Household Consumption Equivalence Scales: Some Estimates from New Zealand Household Expenditure and Income Survey Data

This paper uses data on expenditures and incomes of New Zealand households of different demographic profiles to construct equivalence scales. The scales are useful in estimating the relative levels of spending required by the households to attain a given level of utility. Preference consistent 'complete demand systems' are analyzed to test for the demographic effects on the consumption patterns of households. Equivalence scales for respect to specific items of consumption as well as total consumption are worked out from the observed consumption behaviour of households. The results are largely in line with those found in existing similar studies both in Australia and New Zealand. To the best of our knowledge, this is the first attempt at constructing equivalence scales with New Zealand household budget data in the framework of hypothesized utility maximizing behaviour of households.     

Aggregate Consumer Behavior and Household Equivalence Scales

This article presents estimates of household equivalence scales, broken down by demographic characteristics, of U.S. households. Separate estimates are given by family size, age of head, region, race, and urban versus rural residence. Commodity-specific scales are presented for five separate commodity groups—energy, food, consumer goods, capital services, and other services. The estimates are obtained from an econometric model of aggregate consumer behavior. The parameters of this model are estimated by combining aggregate time series and individual cross-section data.     

De gustibus non est disputandum? A new approach to the estimation of equivalence scales

We argue that when the household composition changes, consumption patterns vary not only because of the cost effect that equivalence scales try to measure, but also because of a “taste” or “style” effect. This effect can be identified and measured, under a few assumptions, with the use of a new methodology, calledDM ² (Decomposition Model of the effects of Demographic Metamorphosis), that can be viewed as a generalisation of Ray's (1983) price-scaling approach to the construction of equivalent scales. An empirical application to data drawn from the Istat 1995 Italian Household Budget Survey suggests that the proposed method improves our understanding of households' consumption patterns and the reliability of the equivalence scales that we derive. We gratefully acknowledge helpful comments from Giorgio Calzolari, Franco Polverini, Ugo Trivellato and an anonymous referee, but retain full responsibility for all errors, and for the processing of Istat (Italian Institute of Statistics) microdata on Household Budgets. Financial support for this research was provided by the italian MURST (Research project on “Equivalence scales” directed by Prof. Guido Ferrari, University of Firenze, Ref. No. 9913105354; and Research project on “Low fertility in Italy: between economic constraints and value changes”, directed by Prof. Massimo Livi Bacci, University of Firenze, Ref. No. MM13107238). Preliminary findings on this research topic have been presented in a few seminars and conferences: cf., e.g., De Santis and Maltagliati (2000 and 2001).     

Tests of independence in incomplete multi-way tables using likelihood functions

Kang (2006) and Kang and Larsen (in press) used the log likelihood function with Lagrangian multipliers for estimation of cell probabilities in two-way incomplete contingency tables. This paper extends results and simulations to three-way and multi-way tables. Numerous studies cross-classify subjects by three or more categorical factors. Constraints on cell probabilities are incorporated through Lagrangian multipliers. Variances of the MLEs are derived from the matrix of second derivatives of the log likelihood with respect to cell probabilities and the Lagrange multiplier. Wald and likelihood ratio tests of independence are derived using the estimates and estimated variances. In simulation results in Kang and Larsen (in press), for data missing at random, maximum likelihood estimation (MLE) produced more efficient estimates of population proportions than either multiple imputation (MI) based on data augmentation or complete case (CC) analysis. Neither MLE nor MI, however, lead to an improvement over CC analysis with respect to power of tests for independence in two-way tables. Results are extended to multidimensional tables with arbitrary patterns of missing data when the variables are recorded on individual subjects. In three-way and higher-way tables, however, there is information relevant for judging independence in partially classified information, as long as two or more variables are jointly observed. Simulations study three-dimensional tables with three patterns of association and two levels of missing information.     

Specification of household engel curves by nonparametric regression

This paper demonstrates the usefulness of nonparametric regression analysis for functional specfication of houshold Engel curves.

After a brief review in section 2 of the literature on demand functions and equivalence scales and the functional specifications used, we first discuss in section 3 the issues of using income versus total expenditure, the origin and nature of the error terms in the light of utility theroy, and the interpretation of empirical demand functions. we shall reach the unorthodox view that household demand functions should be interpreted as conditional expectations relative to prices, household composition and either income or the conditional expectation of total expenditure (rather that total expenditure itself), where the latter conditional expectation is taken relative to income, prices and household composition. these two forms appear to be equivalent. this result also solves the simultaneity problem: the error variance matrix is no longer singular. Moreover, the errors are in general heteroskedastic.

In section 4 we discuss the model and the data, and in section 5 we review the nonparametric kernal regression approach.

In section 6 we derive the functional form of our household engel curves from nonparametric regression results, using the 1980 budget survey for the netherlands, in order to avoid model misspecification. thus the modl is derived directly from the data, without restricting its functional form. the nonparametric regression results are then translated to suitable parametric functional specifications, i.e., we choose parametric functional forms in accordance with the nanparametric regression results. these parametric specification are estimated by least squares, and various parameter restrictions are tested in order to simplify the models. this yields very simple final specifications of the household engel curves involved, namely linear functions of income and the number of children in two age groups.     

Specification of household engel curves by nonparametric regression

This paper demonstrates the usefulness of nonparametric regression analysis for functional specfication of houshold Engel curves.

After a brief review in section 2 of the literature on demand functions and equivalence scales and the functional specifications used, we first discuss in section 3 the issues of using income versus total expenditure, the origin and nature of the error terms in the light of utility theroy, and the interpretation of empirical demand functions. we shall reach the unorthodox view that household demand functions should be interpreted as conditional expectations relative to prices, household composition and either income or the conditional expectation of total expenditure (rather that total expenditure itself), where the latter conditional expectation is taken relative to income, prices and household composition. these two forms appear to be equivalent. this result also solves the simultaneity problem: the error variance matrix is no longer singular. Moreover, the errors are in general heteroskedastic.

In section 4 we discuss the model and the data, and in section 5 we review the nonparametric kernal regression approach.

In section 6 we derive the functional form of our household engel curves from nonparametric regression results, using the 1980 budget survey for the netherlands, in order to avoid model misspecification. thus the modl is derived directly from the data, without restricting its functional form. the nonparametric regression results are then translated to suitable parametric functional specifications, i.e., we choose parametric functional forms in accordance with the nanparametric regression results. these parametric specification are estimated by least squares, and various parameter restrictions are tested in order to simplify the models. this yields very simple final specifications of the household engel curves involved, namely linear functions of income and the number of children in two age groups.     

GIBBS SAMPLERS FOR A SET OF SEEMINGLY UNRELATED REGRESSIONS

Bayesian estimation of a collection of seemingly unrelated regressions, referred to as a ‘set of seemingly unrelated regressions’ is considered. The collection of seemingly unrelated regressions is linked by common coefficients and/or a common error covariance matrix. Gibbs samplers useful for estimating posterior quantities are described and one of them is applied to a set of linear expenditure functions to estimate household equivalent scales.     

Sampling within households in household surveys

Summary.  The number of people to select within selected households has significant consequences for the conduct and output of household surveys. The operational and data quality implications of this choice are carefully considered in many surveys, but the effect on statistical efficiency is not well understood. The usual approach is to select all people in each selected household, where operational and data quality concerns make this feasible. If not, one person is usually selected from each selected household. We find that this strategy is not always justified, and we develop intermediate designs between these two extremes. Current practices were developed when household survey field procedures needed to be simple and robust; however, more complex designs are now feasible owing to the increasing use of computer-assisted interviewing. We develop more flexible designs by optimizing survey cost, based on a simple cost model, subject to a required variance for an estimator of population total. The innovation lies in the fact that household sample sizes are small integers, which creates challenges in both design and estimation. The new methods are evaluated empirically by using census and health survey data, showing considerable improvement over existing methods in some cases.     

Multivariate Three-Parameter Log-Elliptical Distributions

The three-parameter log-elliptical distribution class is developed for the general situation in which the hypothesis of independence for the elements in a sample is not assumed. The parameter estimators are theoretically showed to be invariant under all distributions in the class by considering only a change in the constant of the scale parameter estimator. An estimation procedure based on the three-parameter lognormal distribution is proposed for the parameter estimation problem in any three-parameter log-elliptical distribution. Two classical lognormal data sets are analyzed without assuming independence in the sample in order to illustrate the proposed estimation procedure.     

Three-part model for fractional response variables with application to Chinese household health insurance coverage

A survey on health insurance was conducted in July and August of 2011 in three major cities in China. In this study, we analyze the household coverage rate, which is an important index of the quality of health insurance. The coverage rate is restricted to the unit interval [0, 1], and it may differ from other rate data in that the “two corners” are nonzero. That is, there are nonzero probabilities of zero and full coverage. Such data may also be encountered in economics, finance, medicine, and many other areas. The existing approaches may not be able to properly accommodate such data. In this study, we develop a three-part model that properly describes fractional response variables with non-ignorable zeros and ones. We investigate estimation and inference under two proportional constraints on the regression parameters. Such constraints may lead to more lucid interpretations and fewer unknown parameters and hence more accurate estimation. A simulation study is conducted to compare the performance of constrained and unconstrained models and show that estimation under constraint can be more efficient. The analysis of household health insurance coverage data suggests that household size, income, expense, and presence of chronic disease are associated with insurance coverage.     

Grouping Tests for Misspecification: An Application to Housing Demand

We consider several grouping tests for regression misspecification, with reference to housing-demand function estimation. We compare three existing test procedures, demonstrate modifications necessary in most applications, and propose a fourth test to distinguish between two categories of potential specification error. The test procedures are evaluated in artificial simulations of alternative errors. Finally, we apply the tests to FHA home purchase data. We reject the hypothesis that household and grouped regressions differ only by sampling error or random mismeasurement of household income or price. Our results have implications for choices among test procedures and interpretations of previous housing-demand analysis.     

Nonparametric regression with cross‐classified responses

In this article, we develop regression models with cross‐classified responses. Conditional independence structures can be explored/exploited through the selective inclusion/exclusion of terms in a certain functional ANOVA decomposition, and the estimation is done nonparametrically via the penalized likelihood method. A cohort of computational and data analytical tools are presented, which include cross‐validation for smoothing parameter selection, Kullback–Leibler projection for model selection, and Bayesian confidence intervals for odds ratios. Random effects are introduced to model possible correlations such as those found in longitudinal and clustered data. Empirical performances of the methods are explored in simulation studies of limited scales, and a real data example is presented using some eyetracking data from linguistic studies. The techniques are implemented in a suite of R functions, whose usage is briefly described in the appendix. The Canadian Journal of Statistics 39: 591–609; 2011. © 2011 Statistical Society of Canada     

Robust estimation in generalized linear mixed models

Generalized linear mixed models (GLMMs) are widely used to analyse non-normal response data with extra-variation, but non-robust estimators are still routinely used. We propose robust methods for maximum quasi-likelihood and residual maximum quasi-likelihood estimation to limit the influence of outlying observations in GLMMs. The estimation procedure parallels the development of robust estimation methods in linear mixed models, but with adjustments in the dependent variable and the variance component. The methods proposed are applied to three data sets and a comparison is made with the nonparametric maximum likelihood approach. When applied to a set of epileptic seizure data, the methods proposed have the desired effect of limiting the influence of outlying observations on the parameter estimates. Simulation shows that one of the residual maximum quasi-likelihood proposals has a smaller bias than those of the other estimation methods. We further discuss the equivalence of two GLMM formulations when the response variable follows an exponential family. Their extensions to robust GLMMs and their comparative advantages in modelling are described. Some possible modifications of the robust GLMM estimation methods are given to provide further flexibility for applying the method.     

ESTIMATING BAYESIAN NETWORKS FOR HIGH‐DIMENSIONAL DATA WITH COMPLEX MEAN STRUCTURE AND RANDOM EFFECTS

The estimation of Bayesian networks given high‐dimensional data, in particular gene expression data, has been the focus of much recent research. Whilst there are several methods available for the estimation of such networks, these typically assume that the data consist of independent and identically distributed samples. It is often the case, however, that the available data have a more complex mean structure, plus additional components of variance, which must then be accounted for in the estimation of a Bayesian network. In this paper, score metrics that take account of such complexities are proposed for use in conjunction with score‐based methods for the estimation of Bayesian networks. We propose first, a fully Bayesian score metric, and second, a metric inspired by the notion of restricted maximum likelihood. We demonstrate the performance of these new metrics for the estimation of Bayesian networks using simulated data with known complex mean structures. We then present the analysis of expression levels of grape‐berry genes adjusting for exogenous variables believed to affect the expression levels of the genes. Demonstrable biological effects can be inferred from the estimated conditional independence relationships and correlations amongst the grape‐berry genes.     

Inference for ergodic diffusions plus noise

We research an adaptive maximum‐likelihood–type estimation for an ergodic diffusion process where the observation is contaminated by noise. This methodology leads to the asymptotic independence of the estimators for the variance of observation noise, the diffusion parameter, and the drift one of the latent diffusion process. Moreover, it can lessen the computational burden compared to simultaneous maximum likelihood–type estimation. In addition to adaptive estimation, we propose a test to see if noise exists or not and analyze real data as the example such that the data contain observation noise with statistical significance.     

A Likelihood Based Estimating Equation for the Clayton–Oakes Model with Marginal Proportional Hazards

Multivariate failure time data arise when data consist of clusters in which the failure times may be dependent. A popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption. In this paper, we consider the Clayton–Oakes model with marginal proportional hazards and use the full model structure to improve on efficiency compared with the independence analysis. We derive a likelihood based estimating equation for the regression parameters as well as for the correlation parameter of the model. We give the large sample properties of the estimators arising from this estimating equation. Finally, we investigate the small sample properties of the estimators through Monte Carlo simulations.     

New Zealand Consumption Equivalence Scales From Quasi-unit Record Data

This paper uses quasi-unit record data on expenditures and incomes of New Zealand households of different demographic profiles to estimate consumption equivalence scales which measure the relative levels of spending required by households of different composition to attain given levels of utility. The equivalence scales with respect to specific items of consumption, as well as total consumption, have been obtained from the parameters of the preference-consistent Extended Linear Expenditure System and a few alternative versions of a rank 3 complete demand system with demand shifters. Among the demographic variables included in the models, the type of housing tenure and the actual age of children have special relevance. For the sake of comparison, a few heuristic scales are also computed. The results are largely in line with those found in similar studies of Australia and New Zealand. The analysis is developed within the framework of hypothesized utility-maximizing behaviour of households.     

Semiparametric Spatial Autoregressive Models With Endogenous Regressors: With an Application to Crime Data

This study considers semiparametric spatial autoregressive models that allow for endogenous regressors, as well as the heterogenous effects of these regressors across spatial units. For the model estimation, we propose a semiparametric series generalized method of moments estimator. We establish that the proposed estimator is both consistent and asymptotically normal. As an empirical illustration, we apply the proposed model and method to Tokyo crime data to estimate how the existence of a neighborhood police substation (NPS) affects the household burglary rate. The results indicate that the presence of an NPS helps reduce household burglaries, and that the effects of some variables are heterogenous with respect to residential distribution patterns. Furthermore, we show that using a model that does not adjust for the endogeneity of NPS does not allow us to observe the significant relationship between NPS and the household burglary rate. Supplementary materials for this article are available online.     

Nonlinear mixed effects modelling for estimation of steady state attainment

Assessment of the time needed to attain steady state is a key pharmacokinetic objective during drug development. Traditional approaches for assessing steady state include ANOVA‐based methods for comparing mean plasma concentration values from each sampling day, with either a difference or equivalence test. However, hypothesis‐testing approaches are ill suited for assessment of steady state. This paper presents a nonlinear mixed effects modelling approach for estimation of steady state attainment, based on fitting a simple nonlinear mixed model to observed trough plasma concentrations. The simple nonlinear mixed model is developed and proposed for use under certain pharmacokinetic assumptions. The nonlinear mixed modelling estimation approach is described and illustrated by application to trough data from a multiple dose trial in healthy subjects. The performance of the nonlinear mixed modelling approach is compared to ANOVA‐based approaches by means of simulation techniques. Copyright © 2005 John Wiley & Sons, Ltd.     

Score-based causal learning in additive noise models

Given data sampled from a number of variables, one is often interested in the underlying causal relationships in the form of a directed acyclic graph. In the general case, without interventions on some of the variables it is only possible to identify the graph up to its Markov equivalence class. However, in some situations one can find the true causal graph just from observational data, for example, in structural equation models with additive noise and nonlinear edge functions. Most current methods for achieving this rely on nonparametric independence tests. One of the problems there is that the null hypothesis is independence, which is what one would like to get evidence for. We take a different approach in our work by using a penalized likelihood as a score for model selection. This is practically feasible in many settings and has the advantage of yielding a natural ranking of the candidate models. When making smoothness assumptions on the probability density space, we prove consistency of the penalized maximum likelihood estimator. We also present empirical results for simulated scenarios and real two-dimensional data sets (cause–effect pairs) where we obtain similar results as other state-of-the-art methods.