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Minimax estimation and testing for moment condition models via large deviations
Abstract
This paper studies asymptotically optimal estimation and testing procedures for moment condition models using the theory of large deviations (LD). Minimax risk estimation and testing are discussed in details. The aim of the paper is three-fold. First, it studies a moment condition model by treating it as a statistical experiment in Le Cam's sense, and investigates its large deviation properties. Second, it develops a new minimax estimator for the model by considering Bahadur's large deviation efficiency criterion. The estimator can be regarded as a robustified version of the conventional empirical likelihood estimator. Third, it considers a Chernoff-type risk for parametric testing in the model, which is concerned with the LD probabilities of type I errors and type II errors. It is shown that the empirical likelihood ratio test is asymptotically minimax in this context.
... The minimax bound (4.6) can be achieved by an estimator based on empirical likelihood function (Kitamura and Otsu (2005)). LetˆθLetˆ Letˆθ ld denote the minimizer of the objective function ...
... stimator, while it may be still possible to show that it has an asymptotic optimality property. Such an investigation would involve the theory of moderate deviations, which has been applied to estimation problems; see, for example, Kallenberg (1983). 4.2. Minimax Testing. Section 4.1 applied an asymptotic minimax approach to parameter estima- tion. Kitamura and Otsu (2005) show that the similar approach (cf. Puhalskii and Spokoiny (1998)) leads to a testing procedure that has a large deviation minimax optimality property in the model (2.2). Let Θ 0 be a subset of the parameter space Θ of θ. Consider testing ...
Recent developments in empirical likelihood (EL) methods are reviewed. First, to put the method in perspective, two interpretations of empirical likelihood are presented, one as a nonparametric maximum likelihood estimation method (NPMLE) and the other as a generalized minimum contrast estimator (GMC). The latter interpretation provides a clear connection between EL, GMM, GEL and other related estimators. Second, EL is shown to have various advantages over other methods. The theory of large deviations demonstrates that EL emerges naturally in achieving asymptotic optimality both for estimation and testing. Interestingly, higher order asymptotic analysis also suggests that EL is generally a preferred method. Third, extensions of EL are discussed in various settings, including estimation of conditional moment restriction models, nonparametric specification testing and time series models. Finally, practical issues in applying EL to real data, such as computational algorithms for EL, are discussed. Numerical examples to illustrate the efficacy of the method are presented.
... Let −d ≤ 0 denote the above limit so that Pr{A n } e −nd , which characterizes how fast Pr{A n } decays. The goal is to obtain a procedure that maximizes the speed of decay d. Kitamura and Otsu (2005) study the estimation of models of the form Equation (1) using the LDP. One complication in the application of the LDP to an estimation problem in general is that an estimator that maximizes the limiting decay rate d with A n = 1{|| # n − θ 0 || > c} uniformly in unknown parameters does not exist in general, unless the model belongs to the exponential family. ...
... n = 1{|| # n − θ 0 || > c} depends on θ 0 and F 0 , therefore the worst case scenario is given by the pair (allowed in the model, Equation (1)) that maximizes Pr{A n }. Suppose an estimator # n minimizes this worst case probability, thereby achieving minimaxity. The limit inferior of the minimax probability provides an asymptotic minimax criterion. Kitamura and Otsu (2005) show that an estimator that attains the lower bound of the asymptotic minimax criterion can be obtained from the EL objective function (θ) in Equation (2) as follows, Calculating # ld in practice is straightforward. If the dimension of θ is high, it is also possible to focus on a low dimensional sub-vector of θ and obtain a large deviat ...
Nonparametric likelihood is a natural generalization of parametric likelihood and it offers effective methods for analysing economic models with nonparametric components. This is of great interest, since econometric theory rarely suggests a parametric form of the probability law of data. Being a nonparametric method, nonparametric likelihood is robust to misspecification. At the same time, it often achieves good properties that are analogous to those of parametric likelihood. This paper explores various applications of nonparametric likelihood, with some emphasis on the analysis of biased samples and data combination problems.
... Because of some technical restrictions on higher-order moments, this correction may not be possible for other member of GEL family (e.g. ET, CU-GMM) See Baggerly (1998) Other results (ongoing) Kitamura & Otsu (2006): Minimax large deviation optimal estimation and testing Otsu (2006a): GNP optimality of EL in moment selection Otsu (2006b): GNP optimality of EL in set estimation Otsu (2000c): GNP optimality of EL in parameter hypothesis testing Otsu & Park (2007): Bahadur e¢ ciency of EL in parameter hypothesis testing 5. Topics on GEL 5-1. Comparison with GMM 5-1-3. ...
... For a more basic discussion, within the basics of information theory, see for example Cover and Thomas (1991). For discussion of typical sets and large deviations in econometrics see Kitamura and Stutzer (2002), (Stutzer, 2003a,c), Kitamura (2006) and Kitamura and Otsu (2005). ...
The overall objectives of this review and synthesis are to study the basics of information-theoretic methods in econometrics, to examine the connecting theme among these methods, and to provide a more detailed summary and synthesis of the sub-class of methods that treat the observed sample moments as stochastic. Within the above objectives, this review focuses on studying the inter-connection between information theory, estimation, and inference. To achieve these objectives, it provides a detailed survey of information-theoretic concepts and quantities used within econometrics. It also illustrates
... One may derive optimal results for empirical likelihood ratio tests for structural changes in quantile regression models by using the large deviation principle. For large deviation analysis of empirical likelihood, see Kitamura (2006) and Kitamura and Ostu (2005); for large deviation principle on change-point analysis (Puhalskii and Spokoiny 1998). ...
This paper considers the issues related to the asymptotic properties of estimators and test statistics in linear quantile regression with structural changes. We first address the issue of estimating a single structural change and derive the asymptotic properties of the estimated break point. The rate of convergence of the estimated break point is derived. As a supplementary tool, a smoothed empirical likelihood ratio test is proposed for testing structural changes at the estimated break dates. Furthermore we propose a likelihood-ratio-type test for multiple structural changes in quantile regression. The number of break points can be consistently determined via the test procedure. Finally we construct an algorithm based on the principle of dynamic programming to estimate multiple structural changes occurring at unknown dates. Monte Carlo studies show that our method consistently estimates each break point.
... Special cases of GEL include empirical likelihood (Qin and Lawless, 1994), continuous updating GMM (Hansen et al., 1996) and exponential tilting (Kitamura and Stutzer, 1997, and Imbens et al., 1998). 1 In contrast with the literature on the method of moments estimator, and to the best of our knowledge, there is no theoretical work on the large deviation properties of the GMM and GEL estimators for the over-identified case. 2 The purpose of this paper is to derive some regularity conditions that guarantee exponentially small large deviation error probabilities for the GMM and GEL estimators both when the model is correctly specified (we refer to this case as the model assumption) and also when there exist local 1 See Kitamura (2007) for a review. 2 Kitamura and Otsu (2006) proposed a large deviation minimax optimal estimator for moment restriction models, which is different from the existing GMM or GEL estimator. Our focus is on the large deviation properties of the conventional GMM and GEL estimators. ...
This paper studies moderate deviation behaviors of the generalized method of moments and generalized empirical likelihood estimators for generalized estimating equations, where the number of equations can be larger than the number of unknown parameters. We consider two cases for the data generating probability measure: the model assumption and local contaminations or deviations from the model assumption. For both cases, we characterize the first-order terms of the moderate deviation error probabilities of these estimators. Our moderate deviation analysis complements the existing literature of the local asymptotic analysis and misspecification analysis for estimating equations, and is useful to evaluate power and robust properties of statistical tests for estimating equations which typically involve some estimators for nuisance parameters.
... A nice feature of EL is that imposing moment inequalities preserves the simplicity of the optimization problem from the moment equality case. The only difference lies in the behavior 11 The ELR statistic is also Chernoff-minimax optimal as in Puhalskii and Spokoiny (1998) and Kitamura and Otsu (2005). This result in included in a supplementary appendix available upon request. ...
This paper addresses the issue of optimal inference for parameters that are partially identified in models with moment inequalities. There currently exists a variety of inferential methods for use in this setting. However, the question of choosing optimally among contending procedures is unresolved. In this paper, I first consider a canonical large deviations criterion for optimality and show that inference based on the empirical likelihood ratio statistic is optimal. Second, I introduce a new empirical likelihood bootstrap that provides a valid resampling method for moment inequality models and overcomes the implementation challenges that arise as a result of non-pivotal limit distributions. Lastly, I analyze the finite sample properties of the proposed framework using Monte Carlo simulations. The simulation results are encouraging.
Empirical likelihood (EL) is a method for estimation and inference without making distributional assumptions. Viewed as a nonparametric maximum likelihood estimation procedure (NPMLE), it approximates the unknown distribution function with a discrete distribution, then applies the ML estimation method. Alternatively, EL can be regarded as a minimum divergence estimation procedure. EL works well for estimating moment condition models, though it applies to other models as well. The large deviation principle (LDP) and other techniques show that EL has many optimality properties.
A command economy is one in which the coordination of economic activity, essential to the viability and functioning of a complex social economy, is undertaken through administrative means — commands, directives, targets and regulations — rather than by a market mechanism. A complex social economy is one involving multiple significant interdependencies among economic agents, including significant division of labour and exchange among production units, rendering the viability of any unit dependent on proper coordination with, and functioning of, many others.
This paper studies large deviation optimal properties of the empirical likelihood sequen-tial testing (ELST) procedures for selecting moment restrictions. Since moment selection problems have discrete parameter spaces, the Pitman-type local alternative approach is not very helpful. By the theory of large deviations, we analyze convergence rates of error probabilities under …xed distributions. We propose three optimal properties of the ELST procedures: (i) the generalized Neyman-Pearson optimality, (ii) the overestimation error optimality, and (iii) the minimax misclassi…cation error optimality.
Hansen and Jagannathan (HJ, 1991) provided bounds on the volatility of Stochas-tic Discount Factors (SDF) that proved extremely useful to diagnose and test asset pricing models. This nonparametric bound reflects a duality between the mean-standard deviation frontier for SDFs and the mean-variance frontier for portfolios of asset returns. We extend this fundamental contribution by proposing information bounds that minimize general convex functions of SDFs directly taking into account higher moments of returns. These Minimum Discrepancy bounds reflect a dual-ity with finding the optimal portfolio of asset returns with a general HARA utility function. The maximum utility portfolio implies SDF estimators that are based on implied probabilities associated with the class of Generalized Empirical Likelihood estimators. We analyze the implications of these information bounds for the pricing of size portfolios and the performance evaluation of hedge funds., Address for correspondence: Edhec Business School, 393, Promenade des Anglais, BP 3116, 06202 Nice Cedex 3. We are grateful for comments from seminar participants at the 2007 CIREQ Conference on GMM, EDHEC Business School, Stockholm School of Economics, and Warwick Business School. The first author thanks CNPq-Brazil for financial support. The second author is a research Fellow of CIRANO and CIREQ. He thanks the Fonds québécois de la recherche sur la société et la culture (FQRSC), the Social Sciences and Humanities Research Council of Canada (SSHRC), the Network of Centres of Excellence MITACS and the Bank of Canada for financial support.
This paper studies the Bahadur efficiency of empirical likelihood for testing moment condition models. It is shown that under mild regularity conditions, the empirical likelihood overidentifying restriction test is Bahadur efficient, i.e., its p-value attains the fastest convergence rate under each fixed alternative hypothesis. Analogous results are derived for parameter hypothesis testing and set inference problems.
This paper studies moderate deviation behaviors of the generalized method of moments and generalized empirical likelihood estimators for generalized estimating equations, where the number of equations can be larger than the number of unknown parameters. We consider two cases for the data generating probability measure: the model assumption and local contaminations or deviations from the model assumption. For both cases, we characterize the first-order terms of the moderate deviation error probabilities of these estimators. Our moderate deviation analysis complements the existing literature of the local asymptotic analysis and misspecification analysis for estimating equations, and is useful to evaluate power and robust properties of statistical tests for estimating equations which typically involve some estimators for nuisance parameters.
This paper proposes a robust approach maximizing worst-case utility when both the distributions underlying the uncertain vector
of returns are exactly unknown and the estimates of the structure of returns are unreliable. We introduce concave convex utility
function measuring the utility of investors under model uncertainty and uncertainty structure describing the moments of returns
and all possible distributions and show that the robust portfolio optimization problem corresponding to the uncertainty structure
can be reformulated as a parametric quadratic programming problem, enabling to obtain explicit formula solutions, an efficient
frontier and equilibrium price system.
Best entropy estimation is a technique that has been widely applied in many areas of science. It consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate. This problem can be modelled as a partially-finite convex program, with an integrable function as the variable. A complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular, measure-theoretic solutions. This theory explains the appearance of singular components observed in the literature when the Burg entropy is used. It also provides a unified treatment of existence conditions when the Burg, Boltzmann-Shannon, or some other entropy is used as the objective. Some examples are discussed.
In this paper, we have proved a fundamental property of the characteristic function for the random variable $(\partial/\partial\theta) \log f(x \mid \theta)$. Based on this result, we have proved under regularity conditions different from Bahadur's that certain classes of consistent estimators $\{\theta_n^\ast\}$ are asymptotically efficient in Bahadur's sense $\lim_{\varepsilon \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n\varepsilon^2} \log P\theta\{|\theta_n^\ast - \theta| \geqq \varepsilon\} = -\frac{I(\theta)}{2}.$ Our proof also gives a simple and direct method to verify Bahadur's [2] result.
Introduction.- LDP for Finite Dimensional Space.- Applications - The Finite Dimensional Case.- General Principles.- Sample Path Large Deviations.- The LDP for Abstract Empirical Measures.- Applications of Empirical Measures LDP.
Preliminary Notation and Definitions Modes of Convergence of a Sequence of Random Variables Relationships Among the Modes of Convergence Convergence of Moments; Uniform Integrability Further Discussion of Convergence in Distribution Operations on Sequences to Produce Specified Convergence Properties Convergence Properties of Transformed Sequences Basic Probability Limit Theorems: The WLLN and SLLN Basic Probability Limit Theorems: The CLT Basic Probability Limit Theorems: The LIL Stochastic Process Formulation of the CLT Taylor's Theorem; Differentials Conditions for Determination of a Distribution by Its Moments Conditions for Existence of Moments of a Distribution Asymptotic Aspects of Statistical Inference Procedures Problems
In this paper we make two contributions. First, we show by example that empirical likelihood and other commonly used tests for parametric moment restrictions, including the GMM-based J-test of Hansen (1982), are unable to control the rate at which the probability of a Type I error tends to zero. From this it follows that, for the optimality claim for empirical likelihood in Kitamura (2001) to hold, additional assumptions and qualifications need to be introduced. The example also reveals that empirical and parametric likelihood may have non-negligible differences for the types of properties we consider, even in models in which they are first-order asymptotically equivalent. Second, under stronger assumptions than those in Kitamura (2001), we establish the following optimality result: (i) empirical likelihood controls the rate at which the probability of a Type I error tends to zero and (ii) among all procedures for which the probability of a Type I error tends to zero at least as fast, empirical likelihood maximizes the rate at which probability of a Type II error tends to zero for 'most'' alternatives. This result further implies that empirical likelihood maximizes the rate at which probability of a Type II error tends to zero for all alternatives among a class of tests that satisfy a weaker criterion for their Type I error probabilities.
Let $x_{1}$ , $x_{2}$ ,... be a sequence of independent and identically distributed observations with distribution determined by a real valued parameter θ. For each n = 1, 2,..., let $T_{n}=T_{n}(x_{1},x_{2},\ldots ,x_{n})$ be a statistic such that the sequence { $T_{n}$ } is a consistent estimate of θ. It is shown, under weak regularity conditions on the sample space of a single observation, that the asymptotic effective standard deviation of $T_{n}$ cannot be less than $[nI(\theta)]^{-{\textstyle\frac{1}{2}}}$ . The asymptotic effective standard deviation of $T_{n}$ is defined, roughly speaking, as the solution τ of the equation $P(|T_{n}-\theta|\geq \varepsilon|\theta)=P(|N|\geq \varepsilon /\tau)$ when n is large and ε is a small positive number, where N denotes a standard normal variable. It is also shown, under stronger regularity conditions, that the asymptotic effective standard deviation of the maximum likelihood estimate of θ is $[nI(\theta)]^{-{\textstyle\frac{1}{2}}}$ . These conclusions concerning estimates are derived from certain conclusions concerning the relative efficiency of alternative statistical tests based on large samples.
The presented monograph is devoted to the analysis and calculation of the asymptotic efficiency of
nonparametric tests. The asymptotic efficiency is a fundamental notion of statistics indispensable for comparing and ordering statistical tests in large samples. It is especially useful in nonparametric statistics where there exist numerous tests proposed from the heuristic point of view, such as the Kolmogorov--Smirnov, Cram\'er--von Mises and linear rank tests.
The main feature of the book is the elaboration of powerful methods to evaluate explicitly large deviation probabilities of test statistics based on Sanov's theorem and the techniques of limit theorems, variational calculus and nonlinear analysis. This makes it possible to find the Bahadur, Hodges--Lehmann and Chernoff efficiencies for the majority of nonparametric statistics used for testing goodness-of-fit, homogeneity, symmetry and independence.
The description of domains of the Bahadur local optimality and related characterization problems is of particular interest. This is a new direction initiated by the investigations of the author. The general theory is applied to the classical problem of statistical radiophysics: signal detection in the noises of unknown level.
The publication of this book will for the first time unify and develop rather sparse existing results on
efficiency and acquaint western readers with achievements obtained by the author and his collaborators
and previously published in Russian journals.
We present a general approach to statistical problems with criteria based on probabilities of large deviations. Our main idea, which originates from similarity in the definitions of the large-deviation principle (LDP) and weak convergence, is to develop a large-deviation analogue of asymptotic decision theory. We introduce the concept of the LPD for sequences of statistical experiments, which parallels the concept of weak convergence of experiments, and prove that, in analogy with Le Cam's minimax theorem, the LPD provides an asymptotic lower bound for the sequence of appropriately defined minimax risks. We also show that the bound is tight and give a method of constructing decisions whose asymptotic risk is arbitrarily close to the bound. The construction is further specified for hypothesis testing and estimation problems.
Tests of simple and composite hypothesis for multinomial distributions are considered. It is assumed that the size $\alpha_N$ of a test tends to 0 as the sample size $N$ increases. The main concern of this paper is to substantiate the following proposition: If a given test of size $\alpha_N$ is "sufficiently different" from a likelihood ratio test then there is a likelihood ratio test of size $\leqq\alpha_N$ which is considerably more powerful than the given test at "most" points in the set of alternatives when $N$ is large enough, provided that $\alpha_N \rightarrow 0$ at a suitable rate. In particular, it is shown that chi-square tests of simple and of some composite hypotheses are inferior, in the sense described, to the corresponding likelihood ratio tests. Certain Bayes tests are shown to share the above-mentioned property of a likelihood ratio test.
The performance of a sequence of estimators $\{T_n\}$ of $g(\theta)$ can be measured by its inaccuracy rate $-\lim \inf_{n\rightarrow\infty} n^{-1} \log \mathbb{P}_\theta(\|T_n - g(\theta)\| > \varepsilon)$. For fixed $\varepsilon > 0$ optimality of consistent estimators $\operatorname{wrt}$ the inaccuracy rate is investigated. It is shown that for exponential families in standard representation with a convex parameter space the maximum likelihood estimator is optimal. If the parameter space is not convex, which occurs for instance in curved exponential families, in general no optimal estimator exists. For the location problem the inaccuracy rate of $M$-estimators is established. If the underlying density is sufficiently smooth an optimal $M$-estimator is obtained within the class of translation equivariant estimators. Tail-behaviour of location estimators is studied. A connection is made between gross error and inaccuracy rate optimality.
This paper considers the estimation of a location parameter $\theta$ in a one-sample problem. The asymptotic performance of a sequence of estimates $\{T_n\}$ is measured by the exponential rate of convergence to 0 of $\max \{P_\theta(T_n < \theta - a), P_\theta(T_n > \theta + a)\}, \text{say} e(a).$ This measure of asymptotic performance is equivalent to one considered by Bahadur (1967). The optimal value of $e(a)$ is given for translation invariant estimates. Some computational methods are reviewed for determining $e(a)$ for a general class of estimates which includes $M$-estimates, rank estimates and Hodges-Lehmann estimates. Finally, some numerical work is presented on the asymptotic efficiencies of some standard estimates of location for normal, logistic and double exponential models.
For some time, so-called empirical likelihoods have been used heuristically for purposes of nonparametric estimation. Owen showed that empirical likelihood ratio statistics for various parameters $\theta(F)$ of an unknown distribution $F$ have limiting chi-square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods. Our objective in this paper is twofold: first, to link estimating functions or equations and empirical likelihood; second, to develop methods of combining information about parameters. We do this by assuming that information about $F$ and $\theta$ is available in the form of unbiased estimating functions. Empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood. Efficiency results for estimates of both $\theta$ and $F$ are obtained. The methods are illustrated on several problems, and areas for future investigation are noted.
Let (x,z) be a pair of observable random vectors. We construct a new "smoothed" empirical likelihood-based test for the hypothesis $\E\{ g(z,\break \theta)|x \} = 0$ w.p.1, where g is a vector of known functions and $\theta$ an unknown finite-dimensional parameter. We show that the test statistic is asymptotically normal under the null hypothesis and derive its asymptotic distribution under a sequence of local alternatives. Furthermore, the test is shown to possess an optimality property in large samples. Simulation evidence suggests that it also behaves well in small samples.
The empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution
from which the sample was taken. In this paper the likelihood function for distributions is used to define a likelihood ratio
function for distributions. It is shown that this empirical likelihood ratio function can be used to construct confidence
intervals for the sample mean, for a class of M-estimates that includes quantiles, and for differentiable statistical functionals. The results are nonpara-metric extensions
of Wilks's (1938) theorem for parametric likelihood ratios. The intervals are illustrated on some real data and compared in
a simulation to some bootstrap confidence intervals and to intervals based on Student's t statistic. A hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced.
We prove a large deviation principle for kernel-type empirical distributions. We introduce a metric in the space of distributions on so as to give a simple proof of the principle of large deviation. As an application, we show a smoothed version of the Dvoretzky-Kiefer-Wolfowitz inequality.
For many testing problems several different tests may have optimal exact Bahadur slope. The introduction of Bahadur deficiency provides further information about the performance of such tests. Roughly speaking a sequence of tests is deficient in the sense of Bahadur of order (hn) at a fixed alternative [theta] if the additional number of observations necessary to obtain the same power as the optimal test at [theta] is of order (hn) as the level of significance tends to zero. In this paper it is shown that in typical testing problems in multivariate exponential families the LR test is deficient in the sense of Bahadur of order (log n).
In an effort to improve the small sample properties of generalized method of moments (GMM) estimators, a number of alternative estimators have been suggested. These include empirical likelihood (EL), continuous updating, and exponential tilting estimators. We show that these estimators share a common structure, being members of a class of generalized empirical likelihood (GEL) estimators. We use this structure to compare their higher order asymptotic properties. We find that GEL has no asymptotic bias due to correlation of the moment functions with their Jacobian, eliminating an important source of bias for GMM in models with endogeneity. We also find that EL has no asymptotic bias from estimating the optimal weight matrix, eliminating a further important source of bias for GMM in panel data models. We give bias corrected GMM and GEL estimators. We also show that bias corrected EL inherits the higher order property of maximum likelihood, that it is higher order asymptotically efficient relative to the other bias corrected estimators. Copyright Econometric Society 2004.
This paper develops a variant of one-step efficient GMM based on the KLIC rather than empirical likelihood. As in other one-step methods, the authors introduce M (the number of moments) auxiliary 'tilting' parameters which are used to construct a reweighting of the data so that the reweighted sample obeys all the moment conditions at the parameter estimates. Parameter and overidentification tests can be recast in terms of these tilting parameters; such tests are often startlingly more effective than their conventional counterparts. These performance differences cannot be completely explained by the leading terms of the statistics' asymptotic expansions.
While optimally weighted generalized method of moments (GAM) estimation has desirable large sample properties, its small sample performance is poor in some applications. The authors propose a computationally simple alternative, for weakly dependent data generating mechanisms, based on minimization of the Kullback-Leibler information criterion. Conditions are derived under which the large sample properties of this estimator are similar to GAM, i.e., the estimator will be consistent and asymptotically normal, with the same asymptotic covariance matrix as GAM. In addition, the authors propose overidentifying and parametric restrictions tests as alternatives to analogous GAM procedures.
This paper studies estimators that make sample analogues of population orthogonality conditions close to zero. Strong consistency and asymptotic normality of such estimators is established under the assumption that the observable variables are stationary and ergodic. Since many linear and nonlinear econometric estimators reside within the class of estimators studied in this paper, a convenient summary of the large sample properties of these estimators, including some whose large sample properties have not heretofore been discussed, is provided.
This paper proposes an asymptotically efficient method for estimating models with conditional moment restrictions. Our estimator generalizes the maximum empirical likelihood estimator (MELE) of Qin and Lawless (1994). Using a kernel smoothing method, we efficiently incorporate the information implied by the conditional moment restrictions into our empirical likelihood-based procedure. This yields a one-step estimator which avoids estimating optimal instruments. Our likelihood ratio-type statistic for parametric restrictions does not require the estimation of variance, and achieves asymptotic pivotalness implicitly. The estimation and testing procedures we propose are normalization invariant. Simulation results suggest that our new estimator works remarkably well in finite samples. Copyright The Econometric Society 2004.