Empirical likelihood methods for an AR(1) process with ARCH(1) errors

Measuring the difference between two Gini indexes is important in comparing the inequality between two groups. This paper systematically provides confidence intervals for this difference. First, normal approximation- and bootstrap-based confidence intervals are provided for the difference. Second, empirical likelihood (EL)-based confidence intervals are proposed according to a ratio statistic that is proven to have a weighted Chi-squared distribution with one degree of freedom. Third, two calibration approaches are established to improve the undercoverage issue of the EL method; these two approaches are augmented EL and bootstrap-calibrated EL. Monte Carlo simulations show that bootstrap-calibrated EL generally outperforms other methods in constructing confidence intervals. In the simulations, these methods are compared with the permutation method to test the equality of two Gini indexes. Lastly, these methods are applied to real data.

We study statistical inference for the derivative of the nonparametric regression function using local linear model based empirical likelihood. We first derive a normal equation for the derivative through the local linear model and use this equation to construct an empirical likelihood for the derivative. We show that the limiting distribution of the empirical likelihood ratio is a scaled χ 1 2 distribution rather than the usual (unscaled) χ 1 2 distribution. We use this limiting distribution to construct pointwise confidence intervals for the derivative. Such empirical likelihood ratio confidence intervals are easier to obtain than the normal approximation based confidence intervals. A small simulation study also suggests that they are more accurate.

Pretest-posttest studies are commonly used for assessing the effect of a treatment or an intervention. We propose an empirical likelihood based approach to both testing and estimation of the treatment effect in non-randomized pretest-posttest studies where the posttest outcomes are subject to missingness. The proposed empirical likelihood ratio test and the estimation procedure are multiply robust in the sense that multiple working models are allowed for the propensity score of treatment assignment, the missingness probability and the outcome regression, and the validity of the test and the estimation requires only a certain combination of those multiple working models to be correctly specified. An empirical likelihood ratio confidence interval can be constructed for the treatment effect and has better coverage probabilities than confidence intervals based on the Wald statistic. Simulations are conducted to demonstrate the finite-sample performances of the proposed methods.

Modelling life trajectories and mode choice using Bayesian belief networks

Whether or not a time series is weakly stationary has long been a question of major interest in the field of time series analysis. Stationary time series can be sufficiently described by means of autoregressive moving average (ARMA) processes. When modelling temporal correlations of GNSS observation noise, the applicability of ARMA processes depends on the stationarity of residual time series from GNSS data analysis. According to the property that stationary processes have homogenous variances, statistical inferences on stationarity can be made by testing for homogeneity of variance (HOV). In addition, considering a time series as a realisation of a stochastic process, stationarity can be assessed by testing for stochastic trends using unit root tests. Based on representative data simulations, this paper analyses the empirical size and power of commonly used HOV and unit root tests. The results show that the performance of the HOV test is strongly affected by serial correlations, whereas the unit root test produces high power without significant size distortions.

Introduction Some Methodological Questions Arising from Large Data Sets Finite-Sample Simulation-Based Tests in Seemingly Unrelated Regressions Finding Optimal Penalties for Model Selection in the Linear Regression Model On Bootstrap Coverage Probability with Dependent Data A Comparison of Alternative Causality and Predictive Accuracy Tests in the Presence of Integrated and Cointegrated Economic Variables Finite Sample Performance of the Empirical Likelihood Estimator Under Endogeneity Testing for Unit Roots in Semiannual Data Using Simulation Methods for Bayesian Econometric Models Bayesian Inference in the Seemingly Unrelated Regressions Model Computationally Intensive Methods for Deriving Optimal Trimming Parameters Estimating and Testing Fundamental Stock Prices: Evidence from Simulated Economies Neural Networks: An Econometric Tool Real-Time Forecasting with Vector Autoregressions: Spurious Drift, Structural Change, and Intercept Correction Econometric Modeling Based on Pattern Recognition via the Fuzzy C-Means Clustering Algorithm Nonparametric Bootstrap Specification Testing in Econometric Models The Effect of Economic Growth on Standard of Living: A Semiparametric Analysis Index

This thesis contains three essays on inference in econometric models. Chapter 1 considers the question of bootstrap inference for Propensity Score Matching. Propensity Score Matching, where the propensity scores are estimated in a first step, is widely used for estimating treatment effects. In this context, the naive bootstrap is invalid (Abadie and Imbens, 2008). This chapter proposes a novel bootstrap procedure for this context, and demonstrates its consistency. Simulations and real data examples demonstrate the superior performance of the proposed method relative to using the asymptotic distribution for inference, especially when the degree of overlap in propensity scores is poor. General versions of the procedure can also be applied to other causal effect estimators such as inverse probability weighting and propensity score subclassification, potentially leading to higher order refinements for inference in such contexts. Chapter 2 tackles the question of inference in incomplete econometric models. In many economic and statistical applications, the observed data take the form of sets rather than points. Examples include bracket data in survey analysis, tumor growth and rock grain images in morphology analysis, and noisy measurements on the support function of a convex set in medical imaging and robotic vision. Additionally, nonparametric bounds on treatment effects under imperfect compliance can be expressed by means of random sets. This chapter develops a concept of nonparametric likelihood for random sets and its mean, known as the Aumann expectation, and proposes general inference methods by adapting the theory of empirical likelihood. Chapter 3 considers inference on the cumulative distribution function (CDF) in the classical measurement error model. It proposes both asymptotic and bootstrap based uniform confidence bands for the estimator of the CDF under measurement error. The proposed techniques can also be used to obtain confidence bands for quantiles, and perform various CDF-based tests such as goodness-offit tests for parametric models of densities, two sample homogeneity tests, and tests for stochastic dominance; all for the first time under measurement error.

Empirical likelihood ratio confidence interval estimation of best linear combinations of biomarkers

This article develops empirical likelihood methodology for a class of long range dependent processes driven by a stationary Gaussian process. We consider population parameters that are defined by estimating equations in the time domain. It is shown that the standard block empirical likelihood (BEL) method, with a suitable scaling, has a non‐standard limit distribution based on a multiple Wiener–Itô integral. Unlike the short memory time series case, the scaling constant involves unknown population quantities that may be difficult to estimate. Alternative versions of the empirical likelihood method, involving the expansive BEL (EBEL) methods are considered. It is shown that the EBEL renditions do not require an explicit scaling and, therefore, remove this undesirable feature of the standard BEL. However, the limit law involves the long memory parameter, which may be estimated from the data. Results from a moderately large simulation study on finite sample properties of tests and confidence intervals based on different empirical likelihood methods are also reported.

The purpose of this paper is to incorporate semiparametric alternatives to maximum likelihood estimation and inference in the context of unordered multinomial response data when in practice there is often insufficient information to specify the parametric form of the function linking the observables to the unknown probabilities. We specify the function linking the observables to the unknown probabilities using a very general flexible class of functions belonging to the Pearson system of cumulative distribution equations. In this setting we consider the observations as arising from a multinomial distribution characterized by one of the CDFs in the Pearson system. Given this situation, it is possible to utilize the concept of unbiased estimating functions (EFs), combined with the concept of empirical likelihood (EL) to define an (empirical) likelihood function for the parameter vector based on a nonparametric representation of the sample’s PDF. This leads to the concept of maximum empirical likelihood (MEL) estimation and inference, which is analogous to parametric maximum likelihood methods in many respects.

By introducing the notion of “empirical likelihood function of observing sums”, unequally-spaced time series data and longitudinal data generated from Levy processes can be analyzed. Characteristic function is further incorporated to handle the situations where the density function of the increments is difficult to obtain. In the situations where both characteristic function and the density function are available, it is shown through the simulation examples that the proposed empirical maximum likelihood method does not suffer from significant information loss comparing to the maximum likelihood estimation method. The performances of the proposed method is tested for both equally-spaced and unequally-spaced observations.

This paper studies the measure of symmetry or asymmetry of a continuous variable under the multiplicative distortion measurement errors setting. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. First, two direct plug-in estimation procedures are proposed, and the empirical likelihood based confidence intervals are constructed to measure the symmetry or asymmetry of the unobserved variable. Next, we propose four test statistics for testing whether the unobserved variable is symmetric or not. The asymptotic properties of the proposed estimators and test statistics are examined. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a real dataset for an illustration.

Conditional absolute mean calibration for partial linear multiplicative distortion measurement errors models

Pump-and-treat (PAT) techniques are often applied to the remediation of dissolved \nchemicals from shallow aquifers. A related management problem typically consists of \nthe selection of the pumping strategy and the most appropriate treatment method, in \norder to minimize the total cleanup cost while meeting a set of technical, economic \nand social constraints. However, due to scarcity of information about the \nhydrogeological system, stochastic modeling approaches seem more appropriate. Of \nprimary concern is the inherent spatial variability of hydraulic conductivity. In \ngeneral, the implementation of a remediation strategy assessed based on uncertain \nhydrogeological parameters leads to a decision involving the risk of constraint \nviolations. The decision-making process may then be formulated as a multiobjective \noptimization framework where the optimality of a pumping pattern is traded off \nagainst its reliability. Operations may be structured into a stochastic optimal \ncontrol problem, in which the remediation strategy is sequentially updated based \nupon new measurements collected during the actual cleanup process. The procedure \nrequires the implementation of an inverse simulation model to estimate the \nstochastic hydrogeological parameters based on a set of potential measurements. In \nthis work, we follow a geostatistical conceptual model where the spatial \ndistribution of hydraulic conductivity is represented as a realization of a log-\nnormally distributed stationary process, characterized by an exponential covariance\nfunction. Using the maximum likelihood method, the parameter estimation problem is \nsolved by determining the set of geostatistical parameters -- average, variance, \nand correlation scales. Available data may include direct measurements of hydraulic \nconductivity, water table elevation at a number of monitoring wells, and \ncontaminant mass extracted from active remediation wells. A rigorous solution to \nthis optimization problem would require a stochastic flow and transport model to be \nincluded in the optimization loop to calculate the expected values and the \ncovariance matrix of the available measurements as functions of the decision \nvariables. Because of the overwhelming computational effort involved, a surrogate \nmodel or response surface is introduced to approximate the objective function. The \nsurrogate model is estimated using a multidimensional kriging interpolation over a \nset of data points or measurements obtained from a series of stochastic flow and \ntransport simulations for pre-established combinations of the decision variables. \nSince the goodness of the solution is ultimately determined by the error of \nestimation of the objective function, which in turn depends on the number and the \nlocation of measurements, an optimal search procedure is used in order to optimize \nthe pattern of data points. The method turns out to be computationally efficient \nand produces results that well approximate the actual geostatistical distribution of\nhydraulic conductivity.

In this paper, we consider the empirical likelihood confidence intervals for mean difference under the assumption of missing at random (MAR). We prove that the empirical log-likelihood ratio of mean difference converges to a standard chi-squared distribution. We conduct simulation study to compare the proposed empirical likelihood with the normal approximation method in terms of the coverage accuracies and average lengths of confidence intervals. By Monte Carlo simulations, we verify that the proposed method performs well.