Bayesian Simultaneous Autoregressive Models of Proverty Analysis in East Java Province

Abstract

Simultaneous Autoregressive (SAR) is a spatial model derived from the linear regression equation that the error is designed as an autoregressive model which is the response variable at one location simultaneously observed to the response variable at other locations. In a previous study, Meilisa (2010) used maximum likelihood methods (MLM) without considering the initial information (prior) in the parameter estimates. The Bayesian approach is very good for smallsized samples, but it has not been much used or explored. According to Oliviera and Song (2008), based on simulation results showed that estimate value by Bayesian approach was closer to actual value, whereas by MLM lower than actual value. In this research, the noninformative prior that used for the Bayesian method wass the independence Jeffreys, Jeffreys-rule, and the uniform prior. The dataset that used was percentage poverty of 38 districts/towns in East Java province on the basis of explanatory variables were percentage of household that did not use clean water, residents who get health insurance and subsidized rice. The results showed that the spatial correlation of the three noninformative priors was 0.10. The best prior for SAR Bayes model with Jeffreys-rule prior had a coefficient of determination of 83.89% and it had the smallest value of variance estimator and Bayesian Information Criterion (BIC). In this case study, SAR Bayesian model had almost same coefficient determination with the results of regression and SAR model.

Simultaneous Autoregressive (SAR) adalah model spasial yang berasal dari persamaan regresi linear dengan galatnya dimodelkan dalam bentuk model otoregresif, dengan peubah acak pada satu lokasi diamati secara simultan terhadap peubah acak di lokasi lainnya. Dalam penelitian terdahulu (Meilisa 2010) menggunakan metode kemungkinan maksimum/maximum likelihood (ML) dalam pendugaan parameter tanpa mempertimbangkan informasi awal (prior) dalam pendugaan parameternya. Pendekatan metode Bayes untuk SAR belum banyak digunakan, padahal pendekatan tersebut baik digunakan untuk sampel yang berukuran kecil. Menurut hasil penelitian Oliviera dan Song (2008) berdasarkan hasil simulasi, nilai dugaan parameter menggunakan metode ML memiliki nilai yang dibawah nilai sebenarnya, sedangkan dengan pendekatan Bayes nilai-nilai dugaan yang dihasilkan lebih mendekati nilai yang sebenarnya. Pada penelitian ini digunakan informasi awal yang terdiri dari independence Jeffreys, Jeffreys-rule dan uniform. Pemilihan informasi awal noninformatif ini didasarkan atas pertimbangan bahwa tidak ada informasi berdasarkan penelitian sebelumnya