Some Remarks on Restricted Panel Data Model

Regression analysis and time series analysis are two important applied statistical methods used to analyze data. Regression analysis is a special type of multivariate analysis, where several measurements are taken from each subject. We identify one measurement as a response, or dependent variable; the interest is in making statements about this measurement, controlling for the other variables. With regression analysis, it is customary to analyze data from a crosssection of subjects. In contrast, with time series analysis, we identify one or more subjects and observe them over time. This allows us to study relationships over time, the so-called dynamic aspect of a problem. To employ time series methods, we generally restrict ourselves to a limited number of subjects that have many observations over time. Defining longitudinal and panel data Longitudinal data analysis represents a marriage of regression and time series analysis. As with many regression data sets, longitudinal data are composed of a cross-section of subjects. Unlike regression data, with longitudinal data we observe subjects over time. Unlike time series data, with longitudinal data we observe many subjects. Observing a broad cross-section of subjects over time allows us to study dynamic, as well as cross-sectional, aspects of a problem. The descriptor panel data comes from surveys of individuals. In this context, a “panel” is a group of individuals surveyed repeatedly over time. Historically, panel data methodology within economics had been largely developed through labor economics applications. Now, economic applications of panel data methods are not confined to survey or labor economics problems and the interpretation of the descriptor “panel analysis” is much broader,[6],[9],[11],[12].


INTRODUCTION
Regression analysis and time series analysis are two important applied statistical methods used to analyze data. Regression analysis is a special type of multivariate analysis, where several measurements are taken from each subject. We identify one measurement as a response, or dependent variable; the interest is in making statements about this measurement, controlling for the other variables. With regression analysis, it is customary to analyze data from a crosssection of subjects. In contrast, with time series analysis, we identify one or more subjects and observe them over time. This allows us to study relationships over time, the so-called dynamic aspect of a problem. To employ time series methods, we generally restrict ourselves to a limited number of subjects that have many observations over time. Defining longitudinal and panel data Longitudinal data analysis represents a marriage of regression and time series analysis. As with many regression data sets, longitudinal data are composed of a cross-section of subjects. Unlike regression data, with longitudinal data we observe subjects over time. Unlike time series data, with longitudinal data we observe many subjects. Observing a broad cross-section of subjects over time allows us to study dynamic, as well as cross-sectional, aspects of a problem. The descriptor panel data comes from surveys of individuals. In this context, a "panel" is a group of individuals surveyed repeatedly over time. Historically, panel data methodology within economics had been largely developed through labor economics applications. Now, economic applications of panel data methods are not confined to survey or labor economics problems and the interpretation of the descriptor "panel analysis" is much broader, [6], [9], [11], [12].
The analysis of panel data allows the model builder to learn about economic processes while accounting for both heterogeneity across individuals, firms, countries, and so on and for dynamic effects that are not visible in cross sections. Modeling in this context often calls for complex stochastic specifications, [13]. The panel data model has been investigated by the many researcher as Elhorst in (2001) presented paper surveys panel data models extended to spatial error autocorrelation or spatially lagged dependent variable, [4]. Hurlin in (2004) proposed a simple test of Granger (1969) non causality hypothesis in heterogeneous panel data models with fixed coefficients, [8] . Bun , a. e. in (2005) studied extends earlier results on biascorrected estimators for the fixed effects dynamic panel data model, [3]. Gorgen a. e. in (2008) discussed efficient estimation of nonlinear dynamic panel data models with application to smooth transition models ,they explores estimation of a class of nonlinear dynamic panel data models with additive unobserved individual specific effects, [7]. Feng a. e. in (2015) proposed a panel data Semiparametric varying coefficient model in which covariates (variables affecting the coefficients) are purely categorical, [5]. Ashley and Sun in (2016) proposed subset continuous updating GMM estimators for dynamic panel data models, [1], [2] .
Constrained parameter problems arise in a wide variety of applications, including bioassay, actuarial graduation, ordinal data , response surfaces, reliability development testing and variance component model. The normal linear regression model subject to linear inequality constraints for the coefficients arises commonly in applied econometrics as well as other scientific applications. Typically the motivating economic model restricts the sings of certain coefficients or of known linear combinations of coefficients, [10], [11].
The aim of this paper is to deal with restricted panel data model , we investigate some remarks on panel data model with linear constraints on the coefficients of the model. Furthermore, it investigates the inferences on the model. We explore the estimation of restricted panel data model , likelihood ratio test between restricted and unrestricted models , and prove some properties about the parameters estimation. where, , , ) , = + , thus by using matrix notation the model (4)

3-Constrained maximum likelihood estimator
In this section we consider a set of m linear constraints on the coefficients of the random panel data model (5).Furthermore, it investigates the inferences . The restricted maximum likelihood method is employed to making inferences on the random panel data model.
Consider the model (5) above, we assume that