# MODELING SEEMINGLY UNRELATED REGRESSION (SUR) EQUATIONS WITH HETEROSCEDASTIC ERROR AND NON-NORMAL RESPONSES

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## Date

2018-08

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UNIVERSITY OF ILORIN

## Abstract

The Feasible Generalized Least Square (FGLS) estimator for Seemingly Unrelated Regression (SUR) models is generally efficient when the errors between equations are highly correlated and each equation satisfies all the assumptions of classical linear regression equation.However, the multivariate normal assumption on response variables in SUR model are most often not satisfied in many applications. The aim of the study was to investigate some estimators of SUR equations with heteroscedastic error and non-normal responses. The objectives of the study were to:(i) examine the performances of some estimators under different forms of heteroscedasticity with non-normal responses;(ii) determine the finite and asymptotic sample properties of SUR estimators under different forms of heteroscedasticity and non-normal response; and (iii) identify the efficient estimator of SUR model under different violations.
The Monte Carlo simulation experiment was carried out to generate the data for SUR model. Three distributions namely; Normal, Gamma and Log-Normal were considered for the response, four forms of heteroscedasticity were formulated as: Linear (LN), Quadratic (Quad), Square Root (SQRT) and Exponential (EXP)) and levels of contemporaneous correlation (Cc) were specified as: 0.0, 0.2, 0.5, 0.7 and 0.95. Four estimators; Ordinary Least Squares(OLS),Feasible Generalized Least Square (FGLS), M-Huber (Huber) and MM- Bi-Square (BISQ) were used to estimate the parameters of the models using the simulated data. Root Mean Square error (RMSE), variance and Absolute Bias (AB) criteria were used to assess the relative performances of the estimators under different conditions.
The findings of the study were that:
(i) the performances of estimators depend on the degree of contemporaneous correlation(Cc), form of heteroscedasticity, distribution of the response and sample sizes;
(ii) when there is absence or low contemporaneous correlation (ρ ≤ 0.2) with small sample sizes (n ≤ 30), OLS is the best estimator for normalresponse model except under Quadratic heteroscedasticity where Huber estimator was preferred;
(iii) with moderate or high Cc (ρ≥0.5) irrespective of the form of heteroscedasticity the FGLS outperformed all other estimators for all sample sizes considered under normal responses model;
(iv) when there is low or absence of Cc(ρ ≤ 0.2) irrespective of the heteroscedastic structure and sample sizes, the model parameters are best estimated by BISQ under Gamma or Log-Normal responses model;
(v) FGLS is the most efficient estimator for heteroscedastic Gamma response SUR model when Cc is moderate or high(ρ≥0.5) except in small sample under EXP heteroscedasticity where BISQ is preferred; and
(vi) BISQ estimator is the most efficient in medium (n=50 or 100) and large sample sizes (n=250 or 500) for heteroscedastic lognormal SUR model with moderate or high Cc (ρ≥0.5) while FGLS is preferred in small samples under different forms of heteroscedasticity except Quadratic where BISQ is preferred.
The study concluded that the degrees of Cc and distribution of response variables are key determinants in modeling SUR equations with heteroscedastic non-normal responses. The study therefore recommended that BISQ estimator should be used in estimating parameter of SUR model with non-normal responses.

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## Keywords

MODELING, SEEMINGLY UNRELATED REGRESSION, SUR EQUATIONS, HETEROSCEDASTIC ERROR, NON-NORMAL RESPONSES