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Abstract:
This paper considers flexible conditional (regression) measures of market risk. Value-at-Risk modeling is cast in terms of the quantile regression function - the inverse of the conditional distribution function. A basic specification analysis relates its functional forms to the benchmark models of returns and asset pricing. We stress important aspects of measuring very high and intermediate conditional risk. An empirical application illustrates.

Conditional Quantiles, Quantile Regression, Extreme Quantiles, Extreme Value Theory, Extreme Risk

Abstract:
This paper studies computationally and theoretically attractive estimators referred here as to the Laplace type estimators (LTE). The LTE include means and quantiles of Quasi-posterior distributions defined as transformations of general
(non-likelihood-based) statistical criterion functions, such as those in GMM, nonlinear IV, empirical likelihood, and minimum distance methods. The approach generates an alternative to classical extremum estimation and also falls outside the parametric Bayesian approach. For example, it offers a new attractive estimation method for such important semi-parametric problems as censored and instrumental quantile regression, nonlinear IV, GMM, and value-at-risk, models. The LTE's are computed using Markov Chain Monte Carlo methods, which help circumvent the computational curse of dimensionality. A large sample theory is obtained and illustrated for regular cases.

Laplace, Bayes, Markov Chain Monte Carlo, GMM, Instrumental Regression, Censored Quantile Regression, Instrumental Quantile Regression, Empirical Likelihood, Value-at-Risk

Abstract:
A wide variety of important distributional hypotheses can be assessed using the empirical quantile regression processes. In this paper, a very simple and practical resampling test is offered as an alternative to inference based on Khmaladzation, as developed in Koenker and Xiao (2002). This alternative has better or competitive power, accurate size, and does not require estimation of non-parametric sparsity and score functions. It applies not only to iid but also time series data. Computational experiments and an empirical example that re-examines the effect of re-employment bonus on the unemployment duration strongly support this approach.

bootstrap, subsampling, quantile regression, quantile regression process, Kolmogorov-Smirnov test, unemployment duration

Abstract:
This paper develops a model of quantile treatment effects with treatment endogeneity. The model primarily exploits similarity assumption as a main restriction that handles endogeneity. From this model we derive a Wald IV estimating equation, and show that the model does not require functional form assumptions for identification. We then characterize the quantile treatment function as solving an "inverse" quantile regression problem and suggest its finite-sample analog as a practical estimator. This estimator, unlike generalized method-of-moments, can be easily computed by solving a series of conventional quantile regressions, and does not require grid searches over high-dimensional parameter sets. A properly weighted version of this estimator is also efficient. The model and estimator apply to either continuous or discrete variables. We apply this estimator to characterize the median and other quantile treatment effects in a market demand model and a job training program.

Quantile Regression, Inverse Quantile Regression, Instrumental Quantile Regression, Treatment Effects, Empirical Likelihood,Training, Demand Models.

Abstract:
In this paper we study inference for a conditional model with a jump in the conditional density, where the location and size of the jump are described by regression lines. This interesting structure is shared by several structural econometric models. Two prominent examples are the standard auction model where density jumps from zero to a positive value, and the equilibrium job search model, where the density jumps from one level to another, inducing kinks in the cumulative distribution function. This paper develops the asymptotic inference theory for likelihood based estimators of these models - the Bayes and maximum likelihood estimators. Bayes and ML estimators are useful classical procedures. While MLE is transformation invariant, Bayes estimators offer some theoretic and computational advantages. They also have desirable efficiency properties. We characterize the limit likelihood as a function of a Poisson process that tracks the near-to-jump events and depends on regressors. The approach is applied to an empirical model of a highway procurement auction. We estimated a pareto model of Paarsch (1992) and an alternative flexible parametric model.

Extreme Value Theory, Structural Econometric Model, Auctions, Job Search, Highway Procurement Auction, Likelihood, Point Process, Stochastic Equisemicontinuity

Abstract:
Quantile regression (QR) fits a linear model for conditional quantiles, just as ordinary least squares (OLS) fit a linear model for conditional means. An attractive feature of OLS is that it gives the minimum mean square error linear approximation to the conditional expectation function even when the linear model is misspecified. Empirical research using quantile regression with discrete covariates suggests that QR may have a similar property, but the exact nature of the linear approximation has remained elusive. In this paper, we show that QR can be interpreted as minimizing a weighted mean-squared error loss function for specification error. The weighting function is an average density of the dependent variable near the true conditional quantile. The weighted least squares interpretation of QR is used to derive an omitted variables bias formula and a partial quantile correlation concept, similar to the relationship between partial correlation and OLS. We also derive general asymptotic results for QR processes allowing for misspecification of the conditional quantile function, extending earlier results from a single quantile to the entire process. The approximation properties of QR are illustrated through an analysis of the wage structure and residual inequality in US census data for 1980, 1990, and 2000. The results suggest continued residual inequality growth in the 1990s, primarily in the upper half of the wage distribution and for college graduates.

residual inequality, income distribution, conditional quantiles

Abstract:
Quantile regression(QR) fits a linear model for conditional quantiles, just as ordinary least squares (OLS) fits a linear model for conditional means. An attractive feature of OLS is that it gives the minimum mean square error linear approximation to the conditional expectation function even when the linear model is misspecified. Empirical research using quantile regression with discrete covariates suggests that QR may have a similar property, but the exact nature of the linear approximation has remained elusive. In this paper, we show that QR can be interpreted as minimizing a weighted mean-squared error loss function for specification error. The weighting function is an average density of the dependent variable near the true conditional quantile. The weighted least squares interpretation of QR is used to derive an omitted variables bias formula and a partial quantile correlation concept, similar to the relationship between partial correlation and OLS. We also derive general asymptotic results for QR processes allowing for misspecification of the conditional quantile function, extending earlier results from a single quantile to the entire process. The approximation properties of QR are illustrated through an analysis of the wage structure and residual inequality in US Census data for 1980, 1990, and 2000. The results suggest continued residual inequality growth in the 1990s, primarily in the upper half of the wage distribution and for college graduates.

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Abstract:
This article looks at the theory and empirics of extremal quantiles in economics, in particular value-at-risk. The theory of extremes has gone through remarkable developments and produced valuable empirical findings in the last 20 years. In the discussion, we put a particular focus on conditional extremal quantile models and methods, which have applications in many areas of economic analysis. Examples of applications include the analysis of factors of high risk in finance and risk management, the analysis of socio-economic factors that contribute to extremely low infant birthweights, efficiency analysis in industrial organization, the analysis of reservation rules in economic decisions, and inference in structural auction models.

Extremes, Quantiles, Regression, Value-at-risk, Extremal Bootstrap

Abstract:
This paper develops a theory of high and low (extremal) quantile regression: the linear models, estimation, and inference. In particular, the models coherently combine the convenient, flexible linearity with the extreme-value-theoretic restrictions on tails and the general heteroscedasticity forms. Within these models, the limit laws for extremal quantile regression statistics are obtained under the rank conditions (experiments) constructed to reflect the extremal or rare nature of tail events. An inference framework is discussed. The results apply to cross-section (and possibly dependent) data. The applications, ranging from the analysis of babies' very low birth weights, (S,s) models, tail analysis in heteroscedastic regression models, outlier-robust inference in auction models, and decision-making under extreme uncertainty, provide the motivation and applications of this theory.

Quantile regression, extreme value theory, tail analysis, (S,s) models, auctions, price search, Extreme Risk

Abstract:
Under minimal assumptions finite sample confidence bands for quantile regression models can be constructed. These confidence bands are based on the "conditional pivotal property" of estimating equations that quantile regression methods aim to solve and will provide valid finite sample inference for both linear and nonlinear quantile models regardless of whether the covariates are endogenous or exogenous. The confidence regions can be computed using MCMC, and confidence bounds for single parameters of interest can be computed through a simple combination of optimization and search algorithms. We illustrate the finite sample procedure through a brief simulation study and two empirical examples: estimating a heterogeneous demand elasticity and estimating heterogeneous returns to schooling. In all cases, we find pronounced differences between confidence regions formed using the usual asymptotics and confidence regions formed using the finite sample procedure in cases where the usual asymptotics are suspect, such as inference about tail quantiles or inference when identification is partial or weak. The evidence strongly suggests that the finite sample methods may usefully complement existing inference methods for quantile regression when the standard assumptions fail or are suspect.

Quantile Regression, Extremal Quantile Regression, Instrumental Quantile Regression

Abstract:
This paper suggests simple 3- and 4-step estimators for censored quantile regression models with an envelope or a separation restriction on the censoring probability. The estimators are theoretically attractive (asymptotically as efficient as the celebrated Powell's censored least absolute deviation estimator). At the same time, they are conceptually simple and have trivial computational expenses. They are especially useful in samples of small size or models with many regressors, with desirable finite sample properties and small bias. The envelope restriction costs a small reduction of generality relative to the canonical censored regression quantile model, yet its main plausible features remain intact. The estimator can also be used to estimate a large class of traditional models, including normal Amemiya-Tobin model and many accelerated failure and proportional hazard models. The main empirical example involves a very large data-set on extramarital affairs, with high 68 percent censoring. We estimate 45-90 percent conditional quantiles. Effects of covariates are not representable as location-shifts. Less religious women, with fewer children, and higher status, tend to engage into the matters relatively more than their opposites, especially at the extremes. Marriage longevity effect is positive at moderately high quantiles and negative at high quantiles. Education and marriage happiness effects are negative, especially at the extremes. We also briefly consider the survival quantile regression on the Stanford heart transplant data. We estimate the age and prior surgery effects across survival quantiles.

Quantile regression, median regression, censoring, duration, survival, classification, discriminant analysis

Abstract:
In this paper we describe how quantile regression can be used to evaluate the impact of treatment on the entire distribution of outcomes, when the treatment is endogenous or selected in relation to potential outcomes. We describe an instrumental variable quantile regression process and the set of inferences derived from it, focusing on tests of distributional equality, non-constant treatment effects, conditional dominance, and exogeneity. The inference, which is subject to the Durbin problem, is handled via a method of score resampling. The approach is illustrated with a classical supply-demand and a schooling example. Results from both models demonstrate substantial treatment heterogeneity and serve to illustrate the rich variety of hypotheses that can be tested using inference on the instrumental quantile regression process.

Quantile Regression, Instrumental Quantile Regression, Treatment Effects, Endogeneity, Stochastic Dominance, Hausman Test, Supply-Demand Equations, Returns to Education

Abstract:
Most economic analyses presume that there are limited differences in the prior beliefs of individuals, as assumption most often justified by the argument that sufficient common experiences and observations will eliminate disagreements. We investigate this claim using a simple model of Bayesian learning. Two individuals with different priors observe the same infinite sequence of signals about some underlying parameter. Existing results in the literature establish that when individuals are certain about the interpretation of signals, under very mild conditions there will be asymptotic agreement - their assessments will eventually agree. In contrast, we look at an environment in which individuals are uncertain about the interpretation of signals, meaning that they have non-degenerate probability distributions over the conditional distribution of signals given the underlying parameter. When priors on the parameter and the conditional distribution of signals have full support, we prove the following results: (1) Individuals will never agree, even after observing the same infinite sequence of signals. (2) Before observing the signals, they believe with probability 1 that their posteriors about the underlying parameter will fail to converge. (3) Observing the same sequence of signals may lead to a divergence of opinion rather than the typically presumed convergence. We then characterize the conditions for asymptotic agreement under approximate certainty - i.e., as we look at the limit where uncertainty about the interpretation of the signals disappears. When the family of probability distributions of signals given the parameter has rapidly-varying tails (such as the normal or exponential distributions), approximate certainty restores asymptotic agreement. However, when the family of probability distributions has regularly-varying tails (such as the Pareto, the log-normal, and the t-distributions), asymptotic agreement does not obtain even in the limit as the amount of uncertainty disappears. Lack of common priors has important implications for economic behavior in a range of circumstances. We illustrate how the type of learning outlined in this paper interacts with economic behavior in various different situations, including games of common interest, coordination, asset trading and bargaining.

Abstract:
Most economic analyses presume that there are limited differences in the prior beliefs of individuals, an assumption most often justified by the argument that sufficient common experiences and observations will eliminate disagreements. We investigate this claim using a simple model of Bayesian learning. Two individuals with different priors observe the same infinite sequence of signals about some underlying parameter. Existing results in the literature establish that when individuals are certain about the interpretation of signals, under very mild conditions there will be asymptotic agreement - their assessments will eventually agree. In contrast, we look at an environment in which individuals are uncertain about the interpretation of signals, meaning that they have non-degenerate probability distributions over the conditional distribution of signals given the underlying parameter. When priors on the parameter and the conditional distribution of signals have full support, we prove the following results: (1) Individuals will never agree, even after observing the same infinite sequence of signals. (2) Before observing the signals, they believe with probability 1 that their posteriors about the underlying parameter will fail to converge. (3) Observing the same sequence of signals may lead to a divergence of opinion rather than the typically-presumed convergence. We then characterize the conditions for asymptotic agreement under "approximate certainty" - i.e., as we look at the limit where uncertainty about the interpretation of the signals disappears. When the family of probability distributions of signals given the parameter has "rapidly-varying tails" (such as the normal or the exponential distributions), approximate certainty restores asymptotic agreement. However, when the family of probability distributions has "regularly-varying tails" (such as the Pareto, the log-normal, and the t-distributions), asymptotic agreement does not obtain even in the limit as the amount of uncertainty disappears. Lack of common priors has important implications for economic behavior in a range of circumstances. We illustrate how the type of learning outlined in this paper interacts with economic behavior in various different situations, including games of common interest, coordination, asset trading and bargaining.

asymptotic disagreement, Bayesian learning, merging of opinions

Abstract:
In this paper, we consider simple methods for performing robust inference in linear instrumental variables models with weak instruments. We focus on inference based on the reduced form and show that conventional inference procedures about the relevance of the instruments excluded from the structural equation lead to tests of the structural parameters which are valid even if the instruments are weakly correlated to the endogenous variables. The use of standard heteroskedasticity and autocorrelation consistent covariance matrix estimators in constructing these tests also results in inference which is robust to heteroskedasticity, autocorrelation, and weak instruments. We provide a simulation experiment that demonstrates that the procedures have the correct size and good power in many relevant situations and conclude with an empirical example.

Abstract:
The paper develops estimation and inference methods for econometric models with partial identification, focusing on models defined by moment inequalities and equalities. Main applications of this framework include analysis of game-theoretic models, revealed preference, regression with missing and mismeasured data, auction models, bounds in structural quantile models, bounds in asset pricing, among many others.

Specifically, this paper provides estimators and confidence regions for minima of an econometric criterion function Q(θ). In applications, Q(θ) embodies testable restrictions on economic models. A parameter θ that describes an economic model passes these restrictions if Q(θ) attains the minimum value normalized to be zero. The interest therefore focuses on the set of parameters θI that minimizes Qn(θ), called the identified set. This paper uses the inversion of the sample analog Q(θ) of the population criterion Q(θ) to construct the estimators and confidence regions for θI. We develop consistency, rates of convergence, and inference results for these estimators and regions. The results are shown to hold under general yet simple conditions, and practical procedures are provided to implement the approach. In order to derive these results, the paper also develops methods for analyzing the asymptotics of sample criterion functions under set identification.

Set estimator, contour sets, moment inequalities, moment equalities

Abstract:
This paper provides confidence regions for minima of an econometric criterion function Q(µ). The minima form a set of parameters, £I, called the identified set. In economic applications, £I represents a class of economic models that are consistent with the data. Our inference procedures are criterion function based and so our confidence regions, which cover £I with a prespecified probability, are appropriate level sets of Qn(µ), the sample analog of Q(µ). When £I is a singleton, our confidence sets reduce to the conventional confidence regions based on inverting the likelihood or other criterion functions. We show that our procedure is valid under general yet simple conditions, and we provide feasible resampling procedure for implementing the approach in practice. We then show that these general conditions hold in a wide class of parametric econometric models. In order to verify the conditions, we develop methods of analyzing the asymptotic behavior of econometric criterion functions under set identification and also characterize the rates of convergence of the confidence regions to the identified set. We apply our methods to regressions with in terval data and set identified method of moments problems. We illustrate our methods in an empirical Monte Carlo study based on Current Population Survey data.

Set estimator, level sets, interval regression, subsampling bootsrap

Abstract:
In this paper we study estimation and inference in structural models with a jump in the conditional density, where the location and size of the jump are described by regression lines. Two prominent examples are auction models, where the density jumps from zero to a positive value, and the equilibrium job search model, where the density jumps from one level to another, inducing kinks in the cumulative distribution function. An early model of this kind was introduced by Aigner, Amemiya, and Poirier (19776), but the estimation and inference in such models remained an unresolved problem, with the important exception of the specific cases studied by Donald and Paarsch (1993a) and the univariate case in Ibragimov and Has'minskii (1981a). The main difficulty is the statistical non-regularity of the problem caused by discontinuities in the likelihood function. This difficulty also makes the problem computationally challenging.

This paper develops estimation and inference theory and methods for such models based on likelihood procedures, focusing on the optimal (Bayes) procedures, including the MLEs. We obtain results on convergence rates and distribution theory, and develop Wald and Bayes type inference and confidence intervals. The Bayes procedures are attractive both theoretically and computationally. The Bayes confidence intervals, based on the posterior quantiles, are shown to provide a valid large sample inference method with good small sample properties. This inference result is of independent practical and theoretical interest due to the highly non-regular nature of the likelihood in these models, in which the maximum likelihood statistic or any finite dimensional statistic is not asymptotically sufficient.

Likelihood Principle, Point Process, Frequentist Validity, Posterior, Structural Econometric Model, Auctions, Equilibrium Search, Production Frontier

Abstract:
The paper develops estimation and inference methods for econometric models with partial identification, focusing on models defined by moment inequalities and equalities. Main applications of this framework include analysis of game-theoretic models, regression with missing and mismeasured data, bounds in structural quantile models, and bounds in asset pricing, among others.

Set estimator, contour sets, moment inequalities, moment equalities

Abstract:
In this paper we develop procedures for performing inference in regression models about how potential policy interventions affect the entire marginal distribution of an outcome of interest. These policy interventions consist of either changes in the distribution of covariates related to the outcome holding the conditional distribution of the outcome given covariates fixed, or changes in the conditional distribution of the outcome given covariates holding the marginal distribution of the covariates fixed. Under either of these assumptions, we obtain uniformly consistent estimates and functional central limit theorems for the counterfactual and status quo marginal distributions of the outcome as well as other function-valued effects of the policy, including, for example, the effects of the policy on the marginal distribution function, quantile function, and other related functionals. We construct simultaneous confidence sets for these functions; these sets take into account the sampling variation in the estimation of the relationship between the outcome and covariates. Our procedures rely on, and our theory covers, all main regression approaches for modeling and estimating conditional distributions, focusing especially on classical, quantile, duration, and distribution regressions. Our procedures are general and accommodate both simple unitary changes in the values of a given covariate as well as changes in the distribution of the covariates or the conditional distribution of the outcome given covariates of general form. We apply the procedures to examine the effects of labor market institutions on the U.S. wage distribution.

Policy effects, counterfactual distribution, quantile regression, duration regression, distribution regression

Abstract:
This paper applies a regularization procedure called increasing rearrangement to monotonize Edgeworth and Cornish-Fisher expansions and any other related approximations of distribution and quantile functions of sample statistics. Besides satisfying the logical monotonicity, required of distribution and quantile functions, the procedure often delivers strikingly better approximations to the distribution and quantile functions of the sample mean than the original Edgeworth-Cornish-Fisher expansions.

Edgeworth expansion, Cornish-Fisher expansion, rearrangement

Abstract:
In this work we study the large sample properties of the posterior-based inference in the curved exponential family under increasing dimension. The curved structure arises from the imposition of various restrictions, such as moment restrictions, on the model, and plays a fundamental role in various branches of data analysis. We establish conditions under which the posterior distribution is approximately normal, which in turn implies various good properties of estimation and inference procedures based on the posterior. We also discuss the multinomial model with moment restrictions, which arises in a variety of econometric applications. In our analysis, both the parameter dimension and the number of moments are increasing with the sample size.

Bayesian Infrence, Frequentist Properties

Abstract:
We study the asymptotic distribution of Tikhonov Regularized estimation of quantile structural effects implied by a nonseparable model. The nonparametric instrumental variable estimator is based on a minimum distance principle. We show that the minimum distance problem without regularization is locally ill-posed, and consider penalization by the norms of the parameter and its derivatives. We derive pointwise asymptotic normality and develop a consistent estimator of the asymptotic variance. We study the small sample properties via simulation results, and provide an empirical illustration to estimation of nonlinear pricing curves for telecommunications services in the U.S.

Nonparametric Quantile Regression, Instrumental Variable, Ill-Posed Inverse Problems, Tikhonov Regularization, Nonlinear Pricing Curve.

Abstract:
The most common approach to estimating conditional quantile curves is to fit a curve, typically linear, pointwise for each quantile. Linear functional forms, coupled with pointwise fitting, are used for a number of reasons including parsimony of the resulting approximations and good computational properties. The resulting fits, however, may not respect a logical monotonicity requirement - that the quantile curve be increasing as a function of probability. This paper studies the natural monotonization of these empirical curves induced by sampling from the estimated non-monotone model, and then taking the resulting conditional quantile curves that by construction are monotone in the probability. This construction of monotone quantile curves may be seen as a bootstrap and also as a monotonic rearrangement of the original non-monotone function. It is shown that the monotonized curves are closer to the true curves in finite samples, for any sample size. Under correct specification, the rearranged conditional quantile curves have the same asymptotic distribution as the original non-monotone curves. Under misspecification, however, the asymptotics of the rearranged curves may partially differ from the asymptotics of the original non-monotone curves. An analogous procedure is developed to monotonize the estimates of conditional distribution functions. The results are derived by establishing the compact (Hadamard) differentiability of the monotonized quantile and probability curves with respect to the original curves in discontinuous directions, tangentially to a set of continuous functions. In doing so, the compact differentiability of the rearrangement-related operators is established.

Quantile regression, Monotonicity, Rearrangement, Approximation, Functional Delta Method, Hadamard Differentiability of Rearrangement Operators

Abstract:
Under the assumption that individuals know the conditional distributions of signals given the payoff-relevant parameters, existing results conclude that as individuals observe infinitely many signals, their beliefs about the parameters will eventually merge. We first show that these results are fragile when individuals are uncertain about the signal distributions: given any such model, a vanishingly small individual uncertainty about the signal distributions can lead to a substantial (non-vanishing) amount of differences between the asymptotic beliefs. We then characterize the conditions under which a small amount of uncertainty leads only to a small amount of asymptotic disagreement. According to our characterization, this is the case if the uncertainty about the signal distributions is generated by a family with "rapidly-varying tails" (such as the normal or the exponential distributions). However, when this family has "regularly-varying tails" (such as the Pareto, the log-normal, and the t-distributions), a small amount of uncertainty leads to a substantial amount of asymptotic disagreement.

asymptotic disagreement, Bayesian learning, merging of opinions

Abstract:
Suppose that a target function is monotonic, namely, weakly increasing, and an original estimate of the target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates. We show that these estimates can always be improved with no harm using rearrangement techniques: The rearrangement methods, univariate and multivariate, transform the original estimate to a monotonic estimate, and the resulting estimate is closer to the true curve in common metrics than the original estimate. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts.

Monotone function, improved approximation, multivariate rearrangement, univariate rearrangement, growth chart, quantile regression, mean regression, series, locally linear, kernel methods

Abstract:
This paper studies a model widely used in the weak instruments literature and establishes admissibility of the weighted average power likelihood ratio tests recently derived by Andrews, Moreira, and Stock (2004). The class of tests covered by this admissibility result contains the Anderson and Rubin (1949) test. Thus, there is no conventional statistical sense in which the Anderson and Rubin (1949) test "wastes degrees of freedom". In addition, it is shown that the test proposed by Moreira (2003) belongs to the closure of (i.e., can be interpreted as a limiting case of) the class of tests covered by our admissibility result.

Instrumental Variables, Regression, Inference

Abstract:
This paper studies the computational complexity of Bayesian and quasi-Bayesian estimation in large samples carried out using a basic Metropolis random walk. The framework covers cases where the underlying likelihood or extremum criterion function is possibly non-concave, discontinuous, and of increasing dimension. Using a central limit framework to provide structural restrictions for the problem, it is shown that the algorithm is computationally efficient. Specifically, it is shown that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension d, and in particular is of stochastic order d2 in the leading cases after the burn-in period. The reason is that, in large samples, a central limit theorem implies that the posterior or quasi-posterior approaches a normal density, which restricts the deviations from continuity and concavity in a specific manner, so that the computational complexity is polynomial. An application to exponential and curved exponential families of increasing dimension is given.

Computational Complexity, Metropolis, Large Samples, Sampling, Integration, Exponential family, Moment restrictions

Abstract:
Suppose that a target function is monotonic, namely, weakly increasing, and an original estimate of this target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates. We show that these estimates can always be improved with no harm using rearrangement techniques: The rearrangement methods, univariate and multivariate, transform the original estimate to a monotonic estimate, and the resulting estimate is closer to the true curve in common metrics than the original estimate. The improvement property of the rearrangement also extends to the construction of confidence bands for monotone functions. Suppose we have the lower and upper endpoint functions of a simultaneous confidence interval that covers the target function with a pre-specified probability level, then the rearranged confidence interval, defined by the rearranged lower and upper end-point functions, is shorter in length in common norms than the original interval and covers the target function with probability greater or equal to the pre-specified level. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts.

Monotone function, improved estimation, improved inference, multivariate rearrangement, univariate rearrangement, Lorentz inequalities, growth chart, quantile regression, mean regression, series, locally linear, kernel methods

Abstract:
We consider median regression and, more generally, quantile regression in high-dimensional sparse models. In these models the overall number of regressors p is very large, possibly larger than the sample size n, but only s of these regressors have non-zero impact on the conditional quantile of the response variable, where s grows slower than n. Since in this case the ordinary quantile regression is not consistent, we consider quantile regression penalized by the 1-norm of coefficients (L1-QR). First, we show that L1-QR is consistent, up to a logarithmic factor, at the oracle rate which is achievable when the minimal true model is known. The overall number of regressors p affects the rate only through a logarithmic factor, thus allowing nearly exponential growth in the number of zero-impact regressors. The rate result holds under relatively weak conditions, requiring that s/n converges to zero at a super-logarithmic speed and that regularization parameter satisfies certain theoretical constraints. Second, we propose a pivotal, data-driven choice of the regularization parameter and show that it satisfies these theoretical constraints. Third, we show that L1-QR correctly selects the true minimal model as a valid submodel, when the non-zero coefficients of the true model are well separated from zero. We also show that the number of non-zero coefficients in L1-QR is of same stochastic order as s, the number of non-zero coefficients in the minimal true model. Fourth, we analyze the rate of convergence of a two-step estimator that applies ordinary quantile regression to the selected model. Fifth, we evaluate the performance of L1-QR in a Monte-Carlo experiment, and provide an application to the analysis of the international economic growth.

median regression, quantile regression, sparse models