Estimation and Inference in Two-Step Econometric Models

A commonly used procedure in a wide class of empirical applications is to impute unobserved regressors, such as expectations, from an auxiliary econometric model. This two-step (T-S) procedure fails to account for the fact that imputed regressors are measured with sampling error, so hypothesis tests based on the estimated covariance matrix of the second-step estimator are biased, even in large samples. We present a simple yet general method of calculating asymptotically correct standard errors in T-S models. The procedure may be applied even when joint estimation methods, such as full information maximum likelihood, are inappropriate or computationally infeasible. We present two examples from recent empirical literature in which these corrections have a major impact on hypothesis testing.     

A BAYESIAN INTERPRETATION OF MULTIPLE POINT ESTIMATES

Consider a large number of econometric investigations using different estimation techniques and/or different subsets of all available data to estimate a fixed set of parameters. The resulting empirical distribution of point estimates can be shown - under suitable conditions - to coincide with a Bayesian posterior measure on the parameter space induced by a minimum information procedure. This Bayesian interpretation makes it easier to combine the results of various empirical exercises for statistical decision making. The collection of estimators may be generated by one investigator to ensure the satisfaction of our conditions, or they may be collected from published works, where behavioral assumptions need to be made regarding the dependence structure of econometric studies.     

Estimation under Cox proportional hazards model with covariates missing not at random

This paper considers likelihood-based estimation under the Cox proportional hazards model in the situations where some covariate entries are missing not at random. Assuming the conditional distribution of the missing entries is known, we demonstrate the existence of the semiparametric maximum likelihood estimator of the model parameters, establish the consistency and weak convergence. By simulation, we examine the finite-sample performance of the estimation procedure, and compare the SPMLE with the one resulted from using an estimated conditional distribution of the missing entries. We analyze the data from a tuberculosis (TB) study applying the proposed approach for illustration.     

Two-stage hierarchical modeling for analysis of subpopulations in conditional distributions

In this work, we develop modeling and estimation approach for the analysis of cross-sectional clustered data with multimodal conditional distributions where the main interest is in analysis of subpopulations. It is proposed to model such data in a hierarchical model with conditional distributions viewed as finite mixtures of normal components. With a large number of observations in the lowest level clusters, a two-stage estimation approach is used. In the first stage, the normal mixture parameters in each lowest level cluster are estimated using robust methods. Robust alternatives to the maximum likelihood estimation are used to provide stable results even for data with conditional distributions such that their components may not quite meet normality assumptions. Then the lowest level cluster-specific means and standard deviations are modeled in a mixed effects model in the second stage. A small simulation study was conducted to compare performance of finite normal mixture population parameter estimates based on robust and maximum likelihood estimation in stage 1. The proposed modeling approach is illustrated through the analysis of mice tendon fibril diameters data. Analyses results address genotype differences between corresponding components in the mixtures and demonstrate advantages of robust estimation in stage 1.     

Econometric approaches to the specification of life cycle labour supply and commodity demand behaviour

Many of the recently developed alternative econometric approaches to the construction and estimation of life-cycle consistent models using individual data can be viewed as alternative choices for conditioning variables that summarise past decisions and future anticipations. By ingenious choice of this conditioning variable and by exploitation of the duality relationships between the alternative specifications, many currently available micro-data sets can be used for the estimation of life-cycle consistent models. In reviewing the alternative approaches their stochastic properties and implict preference restrictions are highlighted. Indeed, empirical specifications that are parameterised in a form of direct theoretical interest often can be shown to be unnecessarily restrictive while dual representations may provide more flexible econometric models. These results indicate the particular advantages of different types of data in retrieving life-cycle consistent preference parameters and the appropriate, most flexible, econometric approach for each type of data. A methodology for relaxing the intertemporal separability assumption is developed and the advantages and disadvantages of alternative approaches in this framework are considered.     

Testing for Slope Heterogeneity Bias in Panel Data Models

ABSTRACT

Standard econometric methods can overlook individual heterogeneity in empirical work, generating inconsistent parameter estimates in panel data models. We propose the use of methods that allow researchers to easily identify, quantify, and address estimation issues arising from individual slope heterogeneity. We first characterize the bias in the standard fixed effects estimator when the true econometric model allows for heterogeneous slope coefficients. We then introduce a new test to check whether the fixed effects estimation is subject to heterogeneity bias. The procedure tests the population moment conditions required for fixed effects to consistently estimate the relevant parameters in the model. We establish the limiting distribution of the test and show that it is very simple to implement in practice. Examining firm investment models to showcase our approach, we show that heterogeneity bias-robust methods identify cash flow as a more important driver of investment than previously reported. Our study demonstrates analytically, via simulations, and empirically the importance of carefully accounting for individual specific slope heterogeneity in drawing conclusions about economic behavior.     

VARIABLE SELECTION AND REGRESSION ANALYSIS FOR GRAPH-STRUCTURED COVARIATES WITH AN APPLICATION TO GENOMICS

Graphs and networks are common ways of depicting information. In biology, many different biological processes are represented by graphs, such as regulatory networks, metabolic pathways and protein-protein interaction networks. This kind of a priori use of graphs is a useful supplement to the standard numerical data such as microarray gene expression data. In this paper, we consider the problem of regression analysis and variable selection when the covariates are linked on a graph. We study a graph-constrained regularization procedure and its theoretical properties for regression analysis to take into account the neighborhood information of the variables measured on a graph, where a smoothness penalty on the coefficients is defined as a quadratic form of the Laplacian matrix associated with the graph. We establish estimation and model selection consistency results and provide estimation bounds for both fixed and diverging numbers of parameters in regression models. We demonstrate by simulations and a real dataset that the proposed procedure can lead to better variable selection and prediction than existing methods that ignore the graph information associated with the covariates.     

Modified Profile Likelihood for Fixed-Effects Panel Data Models

We show how modified profile likelihood methods, developed in the statistical literature, may be effectively applied to estimate the structural parameters of econometric models for panel data, with a remarkable reduction of bias with respect to ordinary likelihood methods. Initially, the implementation of these methods is illustrated for general models for panel data including individual-specific fixed effects and then, in more detail, for the truncated linear regression model and dynamic regression models for binary data formulated along with different specifications. Simulation studies show the good behavior of the inference based on the modified profile likelihood, even when compared to an ideal, although infeasible, procedure (in which the fixed effects are known) and also to alternative estimators existing in the econometric literature. The proposed estimation methods are implemented in an R package that we make available to the reader.     

Reply

Many of the recently developed alternative ecocometric approaches to the construction and estimation of life-cycle consistent models using individual data can be viewed as alternative choices for conditioning variables that summarise past decisions and future anticipations. By ingenious choice of this conditioning variable and by exploitation of the duality relationships between the alternative specifications, many currently available micro-data sets can be used for the estimation of life-cycle consistent models. In reviewing the alternative approaches their stochastic properties and implicit preference restrictions are highlighted. Indeed, empirical specifications that are parameterised in a form of direct theoretical interest often can be shown to be unnecessarily restrictive while dual representations may provide more flexible econometric models. These results indicate the particular advantages of different types of data in retrieving life-cycle consistent preference parameters and the appropriate, most flexible, econometric approach for each type of data. A methodology for relaxing the intertemporal separability assumption is developed and the advantages and disadvantages of alternative approaches in this framework are considered.     

Econometric approaches to the specification of life-cycle labour supply and commodity demand behaviour

Many of the recently developed alternative ecocometric approaches to the construction and estimation of life-cycle consistent models using individual data can be viewed as alternative choices for conditioning variables that summarise past decisions and future anticipations. By ingenious choice of this conditioning variable and by exploitation of the duality relationships between the alternative specifications, many currently available micro-data sets can be used for the estimation of life-cycle consistent models. In reviewing the alternative approaches their stochastic properties and implicit preference restrictions are highlighted. Indeed, empirical specifications that are parameterised in a form of direct theoretical interest often can be shown to be unnecessarily restrictive while dual representations may provide more flexible econometric models. These results indicate the particular advantages of different types of data in retrieving life-cycle consistent preference parameters and the appropriate, most flexible, econometric approach for each type of data. A methodology for relaxing the intertemporal separability assumption is developed and the advantages and disadvantages of alternative approaches in this framework are considered.     

Identification and Efficient Estimation of Simultaneous Equations Network Models

This article considers identification and estimation of social network models in a system of simultaneous equations. We show that, with or without row-normalization of the social adjacency matrix, the network model has different equilibrium implications, needs different identification conditions, and requires different estimation strategies. When the adjacency matrix is not row-normalized, the variation in the Bonacich centrality across nodes in a network can be used as an IV to identify social interaction effects and improve estimation efficiency. The number of such IVs depends on the number of networks. When there are many networks in the data, the proposed estimators may have an asymptotic bias due to the presence of many IVs. We propose a bias-correction procedure for the many-instrument bias. Simulation experiments show that the bias-corrected estimators perform well in finite samples. We also provide an empirical example to illustrate the proposed estimation procedure.     

Nonparametric curve estimation with missing data: A general empirical process approach

A general nonparametric imputation procedure, based on kernel regression, is proposed to estimate points as well as set- and function-indexed parameters when the data are missing at random (MAR). The proposed method works by imputing a specific function of a missing value (and not the missing value itself), where the form of this specific function is dictated by the parameter of interest. Both single and multiple imputations are considered. The associated empirical processes provide the right tool to study the uniform convergence properties of the resulting estimators. Our estimators include, as special cases, the imputation estimator of the mean, the estimator of the distribution function proposed by Cheng and Chu [1996. Kernel estimation of distribution functions and quantiles with missing data. Statist. Sinica 6, 63–78], imputation estimators of a marginal density, and imputation estimators of regression functions.     

Beyond support in two-stage variable selection

Numerous variable selection methods rely on a two-stage procedure, where a sparsity-inducing penalty is used in the first stage to predict the support, which is then conveyed to the second stage for estimation or inference purposes. In this framework, the first stage screens variables to find a set of possibly relevant variables and the second stage operates on this set of candidate variables, to improve estimation accuracy or to assess the uncertainty associated to the selection of variables. We advocate that more information can be conveyed from the first stage to the second one: we use the magnitude of the coefficients estimated in the first stage to define an adaptive penalty that is applied at the second stage. We give the example of an inference procedure that highly benefits from the proposed transfer of information. The procedure is precisely analyzed in a simple setting, and our large-scale experiments empirically demonstrate that actual benefits can be expected in much more general situations, with sensitivity gains ranging from 50 to 100 % compared to state-of-the-art.     

Semiparametric model average prediction in panel data analysis

Forecasting in economic data analysis is dominated by linear prediction methods where the predicted values are calculated from a fitted linear regression model. With multiple predictor variables, multivariate nonparametric models were proposed in the literature. However, empirical studies indicate the prediction performance of multi-dimensional nonparametric models may be unsatisfactory. We propose a new semiparametric model average prediction (SMAP) approach to analyse panel data and investigate its prediction performance with numerical examples. Estimation of individual covariate effect only requires univariate smoothing and thus may be more stable than previous multivariate smoothing approaches. The estimation of optimal weight parameters incorporates the longitudinal correlation and the asymptotic properties of the estimated results are carefully studied in this paper.     

Computation of an efficient and robust estimator in a semiparametric mixture model

In this article, we propose an efficient and robust estimation for the semiparametric mixture model that is a mixture of unknown location-shifted symmetric distributions. Our estimation is derived by minimizing the profile Hellinger distance (MPHD) between the model and a nonparametric density estimate. We propose a simple and efficient algorithm to find the proposed MPHD estimation. Monte Carlo simulation study is conducted to examine the finite sample performance of the proposed procedure and to compare it with other existing methods. Based on our empirical studies, the newly proposed procedure works very competitively compared to the existing methods for normal component cases and much better for non-normal component cases. More importantly, the proposed procedure is robust when the data are contaminated with outlying observations. A real data application is also provided to illustrate the proposed estimation procedure.     

Indirect inference for fractional time series models

In this paper we present an indirect estimation procedure for (ARFIMA) fractional time series models.The estimation method is based on an ‘incorrect’criterion which does not directly provide a consistent estimator of the parameters of interest,but leads to correct inference by using simulations.

The main steps are the following. First,we consider an auxiliary model which can be easily estimated.Specifically,we choose the finite lag Autoregressive model.Then, this is estimated on the observations and simulated values drawn from the ARFIMA model associated with a given value of the parameters of interest.Finally,the latter is calibrated in order to obtain close values of the two estimators of the auxiliary parameters.

In this article,we describe the estimation procedure and compare the performance of the indirect estimator with some alternative estimators based on the likelihood function by a Monte Carlo study.     

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.     

Empirical likelihood for nonlinear regression models with nonignorable missing responses

This article develops three empirical likelihood (EL) approaches to estimate parameters in nonlinear regression models in the presence of nonignorable missing responses. These are based on the inverse probability weighted (IPW) method, the augmented IPW (AIPW) method and the imputation technique. A logistic regression model is adopted to specify the propensity score. Maximum likelihood estimation is used to estimate parameters in the propensity score by combining the idea of importance sampling and imputing estimating equations. Under some regularity conditions, we obtain the asymptotic properties of the maximum EL estimators of these unknown parameters. Simulation studies are conducted to investigate the finite sample performance of our proposed estimation procedures. Empirical results provide evidence that the AIPW procedure exhibits better performance than the other two procedures. Data from a survey conducted in 2002 are used to illustrate the proposed estimation procedure. The Canadian Journal of Statistics 48: 386–416; 2020 © 2020 Statistical Society of Canada     

Assessing different uncertainty measures of EBLUP: a resampling-based approach

The empirical best linear unbiased prediction approach is a popular method for the estimation of small area parameters. However, the estimation of reliable mean squared prediction error (MSPE) of the estimated best linear unbiased predictors (EBLUP) is a complicated process. In this paper we study the use of resampling methods for MSPE estimation of the EBLUP. A cross-sectional and time-series stationary small area model is used to provide estimates in small areas. Under this model, a parametric bootstrap procedure and a weighted jackknife method are introduced. A Monte Carlo simulation study is conducted in order to compare the performance of different resampling-based measures of uncertainty of the EBLUP with the analytical approximation. Our empirical results show that the proposed resampling-based approaches performed better than the analytical approximation in several situations, although in some cases they tend to underestimate the true MSPE of the EBLUP in a higher number of small areas.