Journal of Probability and Statistics

Journal of Probability and Statistics / 2012 / Article
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Joint Models and Their Applications

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Research Article | Open Access

Volume 2012 |Article ID 617678 | 26 pages | https://doi.org/10.1155/2012/617678

Bayesian Approach to Zero-Inflated Bivariate Ordered Probit Regression Model, with an Application to Tobacco Use

Academic Editor: Wenbin Lu
Received13 Jul 2011
Revised18 Sep 2011
Accepted02 Oct 2011
Published12 Dec 2011

Abstract

This paper presents a Bayesian analysis of bivariate ordered probit regression model with excess of zeros. Specifically, in the context of joint modeling of two ordered outcomes, we develop zero-inflated bivariate ordered probit model and carry out estimation using Markov Chain Monte Carlo techniques. Using household tobacco survey data with substantial proportion of zeros, we analyze the socioeconomic determinants of individual problem of smoking and chewing tobacco. In our illustration, we find strong evidence that accounting for excess zeros provides good fit to the data. The example shows that the use of a model that ignores zero-inflation masks differential effects of covariates on nonusers and users.

1. Introduction

This paper is concerned with joint modeling of two ordered data outcomes allowing for excess zeros. Economic, biological, and social science studies often yield data on two ordered categorical variables that are jointly dependent. Examples include the relationship between desired and excess fertility [1, 2], helmet use and motorcycle injuries [3], ownership of dogs and televisions [4], severity of diabetic retinopathy of the left and right eyes [5], and self-assessed health status and wealth [6]. The underlying response variables could be measured on an ordinal scale. It is also common in the literature to generate a categorical or grouped variable from an underlying quantitative variable and then use ordinal response regression model (e.g., [4, 5, 7]). The ensuing model is usually analyzed using the bivariate ordered probit model.

Many ordered discrete data sets are characterized by excess of zeros, both in terms of the proportion of nonusers and relative to the basic ordered probit or logit model. The zeros may be attributed to either corner solution to consumer optimization problem or errors in recording. In the case of individual smoking behavior, for example, the zeros may be recorded for individuals who never smoke cigarettes or for those who either used tobacco in the past or are potential smokers. In the context of individual patents applied for by scientists during a period of five years, zero patents may be recorded for scientists who either never made patent applications or for those who do but not during the reporting period [8]. Ignoring the two types of zeros for nonusers or nonparticipants leads to model misspecification.

The univariate as well as bivariate zero-inflated count data models are well established in the literature for example, Lambert [9], Gurmu and Trivedi [10], Mullahy [11], and Gurmu and Elder [12]. The recent literature presents a Bayesian treatment of zero-inflated Poisson models in both cross-sectional and panel data settings (see [13, 14], and references there in). By contrast, little attention has been given to the problem of excess zeros in the ordered discrete choice models. Recently, an important paper by Harris and Zhao [15] developed a zero-inflated univariate ordered probit model. However, the problem of excess zeros in ordered probit models has not been analyzed in the Bayesian framework. Despite recent applications and advances in estimation of bivariate ordered probit models [1–6], we know of no studies that model excess zeros in bivariate ordered probit models.

This paper presents a Bayesian analysis of bivariate ordered probit model with excess of zeros. Specifically, we develop a zero-inflated ordered probit model and carry out the analysis using the Bayesian approach. The Bayesian analysis is carried out using Markov Chain Monte Carlo (MCMC) techniques to approximate the posterior distribution of the parameters. Bayesian analysis of the univariate zero-inflated ordered probit will be treated as a special case of the zero-inflated bivariate order probit model. The proposed models are illustrated by analyzing the socioeconomic determinants of individual choice problem of bivariate ordered outcomes on smoking and chewing tobacco. We use household tobacco prevalence survey data from Bangladesh. The observed proportion of zeros (those identifying themselves as nonusers of tobacco) is about 76% for smoking and 87% for chewing tobacco.

The proposed approach is useful for the analysis of ordinal data with natural zeros. The empirical analysis clearly shows the importance of accounting for excess zeros in ordinal qualitative response models. Accounting for excess zeros provides good fit to the data. In terms of both the signs and magnitudes of marginal effects, various covariates have differential impacts on the probabilities associated with the two types of zeros, nonparticipants and zero-consumption. The usual analysis that ignores excess of zeros masks these differential effects, by just focusing on observed zeros. The empirical results also show the importance of taking into account the uncertainty in the parameter estimates. Another advantage of the Bayesian approach to modeling excess zeros is the flexibility, particularly computational, of generalizing to multivariate ordered response models.

The rest of the paper is organized as follows. Section 2 describes the proposed zero-inflated bivariate probit model. Section 3 presents the MCMC algorithm and model selection procedure for the model. An illustrative application using household tobacco consumption data is given in Section 4. Section 5 concludes the paper.

2. Zero-Inflated Bivariate Ordered Probit Model

2.1. The Basic Model

We consider the basic Bayesian approach to a bivariate latent variable regression model with excess of zeros. To develop notation, let Μƒπ‘¦βˆ—1𝑖 and Μƒπ‘¦βˆ—2𝑖 denote the bivariate latent variables. We consider two observed ordered response variables ̃𝑦1𝑖 and ̃𝑦2𝑖 taking on values 0,1,…,π½π‘Ÿ, for π‘Ÿ=1,2. Define two sets of cut-off parameters π›Όπ‘Ÿ=(π›Όπ‘Ÿ2,π›Όπ‘Ÿ3,…,π›Όπ‘Ÿπ½π‘Ÿ), π‘Ÿ=1,2, where the restrictions π›Όπ‘Ÿ0=βˆ’βˆž, π›Όπ‘Ÿπ½π‘Ÿ+1=∞, and π›Όπ‘Ÿ1=0 have been imposed. We assume that (Μƒπ‘¦βˆ—1𝑖,Μƒπ‘¦βˆ—2𝑖)ξ…žβ‰‘Μƒπ²βˆ—π‘– follows a bivariate regression modelΜƒπ‘¦βˆ—π‘Ÿπ‘–=π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿ+πœ€π‘Ÿπ‘–,π‘Ÿ=1,2,(2.1) where π±π‘Ÿπ‘– is a πΎπ‘Ÿ-variate of regressors for the 𝑖th individual (𝑖=1,…,𝑁) and πœ€π‘Ÿπ‘– are the error terms. For subsequent analysis, let 𝜷=(πœ·ξ…ž1,πœ·ξ…ž2)ξ…ž, 𝝐𝑖=(πœ–1𝑖,πœ–2𝑖)ξ…ž, and 𝐗𝑖=βŽ›βŽœβŽœβŽπ±ξ…ž1𝑖0ξ…ž0ξ…žπ±ξ…ž2π‘–βŽžβŽŸβŽŸβŽ .(2.2) Analogous to the univariate case, the observed bivariate-dependent variables are defined asΜƒπ‘¦π‘Ÿπ‘–=⎧βŽͺ⎨βŽͺ⎩0ifΜƒπ‘¦βˆ—π‘Ÿπ‘–β‰€0,1if0<Μƒπ‘¦βˆ—π‘Ÿπ‘–β‰€π›Όπ‘Ÿ2,𝑗ifπ›Όπ‘Ÿπ‘—<Μƒπ‘¦βˆ—π‘Ÿπ‘–β‰€π›Όπ‘Ÿπ‘—+1,𝑗=2,3,…,π½π‘Ÿπ½βˆ’1,π‘ŸifΜƒπ‘¦βˆ—π‘–β‰€π›Όπ‘Ÿπ½π‘Ÿ,(2.3) where π‘Ÿ=1,2. Let ̃𝐲𝑖=(̃𝑦1𝑖,̃𝑦2𝑖)ξ…ž.

We introduce inflation at the point (̃𝑦1𝑖=0,̃𝑦2𝑖=0), called the zero-zero state. As in the univariate case, define the participation model:π‘ βˆ—π‘–=π³ξ…žπ‘–π›Ύ+πœ‡π‘–,𝑠𝑖𝑠=πΌβˆ—π‘–ξ€Έ.>0(2.4) In the context of the zero-inflation model, the observed response random vector 𝐲𝑖=(𝑦1𝑖,𝑦2𝑖)ξ…ž takes the form𝐲𝑖=𝑠𝑖̃𝐲𝑖.(2.5) We observe 𝐲𝑖=𝟎 when either the individual is a non-participant (𝑠𝑖=0) or the individual is a zero-consumption participant (𝑠𝑖=1 and ̃𝐲𝑖=𝟎). Likewise, we observe positive outcome (consumption) when the individual is a positive consumption participant for at least one good (𝑠𝑖=1 and Μƒπ²π‘–β‰ πŸŽ).

Let Ξ¦(π‘Ž) and πœ™(π‘Ž) denote the respective cumulative distribution and probability density functions of standardized normal evaluated at π‘Ž. Assuming normality and that πœ‡π‘– is uncorrelated with (πœ€1𝑖,πœ€2𝑖), but corr(πœ€1𝑖,πœ€2𝑖)=𝜌12β‰ 0, and each component with unit variance, the zero-inflated bivariate ordered probit (ZIBOP) distribution isπ‘“π‘ξ€·π²βˆ—π‘–,𝐲𝑖,π‘ βˆ—π‘–,π‘ π‘–βˆ£π—π‘–,𝐳𝑖=𝑠,Ξ¨Pr𝑖+𝑠=01βˆ’Pr𝑖=0ξ€Έξ€ΈPr̃𝑦1𝑖=0,̃𝑦2𝑖=0,for̃𝑦1𝑖,̃𝑦2𝑖𝑠=(0,0)1βˆ’Pr𝑖=0ξ€Έξ€ΈPr̃𝑦1𝑖=𝑗,̃𝑦2𝑖=𝑙,for̃𝑦1𝑖,̃𝑦2𝑖≠(0,0),(2.6) where 𝑗=0,1,…,𝐽1,𝑙=0,1,…,𝐽2,Pr(𝑠𝑖=0)=Ξ¦(βˆ’π³ξ…žπ‘–π›Ύ),Pr(𝑠𝑖=1)=Ξ¦(βˆ’π³ξ…žπ‘–π›Ύ). Further, for (̃𝑦1𝑖,̃𝑦2𝑖)=(0,0) in (2.6), we have π›Όπ‘Ÿ0=βˆ’βˆž,π›Όπ‘Ÿ1=0 for π‘Ÿ=1,2 so thatξ€·Pr̃𝑦1𝑖=0,̃𝑦2𝑖=0=Ξ¦2ξ€·βˆ’π±ξ…ž1𝑖𝛽1,βˆ’π±ξ…ž2𝑖𝛽2,𝜌12ξ€Έ,(2.7) where Ξ¦2(β‹…) is the cdf for the standardized bivariate normal. Likewise, Pr(̃𝑦1𝑖=𝑗,̃𝑦2𝑖=𝑙) in (2.6) are given byξ€·Pr̃𝑦1𝑖=𝑗,̃𝑦2𝑖=𝑙=Ξ¦2𝛼1𝑗+1βˆ’π±ξ…ž1𝑖𝛽1,𝛼2𝑙+1βˆ’π±ξ…ž2𝑖𝛽2;𝜌12ξ€Έβˆ’Ξ¦2𝛼1π‘—βˆ’π±ξ…ž1𝑖𝛽1,𝛼2π‘™βˆ’π±ξ…ž2𝑖𝛽2,𝜌12ξ€Έfor𝑗=1,…,𝐽1βˆ’1;𝑙=1,…,𝐽2ξ€·βˆ’1;Pr̃𝑦1𝑖=𝐽1,̃𝑦2𝑖=𝐽2ξ€Έ=1βˆ’Ξ¦2𝛼1𝐽1βˆ’π±ξ…ž1𝑖𝛽1,𝛼2𝐽2βˆ’π±ξ…ž2𝑖𝛽2,𝜌12ξ€Έ.(2.8) The ensuing likelihood contribution for 𝑁-independent observations isβ„’π‘ξ€·π²βˆ—,𝐲,π‘ βˆ—,π‘ βˆ£π—,𝐳,Ψ𝑏=𝑁𝑖=1(𝑗,𝑙)=(0,0)𝑠Pr𝑖+𝑠=01βˆ’Pr𝑖=0ξ€Έξ€ΈPr̃𝑦1𝑖=0,̃𝑦2𝑖=0𝑑𝑖𝑗𝑙×𝑁𝑖=1(𝑗,𝑙)β‰ (0,0)𝑠1βˆ’Pr𝑖=0ξ€Έξ€ΈPr̃𝑦1𝑖=𝑗,̃𝑦2𝑖=𝑙𝑑𝑖𝑗𝑙,(2.9) where 𝑑𝑖𝑗𝑙=1 if ̃𝑦1𝑖=𝑗 and ̃𝑦2𝑖=𝑙, and 𝑑𝑖𝑗𝑙=0 otherwise. Here, the vector Ψ𝑏 consists of 𝜷,𝜸,𝛼1,𝛼2, and the parameters associated with the trivariate distribution of (𝝐,πœ‡).

Regarding identification of the parameters in the model defined by (2.1) through (2.5) with normality assumption, we note that the mean parameter (joint choice probability associate with the observed response vector 𝐲𝑖) depends nonlinearly on the probability of zero inflation (Ξ¦(βˆ’π³ξ…žπ‘–πœΈ)) and choice probability (Pr(̃𝑦1𝑖=𝑗,̃𝑦2𝑖=𝑙)) coming from the BOP submodel. Since the likelihood function for ZIBOP depends separately on the two regression components, the parameters of ZIBOP model with covariates are identified as long as the model is estimated by full maximum likelihood method. The same or different sets of covariates can affect the two components via 𝐳𝑖 and π±π‘Ÿπ‘–. When using quasi-likelihood estimation or generalized estimating equations methods rather than full ML, the class of identifiable zero-inflated count and ordered data models is generally more restricted; see, for example, Hall and Shen [16] and references there in. Although the parameters in the ZIBOP model above are identified through a nonlinear functional form estimated by ML, for more robust identification we can use traditional exclusion restrictions by including instrumental variables in the inflation equation, but excluding them from the ordered choice submodel. We follow this strategy in the empirical section.

About 2/3 of the observations in our tobacco application below have a double-zero-state, (𝑦1=0,𝑦2=0). Consequently, we focused on a mixture constructed from a point mass at (0,0) and a bivariate ordered probit. In addition to allowing for inflation in the double-zero-state, our approach can be extended to allow for zero-inflation in each component.

2.2. Marginal Effects

It is common to use marginal or partial effects to interpret covariate effects in nonlinear models; see, for example, Liu et al. [17]. Due to the nonlinearity in zero-inflated ordered response models and in addition to estimation of regression parameters, it is essential to obtain the marginal effects of changes in covariates on various probabilities of interest. These include the effects of covariates on probability of nonparticipation (zero-inflation), probability of participation, and joint and/or marginal probabilities of choice associated with different levels of consumption.

From a practical point of view, we are less interested in the marginal effects of explanatory variables on the joint probabilities of choice from ZIBOP. Instead, we focus on the marginal effects associated with the marginal distributions of π‘¦π‘Ÿπ‘– for π‘Ÿ=1,2. Define a generic (scalar) covariate 𝑀𝑖 that can be a binary or approximately continuous variable. We obtain the marginal effects of a generic covariate 𝑀𝑖 on various probabilities assuming that the regression results are based on ZIBOP. If 𝑀𝑖 is a binary regressor, then the marginal effect of 𝑀𝑖 on probability, say 𝑃, is the difference in the probability evaluated at 1 and 0, conditional on observable values of covariates: 𝑃(𝑀𝑖=1)βˆ’π‘ƒ(𝑀𝑖=0). For continuous explanatory variables, the marginal effect is given by the partial derivative of the probability of interest with respect to 𝑀𝑖, πœ•π‘ƒ(β‹…)/πœ•π‘€π‘–.

Regressor 𝑀𝑖 can be a common covariate in vectors of regressors π±π‘Ÿπ‘– and 𝐳𝑖 or appears in either π±π‘Ÿπ‘– or 𝐳𝑖. Focusing on the continuous regressor case, the marginal effects of 𝑀𝑖 in each of the three cases are presented below. First, consider the case of common covariate in participation and main parts of the model, that is, 𝑀𝑖 in both π±π‘Ÿπ‘– and 𝐳𝑖. The marginal effect on the probability of participation is given by𝑀𝑖𝑠𝑖=𝑠=1πœ•Pr𝑖=1πœ•π‘€π‘–ξ€·π³=πœ™ξ…žπ‘–πœΈξ€Έπ›Ύπ‘€π‘–,(2.10) where again πœ™(β‹…) is the probability density function (pdf) of the standard normal distribution and 𝛾𝑀𝑖 is the coefficient in the inflation part associated with variable 𝑀𝑖. In terms of the zeros category, the effect on the probability of nonparticipation (zero inflation) is𝑀𝑖𝑠𝑖=𝑠=0πœ•Pr𝑖=0πœ•π‘€π‘–ξ€·=βˆ’πœ™βˆ’π³ξ…žπ‘–πœΈξ€Έπ›Ύπ‘€π‘–,(2.11) while𝑀𝑖𝑠=1,Μƒπ‘¦π‘Ÿπ‘–ξ€Έ=𝑠=0πœ•Pr𝑖=1PrΜƒπ‘¦π‘Ÿπ‘–ξ€Έ=0πœ•π‘€π‘–ξ€·=Ξ¦βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έπœ™ξ€·π³ξ…žπ‘–πœΈξ€Έπ›Ύπ‘€π‘–ξ€·π³βˆ’Ξ¦ξ…žπ‘–πœΈξ€Έπœ™ξ€·βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έπ›½π‘Ÿπ‘€π‘–,π‘Ÿ=1,2,(2.12) represents the marginal effect on the probability of zero-consumption. Here the scalar π›½π‘Ÿπ‘€π‘– is the coefficient in the main part of the model associated with 𝑀𝑖.

Continuing with the case of common covariate, the marginal effects of 𝑀𝑖 on the probabilities of choice are given as follows. First, the total marginal effect on the probability of observing zero-consumption is obtained as a sum of the marginal effects in (2.11) and (2.12); that is,π‘€π‘–ξ€·π‘¦π‘Ÿπ‘–ξ€Έ=ξ€ΊΞ¦ξ€·=0βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€»πœ™ξ€·π³βˆ’1ξ…žπ‘–πœΈξ€Έπ›Ύπ‘€π‘–ξ€·π³βˆ’Ξ¦ξ…žπ‘–πœΈξ€Έπœ™ξ€·βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έπ›½π‘Ÿπ‘€π‘–.(2.13) The effects for the remaining choices for outcomes π‘Ÿ=1,2 are as follows:π‘€π‘–ξ€·π‘¦π‘Ÿπ‘–ξ€Έ=Φ𝛼=1π‘Ÿ2βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€·βˆ’Ξ¦βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿπœ™ξ€·π³ξ€Έξ€»ξ…žπ‘–πœΈξ€Έπ›Ύπ‘€π‘–ξ€·π³βˆ’Ξ¦ξ…žπ‘–πœΈπœ™ξ€·π›Όξ€Έξ€Ίπ‘Ÿ2βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€·βˆ’πœ™βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿπ›½ξ€Έξ€»π‘Ÿπ‘€π‘–;π‘€π‘–ξ€·π‘¦π‘Ÿπ‘–ξ€Έ=Φ𝛼=π‘—π‘Ÿ,𝑗+1βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€·π›Όβˆ’Ξ¦π‘Ÿπ‘—βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿπœ™ξ€·π³ξ€Έξ€»ξ…žπ‘–πœΈξ€Έπ›Ύπ‘€π‘–ξ€·π³βˆ’Ξ¦ξ…žπ‘–πœΈπœ™ξ€·π›Όξ€Έξ€Ίπ‘Ÿ,𝑗+1βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€·π›Όβˆ’πœ™π‘Ÿπ‘—βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿπ›½ξ€Έξ€»π‘Ÿπ‘€π‘–,for𝑗=2,…,π½π‘Ÿπ‘€βˆ’1;π‘–ξ€·π‘¦π‘Ÿπ‘–=π½π‘Ÿξ€Έ=𝛼1βˆ’Ξ¦π‘Ÿ,π½π‘Ÿβˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿπœ™ξ€·π³ξ€Έξ€»ξ…žπ‘–πœΈξ€Έπ›Ύπ‘€π‘–ξ€·π³+Ξ¦ξ…žπ‘–πœΈξ€Έπœ™ξ€·π›Όπ‘Ÿ,π½π‘Ÿβˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έπ›½π‘Ÿπ‘€π‘–.(2.14)

Now consider case 2, where a generic independent variable 𝑀𝑖 is included only in π±π‘Ÿπ‘–, the main part of the model. In this case, covariate 𝑀𝑖 has obviously no direct effect on the inflation part. The marginal effects of 𝑀𝑖 on various choice probabilities can be presented as follows:π‘€π‘–ξ€·π‘¦π‘Ÿπ‘–ξ€Έ=𝑦=π‘—πœ•Prπ‘Ÿπ‘–ξ€Έ=π‘—πœ•π‘€π‘–ξ€·π³=βˆ’Ξ¦ξ…žπ‘–πœΈπœ™ξ€·π›Όξ€Έξ€Ίπ‘Ÿ,𝑗+1βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€·π›Όβˆ’πœ™π‘Ÿπ‘—βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿπ›½ξ€Έξ€»π‘Ÿπ‘€π‘–,for𝑗=0,1,…,π½π‘Ÿ,(2.15) with π›Όπ‘Ÿ0=βˆ’βˆž, π›Όπ‘Ÿ1=0, and π›Όπ‘Ÿ,π½π‘Ÿ+1=∞. The marginal effects in (2.15) can be obtained by simply setting 𝛾𝑀𝑖=0 in (2.13) and (2.14).

For case 3, where 𝑀𝑖 appears only in 𝐳𝑖, its marginal effects on participation components given in (2.10) and (2.11) will not change. Since π›½π‘Ÿπ‘€π‘–=0 in case 3, the partial effects of 𝑀𝑖 on various choice probabilities take the form:π‘€π‘–ξ€·π‘¦π‘Ÿπ‘–ξ€Έ=Φ𝛼=π‘—π‘Ÿ,𝑗+1βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€·π›Όβˆ’Ξ¦π‘Ÿπ‘—βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿπœ™ξ€·π³ξ€Έξ€»ξ…žπ‘–πœΈξ€Έπ›Ύπ‘€π‘–for𝑗=0,1,…,π½π‘Ÿ.(2.16) Again, we impose the restrictions π›Όπ‘Ÿ0=βˆ’βˆž, π›Όπ‘Ÿ1=0 and π›Όπ‘Ÿ,π½π‘Ÿ+1=∞.

As noted by a referee, it is important to understand the sources of covariate effects and the relationship between the marginal effects and the coefficient estimates. Since 𝑦Prπ‘Ÿπ‘–ξ€Έ=𝑠=𝑗Pr𝑖=1PrΜƒπ‘¦π‘Ÿπ‘–=𝑗(2.17) for 𝑗=0,1,…,π½π‘Ÿ, the total effect of a generic covariate 𝑀𝑖 on probability of consumption at level 𝑗 comes from two (weighted) sources: the participation part (Pr(𝑠𝑖=1)) and the main ordered probit part (Pr(Μƒπ‘¦π‘Ÿπ‘–=𝑗)) such thatξ€·π‘ πœ•Pr𝑖=1πœ•π‘€π‘–ξ€·π³=πœ™ξ…žπ‘–πœΈξ€Έπ›Ύπ‘€π‘–;ξ€·(2.18)πœ•PrΜƒπ‘¦π‘Ÿπ‘–ξ€Έ=π‘—πœ•π‘€π‘–ξ€Ίπœ™ξ€·π›Ό=βˆ’π‘Ÿ,𝑗+1βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€·π›Όβˆ’πœ™π‘Ÿπ‘—βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿπ›½ξ€Έξ€»π‘Ÿπ‘€π‘–(2.19) with π›Όπ‘Ÿ0=βˆ’βˆž, π›Όπ‘Ÿ1=0,s and π›Όπ‘Ÿ,π½π‘Ÿ+1=∞. This shows that sign(𝛾𝑀𝑖) is the same as sign(πœ•Pr(𝑠𝑖=1)/πœ•π‘€π‘–)β€”the participation effect in (2.18)β€”but sign(π›½π‘Ÿπ‘€π‘–) is not necessarily the same as the sign of (πœ•Pr(Μƒπ‘¦π‘Ÿπ‘–=𝑗)/πœ•π‘€π‘–). The latter is particularly true in the left tail of the distribution, where the coefficient (π›½π‘Ÿπ‘€π‘–) and the main (unweighted) effect in (2.19) have opposite signs because ξ€½βˆ’ξ€Ίπœ™ξ€·π›Όπ‘Ÿ,𝑗+1βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€·π›Όβˆ’πœ™π‘Ÿπ‘—βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿξ€Έξ€»ξ€Ύβ‰‘πœ›(2.20) is negative. In this case, a positive effect coming from the main part requires π›½π‘Ÿπ‘€π‘– to be negative. By contrast, πœ› is positive in the right tail, but can be positive or negative when the terms (π›Όπ‘Ÿ,π‘—βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿ) and (π›Όπ‘Ÿ,𝑗+1βˆ’π±ξ…žπ‘Ÿπ‘–πœ·π‘Ÿ) are on the opposite sides of the mode of the distribution. This shows that a given covariate can have opposite effects in the participation and main models. Since the total effect of an explanatory variable on probability of choice is a weighted average of (2.18) and (2.19), interpretation of results should focus on marginal effects of covariates rather than the signs of estimated coefficients. This is the strategy adopted in the empirical analysis below.

2.3. A Special Case

Since the zero-inflated univariate ordered probit (ZIOP) model has not been analyzed previously in the Bayesian framework, we provide a brief sketch of the basic framework for ZIOP. The univariate ordered probit model with excess of zeros can be obtained as a special case of the ZIBOP model presented previously. To achieve this, let 𝜌12=0 in the ZIBOP model and focus on the first ordered outcome with π‘Ÿ=1. In the standard ordered response approach, the model for the latent variable Μƒπ‘¦βˆ—1𝑖 is given by (2.1) with π‘Ÿ=1. The observed ordered variable ̃𝑦1𝑖 can be presented compactly as̃𝑦1𝑖=𝐽𝑗=0𝛼𝑗𝐼1𝑗<Μƒπ‘¦βˆ—1𝑖≀𝛼1𝑗+1ξ€Έ,(2.21) where 𝐼(π‘€βˆˆπ΄) is the indicator function equal to 1 or 0 according to whether π‘€βˆˆπ΄ or not. Again 𝛼10,𝛼11,…,𝛼1𝐽1 are unknown threshold parameters, where we set 𝛼10=βˆ’βˆž, 𝛼11=0, and 𝛼1𝐽1+1=∞.

Zero-inflation is now introduced at point ̃𝑦1𝑖=0. Using the latent variable model (2.4) for the zero inflation, the observed binary variable is given by 𝑠𝑖=𝐼(π‘ βˆ—π‘–>0), where 𝐼(π‘ βˆ—π‘–>0)=1 if π‘ βˆ—π‘–>0, and 0 otherwise. In regime 1, 𝑠𝑖=1 or π‘ βˆ—π‘–>0 for participants (e.g., smokers), while, in regime 0, 𝑠𝑖=0 or π‘ βˆ—π‘–β‰€0 for nonparticipants. In the context of the zero-inflation model, the observed response variable takes the form 𝑦1𝑖=𝑠𝑖̃𝑦1𝑖. We observe 𝑦1𝑖=0 when either the individual is a non-participant (𝑠𝑖=0) or the individual is a zero-consumption participant (𝑠𝑖=1 and ̃𝑦1𝑖=0). Likewise, we observe positive outcome (consumption) when the individual is a positive consumption participant (𝑠𝑖=1andΜƒπ‘¦βˆ—1𝑖>0).

Assume that πœ–1 and πœ‡ are independently distributed. Harris and Zhao [15] also consider the case where πœ–1 and πœ‡ are correlated. In the context of our application, the correlated model did not provide improvements over the uncorrelated ZIOP in terms of deviance information criterion. The zero-inflated ordered multinomial distribution, say Pr(𝑦1𝑖), arises as a mixture of a degenerate distribution at zero and the assumed distribution of the response variable ̃𝑦1𝑖 as follows:𝑓1ξ€·π‘¦βˆ—1𝑖,𝑦1𝑖,π‘ βˆ—π‘–,π‘ π‘–βˆ£π±1𝑖,𝐳𝑖,Ξ¨1ξ€Έ=𝑠Pr𝑖𝑠=0+Pr𝑖=1Pr̃𝑦1𝑖𝑠=0,for𝑗=0Pr𝑖=1Pr̃𝑦1𝑖=𝑗,for𝑗=1,2,…,𝐽1,(2.22) where, for any parameter vector Ξ©10 associated with the distribution of (πœ–1,πœ‡), Ξ¨1=(𝜷1,𝜸,𝜢1,Ξ©10) with 𝜢1=(𝛼12,…,𝛼1𝐽1). For simplicity, dependence on latent variables, covariates, and parameters has been suppressed on the right-hand side of (2.22). The likelihood based on 𝑁-independent observations takes the formβ„’1ξ€·π‘¦βˆ—1,𝑦1,π‘ βˆ—,π‘ βˆ£π±1,𝐳,Ξ¨1ξ€Έ=𝑁𝐽𝑖=11𝑗=0𝑦Pr1𝑖=π‘—βˆ£π±1𝑖,𝐳𝑖,Ξ¨1𝑑𝑖𝑗=𝑁𝑖=1𝑗=0𝑠Pr𝑖𝑠=0+Pr𝑖=1Pr̃𝑦1𝑖=𝑗𝑑𝑖𝑗×𝑁𝑖=1𝑗>0𝑠Pr𝑖=1Pr̃𝑦1𝑖=𝑗𝑑𝑖𝑗,(2.23) where, for example, π‘¦βˆ—1=(π‘¦βˆ—1,…,π‘¦βˆ—π‘)ξ…ž, and 𝑑𝑖𝑗=1 if individual 𝑖 chooses outcome 𝑗, or 𝑑𝑖𝑗=0 otherwise.

Different choices of the specification of the joint distribution of (πœ–1𝑖,πœ‡π‘–) give rise to various zero-inflated ordered response models. For example, if the disturbance terms in the latent variable equations are normally distributed, we get the zero-inflated ordered probit model of Harris and Zhao [15]. The zero-inflated ordered logit model can be obtained by assuming that πœ–1𝑖 and πœ‡π‘– are independent, each of the random variables following the logistic distribution with cumulative distribution function defined as Ξ›(π‘Ž)=π‘’π‘Ž/(1+π‘’π‘Ž). Unlike the ordered probit framework, the ordered logit cannot lend itself easily to allow for correlation between bivariate discrete response outcomes. Henceforth, we focus on the ordered probit paradigm in both univariate and bivariate settings.

Assuming that πœ–1𝑖 and πœ‡π‘– are independently normally distributed, each with mean 0 and variance 1, the required components in (2.22) and consequently (2.23) are given by:𝑠Pr𝑖=0=Ξ¦βˆ’π³ξ…žπ‘–πœΈξ€Έ,ξ€·Pr̃𝑦1𝑖=0=Ξ¦βˆ’π‘₯ξ…ž1π‘–πœ·1ξ€Έ,ξ€·Pr̃𝑦1𝑖𝛼=𝑗=Ξ¦1𝑗+1βˆ’π±ξ…ž1π‘–πœ·1ξ€Έξ€·π›Όβˆ’Ξ¦1π‘—βˆ’π±ξ…ž1π‘–πœ·1ξ€Έ,for𝑗=1,…,𝐽1βˆ’1with𝛼10ξ€·=0,Pr̃𝑦1𝑖=𝐽1𝛼=1βˆ’Ξ¦1𝐽1βˆ’π±ξ…ž1π‘–πœ·1ξ€Έ.(2.24) The marginal effects for the univariate ZIOP are given by Harris and Zhao [15]. Bayesian analysis of the univariate ZIOP will be treated as a special case of the zero-inflated bivariate order probit model in the next section.

3. Bayesian Analysis

3.1. Prior Distributions

The Bayesian hierarchical model requires prior distributions for each parameter in the model. For this purpose, we can use noninformative conjugate priors. There are two reasons for adopting noninformative conjugate priors. First, we prefer to let the data dictate the inference about the parameters with little or no influence from prior distributions. Secondly, the noninformative priors facilitate resampling using Markov Chain Monte Carlo algorithm (MCMC) and have nice convergence properties. We assume noninformative (vague or diffuse) normal priors for regression coefficients 𝛽, with mean π›½βˆ— and variance Ω𝛽 which are chosen to make the distribution proper but diffuse with large variances. Similarly, π›ΎβˆΌπ‘(π›Ύβˆ—,Ω𝛾).

In choosing prior distributions for the threshold parameters, 𝛼’s, caution is needed because of the order restriction on them. One way to avoid the order restriction is to reparameterize them. Following Chib and Hamilton [18] treatment in the univariate ordered probit case, we reparameterize the ordered threshold parametersπœπ‘Ÿ2𝛼=logπ‘Ÿ2ξ€Έ;πœπ‘Ÿπ‘—ξ€·π›Ό=logπ‘Ÿπ‘—βˆ’π›Όπ‘Ÿπ‘—βˆ’1ξ€Έ,𝑗=3,…,π½π‘Ÿ;π‘Ÿ=1,2(3.1) with the inverse mapπ›Όπ‘Ÿπ‘—=π‘—ξ“π‘š=2ξ€·πœexpπ‘Ÿπ‘šξ€Έ,𝑗=2,…,π½π‘Ÿ;π‘Ÿ=1,2.(3.2) For π‘Ÿ=1,2, let π‰π‘Ÿ=(πœπ‘Ÿ2,πœπ‘Ÿ3,…,πœπ‘Ÿπ½)ξ…ž so that 𝝉=(𝝉1,𝝉2). We choose normal prior π‰βˆΌπ‘(π‰βˆ—,Ω𝜏) without order restrictions for πœπ‘Ÿβ€™s.

The only unknown parameter associate with the distribution of (𝝐,πœ‡) in (2.1) and (2.4) is 𝜌12, the correlation between πœ–1 and πœ–2. The values of 𝜌12 by definition are restricted to be in the βˆ’1 to 1 interval. Therefore, the choice for prior distribution for 𝜌12 can be uniform (βˆ’1,1) or a proper distribution based on reparameterization. Let 𝜈 denote the hyperbolic arc-tangent transformation of 𝜌12, that is, ξ€·πœŒπœˆ=π‘Žtanh12ξ€Έ,(3.3) and taking hyperbolic tangent transformation of 𝜈 gives us back 𝜌12=tanh(𝜈). Then parameter 𝜈 is asymptotically normal distributed with stabilized variance, 1/(π‘βˆ’3), where 𝑁 is the sample size. We may also assume that πœˆβˆΌπ‘(πœˆβˆ—,𝜎2𝜈).

3.2. Bayesian Analysis via MCMC

For carrying out a Bayesian inference, the joint posterior distribution of the parameters of the ZIBOP model in (2.6) conditional on the data is obtained by combining the likelihood function given in (2.9) and the above-specified prior distributions via Bayes’ theorem, as:π‘“ξ€·Ξ¨π‘ξ€Έβˆβˆ£π±,𝐳𝑁𝑖=1(𝑗,𝑙)=(0,0)ξ€ΊΞ¦ξ€·βˆ’π³ξ…žπ‘–πœΈξ€Έξ€·π³+Ξ¦ξ…žπ‘–πœΈξ€ΈΞ¦2ξ€·βˆ’π±ξ…ž1𝑖𝛽1,βˆ’π±ξ…ž2𝑖𝛽2,𝜌12𝑑𝑖𝑗𝑙×𝑁𝑖=1(𝑗,𝑙)β‰ (0,0)βŽ‘βŽ’βŽ’βŽ£Ξ¦ξ€·π³ξ…žπ‘–πœΈΞ¦ξ€Έξ€Ί2𝛼1𝑗+1βˆ’π±ξ…ž1𝑖𝛽1,𝛼2𝑙+1βˆ’π±ξ…ž2𝑖𝛽2;𝜌12ξ€Έβˆ’Ξ¦2𝛼1π‘—βˆ’π±ξ…ž1𝑖𝛽1,𝛼2π‘™βˆ’π±ξ…ž2𝑖𝛽2,𝜌12⎀βŽ₯βŽ₯βŽ¦ξ€Έξ€»π‘‘π‘–π‘—π‘™ξ€·πšΏΓ—π‘“π‘ξ€Έ,(3.4) where 𝑓(Ψ𝑏)βˆπ‘“(𝜷)𝑓(𝜸)𝑓(𝝉)𝑓(𝜈) and the parameter vector Ψ𝑏 now consists of 𝜷=(πœ·ξ…ž1,πœ·ξ…ž2)ξ…ž, 𝜸, 𝝉=(𝝉1,𝝉2),s and 𝜈=π‘Žtanh(𝜌12). Here 𝑓(𝜷)∝|Ω𝛽|βˆ’1/2exp{βˆ’1/2(πœ·βˆ’πœ·βˆ—)ξ…žΞ©π›½βˆ’1(πœ·βˆ’πœ·βˆ—)};𝑓(𝜸)∝|Ω𝛾|βˆ’1/2exp{βˆ’1/2(πœΈβˆ’πœΈβˆ—)ξ…žΞ©π›Ύβˆ’1(πœΈβˆ’πœΈβˆ—)};𝑓(𝝉)∝|Ω𝜏|βˆ’1/2exp{βˆ’1/2(π‰βˆ’π‰βˆ—)ξ…žΞ©πœβˆ’1(π‰βˆ’π‰βˆ—)};πœπ‘Ÿπ‘— are defined in (3.1), and π›Όπ‘Ÿπ‘— are given via the inverse map (3.2).

Full conditional posterior distributions are required to implement the MCMC algorithm [19–22], and they are given as follows: (1)fixed effects:(a)zero state: π‘“ξ€·πœΈβˆ£π±,𝐳,Ξ¨βˆ’π›Ύξ€Έβˆ||Ω𝛾||βˆ’1/2ξ‚†βˆ’1exp2ξ€·πœΈβˆ’πœΈβˆ—ξ€Έξ…žΞ©π›Ύβˆ’1ξ€·πœΈβˆ’πœΈβˆ—ξ€Έξ‚‡ξ€·Ξ¨Γ—π‘“π‘ξ€Έ;∣𝐱,𝐳(3.5)(b)nonzero state: π‘“ξ€·πœ·βˆ£π±,𝐳,Ξ¨βˆ’π›½ξ€Έβˆ||Ω𝛽||βˆ’1/2ξ‚†βˆ’1exp2ξ€·πœ·βˆ’πœ·βˆ—ξ€Έξ…žΞ©π›½βˆ’1ξ€·πœ·βˆ’πœ·βˆ—ξ€Έξ‚‡ξ€·Ξ¨Γ—π‘“π‘ξ€Έβˆ£π±,𝐳.(3.6)(2)thresholds: π‘“ξ€·π‰βˆ£π±,𝐳,Ξ¨βˆ’πœξ€Έβˆ||Ω𝜏||βˆ’1/2ξ‚†βˆ’1exp2ξ€·π‰βˆ’π‰βˆ—ξ€Έξ…žΞ©πœβˆ’1ξ€·π‰βˆ’π‰βˆ—ξ€Έξ‚‡Γ—π‘ξ‘π‘–=1(𝑗,𝑙)β‰ (0,0)βŽ‘βŽ’βŽ’βŽ£Ξ¦ξ€·π³ξ…žπ‘–πœΈΞ¦ξ€Έξ€Ί2𝛼1𝑗+1βˆ’π±ξ…ž1𝑖𝛽1,𝛼2𝑙+1βˆ’π±ξ…ž2𝑖𝛽2;𝜌12ξ€Έβˆ’Ξ¦2𝛼1π‘—βˆ’π±ξ…ž1𝑖𝛽1,𝛼2π‘™βˆ’π±ξ…ž2𝑖𝛽2,𝜌12⎀βŽ₯βŽ₯βŽ¦ξ€Έξ€»π‘‘π‘–π‘—π‘™.(3.7)(3)bivariate correlation: π‘“ξ€·πœˆβˆ£π±,𝐳,Ξ¨βˆ’πœˆξ€ΈβˆπœŽπœˆβˆ’1ξƒ―βˆ’ξ€·expπœˆβˆ’πœˆβˆ—ξ€Έ22𝜎2πœˆξƒ°ξ€·Ξ¨Γ—π‘“π‘ξ€Έβˆ£π±,𝐳.(3.8)

The MCMC algorithm simulates direct draws from the above full conditionals iteratively until convergence is achieved. A single long chain [23, 24] is used for the proposed model. Geyer [23] argues that using a single longer chain is better than using a number of smaller chains with different initial values. We follow this strategy in our empirical analysis.

The Bayesian analysis of the univariate ZIOP follows as a special case of that of the ZIBOP presented above. In particular, the joint posterior distribution of the parameters of the ZIOP model in (2.22) conditional on the data is obtained by combining the likelihood function given in (2.23) and the above-specified prior distributions (with modified notations) via Bayes' theorem, as follows: 𝑓(Ψ∣𝐱,𝐳,)βˆπ‘ξ‘π‘–=1𝑗=0ξ€ΊΞ¦ξ€·βˆ’π³ξ…žπ‘–πœΈξ€Έξ€·π³+Ξ¦ξ…žπ‘–πœΈξ€ΈΞ¦ξ€·βˆ’π±ξ…žπ‘–πœ·ξ€Έξ€»π‘‘π‘–π‘—Γ—π‘ξ‘π‘–=1𝑗>0ξ€ΊΞ¦ξ€·π³ξ…žπ‘–πœΈΞ¦ξ€·π›Όξ€Έξ€½π‘—+1βˆ’π±ξ…žπ‘–πœ·ξ€Έξ€·π›Όβˆ’Ξ¦π‘—βˆ’π±ξ…žπ‘–πœ·ξ€Έξ€Ύξ€»π‘‘π‘–π‘—Γ—π‘“(𝜷)𝑓(𝜸)𝑓(𝝉),(3.9) where, using notation of Section 2.3 for 𝜷 and the other parameter vectors, 𝑓(𝜷)∝|Ω𝛽|βˆ’1/2exp{βˆ’1/2(πœ·βˆ’πœ·βˆ—)ξ…žΞ©π›½βˆ’1(πœ·βˆ’πœ·βˆ—)};𝑓(𝜸)∝|Ω𝛾|βˆ’1/2exp{βˆ’1/2(πœΈβˆ’πœΈβˆ—)ξ…žΞ©π›Ύβˆ’1(πœΈβˆ’πœΈβˆ—)}; 𝑓(𝝉)∝|Ω𝜏|βˆ’1/2exp{βˆ’1/2(π‰βˆ’π‰βˆ—)ξ…žΞ©πœβˆ’1(π‰βˆ’π‰βˆ—)}, 𝜏2=log(𝛼2) and πœπ‘—=log(π›Όπ‘—βˆ’π›Όπ‘—βˆ’1),𝑗=3,…,𝐽. Apart from dropping the bivariate correlation, we basically replace the bivariate normal cumulative distribution Ξ¦2(β‹…,β‹…;𝜌12) by the univariate counterpart Ξ¦(β‹…). Details are available upon request from the authors.

Apart from Bayesian estimation of the regression parameters, the posterior distributions of other quantities of interest can be obtained. These include posteriors for marginal effects and probabilities for nonparticipation, zero-consumption, and joint outcomes of interest. These will be considered in the application section. Next, we summarize model selection procedure.

The commonly used criteria for model selection like BIC and AIC are not appropriate for the multilevel models (in the presence of random effects), which complicates the counting of the true number of free parameters. To overcome such a hurdle, Spiegelhalter et al. [25] proposed a Bayesian model comparison criterion, called Deviance Information Criterion (DIC). It is given as DIC=goodness-of-fit+penaltyforcomplexity,(3.10) where the β€œgoodness-of-fit” is measured by the deviance for πœƒ=(𝛽,𝛾,𝛼)𝐷(πœƒ)=βˆ’2logβ„’(dataβˆ£πœƒ)(3.11) and complexity is measured by the β€œeffective number of parameters”:𝑝𝐷=πΈπœƒ|𝑦[𝐷]𝐸(πœƒ)βˆ’π·πœƒ|𝑦[πœƒ]ξ€Έ=ξ‚€π·βˆ’π·πœƒξ‚;(3.12) that is, posterior mean deviance minus deviance evaluated at the posterior mean of the parameters. The DIC is then defined analogously to AIC asξ‚€DIC=π·πœƒξ‚=+2𝑝𝐷𝐷+𝑝𝐷.(3.13) The idea here is that models with smaller DIC should be preferred to models with larger DIC. Models are penalized both by the value of 𝐷, which favors a good fit, but also (similar to AIC and BIC) by the effective number of parameters 𝑝𝐷. The advantage of DIC over other criteria, for Bayesian model selection, is that the DIC is easily calculated from the MCMC samples. In contrast, AIC and BIC require calculating the likelihood at its maximum values, which are not easily available from the MCMC simulation.

4. Application

4.1. Data

We consider an application to tobacco consumption behavior of individuals based on the 2001 household Tobacco Prevalence survey data from Bangladesh. The Survey was conducted in two administrative districts of paramount interest for tobacco production and consumption in the country. Data on daily consumption of smoking and chewing tobacco along with other socioeconomic and demographic characteristics and parental tobacco consumption habits were collected from respondents of 10 years of age and above. The data set has been used previously by Gurmu and Yunus [26] in the context of binary response models. Here we focus on a sample consisting of 6000 individual respondents aged between 10 and 101 years.

The ordinal outcomes π‘¦π‘Ÿ=0,1,2,3 used in this paper correspond roughly to zero, low, moderate, and high levels of tobacco consumption in the form of smoking (𝑦1) or chewing tobacco (𝑦2), respectively. The first dependent variable 𝑦1 for an individual's daily cigarette smoking intensities assumes the following 4 choices: 𝑦1=0 if nonsmoker, 𝑦1=1 if smoking up to 7 cigarettes per day, 𝑦=2 if smoking between 8 and 12 cigarettes daily, and 𝑦1=3 if smoking more than 12 cigarettes daily; likewise, for the intensity of chewing tobacco, 𝑦2=0 if reported not chewing tobacco, 𝑦2=1 if uses up to 7 chewing tobacco, and 𝑦2=2 if consuming 7 or more chewing tobacco. The frequency distribution of cigarette smoking and tobacco chewing choices in Table 1 shows that nearly 66% of the respondents identify themselves as nonusers of tobacco. Our modeling strategy recognizes that these self-identified current nonusers of tobacco may include either individuals who never smoke or chew tobacco (genuine nonusers) or those who do, but not during the reporting period (potential users of tobacco). For example, potential tobacco users may include those who wrongly claim to be nonusers, previous tobacco users that are currently nonusers, and those most likely to use tobacco in the future due to changes in, say, prices and income. Table 1 also shows that 76% of the respondents are non-smokers and nearly 87% identify themselves as nonusers of tobacco for chewing. Given the extremely high proportion of observed zeros coupled with sparse cells on the right tail, we employ the zero-inflated bivariate ordered probit framework.


Smoke groupChew groupTotal ( 𝑁 )
012

039313023244557
1265126283
25263537598
34982935562

Total ( 𝑁 )52203784026000

Table 2 gives definition of the explanatory variables as well as their means and standard deviations. The respondents are more likely to be Muslim, married, in early thirties, live in rural area, and have about 7 years of formal schooling. Although the country is mostly agrarian, only around 11% of the respondents were related to agricultural occupation in either doing agricultural operations on their own farms or working as agricultural wage laborers. About 12% of the respondents belong to the service occupation. The benchmark occupational group consists of business and other occupations. More than one-half of the fathers and slightly less than two-thirds of the mothers of the respondents currently use or have used tobacco products in the past.


NameDefinitionMeanbSt. Dev.

AgeaAge in years30.35(14.9)
EducationaNumber of years of formal schooling6.83(4.7)
IncomeMonthly family income in 1000s of Taka7.57(10.3)
Male = 1 if male54.6
Married = 1 if married57.2
Muslim = 1 if religion is Islam78.4
Father use = 1 if father uses tobacco54.0
Mother use = 1 if mother uses tobacco65.1
Region = 1 if Rangpur resident,
= 0 if Chittagong resident
49.7
Urban = 1 if urban resident38.0
Agriservice = 1 if agriculture labor or service occupation23.2
Self-employed = 1 if self-employed or household chores30.7
Student = 1 if student26.8
Other = 1 if business or other occupations (control)19.3

aIn implementation, we also include age squared and education squared.
bThe means for binary variables are in percentage.

Among the variables given in Table 2, the two indicators of parental use of tobacco products are included in 𝐳 as part of the participation equation (2.4). The rest of the variables are included in π±π‘Ÿ and 𝐳 of (2.1) and (2.4). To allow for nonlinear effects, age and education enter all three equations using a quadratic form. Due to lack of data on prices, our analysis is limited to the study of other economic and demographic determinants of participation, smoking, and chewing tobacco.

4.2. Results

We estimate the standard bivariate ordered probit (BOP) and zero-inflated bivariate ordered probit regression models for smoking and chewing tobacco and report estimation results for parameters, marginal effects, and choice probabilities, along measures of model selection. An earlier version of this paper reports results from the standard ordered probit model as well as the uncorrelated and correlated versions of the univariate zero-inflated ordered probit model for smoking tobacco. Convergence of the generated samples is assessed using standard tools (such as trace plots and ACF plots) within WinBUGS software. After initial 10,000 burn-in iterations, every 10th MCMC sample thereafter was retained from the next 100,000 iterations, obtaining 10,000 samples for subsequent posterior inference of the unknown parameters. The slowest convergence is observed for some parameters in the inflation submodel. By contrast, the autocorrelations functions for most of the marginal effects die out quickly relative to those for the associated parameters.

Table 3 reports the goodness-of-fit statistics for the standard bivariate ordered probit model and its zero-inflated version, ZIBOP. The ZIBOP regression model clearly dominates BOP in terms of DIC and its components; compare the DIC of 11330 for the former and 11447 for the latter model. Table 4 gives posterior means, standard deviations, medians, and the 95 percent credible intervals (in terms of the 2.5 and 97.5 percentiles) of the parameters and choice probabilities from ZIBOP model. For comparison, the corresponding results from BOP are shown in Table 6 of the appendix. Both models predict significant negative correlation between the likelihood of smoking and chewing tobacco. The posterior estimates of the cut-off points are qualitatively similar across models. In what follows, we focus on discussion of results from the preferred ZIBOP model. The 95% credible interval for the correlation parameter 𝜌12 from the zero-inflated model is in the range βˆ’0.25 to βˆ’0.12, indicating that smoking and chewing tobacco are generally substitutes. Results of selected predicted choice probabilities (bottom of Table 4) show that the ZIBOP regression model provides very good fit to the data. The posterior mean for the probability of (zero, zero)-inflation is about 24% while the 95% credible interval is [0.15, 0.32], indicating that a substantial proportion of zeros may be attributed to nonparticipants. These results underscore the importance of modeling excess zeros in bivariate ordered probit models.


ModelDbarDhatpDDIC

Bivariate ordered probit (BOP)11417.111386.930.111447.2
Zero-inflated BOP11301.111270.329.811329.9

Dbar: Posterior mean of deviance, Dhat: Deviance evaluated at the posterior mean of the parameters, pD: Dbar-Dhat, the effective number of parameters, and DIC: Deviance information criterion.

Variable MeanSt. dev.2.50%Median97.50%

Main (Ξ² 1,Ξ± 1): smoking ( 𝑦 1 ):
Age/100.6720.1190.4440.685
Age square/100βˆ’0.0700.012βˆ’0.093βˆ’0.071
Educationβˆ’0.0710.014βˆ’0.097βˆ’0.071
Education square0.0010.001βˆ’0.0020.001
Income0.0000.002βˆ’0.0050.000
Male2.0920.0861.9252.091
Married0.2130.0700.0740.213
Muslimβˆ’0.0530.052βˆ’0.157βˆ’0.053
Regionβˆ’0.0070.048βˆ’0.102βˆ’0.007
Urbanβˆ’0.0960.051βˆ’0.198βˆ’0.097
Agriserviceβˆ’0.2340.056βˆ’0.345βˆ’0.233
Self-employedβˆ’0.2460.087βˆ’0.414βˆ’0.247
studentβˆ’0.4760.137βˆ’0.742βˆ’0.478
Ξ± 120.2840.0170.2520.283
Ξ± 130.9870.0300.9280.987

Main (Ξ² 2,Ξ± 2): chewing ( 𝑦 2 )
Age/100.6490.1330.3820.658
Age square/100βˆ’0.0460.013βˆ’0.071βˆ’0.046
Educationβˆ’0.0200.016βˆ’0.052βˆ’0.020
Education squareβˆ’0.0020.001βˆ’0.005βˆ’0.002
Income0.0010.003βˆ’0.0040.002
Maleβˆ’0.4790.081βˆ’0.641βˆ’0.479
Marriedβˆ’0.0250.075βˆ’0.171βˆ’0.025
Muslimβˆ’0.0720.056βˆ’0.181βˆ’0.072
Region0.4170.0510.3170.418
Urbanβˆ’0.0800.058βˆ’0.194βˆ’0.079
Agriservice0.0520.074βˆ’0.0960.052
Self-employed0.1270.092βˆ’0.0580.126
Studentβˆ’0.4500.221βˆ’0.887βˆ’0.448
Ξ± 220.4840.0230.4390.484

Inflation (Ξ³):
Age/10βˆ’0.0122.044βˆ’4.7550.253
Age square/1000.5090.552βˆ’0.1970.398
Educationβˆ’0.2180.115βˆ’0.476βˆ’0.204
Education square0.0280.0110.0100.026
Income0.0060.022βˆ’0.0270.003
Male0.2390.827βˆ’1.5820.417
Married2.3064.478βˆ’0.4160.500
Muslimβˆ’0.5280.356βˆ’1.331βˆ’0.494
Motherβˆ’0.1700.267βˆ’0.716βˆ’0.164
Fatherβˆ’0.1190.330βˆ’0.664βˆ’0.160
Region0.6300.2910.0610.625
Urban0.0400.357βˆ’0.7370.071
Agrservice5.3125.4161.0172.674
Self-employed3.7835.0250.1241.275
Sstudentβˆ’0.3440.411βˆ’1.154βˆ’0.339
ρ 12βˆ’0.1850.033βˆ’0.249βˆ’0.186

Select probabilities:
𝑃 ( 𝑦 1 = 0 )0.7600.0040.7520.760
𝑃 ( 𝑦 2 = 0 )0.8710.0040.8640.871
𝑃 ( 𝑦 1 = 0 , 𝑦 2 = 0 )0.6620.0050.6520.662
𝑃 (zero-inflation)0.2420.0480.1510.243

Results for the constant terms in the main and inflation parts have been suppressed for brevity.

To facilitate interpretation of results, we report in Tables 5 and 7 the same set of posterior estimates for the marginal effects from ZIBOP and BOP models, respectively. Since age and education enter the three equations non-linearly, we report the total marginal effects coming from the linear and quadratic parts. We examine closely the marginal effects on the unconditional marginal probabilities at all levels of smoking and chewing tobacco (𝑦1=0,1,2,3; 𝑦2=0,1,2). The marginal effects reported in Table 5 show that the results for covariates are generally plausible. Age has a negative impact on probabilities of moderate and heavy use of tobacco. For heavy smokers, education has a significant negative impact on the probability of smoking cigarettes. An additional year of schooling on average decreases probability of smoking by about 6.9% for heavy smokers. Among participants, being male or married has positive impact on probability of smoking, while the effects for being Muslim, urban resident, and student are largely negative. Male respondents are more likely to smoke cigarettes while women respondents are more likely to use chewing tobacco with heavy intensity, a result which is in line with custom of the country [26].


VariableProbability MeanSt. dev.2.50%Median97.50%

AgeNonparticipationβˆ’0.02590.0129βˆ’0.0556βˆ’0.0236βˆ’0.0078
Zero-consumption, 𝑦 1 0.04630.01020.02940.04530.0687
All zeros, 𝑦 1 = 0 0.02040.00590.00780.02130.0304
𝑦 1 = 1 0.00580.00350.00090.00530.0138
𝑦 1 = 2 βˆ’0.00140.0029βˆ’0.0057βˆ’0.00190.0055
𝑦 1 = 3 βˆ’0.06900.0235βˆ’0.1223βˆ’0.0658βˆ’0.0344
Zero-consumption, 𝑦 2 0.04030.01160.01950.03860.0675
All zeros, 𝑦 2 = 0 0.01450.00640.00180.01490.0264
𝑦 2 = 1 βˆ’0.00340.0021βˆ’0.0071βˆ’0.00350.0008
𝑦 2 = 2 βˆ’0.00190.0014βˆ’0.0043βˆ’0.00200.0011
EducationNonparticipationβˆ’0.28230.0768βˆ’0.4260βˆ’0.2837βˆ’0.1252
Zero-consumption, 𝑦 1 0.24470.07490.09170.24590.3851
All zeros, 𝑦 1 = 0 βˆ’0.03770.0241βˆ’0.0853βˆ’0.03740.0094
𝑦 1 = 1 0.04980.01410.02310.04940.0789
𝑦 1 = 2 0.02410.01020.00450.02390.0444
𝑦 1 = 3 βˆ’0.55570.1536βˆ’0.8415βˆ’0.5588βˆ’0.2417
Zero-consumption, 𝑦 2 0.31360.07720.15610.31590.4546
All zeros, 𝑦 2 = 0 0.03130.0161βˆ’0.00090.03150.0618
𝑦 2 = 1 βˆ’0.01340.0080βˆ’0.0288βˆ’0.01350.0027
𝑦 2 = 2 βˆ’0.02220.0119βˆ’0.0455βˆ’0.02210.0009
IncomeNonparticipationβˆ’0.00040.0015βˆ’0.0038βˆ’0.00020.0022
Zero-consumption, 𝑦 1 0.00030.0014βˆ’0.00220.00020.0035
All zeros, 𝑦 1 = 0 βˆ’0.00010.0004βˆ’0.0009βˆ’0.00010.0008
𝑦 1 = 1 0.00010.0003βˆ’0.00050.00000.0008
𝑦 1 = 2 0.00000.0002βˆ’0.00030.00000.0004
𝑦 1 = 3 βˆ’0.00070.0030βˆ’0.0075βˆ’0.00040.0044
Zero-consumption, 𝑦 2 0.00010.0016βˆ’0.00250.00000.0036
All zeros, 𝑦 2 = 0 βˆ’0.00020.0005βˆ’0.0011βˆ’0.00020.0007
𝑦 2 = 1 0.00010.0002βˆ’0.00030.00010.0004
𝑦 2 = 2 0.00010.0001βˆ’0.00020.00010.0003
MaleNonparticipationβˆ’0.02540.0599βˆ’0.1268βˆ’0.03050.1012
Zero-consumption, 𝑦 1 βˆ’0.35950.0611βˆ’0.4900βˆ’0.3540βˆ’0.2565
All zeros, 𝑦 1 = 0 βˆ’0.38490.0116βˆ’0.4078βˆ’0.3849βˆ’0.3618
𝑦 1 = 1 0.06300.00400.05550.06300.0711
𝑦 1 = 2 0.15600.00650.14350.15590.1689
𝑦 1 = 3 0.16590.00830.15030.16570.1829
Zero-consumption, 𝑦 2 0.10120.0623βˆ’0.03090.10640.2075
All zeros, 𝑦 2 = 0 0.07580.01260.05110.07590.1004
𝑦 2 = 1 0.05010.00330.04380.05000.0567
𝑦 2 = 2 βˆ’0.12580.0112βˆ’0.1478βˆ’0.1258βˆ’0.1040
MarriedNonparticipationβˆ’0.06800.0777βˆ’0.2274βˆ’0.04330.0346
Zero-consumption, 𝑦 1 0.02000.0705βˆ’0.0778βˆ’0.00010.1692
All zeros, 𝑦 1 = 0 βˆ’0.04800.0149βˆ’0.0796βˆ’0.0472βˆ’0.0207
𝑦 1 = 1 0.00560.00350.00060.00470.0132
𝑦 1 = 2 0.01610.00610.00600.01540.0296
𝑦 1 = 3 0.02630.00730.01160.02640.0406
Zero-consumption, 𝑦 2 0.07090.0791βˆ’0.03710.04740.2349
All zeros, 𝑦 2 = 0 0.00280.0119βˆ’0.02000.00260.0269
𝑦 2 = 1 0.06280.00320.05660.06270.0693
𝑦 2 = 2 βˆ’0.06560.0115βˆ’0.0888βˆ’0.0654βˆ’0.0434
MuslimNonparticipation0.03930.0243βˆ’0.00500.03840.0900
Zero-consumption, 𝑦 1 βˆ’0.02390.0247βˆ’0.0752βˆ’0.02310.0216
All zeros, 𝑦 1 = 0 0.01540.0090βˆ’0.00160.01530.0334
𝑦 1 = 1 βˆ’0.00230.0011βˆ’0.0044βˆ’0.0022βˆ’0.0002
𝑦 1 = 2 βˆ’0.00530.0027βˆ’0.0106βˆ’0.0053βˆ’0.0001
𝑦 1 = 3 βˆ’0.00780.0060βˆ’0.0200βˆ’0.00770.0036
Zero-consumption, 𝑦 2 βˆ’0.02600.0258βˆ’0.0797βˆ’0.02530.0222
All zeros, 𝑦 2 = 0 0.01330.0092βˆ’0.00460.01330.0315
𝑦 2 = 1 0.06130.00300.05540.06130.0674
𝑦 2 = 2 βˆ’0.07460.0091βˆ’0.0926βˆ’0.0746βˆ’0.0569
Father useNonparticipation0.01220.0187βˆ’0.02510.01240.0487
Zero-consumption, 𝑦 1 βˆ’0.01020.0158βˆ’0.0411βˆ’0.01040.0214
All zeros, 𝑦 1 = 0 0.00200.0030βˆ’0.00400.00190.0082
𝑦 1 = 1 βˆ’0.00050.0008βˆ’0.0022βˆ’0.00050.0011
𝑦 1 = 2 βˆ’0.00090.0014βˆ’0.0037βˆ’0.00090.0018
𝑦 1 = 3 βˆ’0.00050.0008βˆ’0.0023βˆ’0.00050.0011
Zero-consumption, 𝑦 2 βˆ’0.01160.0179βˆ’0.0464βˆ’0.01180.0240
All zeros, 𝑦 2 = 0 0.00060.0011βˆ’0.00120.00030.0033
𝑦 2 = 1 βˆ’0.00030.0006βˆ’0.0019βˆ’0.00020.0007
𝑦 2 = 2 βˆ’0.00020.0005βˆ’0.0014βˆ’0.00010.0005
Mother useNonparticipation0.01290.0257βˆ’0.03430.01230.0634
Zero-consumption, 𝑦 1 βˆ’0.01060.0215βˆ’0.0527βˆ’0.01030.0298
All zeros, 𝑦 1 = 0 0.00240.0043βˆ’0.00470.00200.0115
𝑦 1 = 1 βˆ’0.00060.0012βˆ’0.0031βˆ’0.00060.0014
𝑦 1 = 2 βˆ’0.00110.0019βˆ’0.0051βˆ’0.00090.0022
𝑦 1 = 3 βˆ’0.00070.0012βˆ’0.0033βˆ’0.00050.0012
Zero-consumption, 𝑦 2 βˆ’0.01190.0242βˆ’0.0587βˆ’0.01180.0338
All zeros, 𝑦 2 = 0 0.00100.0016βˆ’0.00070.00040.0053
𝑦 2 = 1 βˆ’0.00060.0009βˆ’0.0030βˆ’0.00020.0005
𝑦 2 = 2 βˆ’0.00040.0007βˆ’0.0023βˆ’0.00010.0002
RegionNonparticipationβˆ’0.04800.0240βˆ’0.0963βˆ’0.0470βˆ’0.0040
Zero-consumption, 𝑦 1 0.04120.0237βˆ’0.00390.04060.0889
All zeros, 𝑦 1 = 0 βˆ’0.00680.0079βˆ’0.0222βˆ’0.00680.0086
𝑦 1 = 1 0.00210.00110.00010.00210.0046
𝑦 1 = 2 0.00330.0025βˆ’0.00160.00330.0083
𝑦 1 = 3 0.00130.0052βˆ’0.00870.00140.0114
Zero-consumption, 𝑦 2 βˆ’0.02060.0252βˆ’0.0672βˆ’0.02170.0301
All zeros, 𝑦 2 = 0 βˆ’0.06860.0078βˆ’0.0840βˆ’0.0686βˆ’0.0533
𝑦 2 = 1 0.07560.00380.06820.07550.0832
𝑦 2 = 2 βˆ’0.00700.0070βˆ’0.0207βˆ’0.00700.0072
UrbanNonparticipationβˆ’0.00620.0261βˆ’0.0595βˆ’0.00540.0428
Zero-consumption, 𝑦 1 0.02170.0258βˆ’0.02710.02110.0733
All zeros, 𝑦 1 = 0 0.01550.0088βˆ’0.00180.01550.0324
𝑦 1 = 1 βˆ’0.00070.0012βˆ’0.0029βˆ’0.00080.0017
𝑦 1 = 2 βˆ’0.00420.0028βˆ’0.0096βˆ’0.00420.0014
𝑦 1 = 3 βˆ’0.01060.0056βˆ’0.0215βˆ’0.01060.0006
Zero-consumption, 𝑦 2 0.01810.0275βˆ’0.03370.01780.0739
All zeros, 𝑦 2 = 0 0.01190.0090βˆ’0.00620.01200.0295
𝑦 2 = 1 0.05970.00360.05280.05970.0668
𝑦 2 = 2 βˆ’0.07160.0075βˆ’0.0864βˆ’0.0717βˆ’0.0566
AgriserviceNonparticipationβˆ’0.19890.0521βˆ’0.3092βˆ’0.1960βˆ’0.1062
Zero-consumption, 𝑦 1 0.21020.05060.12020.20750.3161
All zeros, 𝑦 1 = 0 0.01130.0098βˆ’0.00840.01150.0297
𝑦 1 = 1 0.00580.00180.00260.00570.0097
𝑦 1 = 2 0.00230.0033βˆ’0.00390.00210.0092
𝑦 1 = 3 βˆ’0.01940.0060βˆ’0.0311βˆ’0.0194βˆ’0.0077
Zero-consumption, 𝑦 2 0.18380.05300.08710.18110.2940
All zeros, 𝑦 2 = 0 βˆ’0.01510.0126βˆ’0.0400βˆ’0.01500.0091
𝑦 2 = 1 0.06800.00490.05880.06780.0782
𝑦 2 = 2 βˆ’0.05290.0096βˆ’0.0716βˆ’0.0530βˆ’0.0338
Self-employedNonparticipationβˆ’0.12870.0693βˆ’0.2542βˆ’0.1191βˆ’0.0122
Zero-consumption, 𝑦 1 0.15900.06860.04310.15080.2845
All zeros, 𝑦 1 = 0 0.03030.0166βˆ’0.00340.03050.0627
𝑦 1 = 1 0.00050.0025βˆ’0.00420.00050.0058
𝑦 1 = 2 βˆ’0.00750.0060βˆ’0.0192βˆ’0.00750.0043
𝑦 1 = 3 βˆ’0.02330.0089βˆ’0.0398βˆ’0.0237βˆ’0.0046
Zero-consumption, 𝑦 2 0.10340.0704βˆ’0.01790.09410.2327
All zeros, y2=0βˆ’0.02540.0147βˆ’0.0546βˆ’0.02510.0035
𝑦 2 = 1 0.06840.00470.05940.06810.0781
𝑦 2 = 2 βˆ’0.04300.0118βˆ’0.0660βˆ’0.0431βˆ’0.0195
StudentNonparticipation0.03050.0357βˆ’0.03120.02700.1076
Zero-consumption, 𝑦 1 0.05480.0434βˆ’0.03530.05640.1354
All zeros, 𝑦 1 = 0 0.08520.02060.04370.08550.1247
𝑦 1 = 1 βˆ’0.00900.0027βˆ’0.0149βˆ’0.0089βˆ’0.0041
𝑦 1 = 2 βˆ’0.02950.0079βˆ’0.0455βˆ’0.0294βˆ’0.0143
𝑦 1 = 3 βˆ’0.04680.0106βˆ’0.0657βˆ’0.0475βˆ’0.0244
Zero-consumption, 𝑦 2 0.02840.0448βˆ’0.06860.03130.1073
All zeros, 𝑦 2 = 0 0.05880.02390.00650.06100.0995
𝑦 2 = 1 0.03900.01020.02070.03830.0604
𝑦 2 = 2 βˆ’0.09790.0142βˆ’0.1211βˆ’0.0994βˆ’0.0659


VariableMeanSt. Dev.2.50%Median97.50%

Smoking ( 𝑦 1 ) equation, (Ξ² 1,Ξ± 1)
Age/10 1.029 0.095 0.828 1.030
Age square/100 βˆ’0.104 0.010 βˆ’0.123 βˆ’0.105
Education βˆ’0.078 0.014 βˆ’0.105 βˆ’0.078
Education square 0.002 0.001 0.000 0.002
Income 0.000 0.002 βˆ’0.004 0.000
Male 2.066 0.091 1.888 2.067
Married 0.221 0.064 0.093 0.220
Muslim βˆ’0.083 0.049 βˆ’0.177 βˆ’0.083
Region 0.041 0.043 βˆ’0.044 0.041
Urban βˆ’0.091 0.048 βˆ’0.186 βˆ’0.091
Agriservice βˆ’0.121 0.050 βˆ’0.219 βˆ’0.122
Self-employed βˆ’0.149 0.087 βˆ’0.318 βˆ’0.150
Sstudent βˆ’0.720 0.093 βˆ’0.905 βˆ’0.719
Ξ± 12 0.270 0.015 0.241 0.270
Ξ± 13 0.956 0.028 0.901 0.956

Chewing ( 𝑦 2 ) equation, (Ξ² 2,Ξ± 2)
Age/10 0.797 0.091 0.609 0.801
Age square/100 βˆ’0.059 0.010 βˆ’0.079 βˆ’0.059
Education βˆ’0.023 0.016 βˆ’0.055 βˆ’0.023
Education square βˆ’0.002 0.001 βˆ’0.005 βˆ’0.002
Income 0.002 0.003 βˆ’0.004 0.002
Male βˆ’0.441 0.074 βˆ’0.586 βˆ’0.441
Married βˆ’0.010 0.073 βˆ’0.153 βˆ’0.011
Muslim βˆ’0.077 0.056 βˆ’0.187 βˆ’0.077
Region 0.430 0.049 0.334 0.430
Urban βˆ’0.082 0.056 βˆ’0.193 βˆ’0.081
Agriservice 0.078 0.073 βˆ’0.067 0.078
Self employed 0.177 0.087 0.010 0.176
Student βˆ’0.715 0.177 βˆ’1.070 βˆ’0.710
Ξ± 22 0.480 0.023 0.436 0.480
ρ 12 βˆ’0.178 0.034 βˆ’0.244 βˆ’0.179

Each equation includes father use and mother use variables as well as a constant term.

VariableProbabilityMeanSt. dev.2.50%Median97.50%

AgeAll zeros, 𝑦 1 = 0 0.0368 0.0038 0.0288 0.0369 0.0438
𝑦 1 = 1 βˆ’0.0004 0.0003 βˆ’0.0009 βˆ’0.0004 0.0002
𝑦 1 = 2 βˆ’0.0073 0.0010 βˆ’0.0093 βˆ’0.0073 βˆ’0.0054
𝑦 1 = 3 βˆ’0.0292 0.0029 βˆ’0.0345 βˆ’0.0292 βˆ’0.0230
All zeros, 𝑦 2 = 0 0.0213 0.0043 0.0125 0.0215 0.0298
𝑦 2 = 1 βˆ’0.0056 0.0013 βˆ’0.0082 βˆ’0.0057 βˆ’0.0031
𝑦 2 = 2 βˆ’0.0030 0.0008 βˆ’0.0047 βˆ’0.0030 βˆ’0.0014
EducationAll zeros, 𝑦 1 = 0 βˆ’0.0342 0.0236 βˆ’0.0803 βˆ’0.0340 0.0126
𝑦 1 = 1 0.0038 0.0025 βˆ’0.0011 0.0039 0.0086
𝑦 1 = 2 0.0130 0.0084 βˆ’0.0039 0.0129 0.0293
𝑦 1 = 3 0.0174 0.0128 βˆ’0.0076 0.0172 0.0428
All zeros, 𝑦 2 = 0 0.0322 0.0156 0.0002 0.0326 0.0616
𝑦 2 = 1 βˆ’0.0150 0.0077 βˆ’0.0296 βˆ’0.0151 0.0009
𝑦 2 = 2 βˆ’0.0201 0.0113 βˆ’0.0418 βˆ’0.0203 0.0024
IncomeAll zeros, 𝑦 1 = 0 βˆ’0.0001 0.0004 βˆ’0.0009 βˆ’0.0001 0.0007
𝑦 1 = 1 0.0000 0.0000 βˆ’0.0001 0.0000 0.0001
𝑦 1 = 2 0.0000 0.0001 βˆ’0.0002 0.0000 0.0002
𝑦 1 = 3 0.0000 0.0002 βˆ’0.0004 0.0000 0.0005
All zeros, 𝑦 2 = 0 βˆ’0.0003 0.0005 βˆ’0.0012 βˆ’0.0003 0.0007
𝑦 2 = 1 0.0001 0.0002 βˆ’0.0002 0.0001 0.0004
𝑦 2 = 2 0.0001 0.0002 βˆ’0.0002 0.0001 0.0004
MaleAll zeros, 𝑦 1 = 0 βˆ’0.3824 0.0121 βˆ’0.4064 βˆ’0.3826 βˆ’0.3586
𝑦 1 = 1 0.0641 0.0040 0.0567 0.0640 0.0722
𝑦 1 = 2 0.1540 0.0065 0.1416 0.1540 0.1667
𝑦 1 = 3 0.1643 0.0083 0.1487 0.1641 0.1807
All zeros, 𝑦 2 = 0 0.0721 0.0123 0.0481 0.0721 0.0962
𝑦 2 = 1 0.0500 0.0032 0.0438 0.0500 0.0565
𝑦 2 = 2 βˆ’0.1222 0.0108 βˆ’0.1430 βˆ’0.1220 βˆ’0.1014
MarriedAll zeros, 𝑦 1 = 0 βˆ’0.0416 0.0124 βˆ’0.0666 βˆ’0.0415 βˆ’0.0174
𝑦 1 = 1 0.0039 0.0013 0.0015 0.0038 0.0067
𝑦 1 = 2 0.0131 0.0042 0.0053 0.0130 0.0218
𝑦 1 = 3 0.0246 0.0070 0.0106 0.0247 0.0385
All zeros, 𝑦 2 = 0 0.0018 0.0118 βˆ’0.0207 0.0018 0.0254
𝑦 2 = 1 0.0622 0.0031 0.0563 0.0622 0.0685
𝑦 2 = 2 βˆ’0.0640 0.0114 βˆ’0.0873 βˆ’0.0640 βˆ’0.0420
MuslimAll zeros, 𝑦 1 = 0 0.0154 0.0092 βˆ’0.0029 0.0154 0.0331
𝑦 1 = 1 βˆ’0.0013 0.0008 βˆ’0.0028 βˆ’0.0013 0.0002
𝑦 1 = 2 βˆ’0.0044 0.0026 βˆ’0.0093 βˆ’0.0044 0.0008
𝑦 1 = 3 βˆ’0.0097 0.0058 βˆ’0.0211 βˆ’0.0097 0.0018
All zeros, 𝑦 2 = 0 0.0126 0.0093 βˆ’0.0051 0.0125 0.0313
𝑦 2 = 1 0.0613 0.0031 0.0555 0.0613 0.0675
𝑦 2 = 2 βˆ’0.0739 0.0091 βˆ’0.0922 βˆ’0.0739 βˆ’0.0562
Father useAll zeros, 𝑦 1 = 0 0.7604 0.0042 0.7521 0.7604 0.7684
𝑦 1 = 1 0.0477 0.0027 0.0426 0.0477 0.0531
𝑦 1 = 2 0.0982 0.0035 0.0915 0.0982 0.1051
𝑦 1 = 3 0.0937 0.0032 0.0874 0.0936 0.1000
All zeros, 𝑦 2 = 0 0.8713 0.0039 0.8635 0.8713 0.8789
𝑦 2 = 1 0.0623 0.0030 0.0566 0.0623 0.0684
𝑦 2 = 2 0.0664 0.0030 0.0607 0.0664 0.0724
Mother useAll zeros, 𝑦 1 = 0 0.7604 0.0042 0.7521 0.7604 0.7684
𝑦 1 = 1 0.0477 0.0027 0.0426 0.0477 0.0531
𝑦 1 = 2 0.0982 0.0035 0.0915 0.0982 0.1051
𝑦 1 = 3 0.0937 0.0032 0.0874 0.0936 0.1000
All zeros, 𝑦 2 = 0 0.8713 0.0039 0.8635 0.8713 0.8789
𝑦 2 = 1 0.0623 0.0030 0.0566 0.0623 0.0684
𝑦 2 = 2 0.0664 0.0030 0.0607 0.0664 0.0724
RegionAll zeros, 𝑦 1 = 0 βˆ’0.0075 0.0079 βˆ’0.0229 βˆ’0.0075 0.0080
𝑦 1 = 1 0.0006 0.0007 βˆ’0.0007 0.0006 0.0020
𝑦 1 = 2 0.0022 0.0023 βˆ’0.0023 0.0022 0.0067
𝑦 1 = 3 0.0047 0.0049 βˆ’0.0050 0.0047 0.0144
All zeros, 𝑦 2 = 0 βˆ’0.0691 0.0078 βˆ’0.0846 βˆ’0.0691 βˆ’0.0539
𝑦 2 = 1 0.0756 0.0038 0.0684 0.0755 0.0832
𝑦 2 = 2 βˆ’0.0065 0.0070 βˆ’0.0200 βˆ’0.0065 0.0072
UrbanAll zeros, 𝑦 1 = 0 0.0167 0.0087 βˆ’0.0003 0.0167 0.0339
𝑦 1 = 1 βˆ’0.0014 0.0008 βˆ’0.0030 βˆ’0.0014 0.0000
𝑦 1 = 2 βˆ’0.0049 0.0026 βˆ’0.0100 βˆ’0.0049 0.0001
𝑦 1 = 3 βˆ’0.0104 0.0054 βˆ’0.0210 βˆ’0.0104 0.0002
All zeros, 𝑦 2 = 0 0.0130 0.0088 βˆ’0.0041 0.0129 0.0303
𝑦 2 = 1 0.0592 0.0036 0.0524 0.0591 0.0664
𝑦 2 = 2 βˆ’0.0721 0.0074 βˆ’0.0866 βˆ’0.0721 βˆ’0.0576
AgriserviceAll zeros, 𝑦 1 = 0 0.0218 0.0088 0.0043 0.0219 0.0390
𝑦 1 = 1 βˆ’0.0018 0.0007 βˆ’0.0032 βˆ’0.0018 βˆ’0.0004
𝑦 1 = 2 βˆ’0.0062 0.0025 βˆ’0.0110 βˆ’0.0062 βˆ’0.0013
𝑦 1 = 3 βˆ’0.0138 0.0057 βˆ’0.0250 βˆ’0.0139 βˆ’0.0027
All zeros, 𝑦 2 = 0 βˆ’0.0127 0.0119 βˆ’0.0366 βˆ’0.0127 0.0106
𝑦 2 = 1 0.0656 0.0043 0.0572 0.0655 0.0742
𝑦 2 = 2 βˆ’0.0528 0.0094 βˆ’0.0711 βˆ’0.0529 βˆ’0.0338
Self employedAll zeros, 𝑦 1 = 0 0.0277 0.0162 βˆ’0.0039 0.0277 0.0592
𝑦 1 = 1 βˆ’0.0028 0.0018 βˆ’0.0065 βˆ’0.0027 0.0003
𝑦 1 = 2 βˆ’0.0087 0.0053 βˆ’0.0194 βˆ’0.0086 0.0012
𝑦 1 = 3 βˆ’0.0163 0.0093 βˆ’0.0335 βˆ’0.0165 0.0025
All zeros, 𝑦 2 = 0 βˆ’0.0290 0.0144 βˆ’0.0578 βˆ’0.0286 βˆ’0.0017
𝑦 2 = 1 0.0686 0.0046 0.0600 0.0685 0.0779
𝑦 2 = 2 βˆ’0.0396 0.0116 βˆ’0.0617 βˆ’0.0398 βˆ’0.0162
StudentAll zeros, 𝑦 1 = 0 0.1287 0.0155 0.0980 0.1286 0.1588
𝑦 1 = 1 βˆ’0.0173 0.0030 βˆ’0.0235 βˆ’0.0171 βˆ’0.0118
𝑦 1 = 2 βˆ’0.0475 0.0069 βˆ’0.0614 βˆ’0.0475 βˆ’0.0343
𝑦 1 = 3 βˆ’0.0639 0.0063 βˆ’0.0758 βˆ’0.0640 βˆ’0.0510
All zeros, 𝑦 2 = 0 0.0855 0.0151 0.0531 0.0866 0.1117
𝑦 2 = 1 0.0278 0.0071 0.0154 0.0274 0.0428
𝑦 2 = 2 βˆ’0.1133 0.0088 βˆ’0.1288 βˆ’0.1140 βˆ’0.0944

Using (2.13), we decompose the marginal effect on probability of observing zero-consumption into two components: the effect on nonparticipation (zero inflation) and zero-consumption. For each explanatory variable, this decomposition is shown in Table 5 in the first three rows for smoking and in rows 1, 7, and 8 for chewing tobacco. For most variables, the effects on probabilities of nonparticipation and zero-consumption are on average opposite in sign, but this difference seems to diminish at the upper tail of the distribution. For example, looking at the posterior mean for age under smoking, getting older by one more year decreases probability of nonparticipation by about 2.6% but increases probability of zero-consumption by 4.6%, implying a net increase of 2.0% in predicted probability of observing zero. The effect of age in the case of chewing tobacco is qualitatively similar, negative effect on genuine nonusers and positive effect on potential tobacco users, with the latter dominating in the overall effect.

Income has opposite effects on probability of nonparticipation and zero-consumption, predicting on average that tobacco is an inferior good for nonparticipants and a normal good for participants. However, the 95% credible interval contains zero, suggesting that the effect of income is weak. Generally, the opposing effects on probabilities of nonparticipation and zeroconsumption would have repercussions on both the magnitude and the statistical significance of the full effect of observing zero-consumption. Similar considerations apply to positive levels of consumption since the marginal effect on probability of observing consumption level 𝑗(𝑗=1,2,…) can be decomposed into the marginal effects on (i) participation 𝑃(𝑠𝑖=1) and (ii) levels of consumption conditional on participation, 𝑃(π‘¦π‘Ÿπ‘–=π‘—βˆ£π‘ π‘–=1). These results show that policy recommendations that ignore excess zeros may lead to misleading conclusions.

5. Conclusion

In this paper we analyze the zero-inflated bivariate ordered probit model in a Bayesian framework. The underlying model arises as a mixture of a point mass distribution at (0,0) for nonparticipants and the bivariate ordered probit distribution for participants. The Bayesian analysis is carried out using MCMC techniques to approximate the posterior distribution of the parameters. Using household tobacco survey data with substantial proportion of zeros, we analyze the socioeconomic determinants of individual problem of smoking and chewing tobacco. In our illustration, we find evidence that accounting for excess zeros provides very good fit to the data. The use of a model that ignores zero-inflation masks differential effects of covariates on nonusers and users at various levels of consumption, including zeros. The Bayesian approach to modeling excess zeros provides computational flexibility of generalizing to multivariate ordered response models as well as ordinal panel data models.

The proposed zero-inflated bivariate model is particularly useful when most of the bivariate ordered outcomes are zero (𝑦1=0,𝑦2=0). In addition to allowing for inflation in the double-zero state, our approach can be extended to allow for zero inflation in each component. If needed, other states in an ordered regression model may be inflated as well. These extensions need to be justified empirically on a case-by-case basis and are beyond the scope of this paper.

Appendices

A.

For more details see Tables 6 and 7.

B.

WinBUGS Code for Fitting the Proposed Models (see Algorithm 1).

617678.fig.001

Acknowledgments

The authors thank Alfonso Flores-Lagunes, the editor, two anonymous referees and seminar participants at the Conference on Bayesian Inference in Econometrics and Statistics, the Joint Statistical Meetings, the Southern Economics Association Conference, and Syracuse University for useful comments. Mohammad Yunus graciously provided the data used in this paper.

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Copyright © 2012 Shiferaw Gurmu and Getachew A. Dagne. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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