $e'_2\widehat\theta<-\ddot c$, the inequality is estimated to be sufficiently slack, and $IICI=UCI$. When $e'_2\widehat\theta\in[-\ddot c, \ddot c]$, the upper bound of the IICI is equal to the upper bound of the EICI and the lower bound of the IICI is equal to the lower bound of the UCI. When $e'_2\widehat\theta>\ddot c$, the inequality is estimated to be sufficiently violated, and $IICI=EICI$.

(2) In Panel A, the EIE is depicted by the dotted red line. The EIE is negative for values of $e'_2\widehat\theta>0$. This makes sense because $e'_1\widehat\theta$ and $e'_2\widehat\theta$ are positively correlated. When $e'_2\widehat\theta>0$, then $e'_2\theta$ must be over-estimated. This indicates that $e'_1\widehat\theta=0$ also over-estimates $e'_1\theta$, and so the EIE is negative. This explains why the EIE has a negative slope as a function of $e'_2\widehat\theta$.

(3) Panel B depicts the coverage probabilities of the UCI, EICI, and IICI as functions of $e'_2\theta$. The UCI has $95\%$ coverage probability for any value of $e'_2\theta$. The EICI only has $95\%$ coverage probability for $e'_2\theta=0$. The EICI is not designed to cover $e'_1\theta$ when the inequality is slack. The IICI has coverage probability greater than $95\%$ for any value of $e'_2\theta< 0$, with equality when $e'_2\theta=0$ and in the limit as $e'_2\theta\rightarrow -\infty$.

(4) Panel C depicts the average lengths of the UCI, EICI, and IICI as functions of $e'_2\theta$. The UCI always has length $2\times z_{1-\alpha/2}\times \hat s\approx 3.92$ (because $\hat s=1$). The EICI always has length $2\times z_{1-\alpha/2}\times\ddot s\approx 2.78$ (because $\ddot s\approx 0.71$). The average length of the IICI is equal to the length of the UCI in the limit as $e'_2\theta\rightarrow-\infty$ and decreases as a function of $e'_2\theta$. When $e'_2\theta=0$, the average length of the IICI is exactly halfway between the length of the UCI and the length of the EICI.

(5) If $e'_2\widehat\theta$ is very large, then the inequality is estimated to be very violated. In that case, it is possible the IICI is disjoint from the UCI. That is, imposing the inequality gives a completely distinct interval from ignoring the inequality. This is very unlikely if the inequality holds. In this case, the one-sided test for $H_0: e'_2\theta\le 0$ would reject if $e'_2\widehat\theta>z_{1-\alpha}\approx 1.64$. \qed

Validity of the IICI

Most of the attractive properties that have been claimed to hold for the IICI follow directly from the definition. The surprising property of the IICI is its validity. We prove validity in two cases. The first case covers the finite-sample model with normally distributed $\widehat\theta$, known covariance matrix, and a linear inequality. The second case covers asymptotically normal $\widehat\theta$, consistently estimated asymptotic covariance matrix, and a possibly nonlinear inequality.

Finite-Sample Validity

Let $\theta_0\in\mathbb{R}^k$ denote the true value. The following assumption specifies the finite-sample model.

assumption\begin{enumerate}[label=(\alph*)] • $\sqrt{n}(\widehat\theta-\theta_0)\sim N(0,V)$ for some positive definite and symmetric matrix, $V$. • $\widehat V=V$ almost surely. • $g(\theta)$ is linear in $\theta$. Let $g(\theta)=a'\theta+b$ for some $a\in\mathbb{R}^k$ and $b\in\mathbb{R}$. • $e_1e'_1a\neq a$. \end{enumerate}

\noindentRemark. Part (a) specifies the multivariate normal distribution for the estimator. Part (b) specifies that the covariance matrix is known. Part (c) focuses on a linear inequality. Part (d) requires $a$ to not be a scalar multiple of $e_1$. This ensures the inequality is (at least partly) a function of the nuisance parameters. The parameter of interest can have a nonzero coefficient, but it cannot be the only parameter with a nonzero coefficient. This is somewhat more general than the title of this paper suggests.\footnote{The title of this paper suggests the inequality is only on the nuisance parameters, but actually we can allow the inequality to be a function of all the parameters as long as it is not exclusively a function of the parameter of interest.} \qed

Let $P(\cdot)$ denote the probability calculated with respect to the distribution of $\widehat\theta$ specified in Assumption (ref)(a).

theoremUnder Assumption (ref), if $g(\theta_0)\le 0$, then $P(e'_1\theta_0\in IICI)\ge 1-\alpha$. Furthermore, if $g(\theta_0)=0$, then $P(e'_1\theta_0\in IICI)=1-\alpha$.

\noindentRemarks. (1) Theorem (ref) establishes finite-sample validity of the IICI for any value of $\theta_0$ that satisfies the inequality. If $\theta_0$ satisfies the inequality with equality, then the IICI has exact coverage probability.

(2) The proof of Theorem (ref) uses a new argument based on translating the normal distribution; see Lemma (ref) in the appendix. \qed

Asymptotic Uniform Validity

We next show that the IICI is asymptotically uniformly valid in an asymptotically normal model. We state two theorems, one with linear $g(\cdot)$ and one with nonlinear $g(\cdot)$. Let $F\in\mathcal{F}_n$ specify the distribution of $\widehat\theta$ and $\widehat V$, where $\mathcal{F}_n$ is a parameter space that is allowed to depend on $n$.\footnote{It is helpful to allow $\mathcal{F}_n$ to depend on $n$ because, in general, the distribution of $\widehat\theta$ is determined by the distribution of a sample, $\{W_i\}_{i=1}^n$, which depends on $n$.} Also suppose the true value of $\theta$ depends on $F$, which we denote by $\theta_F$. The following assumption specifies the asymptotic model.

assumptionFor every sequence $\{F_n\}_{n=1}^{\infty}$ such that $F_n\in\mathcal{F}_n$, and for every subsequence, $n_m$, there exists a further subsequence, $n_q$, and there exists a positive definite and symmetric matrix, $V$, such that: $\sqrt{n_q}(\widehat\theta-\theta_{F_{n_q}})\rightarrow_d N(0,V)$ and $\widehat V\rightarrow_p V$ as $q\rightarrow\infty$.

\noindentRemark. Assumption (ref) is essentially the requirement that $\widehat\theta$ is asymptotically normal and $\widehat{V}$ is a consistent estimator of the asymptotic covariance matrix. Assumption (ref) is stated using an extra subsequencing step that makes the convergence uniform over $F\in\mathcal{F}_n$.\footnote{Papers on uniform inference commonly require subsequencing conditions; see AndrewsGuggenberger2009ET and AndrewsChengGuggenberger2020.} The extra subsequencing step is innocuous in practice because typical arguments for asymptotic normality, including central limit theorems that allow for dependence and/or triangular arrays, can also be applied along subsequences. \qed

Let $P_{F}(\cdot)$ denote the probability calculated with respect to the distribution of $\widehat\theta$ and $\widehat V$ specified by $F$.

theoremUnder Assumption (ref)(c,d) and Assumption (ref), \[ \liminf_{n\rightarrow\infty}\inf_{\{F\in\mathcal{F}_n: g(\theta_F)\le 0\}}P_{F}(e'_1\theta_F\in IICI)\ge 1-\alpha. \] Furthermore, if $\{F_n\}_{n=1}^\infty$ is a sequence such that $F_n\in\mathcal{F}_n$ and $\sqrt{n}g(\theta_{F_n})\rightarrow 0$, then $P_{F_n}(e'_1\theta_{F_n}\in IICI)\rightarrow1-\alpha$ as $n\rightarrow\infty$.

\noindentRemark. Theorem (ref) establishes asymptotic uniform validity of the IICI over values of $F$ for which $\theta_F$ satisfies the inequality. If $g(\theta_F)$ is sufficiently close to zero, then the IICI has exact asymptotic uniform coverage probability. \qed

Theorem (ref) requires $g(\theta)$ to be linear. We next generalize Theorem (ref) to allow $g(\theta)$ to be nonlinear. The following assumption states restrictions on $g(\theta)$.

assumption\begin{enumerate}[label=(\alph*)] • There exists a set, $\mathcal{U}\subset\mathbb{R}^k$, such that $g:\mathcal{U}\rightarrow\mathbb{R}$ is differentiable with derivative $G: \mathcal{U}\rightarrow\mathbb{R}^k$ and such that $G(\theta)$ is uniformly continuous on $\mathcal{U}$. • For every sequence, $\{F_n\}_{n=1}^{\infty}$, such that $F_n\in\mathcal{F}_n$, there exists an $\epsilon>0$ such that $B(\theta_{F_n},\epsilon)\subset\mathcal{U}$ eventually as $n\rightarrow\infty$, where $B(\theta,\epsilon)=\{x\in\mathbb{R}^k: \|x-\theta\|<\epsilon\}$. • $0<\inf_{\theta\in\mathcal{U}}\|e_1e'_1G(\theta)-G(\theta)\|$ and $\sup_{\theta\in\mathcal{U}}\|G(\theta)\|<\infty$. \end{enumerate}

\noindentRemarks. (1) Part (a) requires $g(\theta)$ to be defined and differentiable on a set, $\mathcal{U}$, where the derivative is uniformly continuous. Part (b) requires $\mathcal{U}$ to contain $\theta_F$ in its interior, where the interiority requirement is uniform over $F\in\mathcal{F}_n$. Part (c) requires $G(\theta)$ to not be a scalar multiple of $e_1$ and $G(\theta)$ to be bounded in a way that is uniform over $\theta\in\mathcal{U}$.

(2) Assumption (ref) is trivially satisfied under Assumption (ref)(c,d) by setting $\mathcal{U}=\mathbb{R}^k$ because, in that case, $G(\theta)=a$ does not depend on $\theta$.

(3) In nonlinear models, it is common to assume the parameter space for $\theta$, say $\Theta$, is compact. In that case, Assumption (ref) is satisfied by taking $\mathcal{U}$ to be a small expansion of $\Theta$, assuming $g(\theta)$ is continuously differentiable on $\mathcal{U}$, and assuming $e_1e'_1G(\theta)\neq G(\theta)$ for all $\theta\in\Theta$. \qed

theoremUnder Assumptions (ref) and (ref), \[ \liminf_{n\rightarrow\infty}\inf_{\{F\in\mathcal{F}_n: g(\theta_F)\le 0\}}P_{F}(e'_1\theta_F\in IICI)\ge 1-\alpha. \] Furthermore, if $\{F_n\}_{n=1}^\infty$ is a sequence such that $F_n\in\mathcal{F}_n$ and $\sqrt{n}g(\theta_{F_n})\rightarrow 0$, then $P_{F_n}(e'_1\theta_{F_n}\in IICI)\rightarrow1-\alpha$ as $n\rightarrow\infty$.

\noindentRemarks. (1) Theorem (ref) establishes asymptotic uniform validity of the IICI for nonlinear $g(\theta)$. As in Theorem (ref), if $g(\theta_F)$ is sufficiently close to zero, then the IICI has exact asymptotic uniform coverage probability.

(2) The proof of Theorem (ref) shows that the event $\{e'_1\theta_F\in IICI\}$ converges to a limiting event based on the finite-sample normal model. The limiting coverage probability can then be evaluated using Theorem (ref). The proof of Theorem (ref) is omitted because it follows from Theorem (ref), using the fact that Assumption (ref)(c,d) implies Assumption (ref) with $\mathcal{U}=\mathbb{R}^k$. \qed

Comparison to Other Confidence Intervals

This section compares the IICI to other CIs available in the literature. We consider (1a) a likelihood-ratio CI (LRCI), calculated by inverting the likelihood-ratio test with a chi-squared critical value, (1b) a size-corrected version of the LRCI (SCLRCI) that simulates a fixed critical value, (2) a conditional likelihood-ratio CI (CLRCI), calculated by inverting the conditional likelihood-ratio (CLR) test proposed in Ketz2018, (3) the CI defined using the inequality-imposed t-statistic (IITCI) considered in AndrewsGuggenberger2010ET, (4) a CI calculated by inverting a test from ElliottMullerWatson2015, called EMWCI, and (5) the simple and short CI (SSCI) recommended by KetzMcCloskey2023.

We find that the only CIs that are valid and never longer than the UCI are the IICI and the IITCI. (The LRCI is slightly invalid and the SCLRCI can be slightly longer than the UCI.) The IITCI is never shorter than the UCI. This is a persuasive reason to recommend the IICI, in addition to the fact that the IICI is simpler and more adaptive than the other CIs.

Throughout this section, we use the example from Figure (ref), where $\theta$ is two-dimensional, $g(\theta)=e'_2\theta$ is assumed to be nonpositive, $\widehat\theta$ is normally distributed, and $\widehat V=n\left[

smallmatrix1&0.7\\0.7&1

\right]$. Figures (ref) and (ref) depict these CIs, together with their simulated coverage probabilities and average lengths as a function of the slackness of the inequality.

figure[figure omitted — 3,517 chars of source]

(1) LRCI/SCLRCI. A natural approach to define a CI is to start with the likelihood-ratio statistic for testing $H_0: e'_1\theta=r, e'_2\theta\le 0$ against the alternative $H_1: e'_1\theta\neq r, e'_2\theta\le 0$, where $r$ is a hypothesized value of $e'_1\theta$. The LRCI is the set of values of $r$ for which the likelihood-ratio test fails to reject using the $1-\alpha$ quantile of the chi-squared distribution with one degree of freedom as the critical value. The LRCI is an attractive CI because it is equal to an upper-level set of the likelihood. The LRCI also reduces to the UCI when the inequality is sufficiently slack.

The problem with the LRCI is that the distribution of the likelihood-ratio statistic is not chi-squared when there is an inequality on the nuisance parameters; the LRCI has not been shown to be valid. Simulations suggest the under-coverage is positive but small. A size-corrected version of the LRCI, SCLRCI, can be defined by taking the critical value to be the $1-\alpha$ quantile of the distribution of the likelihood-ratio statistic when the inequality is binding, calculated with $10^5$ simulations.\footnote{The SCLRCI is related to the second step of the two-step quasi-likelihood-ratio test in FanShi2023. In FanShi2023, the first step constructs a confidence set for a local slackness parameter and the second step estimates quantiles of the asymptotic distribution of the QLR statistic. The SCLRCI is also related to the test in CNPR2022, which uses shrinkage in the bootstrap draws to approximate the quantile of the asymptotic distribution of the likelihood-ratio statistic when the inequality is binding.}

The LRCI and SCLRCI are depicted in Figure (ref) with dotted blue lines and dashed red lines, respectively. They are practically indistinguishable because the size-correction is very small, although the SCLRCI is slightly longer. Compared to the IICI, the LRCI is equal to the IICI for most values of $e'_2\widehat\theta$. It is only when $e'_2\widehat\theta\approx\pm\ddot c$ that they differ. The differences in coverage probability and average lengths are barely visible in Panels B and C.

The IICI can be viewed as a more nuanced size-correction to the LRCI than the SCLRCI. (Since the LRCI is only slightly invalid, only a minor adjustment should be needed.) The SCLRCI uses a different critical value that does not reduce to the UCI when the inequality is sufficiently slack---a blunt way to guarantee validity. It distorts the LRCI for all values of the slackness and ruins the “never longer” property of the LRCI. In contrast, the IICI modifies the LRCI only for values of $e'_2\hat\theta\approx\pm\ddot c$ and preserves the adaptation to the UCI/EICI when the inequality is estimated to be sufficiently slack/violated.

figure[figure omitted — 3,473 chars of source]

(2) CLRCI. Ketz2018 recommends testing $H_0$ by simulating the conditional distribution of the likelihood-ratio statistic given a sufficient statistic for the slackness of the inequality. The CLRCI, implemented with $10^5$ simulations, is depicted in Figure (ref) with green dash-dotted lines. The CLRCI agrees with the UCI when the inequality is estimated to be sufficiently slack. As the inequality becomes less slack, the upper bound decreases until it is equal to $z_{1-\alpha}$. When the inequality is violated, the lower bound on the CLRCI decreases. The CLRCI is usually (weakly) shorter than the UCI, but it can be longer. The CLR test for $H_0$ is essentially transitioning from a two-sided t-test to a one-sided t-test. Looking at Panel B, the coverage probability is always $95\%$, indicating that the CLRCI is similar. Looking at Panel C, the CLRCI has shorter average length than the IICI for some values of $e'_2\theta$ and longer average length for other values.

(3) IITCI. Another CI is based on the inequality-imposed estimator (IIE), which is defined by

equation[equation omitted — 141 chars of source]

The inequality-imposed t-statistic (IIT) is formed using the IIE and the usual standard error ($\hat s$). The IITCI is depicted in Figure (ref) with blue dotted lines. The IITCI is equal to the UCI when the inequality is estimated to be slack. When the inequality is estimated to be violated, the IITCI is a translation of the UCI. Thus, the IITCI is always the same length as the UCI (also visible in Panel C). Panel A also shows how to analytically prove that the IITCI is valid because it contains the IICI. AndrewsGuggenberger2010ET show using simulations that the maximum null rejection probability of the t-test using the IIT is $5\%$. This fact can be seen in Panel B because the coverage is monotonically increasing in $e'_2\theta$, so the minimum coverage is achieved in the limit when the inequality is infinitely slack.

(4) EMWCI. ElliottMullerWatson2015 recommend a general strategy for calculating nearly weighted average power (WAP) optimal tests for a given hypothesis by specifying a WAP criterion, discretizing the null parameter space, and solving numerically for a least-favorable distribution. They demonstrate the general strategy in this setting---testing a hypothesis when a nuisance parameter is subject to an inequality. Figure (ref) depicts the EMWCI with red dashed lines.\footnote{We use the switching version of the EMWCI, where the test is equal to the usual two-sided t-test when the inequality is estimated to be sufficiently slack.}

In Panel A, the EMWCI is approximately the same as the UCI when the inequality is very slack. As the slackness decreases, the upper bound decreases slightly (although not monotonically). As the inequality becomes more violated, the EMWCI approaches the IITCI. In Panel B, the coverage probabilities of the EMWCI and IICI are similar. In Panel C, there is a range of slackness values for which the EMWCI is shorter on average than the IICI, with the IICI shorter outside this range.

The WAP of the IICI is $53.1\%$ with $2.5\times 10^5$ simulations, calculated with the WAP function specified in ElliottMullerWatson2015.\footnote{Technically, this WAP is for the test that rejects when the CI does not include zero. Following ElliottMullerWatson2015, the number of simulations is chosen so the simulation standard error is $0.1\%$.} This is the same as the WAP for the EMWCI reported in ElliottMullerWatson2015 and is within $\epsilon=0.5\%$ of their reported power bound of $53.5\%$. Thus, the IICI can be considered nearly optimal for their WAP function.

(5) SSCI. KetzMcCloskey2023 recommend a simple and short CI (SSCI). The SSCI is calculated by widening the UCI by a small amount. Then, an upper/lower bound makes the CI shorter when the inequality is violated. A constant is added to the bound to ensure the SSCI is valid. The SSCI is designed to trade off a shorter CI when the inequality is violated with a longer CI when the inequality is slack. Thus, the researcher should have a reasonable expectation that the inequality is binding or close to binding before implementing the SSCI. Figure (ref) depicts the SSCI with dash-dotted green lines. The SSCI is longer than the UCI when the inequality is sufficiently slack. When the inequality is violated, then the SSCI can be very short and even empty. In Panel B, the SSCI is conservative when the inequality is sufficiently slack. In Panel C, the average length of the SSCI is longer than the IICI over this range.

Empirical Applications

Cognitive Behavioral Therapy for Crime and Violence

BlattmanJamisonSheridan2017 analyze an experiment designed to evaluate the effects of cognitive behavioral therapy on crime and violence. The experiment selected a sample of poor young men in Liberia that are high risk for crime and violence. The participants were randomized into one or both (or neither) of two treatments. The first treatment was eight weeks of cognitive behavioral therapy designed to formulate a law-abiding identity, encourage forward-looking reasoning and planning, and practice acceptable social behaviors. The second treatment was a cash grant of \$200 (USD, equivalent to about 3 months' wages). Participants were surveyed 2-5 weeks after treatment and again 12-13 months after treatment.

BJS use an index of “antisocial activity” that is calculated from survey questions on crime (drug selling, theft, arrests) and violence (fights, owning weapons, aggressive/hostile behavior, intimate partner abuse). The survey responses were validated using in-depth interviews of a subsample of participants. BJS estimate treatment effects\footnote{Technically, the treatment effects are for the “intent-to-treat” because not everyone completed the therapy sessions, although about two-thirds completed 80% of the sessions.} by regressing the index of antisocial activity on treatment status, together with baseline characteristics and randomization-block fixed effects. The short-term effects are estimated using the surveys 2-5 weeks after treatment and the long-term effects are estimated using the surveys 12-13 months after treatment.

Table (ref) reports estimates of the treatment effects, heteroskedasticity-robust standard errors, and UCIs (c.f. Table 2 in BJS). For the cash-only treatment, the short-term and long-term treatment effects are statistically insignificant. For the therapy-only treatment, the short-term treatment effect is large and statistically significant---a 0.25 standard deviation decrease in antisocial activity---while the long-term treatment effect is diminished and statistically insignificant. For both treatments combined, the short-term and long-term treatment effects are both large and statistically significant. BJS conclude that receiving the cash is equivalent to extending the effect of the therapy.

table[table omitted — 1,773 chars of source]

There are several reasons to think the cash-only treatment effect is nonpositive (does not increase antisocial activity). Survey data suggests participants spend little of the cash on temptation goods, including drugs, alcohol, gambling, or prostitution.\footnote{More generally, EvansPopova2017 conclude that cash transfers to poor individuals decreases expenditures on temptation goods.} This is validated by reported business investments/expenses, which accounts for more than half of the cash grant.\footnote{BlattmanFialaMartinez2014 and HaushoferShapiro2016 find that poor and unemployed individuals in East Africa have a high return to cash, possibly because they are credit-constrained.} In addition, a pilot study found that the treatments were not harmful.

Table (ref) reports the IICIs for the treatment effects assuming the cash-only treatment effect is nonpositive.\footnote{Analogous IICIs can be calculated assuming the therapy-only treatment effect is nonpositive. In this example, those IICIs would be equal to the UCI because the treatment effect is not estimated to be close enough to zero. Combining the assumptions that the cash-only and therapy-only treatment effects are nonpositive is beyond the scope of this paper because it requires more than one inequality.} The length ratio is calculated by dividing the length of the IICI by the length of the UCI. We make the following remarks on Table (ref).

Remarks. (1) The IICI is not calculated for the cash-only treatment. This is because the IICI is only designed for inequalities on the nuisance parameters. A simple option for imposing the inequality on the UCI for the cash-only treatment effect is to intersect the UCI with $(-\infty,0]$, although MullerNorets2016 criticize the possibility that the resulting CI can be unreasonably short or empty.

(2) The IICIs for the short-term effects are equal to the UCIs, demonstrating the adaptation of the IICI. (If an inequality is estimated to be sufficiently slack, then the IICI reduces to the UCI.) The IICIs for the long-term effects differ from the UCIs because the cash-only treatment effect is estimated to be positive. Under the assumption that the true cash-only treatment effect is nonpositive, this positive estimate is noise that can be removed by imposing the inequality.

(3) The IICI for the long-term treatment effect for therapy is about 15% shorter and barely covers zero. The IICI for the long-term treatment effect for both treatments combined is about 17% shorter and more negative than the UCI. These IICIs indicate larger and longer-term treatment effects (less attenuation) of the therapy-only treatment and both treatments combined. These IICIs also suggest less importance for the cash follow-up in prolonging the effect of the therapy. \qed

The Environmental Bias of Trade Policy

Shapiro2021 analyzes the relationship between tariffs and NTBs and the carbon emissions of various industries. S21 investigates whether industries with higher carbon emissions, or “dirty” industries, receive an implicit subsidy from trade policy because they face lower tariffs and NTBs. S21 estimate a large and significant implicit subsidy for dirty industries using an IV regression.

The data includes $n=2{,}021$ industry-country pairs. The dependent variables are tariffs or NTBs, measured as the average tax rate on imports of the output of a given industry into a given country.\footnote{NTBs are barriers to trade that are not tariffs, including price regulations, product standards, and quotas. S21 uses the ad-valorem equivalents of NTBs that are estimated in KeeNicitaOlarraga2009.} Total carbon emissions includes direct carbon emissions, which are carbon emissions used to transform the input goods to the output good, and indirect carbon emissions, which are carbon emissions used to produce the input goods. Total carbon emissions are measured using a global input-output table that records, for each industry, how much of each input is used to produce \$1 of output. The fossil fuel inputs are converted to a measure of tons of carbon emissions. By inverting the input-output table, one can calculate a measure of total carbon emissions that includes direct and indirect emissions.

As discussed in S21, this way of measuring total carbon emissions may include classical measurement error for two reasons. (1) The input-output table itself may include errors in the amount of input used to create the output. (2) There may be industry-specific prices paid for fossil fuels that is not accounted for in this measure. To deal with measurement error in the carbon emissions of an industry, S21 instrument for it using the direct emissions of that industry for the 10 other smallest countries. The smallest countries are more likely to take conditions in the rest of the world as given.

The endogeneity of a regressor measured with error has the opposite sign as the coefficient. S21 discuss several reasons why one might expect the sign of the coefficient to be negative. Three reasons are: (a) The optimal tariff a large country places on imports from a small country is inversely proportional to the supply elasticity. This, combined with the fact that more differentiated industries may have lower elasticities and tend to be cleaner, would imply a negative coefficient. (b) Local politicians may impose lower tariffs or NTBs on dirty industries to try to relocate them to other countries to decrease local air pollution. (c) Firms lobby for lower tariffs on inputs and higher tariffs on outputs, combined with the disorganization of final consumers, implies that downstream industries will have higher tariffs. This, combined with the fact that upstream firms tend to be dirtier, would imply a negative coefficient. S21 ultimately conclude that the third reason best explains the implicit subsidy for dirty industries. (This conclusion overturns anecdotal evidence that dirty firms have outsized political influence.) For these reasons, it is reasonable to assume $\gamma$, the endogeneity of total carbon emissions, is nonnegative.

table[table omitted — 2,250 chars of source]

We can estimate $\gamma$ jointly with the coefficients in the IV regression using GMM by adding a moment, as described in the last paragraph of Example (ref). S21 implement two versions of the IV regression, one with unweighted observations and one with observations weighted by trade flow. Table (ref) reports results from these IV regressions. The first two rows of Table (ref) report the coefficient estimates, together with their standard error, clustered at the industry level. The next three rows compare the UCI and the IICI. The final three rows report the estimates of the inequality and the value of $\ddot c$ used to calculate the IICI. We make three remarks on Table (ref).

Remarks. (1) The first two rows in Table (ref) replicate findings in Tables 2 and 3 in S21. The point estimates indicate a significant implicit subsidy for dirty industries through both tariffs and NTBs. As a point of comparison, the social cost of carbon emissions is usually estimated to be about \$40 per ton; see IWG2016. The implicit subsidies from NTBs alone approximately double this amount.

(2) Rows 3-5 in Table (ref) compare the UCI and the IICI. For tariffs, the IICIs are equal to the UCIs. This again demonstrates the adaptation of the IICIs when the inequality is estimated to be sufficiently slack. For NTBs, the IICIs are shorter than the UCIs and exclude the smallest values of the implicit subsidy. Rows 6-8 in Table (ref) explain why the IICIs are only slightly shorter than the UCIs (1.8% in column 2 and 0.1% in column 4). (The negative of $\widehat\gamma$ is reported in Row 6 in order to satisfy the setting in Section (ref), where the inequality is assumed to be nonpositive.) In all columns, the inequality is not violated, and it is only in columns 2 and 4 that the inequality is close enough to make a difference. The IICI gradually adjusts to the influence of the inequality, and therefore the IICI is only slightly shorter. This example demonstrates that the inequality need not be violated in order for the IICI to be shorter than the UCI. It also demonstrates that the IICI is a subset of the UCI whenever $g(\widehat\theta)\le|\ddot c|$.

(3) In the fourth column, the IICI is essentially equal to the UCI. This is also because $\ddot s$ is very close to $\hat s$, or, equivalently, the estimator of the endogeneity is approximately uncorrelated with the estimator of the coefficient. This demonstrates another adaptation property of the IICI: when $a'\widehat\theta$ is approximately uncorrelated with $e'_1\widehat\theta$, the IICI is approximately equal to the UCI. Even with this being true, unreported calculations show that the 99% IICI for the fourth column excludes zero, while the 99% UCI does not. \qed

Conclusion

This paper proposes a new CI, the IICI, for models with an inequality. The IICI has many attractive features, including being simple, adaptive, and never longer than the UCI. This paper proves finite-sample validity and asymptotic uniform validity and compares the IICI to alternatives available in the literature. This paper also demonstrates the IICI in two empirical applications.