Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
130,378 characters · 30 sections · 76 citation commands
Confidence Sets under Weak Identification: Theory and Practice
Weak instruments can distort statistical inference in instrumental variables (IV) models. When identification is weak, conventional $t$-based confidence intervals may have zero asymptotic coverage probability. Dufour97 shows that any confidence interval that is bounded with probability one, such as the usual interval $[\hat{{\Greekmath 010C}}\pm 1.96\cdot \text{se}(\hat{{\Greekmath 010C}} )] $, has zero confidence level under weak identification. In practical terms, intervals that appear informative may fail to contain the true structural parameter altogether. Reliable inference therefore requires tests that control size regardless of instrument strength, together with confidence sets obtained by inverting such tests.
A central but often overlooked issue is that even when valid tests are available, empirical inference depends on how these tests are inverted. Standard numerical procedures used in applied work can produce incorrect confidence sets. In particular, commonly used grid-search methods may include parameter values that are rejected by the underlying test and exclude values that should belong to the confidence set. As a result, empirical conclusions can depend on the numerical inversion method rather than on the statistical procedure itself.
Consider the linear IV model
where $y_{1i}$ is the outcome, $y_{2i}$ is the endogenous regressor of interest, $z_{i}$ is a vector of instruments, and $x_{i}$ contains exogenous covariates. A leading example is the Euler equation design of Yogo04 , where $y_{1i}$ represents consumption growth, $y_{2i}$ is an asset return or interest rate, and ${\Greekmath 010C}$ corresponds to the elasticity of intertemporal substitution. The instrument vector consists of lagged financial predictors such as interest rates, dividend--price ratios, and inflation. In this environment the instruments forecast returns only weakly, making conventional inference unreliable.
A large literature has developed tests that remain valid under weak identification. In the classical homoskedastic IV model with one endogenous regressor and $k$ instruments, AndersonRubin49 introduce the AR test with a ${\Greekmath 011F}_k^2$ distribution that is robust to weak instruments. Moreira02 shows that this test is optimal in the just-identified case and that additional instruments can improve power. Building on this idea, he demonstrates that several tests, including a score (LM) test, are robust to weak identification. Moreira03 proposes a conditional approach that replaces fixed critical values with conditional quantiles and introduces the conditional likelihood ratio (CLR) test. AndrewsMoreiraStock06 show that the CLR test satisfies natural invariance properties and is nearly optimal. These advances establish valid tests, but they do not resolve a key practical issue: how to reliably invert these tests to obtain confidence sets.
In parallel with the development of these tests, researchers have studied the geometry and computation of the associated confidence sets. Although confidence sets need not be intervals, associated confidence intervals can be defined as the smallest interval that contains the set (convex hull). DufourTaamouti05 highlight the quadratic structure of AR and LM confidence sets and provide conditions for boundedness and projection methods for restricted parameters. Mikusheva10 introduces an algorithm for inverting the CLR test under homoskedasticity. These results show that exact inversion is possible in important special cases, but they do not provide general procedures applicable in the settings most commonly encountered in empirical work.
Most empirical applications allow for heteroskedasticity, autocorrelation, or clustering. Extensions of the AR, LM, and CLR tests to general HAC settings have been developed by StockWright00, AndrewsMoreiraStock04, Kleibergen05, AndrewsMikusheva16, and MoreiraMoreira19, among others. MoreiraRidderSharifvaghefi25 also proposes the conditional integrated likelihood (CIL) test, which has a Bayesian interpretation. These results establish valid tests under weak identification and general error structures, making weak-instrument-robust inference feasible in a wide range of empirical environments.
However, validity of the test does not guarantee reliable inversion. Outside the homoskedastic case, both the test statistics and their critical value functions become complicated functions of the structural parameter. The algebra that allows closed-form inversion in simple settings no longer applies directly, and constructing the confidence set becomes a problem that can materially distort inference if handled poorly.
In practice, researchers typically resort to grid search. This step is often treated as a routine implementation detail, but it can materially affect the resulting inference. Using 158 empirical specifications from five well-known IV applications in macroeconomics, labor, and public finance, we show that these failures are not rare. Grid inversion frequently fails to recover the true confidence set and often produces materially different regions relative to exact inversion. In a substantial fraction of cases, grid methods miss disconnected components or fail to detect sharp features of the confidence set. In others, they distort its qualitative shape, for example by reporting bounded intervals when the true confidence set is unbounded, or by reporting nonempty intervals when the exact procedure yields an empty set. As a result, reported confidence intervals may include values that are rejected by the underlying test or omit values that belong to the confidence set. These discrepancies can be economically large and arise solely from the numerical inversion step rather than from the underlying statistical procedure. In contrast, our methods recover the full confidence set with correct nominal coverage under general HAC errors.
We develop new methods for constructing confidence sets and intervals for weak-identification robust tests. For the AR and LM tests, we exploit the fact that the test statistics are rational polynomial functions of the structural parameter to obtain exact confidence sets by solving for all polynomial roots. For the conditional quasi-likelihood ratio (CQLR) test, we use the statistic's monotonicity and convexity properties to derive an exact inversion algorithm.
For more general conditional tests, including CLR and CIL, we propose a simple approximation procedure with two steps. First, we compactify the parameter space to a bounded interval and evaluate the test at Chebyshev nodes, which allows for uniform control of the approximation error. Second, we approximate the test procedure as a function of the hypothesized parameter by a polynomial whose degree controls numerical accuracy.\footnote{ Chebyshev approximations are widely used in economics. For example, Heckman74 employ Chebyshev--Hermite expansions in structural labor-supply estimation, RennerSchmedders15 use Chebyshev polynomials to transform non-polynomial expected-utility problems into polynomial optimization problems, and TaylorUhlig16 survey projection methods in macroeconomics built on Chebyshev approximations.} Confidence sets are then obtained by solving a polynomial inequality, which allows us to recover all components of the set, including unbounded regions, without relying on a grid.
We compare three numerical approaches: evenly spaced grids commonly used in empirical work, such as those implemented by the weakiv command in Stata; grids based on Chebyshev nodes; and our Chebyshev approximation method. Changing the grid alone improves node placement, but it is not enough to recover the full confidence set reliably. The approximation step is essential.
Across 158 empirical specifications from five well-known IV applications, standard grid procedures frequently fail. For example, the weakiv command can produce qualitatively incorrect inference by misclassifying whether the confidence set is bounded or unbounded. Such failures occur in more than 40% of specifications. Our methods recover confidence sets with correct nominal coverage. Numerical inversion is therefore not a secondary implementation detail, but a central component of valid weak-identification robust inference in practice.
Although the paper focuses on the linear IV model, the same algebraic structure applies to any model with piecewise polynomial or rational moment conditions. Our methodology therefore extends beyond IV to a broader class of econometric models with nonlinear identification features.
The remainder of the paper proceeds as follows. Section (ref) illustrates the empirical importance of reliable inversion. Section (ref) compares grid search with our methods. Section (ref) develops exact and approximate inversion algorithms. Section (ref) presents theoretical details. Section (ref) extends the framework to more general models. Section (ref) concludes.
We illustrate the empirical relevance of our methods using the Euler equation application of Yogo04, a leading example in applied work on the equity premium. This setting highlights why confidence sets based on tests that are robust to weak identification and HAC errors are essential. It also reveals the limitations of standard grid-search procedures, which can miss components of the confidence set and produce intervals that are too wide or even qualitatively incorrect. In some cases, grid methods even fail to detect that the confidence set is empty or unbounded.
The Euler equation implies the linear relationship
where $\Delta c_{t}$ is the log of the consumption at time $t$, $ r_{i,t}=\log (1+R_{i,t})$, $R_{i,t}$ is the gross real return of asset $i$ at time $t$, and ${\Greekmath 0120} $ is the elasticity of intertemporal substitution. Under power utility, it represents the inverse of the relative risk aversion. Up to a linear transformation, Equation (ref) is equivalent to
Yogo04 uses four instruments: the twice-lagged nominal interest rate, inflation, consumption growth, and the log dividend-price ratio.
The Euler equation application of Yogo04 provides a canonical setting where weak identification and heteroskedastic errors arise simultaneously. The instruments have limited predictive power for consumption growth and asset returns, so identification is often weak, while macroeconomic data are heteroskedastic. This combination makes it a particularly demanding environment for inference and a natural setting to evaluate procedures that claim robustness to both features.
Inference procedures differ sharply along these two dimensions. The conventional t ratio interval under homoskedasticity is robust to neither weak identification nor heteroskedasticity. The heteroskedasticity-robust t ratio corrects only the second dimension, but remains invalid under weak identification. In contrast, the AR, LM, CQLR, CLR, and CIL tests are constructed to control size under weak identification. When implemented under homoskedasticity, they are robust to weak instruments, but not to heteroskedasticity. Their heteroskedasticity-robust versions correct both dimensions simultaneously.
Yogo04 reports confidence intervals rather than general confidence sets. To facilitate comparison, we also report confidence intervals. These intervals are obtained by inverting the corresponding tests using our methods. For the AR, LM, and CQLR tests, inversion is exact. For the CLR and CIL tests, which do not admit an exact algebraic characterization of the acceptance region, we compute approximate intervals using the Chebyshev-based procedures developed in this paper. Although the derivations rely on the algebraic structure of the test statistics, the final output is conventional: a confidence interval or confidence set that can be reported and interpreted in the usual way.
Table (ref) reports confidence intervals for the elasticity of intertemporal substitution in Equation (ref), using interest rates as the endogenous regressor, for eleven developed countries: Australia (AUL), Canada (CAN), France (FRA), Germany (GER), Italy (ITA), Japan (JAP), Netherlands (NTH), Sweden (SWE), Switzerland (SWI), United Kingdom (UK), and United States (USA). The countries discussed explicitly in the text are highlighted with gray shading in the table.
In Section (ref) in the appendix, we repeat the analysis using stock returns rather than interest rates as the endogenous variable. Because stock returns are less predictable, weak identification is more severe in these specifications. As a result, weak-IV robust procedures often yield unbounded confidence sets.
Table (ref) shows that allowing for heteroskedasticity can materially alter inference. The key pattern is that robust procedures can substantially change both the location and the shape of the confidence sets, sometimes dramatically.
Germany provides a clear example. The heteroskedasticity-robust t ratio interval shifts from [-1.09, 0.25] to [-1.45, 0.61], indicating a sizable change in uncertainty. The effect is even more pronounced for weak-IV robust procedures. The LM interval changes from [-1.18, 15.91] to [-110.06, 0.34], reflecting a drastic reallocation of mass toward extreme negative values and a sharp contraction on the upper end. Similar, though less extreme, shifts occur for the AR and CQLR intervals. Among conditional procedures, the effects are more nuanced. The CLR interval widens under heteroskedasticity, whereas the CIL interval changes asymmetrically: its lower bound shifts outward while its upper bound contracts. These differences arise solely from the choice of covariance estimator, highlighting that inference can be highly sensitive to how sampling uncertainty is modeled in weakly identified settings.
The contrast between t ratio intervals and weak IV robust intervals is also pronounced. For the United States the heteroskedasticity-robust t ratio interval ranges from -0.09 to 0.20 and is bounded and nonempty. In contrast, the heteroskedasticity-robust AR interval is empty and the LM interval is unbounded. In particular, the t ratio procedure never signals lack of identification, since it always produces a bounded interval.
Even among weak IV robust procedures, behavior differs. For Canada the heteroskedasticity-robust LM interval ranges from -0.85 to 250.88, much wider than the corresponding CLR and CIL intervals. For the United Kingdom the heteroskedasticity-robust CQLR interval ranges from -0.68 to 9.45, again much wider than the CLR and CIL intervals.
We do not take a normative stand on which test researchers should adopt. Each procedure embodies a different tradeoff between robustness, power, and computational complexity. The purpose of this example is to illustrate that, once weak identification and heteroskedastic errors are taken seriously, inference can differ substantially across procedures even in familiar empirical applications. Our contribution is to compute these objects exactly or with controlled approximation error, ensuring that the reported confidence regions faithfully reflect the underlying test.
That said, existing theory provides guidance in over-identified settings. It is well-known that the AR test can be inefficient when there is more than one instrument (Moreira02, Moreira09a). Recent work also documents non-trivial power losses for LM and CQLR procedures in over-identified models (MoreiraRidderSharifvaghefi23). Conditional procedures such as CLR and CIL are designed to address these efficiency concerns while preserving size control under weak identification. For this reason, in empirical applications with multiple instruments, CLR and CIL often provide an attractive balance between robustness and precision. Our methods make their implementation as reliable as that of AR and LM.
In Section 2.2 we compare these intervals with those obtained using grid search in the same application. The eleven country specifications of Yogo04 provide a transparent setting in which we can directly contrast the confidence intervals produced by our exact and controlled approximation methods with those reported by Yogo04 using grid search.
In applied work, confidence sets are often obtained by evaluating a test statistic on a grid over the parameter space and collecting the parameter values that are not rejected. This approach requires two arbitrary choices: a compact interval over which the parameter is searched and the number of grid points used within that interval. In the linear IV model, the parameter space is the entire real line. Any finite grid therefore imposes truncation, and its resolution determines whether narrow components of the confidence set are detected.
Table (ref) compares the confidence intervals reported by Yogo04, which are based on grid inversion, with those obtained using our exact and controlled approximation methods. The comparison is conducted for the same eleven country specifications analyzed in Section 2.1. The countries discussed explicitly in the text are highlighted with gray shading in the table.
At the time Yogo04 was published, heteroskedasticity-robust implementations were available for the AR test, but the LM and CLR procedures were typically implemented under homoskedasticity. For this reason, comparisons involving heteroskedasticity-robust procedures focus on the AR test, while comparisons for LM and CLR are conducted under homoskedasticity to match the specification used by Yogo04.
Several patterns emerge.
First, grid inversion mechanically can enlarge intervals because it includes entire grid cells containing boundary points rather than precisely locating the roots of the test statistic. This is visible for the AR test under heteroskedasticity. For Germany, Yogo04 reports a robust AR interval from -1.95 to 1.63, whereas our exact inversion yields -1.73 to 0.66. Similarly, for Australia under heteroskedasticity, Yogo04 reports -0.17 to 0.30, while we obtain -0.11 to 0.22. A comparable phenomenon appears for the homoskedastic CLR test. For Germany, Yogo04 reports -1.23 to 0.28, whereas we obtain -1.18 to 0.24.
Second, grid search may fail to detect sharp or highly localized features of the confidence interval. Under heteroskedasticity, the AR test for the United Kingdom illustrates this problem. Yogo04 reports an interval from -0.45 to 0.51, whereas our exact inversion yields 0.19 to 0.28. The grid procedure merges disconnected components and includes values that are rejected under exact inversion. A similar issue arises for the homoskedastic LM test in Germany. Yogo04 reports an interval from -1.21 to 0.26, while our inversion reveals sharp behavior of the LM statistic near 15.9 and yields a much wider interval, from -1.18 to 15.91. The coarse grid masks this nonlinearity.
Third, grid search can misrepresent qualitative properties of the confidence set, such as emptiness or unboundedness. For the heteroskedastic-robust AR test in the United States, our exact inversion yields an empty confidence set, whereas Yogo04 reports a nonempty interval from -0.14 to -0.02. In the German homoskedastic LM case discussed above, the grid-based interval fails to reveal the full extent of the acceptance region. These discrepancies do not reflect differences in the underlying tests. They arise solely from numerical inversion.
The Euler equation application is particularly informative because it was one of the earliest and most influential empirical implementations of grid inversion for weak identification robust tests. The example therefore shows that grid-related distortions can arise even in widely studied and carefully implemented designs.
At the same time, a single application cannot determine whether this behavior is exceptional or representative. Section 3 revisits grid methods in a more systematic way, examining 158 empirical specifications spanning macroeconomics, labor, and public finance to assess how often such failures occur in practice.
This section evaluates how the Chebyshev approximation performs in applied settings and compares it to grid-based procedures commonly used in empirical work. We consider three approaches: (i) the evenly spaced grid search implemented in the weakiv package, which reflects standard empirical practice; (ii) a grid based on Chebyshev nodes, which improves node placement but retains a pure grid-search inversion; and (iii) our Chebyshev approximation method, which replaces grid search with a global polynomial approximation.
Whenever exact confidence sets are available, as in the AR, LM, and CQLR tests, we use them as benchmarks. Our goal is not to reassess the theoretical properties of these tests, but to study how closely our approximation reproduces the exact regions and how the two grid-based methods deviate from them in practice. Coverage comparisons across methods are not the focus here. Instead, we study numerical accuracy relative to exact inversion.
For comparability, all grid-based procedures use 501 evaluation points, a choice that reflects common empirical practice. This ensures that differences across methods are driven by the inversion approach rather than by arbitrary choices of grid density. For tests such as CLR and CIL, where exact inversion is not available and is treated elsewhere in the paper, this exercise provides reassurance that the approximation behaves as intended, even though exact confidence sets are not displayed in this section.
We implement the approximation using Chebyshev nodes of the second kind and a 500-degree polynomial. These nodes are chosen to include the endpoints and improve approximation accuracy over the entire domain. After compactifying the parameter space, this ensures that ${\Greekmath 010C} =0$ and the tails corresponding to ${\Greekmath 010C} \rightarrow \pm \infty $ are well represented. This feature is important because weak identification often produces unbounded confidence sets, and the behavior of the statistic in the tails determines whether the region is infinite in each direction.
Maintaining approximate nominal coverage does not require the approximated confidence set to match the exact analytical form point by point. Different procedures can achieve the same coverage probability while producing regions with slightly different shapes. In our empirical designs, such discrepancies are rare and small, and they do not affect coverage. The approximation therefore behaves as a controlled perturbation of the exact procedure. For completeness, we nevertheless quantify how closely the approximation reproduces the exact geometry whenever a benchmark is available.
To quantify numerical accuracy, we use the Hausdorff distance between approximate and exact confidence sets. This metric captures the largest gap between two regions and indicates how much one set must expand to include the other. A small distance therefore means the approximation closely matches the exact region, while a large distance signals a meaningful difference in shape. To make distances comparable across designs and to keep the scale bounded, all figures report the normalized distance $d/(1+d)$, which maps $d \in [0,\infty)$ into the unit interval $[0,1]$. Values close to zero indicate near-perfect agreement with the exact set.
For two sets $A,B\subset \mathbb{R}^{k}$,
The first term measures the distance from points in $A$ to the closest point in $B$. The second does the reverse. Taking the maximum makes the measure symmetric.
We evaluate performance across 158 empirical specifications drawn from five well-known instrumental variables applications spanning macroeconomics, labor, and public finance. Three come from the database compiled by AndrewsStockSun19. AcconciaCorsettiSimonelli14 study the effect of public spending on local economic activity, contributing 20 designs. StephensYang14 estimate returns to schooling using compulsory schooling laws, contributing 30 designs. Young14 studies sectoral employment and productivity using exposure to defense spending, contributing 40 designs. We also include two classic applications widely used in the weak-IV literature. AngristKrueger91 contribute 24 designs based on quarter of birth as an instrument for schooling, and Yogo04 contributes 44 designs estimating the elasticity of intertemporal substitution across countries and instrument sets.
Figure (ref) compares distances design by design using the full confidence sets. Studying the full sets is important because they contain more information than their associated confidence intervals, and differences in shape can reveal numerical errors that are masked once the region is reduced to an interval. Each point corresponds to one empirical specification. In the left panel, the horizontal axis reports the normalized distance $d/(1+d)$ between our Chebyshev approximation and the exact confidence set, while the vertical axis reports the same distance for a Chebyshev grid that uses the same nodes but does not apply the polynomial approximation. Points above the 45-degree line indicate designs for which the approximation is closer to the exact region. The heavy concentration of points near zero on the horizontal axis shows that, with only a few rare exceptions, the approximation is nearly identical to the exact confidence set. By contrast, the Chebyshev grid alone can exhibit large discrepancies. The right panel compares grid procedures. The horizontal axis reports the distance for the Chebyshev grid, while the vertical axis reports the distance for the evenly-spaced grid implemented in the Stata command weakiv. Most points above the 45-degree line show that the Chebyshev grid improves substantially on the standard Stata grid. Although the Chebyshev grid remains dominated by the full approximation, it performs considerably better than evenly-spaced grids.
Confidence sets robust to weak instruments are not always intervals. The full region can reveal identification features and show how the data restrict different parts of the parameter space. In practice, applied researchers often prefer intervals for communication. We therefore also examine convex hulls, the smallest intervals containing each confidence set. Figure (ref) repeats the comparison for convex hulls, which correspond to confidence intervals. Confidence intervals are often the primary object reported in applied work because they summarize uncertainty in a way that is easy to communicate and compare across specifications. Each point again represents a design, and the axes have the same interpretation as in Figure (ref). The pattern is similar: the Chebyshev approximation remains closest to the exact intervals in most designs, while grid methods frequently produce larger deviations.
Figure (ref) summarizes performance using empirical cumulative distribution functions of normalized distances for all three tests. First-order stochastic dominance shows that the approximation delivers smaller numerical errors more often. The Chebyshev grid improves on evenly-spaced grids but remains dominated by the approximation.
The evidence points to clear patterns. The Stata grid method frequently fails due to how the grid is constructed around the two-stage least squares (TSLS) estimator, with the search region typically set to $\pm 2$ standard deviations. When an endpoint of this fixed grid lies inside the confidence set, the procedure incorrectly extends the set to be unbounded, even when the true set is bounded. Conversely, when the true confidence set is unbounded but the grid endpoints fall outside it, the method incorrectly returns a bounded set. These boundary-driven errors explain why more than 40% of the specifications in the empirical applications exhibit infinite Hausdorff distance, corresponding to values equal to one on the x-axis in Figure (ref): the issue is not small numerical inaccuracy, but a systematic misclassification of whether the confidence set is informative.
Furthermore, grid methods depend on arbitrary discretization choices and provide no theoretical bound on approximation error. Replacing evenly-spaced grids with Chebyshev nodes improves performance, but the choice of nodes alone does not eliminate large discrepancies. The Chebyshev approximation adds an additional layer of control and delivers much more reliable recovery of confidence regions. Whenever exact inversion is available, as in the AR, LM, and CQLR tests, it should be used in practice. When exact inversion is not available, as in the CLR and CIL tests, the Chebyshev approximation provides a reliable and practical alternative that accurately recovers confidence sets in our empirical applications and avoids the failures of standard grid methods.
In this section we derive exact confidence sets (CSs) and confidence intervals (CIs) based on the AR, LM, and CQLR tests. We then construct approximate CSs and CIs for general conditional tests, including CLR and CIL. Our goal is to obtain exact confidence regions for the structural parameter ${\Greekmath 010C} $. A confidence region is the set of values ${\Greekmath 010C} _{0}$ for which we do not reject the null hypothesis $H_{0}:{\Greekmath 010C} ={\Greekmath 010C} _{0}$ against the two-sided alternative.
These tests depend on the data through the sample second moments. Recall from Equation ((ref)) that $y_{1i}$ denotes the outcome variable, $ y_{2i}$ the endogenous regressor of interest, $z_i$ the $k\times 1$ vector of instruments, and $x_i$ the $d\times 1$ vector of included exogenous covariates. Inference is based on low-dimensional functions of these observable variables.
as well as the scaled sample moments
The moments in ((ref)) and ((ref)) arise directly from the IV orthogonality condition. Under the null hypothesis ${\Greekmath 010C}={\Greekmath 010C}_0$, the structural residual $y_{1i}-{\Greekmath 010C}_0 y_{2i}-x_i^{\prime }{\Greekmath 010D}_1$ must be uncorrelated with the instruments. The scaled sample moments in ((ref)) therefore collect the empirical covariances between the instruments and the outcome, the endogenous regressor, and the included controls. The second moments in ((ref)) provide the normalization and covariance structure required to partial out the controls and to form quadratic test statistics. Together, these objects contain all information in the sample that is relevant for inference on ${\Greekmath 010C}$.
Inference relies on laws of large numbers for the averages in ((ref) ) and central limit theorems for the statistics in ((ref)), allowing for heteroskedasticity, autocorrelation, or general dependence. Under standard regularity conditions, the statistics in ((ref)), properly centered and scaled, are asymptotically normal with well-defined covariance matrices. We take these asymptotic results as given and focus on expressing the tests in terms of a lower-dimensional sufficient statistic.
The tests depend on ((ref)) and ((ref)) only through a much smaller set of transformed statistics. This reduction removes the nuisance coefficients on the covariates by projecting the instruments onto the space orthogonal to the exogenous regressors and standardizing the result. After these two steps, inference depends only on a low-dimensional statistic.
To describe the reduction precisely, consider the linear IV model ((ref)) written in matrix form
where $y_{1}$ and $y_{2}$ are $n\times 1$ vectors, $Z$ and $X$ are $n\times k $ and $n\times d$ matrices of instruments and exogenous covariates with full column rank, and $u$ and $v_{2}$ are zero mean errors. The reduced form for $Y=[y_{1},y_{2}]$ can be written as
where $a=({\Greekmath 010C},1)^{\prime}$, $\Gamma=[{\Greekmath 010D}_{1},{\Greekmath 010D}_{2}]$, and $ V=[v_{1},v_{2}]=[u+{\Greekmath 010C} v_{2},v_{2}]$.
We partial out the covariates by regressing the instruments on $X$ and working with the residuals. Let
and write $z_{i}^{\perp \prime }$ for the $i$th row of $Z_{\perp }$. The relevant orthonormalized moments are
These moments can be written compactly as
The statistic $R$ summarizes all information in ((ref)) and ((ref)) that is relevant for inference on ${\Greekmath 010C}$. This transformation eliminates dependence on nuisance coefficients and yields a low-dimensional sufficient statistic. Under standard conditions, a central limit theorem implies that $R$ is asymptotically normal with covariance matrix $\Sigma$. A consistent estimator $\widehat{\Sigma}_{n}$ can be constructed using standard variance estimators. See White80 for heteroskedastic data, NeweyWest87 and Andrews91 for heteroskedastic and autocorrelated data, and CameronGelbachMiller11 for clustered data. A general textbook overview is given by Hansen22.
We study confidence sets based on tests that are robust to weak identification: Anderson Rubin (AR), Lagrange multiplier (LM), conditional quasi likelihood ratio (CQLR), conditional likelihood ratio (CLR), and conditional integrated likelihood (CIL). We first develop exact numerical methods for AR, LM, and CQLR. These tests control size at level ${\Greekmath 010B}$ under weak identification and therefore generate confidence sets with exact coverage. The resulting confidence sets can be unbounded with positive probability, an important empirical feature of weak identification. From now on, $1-{\Greekmath 010B}$ denotes the target coverage probability, typically 95 percent. We then propose an approximation method for more general conditional tests, including CLR and CIL, that achieves coverage arbitrarily close to $1-{\Greekmath 010B}$.
The Anderson-Rubin (AR) test, introduced by AndersonRubin49 under homoskedasticity and extended to general GMM settings by StockWright00, is based on the moment condition $\mathbb{E} [Z^{\prime }u]=0$ evaluated under the null hypothesis. Under the null hypothesis, the AR statistic has an asymptotic ${\Greekmath 011F} _{k}^{2}$ pivotal distribution regardless of the strength of the instruments.
For a candidate value ${\Greekmath 010C} _{0}$, the AR statistic is
where $b_{0}=(1,-{\Greekmath 010C} _{0})^{\prime }$. This quadratic form corresponds to the sample moment
that is, the structural residual evaluated at ${\Greekmath 010C} _{0}$. The covariance matrix in ((ref)) is estimated at the same parameter value at which the moment in ((ref)) is evaluated, so the weighting matrix depends on ${\Greekmath 010C} _{0}$. The criterion is therefore continuously updating.
MoreiraNeweySharifvaghefi24 apply the Sherman-Morrison formula to show that the matrix
is a ratio of matrix valued polynomials in ${\Greekmath 010C} _{0}$. The numerator is a degree $(2k-2)$ matrix polynomial and the denominator is a degree $2k$ scalar polynomial. Because $Rb_{0}$ is linear in ${\Greekmath 010C} _{0}$, the AR statistic itself can be written as a ratio of degree $2k$ polynomials,
Therefore inversion of the AR test reduces to solving a polynomial inequality. This algebraic structure is the key computational simplification. Without loss of generality we normalize ${\Greekmath 010E} _{0}=1$. The coefficients of these polynomials are obtained numerically by evaluating the statistic at sufficiently many values of ${\Greekmath 010C} _{0}$ and solving a linear system.
Let $c_{{\Greekmath 010B} }(k)$ denote the $(1-{\Greekmath 010B} )$ quantile of the ${\Greekmath 011F} _{k}^{2}$ distribution. The AR confidence set is
The boundary of the confidence set is obtained by solving the degree 2k polynomial equation
All roots can be computed using standard numerical routines. Figure (ref) illustrates a typical realization of the AR statistic for $ k=10$. After computing all boundary points, we evaluate the statistic at midpoints of the induced intervals to determine exactly which intervals belong to the confidence set.
While Figure (ref) appears to show only two confidence intervals, the test statistic actually fluctuates sharply in the middle. Because our method is exact, it identifies a third, very small interval that would typically be missed by a standard grid search or simple visual inspection. Figure (ref) zooms in on this area to reveal this hidden component, with all numerical bounds detailed in Table (ref).
The LM statistic can be interpreted as a score test. Intuitively, it measures how sensitive the likelihood of the statistic $R$ is to small deviations from the null value ${\Greekmath 010C}_0$. The statistic is obtained from the derivative of the Gaussian log likelihood of $R$ with respect to ${\Greekmath 010C}$, evaluated at ${\Greekmath 010C}_0$, and normalized by an estimate of its variance.
AndrewsMoreiraStock04 and Kleibergen05 show that the LM statistic asymptotically has a pivotal ${\Greekmath 011F} _{1}^{2}$ distribution under the null hypothesis, even under weak identification. Inverting the LM test therefore yields valid confidence sets based on the same sufficient statistic $R$ used in the AR case.
Formally, the LM statistic at a candidate value ${\Greekmath 010C} _{0}$ is
where the numerator
is the profile score of the Gaussian log likelihood for $R$ evaluated at $ {\Greekmath 010C} _{0}$, and the denominator
is an estimator of the asymptotic variance of that score.
Although the expression is algebraically more involved than in the AR case, the computational structure is the same. The key computational point is that, like the AR statistic, the LM statistic is a rational function of $ {\Greekmath 010C} _{0}$. The Sherman Morrison formula implies that all matrix terms above can be written as ratios of matrix polynomials in ${\Greekmath 010C} _{0}$, and all linear forms in $\text{vec}(R)$ depend linearly on ${\Greekmath 010C} _{0}$. As a result, the LM statistic itself can be written as a ratio of finite degree polynomials in ${\Greekmath 010C} _{0}$. The maximal degree is $8k-4$. Finite degree implies a finite number of real boundary points, so the confidence set can be recovered exactly by enumerating all roots.
Exactly as in the AR case, this algebraic structure reduces inversion to solving a polynomial inequality in ${\Greekmath 010C}_0$. The coefficients of these polynomials are obtained numerically by evaluating the statistic at sufficiently many values of ${\Greekmath 010C}_0$ and solving a linear system. No symbolic manipulation is required.
Let $c_{\Greekmath 010B}(1)$ denote the $(1-{\Greekmath 010B})$ quantile of the ${\Greekmath 011F}^2_1$ distribution. The LM confidence set is
The boundary of this set is obtained by solving the polynomial equation
All real roots of this polynomial are computed numerically. We then evaluate the statistic at midpoints between consecutive roots to determine which intervals belong to the confidence set.
Figure (ref) illustrates a typical realization for $k=10$. The LM statistic can exhibit highly non-monotonic behavior in parts of the parameter space, generating many disjoint components. Table (ref) shows that the confidence set contains arbitrarily small intervals near zero and also a large interval far from the true value ${\Greekmath 010C} =0$. Any grid based inversion with a finite resolution would necessarily miss some of these components, especially the arbitrarily small ones. Our method recovers all of them by construction.
AndrewsMoreiraStock04 and Kleibergen05 adapt the likelihood ratio statistic from the homoskedastic linear IV model to settings with HAC errors and to the general GMM framework, respectively. The QLR statistic specialized for the HAC-IV model becomes
where $AR({\Greekmath 010C} _{0})$ is defined in (ref), $LM({\Greekmath 010C} _{0})$ is defined in (ref), and $r({\Greekmath 010C} _{0})$ is a rank statistic.
Rank statistics depend on a weighted orthogonalization of the sample Jacobian of the moment conditions in such a way that the orthogonalization creates a statistic asymptotically independent of the sample moments (see AndrewsGuggenberger17). Intuitively, the rank statistic isolates the information provided by the instruments about the endogenous regressor, netting out the effect of the structural parameter being tested. These rank statistics are suitable for testing rank conditions. In the linear IV setting, the rank restriction can be written as $\text{rank}\,\mathbb{E} (z_{i}y_{2i})=0$, which is equivalent to testing ${\Greekmath 0119} =0$. Different rank statistics generate different CQLR tests.
One particular choice of rank statistic is
where $a_{0}=({\Greekmath 010C} _{0},1)^{\prime }$. Under Gaussian $R$, this rank statistic is related to an estimator of ${\Greekmath 0119} $ under the null ${\Greekmath 010C} ={\Greekmath 010C} _{0}$. This rank statistic has an important property on which we rely to develop our method: it is a ratio of polynomials in ${\Greekmath 010C} _{0}$. This algebraic structure is central because it allows us to map sets in the rank domain back into the parameter domain by solving polynomial equations. We proceed using (ref), but the method generalizes to any rational rank statistic.
Unlike the AR and LM statistics, the likelihood ratio (LR), quasi-likelihood ratio (QLR), and integrated likelihood (IL) statistics are not pivotal, so Moreira03 proposes replacing the fixed chi-square critical value by a conditional critical value. For the CQLR test, Moreira03 and Kleibergen05 show that the conditional critical values depend on the data only through the rank statistic. If we denote by ${\Greekmath 0114} _{{\Greekmath 010B} }(r({\Greekmath 010C} _{0}))$ the critical value function (CVF) when we observe $r({\Greekmath 010C} _{0})$, the confidence set is given by the solution of
In this case, we cannot solve for the boundary points of the confidence set by solving a single polynomial equation. Because the critical value varies with the data, we cannot use a flat horizontal threshold as in the pivotal chi-square case. The difficulty in inverting conditional tests is that their CVFs do not admit a simple algebraic representation in ${\Greekmath 010C}_0$.
To motivate our algorithm, it is useful to recall the structure under homoskedasticity. In this case, $\widehat{\Sigma}_n$ has a particularly simple form: it is built from a $2\times 2$ matrix $\widehat{\Omega}_n$ that captures the covariance of the reduced-form errors, repeated in the same way across all instruments. The QLR statistic simplifies to a function of the rank statistic,
where ${\Greekmath 0115} _{\max }$ is the largest eigenvalue of $\widehat{\Omega } _{n}^{-1/2}R^{\prime }R\widehat{\Omega }_{n}^{-1/2}$ and depends only on the data, while $r({\Greekmath 010C} _{0})$ is a ratio of quadratic polynomials in ${\Greekmath 010C} _{0}$.
Mikusheva10 exploits this simplification to construct a threshold method that exactly recovers the confidence set. The key idea is to view the inequality defining the confidence set as an inequality between two functions of the rank statistic, rather than as a direct inequality in $ {\Greekmath 010C}_0$. The reason is that the behavior of the CVF as a function of $ {\Greekmath 010C}_0$ is complicated, while both the statistic and the CVF are well behaved as functions of $r$: $QLR(r)$ is linear, and ${\Greekmath 0114}_{\Greekmath 010B}(r)$ is strictly decreasing (Moreira03 and Mikusheva10) and strictly convex (Figure (ref) illustrates these properties for $ k=10 $, and Appendix Section (ref) extends the analysis to other values of $k$.). The algorithm becomes: (i) solve for the values of $r$ satisfying ${\Greekmath 0115}_{\max}-r\leq {\Greekmath 0114}_{\Greekmath 010B}(r)$; (ii) recover the values of ${\Greekmath 010C}_0$ using the rational structure of $r({\Greekmath 010C}_0)$. Under homoskedasticity, the function ${\Greekmath 0114}_{\Greekmath 010B}(r)+r$ is strictly increasing. Therefore, the confidence set can be characterized by $r({\Greekmath 010C}_0)\geq r^*$, where $r^*$ solves $r^*+{\Greekmath 0114}_{\Greekmath 010B}(r^*)={\Greekmath 0115}_{\max}$. The threshold $ r^*$ (if it exists) can be found by bisection, so inversion reduces to a simple intersection problem.\footnote{ Practitioners need to be careful, because the value $r^*$ does not always exist. Moreira03 shows that the CVF is bounded above by $ q_{{\Greekmath 010B}}(k)$, which implies that a threshold $r^*$ exists if and only if $ {\Greekmath 0115}_{\max}\geq q_{{\Greekmath 010B}}(k)$. Otherwise, the confidence set is the whole real line. Other possible shapes for the confidence set are $ (-\infty,x_{1}]\cup[x_{2},+\infty)$ and $[x_{1},x_{2}]$.}
This approach cannot be directly generalized to HAC settings. Under HAC errors, the QLR statistic does not admit a global linear representation in $ r({\Greekmath 010C}_0)$, and the geometric symmetry of the homoskedastic case breaks down. We overcome this difficulty with a piecewise approach. We split the parameter space into intervals on which $r({\Greekmath 010C}_0)$ is a bijection. Within each such interval, the QLR statistic becomes an implicit function of the rank statistic. We then solve for the values of $r$ satisfying the confidence set inequality $QLR(r)\leq {\Greekmath 0114}_{\Greekmath 010B}(r)$, obtaining a finite union of intervals for $r$. Finally, for each valid interval for $r$, we map back to ${\Greekmath 010C}_0$ using the rational structure of $r({\Greekmath 010C}_0)$.
A remaining difficulty is that solving $QLR(r)\leq {\Greekmath 0114}_{\Greekmath 010B}(r)$ for $r$ can be nontrivial when errors are not homoskedastic. We can find all roots of the difference between two functions if we know the points at which they change their monotonicity or curvature. Mapping out these shape changes allows us to bound the functions and locate all intersections without missing hidden dips or disconnected components. This is another reason for working in the rank domain: although we do not know much about the composition ${\Greekmath 0114}_{\Greekmath 010B}(r({\Greekmath 010C}))$, we do know that ${\Greekmath 0114}_{\Greekmath 010B}(r)$ is strictly decreasing and strictly convex. The homoskedastic case can be viewed as a highly symmetric benchmark in which the QLR is globally linear in the rank statistic. Our method extends the reliability of that case to HAC settings by handling the nonlinearities piecewise.
To implement exact inversion under HAC errors, we must address two difficulties: (i) the mapping $r({\Greekmath 010C}_0)$ is only piecewise invertible in $ {\Greekmath 010C}_0$, and (ii) the QLR statistic may change monotonicity or curvature within each piece. Exact inversion therefore proceeds by first decomposing the parameter space into regions where the geometry is well-behaved and then locating all intersections in the rank domain before mapping back to $ {\Greekmath 010C}_0 $. Formally, the procedure consists of four steps:
The appendix provides a complete algorithmic implementation of these steps, including proofs that all boundary points are recovered and no components of the confidence set are missed.
We now extend our analysis to general conditional tests under HAC errors. Examples include the Conditional Likelihood Ratio (CLR) test and the Conditional Integrated Likelihood (CIL) test. These procedures often deliver substantial power gains relative to AR in over-identified settings, but they are considerably more difficult to invert. Our goal in this section is to provide a general, numerically stable method for constructing reliable confidence sets based on such tests.
\paragraph{The CLR Test.}
The likelihood ratio statistic underlying the CLR test is
where $r({\Greekmath 010C})$ is the rank statistic defined in ((ref)).
Under homoskedasticity, the CQLR and the CLR are numerically identical. Under HAC errors, however, the CLR allows for a more flexible treatment of the covariance structure. AndrewsMikusheva16 develop the CLR test for general nonlinear GMM settings. In the linear IV model, MoreiraMoreira19 analyze the CLR but do not provide a complete computational implementation, as the statistic requires solving a supremum over the parameter space at each evaluation point. This nested optimization makes direct inversion numerically demanding. MoreiraNeweySharifvaghefi24 address this difficulty by exploiting the algebraic structure of the continuously updating GMM criterion to compute the CU-GMM estimator.
The intuition behind the CLR test is straightforward. The AR test fixes the null value ${\Greekmath 010C}_0$ and evaluates whether the instruments are consistent with that value. In contrast, the CLR test compares the null to the most favorable alternative in the entire parameter space. It asks how much larger the rank statistic can become if we are allowed to choose the value of $ {\Greekmath 010C} $ that best aligns the reduced form with the structural equation. If the null is true, the rank at ${\Greekmath 010C}_0$ should already be near its maximum and the difference in ((ref)) will be small. If the null is false, there exists some ${\Greekmath 010C}$ that fits the reduced-form evidence better, and the supremum will substantially exceed $r({\Greekmath 010C}_0)$. By benchmarking the null against the strongest possible alternative, the CLR typically improves power in over-identified designs.
\paragraph{The CIL Test.}
Another robust conditional test is the CIL test proposed by MoreiraRidderSharifvaghefi25. They exploit symmetries in the linear IV model with HAC errors and construct a test based on integrating out uncertainty via the integrated likelihood statistic
where $a=({\Greekmath 010C} ,1)^{\prime }$.
The intuition behind the CIL differs from that of the CLR. Instead of comparing the null to a single best-fitting alternative, the CIL aggregates evidence across all structural values. Each ${\Greekmath 010C}$ contributes according to how well it aligns with the rank statistic. Alternatives that generate larger rank statistics receive greater weight through the exponential term in ((ref)). This integration smooths local irregularities and efficiently combines information from multiple instruments. When the model is just-identified ($k=1$), the AR test already performs well and little is gained from aggregation. When the model is over-identified ($k>1$), however, the CIL can substantially outperform AR because it systematically pools identifying information across instruments.
\paragraph{Why Inversion Becomes Difficult.}
For the CQLR test, the critical value function depends on the null only through the one-dimensional rank statistic $r({\Greekmath 010C} _{0})$. In contrast, the CVFs for the CLR and CIL tests are much more complex. They may depend on the entire vector
and directly on ${\Greekmath 010C} _{0}$ itself. Consequently, inversion no longer reduces to the intersection of two well-behaved one-dimensional curves. Neither the geometric threshold argument of Mikusheva10 nor the piecewise algebraic method developed for CQLR extends directly to this multidimensional setting.
We therefore adopt a different strategy.
\paragraph{General Approximation Framework.}
Let ${\Greekmath 0127}({\Greekmath 010C}_0)$ denote a general conditional test statistic and $ c_{{\Greekmath 010B}}({\Greekmath 010C}_0)$ its associated critical value function. The confidence set is defined by
Our strategy is to approximate both sides of ((ref)) uniformly by functions that admit exact numerical root finding, such as polynomials. The main obstacle is that the domain of ${\Greekmath 010C}_0$ is the entire real line. Polynomial approximation methods provide reliable uniform approximations only on compact sets. Direct approximation over $ (-\infty,+\infty)$ leads to instability in the tails.
We resolve this issue by exploiting the known asymptotic behavior of the test statistics and reparametrizing the parameter space through a smooth bijection
Precise conditions on allowable compactification functions are provided in Section (ref). Because ${\Greekmath 010C} _{0}=\pm \infty $ if and only if ${\Greekmath 0112} _{0}=\pm 1$, unbounded confidence sets can be detected by checking whether
where $\overline{{\Greekmath 0127} }$ and $\overline{c}_{{\Greekmath 010B} }$ denote the reparametrized functions. In the weak-instrument environment, this feature is essential: unbounded confidence sets occur with positive probability and must be reported accurately.
Once mapped to the compact domain $[-1,1]$, we approximate both $\overline{ {\Greekmath 0127}}({\Greekmath 0112}_0)$ and $\overline{c}_{{\Greekmath 010B}}({\Greekmath 0112}_0)$ using Chebyshev polynomial interpolation. Chebyshev nodes place greater weight near the boundaries, preventing oscillations and ensuring stable approximation even in regions where the statistic changes rapidly. Unlike evenly spaced grids, Chebyshev interpolation admits explicit uniform error bounds that shrink as the approximation degree increases.
After approximating both sides of ((ref)) by polynomials, we solve the resulting polynomial inequality exactly to recover all boundary points in ${\Greekmath 0112}_0$. We then map these points back to ${\Greekmath 010C}_0$ using the inverse reparametrization ${\Greekmath 010C}_0={\Greekmath 0112}^{-1}({\Greekmath 0112}_0)$.
This procedure has three key advantages over grid search. First, it provides explicit control over numerical error through the degree of approximation. Second, it guarantees detection of unbounded confidence sets. Third, it ensures that disconnected or narrow components are not missed due to discretization.
Although we emphasize Chebyshev interpolation for stability and simplicity, any uniform approximation method based on functions with exact root-finding properties can be embedded in this framework. The central idea is to replace arbitrary discretization with controlled uniform approximation. As the degree of approximation increases, the coverage error induced by numerical inversion converges to zero.
In this section we formalize the statistical structure underlying the inversion methods developed in Section (ref). Our goal is to characterize a pair of statistics that jointly govern weak-identification robust inference under general HAC errors.
The first statistic, denoted $S({\Greekmath 010C} _{0})$, is the component of the continuously updating GMM objective that evaluates the structural moment condition at the null. The second statistic, denoted $T({\Greekmath 010C} _{0})$, is a sufficient and complete statistic for the first-stage coefficients under the null. Importantly, $T({\Greekmath 010C} _{0})$ is asymptotically independent of $S({\Greekmath 010C} _{0})$ under standard regularity conditions. This separation allows us to construct conditional tests whose critical value functions depend only on $ T({\Greekmath 010C} _{0})$.
We address two theoretical issues:
These results provide the formal justification for the numerical procedures introduced earlier and establish that computational tractability does not come at the expense of statistical validity.
To isolate the information relevant for testing the structural parameter $ {\Greekmath 010C} $ from the nuisance parameter ${\Greekmath 0119} $ (the strength of the instruments), it is mathematically convenient to construct a one-to-one transformation between the reduced-form statistic $R$ and a pair of independent random vectors. All results in this section hold exactly under normal errors with known $\Sigma $. More generally, when $\Sigma $ is unknown and estimated, the same arguments go through using standard asymptotic approximations. Define
where $a_{0}=({\Greekmath 010C} _{0},1)^{\prime }$ and $b_{0}=(1,-{\Greekmath 010C} _{0})^{\prime }$.
The statistic $S^{\prime }S$ coincides exactly with the Anderson--Rubin statistic, while $T$ measures the strength of identification. Following Moreira02, Moreira09a, the pair $(S,T)$ satisfies three key properties:
The third property is central for conditional inference. Because $T$ is complete and sufficient for the nuisance parameter, any similar test must also be conditionally similar given $T$. Conditioning on $T$ therefore allows exact size control without sacrificing power. This insight underlies the conditional approach developed by Moreira03.
To explicitly characterize conditional rejection probabilities, we use the inverse of the transformation in (ref), derived by MoreiraMoreira19.
Lemma (ref) allows us to express any weak-IV robust test statistic as a function of $({\Greekmath 010C}_0,S,T)$. We therefore write $ {\Greekmath 0127}({\Greekmath 010C}_0,S,T)$ interchangeably with ${\Greekmath 0127}({\Greekmath 010C}_0,R)$.
Under the null hypothesis and conditional on $T$, we have
Hence the conditional distribution of any test statistic depends only on the known Gaussian distribution of $S$.
Let $G(y,{\Greekmath 010C}_0,T)$ denote the conditional cumulative distribution function of ${\Greekmath 0127}({\Greekmath 010C}_0,S,T)$ given $T$ when ${\Greekmath 010C}={\Greekmath 010C}_0$. The conditional critical value function $c_{\Greekmath 010B}({\Greekmath 010C}_0,T)$ is defined by
This threshold leaves probability ${\Greekmath 010B}$ in the rejection region, conditional on the observed identification strength.
Using the Gaussian distribution of $S$, the conditional CDF can be written explicitly as
where ${\Greekmath 011E}_k$ is the $k$-dimensional standard normal density.
For the CQLR test, the conditional distribution depends on $({\Greekmath 010C}_0,T)$ only through the scalar statistic $T^{\prime }T$. AndrewsMoreiraStock07 provide an explicit representation of this distribution, which allows computation of $c_{\Greekmath 010B}$ by bisection between the $(1-{\Greekmath 010B})$ quantiles of ${\Greekmath 011F}^2_1$ and ${\Greekmath 011F}^2_k$.
For more complex conditional tests such as CLR and CIL, analytical evaluation of (ref) is generally infeasible. In practice, we approximate the conditional CDF by simulation: draw $S_j \sim N(0,I_k)$ independently, compute ${\Greekmath 0127}({\Greekmath 010C}_0,S_j,T)$ for each draw, and estimate $ c_{\Greekmath 010B}({\Greekmath 010C}_0,T)$ as the empirical $(1-{\Greekmath 010B})$ quantile.
For general conditional tests where exact algebraic inversion is impossible, we must uniformly approximate the inequality that defines the confidence set and then solve the resulting polynomial inequality to obtain all distinct interval components of the approximated set.
Traditionally, the test statistic and its CVF are treated as direct functions of ${\Greekmath 010C} _{0}$. However, because ${\Greekmath 010C} _{0}\in (-\infty ,+\infty )$, the parameter space is not compact. Uniform polynomial approximation over an unbounded domain is numerically unstable and can lead to uncontrolled oscillations in the tails. To resolve this problem permanently, we introduce a geometric reparametrization that compactifies the parameter space into a bounded, closed interval.
\paragraph{A Geometric Compactification.}
To motivate the compactification, consider the rank statistic $r({\Greekmath 010C} _{0})$ . It can be written as
where
This normalization suggests a trigonometric substitution. Define
where $pi=3.1415926...$ This bijection maps the entire real line onto the bounded interval $(-1,1)$. Under this transformation,
The extreme values ${\Greekmath 010C}_0 = \pm\infty$ correspond exactly to ${\Greekmath 0112}_0 = \pm 1$, so the compactified parameter space becomes the closed interval $ [-1,1]$.
Let $\overline r({\Greekmath 0112}_0)$ denote the rank statistic written as a function of the compactified parameter.
\paragraph{Regularity of the Test Statistic.}
To ensure well-defined limits at the boundary, we impose the following mild regularity condition. This condition is satisfied by the LR and IL test statistics.
Under Assumption (ref), the statistic does not diverge in the tails but instead converges to finite limits. We therefore define the compactified test statistic
This produces a continuous function on the compact support $[-1,1]$. The associated CVF becomes $\overline c_{\Greekmath 010B}({\Greekmath 0112}_0,T)$.
\paragraph{Equivalent Characterization of the Confidence Set.}
The exact confidence set for the compactified parameter solves
Rather than approximating both sides of (ref) separately, we instead exploit the definition of the conditional CDF. Let $\overline G(\cdot,{\Greekmath 0112}_0,T)$ denote the CDF of $\overline{{\Greekmath 0127}}({\Greekmath 0112}_0,R)$ conditional on $T$. By definition,
Therefore, Inequality (ref) is exactly equivalent to
Thus inversion reduces to studying the single composite function
\paragraph{Uniform Approximation and Coverage Error.}
Let $\widehat G_{\Greekmath 0122}$ be a polynomial approximation satisfying
We now quantify the coverage distortion.
\paragraph{Implications.}
Because the compactified domain $[-1,1]$ is closed and bounded, the composite function is uniformly continuous. Standard Chebyshev approximation theory guarantees that the uniform error ${\Greekmath 0122}$ converges to zero as the polynomial degree increases.
Therefore, the coverage distortion of the approximated confidence set vanishes at the same rate.
This provides an explicit and transparent link between numerical approximation error and statistical coverage, ensuring computational reliability without sacrificing weak-identification robustness.
This section provides extensions from the leading linear IV model to more general moment conditions often encountered in applied empirical work.
We now extend our exact algebraic approach to construct AR, LM, and CQLR confidence sets for GMM models featuring polynomial or rational moment conditions. These models are not theoretical curiosities. They arise in many standard empirical settings, including dynamic panel estimators such as Arellano-Bond moment conditions.
\paragraph{A motivating example.}
To illustrate the algebraic structure clearly, consider a simple example. Suppose we observe $n$ iid draws $X_{i}\sim N({\Greekmath 0116} ,1)$. Inference on ${\Greekmath 0116} $ can be based on the first two moments of the normal distribution, which generate the simultaneous moment conditions
The second component contains a quadratic term in ${\Greekmath 0116}$. Therefore each component of $g(X_i,{\Greekmath 0116})$ is a polynomial in ${\Greekmath 0116}$ of degree at most two.
The Anderson-Rubin statistic extends directly to this general GMM setting as shown by StockWright00. The generalized AR statistic is
where
\paragraph{Preservation of polynomial structure.}
Because $g(X_i,{\Greekmath 0116})$ contains terms up to degree two in ${\Greekmath 0116}$, each entry of $W_n({\Greekmath 0116})$ is a polynomial of degree at most four. Indeed, each entry is a sample average of products of two degree-two polynomials.
Writing
each ${\Greekmath 0121}_{ij}({\Greekmath 0116})$ is degree at most four. By the standard formula for matrix inversion,
The determinant in the denominator is a polynomial of degree at most eight. Therefore every entry of $W_n({\Greekmath 0116})^{-1}$ is a rational function whose numerator has degree at most four and whose denominator has degree at most eight.
Since $g_n({\Greekmath 0116})$ is degree two in ${\Greekmath 0116}$, the quadratic form $ g_n({\Greekmath 0116})^{\prime }W_n({\Greekmath 0116})^{-1} g_n({\Greekmath 0116})$ is a rational function whose numerator and denominator are both finite-degree polynomials. In this example, both have degree at most eight.
The crucial point is structural: quadratic forms in polynomial moments, weighted by the inverse of their covariance matrix, preserve rationality.
\paragraph{General polynomial moment conditions.}
We now formalize this property.
Suppose the model contains $k$ moment conditions and each component of $ g(X_i,{\Greekmath 0112})$ is a polynomial in the scalar parameter ${\Greekmath 0112}$ of degree at most $d$.
Then,
1. Each entry of the sample covariance matrix $W_n({\Greekmath 0112})$ is a polynomial of degree at most $2d$.
2. The determinant of $W_n({\Greekmath 0112})$ is a polynomial of degree at most $2dk$.
3. By Cramer's rule or the Sherman-Morrison formula as demonstrated by MoreiraNeweySharifvaghefi24, each entry of $W_n({\Greekmath 0112})^{-1}$ is a ratio of polynomials whose numerator has degree at most $2d(k-1)$ and whose denominator has degree at most $2dk$.
4. Since $g_n({\Greekmath 0112})$ is degree $d$, the AR statistic
is a rational function whose numerator and denominator have degree at most $ 2dk$.
Hence the AR statistic is always a rational function of finite degree. Finite degree implies finitely many real roots of the defining polynomial inequality, so exact confidence sets can be obtained via polynomial root finding, exactly as in the linear IV case.
\paragraph{LM and CQLR statistics.}
The same algebraic preservation principle applies to the LM statistic. Although its expression involves additional matrix products and inversions (see Kleibergen05), all building blocks are polynomial or rational functions of ${\Greekmath 0112}$. Repeated application of the Sherman-Morrison formula implies that the LM statistic is also a rational function of finite degree. The maximal degree increases relative to AR but remains finite, which guarantees exact inversion via root enumeration.
For the CQLR statistic, provided the rank statistic is itself a rational function of ${\Greekmath 0112}$, the entire statistic becomes a composition of rational functions\footnote{ The expression also involves a square root, which can be eliminated by squaring, yielding a rational representation; see Section (ref)}. In that case we can apply the same partition method developed for the linear IV model: work segment by segment where the rank statistic is monotone, identify all boundary points in the rank domain, and map them back into ${\Greekmath 0112}$ via polynomial equations.
\paragraph{Piecewise rational moment conditions.}
Many empirical models involve piecewise polynomial or rational moment conditions, such as models with structural breaks or threshold effects. Within each regime the moments are polynomial or rational, so the AR and LM statistics remain piecewise rational. We can therefore isolate each regime, compute all boundary points within that regime using root finding, and then assemble the global confidence set by taking unions across regimes.
\paragraph{Implication for practice.}
The central message is that polynomial GMM models inherit the same algebraic structure as the linear IV model. Test statistics are rational functions of finite degree. Exact confidence sets can therefore be constructed without grid search and without numerical approximation error.
Whenever the moment conditions are polynomial or rational, confidence regions can be computed exactly, with no risk of missing disconnected components or truncating unbounded regions.
We now consider empirical models with fully nonlinear and non-polynomial moment conditions:
where ${\Greekmath 0112} \in \Theta \subset \mathbb{R}^{k}$ is the structural parameter of interest and ${\Greekmath 010D} \in \Gamma \subset \mathbb{R}^{p}$ is a nuisance parameter that must be profiled or partialled out.
Weak-identification robust tests extend to this setting. StockWright00 provide the nonlinear extension of the AR test, Kleibergen05 extend the LM and CQLR statistics, and AndrewsMikusheva16 develop the nonlinear CLR test. The inferential logic remains unchanged: confidence sets are obtained by inverting tests that control size under weak identification.
However, because these models lack exact polynomial structure, we cannot rely on algebraic root-finding. Instead, we must use the uniform approximation framework developed in Section (ref). To apply uniform approximation safely in highly nonlinear settings, two mathematical conditions are required: continuity and compactness.
1. Continuity.
First, the moment function $g(X_i,{\Greekmath 0112},{\Greekmath 010D})$ must be continuous in $ ({\Greekmath 0112},{\Greekmath 010D})$. This condition is mild and satisfied in standard nonlinear GMM applications.
Second, when the nuisance parameter ${\Greekmath 010D}$ is profiled out by minimizing a GMM objective function for each ${\Greekmath 0112}$, we must ensure that this minimization step does not introduce discontinuities. This is guaranteed under the conditions of Berge's Maximum Theorem. If the criterion function is continuous and the nuisance parameter space $\Gamma$ is compact, then the profiled objective remains continuous in ${\Greekmath 0112}$. This rules out sudden jumps in the test statistic.
2. Compact support.
Uniform polynomial approximation requires the parameter domain to be compact. In many applications, however, $\Theta$ and $\Gamma$ are unbounded, for example $\mathbb{R}$ or $\mathbb{R}_+$.
This is not a fundamental obstacle. If the test statistic exhibits stable asymptotic behavior as $({\Greekmath 0112},{\Greekmath 010D})$ diverge, the infinite domain can be smoothly mapped onto a bounded domain without creating discontinuities. The next proposition formalizes this idea.
This proposition formalizes a practical principle. If the test statistic flattens out and converges smoothly to well-defined limits when parameters diverge, then a trigonometric transformation maps the infinite parameter space onto a bounded domain while preserving continuity.
Once continuity on a compact domain is established, we approximate the nonlinear test inequality uniformly by a polynomial inequality. By Proposition (ref), a uniform bound on approximation error implies a strict bound on coverage distortion. The resulting confidence set is semi-algebraic, meaning it is defined by finitely many polynomial equalities and inequalities.
In the one-dimensional case, all boundary points are roots of a single polynomial, which can be computed rapidly using standard numerical eigenvalue routines such as the QR algorithm.
When ${\Greekmath 0112}$ is multidimensional, the confidence region may have a complex geometry. Nevertheless, it remains semi-algebraic. Cylindrical Algebraic Decomposition (CAD) decomposes such sets into finitely many simple cells. By the Tarski-Seidenberg theorem, projections of semi-algebraic sets remain semi-algebraic. Hence even after projecting onto a lower-dimensional parameter of interest, the bounds of the confidence set can still be computed using polynomial root-finding.
The main implication for empirical practice is that the uniform approximation framework developed for the linear IV model extends naturally to nonlinear GMM models. Under mild continuity and asymptotic regularity conditions, researchers can construct confidence sets with arbitrarily small coverage distortion, avoiding the numerical failures and hidden omissions that arise from standard grid search methods.
This paper develops new methods for constructing confidence sets for structural parameters in linear IV models when instruments may be weak and errors may be heteroskedastic, autocorrelated, or clustered. While the literature has established tests that remain valid under weak identification, empirical practice typically relies on grid search to invert those tests. We show that this numerical step is not innocuous. Grid procedures often miss disconnected components, truncate unbounded regions, and generate confidence sets that are wider or qualitatively different from the true acceptance region. These distortions arise from arbitrary discretization choices rather than from the underlying statistical theory.
Our approach replaces grid inversion with exact and approximation-based methods that respect the algebraic structure of the test statistics. For the AR and LM tests, we exploit their rational form to characterize the confidence set as the solution to a polynomial inequality and recover all boundary points via polynomial root finding. The same logic extends to models with polynomial or rational moment conditions, since the algebraic structure of the statistics is preserved.
For the CQLR test, we use the geometry of the statistic and the monotonicity and convexity properties of the critical value function to derive an exact inversion algorithm. For more general conditional tests, including CLR and CIL, we construct uniform polynomial approximations to the inequality that defines the confidence set. The approximation error for the test statistic translates directly into coverage error, and both can be made arbitrarily small by increasing the degree of approximation.
Across a wide range of empirical specifications drawn from the literature, our exact and approximation-based methods reliably recover the true confidence sets, while grid search frequently fails. The discrepancies are particularly pronounced in designs where the confidence region contains narrow components, exhibits sharp curvature, or extends far into the tails. In these cases, coarse grids can materially distort inference.
The methods developed here are straightforward to implement and apply to a broad class of models. They provide researchers with practical tools for reporting confidence sets that remain valid under weak identification and complex error structures, without relying on arbitrary numerical choices. More broadly, the results highlight that numerical inversion is not a minor computational detail but a central component of valid weak-identification robust inference. By replacing grid search with algebraically grounded procedures, we offer a transparent and theoretically disciplined framework for empirical practice.
To implement exact inversion under HAC errors, we must address two primary difficulties: (i) the mapping $r({\Greekmath 010C}_0)$ is only piecewise invertible in $ {\Greekmath 010C}_0$, and (ii) the QLR statistic may change monotonicity or curvature within each piece. Exact inversion therefore proceeds by first decomposing the parameter space into regions where the geometry is well behaved, and then locating all intersections in the rank domain before mapping back to $ {\Greekmath 010C}_0$.
Formally, this chapter executes the following sequence:
Our first job is to write the QLR statistic as a function of $r$. Because we can only do this over intervals where the function ${\Greekmath 010C}\mapsto r({\Greekmath 010C})$ is injective, we must locate these intervals and subsequently determine the shape of the test statistic within them.
Since the rank statistic is rational, its derivative is also rational, so we are able to numerically find all the real roots ${\Greekmath 010C}_1<\cdots<{\Greekmath 010C}_{m-1}$ of $r^{\prime}({\Greekmath 010C})$. Defining ${\Greekmath 010C}_0=-\infty$, ${\Greekmath 010C}_m=+\infty$, the function $r({\Greekmath 010C})$ is injective on $[{\Greekmath 010C}_{i-1},{\Greekmath 010C}_{i}]$, for $ i=1,\cdots,m$.
For each of these intervals, we can define the inverse function ${\Greekmath 010C}^i(r)$ and write the test statistic as an implicit function of $r$:
where the boundaries in the rank domain are:
We now provide an algorithm to find increasing/decreasing and convex/concave intervals for the function $g_i$ on $[r^i_{0},r^i_{1}]$, by finding all the roots of $g_i^{\prime }$ and $g_i^{\prime \prime }$. To simplify notation, define:
By the Chain Rule, the first and second derivatives of $g_i$ are given by:
Using the Inverse Mapping Theorem to find $({\Greekmath 010C}^i)^{\prime }(\cdot)$ and $ ({\Greekmath 010C}^i)^{\prime \prime }(\cdot)$, and setting the derivatives of $g_i$ to zero, we get the following equations in terms of the structural parameter $ {\Greekmath 010C}$
Since $AR(\cdot)$, $r(\cdot)$ and $\Delta(\cdot)$ are rational functions, we can isolate the term $\Delta({\Greekmath 010C})^{1/2}$ in equations (ref) and (ref) and take the square to get rational equations in ${\Greekmath 010C}$, that we can solve numerically. Although taking the square of the equations may introduce new roots, we can rule them out by checking if indeed derivatives change sign at these points. This step also rules out points of inflection.
Let $\{{\Greekmath 010C}_j\}_{j\in J}$ be the union of all ${\Greekmath 010C}$'s found with the procedures in Section (ref). The intervals generated by these points are such that the test statistic can be implicitly defined as a functions of $r$, and these functions change neither their monotonicity nor their convexity on these intervals.
For each interval $[{\Greekmath 010C}_0,{\Greekmath 010C}_1]$ generated by these points, we view the test statistic $g$ as a function of $r\in[r_0,r_1]$. We already know $ {\Greekmath 0114}(r)$ is strictly decreasing and convex. We evaluate the roots of $g(r) = {\Greekmath 0114}(r)$ based on the geometric behavior of $g$ on $[r_0,r_1]$.
In this case, there is at most one $\overline{r }\in [r_{0},r_{1}]$ such that $g(\overline{r })={\Greekmath 0114} (\overline{r })$.
This is the most difficult case to handle. Finding intersection points requires Algorithm 1, introduced below, which systematically recovers all solutions. In order to intuitively understand the mechanism behind this algorithm, let us consider a particular example where $g$ is linear, which is the case under homoskedasticity (see Figure (ref)). You can see the graphs of $g$ and ${\Greekmath 0114} $ intercept each other at two points.
In order to find the first intersection point, we draw a red line starting from $(r _{0},g(r _{0}))$ with the largest slope in such a way the whole line is bellow the graph of ${\Greekmath 0114} $, and we determine $r ^{\ast }$, the point at which the red line and the graph of ${\Greekmath 0114} $ are tangent. Because of the convexity and monotonicity of both ${\Greekmath 0114}$ and $g$, we ensure there is at most one intersection point between $r_0$ and the tangency point $ r^{\ast}$. Once found, we restart the algorithm for the remainder of the interval with $r^*$ as the updated value for the lower bound $r_0$.
Our algorithm uses the following generalization of the bisection method for monotone, but not necessarily strictly monotone, functions.
\noindentGeneralized Bisection Method:\newline Suppose we know the function $h(x)$ is strictly positive for $x<x^*$ and non-positive for $x\geq x^*$, then we can use the following algorithm to find $x^*\in(a,b)$ with a tolerance ${\Greekmath 0122}>0$:
A symmetric algorithm applies if $h$ is strictly positive after $x^*$ and non-positive before $x^*$.
\noindentAlgorithm 1: Here we provide a step-by-step procedure to find all the roots of the equation ${\Greekmath 0114}-g=0$ over $[r_0,r_1]$ , when $g$ in decreasing and convex. Proposition (ref) in Appendix (ref) formalizes the claims presented in the description of this algorithm. We have five possibilities of ordering for the values of the functions at $r_0$ and $r_1$:
The algorithm stops if $r_{1}-r_{0} \leq {\Greekmath 0122}$.
For each solution $r^*$ of equation $g(r)={\Greekmath 0114} (r)$ on the interval $ [r_{0},r_{1}]$, we map them back to the structural parameter ${\Greekmath 010C}$ using the rational representation of the rank statistic. We solve the equation
for ${\Greekmath 010C}\in[{\Greekmath 010C}_0,{\Greekmath 010C}_1]$ numerically using polynomial root finding algorithms.
For completeness, we revisit the empirical application from Section (ref), using stock returns rather than interest rates as the endogenous variable. Because stock returns are less predictable, weak identification is more severe. Consequently, Table (ref) shows that weak-IV robust tests usually yield unbounded sets, unlike the mechanically bounded t-ratio. Furthermore, Table (ref) confirms our earlier results: grid confidence sets are wider and can miss interval components, as demonstrated by the LM test for Canada.
{
}
{
}
AndrewsMoreiraStock07 provide an explicit formula for the conditional CDF of the QLR statistic:
The CVF, ${\Greekmath 0114}_{\Greekmath 010B}(r)$, is implicitly defined by the integral equation:
We obtain formulas for its derivatives via implicit differentiation. Because the conditional CDF does not admit a closed-form expression, the integrals involved in $G(x,r)$ and its derivatives are evaluated numerically.
We verify the shape of the CVF numerically. In particular, for $ k=1,\ldots,150$, we find that the first derivative of ${\Greekmath 0114}_{\Greekmath 010B}(r)$ is strictly negative and the second derivative is strictly positive over the relevant domain. These properties confirm that the CVF is strictly decreasing and strictly convex in $r$.
Figure (ref) illustrates these properties for six representative values of $k$.