PROS Elok FF El Fahmi, Vita R, Santi PR Estimation of parameter fulltext
Proceedings of the IConSSE FSM SWCU (2015), pp. MA.5–7
MA.5
ISBN: 978-602-1047-21-7
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 autoregressive model, 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):
*
Corresponding author. E-mail address: elok_faiz@yahoo.co.id
SWUP
Estimation of parameter in spatial probit regression model
MA.6
= q + r, + ,
(1)
= Gq + .
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 r7×! associated with the
parameter vector ,!× . The matrix q7×7 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
0 , if ∗ ≤ 0,
=S
1 , if ∗ > 0.
Eq. (1) can be written in a reducted form as:
= N − q o r, + ,
S
= N− q o .
p
3. Results and discussion
The probability for = 0 and = 1 equals to
x = 0 = Φ z − N − q o r, = { ,
(2)
S
x = 1 = 1 − Φ z − N − q o r, = 1 − {
=
.
The parameter | is determined using MLE method where the ln-likelihood function is
formed from Eq. (2). Below is the ln-likelihood function for spatial probit model:
W
ln < , = ∑7# V ln
ln 1 −
+ 1−
ln-Φ z − N − q o r, .5
= ∑7# 4 ln-1 − Φ z − N − q o r, . + 1 −
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 ,
H
W
= ∑7# V ln
+ 1−
ln 1 −
H,
=
=
since
H‚ ƒ„
H,
=− † • .
=
H,
H
∑7# V ln 1 − Φ • + 1 −
H,
H‚ ƒ„
…
o…
∑7#
Q− o‚„ ƒ + ‚ ƒ „ R
H,
„
„
…„
…„ o
∑7# † • Q
+
R
o‚ ƒ
‚ ƒ
„
„
To obtain estimation of |, Eq. (3) is equated with zero:
…„
…o
∑7# † • Q
+ „ R = 0.
o‚ ƒ„
‚ ƒ„
ln Φ • W
(3)
(4)
Eq. (4) presents an equation which is not in a closed form. 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):
‰
‡ ln < ,
1
‡ ln < ,
/ ‡ ln < ,
0=
+ -, − ,.
+
−
,.-,
−
,.
+ ⋯,
-,
2!
‡,
‡,‡,n
‡,n ‡,‡,n
where vector | is initial value. If it is assumed that Š, 1 − ,Š is very small, then the third
SWUP
E.F.F.E Fahmi, V. Ratnasari, S.P. Rahayu
MA.7
term and such that can be ignored. So the expansion of Taylor series into
‡ ln < ,
‡ ln < ,
+ -, − ,.
= 0.
‡,‡,n
‡,
Thus the value of , obtained from
o•
‡ ln < ,
‡ ln < ,
.
, = ,−‹
Œ
n
‡,‡,
‡,
So Ž iteration Newton Raphson method is, (Hardin & Hibe, 2007),
o•
‡ ln < ,
‡ ln < ,
, • = , •"• − ‹ •o•
.
Œ
‡,
‡, •o•
‡ , •"• n
The proses will stop, if β ( m ) − β ( m−1) ≤ Θ , where Θ is very small number. A required
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
matrix ’ , is
‘ , =Q
H •€ I ,
H,
R
“"• ו
and ’ , = Q
H” •€ I ,
is obtained from Eq. (3):
H,H,n
H” •€ I ,
H
…„
…o
= n •∑7# † • Q
+ „ R–
H,
o‚ ƒ„
‚ ƒ„
H,H,n
H
H
— ƒ„
– + ∑7# H,n •
= ∑7# H,n •
o‚ ƒ„
H” •€ I ,
H,H,n
R
“"• × “"•
,
where
=
∑7#
− ∑7#
Since
/
H
•
H, ‚ ƒ„
o‚ ƒ„
‚
˜™ š„
ƒ„
˜,n
– = •‚
ƒ„
–
˜™ š„
˜,n
o—
o— ƒ„
o‚ ƒ„ ”
˜• š„
ƒ„
˜,n
‚ ƒ„ ”
† •
,
˜-›œ• š„ .
˜,n
+
— ƒ„
‚ ƒ„
– − ∑7#
∑7#
‚ ƒ„
H
•
H,n
˜™ š„
˜,n
— ƒ„
o‚ ƒ„
o— ƒ„
‚ ƒ„ ”
–
˜• š„
˜,n
.
H
•
H, o‚ ƒ„
– = −•
o‚ ƒ„
–
† •
, and
• † • , the second derivative of ln-likelihood function of the parameter , is
H” •€ I ,
H,H,n
= − ∑7#
/
+ ∑7# 1 −
H” ‚ ƒ„
H,H,n
=
o‚ ƒ„ ƒ„ — ƒ„ "— ƒ„ — ƒ„
o‚ ƒ„ ”
/ ‚ ƒ„ ƒ„ — ƒ„ o— ƒ„ — ƒ„
‚ ƒ„ ”
4. Conclusion and remarks
With acquisition of vector elemen ‘ , and matrix elemen ’ , , then will be
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
Le Sage, J., & Pace, R.K. (2009). Introduction to Spatial Econometrics. CRC Press, New York.
SWUP
MA.5
ISBN: 978-602-1047-21-7
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 autoregressive model, 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):
*
Corresponding author. E-mail address: elok_faiz@yahoo.co.id
SWUP
Estimation of parameter in spatial probit regression model
MA.6
= q + r, + ,
(1)
= Gq + .
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 r7×! associated with the
parameter vector ,!× . The matrix q7×7 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
0 , if ∗ ≤ 0,
=S
1 , if ∗ > 0.
Eq. (1) can be written in a reducted form as:
= N − q o r, + ,
S
= N− q o .
p
3. Results and discussion
The probability for = 0 and = 1 equals to
x = 0 = Φ z − N − q o r, = { ,
(2)
S
x = 1 = 1 − Φ z − N − q o r, = 1 − {
=
.
The parameter | is determined using MLE method where the ln-likelihood function is
formed from Eq. (2). Below is the ln-likelihood function for spatial probit model:
W
ln < , = ∑7# V ln
ln 1 −
+ 1−
ln-Φ z − N − q o r, .5
= ∑7# 4 ln-1 − Φ z − N − q o r, . + 1 −
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 ,
H
W
= ∑7# V ln
+ 1−
ln 1 −
H,
=
=
since
H‚ ƒ„
H,
=− † • .
=
H,
H
∑7# V ln 1 − Φ • + 1 −
H,
H‚ ƒ„
…
o…
∑7#
Q− o‚„ ƒ + ‚ ƒ „ R
H,
„
„
…„
…„ o
∑7# † • Q
+
R
o‚ ƒ
‚ ƒ
„
„
To obtain estimation of |, Eq. (3) is equated with zero:
…„
…o
∑7# † • Q
+ „ R = 0.
o‚ ƒ„
‚ ƒ„
ln Φ • W
(3)
(4)
Eq. (4) presents an equation which is not in a closed form. 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):
‰
‡ ln < ,
1
‡ ln < ,
/ ‡ ln < ,
0=
+ -, − ,.
+
−
,.-,
−
,.
+ ⋯,
-,
2!
‡,
‡,‡,n
‡,n ‡,‡,n
where vector | is initial value. If it is assumed that Š, 1 − ,Š is very small, then the third
SWUP
E.F.F.E Fahmi, V. Ratnasari, S.P. Rahayu
MA.7
term and such that can be ignored. So the expansion of Taylor series into
‡ ln < ,
‡ ln < ,
+ -, − ,.
= 0.
‡,‡,n
‡,
Thus the value of , obtained from
o•
‡ ln < ,
‡ ln < ,
.
, = ,−‹
Œ
n
‡,‡,
‡,
So Ž iteration Newton Raphson method is, (Hardin & Hibe, 2007),
o•
‡ ln < ,
‡ ln < ,
, • = , •"• − ‹ •o•
.
Œ
‡,
‡, •o•
‡ , •"• n
The proses will stop, if β ( m ) − β ( m−1) ≤ Θ , where Θ is very small number. A required
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
matrix ’ , is
‘ , =Q
H •€ I ,
H,
R
“"• ו
and ’ , = Q
H” •€ I ,
is obtained from Eq. (3):
H,H,n
H” •€ I ,
H
…„
…o
= n •∑7# † • Q
+ „ R–
H,
o‚ ƒ„
‚ ƒ„
H,H,n
H
H
— ƒ„
– + ∑7# H,n •
= ∑7# H,n •
o‚ ƒ„
H” •€ I ,
H,H,n
R
“"• × “"•
,
where
=
∑7#
− ∑7#
Since
/
H
•
H, ‚ ƒ„
o‚ ƒ„
‚
˜™ š„
ƒ„
˜,n
– = •‚
ƒ„
–
˜™ š„
˜,n
o—
o— ƒ„
o‚ ƒ„ ”
˜• š„
ƒ„
˜,n
‚ ƒ„ ”
† •
,
˜-›œ• š„ .
˜,n
+
— ƒ„
‚ ƒ„
– − ∑7#
∑7#
‚ ƒ„
H
•
H,n
˜™ š„
˜,n
— ƒ„
o‚ ƒ„
o— ƒ„
‚ ƒ„ ”
–
˜• š„
˜,n
.
H
•
H, o‚ ƒ„
– = −•
o‚ ƒ„
–
† •
, and
• † • , the second derivative of ln-likelihood function of the parameter , is
H” •€ I ,
H,H,n
= − ∑7#
/
+ ∑7# 1 −
H” ‚ ƒ„
H,H,n
=
o‚ ƒ„ ƒ„ — ƒ„ "— ƒ„ — ƒ„
o‚ ƒ„ ”
/ ‚ ƒ„ ƒ„ — ƒ„ o— ƒ„ — ƒ„
‚ ƒ„ ”
4. Conclusion and remarks
With acquisition of vector elemen ‘ , and matrix elemen ’ , , then will be
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
Le Sage, J., & Pace, R.K. (2009). Introduction to Spatial Econometrics. CRC Press, New York.
SWUP