On Taylor Series and Chebyshev Polynomial Approximation in Parameter Estimation for the Spatial Autoregressive Model

Abstract

Parameter estimation for the spatial autoregressive (SAR) model by using maximum likelihood (MLE) method involve log determinant of spatial weight matrix where its dimension is large.  Therefore, to solve log determinant of this matrix often used approximation.  The paper studied performances of Taylor series and Chebyshev polynomial methods in parameter estimation of SAR model. In this paper, we also studied performance of two types of spatial matrices, W-AMOEBA and W-contiguity, to choose the best of spatial weighted matrix in SAR model.  Evaluation of approximation methods to solve log determinant and to choose the best performance model, we used data simulation and root mean square error (RMSE) criteria.  The data simulation is generated by Monte Carlo simulation methods.  Furthermore, the best model (the best performances of approximation method and spatial weight matrix) is implemented to model human development index (HDI) and its factors in Central Java Province.  The HDI factors are population, gross enrolment rate, district minimum wage, the number of poor people and poverty line.  The results showed that RMSE of models used to Chebyshev polynomial is smaller than Taylor series.  Therefore, Chebyshev polynomial approximation is more accurate than Taylor series approximation.  Furthermore, the Chebyshev polynomial is used to analysis the human development index (HDI) and its factors by using SAR model.  The result showed that the gross enrollment rate, district minimum wage, and poverty line then the HDI have positive impact.  It means that increasing of theirs factors can improve HDI.