Generalized Method of Moments for Additive Hazards Model with Clustered Dental Survival Data

For multivariate survival data, we study the generalized method of moments (GMM) approach to estimation and inference based on the marginal additive hazards model. We propose an efficient iterative algorithm using closed‐form solutions, which dramatically reduces the computational burden. Asymptotic normality of the proposed estimators is established, and the corresponding variance–covariance matrix can be consistently estimated. Inference procedures are derived based on the asymptotic chi‐squared distribution of the GMM objective function. Simulation studies are conducted to empirically examine the finite sample performance of the proposed method, and a real data example from a dental study is used for illustration.     

Oracle GMM estimation for misspecified models via thresholding

We propose a thresholding generalized method of moments (GMM) estimator for misspecified time series moment condition models. This estimator has the following oracle property: its asymptotic behavior is the same as of any efficient GMM estimator obtained under the a priori information that the true model were known. We propose data adaptive selection methods for thresholding parameter using multiple testing procedures. We determine the limiting null distributions of classical parameter tests and show the consistency of the corresponding block-bootstrap tests used in conjunction with thresholding GMM inference. We present the results of a simulation study for a misspecified instrumental variable regression model and for a vector autoregressive model with measurement error. We illustrate an application of the proposed methodology to data analysis of a real-world dataset.     

Further Results on the Limiting Distribution of GMM Sample Moment Conditions

In this article, we examine the limiting behavior of generalized method of moments (GMM) sample moment conditions and point out an important discontinuity that arises in their asymptotic distribution. We show that the part of the scaled sample moment conditions that gives rise to degeneracy in the asymptotic normal distribution is T-consistent and has a nonstandard limiting distribution. We derive the appropriate asymptotic (weighted chi-squared) distribution when this degeneracy occurs and show how to conduct asymptotically valid statistical inference. We also propose a new rank test that provides guidance on which (standard or nonstandard) asymptotic framework should be used for inference. The finite-sample properties of the proposed asymptotic approximation are demonstrated using simulated data from some popular asset pricing models.     

Generalized Method of Moments and Empirical Likelihood

Generalized method of moments (GMM) estimation has become an important unifying framework for inference in econometrics in the last 20 years. It can be thought of as encompassing almost all of the common estimation methods, such as maximum likelihood, ordinary least squares, instrumental variables, and two-stage least squares, and nowadays is an important part of all advanced econometrics textbooks. The GMM approach links nicely to economic theory where orthogonality conditions that can serve as such moment functions often arise from optimizing behavior of agents. Much work has been done on these methods since the seminal article by Hansen, and much remains in progress. This article discusses some of the developments since Hansen's original work. In particular, it focuses on some of the recent work on empirical likelihood–type estimators, which circumvent the need for a first step in which the optimal weight matrix is estimated and have attractive information theoretic interpretations.     

Robust estimators and robust tests for the slightly contaminated stochastic logistic population models

This article introduces the robust indirect technique for the slightly contaminated stochastic logistic population models. Based on discrete sampled data with a fixed unit of time between two consecutive observations, we not only construct the robust indirect inference generalized method of moments (GMM) estimator for the model parameters, but also propose a likelihood-ratio-type indirect statistic and a robust indirect GMM saddle-point statistic for testing the parameters of interest. In addition, we develop the robust exponential tilting estimator and the robust exponential tilting test to improve their small sample performances. Finally, their finite-sample properties are studied through Monte Carlo experiments.     

Alternative GMM estimators for first-order autoregressive panel model: An improving efficiency approach

This article considers first-order autoregressive panel model that is a simple model for dynamic panel data (DPD) models. The generalized method of moments (GMM) gives efficient estimators for these models. This efficiency is affected by the choice of the weighting matrix that has been used in GMM estimation. The non-optimal weighting matrices have been used in the conventional GMM estimators. This led to a loss of efficiency. Therefore, we present new GMM estimators based on optimal or suboptimal weighting matrices. Monte Carlo study indicates that the bias and efficiency of the new estimators are more reliable than the conventional estimators.     

Inference in the presence of redundant moment conditions and the impact of government health expenditure on health outcomes in England

In his 1999 article with Breusch, Qian, and Wyhowski in the Journal of Econometrics, Peter Schmidt introduced the concept of “redundant” moment conditions. Such conditions arise when estimation is based on moment conditions that are valid and can be divided into two subsets: one that identifies the parameters and another that provides no further information. Their framework highlights an important concept in the moment-based estimation literature, namely, that not all valid moment conditions need be informative about the parameters of interest. In this article, we demonstrate the empirical relevance of the concept in the context of the impact of government health expenditure on health outcomes in England. Using a simulation study calibrated to this data, we perform a comparative study of the finite performance of inference procedures based on the Generalized Method of Moment (GMM) and info-metric (IM) estimators. The results indicate that the properties of GMM procedures deteriorate as the number of redundant moment conditions increases; in contrast, the IM methods provide reliable point estimators, but the performance of associated inference techniques based on first order asymptotic theory, such as confidence intervals and overidentifying restriction tests, deteriorates as the number of redundant moment conditions increases. However, for IM methods, it is shown that bootstrap procedures can provide reliable inferences; we illustrate such methods when analysing the impact of government health expenditure on health outcomes in England.     

Finite-Sample Properties of Some Alternative GMM Estimators

We investigate the small-sample properties of three alternative generalized method of moments (GMM) estimators of asset-pricing models. The estimators that we consider include ones in which the weighting matrix is iterated to convergence and ones in which the weighting matrix is changed with each choice of the parameters. Particular attention is devoted to assessing the performance of the asymptotic theory for making inferences based directly on the deterioration of GMM criterion functions.     

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.     

Sparse moving maxima models for tail dependence in multivariate financial time series

The multivariate maxima of moving maxima (M4) model has the potential to model both the cross-sectional and temporal tail-dependence for a rich class of multivariate time series. The main difficulty of applying M4 model to real data is due to the estimation of a large number of parameters in the model and the intractability of its joint likelihood. In this paper, we consider a sparse M4 random coefficient model (SM4R), which has a parsimonious number of parameters and it can potentially capture the major stylized facts exhibited by devolatized asset returns found in empirical studies. We study the probabilistic properties of the newly proposed model. Statistical inference can be made based on the Generalized Method of Moments (GMM) approach. We also demonstrate through real data analysis that the SM4R model can be effectively used to improve the estimates of the Value-at-Risk (VaR) for portfolios consisting of multivariate financial returns while ignoring either temporal or cross-sectional tail dependence could potentially result in serious underestimate of market risk.     

GMM estimation of spatial autoregressive models in a system of simultaneous equations with heteroskedasticity

This paper proposes a GMM estimation framework for the SAR model in a system of simultaneous equations with heteroskedastic disturbances. Besides linear moment conditions, the proposed GMM estimator also utilizes quadratic moment conditions based on the covariance structure of model disturbances within and across equations. Compared with the QML approach, the GMM estimator is easier to implement and robust under heteroskedasticity of unknown form. We derive the heteroskedasticity-robust standard error for the GMM estimator. Monte Carlo experiments show that the proposed GMM estimator performs well in finite samples.     

Asymptotic variance approximations for invariant estimators in uncertain asset-pricing models

This article derives explicit expressions for the asymptotic variances of the maximum likelihood and continuously-updated GMM estimators in models that may not satisfy the fundamental asset-pricing restrictions in population. The proposed misspecification-robust variance estimators allow the researcher to conduct valid inference on the model parameters even when the model is rejected by the data. While the results for the maximum likelihood estimator are only applicable to linear asset-pricing models, the asymptotic distribution of the continuously-updated GMM estimator is derived for general, possibly nonlinear, models. The large corrections in the asymptotic variances, that arise from explicitly incorporating model misspecification in the analysis, are illustrated using simulations and an empirical application.     

GMM nonparametric correction methods for logistic regression with error‐contaminated covariates and partially observed instrumental variables

We consider logistic regression with covariate measurement error. Most existing approaches require certain replicates of the error‐contaminated covariates, which may not be available in the data. We propose generalized method of moments (GMM) nonparametric correction approaches that use instrumental variables observed in a calibration subsample. The instrumental variable is related to the underlying true covariates through a general nonparametric model, and the probability of being in the calibration subsample may depend on the observed variables. We first take a simple approach adopting the inverse selection probability weighting technique using the calibration subsample. We then improve the approach based on the GMM using the whole sample. The asymptotic properties are derived, and the finite sample performance is evaluated through simulation studies and an application to a real data set.     

Inference for Income Distributions Using Grouped Data

We develop a general approach to estimation and inference for income distributions using grouped or aggregate data that are typically available in the form of population shares and class mean incomes, with unknown group bounds. We derive generic moment conditions and an optimal weight matrix that can be used for generalized method-of-moments (GMM) estimation of any parametric income distribution. Our derivation of the weight matrix and its inverse allows us to express the seemingly complex GMM objective function in a relatively simple form that facilitates estimation. We show that our proposed approach, which incorporates information on class means as well as population proportions, is more efficient than maximum likelihood estimation of the multinomial distribution, which uses only population proportions. In contrast to the earlier work of Chotikapanich, Griffiths, and Rao, and Chotikapanich, Griffiths, Rao, and Valencia, which did not specify a formal GMM framework, did not provide methodology for obtaining standard errors, and restricted the analysis to the beta-2 distribution, we provide standard errors for estimated parameters and relevant functions of them, such as inequality and poverty measures, and we provide methodology for all distributions. A test statistic for testing the adequacy of a distribution is proposed. Using eight countries/regions for the year 2005, we show how the methodology can be applied to estimate the parameters of the generalized beta distribution of the second kind (GB2), and its special-case distributions, the beta-2, Singh–Maddala, Dagum, generalized gamma, and lognormal distributions. We test the adequacy of each distribution and compare predicted and actual income shares, where the number of groups used for prediction can differ from the number used in estimation. Estimates and standard errors for inequality and poverty measures are provided. Supplementary materials for this article are available online.     

Efficient M-estimators with auxiliary information

This paper introduces a new class of M-estimators based on generalised empirical likelihood (GEL) estimation with some auxiliary information available in the sample. The resulting class of estimators is efficient in the sense that it achieves the same asymptotic lower bound as that of the efficient generalised method of moment (GMM) estimator with the same auxiliary information. The paper also shows that in case of smooth estimating equations the proposed estimators enjoy a small second order bias property compared to both efficient GMM and full GEL estimators. Analytical formulae to obtain bias corrected estimators are also provided. Simulations show that with correctly specified auxiliary information the proposed estimators and in particular those based on empirical likelihood outperform standard M and efficient GMM estimators both in terms of finite sample bias and efficiency. On the other hand with moderately misspecified auxiliary information estimators based on the nonparametric tilting method are typically characterised by the best finite sample properties.     

A study of the properties of Gaussian mixture model for stable isotope standard quantification in MALDI-TOF MS

The quantification of peptides in Matrix assisted laser desorption/ionization time-of-flight mass spectrum analysis coupled with stable isotope standards has been used to quantify native peptides under many experimental conditions. This approach has difficulties quantifying samples containing peptides with ion currents in overlapping (convolved) spectra. In a previous article we proposed a reparametrized Gaussian mixture model based on the known characteristics of the peptides that could also accommodate overlapping spectra. We demonstrated the application of our model in a series of single and overlapping peptides quantification experiments. Here, we focus solely on studying the properties of our approach and examine the characteristics of the GMM approach in convolved peptides using simulated spectra and provide a method for simulating these spectra.     

Structural change tests for GEL criteria

This article examines structural change tests based on generalized empirical likelihood methods in the time series context, allowing for dependent data. Standard structural change tests for the Generalized method of moments (GMM) are adapted to the generalized empirical likelihood (GEL) context. We show that when moment conditions are properly smoothed, these test statistics converge to the same asymptotic distribution as in the GMM, in cases with known and unknown breakpoints. New test statistics specific to GEL methods, and that are robust to weak identification, are also introduced. A simulation study examines the small sample properties of the tests and reveals that GEL-based robust tests performed well, both in terms of the presence and location of a structural change and in terms of the nature of identification.     

Comparative GMM and GQL logistic regression models on hierarchical data

We often rely on the likelihood to obtain estimates of regression parameters but it is not readily available for generalized linear mixed models (GLMMs). Inferences for the regression coefficients and the covariance parameters are key in these models. We presented alternative approaches for analyzing binary data from a hierarchical structure that do not rely on any distributional assumptions: a generalized quasi-likelihood (GQL) approach and a generalized method of moments (GMM) approach. These are alternative approaches to the typical maximum-likelihood approximation approach in Statistical Analysis System (SAS) such as Laplace approximation (LAP). We examined and compared the performance of GQL and GMM approaches with multiple random effects to the LAP approach as used in PROC GLIMMIX, SAS. The GQL approach tends to produce unbiased estimates, whereas the LAP approach can lead to highly biased estimates for certain scenarios. The GQL approach produces more accurate estimates on both the regression coefficients and the covariance parameters with smaller standard errors as compared to the GMM approach. We found that both GQL and GMM approaches are less likely to result in non-convergence as opposed to the LAP approach. A simulation study was conducted and a numerical example was presented for illustrative purposes.     

A new variational Bayesian algorithm with application to human mobility pattern modeling

A new variational Bayesian (VB) algorithm, split and eliminate VB (SEVB), for modeling data via a Gaussian mixture model (GMM) is developed. This new algorithm makes use of component splitting in a way that is more appropriate for analyzing a large number of highly heterogeneous spiky spatial patterns with weak prior information than existing VB-based approaches. SEVB is a highly computationally efficient approach to Bayesian inference and like any VB-based algorithm it can perform model selection and parameter value estimation simultaneously. A significant feature of our algorithm is that the fitted number of components is not limited by the initial proposal giving increased modeling flexibility. We introduce two types of split operation in addition to proposing a new goodness-of-fit measure for evaluating mixture models. We evaluate their usefulness through empirical studies. In addition, we illustrate the utility of our new approach in an application on modeling human mobility patterns. This application involves large volumes of highly heterogeneous spiky data; it is difficult to model this type of data well using the standard VB approach as it is too restrictive and lacking in the flexibility required. Empirical results suggest that our algorithm has also improved upon the goodness-of-fit that would have been achieved using the standard VB method, and that it is also more robust to various initialization settings.     

Simple heterogeneity variance estimation for meta-analysis

Summary.  A simple method of estimating the heterogeneity variance in a random-effects model for meta-analysis is proposed. The estimator that is presented is simple and easy to calculate and has improved bias compared with the most common estimator used in random-effects meta-analysis, particularly when the heterogeneity variance is moderate to large. In addition, it always yields a non-negative estimate of the heterogeneity variance, unlike some existing estimators. We find that random-effects inference about the overall effect based on this heterogeneity variance estimator is more reliable than inference using the common estimator, in terms of coverage probability for an interval estimate.