Testing exogeneity in nonparametric instrumental variables models identified by conditional quantile restrictions

2020 ◽  
Author(s):  
Jia-Young Michael Fu ◽  
Joel L Horowitz ◽  
Matthias Parey
Keyword(s):  
Quantile Regression ◽  
Instrumental Variables ◽  
Regression Models ◽  
Quantile Function ◽  
Structural Function ◽  
Conditional Quantile ◽  
Explanatory Variables ◽  
Monte Carlo Experiments ◽  
Nonparametric Estimates ◽  
Conditional Quantile Function

Summary This paper presents a test for exogeneity of explanatory variables in a nonparametric instrumental variables (IV) model whose structural function is identified through a conditional quantile restriction. Quantile regression models are increasingly important in applied econometrics. As with mean-regression models, an erroneous assumption that the explanatory variables in a quantile regression model are exogenous can lead to highly misleading results. In addition, a test of exogeneity based on an incorrectly specified parametric model can produce misleading results. This paper presents a test of exogeneity that does not assume that the structural function belongs to a known finite-dimensional parametric family and does not require estimation of this function. The latter property is important because nonparametric estimates of the structural function are unavoidably imprecise. The test presented here is consistent whenever the structural function differs from the conditional quantile function on a set of nonzero probability. The test has nontrivial power uniformly over a large class of structural functions that differ from the conditional quantile function by $O({n^{ - 1/2}})$. The results of Monte Carlo experiments and an empirical application illustrate the performance of the test.

scholarly journals What Quantile Regression Does and Doesn't Do

2018 ◽  
Author(s):  
Sebastian Ernst Wenz
Keyword(s):  
Linear Regression ◽  
Quantile Regression ◽  
Regression Models ◽  
Simulated Data ◽  
Quantile Function ◽  
Linear Regression Models ◽  
Conditional Quantile ◽  
Correct Understanding ◽  
Mean Function ◽  
Conditional Quantile Function

Petscher and Logan (2014)’s description of quantile regression might mislead readers to believe it would estimate the relation between an outcome, y, and one or more predictors, x, at different quantiles of the unconditional distribution of y. However, quantile regression models the conditional quantile function of y given x just as linear regression models the conditional mean function. This article’s contribution is twofold: First, it discusses potential consequences of methodological misconceptions and formulations of Petscher and Logan (2014)’s presentation by contrasting features of quantile regression and linear regression. Secondly, it reinforces the importance of correct understanding of quantile regression in empirical research by illustrating similarities and differences of various quantile regression estimators and linear regression using simulated data.

qmodel: A command for fitting parametric quantile models

2019 ◽  
Vol 19 (2) ◽  
pp. 261-293 ◽  
Author(s):  
Matteo Bottai ◽  
Nicola Orsini
Keyword(s):  
Quantile Regression ◽  
Simulated Data ◽  
Quantile Function ◽  
Parametric Models ◽  
Outcome Variable ◽  
Simple Type ◽  
Conditional Quantile ◽  
Quantile Functions ◽  
Conditional Quantile Function ◽  
Insight Into

In this article, we introduce the qmodel command, which fits parametric models for the conditional quantile function of an outcome variable given covariates. Ordinary quantile regression, implemented in the qreg command, is a popular, simple type of parametric quantile model. It is widely used but known to yield erratic estimates that often lead to uncertain inferences. Parametric quantile models overcome these limitations and extend modeling of conditional quantile functions beyond ordinary quantile regression. These models are flexible and efficient. qmodel can estimate virtually any possible linear or nonlinear parametric model because it allows the user to specify any combination of qmodel-specific built-in functions, standard mathematical and statistical functions, and substitutable expressions. We illustrate the potential of parametric quantile models and the use of the qmodel command and its postestimation commands through realand simulated-data examples that commonly arise in epidemiological and pharmacological research. In addition, this article may give insight into the close connection that exists between quantile functions and the true mathematical laws that generate data.

scholarly journals ON USING LINEAR QUANTILE REGRESSIONS FOR CAUSAL INFERENCE

2016 ◽  
Vol 33 (3) ◽  
pp. 664-690 ◽  
Author(s):  
Ryutah Kato ◽  
Yuya Sasaki
Keyword(s):  
Causal Inference ◽  
Unobserved Heterogeneity ◽  
Weighted Average ◽  
Quantile Function ◽  
Structural Function ◽  
Slope Parameter ◽  
Quantile Regressions ◽  
Conditional Quantile ◽  
Conditional Quantile Function ◽  
Linear Regressions

We show that the slope parameter of the linear quantile regression measures a weighted average of the local slopes of the conditional quantile function. Extending this result, we also show that the slope parameter measures a weighted average of the partial effects for a general structural function. Our results support the use of linear quantile regressions for causal inference in the presence of nonlinearity and multivariate unobserved heterogeneity. The same conclusion applies to linear regressions.

scholarly journals Asymptotic Normality of a Nonparametric Conditional Quantile Estimator for Random Fields

2011 ◽  
Vol 2011 ◽  
pp. 1-35
Author(s):  
Sidi Ali Ould Abdi ◽  
Sophie Dabo-Niang ◽  
Aliou Diop ◽  
Ahmedoune Ould Abdi
Keyword(s):  
Asymptotic Normality ◽  
Random Fields ◽  
Quantile Function ◽  
Spatial Process ◽  
Kernel Estimate ◽  
Conditional Quantile ◽  
Conditional Quantile Estimator ◽  
Conditional Quantile Function ◽  
Mixing Sequence ◽  
Explicative Variable

Given a stationary multidimensional spatial process , we investigate a kernel estimate of the spatial conditional quantile function of the response variable given the explicative variable . Asymptotic normality of the kernel estimate is obtained when the sample considered is an -mixing sequence.

scholarly journals Nonparametric Estimation of a Conditional Quantile Function in a Fixed Effects Panel Data Model

2018 ◽  
Vol 11 (3) ◽  
pp. 44 ◽  
Author(s):  
Karen Yan ◽  
Qi Li
Keyword(s):  
Panel Data ◽  
Data Model ◽  
Numerical Optimization ◽  
Fixed Effects ◽  
Quantile Function ◽  
Panel Data Model ◽  
Nonparametric Method ◽  
Conditional Quantile ◽  
Finite Samples ◽  
Conditional Quantile Function

This paper develops a nonparametric method to estimate a conditional quantile function for a panel data model with an additive individual fixed effects. The proposed method is easy to implement, it does not require numerical optimization and automatically ensures quantile monotonicity by construction. Monte Carlo simulations show that the proposed estimator performs well in finite samples.

scholarly journals NONPARAMETRIC WEIGHTED AVERAGE QUANTILE DERIVATIVE

2021 ◽  
pp. 1-39
Author(s):  
Ying-Ying Lee
Keyword(s):  
Linear Models ◽  
Weighted Average ◽  
Linear Representation ◽  
Quantile Function ◽  
Structural Effect ◽  
Compact Set ◽  
Conditional Quantile ◽  
Partially Linear ◽  
Conditional Quantile Function ◽  
Quantile Processes

The weighted average quantile derivative (AQD) is the expected value of the partial derivative of the conditional quantile function (CQF) weighted by a function of the covariates. We consider two weighting functions: a known function chosen by researchers and the density function of the covariates that is parallel to the average mean derivative in Powell, Stock, and Stoker (1989, Econometrica 57, 1403–1430). The AQD summarizes the marginal response of the covariates on the CQF and defines a nonparametric quantile regression coefficient. In semiparametric single-index and partially linear models, the AQD identifies the coefficients up to scale. In nonparametric nonseparable structural models, the AQD conveys an average structural effect under certain independence assumptions. Including a stochastic trimming function, the proposed two-step estimator is root-n-consistent for the AQD defined by the entire support of the covariates. To facilitate tractable asymptotic analysis, a key preliminary result is a new Bahadur-type linear representation of the generalized inverse kernel-based CQF estimator uniformly over the covariates in an expanding compact set and over the quantile levels. The weak convergence to Gaussian processes applies to the differentiable nonlinear functionals of the quantile processes.

scholarly journals Quantile Regression Models and the Crucial Difference Between Individual-Level Effects and Population-Level Effects

2020 ◽  
Author(s):  
Nicolai T. Borgen ◽  
Andreas Haupt ◽  
Øyvind N. Wiborg
Keyword(s):  
Quantile Regression ◽  
Regression Models ◽  
Treatment Effects ◽  
Population Level ◽  
Conditional Quantile ◽  
Individual Level ◽  
Quantile Treatment Effects ◽  
Crucial Difference ◽  
Practical Guidelines ◽  
Conceptual Distinction

The unconditional quantile regression (UQR) model – which has gained increasing popularity in the 2010s and is regularly applied in top-rated academic journals within sociology and other disciplines – is poorly understood and frequently misinterpreted. The main reason for its increased popularity is that the UQR model seemingly tackles an issue with the traditional conditional quantile regression (CQR) model: the interpretation of coefficients as quantile treatment effects changes whenever control variables are included. However, the UQR model was not developed to solve this issue but to study influences on quantile values of the overall outcome distribution. This paper clarifies the crucial conceptual distinction between influences on overall distributions, which we term population-level influences, and individual-level quantile treatment effects. Further, we use data simulations to illustrate that various classes of quantile regression models may, in some instances, give entirely different conclusions (to different questions). The conceptual and empirical distinctions between various quantile regression models underline the need to match the correct quantile regression model to the specific research questions. We conclude the paper with some practical guidelines for researchers.

RISK MARGIN QUANTILE FUNCTION VIA PARAMETRIC AND NON-PARAMETRIC BAYESIAN APPROACHES

2015 ◽  
Vol 45 (3) ◽  
pp. 503-550 ◽  
Author(s):  
Alice X.D. Dong ◽  
Jennifer S.K. Chan ◽  
Gareth W. Peters
Keyword(s):  
Quantile Regression ◽  
Regression Model ◽  
Loss Function ◽  
Regression Models ◽  
Quantile Function ◽  
Parametric Model ◽  
Parametric Models ◽  
Risk Margin ◽  
Quantile Functions ◽  
Non Parametric

AbstractWe develop quantile functions from regression models in order to derive risk margin and to evaluate capital in non-life insurance applications. By utilizing the entire range of conditional quantile functions, especially higher quantile levels, we detail how quantile regression is capable of providing an accurate estimation of risk margin and an overview of implied capital based on the historical volatility of a general insurers loss portfolio. Two modeling frameworks are considered based around parametric and non-parametric regression models which we develop specifically in this insurance setting. In the parametric framework, quantile functions are derived using several distributions including the flexible generalized beta (GB2) distribution family, asymmetric Laplace (AL) distribution and power-Pareto (PP) distribution. In these parametric model based quantile regressions, we detail two basic formulations. The first involves embedding the quantile regression loss function from the nonparameteric setting into the argument of the kernel of a parametric data likelihood model, this is well known to naturally lead to the AL parametric model case. The second formulation we utilize in the parametric setting adopts an alternative quantile regression formulation in which we assume a structural expression for the regression trend and volatility functions which act to modify a base quantile function in order to produce the conditional data quantile function. This second approach allows a range of flexible parametric models to be considered with different tail behaviors. We demonstrate how to perform estimation of the resulting parametric models under a Bayesian regression framework. To achieve this, we design Markov chain Monte Carlo (MCMC) sampling strategies for the resulting Bayesian posterior quantile regression models. In the non-parametric framework, we construct quantile functions by minimizing an asymmetrically weighted loss function and estimate the parameters under the AL proxy distribution to resemble the minimization process. This quantile regression model is contrasted to the parametric AL mean regression model and both are expressed as a scale mixture of uniform distributions to facilitate efficient implementation. The models are extended to adopt dynamic mean, variance and skewness and applied to analyze two real loss reserve data sets to perform inference and discuss interesting features of quantile regression for risk margin calculations.

QUANTILE REGRESSION WITH MISMEASURED COVARIATES

2008 ◽  
Vol 24 (4) ◽  
pp. 1010-1043 ◽  
Author(s):  
Susanne M. Schennach
Keyword(s):  
Measurement Error ◽  
Quantile Regression ◽  
Instrumental Variables ◽  
Regression Models ◽  
Compact Set ◽  
Consistent Estimation ◽  
Nonlinear Functional ◽  
Conditional Expectations ◽  
Quantile Functions ◽  
Derivatives Of

This paper establishes that the availability of instrumental variables enables the identification and the consistent estimation of nonparametric quantile regression models in the presence of measurement error in the regressors. The proposed estimator takes the form of a nonlinear functional of derivatives of conditional expectations and is shown to provide estimated quantile functions that are uniformly consistent over a compact set.

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