The general aim of this paper is to deal with problems of estimation , prediction, and model building for panel data model .Bayesian approach based on Markov chain Monte Carlo (MCMC) employed to make inferences on panel data model coefficients under some conditions on the prior distribution . We investigate the posterior density and identify the analytic form of the Bayes factor for checking the model.   Keywords: Panel Data Model , Likelihood function , Bayesian approach , Markov chain Monte Carlo (MCMC), Prior distribution, Posterior distribution , Bayes factor

Abdulsamad, Saja Yaseen

Mohaisen, Ameera Jaber

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  12 Bayesian Panel Data Model Based on Markov Chain Monte Carlo Ameera Jaber Mohaisen   and    Saja Yaseen  Abdulsamad Mathematics Department College of Education for Pure Science AL-Basrah University-Iraq E-mail:ameera.jaber@yahoo.com E-mail:saja.shareeda@yahoo.com Abstract The general aim of this paper is to deal with problems of estimation , prediction, and model building for panel data model .Bayesian approach based on Markov chain Monte Carlo (MCMC) employed to make inferences on panel data model coefficients under some conditions on the prior distribution . We investigate the posterior density and identify the analytic form of the Bayes factor for checking the model.   Keywords: Panel Data Model , Likelihood function , Bayesian approach , Markov chain Monte Carlo (MCMC), Prior distribution, Posterior distribution , Bayes factor.  1. Introduction  One of the aims of science is to describe and predict events in the world in which we live. One way this is accomplished is by finding a formula or equation that relates quantities  in the world, in linear model we attempt to model the relationship between  variables,[5],[10].  Linear models play a central part in modern statistical methods. On the one hand ,these models are able to approximate a large amount of metric data structures in their entire range of definition or at least piecewise. The theory of generalized models enables us, through appropriate link function , to apprehend error structures that deviate from the normal distribution , hence ensuring that a linear model is maintained in principle,[9]. Linear statistical methods are widely used as part of this learning process. In the biological, physical, and social sciences , as well as in business and engineering ,linear models are used in both the planning stages of research and analysis of the resulting data,[10],[12] .  In spite of the availability of highly innovative tools in statistics , the main tool of  the applied statistician remains the linear model. The linear model involves the simplest and seemingly most restrictive statistical properties: independence, normality, constancy of variance , and linearity. However, the model and the statistical methods associated with it are surprisingly versatile and robust. More importantly, mastery of the  linear model is a prerequisite to work with advanced statistical tools because most advanced tools are   generalizations of the linear model. The linear model is thus central to the training of any statistician, applied or theoretical,[10].   Panel (or longitudinal) data are cross – sectional and time – series . There are multiple entities, each of which has repeated measurements at different time periods . Panel data have a cross – sectional (entity or subject) variable and  a time series variable . In Stata , this arrangement is called the long form (as opposed to the wide form) . While the long form has both group (individual level) and time variables ,the wide form includes either group or time variables. Panel data usually give the researcher a large number of data points, increasing the degrees of Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  13 freedom and reducing the collinearity among explanatory variables. Panel data models have becomes increasingly popular among applied researchers due to their heightened capacity for capacity for capturing the complexity of human behaviour as compared to cross-sectional or time –series data models. As a consequence , more and richer panel data sets also have become increasingly available,[1],[6],[13]. Bayesian estimation and inference has a number of advantages in statistical modelling and data analysis. The Bayesian statistical paradigm is conceptually simple and general because inference involve only probability calculation as opposed to maximization of a function like the log likelihood .On the other hand , the probability calculations usually entail complicated or even intractable integrals. In Bayesian statistics, uncertainty about the value of a parameter is expressed using the tools of the probability theory . Density functions of the parameters reflect the current credibility of possible update the uncertainty distributions for parameters, and then draw sensible conclusions using theses updated distribution,[3],[7],[8],[11],[14]. Classical statistics treats parameters as fixed unknown quantities , Bayesian statistics is based on different philosophy; parameters are treated as random variables . The probability distribution changes as new data are acquired. Bayesian statistical differs from classical statistics in two important respects: the use of prior distribution to characterize knowledge of the parameter values prior to data collection and the use of posterior distribution – that is, the conditional distribution of the parameters given the data- as the basis of inference. Some statisticians are uneasy about the use of priors, but when done with care, the use of priors is quite  sensible. In some situations, we might have strong prior beliefs that will influence our analysis. In the other situations, where we feel  that know little or nothing about the parameters, we can chose  " noninformative " prior , one could also use an " improper prior" (i.e., one with infinite mass) such as the uniform distribution an real line. By stating a prior, we make clear how much or how little, we believe we know a prior about the parameter. In constant to the controversy  about the use of priors, there is little argument that the posterior provides a powerful inferential machinery,[3],[7],[8],[11],[14].  The calculation required by the Bayesian inference – in particular , the need to integrate the parameters out of joint density – have been a serious  obstacle on the application of Bayesian methods. However, new Monte Carlo techniques such as Markov chain Monte Carlo and importance sampling have provided powerful methods for attacking this problem,[2],[4],[11].  The general aim of this paper is to deal with problems of estimation , prediction, and model building for panel data model .Bayesian approach based on Markov chain Monte Carlo (MCMC) employed to make inferences on panel data model coefficients under some conditions on the prior distribution . We investigate  the posterior density and identify the analytic form of the Bayes factor for checking the model    2.Panel Data Model and The Prior Distribution      Consider the model:    where,  the value of response variable for  unit at time t , the explanatory variables ,  are fixed  parameters and  is an error term with   .  Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  14 Now , if the parameter  is specified as :   where,  then, the model (1) is   +                                                             The model (3) is rewrite as follows :  where,  ,  ,  ) ,  =  +  , thus by using matrix notation the model (4) is  where,   ,  has length   ,  has length  ,  is  a  design matrix of fixed effects ,   has length , and  has length  .From  model ( 5 ) , we have  , where                                                       ), replace  by (  and  by  , where  and   = , then         (   (  ,  by collecting terms with the same matrices,  we get    (   ( , where , and    ,  product of its characteristic roots, [1]  . The likelihood function is the joint density of the  that is  ) =    exp {     } Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  15              = (2π  (   . Then, the likelihood estimators of parameters θ ,  are   ,    and   . To specify a complete Bayesian model , we need a prior distribution on  .we will use the uniform distribution   of the vector parameters  , as well as we will assume that  the prior distribution on  and  are  invers gamma with of  parameters  ,  , and  respectively .i.e.  and   where ,  ,  and  are hyperparameters that determine the priors and must be chosen by the statistician.       3 . The Posterior Distribution From the model (5) we have . Then ,the  likelihood function  is   .   In the exponent, we add and subtract  to obtain                                                               Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  16                                            ,  since, , then   The joint  posterior  density of the coefficients θ and the  variances    and   given  by the expression                                                                                       From this expression , we can deduce the following conditional and   marginal  posterior distributions   and                                                                                                 Therefore , it follows that            Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  17 4. Bayes factor We would like to choose between a fully Bayesian panel data model with (K+1) of parameters against a Bayesian panel data model with  of parameters ,where  , by using Bayes factor for two hypotheses  where ,  is  vectors of  parameters ,  is an  design matrix and . We compute the Bayes factor,  of  relative to  for testing problem(13) as follows    where  is the marginal density of  under model  We have:                                                                                                                     Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  18                                                                                                                                                                                                                                                                                                                                                                                                                                                                           =   Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  19                  d                                                                                                                                                            Γ . d                                                                              Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  20                              d .                                                                                                                                                                         Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  21                                                                     =  Γ                           Γ                                                                 d        Γ                                   Γ  ×                                         Γ  ×  Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  22                                               .       5. Conclusion       The conclusions which are  obtained throughout this paper are given as follows : 1. The likelihood function of the panel data model is                       2. The posterior of  ,  and  are respectively :            ,           and           3. The marginal density of  under model  are Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  23              and            Γ     4. The Bayes factor for testing  against    is given by the following form:   .  References [1] Baltagi , badi, ” Econometric Analysis of panel data “ , John Wily & Sons Inc. third           edition,(2005). [2]  Catecki, Andrzej and Burzykowski," Linear Mixed –effect Models using R",A step by        step approach, springer,(2013). [3] Congdon, Peter," Bayesian Statistical Modelling",  second edition, Wiley Jon & Sons, Ltd,(2006).   [4] Davision , A. C., "Statistical Model ",Cambridge university Press,(2008). [5] Graybill, F.A. “ Theory and Application of the  Linear Model “North Scituate, MA; Duxbury       Press,(1976).   [6] Husio , Cheng, “ Analysis of panel data “ ,second edition, Cambridge university Press,       (2003).  [7] Lavielle, Marc, "Mixed Effects Models for Population Approach", CRC Press,(2015).   [8] Marin, Jean-Michel ,Robert, Christian P., "Bayesian Essentials With R", second            edition  ,Springer,(2014). [9] Rao, C. Rad hakrishna and Toutenburg,Helge,"Linear Models Least Squares and Alternatives", Second   Mathematical Theory and Modeling                                                                                           www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.5, 2017  24     edition, springer,(1999). [10] Rencher , Alvin C. and Schaalje, G. Bruce, “ Linear Models in Statistics “ ,second edition,       Wiley-Interscience, A John Wiley & Sons ,Inc., Publication,(2008). [11] Ruppert , David ,Wand, M. P., and Carroll, R. J.," Semiparametric Regression",  Cambridge University       Press,(2003). [12] Stapleton, James H.," Linear Statistical Models", John Wily & Sons, Inc.,(1995). [13] Sun, Jianguo & Zhao, Xingqiu , “ Statistical Analysis of Panel Count Data “, Springer , (2013). [14] Waskefield ,Jon," Bayesian and Frequentist Regression Methods", Springer,(2013).  

Bayesian Panel Data Model Based on Markov Chain Monte Carlo

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Bayesian Panel Data Model Based on Markov Chain Monte Carlo

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