Small Area Estimation-New Developments and Directions

The purpose of this paper is to provide a critical review of the main advances in small area estimation (SAE) methods in recent years. We also discuss some of the earlier developments, which serve as a necessary background for the new studies. The review focuses on model dependent methods with special emphasis on point prediction of the target area quantities, and mean square error assessments. The new models considered are models used for discrete measurements, time series models and models that arise under informative sampling. The possible gains from modeling the correlations among small area random effects used to represent the unexplained variation of the small area target quantities are examined. For review and appraisal of the earlier methods used for SAE, see Ghosh & Rao (1994).     

Multivariate Small Area Estimation of Multidimensional Latent Economic Well-being Indicators

Factor analysis models are used in data dimensionality reduction problems where the variability among observed variables can be described through a smaller number of unobserved latent variables. This approach is often used to estimate the multidimensionality of well-being. We employ factor analysis models and use multivariate empirical best linear unbiased predictor (EBLUP) under a unit-level small area estimation approach to predict a vector of means of factor scores representing well-being for small areas. We compare this approach with the standard approach whereby we use small area estimation (univariate and multivariate) to estimate a dashboard of EBLUPs of the means of the original variables and then averaged. Our simulation study shows that the use of factor scores provides estimates with lower variability than weighted and simple averages of standardised multivariate EBLUPs and univariate EBLUPs. Moreover, we find that when the correlation in the observed data is taken into account before small area estimates are computed, multivariate modelling does not provide large improvements in the precision of the estimates over the univariate modelling. We close with an application using the European Union Statistics on Income and Living Conditions data.     

A Comparison of Methods for Poverty Estimation in Developing Countries

Small area estimation is a widely used indirect estimation technique for micro‐level geographic profiling. Three unit level small area estimation techniques—the ELL or World Bank method, empirical best prediction (EBP) and M‐quantile (MQ) — can estimate micro‐level Foster, Greer, & Thorbecke (FGT) indicators: poverty incidence, gap and severity using both unit level survey and census data. However, they use different assumptions. The effects of using model‐based unit level census data reconstructed from cross‐tabulations and having no cluster level contextual variables for models are discussed, as are effects of small area and cluster level heterogeneity. A simulation‐based comparison of ELL, EBP and MQ uses a model‐based reconstruction of 2000/2001 data from Bangladesh and compares bias and mean square error. A three‐level ELL method is applied for comparison with the standard two‐level ELL that lacks a small area level component. An important finding is that the larger number of small areas for which ELL has been able to produce sufficiently accurate estimates in comparison with EBP and MQ has been driven more by the type of census data available or utilised than by the model per se.     

Modeling Clustered Ordered Categorical Data: A Survey

This article surveys various strategies for modeling ordered categorical (ordinal) response variables when the data have some type of clustering, extending a similar survey for binary data by Pendergast, Gange, Newton, Lindstrom, Palta & Fisher (1996). An important special case is when repeated measurement occurs at various occasions for each subject, such as in longitudinal studies. A much greater variety of models and fitting methods are available than when a similar survey for repeated ordinal response data was prepared a decade ago (Agresti, 1989). The primary emphasis of the review is on two classes of models, marginal models for which effects are averaged over all clusters at particular levels of predictors, and cluster-specific models for which effects apply at the cluster level. We present the two types of models in the ordinal context, review the literature for each, and discuss connections between them. Then, we summarize some alternative modeling approaches and ways of estimating parameters, including a Bayesian approach. We also discuss applications and areas likely to be popular for future research, such as ways of handling missing data and ways of modeling agreement and evaluating the accuracy of diagnostic tests. Finally, we review the current availability of software for using the methods discussed in this article.     

BAYESIAN INFERENCE METHODS FOR UNIVARIATE AND MULTIVARIATE GARCH MODELS: A SURVEY

This survey reviews the existing literature on the most relevant Bayesian inference methods for univariate and multivariate GARCH models. The advantages and drawbacks of each procedure are outlined as well as the advantages of the Bayesian approach versus classical procedures. The paper makes emphasis on recent Bayesian non‐parametric approaches for GARCH models that avoid imposing arbitrary parametric distributional assumptions. These novel approaches implicitly assume infinite mixture of Gaussian distributions on the standardized returns which have been shown to be more flexible and describe better the uncertainty about future volatilities. Finally, the survey presents an illustration using real data to show the flexibility and usefulness of the non‐parametric approach.     

Variational Bayes approximation of factor stochastic volatility models

Estimation and prediction in high dimensional multivariate factor stochastic volatility models is an important and active research area, because such models allow a parsimonious representation of multivariate stochastic volatility. Bayesian inference for factor stochastic volatility models is usually done by Markov chain Monte Carlo methods (often by particle Markov chain Monte Carlo methods), which are usually slow for high dimensional or long time series because of the large number of parameters and latent states involved. Our article makes two contributions. The first is to propose a fast and accurate variational Bayes methods to approximate the posterior distribution of the states and parameters in factor stochastic volatility models. The second is to extend this batch methodology to develop fast sequential variational updates for prediction as new observations arrive. The methods are applied to simulated and real datasets, and shown to produce good approximate inference and prediction compared to the latest particle Markov chain Monte Carlo approaches, but are much faster.     

Penalized Regression with Ordinal Predictors

Ordered categorial predictors are a common case in regression modelling. In contrast to the case of ordinal response variables, ordinal predictors have been largely neglected in the literature. In this paper, existing methods are reviewed and the use of penalized regression techniques is proposed. Based on dummy coding two types of penalization are explicitly developed; the first imposes a difference penalty, the second is a ridge type refitting procedure. Also a Bayesian motivation is provided. The concept is generalized to the case of non-normal outcomes within the framework of generalized linear models by applying penalized likelihood estimation. Simulation studies and real world data serve for illustration and to compare the approaches to methods often seen in practice, namely simple linear regression on the group labels and pure dummy coding. Especially the proposed difference penalty turns out to be highly competitive.     

Modelling Group Heterogeneity for Small Area Estimation Using M‐Quantiles

Small area estimation typically requires model‐based methods that depend on isolating the contribution to overall population heterogeneity associated with group (i.e. small area) membership. One way of doing this is via random effects models with latent group effects. Alternatively, one can use an M‐quantile ensemble model that assigns indices to sampled individuals characterising their contribution to overall sample heterogeneity. These indices are then aggregated to form group effects. The aim of this article is to contrast these two approaches to characterising group effects and to illustrate them in the context of small area estimation. In doing so, we consider a range of different data types, including continuous data, count data and binary response data.     

Estimation Methods and Related Systems at Statistics Canada

This paper provides an overview of research in estimation techniques, their application, and the development of generalized estimation systems at Statistics Canada. In Canada, the demand for more detailed and better quality cross-sectional data related to various sodo-economic issues has increased significantly in recent years. Also, there has been increasing interest in longitudinal data to better understand and interpret the relationships among variables, necessitating the implementation of a number of large scale panel surveys by Statistics Canada. The paper briefly discusses estimation for longitudinal data and a weighting approach developed for cross-sectional data Prom these surveys. For cross-sectional household and business surveys, as well as the census of population, appropriate dibration estimators developed for each situation are briefly discussed. In addition, regression composite estimation, a method developed to improve the quality of cross-sectional estimates from rotating panel surveys such as the Canadian Labour Force Survey, is presented. With regard to more detailed cross-sectional estimates at sub-provincial levels, different approaches to small area estimation developed for various programs are also presented. We SUmmarize the various modules developed lor the GeneraIiized Ektimation System. important new developments within the system include two-phase estimation as well as the estimation of variance for a number of imputation procedures. We briefly review the status of current estimation research on selected topics as well as the direction of future research.     

Alternative Approaches to Multilevel Modelling of Survey Non‐Contact and Refusal

We review three alternative approaches to modelling survey non‐contact and refusal: multinomial, sequential, and sample selection (bivariate probit) models. We then propose a multilevel extension of the sample selection model to allow for both interviewer effects and dependency between non‐contact and refusal rates at the household and interviewer level. All methods are applied and compared in an analysis of household non‐response in the United Kingdom, using a data set with unusually rich information on both respondents and non‐respondents from six major surveys. After controlling for household characteristics, there is little evidence of residual correlation between the unobserved characteristics affecting non‐contact and refusal propensities at either the household or the interviewer level. We also find that the estimated coefficients of the multinomial and sequential models are surprisingly similar, which further investigation via a simulation study suggests is due to non‐contact and refusal having largely different predictors.     

Cumulative and CUB Models for Rating Data: A Comparative Analysis

Ordinal measurements as ratings, preference and evaluation data are very common in applied disciplines, and their analysis requires a proper modelling approach for interpretation, classification and prediction of response patterns. This work proposes a comparative discussion between two statistical frameworks that serve these goals: the established class of cumulative models and a class of mixtures of discrete random variables, denoted as CUB models, whose peculiar feature is the specification of an uncertainty component to deal with indecision and heterogeneity. After surveying their definition and main features, we compare the performances of the selected paradigms by means of simulation experiments and selected case studies. The paper is tailored to enrich the understanding of the two approaches by running an extensive and comparative analysis of results, relative advantages and limitations, also at graphical level. In conclusion, a summarising review of the key issues of the alternative strategies and some final remarks are given, aimed to support a unifying setting.     

NONPARAMETRIC AND SEMIPARAMETRIC PANEL DATA MODELS: RECENT DEVELOPMENTS

In this paper, we provide an intensive review of the recent developments for semiparametric and fully nonparametric panel data models that are linearly separable in the innovation and the individual-specific term. We analyze these developments under two alternative model specifications: fixed and random effects panel data models. More precisely, in the random effects setting, we focus our attention in the analysis of some efficiency issues that have to do with the so-called working independence condition. This assumption is introduced when estimating the asymptotic variance–covariance matrix of nonparametric estimators. In the fixed effects setting, to cope with the so-called incidental parameters problem, we consider two different estimation approaches: profiling techniques and differencing methods. Furthermore, we are also interested in the endogeneity problem and how instrumental variables are used in this context. In addition, for practitioners, we also show different ways of avoiding the so-called curse of dimensionality problem in pure nonparametric models. In this way, semiparametric and additive models appear as a solution when the number of explanatory variables is large.     

Modelling factor demands with SEM and VAR: an empirical comparison

The empirical analysis of the economic interactions between factors of production, output and corresponding prices has received much attention over the last two decades. Most contributions in this area have agreed on the neoclassical principle of a representative optimizing firm and typically use theory-based structural equation models (SEM). A popular alternative to SEM is the vector autoregression (VAR) methodology. The most recent attempts to link the SEM approach with VAR analysis in the area of factor demands concentrate on single-equation models, whereas no effort has been devoted to compare these alternative approaches when a firm is assumed to face a multi-factor technology and to decide simultaneously the optimal quantity for each input. This paper bridges this gap. First, we illustrate how the SEM and the VAR approaches can both represent valid alternatives to model systems of dynamic factor demands. Second, we show how to apply both methodologies to estimate dynamic factor demands derived from a cost-minimizing capital-labour-energy-materials (KLEM) technology with adjustment costs (ADC) on the quasi-fixed capital factor. Third, we explain how to use both models to calculate some widely accepted indicators of the production structure of an economic sector, such as price and quantity elasticities, and alternative measures of ADC. In particular, we propose and discuss some theoretical and empirical justifications of the differences between observed elasticities, measures of ADC, and the assumption of exogeneity of output and/or input prices. Finally, we offer some suggestions for the applied researcher.      

Bayesian prior elicitation in DSGE models: Macro- vs micropriors

Bayesian approaches to the estimation of DSGE models are becoming increasingly popular. Prior knowledge is normally formalized either directly on deep parameters' values (‘microprior’) or indirectly, on macroeconomic indicators, e.g. moments of observable variables (‘macroprior’). We introduce a non-parametric macroprior which is elicited from impulse response functions and assess its performance in shaping posterior estimates. We find that using a macroprior can lead to substantially different posterior estimates. We probe into the details of our result, showing that model misspecification is likely to be responsible of that. In addition, we assess to what extent the use of macropriors is impaired by the need of calibrating some hyperparameters.     

Holt’s exponential smoothing and neural network models for forecasting interval-valued time series

Interval-valued time series are interval-valued data that are collected in a chronological sequence over time. This paper introduces three approaches to forecasting interval-valued time series. The first two approaches are based on multilayer perceptron (MLP) neural networks and Holt’s exponential smoothing methods, respectively. In Holt’s method for interval-valued time series, the smoothing parameters are estimated by using techniques for non-linear optimization problems with bound constraints. The third approach is based on a hybrid methodology that combines the MLP and Holt models. The practicality of the methods is demonstrated through simulation studies and applications using real interval-valued stock market time series.     

Varying Coefficient Regression Models: A Review and New Developments

Varying coefficient regression models are known to be very useful tools for analysing the relation between a response and a group of covariates. Their structure and interpretability are similar to those for the traditional linear regression model, but they are more flexible because of the infinite dimensionality of the corresponding parameter spaces. The aims of this paper are to give an overview on the existing methodological and theoretical developments for varying coefficient models and to discuss their extensions with some new developments. The new developments enable us to use different amount of smoothing for estimating different component functions in the models. They are for a flexible form of varying coefficient models that requires smoothing across different covariates' spaces and are based on the smooth backfitting technique that is admitted as a powerful technique for fitting structural regression models and is also known to free us from the curse of dimensionality.     

Statistical Inference in Nonparametric Frontier Models: The State of the Art

Efficiency scores of firms are measured by their distance to an estimated production frontier. The economic literature proposes several nonparametric frontier estimators based on the idea of enveloping the data (FDH and DEA-type estimators). Many have claimed that FDH and DEA techniques are non-statistical, as opposed to econometric approaches where particular parametric expressions are posited to model the frontier. We can now define a statistical model allowing determination of the statistical properties of the nonparametric estimators in the multi-output and multi-input case. New results provide the asymptotic sampling distribution of the FDH estimator in a multivariate setting and of the DEA estimator in the bivariate case. Sampling distributions may also be approximated by bootstrap distributions in very general situations. Consequently, statistical inference based on DEA/FDH-type estimators is now possible. These techniques allow correction for the bias of the efficiency estimators and estimation of confidence intervals for the efficiency measures. This paper summarizes the results which are now available, and provides a brief guide to the existing literature. Emphasizing the role of hypotheses and inference, we show how the results can be used or adapted for practical purposes.     

The emperor's new clothes: a critique of the multivariate t regression model

Zellner (1976) proposed a regression model in which the data vector (or the error vector) is represented as a realization from the multivariate Student t distribution. This model has attracted considerable attention because it seems to broaden the usual Gaussian assumption to allow for heavier-tailed error distributions. A number of results in the literature indicate that the standard inference procedures for the Gaussian model remain appropriate under the broader distributional assumption, leading to claims of robustness of the standard methods. We show that, although mathematically the two models are different, for purposes of statistical inference they are indistinguishable. The empirical implications of the multivariate t model are precisely the same as those of the Gaussian model. Hence the suggestion of a broader distributional representation of the data is spurious, and the claims of robustness are misleading. These conclusions are reached from both frequentist and Bayesian perspectives.     

A Bayesian approach for predicting food and beverage sales in staff canteens and restaurants

Accurate demand forecasting is one of the key aspects for successfully managing restaurants and staff canteens. In particular, properly predicting future sales of menu items allows for a precise ordering of food stock. From an environmental point of view, this ensures a low level of pre-consumer food waste, while from the managerial point of view, this is critical to the profitability of the restaurant. Hence, we are interested in predicting future values of the daily sold quantities of given menu items. The corresponding time series show multiple strong seasonalities, trend changes, data gaps, and outliers. We propose a forecasting approach that is solely based on the data retrieved from point-of-sale systems and allows for a straightforward human interpretation. Therefore, we propose two generalized additive models for predicting future sales. In an extensive evaluation, we consider two data sets consisting of multiple time series collected at a casual restaurant and a large staff canteen and covering a period of 20 months. We show that the proposed models fit the features of the considered restaurant data. Moreover, we compare the predictive performance of our method against the performance of other well-established forecasting approaches.