On Seemingly Unrelated Semiparametric Models
Mahdi Roozbeh1, Mohammad Arashi2
1Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Khorasan, Islamic Republic of Iran; 2Faculty of Mathematics, Shahrood University of Technology, Shahrood, Semnan, Islamic Republic of Iran

This article considers estimation in the seemingly unrelated semiparametric (SUS) models, when the explanatory variables are affected by multicollinearity. It is also suspected that some additional linear constraints may hold on the whole parameter space. In sequel we propose difference-based and difference-based ridge type estimators combining the restricted least squares method in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their biases and covariances are derived. Also, necessary and sufficient conditions for the superiority of the ridge type estimator over the non-ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and non-parametric parts. In this regard, local linear regression method for estimating the non-parametric function are used.

A seemingly unrelated regression (SUR) system proposed by Zellner (1962) comprises several individual relationships that are linked by the fact that their disturbances are correlated. Such models have found many applications. For example, demand functions can be estimated for different households (or household types) for a given commodity. The correlation among the equation disturbances could come from several sources such as correlated shocks to household income. There are two main motivations for use of SUR. The first one is to gain efficiency in estimation by combining information on different equations. The second motivation is to impose and/or test restrictions that involve parameters in different equations.

The difference-based estimation procedure is optimal in the sense that the estimator of the linear component is asymptotically efficient and the estimator of the nonparametric component is asymptotically minimax rate optimal for the semiparametric model (Wang et al, 2007).

In most of the empirical works people are often concerned about problems with the specification of the model or problems with the data. This problem arises in situations when the explanatory variables are highly inter-correlated. Multicollinearity is defined as the existence of nearly linear dependency among column vectors of the design matrix in the linear parts of the model. The existence of multicollinearity may lead to wide confidence intervals for individual parameters or linear combination of the parameters and may produce estimates with wrong signs, etc.

In this paper, we apply differencing method to remove the nonparametric parts in SUS model and then estimate the linear parts to accelerate estimating the parametric parts. At the second step, nonparametric technique is applied to estimate the nonparametric parts. In this regard we deal with SUS model applying differencing methodology, under multicollinearity setting. Thus we use ridge regression concept that was proposed in the 1970's to combat the multicollinearity in regression problems.

Keywords: Differencing estimator; Multicollinearity; Ridge estimator; Seemingly unrelated Semiparametric model

Biography: Research interests: Partial Linear Models; Seemingly Unrelated Regression; Fisher Information in Order Statistics

Bachelor of Science in Statistics: Ferdowsi University of Mashhad. MASHHAD–IRAN, 2000-2004, Total GPA:18.32/20

B.Sc. Thesis: An introduction on multivariate analysis with SAS

Master of Science in Mathematical Statistics: Ferdowsi University of Mashhad. MASHHAD–IRAN, 2004-2007, Total GPA:18.27/20

M.Sc. Thesis: On Calculating The Fisher Information In Order Statistics And Considering Some Statistical Classic Distributions.

Ph.D. Student in Statistics: Ferdowsi University of Mashhad. MASHHAD–IRAN, 2007–present, Total GPA:18.33/20

Ph.D Thesis: Estimation of Semiparametric Models.