PROS Elok FF El Fahmi, Vita R, Santi PR Estimation of parameter fulltext

Proceedings of the IConSSE FSM SWCU (2015), pp. MA.5–7 ISBN: 978-602-1047-21-7

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Estimation of parameter in spatial probit regression model

Elok Faiz Fatma El Fahmi*_{, Vita Ratnasari, Santi Puteri Rahayu}

Department of Statistics, Sepuluh Nopember Institute of Technology, Surabaya, Indonesia

Abstract

Probit model is a non linear model used to analyze a relationship between a dependent variable (response) and some independent variables where the response is dichotomy qualitative data in which the value is equal to 1 for expressing the presence of a characteristic and 0 for expressing the absence of a characteristic. A data modeling associated with region or area is usually called as spatial. In spatial data, there is a spatial correlation effect which refers to spatial autocorrelation. An estimation using Ordinary Least Square (OLS) can not be applied in this condition because of this spatial autocorrelation effect, and Maximum Likelihood Estimation (MLE) is used as the alternative. In qualitative data involving an aspect of connection between one region to another needs a special method which combines probit regression method and spatial aspect, i.e. spatial probit regression with SAR (Spatial Autoregressive) model.

Keywords maximum likelihood estimation, spatial autoregressivemodel, spatial probit regression

1.

Introduction

Regression is a method used to determine a relationship between a dependent variable and one or more independent variables. In the case of regression models often encountered with a dependent variable is qualitative. Probit model is a non linear model used to analyze a relationship between a dependent variable (response) and some independent variables where the response is dichotomy qualitative data in which the value is equal to 1 for expressing the presence of a characteristic and 0 for expressing the absence of a characteristic. A data modeling associated with region or area is usually called as spatial. In spatial data, there is a spatial correlation effect which refers to spatial autocorrelation. An estimation using OLS can not be applied in this condition because of this spatial autocorrelation effect, and MLE is used as the alternative. In qualitative data involving an aspect of connection between one region to another needs a special method which combines probit regression method and spatial aspect, i.e. spatial probit regression with SAR model.

2.

Spatial probit models

Probit model is one of statistical modeling of which response variable is qualitative (categorical). A univariate probit model is a probit model involving only one response variable. If the qualitative response has two categories then the model is binary probit model. A research which is associated with region or area is often called as spatial. The general spatial regression model is shown in the following equation (Le Sage, 1999):

*

Estimation of parameter in spatial probit regression model

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p = q + r, + , _{= Gq + .} (1)

In LeSage & Pace (2009), SAR with a limitation for the spatial dependent variable is generally shown as

= q + r, + ,

for = , … , ′ with some fixed matrix of covariates r_7×! associated with the

parameter vector ,_!× . The matrix q_7×7 is called the spatial weight matrix and captures the dependence structure between neighboring observations such as friends or nearby locations. The term q is a linear combination of neighboring observations. The scalar ρ is the dependence parameter and will assumed | | < 1. The u + 1 model parameters to be estimated are the parameter vector ,and the scalarρ.

In spatial probit model, y is considered as latent variable, so the observed variable is only binary variable 0,1 as follows

= S0 , if _{1 , if} ∗∗≤ 0,_{> 0.} formed from Eq. (2). Below is the ln-likelihood function for spatial probit model:

ln < , = ∑ V ln7 + 1 − ln 1 − W

#

= ∑ 4 ln-1 − Φ z − N − q7 o _{r, . + 1 − ln-Φ z − N − q} o _{r, .5}

#

The ln-likelihood function is maximized by determining the first derivation of parameter

|

, then it is considered to be equal to zero. By assuming z − N − q o r, to be ~, then the first derivation of parameter

|

is obtained as follows:

H •€ I ,

Eq. (4) presents an equation which is not in a closedform. Hence, an iteration procedure of Newton Raphson method is used to obtain the estimator of

|

. This method is obtained from approach taylor series (Agresti, 2002):

E.F.F.E Fahmi, V. Ratnasari, S.P. Rahayu

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term and such that can be ignored. So the expansion of Taylor series into ‡ ln < , So Žiteration Newton Raphson method is, (Hardin & Hibe, 2007),

,• _{= ,}•"• _{− ‹} ‡ ln < ,

component in process of iteration Newton Raphson method is determine first derivative vector likelihood function of parameter , or vector ‘ , and determine a matrix of second derivation of Likelihood function of parameter , or ’ , . Mathematically vector ‘ , and

/_{• † • , the second derivative of ln-likelihood function of the parameter , is}

H”_{•€ I ,} obtained estimation of , with iteration of Newton Raphson method.

References

Agresti, A. (2002). Categorical Data Analysis (2nd ed.). John Wiley & Sons, Inc.

Hardin, J.W. , & Hilbe, J.M. (2007). Generalized Linear Models and Extensions (2nd ed.) A Stata Press Publications, Texas.

LeSage, J. (1999). The Theory and Practice of Spatial Econometrics. Retrieved from http://www.econ.utoledo.edu