Constructing Confidence Sets Research Articles

SECOND ORDER EXPANSION OF THE T-STATISTIC IN AR(1) MODELS

The purpose of this paper is to differentiate between several asymptotically valid methods for confidence set construction for the autoregressive coefficient in AR(1) models. We show that the nonparametric grid bootstrap procedure suggested by Hansen (1999, Review of Economics and Statistics 81, 594–607) achieves a second order refinement in the local-to-unity asymptotic approach when compared with a modified version of Stock’s (1991, Journal of Monetary Economics 28, 435–459) and Andrews’ (1993, Econometrica 61, 139–165) grid testing procedures. We establish a second order expansion of the t-statistic in an AR(1) model in the local-to-unity asymptotic approach, which differs drastically from the usual Edgeworth-type expansions by approximating the statistic around a nonstandard and nonpivotal limit.

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.

Confidence sets in sparse regression

The problem of constructing confidence sets in the high-dimensional linear model with $n$ response variables and $p$ parameters, possibly $p\ge n$, is considered. Full honest adaptive inference is possible if the rate of sparse estimation does not exceed $n^{-1/4}$, otherwise sparse adaptive confidence sets exist only over strict subsets of the parameter spaces for which sparse estimators exist. Necessary and sufficient conditions for the existence of confidence sets that adapt to a fixed sparsity level of the parameter vector are given in terms of minimal $\ell^2$-separation conditions on the parameter space. The design conditions cover common coherence assumptions used in models for sparsity, including (possibly correlated) sub-Gaussian designs.

Bootstrap Confidence Sets with Weak Instruments

We study several methods of constructing confidence sets for the coefficient of the single right-hand-side endogenous variable in a linear equation with weak instruments. Two of these are based on conditional likelihood ratio (CLR) tests, and the others are based on inverting t statistics or the bootstrap P values associated with them. We propose a new method for constructing bootstrap confidence sets based on t statistics. In large samples, the procedures that generally work best are CLR confidence sets using asymptotic critical values and bootstrap confidence sets based on limited-information maximum likelihood (LIML) estimates.

Identifying common normal distributions

Consider a one-way layout where random samples of data are available from k populations, where the distributions of the data from each population are considered to be completely unknown. This paper discusses a methodology for investigating whether it can be concluded that the k unknown distributions, or any subsets of these distributions, can be taken to be equal to a common normal distribution, and if so it is shown how to identify these common normal distributions. This is accomplished with an exact specified error rate by constructing confidence sets for the parameters of the common normal distributions using Kolmogorov’s (G. Ist. Ital. Attuari 4:83–91, 1933) procedure. The relationship of this methodology to standard tests of normality and to standard procedures for constructing confidence sets for the parameters of a normal distribution are discussed, together with its relationship to functional data analysis and other standard test procedures for data of this kind. Some examples of the implementation of the methodology are provided.

Non-Standard Rates of Convergence of Criterion-Function-Based Set Estimators

This paper establishes consistency and non‐standard rates of convergence for set estimators based on contour sets of criterion functions for a semi‐parametric binary response model under a conditional median restriction. The model can be partially identified due to potentially limited‐support regressors and an unknown distribution of errors. A set estimator analogous to the maximum score estimator is essentially cube‐root consistent for the identified set when a continuous but possibly bounded regressor is present. Arbitrarily fast convergence occurs when all regressors are discrete. We also establish the validity of a subsampling procedure for constructing confidence sets for the identified set. As a technical contribution, we provide more convenient sufficient conditions on the underlying empirical processes for cube‐root convergence and a sufficient condition for arbitrarily fast convergence, both of which can be applied to other models. Finally, we carry out a series of Monte Carlo experiments, which verify our theoretical findings and shed light on the finite‐sample performance of the proposed procedures.

Ryszard Zieliński’s contribution to statistical optimization and fixed-precision estimation

Professor Ryszard Zielinski's results in stochastic approximation, extremal experimental design in the framework of response surface analysis and fixed-precision set estimation are outlined. First, he proposed a randomized version of Fabian's (1967) gradient estimate in the Kiefer-Wolfowitz procedure, which reduced the number of required observations and improved the rate of convergence. Second, when considering response surface analysis and experimental designs for the gradient estimation, he constructed a randomized simplex design which resulted in the unbiased estimator. Third, he gave a method to construct confidence sets with prescribed accuracy (i. e. the width and the confidence level) by sampling independent copies of a process of interest. Professor Ryszard Zielinski's results in stochastic approximation, extremal experimental design in the framework of response surface analysis and fixed precision set estimation are outlined. First, he proposed a randomized version of Fabian's (1967) gradient estimate in the Kiefer-Wolfowitz procedure, which reduced the number of required observations and improved the rate of convergence. Second, when considering response surface analysis and experimental designs for the gradient estimation, he constructed a randomized simplex design which resulted in the unbiased estimator. Third, he gave a method to construct confidence sets with prescribed accuracy (i. e. the width and the confidence level) by sampling independent copies of a process of interest.

Adaptive confidence sets in $$L^2$$

The problem of constructing confidence sets that are adaptive in $$L^2$$ -loss over a continuous scale of Sobolev classes of probability densities is considered. Adaptation holds, where possible, with respect to both the radius of the Sobolev ball and its smoothness degree, and over maximal parameter spaces for which adaptation is possible. Two key regimes of parameter constellations are identified: one where full adaptation is possible, and one where adaptation requires critical regions be removed. Techniques used to derive these results include a general nonparametric minimax test for infinite-dimensional null- and alternative hypotheses, and new lower bounds for $$L^2$$ -adaptive confidence sets.

Distortions of Asymptotic Confidence Size in Locally Misspecified Moment Inequality Models

This paper studies the behavior, under local misspecification, of several confidence sets (CSs) commonly used in the literature on inference in moment (in)equality models. We propose the amount of asymptotic confidence size distortion as a criterion to choose among competing inference methods. This criterion is then applied to compare across test statistics and critical values employed in the construction of CSs. We find two important results under weak assumptions. First, we show that CSs based on subsampling and generalized moment selection (Andrews and Soares (2010)) suffer from the same degree of asymptotic confidence size distortion, despite the fact that asymptotically the latter can lead to CSs with strictly smaller expected volume under correct model specification. Second, we show that the asymptotic confidence size of CSs based on the quasi-likelihood ratio test statistic can be an arbitrary small fraction of the asymptotic confidence size of CSs based on the modified method of moments test statistic.

Inference Based on Conditional Moment Inequalities

In this paper, we construct confidence sets for models defined by many conditional moment inequalities/equalities. The conditional moment restrictions in the models can be finite, countably infinite, or uncountably infinite. To deal with the complication brought about by the vast number of moment restrictions, we exploit the manageability (Pollard (1990)) of the class of moment functions. We verify the manageability condition in five examples from the recent partial identification literature. The proposed confidence sets are shown to have correct asymptotic size in a uniform sense and to exclude parameter values outside the identified set with probability approaching one. Monte Carlo experiments for a conditional stochastic dominance example and a random-coefficients binary-outcome example support the theoretical results.

One-Dimensional Inference in Autoregressive Models With the Potential Presence of a Unit Root

This paper examines the problem of testing and confidence set construction for one-dimensional functions of the coefficients in autoregressive (AR(p)) models with potentially persistent time series. The primary example concerns inference on impulse responses. A new asymptotic framework is suggested and some new theoretical properties of known procedures are demonstrated. I show that the likelihood ratio (LR) and LR± statistics for a linear hypothesis in an AR(p) can be uniformly approximated by a weighted average of local-to-unity and normal distributions. The corresponding weights depend on the weight placed on the largest root in the null hypothesis. The suggested approximation is uniform over the set of all linear hypotheses. The same family of distributions approximates the LR and LR± statistics for tests about impulse responses, and the approximation is uniform over the horizon of the impulse response. I establish the size properties of tests about impulse responses proposed by Inoue and Kilian (2002) and Gospodinov (2004), and theoretically explain some of the empirical findings of Pesavento and Rossi (2007). An adaptation of the grid bootstrap for impulse response functions is suggested and its properties are examined.

Confidence sets in a linear regression model for interval data

The construction of confidence sets for the parameters of a flexible simple linear regression model for interval-valued random sets is addressed. For that purpose, the asymptotic distribution of the least-squares estimators is analyzed. A simulation study is conducted to investigate the performance of those confidence sets. In particular, the empirical coverages are examined for various interval linear models. The applicability of the procedure is illustrated by means of a real-life case study.

Improved Least Squares Topology Testing and Estimation

Generalized least squares (GLS) methods provide a relatively fast means of constructing a confidence set of topologies. Because they utilize information about the covariances between distances, it is reasonable to expect additional efficiency in estimation and confidence set construction relative to other least squares (LS) methods. Difficulties have been found to arise in a number of practical settings due to estimates of covariance matrices being ill conditioned or even noninvertible. We present here new ways of estimating the covariance matrices for distances that are much more likely to be positive definite, as the actual covariance matrices are. A thorough investigation of performance is also conducted. An alternative to GLS that has been proposed for constructing confidence sets of topologies is weighted least squares (WLS). As currently implemented, this approach is equivalent to the use of GLS but with covariances set to zero rather than being estimated. In effect, this approach assumes normality of the estimated distances and zero covariances. As the results here illustrate, this assumption leads to poor performance. A 95% confidence set is almost certain to contain the true topology but will contain many more topologies than are needed. On the other hand, the results here also indicate that, among LS methods, WLS performs quite well at estimating the correct topology. It turns out to be possible to improve the performance of WLS for confidence set construction through a relatively inexpensive normal parametric bootstrap that utilizes the same variances and covariances of GLS. The resulting procedure is shown to perform at least as well as GLS and thus provides a reasonable alternative in cases where covariance matrices are ill conditioned.

Non-Asymptotic Confidence Sets for the Parameters of Linear Transfer Functions

We consider the problem of constructing confidence sets for the parameters of input-output transfer functions based on observed data. The assumptions on the noise affecting the system are reduced to a minimum; the noise can virtually be anything, but in return the user must be able to select the input signal. In this paper a procedure for solving this problem is developed in the general framework of leave-out sign-dominant confidence regions. The procedure returns confidence regions that are guaranteed to contain the true transfer function with a user-chosen probability for any finite data set.

Partial Identification in Econometrics

Identification in econometric models maps prior assumptions and the data to information about a parameter of interest. The partial identification approach to inference recognizes that this process should not result in a binary answer that consists of whether the parameter is point identified. Rather, given the data, the partial identification approach characterizes the informational content of various assumptions by providing a menu of estimates, each based on different sets of assumptions, some of which are plausible and some of which are not. Of course, more assumptions beget more information, so stronger conclusions can be made at the expense of more assumptions. The partial identification approach advocates a more fluid view of identification and hence provides the empirical researcher with methods to help study the spectrum of information that we can harness about a parameter of interest using a menu of assumptions. This approach links conclusions drawn from various empirical models to sets of assumptions made in a transparent way. It allows researchers to examine the informational content of their assumptions and their impacts on the inferences made. Naturally, with finite sample sizes, this approach leads to statistical complications, as one needs to deal with characterizing sampling uncertainty in models that do not point identify a parameter. Therefore, new methods for inference are developed. These methods construct confidence sets for partially identified parameters, and confidence regions for sets of parameters, or identifiable sets.

Bootstrapping Semiparametric Models with Single-Index Nuisance Parameters, Second Version

This paper considers models of conditional moment restrictions that involve non-parametric functions of single-index nuisance parameters. This paper proposes a bootstrap method of constructing confidence sets which has the following three merits. First, the bootstrap is valid even when the single-index estimator follows cube-root asymptotics. Second, the bootstrap method accommodates conditional heteroskedasticity. Third, the bootstrap does not require re-estimation of the single-index component for each bootstrap sample. The method is built on this paper’s general finding that as far as the single-index is a conditioning variable of a conditional expectation, the influence of the estimated single-indices in these models is asymptotically negligible. This finding is shown to have a generic nature through an analysis of Frechet derivatives of linear functionals of conditional expectations. Some results from Monte Carlo simulations are presented and discussed.

Inference Based on Many Conditional Moment Inequalities

In this paper, we construct confidence sets for models defined by many conditional moment inequalities/equalities. The conditional moment restrictions in the models can be finite, countably in finite, or uncountably in finite. To deal with the complication brought about by the vast number of moment restrictions, we exploit the manageability (Pollard (1990)) of the class of moment functions. We verify the manageability condition in five examples from the recent partial identification literature. The proposed confidence sets are shown to have correct asymptotic size in a uniform sense and to exclude parameter values outside the identified set with probability approaching one. Monte Carlo experiments for a conditional stochastic dominance example and a random-coefficients binary-outcome example support the theoretical results.

Robust confidence sets in the presence of weak instruments

This paper considers instrumental variable regression with a single endogenous variable and the potential presence of weak instruments. I construct confidence sets for the coefficient on the single endogenous regressor by inverting tests robust to weak instruments. I suggest a numerically simple algorithm for finding the Conditional Likelihood Ratio (CLR) confidence sets. Full descriptions of possible forms of the CLR, Anderson–Rubin (AR) and Lagrange Multiplier (LM) confidence sets are given. I show that the CLR confidence sets have nearly the shortest expected arc length among similar symmetric invariant confidence sets in a circular model. I also prove that the CLR confidence set is asymptotically valid in a model with non-normal errors.

Asymptotic equivalence and adaptive estimation for robust nonparametric regression

Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions. The results imply that all the Gaussian nonparametric regression procedures can be robustified in a unified way. A key step in our equivalence argument is to bin the data and then take the median of each bin. The asymptotic equivalence results have significant practical implications. To illustrate the general principles of the equivalence argument we consider two important nonparametric inference problems: robust estimation of the regression function and the estimation of a quadratic functional. In both cases easily implementable procedures are constructed and are shown to enjoy simultaneously a high degree of robustness and adaptivity. Other problems such as construction of confidence sets and nonparametric hypothesis testing can be handled in a similar fashion.

Exponential bounds for minimum contrast estimators

The paper focuses on general properties of parametric mini- mum contrast estimators. The quality of estimation is measured in terms of the rate function related to the contrast, thus allowing to derive ex- ponential risk bounds invariant with respect to the detailed probabilistic structure of the model. This approach works well for small or moderate samples and covers the case of a misspecified parametric model. Another important feature of the presented bounds is that they may be used in the case when the parametric set is unbounded and non-compact. These bounds do not rely on the entropy or covering numbers and can be easily computed. The most important statistical fact resulting from the exponential bonds is a concentration inequality which claims that minimum contrast estimators concentrate with a large probability on the level set of the rate function. In typical situations, every such set is a root-n neighborhood of the parameter of interest. We also show that the obtained bounds can help for bounding the estimation risk, constructing confidence sets for the underlying param- eters. Our general results are illustrated for the case of an i.i.d. sample. We also consider several popular examples including least absolute deviation estimation and the problem of estimating the location of a change point. What we obtain in these examples slightly differs from the usual asymp- totic results presented in statistical literature. This difference is due to the unboundness of the parameter set and a possible model misspecification.