Generalized linear mixed models with informative dropouts and missing covariates

Generalized linear mixed models (GLMM) are useful in many longitudinal data analyses. In the presence of informative dropouts and missing covariates, however, standard complete-data methods may not be applicable. In this article, we consider a likelihood method and an approximate method for GLMM with informative dropouts and missing covariates. The methods are implemented by Monte–Carlo EM algorithms combined with Gibbs sampler. The approximate method may lead to inconsistent estimators but is computationally more efficient than the likelihood method. The two methods are evaluated via a simulation study for longitudinal binary data, and appear to perform reasonably well. A dataset on mental distress is analyzed in details.     

Methods for Handling Dropouts in Longitudinal Clinical Trials

This paper focuses on the monotone missing data patterns produced by dropouts and presents a review of the statistical literature on approaches for handling dropouts in longitudinal clinical trials. A variety of ad hoc procedures for handling dropouts are widely used. The rationale for many of these procedures is not well-founded and they can result in biased estimates of treatment comparisons. A fundamentally difficult problem arises when the probability of dropout is thought to be related to the specific value that in principle should have been obtained; this is often referred to as informative or non-ignorable dropout. Joint models for the longitudinal outcomes and the dropout times have been proposed in order to make corrections for non-ignorable dropouts. Two broad classes of joint models are reviewed: selection models and pattern-mixture models. Finally, when there are dropouts in a longitudinal clinical trial the goals of the analysis need to be clearly specified. In this paper we review the main distinctions between a 'pragmatic' and an 'explanatory' analysis. We note that many of the procedures for handling dropouts that are widely used in practice come closest to producing an explanatory rather than a pragmatic analysis.     

Model‐Assisted Regression Estimators for Longitudinal Data with Nonignorable Dropout

Estimation with longitudinal Y having nonignorable dropout is considered when the joint distribution of Y and covariate X is nonparametric and the dropout propensity conditional on (Y,X) is parametric. We apply the generalised method of moments to estimate the parameters in the nonignorable dropout propensity based on estimating equations constructed using an instrument Z, which is part of X related to Y but unrelated to the dropout propensity conditioned on Y and other covariates. Population means and other parameters in the nonparametric distribution of Y can be estimated based on inverse propensity weighting with estimated propensity. To improve efficiency, we derive a model‐assisted regression estimator making use of information provided by the covariates and previously observed Y‐values in the longitudinal setting. The model‐assisted regression estimator is protected from model misspecification and is asymptotically normal and more efficient when the working models are correct and some other conditions are satisfied. The finite‐sample performance of the estimators is studied through simulation, and an application to the HIV‐CD4 data set is also presented as illustration.     

Weak Axiom of Cost Dominance: A Nonparametric Test of Cost Efficiency Without Input Quantity Data

Varian's Weak Axiom of Cost Minimization provides a nonparametric test of cost minimization, which can be applied only when both input price and quantity data are available for individual firms. In this paper we propose a Weak Axiom of Cost Dominance (WACD), which serves as the basis of an alternative test of cost-minimization applicable in situations where input quantity data are missing. Unlike a previous test developed by Diewert and Parkan, the proposed test does identify individual firms that violate the assumption of cost-minimizing behavior. It also provides an upper bound of the cost-efficiency of any observed firm. The test procedure is shown to be equivalent to applying dominance analysis using normalized input prices with reference to the Cost-indirect technology. The proposed method is applied to Nerlove's electrical utility data. The nonparametric results are also compared with parametric efficiency levels computed from a stochastic frontier cost function.     

Continuous-time modelling of irregularly spaced panel data using a cubic spline model

Continuous-time modelling remains a somewhat 'idealized' representation tool. Even though conceptualizing a dynamic process as a continuous process has clear appeal from a theoretical standpoint, practical tools that allow researchers to effectively map an idealized continuous model onto a set of discrete-time observed data are still lacking observed data. Irregularly spaced longitudinal data frequently arise in empirical settings because of the prevalence of longitudinal studies with partially randomized measurement intervals and other related designs. We present a practical approach that capitalizes on a nonparametric spline interpolation approach to impute the gaps in irregularly spaced panel data. Simulated and empirical examples are provided to demonstrate the applicability of the proposed approach to studies of group-based dynamics using panel data.     

A consistent model specification test with mixed discrete and continuous data

In this paper we propose a nonparametric kernel-based model specification test that can be used when the regression model contains both discrete and continuous regressors. We employ discrete variable kernel functions and we smooth both the discrete and continuous regressors using least squares cross-validation (CV) methods. The test statistic is shown to have an asymptotic normal null distribution. We also prove the validity of using the wild bootstrap method to approximate the null distribution of the test statistic, the bootstrap being our preferred method for obtaining the null distribution in practice. Simulations show that the proposed test has significant power advantages over conventional kernel tests which rely upon frequency-based nonparametric estimators that require sample splitting to handle the presence of discrete regressors.     

A consistent nonparametric test of affiliation in auction models

We propose a new nonparametric test of affiliation, a strong form of positive dependence with independence as a special, knife-edge, case. The test is consistent against all departures from the null of affiliation, and its null distribution is standard normal. Like most nonparametric tests, a sample-size dependent input parameter is needed. We provide an informal procedure for choosing the input parameter and evaluate the test’s performance using a simulation study. Our test can be used to test the fundamental assumptions of the auctions literature. We implement our test empirically using the Outer Continental Shelf (OCS) auction data.     

Semiparametric regression models and sensitivity analysis of longitudinal data with nonrandom dropouts

We propose a family of regression models to adjust for nonrandom dropouts in the analysis of longitudinal outcomes with fully observed covariates. The approach conceptually focuses on generalized linear models with random effects. A novel formulation of a shared random effects model is presented and shown to provide a dropout selection parameter with a meaningful interpretation. The proposed semiparametric and parametric models are made part of a sensitivity analysis to delineate the range of inferences consistent with observed data. Concerns about model identifiability are addressed by fixing some model parameters to construct functional estimators that are used as the basis of a global sensitivity test for parameter contrasts. Our simulation studies demonstrate a large reduction of bias for the semiparametric model relatively to the parametric model at times where the dropout rate is high or the dropout model is misspecified. The methodology's practical utility is illustrated in a data analysis.     

Nonparametric estimation and testing of fixed effects panel data models

In this paper we consider the problem of estimating nonparametric panel data models with fixed effects. We introduce an iterative nonparametric kernel estimator. We also extend the estimation method to the case of a semiparametric partially linear fixed effects model. To determine whether a parametric, semiparametric or nonparametric model is appropriate, we propose test statistics to test between the three alternatives in practice. We further propose a test statistic for testing the null hypothesis of random effects against fixed effects in a nonparametric panel data regression model. Simulations are used to examine the finite sample performance of the proposed estimators and the test statistics.     

Nonparametric tests for Optimal Predictive Ability

A nonparametric method for comparing multiple forecast models is developed and implemented. The hypothesis of Optimal Predictive Ability generalizes the Superior Predictive Ability hypothesis from a single given loss function to an entire class of loss functions. Distinction is drawn between General Loss functions, Convex Loss functions, and Symmetric Convex Loss functions. The research hypothesis is formulated in terms of moment inequality conditions. The empirical moment conditions are reduced to an exact and finite system of linear inequalities based on piecewise-linear loss functions. The hypothesis can be tested in a statistically consistent way using a blockwise Empirical Likelihood Ratio test statistic. A computationally feasible test procedure computes the test statistic using Convex Optimization methods, and estimates conservative, data-dependent critical values using a majorizing chi-square limit distribution and a moment selection method. An empirical application to inflation forecasting reveals that a very large majority of thousands of forecast models are redundant, leaving predominantly Phillips Curve-type models, when convexity and symmetry are assumed.     

Detections of changes in return by a wavelet smoother with conditional heteroscedastic volatility

In this paper, we propose two estimators, an integral estimator and a discretized estimator, for the wavelet coefficient of regression functions in nonparametric regression models with heteroscedastic variance. These estimators can be used to test the jumps of the regression function. The model allows for lagged-dependent variables and other mixing regressors. The asymptotic distributions of the statistics are established, and the asymptotic critical values are analytically obtained from the asymptotic distribution. We also use the test to determine consistent estimators for the locations of change points. The jump sizes and locations of change points can be consistently estimated using wavelet coefficients, and the convergency rates of these estimators are derived. We perform some Monte Carlo simulations to check the powers and sizes of the test statistics. Finally, we give practical examples in finance and economics to detect changes in stock returns and short-term interest rates using the empirical wavelet method.     

Empirical Bayesball Remixed: Empirical Bayes Methods for Longitudinal Data

Empirical Bayes methods for Gaussian and binomial compound decision problems involving longitudinal data are considered. A recent convex optimization reformulation of the nonparametric maximum likelihood estimator of Kiefer and Wolfowitz (Annals of Mathematical Statistics 1956; 27 : 887–906) is employed to construct nonparametric Bayes rules for compound decisions. The methods are illustrated with an application to predict baseball batting averages, and the age profile of batting performance. An important aspect of the empirical application is the general bivariate specification of the distribution of heterogeneous location and scale effects for players that exhibits a weak positive association between location and scale attributes. Prediction of players' batting averages for 2012 based on performance in the prior decade using the proposed methods shows substantially improved performance over more naive methods with more restrictive treatment of unobserved heterogeneity. Comparisons are also made with nonparametric Bayesian methods based on Dirichlet process priors, which can be viewed as a regularized, or smoothed, version of the Kiefer–Wolfowitz method. Copyright © 2016 John Wiley & Sons, Ltd.     

Using nonparametric methods in social surveys: an empirical study

The most common form of data for socio-economic studies comes from survey sampling. Often the designs of such surveys are complex and use stratification as a method for selecting sample units. A parametric regression model is widely employed for the analysis of such survey data. However the use of a parametric model to represent the relationship between the variables can be inappropriate. A natural alternative is to adopt a nonparametric approach. In this article we address the problem of estimating the finite population mean under stratified sampling. A new stratified estimator based on nonparametric regression is proposed for stratification with proportional allocation, optimum allocation and post-stratification. We focus on an educational and labor-related context with natural populations to test the proposed nonparametric method. Simulated populations have also been considered to evaluate the practical performance of the proposed method.     

Nonparametric Varying-Coefficient Models for the Analysis of Longitudinal Data

Longitudinal methods have been widely used in biomedicine and epidemiology to study the patterns of time-varying variables, such as disease progression or trends of health status. Data sets of longitudinal studies usually involve repeatedly measured outcomes and covariates on a set of randomly chosen subjects over time. An important goal of statistical analyses is to evaluate the effects of the covariates, which may or may not depend on time, on the outcomes of interest. Because fully parametric models may be subject to model misspecification and completely unstructured nonparametric models may suffer from the drawbacks of "curse of dimensionality", the varying-coefficient models are a class of structural nonparametric models which are particularly useful in longitudinal analyses. In this article, we present several important nonparametric estimation and inference methods for this class of models, demonstrate the advantages, limitations and practical implementations of these methods in different longitudinal settings, and discuss some potential directions of further research in this area. Applications of these methods are illustrated through two epidemiological examples.     

Partial parametric estimation for nonstationary nonlinear regressions

This paper proposes an estimation method for a partial parametric model with multiple integrated time series. Our estimation procedure is based on the decomposition of the nonparametric part of the regression function into homogeneous and integrable components. It consists of two steps: In the first step we parameterize and fit the homogeneous component of the nonparametric part by the nonlinear least squares with other parametric terms in the model, and use in the second step the standard kernel method to nonparametrically estimate the integrable component of the nonparametric part from the residuals in the first step. We establish consistency and obtain the asymptotic distribution of our estimator. A simulation shows that our estimator performs well in finite samples. For the empirical illustration, we estimate the money demand functions for the US and Japan using our model and methodology.     

A test for multimodality of regression derivatives with application to nonparametric growth regressions

This paper presents a method to test for multimodality of an estimated kernel density of derivative estimates from a nonparametric regression. The test is included in a study of nonparametric growth regressions. The results show that in the estimation of unconditional β‐convergence the distribution of the partial effects is multimodal, with one mode in the negative region (primarily OECD economies) and possibly two modes in the positive region (primarily non‐OECD economies) of the estimates. The results for conditional β‐convergence show that the density is predominantly negative and there is mixed evidence that the distribution is unimodal. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonparametric cointegration analysis of fractional systems with unknown integration orders

In this paper a nonparametric variance ratio testing approach is proposed for determining the cointegration rank in fractionally integrated systems. The test statistic is easily calculated without prior knowledge of the integration order of the data, the strength of the cointegrating relations, or the cointegration vector(s). The latter property makes it easier to implement than regression-based approaches, especially when examining relationships between several variables with possibly multiple cointegrating vectors. Since the test is nonparametric, it does not require the specification of a particular model and is invariant to short-run dynamics. Nor does it require the choice of any smoothing parameters that change the test statistic without being reflected in the asymptotic distribution. Furthermore, a consistent estimator of the cointegration space can be obtained from the procedure. The asymptotic distribution theory for the proposed test is non-standard but easily tabulated or simulated. Monte Carlo simulations demonstrate excellent finite sample properties, even rivaling those of well-specified parametric tests. The proposed methodology is applied to the term structure of interest rates, where, contrary to both fractional- and integer-based parametric approaches, evidence in favor of the expectations hypothesis is found using the nonparametric approach.     

Nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters

We consider nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters. Most of the existing works on asymptotic distributions of a nonparametric/semiparametric estimator or a test statistic are based on some deterministic smoothing parameters, while in practice it is important to use data-driven methods to select the smoothing parameters. In this paper we give a simple sufficient condition that can be used to establish the first order asymptotic equivalence of a nonparametric estimator or a test statistic with stochastic smoothing parameters to those using deterministic smoothing parameters. We also allow for general weakly dependent data.     

Breaking the curse of dimensionality in nonparametric testing

For tests based on nonparametric methods, power crucially depends on the dimension of the conditioning variables, and specifically decreases with this dimension. This is known as the “curse of dimensionality”. We propose a new general approach to nonparametric testing in high dimensional settings and we show how to implement it when testing for a parametric regression. The resulting test behaves against directional local alternatives almost as if the dimension of the regressors was one. It is also almost optimal against classes of one-dimensional alternatives for a suitable choice of the smoothing parameter. The test performs well in small samples compared to several other tests.     

On nonparametric estimation of a hedonic price function

Recently, using mixed data on Canadian housing, Parmeter, Henderson, and Kumbhakar (Journal of Applied Econometrics 2007; 22 : 695–699) found that a nonparametric approach for estimating a hedonic house price function is superior to formerly suggested parametric and semiparametric specifications. We carefully reanalyze these specifications for this dataset by applying a recent nonparametric specification test and simulation‐based prediction comparisons. For the case at issue our results suggest that a previously proposed parametric specification does not have to be rejected and we illustrate how nonparametric methods provide valuable insights during all modeling steps. Copyright © 2010 John Wiley & Sons, Ltd.