Conditional Moment Restrictions Research Articles

Semiparametric Efficiency Bound for Models of Sequential Moment Restrictions Containing Unknown Functions

This paper computes the semiparametric efficiency bound for finite dimensional parameters identified by models of sequential moment restrictions containing unknown functions. Our results extend those of Chamberlain (1992b) and Ai and Chen (2003) for semiparametric conditional moment restriction models with identical information sets to the case of nested information sets, and those of Chamberlain (1992a) and Brown and Newey (1998) for models of sequential moment restrictions without unknown functions to cases with unknown functions of possibly endogenous variables. Our bound results are applicable to semiparametric panel data models and semiparametric two stage plug-in problems. As an example, we compute the efficiency bound for a weighted average derivative of a nonparametric instrumental variables (IV) regression, and find that the simple plug-in estimator is not efficient. Finally, we present an optimally weighted, orthogonalized, sieve minimum distance estimator that achieves the semiparametric efficiency bound.

Optimal Inference in Dynamic Models with Conditional Moment Restrictions

By an application of the theory of optimal estimating function, optimal instruments for dynamic models with conditional moment restrictions are derived. The general efficiency bound is provided, along with estimators attaining the bound. It is demonstrated that the optimal estimators are always at least as efficient as the traditional optimal generalized method of moments estimator, and usually more efficient. The form of our optimal instruments resembles that from Newey (1990), but involves conditioning on the history of the stochastic process. In the special case of i.i.d. observations, our optimal estimator reduces to Newey’s. Specification and hypothesis testing in our framework are introduced. We derive the theory of optimal instruments and the associated asymptotic distribution theory for general cases including non-martingale estimating functions and general history dependence. Examples involving time-varying conditional volatility and stochastic volatility are offered.

A semiparametric regression estimator under left truncation and right censoring

An estimator is proposed for semiparametric linear regression models with left truncated and right censored dependent variables. The estimator is derived from a moment condition following the principles of Newey [Newey, W.K., 2001. Conditional moment restrictions in censored and truncated regression models. Econometric Theory 17, 863–888] on conditional moment conditions. Consistency of the estimator is shown and simulation is used for illustration of the small sample properties.

Inference in panel data models under attrition caused by unobservables

This paper concerns identification and estimation of a finite-dimensional parameter in a panel data-model under nonignorable sample attrition. Attrition can depend on second period variables which are unobserved for the attritors but an independent refreshment sample from the marginal distribution of the second period values is available. This paper shows that under a quasi-separability assumption, the model implies a set of conditional moment restrictions where the moments contain the attrition function as an unknown parameter. This formulation leads to (i) a simple proof of identification under strictly weaker conditions than those in the existing literature and, more importantly, (ii) a sieve-based root-n consistent estimate of the finite-dimensional parameter of interest. These methods are applicable to both linear and nonlinear panel data models with endogenous attrition and analogous methods are applicable to situations of endogenously missing data in a single cross-section. The theory is illustrated with a simulation exercise, using Current Population Survey data where a panel structure is introduced by the rotation group feature of the sampling process.

Asset Prices Under Habit Formation and Reference-Dependent Preferences

This article explains the high level and the countercyclical variation of the equity premium in a consumption-based asset pricing model with low large-scale risk aversion. Investors have gain-loss utility over consumption relative to slowly time-varying habit. Stocks deliver low returns in recessions when consumption falls below habit; investors therefore require a high premium for holding stocks. The model's conditional moment restrictions are tested on consumption and asset returns data. The empirical estimate of large-scale risk aversion is low, whereas the estimate of loss aversion agrees with prior experimental evidence.

Maximum empirical likelihood estimation of continuous-time models with conditional characteristic functions

For some popular financial continuous-time models, tractable expressions of likelihood functions are unknown. For that reason, the maximum likelihood estimation method is infeasible. Fortunately, closed functional forms of conditional characteristic functions of some of these models are known. We construct an empirical likelihood estimation method using tractable conditional characteristic functions to estimate such a model. This method resolves the problem of covariance matrix singularity in the standard generalized method of moments and fully utilizes information in conditional moment restrictions. It is applicable to many popular financial models such as some diffusion models, jump diffusion models, and stochastic volatility models. Using a Monte Carlo comparison, we show that this method provides superior performance compared to other methods in some situations.

SEMINONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION OF CONDITIONAL MOMENT RESTRICTION MODELS*

This article studies estimation of a conditional moment restriction model with the seminonparametric maximum likelihood approach proposed by Gallant and Nychka (Econometrica 55 (March 1987), 363–90). Under some sufficient conditions, we show that the estimator of the finite dimensional parameter θ is asymptotically normally distributed and attains the semiparametric efficiency bound and that the estimator of the density function is consistent under L2 norm. Some results on the convergence rate of the estimated density function are derived. An easy to compute covariance matrix for the asymptotic covariance of the θ estimator is presented.

Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves

This paper studies a shape-invariant Engel curve system with endogenous total expenditure, in which the shape-invariant specification involves a common shift parameter for each demographic group in a pooled system of nonparametric Engel curves. We focus on the identification and estimation of both the nonparametric shapes of the Engel curves and the parametric specification of the demographic scaling parameters. The identification condition relates to the bounded completeness and the estimation procedure applies the sieve minimum distance estimation of conditional moment restrictions, allowing for endogeneity. We establish a new root mean squared convergence rate for the nonparametric instrumental variable regression when the endogenous regressor could have unbounded support. Root-n asymptotic normality and semiparametric efficiency of the parametric components are also given under a set of low-level sufficient conditions. Our empirical application using the U.K. Family Expenditure Survey shows the importance of adjusting for endogeneity in terms of both the nonparametric curvatures and the demographic parameters of systems of Engel curves.

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.

A simple specification test for conditional moment restrictions

In this article we extend the martingale difference test of Kuan and Lee (2004) to tests of conditional moment restrictions involving unknown parameters. The proposed test is asymptotically pivotal, easy to implement and capable of diagnostic checking misspecifications of quantile regressions. The simulation results show that the proposed test performs quite well in finite samples.

A Specification Test for Nonparametric Instrumental Variable Regression

We consider testing for correct specification of a nonparametric instrumental variable regression. In this ill-posed inverse problem setting, the test statistic is based on the empirical minimum distance criterion corresponding to the conditional moment restriction evaluated with a Tikhonov Regularized estimator of the functional parameter. Its asymptotic distribution is normal under the null hypothesis, and a consistent bootstrap is available to get simulation based critical values. We explore the finite sample behavior with Monte Carlo experiments. Finally, we provide an empirical application for an estimated Engel curve.

Conditional Moment Restrictions and Triangular Simultaneous Equations

We show that in the nonparametric triangular simultaneous equations model, the mean independence conditional moment restriction (CMR) identifies a causal relation between the dependent variable and an endogenous covariate, only if the model is structurally separable in observable covariates and unobservable random errors. If the CMR is used in a nonseparable population model, the average structural function (ASF) is not recovered. However, the ASF is recovered from the average CMR (ACMR) based on independence of the endogenous covariate and the random error given a control variate. We also provide a condition under which the nonseparable model is nonparametrically just identified from the population distribution of the observables, so that under this assumption the nonseparable triangular simultaneous equations model does not restrict their population distribution.

Estimation of possibly misspecified semiparametric conditional moment restriction models with different conditioning variables

Abstract Newey and Powell [2003. Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578] and Ai and Chen [2003. Efficient estimation of conditional moment restrictions models containing unknown functions. Econometrica 71, 1795–1843] propose sieve minimum distance (SMD) estimation of both finite dimensional parameter ( θ ) and infinite dimensional parameter (h) that are identified through a conditional moment restriction model, in which h could depend on endogenous variables. This paper modifies their SMD procedure to allow for different conditioning variables to be used in different equations, and derives the asymptotic properties when the model may be misspecified. Under low-level sufficient conditions, we show that: (i) the modified SMD estimators of both θ and h converge to some pseudo-true values in probability; (ii) the SMD estimators of smooth functionals, including the θ estimator and the average derivative estimator, are asymptotically normally distributed; and (iii) the estimators for the asymptotic covariances of the SMD estimators of smooth functionals are consistent and easy to compute. These results allow for asymptotically valid tests of various hypotheses on the smooth functionals regardless of whether the semiparametric model is correctly specified or not.

Efficient Estimation in Marginal Partially Linear Models for Longitudinal/Clustered Data Using Splines

Abstract. We consider marginal semiparametric partially linear models for longitudinal/clustered data and propose an estimation procedure based on a spline approximation of the non‐parametric part of the model and an extension of the parametric marginal generalized estimating equations (GEE). Our estimates of both parametric part and non‐parametric part of the model have properties parallel to those of parametric GEE, that is, the estimates are efficient if the covariance structure is correctly specified and they are still consistent and asymptotically normal even if the covariance structure is misspecified. By showing that our estimate achieves the semiparametric information bound, we actually establish the efficiency of estimating the parametric part of the model in a stronger sense than what is typically considered for GEE. The semiparametric efficiency of our estimate is obtained by assuming only conditional moment restrictions instead of the strict multivariate Gaussian error assumption.

Root-N consistent semiparametric estimators of a dynamic panel-sample-selection model

Abstract This paper considers the problem of identification and estimation in panel-data sample-selection models with a binary selection rule when the latent equations contain possibly predetermined variables, lags of the dependent variables, and unobserved individual effects. The selection equation contains lags of the dependent variables from both the latent and the selection equations as well as other possibly predetermined variables relative to the latent equations. We derive a set of conditional moment restrictions that are then exploited to construct a three-step sieve estimator for the parameters of the main equation including a nonparametric estimator of the sample-selection term. In the second step the unknown parameters of the selection equation are consistently estimated using a transformation approach in the spirit of Berkson's minimum chi-square sieve method and a first-step kernel estimator for the selection probability. This second-step estimator is of interest in its own right. It can be used to semiparametrically estimate a panel-data binary response model with a nonparametric individual-specific effect without making any other distributional assumptions. We show that both estimators (second and third stage) are n -consistent and asymptotically normal.

A local generalized method of moments estimator

A local Generalized Method of Moments Estimator is proposed for nonparametrically estimating unknown functions that are defined by conditional moment restrictions.

Factor Stochastic Volatility in Mean Models: A GMM Approach

This paper provides a semiparametric framework for modeling multivariate conditional heteroskedasticity. We put forward latent stochastic volatility (SV) factors as capturing the commonality in the joint conditional variance matrix of asset returns. This approach is in line with common features as studied by Engle and Kozicki (1993), and it allows us to focus on identication of factors and factor loadings through first- and second-order conditional moments only. We assume that the time-varying part of risk premiums is based on constant prices of factor risks, and we consider a factor SV in mean model. Additional specification of both expectations and volatility of future volatility of factors provides conditional moment restrictions, through which the parameters of the model are all identied. These conditional moment restrictions pave the way for instrumental variables estimation and GMM inference.

On the efficient use of the informational content of estimating equations: Implied probabilities and Euclidean empirical likelihood

Abstract A number of information-theoretic alternatives to GMM have recently been proposed in the literature. For practical use and general interpretation, the main drawback of these alternatives, particularly in the case of conditional moment restrictions, is that they give up the computational and interpretational simplicity of quadratic optimization. The main contribution of this paper is to analyze the informational content of estimating equations within the unified framework of Chi-square distance. Improved inference by control variables, closed form formulae for implied probabilities and information-theoretic interpretations of continuously updated GMM are discussed in the two cases of unconditional and conditional moment restrictions.

SOME IDENTIFICATION ISSUES IN NONPARAMETRIC LINEAR MODELS WITH ENDOGENOUS REGRESSORS

In applied work economists often seek to relate a given response variable y to some causal parameter μ* associated with it. This parameter usually represents a summarization based on some explanatory variables of the distribution of y, such as a regression function, and treating it as a conditional expectation is central to its identification and estimation. However, the interpretation of μ* as a conditional expectation breaks down if some or all of the explanatory variables are endogenous. This is not a problem when μ* is modeled as a parametric function of explanatory variables because it is well known how instrumental variables techniques can be used to identify and estimate μ*. In contrast, handling endogenous regressors in nonparametric models, where μ* is regarded as fully unknown, presents difficult theoretical and practical challenges. In this paper we consider an endogenous nonparametric model based on a conditional moment restriction. We investigate identification-related properties of this model when the unknown function μ* belongs to a linear space. We also investigate underidentification of μ* along with the identification of its linear functionals. Several examples are provided to develop intuition about identification and estimation for endogenous nonparametric regression and related models.We thank Jeff Wooldridge and two anonymous referees for comments that greatly improved this paper.

Consistent Tests of Conditional Moment Restrictions

We propose two classes of consistent tests in parametric econometric models defi ned through multiple conditional moment restrictions. The fi rst type of tests relies on nonparametric estimation, while the second relies on a functional of a marked empirical process. For both tests, a simulation procedure for obtaining critical values is shown to be asymptotically valid. Finite sample performances of the tests are investigated by means of several Monte-Carlo experiments. 234