Tikhonov regularization for nonparametric instrumental variable estimators

We study a Tikhonov Regularized (TiR) estimator of a functional parameter identified by conditional moment restrictions in a linear model with both exogenous and endogenous regressors. The nonparametric instrumental variable estimator is based on a minimum distance principle with penalization by the norms of the parameter and its derivatives. After showing its consistency in the Sobolev norm and uniform consistency under an embedding condition, we derive the expression of the asymptotic Mean Integrated Square Error and the rate of convergence. The optimal value of the regularization parameter is characterized in two examples. We illustrate our theoretical findings and the small sample properties with simulation results. Finally, we provide an empirical application to estimation of an Engel curve, and discuss a data driven selection procedure for the regularization parameter.     

Convergence in distribution of minimum-distance estimators

It is shown that (under some regularity conditions) minimum distance estimators for a (possibly multidimensional) real parameter of a family of univariate continuous distribution functions have an asymptotic distribution. If the distance is derived from the mean-square norm it is proved that the asymptotic distribution is normal. Weak convergence of empirical distribution to the Brownian bridge is the essential tool for the proof.     

Estimating Euler equations with noisy data: two exact GMM estimators

In this paper we exploit the specific structure of the Euler equation and develop two alternative GMM estimators that deal explicitly with measurement error. The first estimator assumes that the measurement error is log‐normally distributed. The second estimator drops the distributional assumption at the cost of less precision. Our Monte Carlo results suggest that both proposed estimators perform much better than conventional alternatives based on the exact Euler equation or its log‐linear approximation, especially with short panels. An empirical application to the PSID yields plausible and precise estimates of the coefficient of relative risk aversion and the discount rate. Copyright © 2008 John Wiley & Sons, Ltd.     

Edgeworth approximations for semiparametric instrumental variable estimators and test statistics

We establish the validity of higher order asymptotic expansions to the distribution of a version of the nonlinear semiparametric instrumental variable estimator considered in Newey (Econometrica 58 (1990) 809) as well as to the distribution of a Wald statistic derived from it. We employ local polynomial smoothing with variable bandwidth, which includes local linear, kernel, and (a version of) nearest neighbor estimates as special cases. Our expansions are valid to order n^−2ε for some 0<ε<1/2, where ε depends on the smoothness and dimensionality of the data distribution and on the order of the polynomial chosen by the practitioner. We use the expansions to define optimal bandwidth selection methods for both estimation and testing problems and apply our methods to simulated data.     

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.     

基尼系数与收入分配

一、基尼系数与贫富差距 自古以来,有没有一种社会形态是贫富完全平均没有任何差异的呢?有,那就是原始社会早期,那时候物质财富极度贫乏,什么东西都是平均分配的。而从原始社会后期到现在,贫富差距就不同程度地存在着,“朱门酒肉臭,路有冻死骨”,绝不仅仅是中国古代贫富差距的写照。在现代发达国家,“富裕中的贫困”仍是普遍现象。在美国,一个普通医生年薪约20万美元,一个普通警官年薪为5万美元,而一个普通农场工人一年只能挣2万美元。即使同是医生,收入差距也相当大,横向和纵向不平等无处不在。     

Asymptotic distribution of factor augmented estimators for panel regression

In this paper we derive an asymptotic theory for linear panel regression augmented with estimated common factors. We give conditions under which the estimated factors can be used in place of the latent factors in the regression equation. For the principal components estimate of the factor space it is shown that these conditions are satisfied when T/N→0$T/N\to 0$ and N/T³→0$N/T^3\to 0$ under regularity. Monte Carlo studies verify the asymptotic theory.     

Comparing forecasts from fixed and variable coefficient models: The case of money demand

In this paper we introduce a class of tentatively plausible, fixed-coefficient models of money demand and evaluate their forecast performance. When these models are reestimated allowing all coefficients to vary over time, the forecasting performance improves dramatically. Aside from offering insights about improved methods of analyzing time series data, the most promising direct use for point estimates derived from time-varying coefficients is as an aid in calibrating proposed models of the kind discussed here.     

Best unbiased estimators for the parameters of a two-parameter Pareto distribution

Summary For a two-parameter Pareto distributionMalik [1970] has shown that the maximum likelihood estimators of the parameters are jointly sufficient. In this article the maximum likelihood estimators are shown to be jointly complete. Furthermore, unbiased estimators for the two parameters are obtained and are shown to be functions of the jointly complete sufficient statistics, thereby establishing them as the best unblased estimators of the two parameters.This research is a part of the first author's Ph.D. dissertation. The authors wish to thank Dr. Kenny S. Crump, for many helpful suggestions and a referee for improvements in the proofs.     

Isotonic estimators of monotone densities and distribution functions: basic facts

We present short proofs of some basic results from isotonic regression theory. A straightforward argument is given to show that the left continuous version of the concave majorant of the empirical distribution function maximizes the likelihood function f↦f (X,)… f (X _n ) within the class of non-increasing densities. Similarly, it is shown that the nonparametric maximum likelihood estimator (NPMLE) of the distribution function of interval censored data has an interpretation in terms of the left derivative of a convex minor ant. Finally, a short proof is given to show that the number of vertices of the concave major ant of the uniform empirical distribution function is asymptotically normal with asymptotic mean and variance both equal to log n .     

Exponential convergence properties of linear estimators under exponential and uniform distribution

Summary Certain measures of asymptotic efficiency of test statistics are based on exponential convergence properties of the underlying error probabilities (Bahadur-, Hodges-Lehmann-efficiency). From a general large deviation theorem, that is specified to weighted sums of independent identically distributed (i.i.d.) random variables, such exponential convergence properties are derived for test statistics which are linear functions of order statistics of i.i.d. random variables under exponential and uniform distribution. For that purpose some smoothness-conditions for the weights have to be established. In a series of examples it is shown that these conditions are fulfilled for certain robust linear estimators of location or scale parameters. With the help of some numerical results two of them, namely Winsorized and trimmed mean, are compared with regard to the asymptotic relative efficiency against each other.

---

Zusammenfassung Bestimmte asymptotische Effizienzbegriffe für Tests basieren auf einem exponentiellen Konvergenzverhalten der zugrundeliegenden Fehlerwahrscheinlichkeiten (Bahadur-, Hodges-Lehmann-Effizienz). Mit Hilfe eines allgemeinen Satzes üver Wahrscheinlichkeiten großer Abweichungen, der spezialisiert wird auf gewichtete Summen unabhängiger, identisch verteilter (i.i.d.) Zufallsvariablen mit momenterzeugenden Funktionen, wird ein solches exponentielles Konvergenzverhalten nachgewiesen für Linearkombinationen von order statistics von i.i.d. Zufallsvariablen unter Exponential- und Rechteckverteilung. Dazu sind bestimmte Bedingungen an die Gewichte zu stellen. In einigen Beispielen wird gezeigt, daß solche Gewichtsbedingungen für eine Reihe von robusten Schätzern erfüllt sind. Zwei spezielle, nämlich das Winsorisierte und getrimmte Mittel, werden mit Hilfe einiger numerischer Ergebnisse hinsichtlich ihrer asymptotischen Effizienz miteinander verglichen.

     

Relative efficiency and deficiency of kernel type estimators of smooth distribution functions

Abstract  The problem is investigated whether a given kernel type estimator of a distribution function at a single point has asymptotically better performance than the empirical estimator. A representation of the relative deficiency of the empirical distribution function with respect to a kernel type estimator is established which gives a complete solution to this problem. The problem of finding optimal kernels is studied in detail.     

The structure of simultaneous equations estimators

The formula for the Full Information Maximum Likelihood Estimator for a linear simultaneous system (with finite variance, serially independent errors) is demonstrated to be an estimator generating equation for econometrics in that all presently known estimators are readily derivable from that formula if they are considered as numerical approximations to its solution. Further, the approach immediately classifies the resulting estimators into asymptotically equivalent groups. The method is then generalised to encompass the large class of estimators for dynamic systems with (vector) autoregressive errors. The very close relationship between estimation rules and non-linear optimisation algorithms is highlighted.     

On multidimensional inequality with variable distribution mean

We compare inequality of alternative populations of individuals who differ in many characteristics besides income. To do so, we introduce a new criterion for ranking multivariate distributions with different means of marginals (hereafter marginal means) and show that some conditions, relevant in the analysis of multidimensional inequality, are equivalent to the ordering we propose.     

Regular estimators for subclasses of linear estimators

Summary The concept of regular estimator is due toRoy/Chakravarti. For its application they confined to the most general class of linear estimators. The present paper considers some subclasses of linear estimators.     

Consistent estimation when the left-hand variable is exogenous over part of the sample period

This paper proposes a consistent estimation method for regression equations with a left-hand variable that is endogenous for some observations, and exogenous for others. This method is applied to the estimation of a demand-for-money function for Switzerland over a time interval which includes periods of monetary control; that is periods when the quantity of money can best be viewed as exogenous.