Nonresponse in panel data: The impact on estimates of a life cycle consumption function

If missing observations in a panel data set are not missing at random, many widely applied estimators may be inconsistent. In this paper we examine empirically several ways to reveal the nature and severity of the selectivity problem due to nonresponse, as well as a number of methods to estimate the resulting models. Using a life cycle consumption function and data from the Expenditure Index Panel from the Netherlands, we discuss simple procedures that can be used to assess whether observations are missing at random, and we consider more complicated estimation procedures that can be used to obtain consistent or efficient estimates in case of selectivity of attrition bias. Finally, some attention is paid to the differences in identification, consistency, and efficiency between inferences from a single wave of the panel, a balanced sub-panel, and an unbalanced panel.     

Bayesian Analysis of a Sensitive Proportion for a Small Area

Without accounting for sensitive items in sample surveys, sampled units may not respond (nonignorable nonresponse) or they respond untruthfully. There are several survey designs that address this problem and we will review some of them. In our study, we have binary data from clusters within small areas, obtained from a version of the unrelated‐question design, and the sensitive proportion is of interest for each area. A hierarchical Bayesian model is used to capture the variation in the observed binomial counts from the clusters within the small areas and to estimate the sensitive proportions for all areas. Both our example on college cheating and a simulation study show significant reductions in the posterior standard deviations of the sensitive proportions under the small‐area model as compared with an analogous individual‐area model. The simulation study also demonstrates that the estimates under the small‐area model are closer to the truth than for the corresponding estimates under the individual‐area model. Finally, for small areas, we discuss many extensions to accommodate covariates, finite population sampling, multiple sensitive items and optional designs.     

Estimation of functions of population means and regression coefficients including structural coefficients: A minimum expected loss (MELO) approach

Parameter estimates that minimize the posterior expectation of generalized quadratic loss functions are derived for a wide range of estimation problems encountered in econometrics and statistics including estimation of structural coefficients of linear structural econometric models and reciprocals and ratios of population means and regression or reduced form coefficients. These MELO estimates are simple in form and possess finite sampling moments and risk in contrast to other estimators, including maximum likelihood estimators, that possess no finite moments and have infinite risk relative to quadratic loss functions. Additional finite and large sample properties of MELO estimates are derived and discussed.     

Nonparametric trending regression with cross-sectional dependence

Panel data, whose series length T$T$ is large but whose cross-section size N$N$ need not be, are assumed to have common time trend, of unknown form. The model includes additive, unknown, individual-specific components and allows for spatial or other cross-sectional dependence and/or heteroscedasticity. A simple smoothed nonparametric trend estimate is shown to be dominated by an estimate which exploits availability of cross-sectional data. Asymptotically optimal bandwidth choices are justified for both estimates. Feasible optimal bandwidths, and feasible optimal trend estimates, are asymptotically justified, finite sample performance of the latter being examined in a Monte Carlo study. Potential extensions are discussed.     

On selectivity of nonresponse in discrete-time multi-wave panel studies

The current paper deals with the issue of the detection of selective nonresponse in discrete-time, multi-wave panel studies. If groups in a population differ with respect to the chances that they will be (and remain) in a longitudinal sample, we speak of selective nonresponse. Ultimately, selective nonresponse may lead to a sample that is very different from the target population. We discuss ways to detect and quantify the amount of selectiveness by means of discrete-time Markov models. Then we proceed by addressing how a researcher may gain understanding of how to solve the problems caused by selective nonresponse, and the degree to which these solutions will be effective, by means of data on the nonresponse during a three-wave panel study involving 2800 young Dutch adults.     

Estimation in Two-Stage Models with Heteroscedasticity

A surprising number of important problems can be cast in the framework of estimating a mean and variance using data arising from a two-stage structure. The first stage is a random sampling of "units" with some quantity of interest associated with the unit. The second stage produces an estimate of that quantity and usually, but not always, an estimated standard error, which may change considerably across units. Heteroscedasticity in the estimates over different units can arise for a number of reasons, including variation associated with the unit and changing sampling effort over units. This paper presents a broad discussion of the problem of making inferences for the population mean and variance associated with the unobserved true values at the first stage of sampling. A careful discussion of the causes of heteroscedasticity is given, followed by an examination of ways in which inferences can be carried out in a manner that is robust to the nature of the within unit heteroscedasticity. Among the conclusions are that under any type of heteroscedasticity, an unbiased estimate of the mean and the variance of the estimated mean can be obtained by using the estimates as if they were true unobserved values from the first stage. The issue of using the mean versus a weighted average which tries to account for the heteroscedasticity is also discussed. An unbiased estimate of the population variance is given and the variance of this estimate and its covariance with the estimated mean is provided under various types of heteroscedasticity. The two-stage setting arises in many contexts including the one-way random effects models with replication, meta-analysis, multi-stage sampling from finite populations and random coefficients models. We will motivate and illustrate the problem with data arising from these various contexts with the goal of providing a unified framework for addressing such problems.     

Measuring the impact of nonignorability in panel data with non‐monotone nonresponse

The analysis of panel data with non‐monotone nonresponse often relies on the critical and untestable assumption of ignorable missingness. It is important to assess the consequences of departures from the ignorability assumption. Non‐monotone nonresponse, however, can often make such sensitivity analysis infeasible because the likelihood functions for alternative models involve high‐dimensional and difficult‐to‐evaluate integrals with respect to missing outcomes. We develop an extension of the local sensitivity method that overcomes computational difficulty and completely avoids fitting alternative models and evaluating these high‐dimensional integrals. The proposed method is applicable to a wide range of panel outcomes. We apply the method to a Smoking Trend dataset where we relax the standard ignorability assumption and evaluate how smoking‐trend estimates in different groups of US young adults are affected by alternative assumptions about the missing‐data mechanism. The main finding is that the standard estimate in the black male group is sensitive to nonignorable missingness but those in other groups are reasonably robust. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimating moments in ANOVA-type mixed models

In the paper, a simple projection-based method is systematically developed to estimate the qth (\(q\ge 2\)) order moments of random effects and errors in the ANOVA type mixed model (ANOVAMM), where the response may not be divided into independent sub-vectors. All the estimates are weakly consistent and the second-order moment estimates are strongly consistent. Besides, the derived estimates are different from those in mixed models with cluster design. Simulation studies are conducted to examine the finite sample performance of the estimates and two real data examples are analyzed for illustration.     

Randomized response: Estimating mean square errors of linear estimators and finding optimal unbiased strategies

Summary General procedures are described to generate quantitative randomized response (RR) required to estimate the finite population total of a sensitive variable. Permitting sample selection with arbitrary probabilities a formula for the mean square error (MSE) of a linear estimator of total based on RR is noted indicating the simple modification over one that might be based on direct response (DR) if the latter were available. A general formula for an unbiased estimator of the MSE is presented. A simple approximation is proposed in case the RR ratio estimator is employed based on a simple random sample (SRS) taken without replacement (WOR). Among sampling strategies employing unbiased but not necessarily linear estimators based on RR, certain optimal ones are identified under two alternative models analogously to well-known counterparts based on DR, if available. Unlike Warner’s (1965) treatment of categorical RR we consider quantitative RR here.     

Spline-based quasi-likelihood estimation of mixed Poisson regression with single-index models

We consider spline-based quasi-likelihood estimation for mixed Poisson regression with single-index models. The unknown smooth function is approximated by B-splines, and a modified Fisher scoring algorithm is employed to compute the estimates. The spline estimate of the nonparametric component is shown to achieve the optimal rate of convergence, and the asymptotic normality of the regression parameter estimates is still valid even if the variance function is misspecified. The semiparametric efficiency of the model can be established if the variance function is correctly specified. The variance of the regression parameter estimates can be consistently estimated by a simple procedure based on the least-squares estimation. The proposed method is evaluated via an extensive Monte Carlo study, and the methodology is illustrated on an air pollution study.     

Respondent Related Correlates of Response Behaviour in Audience Research

Audience research by means of surveys has a long tradition, certainly withinarts and humanities oriented research. Yet, due to selective sampling and unitnonresponse it frequently lacks the methodological rigour to make scientificallyvalid statements based on sample estimates. This is one of the first attempts toexplore unit nonresponse in audience research. More specifically, it focuses onthe explanation of nonresponse by the socio-demographic and more topic relatedcharacteristics of a theatre audience. Using a two-step procedure for the on-sitecollection of data, the characteristics of respondents are compared with those ofnonrespondents. In step 1 the composition of the theatre audience is comparedto a proxy of a theatre population benchmark based on a weighted sample of theFlemish population (APS-2000). The validity of this best available method isdiscussed. Step 2 compares respondents with nonrespondents on a micro-level:ignoring unit nonresponse in step 1, we use logistic regression to map selectionin step 2. The chance of cooperating with the survey has been found to increasewith educational attainment and vary according to occupational category. Moreover,involvement with survey topic is confirmed as a strong predictor of survey participation.Gender, age and experience with theatre remain insignificant in predicting responsebehaviour. These findings are compared with the socio-demographic correlates ofresponse behaviour in general populations. Implications for statistically controllingfor nonresponse bias in audience research are discussed. Suggestions for furtherresearch are presented.     

Estimation in sampling on two successive occasions

For estimating the mean of a finite population on the second of two successive occasions from a simple random sample, the authors [1] have proposedelsewhere an estimate which utilizes the data obtained from the sample on the first occasion as ancillary information. In this paper, it is shown that this estimate is more efficient than the one similar to that of P athak and R ao [2] in all situations where the well known ratio estimate in simple random sampling is no less efficient than the usual regression estimate or the R ao -H artley -C ochran estimate in sampling with varying probabilities and without replacement.     

Logit based parameter estimation in the Rasch model

The similarities between the logistic regression model and the Rasch model (used in psychometric item response theory) are used to derive several methods based on logits that produce parameter estimates for the Rasch model. A result from LeCam and Dzhaparidze is used by which an initial consistent estimate is transformed by one scoring method iteration into an estimate that has the same asymptotic efficiency as the (in this case conditional) maximum likelihood estimate of the item parameters. Indirect evidence about the bias of this CML estimator is produced by studying the (more easily derived) bias of the estimator based on the unweighted logits. Finally, some simple weighted least squares logit-based estimates are presented, and their performance is assessed. On the whole, the computationally simpler logit-based estimates give a fairly good approximation to the CML estimates.     

Conditional feature screening for mean and variance functions in models with multiple-index structure

The existing methods for feature screening focus mainly on the mean function of regression models. The variance function, however, plays an important role in statistical theory and application. We thus investigate feature screening for mean and variance functions with multiple-index framework in high dimensional regression models. Notice that some information about predictors can be known in advance from previous investigations and experience, for example, a certain set of predictors is related to the response. Based on the conditional information, together with empirical likelihood, we propose conditional feature screening procedures. Our methods can consistently estimate the sets of active predictors in the mean and variance functions. It is interesting that the proposed screening procedures can avoid estimating the unknown link functions in the mean and variance functions, and moreover, can work well in the case of high correlation among the predictors without iterative algorithm. Therefore, our proposal is of computational simplicity. Furthermore, as a conditional method, our method is robust to the choice of the conditional set. The theoretical results reveal that the proposed procedures have sure screening properties. The attractive finite sample performance of our method is illustrated in simulations and a real data application.     

The power of online panel paradata to predict unit nonresponse and voluntary attrition in a longitudinal design

The objective of this study is to identify factors affecting participation rates, i.e., nonresponse and voluntary attrition rates, and their predictive power in a probability-based online panel. Participation for this panel had already been investigated in the literature according to the socio-demographic and socio-psychological characteristics of respondents and different types of paradata, such as device type or questionnaire navigation, had also been explored. In this study, the predictive power of online panel participation paradata was instead evaluated, which was expected (at least in theory) to offer even more complex insight into respondents’ behavior over time. This kind of paradata would also enable the derivation of longitudinal variables measuring respondents’ panel activity, such as survey outcome rates and consecutive waves with a particular survey outcome prior to a wave (e.g., response, noncontact, refusal), and could also be used in models controlling for unobserved heterogeneity. Using the Life in Australia? participation data for all recruited members for the first 30 waves, multiple linear, binary logistic and panel random-effect logit regression analyses were carried out to assess socio-demographic and online panel paradata predictors of nonresponse and attrition that were available and contributed to the accuracy of prediction and the best statistical modeling. The proposed approach with the derived paradata predictors and random-effect logistic regression proved to be reasonably accurate for predicting nonresponse—with just 15 waves of online panel paradata (even without sociodemographics) and logit random-effect modeling almost four out of five nonrespondents could be correctly identified in the subsequent wave.

     

Modeling Item Nonresponse in Questionnaires

The statistical analysis of empirical questionnaire data can be hampered by the fact that not all questions are answered by all individuals. In this paper we propose a simple practical method to deal with such item nonresponse in case of ordinal questionnaire data, where we assume that item nonresponse is caused by an incomplete set of answers between which the individuals are supposed to choose. Our statistical method is based on extending the ordinal regression model with an additional category for nonresponse, and on investigating whether this extended model describes and forecasts the data well. We illustrate our approach for two questions from a questionnaire held amongst a sample of clients of a financial investment company.     

One-Sided Coverage Intervals for a Proportion Estimated from a Stratified Simple Random Sample

Using an Edgeworth expansion to speed up the asymptotics, we develop one-sided coverage intervals for a proportion based on a stratified simple random sample. To this end, we assume the values of the population units are generated from independent random variables with a common mean within each stratum. These stratum means, in turn, may either be free to vary or are assumed to be equal. The more general assumption is equivalent to a model-free randomization-based framework when finite population correction is ignored. Unlike when an Edgeworth expansion is used to construct one-sided intervals under simple random sampling, it is necessary to estimate the variance of the estimator for the population proportion when the stratum means are allowed to differ. As a result, there may be accuracy gains from replacing the normal  z -score in the Edgeworth expansion with a  t -score.     

Optimal allocation in J-stage sampling with stratification from finite population

Summary  The known results of optimal allocation in two-stage simple random sampling with stratification from finite population are generalized for J-stages. The optimalization is treated by minimizing the cost function for given variance as well as by minimizing the variance for given cost. It turns out that for both cases the optimal values differ only for the first-stage units and are the same for all the subsequent. The general formula for minimum variance of the unbiased estimator of the population mean per element is given.     

Optimal allocation in J-stage sampling with stratification from finite population

Summary The known results of optimal allocation in two-stage simple random sampling with stratification from finite population are generalized for J-stages. The optimalization is treated by minimizing the cost function for given variance as well as by minimizing the variance for given cost. It turns out that for both cases the optimal values differ only for the first-stage units and are the same for all the subsequent. The general formula for minimum variance of the unbiased estimator of the population mean per element is given.     

Optimal growth models as Economic Planning Tools

Conclusion In the present paper, I have carried out an investigation into the form of models which would be suitable for economy-wide planning. Optimal growth models have usually been constructed in an abstract framework. Therefore, the economies for which optimal growth models would be suitable are not identified. I have shown that, because these models only embody purely physical constraints, they will, in general, only be suitable for totally controlled economies. The conclusion, establishing the general inapplicability of optimal growth models to non-controlled economies, is dependent on the view of economic planning which regards the economic structure, particularly the scope of government policy, as fixed prior to plan construction.The second half of the paper elucidates the nature of the planning process when planners take into account the economic structure of the economy which they are planning. I have deliberately chosen to show the planning process in an abstract context for a simple economy, because the object is not to describe how to plan a particular type of economy, but to describe a methodology of planning. In that methodology, there is a place for optimal growth methods but only after the planners have given utmost consideration to the particular features of the economy they are planning. In using that methodology, one finds that problems must be confronted which are hidden when optimal growth models are formulated in the usual manner. Thus, planning for an infinite horizon may be the planners' preferred method on some basic philosophical grounds but in fact in order to construct an implementable plan they may need to follow the uncontrolled sector and adopt a pragmatic finite horizon approach.This paper is a revised version of a chapter of my Ph.D. thesis at the University of Pennsylvania. I would like to thank my adviser, Don Green, for many helpful comments.