85528833
Smoothed GMM for quantile models
This paper develops theory for feasible estimators of finite-dimensional parameters identified by general conditional quantile restrictions, under much weaker assumptions than previously seen in the literature. This includes instrumental variables nonlinear quantile regression as a special case. More specifically, we consider a set of unconditional moments implied by the conditional quantile restrictions, providing conditions for local identification. Since estimators based on the sample moments are generally impossible to compute numerically in practice, we study feasible estimators based on smoothed sample moments. We propose a method of moments estimator for exactly identified models, as well as a generalized method of moments estimator for over-identified models. We establish consistency and asymptotic normality of both estimators under general conditions that allow for weakly dependent data and nonlinear structural models. Simulations illustrate the finite-sample properties of the methods. Our in-depth empirical application concerns the consumption Euler equation derived from quantile utility maximization. Advantages of the quantile Euler equation include robustness to fat tails, decoupling of risk attitude from the elasticity of intertemporal substitution, and log-linearization without any approximation error. For the four countries we examine, the quantile estimates of discount factor and elasticity of intertemporal substitution are economically reasonable for a range of quantiles above the median, even when two-stage least squares estimates are not reasonable.
16793469
An IV Model of Quantile Treatment Effects
The ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications. However, in observational studies, the variables of interest (e.g., education, prices) are often endogenous, making conventional quantile regression inconsistent and hence inappropriate for recovering the causal effects of these variables on the quantiles of economic outcomes. In order to address this problem, we develop a model of quantile treatment effects (QTE) in the presence of endogeneity and obtain conditions for identification of the QTE without functional form assumptions. The principal feature of the model is the imposition of conditions that restrict the evolution of ranks across treatment states. This feature allows us to overcome the endogeneity problem and recover the true QTE through the use of instrumental variables. The proposed model can also be equivalently viewed as a structural simultaneous equation model with nonadditive errors, where QTE can be interpreted as the structural quantile effects (SQE). Copyright The Econometric Society 2005.
121283197
Conditional empirical likelihood estimation and inference for quantile regression models
Abstract This paper considers two empirical likelihood-based estimation, inference, and specification testing methods for quantile regression models. First, we apply the method of conditional empirical likelihood (CEL) by Kitamura et al. [2004. Empirical likelihood-based inference in conditional moment restriction models. Econometrica 72, 1667–1714] and Zhang and Gijbels [2003. Sieve empirical likelihood and extensions of the generalized least squares. Scandinavian Journal of Statistics 30, 1–24] to quantile regression models. Second, to avoid practical problems of the CEL method induced by the discontinuity in parameters of CEL, we propose a smoothed counterpart of CEL, called smoothed conditional empirical likelihood (SCEL). We derive asymptotic properties of the CEL and SCEL estimators, parameter hypothesis tests, and model specification tests. Important features are (i) the CEL and SCEL estimators are asymptotically efficient and do not require preliminary weight estimation; (ii) by inverting the CEL and SCEL ratio parameter hypothesis tests, asymptotically valid confidence intervals can be obtained without estimating the asymptotic variances of the estimators; and (iii) in contrast to CEL, the SCEL method can be implemented by some standard Newton-type optimization. Simulation results demonstrate that the SCEL method in particular compares favorably with existing alternatives.
41491047
SMOOTHED ESTIMATING EQUATIONS FOR INSTRUMENTAL VARIABLES QUANTILE REGRESSION
The moment conditions or estimating equations for instrumental variables quantile regression involve the discontinuous indicator function. We instead use smoothed estimating equations (SEE), with bandwidth h. We show that the mean squared error (MSE) of the vector of the SEE is minimized for some h > 0, leading to smaller asymptotic MSE of the estimating equations and associated parameter estimators. The same MSE-optimal h also minimizes the higher-order type I error of a SEE-based χ2 test and increases size-adjusted power in large samples. Computation of the SEE estimator also becomes simpler and more reliable, especially with (more) endogenous regressors. Monte Carlo simulations demonstrate all of these superior properties in finite samples, and we apply our estimator to JTPA data. Smoothing the estimating equations is not just a technical operation for establishing Edgeworth expansions and bootstrap refinements; it also brings the real benefits of having more precise estimators and more powerful tests.
127457133
A closed-form estimator for quantile treatment effects with endogeneity
Abstract This paper studies the estimation of quantile treatment effects based on the instrumental variable quantile regression (IVQR) model ( Chernozhukov and Hansen, 2005 ). I develop a class of flexible plug-in estimators based on closed-form solutions derived from the IVQR moment conditions. The proposed estimators remain tractable and root-n-consistent, while allowing for rich patterns of effect heterogeneity. Functional central limit theorems and bootstrap validity results for the estimators of the quantile treatment effects and other functionals are provided. Monte Carlo simulations demonstrate favorable finite sample properties of the proposed approach. I apply my method to reanalyze the causal effect of 401(k) plans.
6676830
Estimation of Semiparametric Models When the Criterion Function is Not Smooth
We provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated. Our results extend existing theories like those of Pakes and Pollard (1989), Andrews (1994a) and Newey (1994). We also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters. We apply our results to two examples: a 'hit rate' and a partially linear median regression with some endogenous regressors.
2447122
Efficient Estimation of Semiparametric Conditional Moment Models with Possibly Nonsmooth Residuals
This paper considers semiparametric efficient estimation of conditional moment models with possibly nonsmooth residuals in unknown parametric components (theta) and unknown functions (h) of endogenous variables. We show that: (1) the penalized sieve minimum distance (PSMD) estimator (theta\hat,h\hat) can simultaneously achieve root-n asymptotic normality of theta\hat and nonparametric optimal convergence rate of h\hat, allowing for noncompact function parameter spaces; (2) a simple weighted bootstrap procedure consistently estimates the limiting distribution of the PSMD theta\hat; (3) the semiparametric efficiency bound formula of Ai and Chen (2003) remains valid for conditional models with nonsmooth residuals, and the optimally weighted PSMD estimator achieves the bound; (4) the centered, profiled optimally weighted PSMD criterion is asymptotically chi-square distributed. We illustrate our theories using a partially linear quantile instrumental variables (IV) regression, a Monte Carlo study, and an empirical estimation of the shape-invariant quantile IV Engel curves.
52220432
Sieve Semiparametric Two-Step GMM Under Weak Dependence
This paper considers semiparametric two-step GMM estimation and inference with weakly dependent data, where unknown nuisance functions are estimated via sieve extremum estimation in the first step. We show that although the asymptotic variance of the second-step GMM estimator may not have a closed form expression, it can be well approximated by sieve variances that have simple closed form expressions. We present consistent or robust variance estimation, Wald tests and Hansen’s (1982) over-identification tests for the second step GMM that properly reflect the first-step estimated functions and the weak dependence of the data. Our sieve semiparametric two-step GMM inference procedures are shown to be numerically equivalent to the ones computed as if the first step were parametric. A new consistent random-perturbation estimator of the derivative of the expectation of the non-smooth moment function is also provided.
119518425
Ordinal utility models of decision making under uncertainty
This paper studies two models of rational behavior under uncertainty whose predictions are invariant under ordinal transformations of utility. The ‘quantile utility’ model assumes that the agent maximizes some quantile of the distribution of utility. The ‘utility mass’ model assumes maximization of the probability of obtaining an outcome whose utility is higher than some fixed critical value. Both models satisfy weak stochastic dominance. Lexicographic refinements satisfy strong dominance.The study of these utility models suggests a significant generalization of traditional ideas of riskiness and risk preference. We define one action to be riskier than another if the utility distribution of the latter crosses that of the former from below. The single crossing property is equivalent to a ‘minmax spread’ of a random variable. With relative risk defined by the single crossing criterion, the risk preference of a quantile utility maximizer increases with the utility distribution quantile that he maximizes. The risk preference of a utility mass maximizer increases with his critical utility value.
10657976
An Axiomatization of Quantiles on the Domain of Distribution Functions
In an environment in which the primitive is the space of distribution functions, we characterize the quantile functions by the axioms ordinal covariance, monotonicity with respect to first-order stochastic dominance, and upper semicontinuity. We show how to characterize the VaR in a similar manner.
607334
Quantile Maximization in Decision Theory
This paper introduces a model of preferences, in which, given beliefs about uncertain outcomes, an individual evaluates an action by a quantile of the induced distribution. The choice rule of Quantile Maximization unifies maxmin and maxmax as maximizing the lowest and the highest quantiles of beliefs distributions, respectively, and offers a family of less extreme preferences. Taking preferences over acts as a primitive, we axiomatize Quantile Maximization in a Savage setting. Our axiomatization also provides a novel derivation of subjective beliefs, which demonstrates that neither the monotonicity nor the continuity conditions assumed in the literature are essential for probabilistic sophistication. We characterize preferences of quantile maximizers towards downside risk. We discuss how the distinct properties of the model, robustness and ordinality, can be useful in studying choice behaviour for categorical variables and in economic policy design. We also offer applications to poll design and insurance problems. Copyright , Wiley-Blackwell.
214191503
Dynamic Quantile Models of Rational Behavior
This paper develops a dynamic model of rational behavior under uncertainty, in which the agent maximizes the stream of future 
 τ‐quantile utilities, for 
 τ ∈ (0,1). That is, the agent has a quantile utility preference instead of the standard expected utility. Quantile preferences have useful advantages, including the ability to capture heterogeneity and allowing the separation between risk aversion and elasticity of intertemporal substitution. Although quantiles do not share some of the helpful properties of expectations, such as linearity and the law of iterated expectations, we are able to establish all the standard results in dynamic models. Namely, we show that the quantile preferences are dynamically consistent, the corresponding dynamic problem yields a value function, via a fixed point argument, this value function is concave and differentiable, and the principle of optimality holds. Additionally, we derive the corresponding Euler equation, which is well suited for using well‐known quantile regression methods for estimating and testing the economic model. In this way, the parameters of the model can be interpreted as structural objects. Therefore, the proposed methods provide microeconomic foundations for quantile regression methods. To illustrate the developments, we construct an intertemporal consumption model and estimate the discount factor and elasticity of intertemporal substitution parameters across the quantiles. The results provide evidence of heterogeneity in these parameters.

145926581
Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns
This paper studies the time-series behavior of asset returns and aggregate consumption. Using a representative consumer model and imposing restrictions on preferences and the joint distribution of consumption and returns, we deduce a restricted log-linear time-series representation. Preference parameters for the representative agent are estimated and the implied restrictions are tested using postwar data.
7940245
Consumption, Income, and Interest Rates: Reinterpreting the Time Series Evidence
This paper proposes that the time-series data on consumption, income, and interest rates are best viewed as generated not by a single representative consumer but by two groups of consumers. Half the consumers are forward-looking and consume their permanent income, but are extremely reluctant to substitute consumption intertemporally. Half the consumers follow the "rule of thumb" of consuming their current income. The paper documents three empirical regularities that, it argues, are best explained by this model. First, expected changes in income are associated with expected changes in consumption. Second, expected real interest rates are not associated with expected changes in consumption. Third, periods in which consumption is high relative to income are typically followed by high growth in income. The paper concludes by briefly discussing the implications of these findings for economic policy and economic research.
55031289
Measuring Intertemporal Substitution: The Role of Durable Goods
In estimating, the intertemporal elasticity of substitution, Hall finds that, when one takes account of time aggregation, point estimates are small and not significantly different from zero. He concludes that the clasticity is unlikely to be much above 0.1 and may well be zero. Applying improved inference methods to an economic model similar to Hall's, Hansen and Singleton show that there is considerably less precision in the estimation. We argue that the model used by these authors is misspecified because the intratemporal substitution between nondurable consumption goods and durable consumption goods is ignored. We use a two‐step procedure that combines a cointegration approach to preference parameter estimation with generalized method of moments to take these effects into account. Our estimates for the intertemporal elasticity of substitution are positive and significantly different from zero, even when time aggregation is taken into account.
18549143
Estimating the Elasticity of Intertemporal Substitution When Instruments Are Weak
In the instrumental variables (IV) regression model, weak instruments can lead to bias in estimators and size distortion in hypothesis tests. This paper examines how weak instruments affect the identification of the elasticity of intertemporal substitution (EIS) through the linearized Euler equation. Conventional IV methods result in an empirical puzzle that the EIS is significantly less than 1 but its reciprocal is not different from 1. This paper shows that weak instruments can explain the puzzle and reports valid confidence intervals for the EIS using pivotal statistics. The EIS is less than 1 and not significantly different from 0 for eleven developed countries.
14574304
An MCMC Approach to Classical Estimation
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.
31512770
ASYMPTOTIC THEORY FOR NONLINEAR QUANTILE REGRESSION UNDER WEAK DEPENDENCE
This paper studies the asymptotic properties of the nonlinear quantile regression model under general assumptions on the error process, which is allowed to be heterogeneous and mixing. We derive the consistency and asymptotic normality of regression quantiles under mild assumptions. First-order asymptotic theory is completed by a discussion of consistent covariance estimation.
44198279
Instrumental Variable Quantile Estimation of Spatial Autoregressive Models
We propose an instrumental variable quantile regression (IVQR) estimator for spatial autoregressive (SAR) models. Like the GMM estimators of Lin and Lee (2006) and Kelejian and Prucha (2006), the IVQR estimator is robust against heteroscedasticity. Unlike the GMM estimators, the IVQR estimator is also robust against outliers and requires weaker moment conditions. More importantly, it allows us to characterize the heterogeneous impact of variables on different points (quantiles) of a response distribution. We derive the limiting distribution of the new estimator. Simulation results show that the new estimator performs well in finite samples at various quantile points. In the special case of median restriction, it outperforms the conventional QML estimator without taking into account of heteroscedasticity in the errors; it also outperforms the GMM estimators with or without considering the heteroscedasticity.
54901949
Exact computation of GMM estimators for instrumental variable quantile regression models
We show that the generalized method of moments (GMM) estimation problem in instrumental variable quantile regression (IVQR) models can be equivalently formulated as a mixed integer quadratic programming problem. This enables exact computation of the GMM estimators for the IVQR models. We illustrate the usefulness of our algorithm via Monte Carlo experiments and an application to demand for fish.
3965047
Smoothed Quantile Regression for Panel Data
This paper studies fixed effects estimation of quantile regression models for panel data. Under an asymptotic framework where both the numbers of individuals and time periods grow at the same rate, we show that the fixed-effects estimator for the smoothed objective function has a limiting normal distribution with a bias in the mean, and provide the analytic form of the asymptotic bias. We propose a one-step bias correction estimator based on the analytic bias formula obtained from the asymptotic analysis. Importantly, our results cover the case that observations are dependent over time. We illustrate the effects of the bias correction through simulations.
122867679
A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix
This paper describes a simple method of calculating a heteroskedasticity and autocorrelation consistent covariance matrix that is positive semi-definite by construction. It also establishes consistency of the estimated covariance matrix under fairly general conditions. (This abstract was borrowed from another version of this item.)
10302429
Generalized Simulated Annealing for Global Optimization: The GenSA Package
Many problems in statistics, finance, biology, pharmacology, physics, mathematics, economics, and chemistry involve determination of the global minimum of multidimensional functions. R packages for different stochastic methods such as genetic algorithms and differential evolution have been developed and successfully used in the R community. Based on Tsallis statistics, the R package GenSA was developed for generalized simulated annealing to process complicated non-linear objective functions with a large number of local minima. In this paper we provide a brief introduction to the R package and demonstrate its utility by solving a non-convex portfolio optimization problem in finance and the Thomson problem in physics. GenSA is useful and can serve as a complementary tool to, rather than a replacement for, other widely used R packages for optimization.
12372146
Consistency of Kernel Estimators of Heteroscedastic and Autocorrelated Covariance Matrices
Conditions are derived for the consistency of kernel estimators of the covariance matrix of a sum of vectors of dependent heterogeneous random variables, which match those of the currently best-known conditions for the central limit theorem, as required for a unified theory of asymptotic inference. These include finite moments of order no more than 2 + for > 0, trending variances, and variables which are near-epoch dependent on a mixing process, but not necessarily mixing. The results are also proved for the case of sample-dependent bandwidths.
122987368
The Existence of Moments of k-Class Estimators
For the regression equation C = A,B + e where A is a p x q stochastic matrix whose elements are independently distributed and contemporaneously correlated with the elements of 8, the lth moment of the least squares estimator of (3 exists if and only if l < p - q + 1. In particular, this implies that the lth moment of the k-class estimator of the coefficients of the G1 -1 non-normalizing endogenous variables of an equation with K1 included and K2 excluded exogenous variables in a simultaneous system with N observations exists if and only if 1 < M where
29051333
Should We Go One Step Further? An Accurate Comparison of One-Step and Two-Step Procedures in a Generalized Method of Moments Framework
According to the conventional asymptotic theory, the two-step Generalized Method of Moments (GMM) estimator and test perform as least as well as the one-step estimator and test in large samples. The conventional asymptotic theory, as elegant and convenient as it is, completely ignores the estimation uncertainty in the weighting matrix, and as a result it may not reflect finite sample situations well. In this paper, we employ the fixed-smoothing asymptotic theory that accounts for the estimation uncertainty, and compare the performance of the one-step and two-step procedures in this more accurate asymptotic framework. We show the two-step procedure outperforms the one-step procedure only when the benefit of using the optimal weighting matrix outweighs the cost of estimating it. This qualitative message applies to both the asymptotic variance comparison and power comparison of the associated tests. A Monte Carlo study lends support to our asymptotic results.
154473095
Consumption-Based Asset Pricing
This chapter reviews the behavior of financial asset prices in relation to consumption. The chapter lists some important stylized facts that characterize U.S. data, and relates them to recent developments in equilibrium asset pricing theory. Data from other countries are examined to see which features of the U.S. experience apply more generally. The chapter argues that to make sense of asset market behavior one needs a model in which the market price of risk is high, time-varying, and correlated with the state of the economy. Models that have this feature, including models with habit-formation in utility, heterogeneous investors, and irrational expectations, are discussed. The main focus is on stock returns and short-term real interest rates, but bond returns are also considered.
153555594
Recursive Macroeconomic Theory
Recursive methods offer a powerful approach in dynamic macroeconomics. This book contains both an introduction to recursive tools, including standard applications such as asset pricing, and advanced material, including analyses of reputational mechanisms and contract design. The tools are presented with enough technical sophistication to get the reader started working on practical problems. When numerical simulations are called for, the book provides suggestions for how to proceed, as well as references for further reading. The applications cover many substantive issues in macroeconomics, such as equilibrium asset prices, market incompleteness, wealth distribution, fiscal-monetary theories of inflation, government debt, optimal labour and capital taxation, time consistency and credible government policies, optimal social insurance, economic growth and labour market dynamics.
54628310
CONSUMPTION AND PORTFOLIO DECISIONS WHEN EXPECTED RETURNS ARE TIME VARYING *
This paper presents an approximate analytical solution to the optimal consumption and portfolio choice problem of an innitely lived investor with Epstein-Zin-Weil utility who faces a constant riskless interest rate and a timevarying equity premium. When the model is calibrated to U. S. stock market data, it implies that intertemporal hedging motives greatly increase, and may even double, the average demand for stocks by investors whose risk-aversion coefficients exceed one. The optimal portfolio policy also involves timing the stock market. Failure to time or to hedge can cause large welfare losses relative to the optimal policy.
154923121
Asset Pricing Under Quantile Utility Maximization
“Focus on the downside, and the upside will take care of itself” is a famous quote among professional investors. By considering an agent who follows this advice, we reproduce the first and second moments of stock returns, risk-free rate and consumption growth. The agent's behavior toward risk is analogous to a relative risk aversion of about 3 under expected utility, the elasticity of intertemporal substitution is about 0.5 and the time discount factor is below 1. In particular, the proposed model separates time and risk preferences in an innovative way.
88519089
Local Identification of Nonparametric and Semiparametric Models
In parametric models a sufficient condition for local identification is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix. We show that there are corresponding sufficient conditions for nonparametric models. A nonparametric rank condition and differentiability of the moment conditions with respect to a certain norm imply local identification. It turns out these conditions are slightly stronger than needed and are hard to check, so we provide weaker and more primitive conditions. We extend the results to semiparametric models. We illustrate the sufficient conditions with endogenous quantile and single index examples. We also consider a semiparametric habit-based, consumption capital asset pricing model. There we find the rank condition is implied by an integral equation of the second kind having a one-dimensional null space.
122662932
CONSISTENCY IN NONLINEAR ECONOMETRIC MODELS: A GENERIC UNIFORM LAW OF LARGE NUMBERS
A basic tool of modern econometrics is a uniform law of large numbers (LLN). It is a primary ingredient used in proving consistency and asymptotic normality of parametric and nonparametric estimators in nonlinear econometric models. Thus, in a well-known review article, Burguete, Gallant, and Sousa [8, p. 162] introduce a uniform LLN with the statement: "The following theorem is the result upon which the asymptotic theory of nonlinear econometrics rests. "So pervasive is the use of uniform LLNs, that numerous authors appeal to an unspecified generic uniform LLN. Others appeal to some specific result. The purpose of this paper is to provide a generic uniform LLN that is sufficiently general to incorporate most applications of uniform LLNs in the nonlinear econometrics literature. In summary, the paper presents a result that can be used to turn state of the art pointwise LLNs into uniform LLNs over compact sets, with the addition of a single smoothness condition -- either a Lipschitz condition or a derivative condition. The latter is particularly easy to verify, and is implied by common assumptions used to prove asymptotic normality of estimators. Thus, the additional condition is not particularly restrictive. In contrast to other uniform LLNs that appear in the literature, the one given here allows the full range of heterogeneity of summands (i.e., non-identical distributions), and temporal dependence, that is available with pointwise LLNs.
29064974
Asymptotic normality of Powell’s kernel estimator
We establish asymptotic normality of Powell’s kernel estimator for the asymptotic covariance matrix of the quantile regression estimator for both i.i.d. and weakly dependent data. As an application, we derive the optimal bandwidth that minimizes the approximate mean squared error of the kernel estimator. We also derive the corresponding results to censored quantile regression.