Estimation and Test Statistic in Bivariate Probit Model ( r c)

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1 J Basic Appl Sci Res, (3)78-88, 0 00, extroad Publication ISSN X Journal of Basic and Applied Scientific Research wwwtextroadcom Estimation and est Statistic in Bivariate Probit Model ( r c) Vita Ratnasari, Purhadi, Ismaini, Suhartono Doctorate Program in Statistics, Facult of Mathematics and Natural Science, Sepuluh Nopember Institute of echnolog (IS), Surabaa, Indonesia Lecturer at Statistics Department, Facult of Mathematics and Natural Science, Sepuluh Nopember Institute of echnolog (IS), Surabaa, Indonesia ABSRAC One of statistics models that could be used to anale categorical response data is probit model his paper focuses to discuss about estimation and test statistic in bivariate probit model ( r c) Bivariate probit model ( r c) is a probit model which involves two response variables, ie the first variable has r categor and the second has c categor Both response variables are correlated In this model, the estimation of model parameters is done b Maximum Likelihood Estimation method with the Newton-Raphson iteration Furthermore, test statistic to evaluate the significance of model parameters is obtained b using Maximum Likelihood Ratio est, particularl the G test for overall test and t test for partial test KEY WORDS: Bivariate probit, Maximum Likelihood Estimation, Maximum Likelihood Ratio est INRODUCION Probit model is a statistics model that can explain the relationship between the discrete response variable and continuous, discrete or mix predictor variables here are man researches about probit model, particularl univariate probit model such as (Aitchison and Silve,957), (Boes and Winkelmann, (005), (Jackman, 000), (McKelve and Zavoina, 975), (Ronning and Kukuk, 996), and (Snapinn and Small, 986) In fact, the response variables in man cases could be more than one and correlated each others Up to now, research on bivariate probit model mostl focused on the application, such as (Bokosi 007), (Moumder, et al, 008) and (Yamamoto and Shankar, 004) herefore, this paper will discuss about theor of probit models with two response variables, especiall the estimation and test statistic Bivariate probit model is a probit model which involves two response variables, ie the first variable has r categor and the second has c categor In this case, both response variables are correlated In general, the initial step for building bivariate probit model is to estimate the model parameters and determine the test statistic to validate significance of the parameters One of estimation methods that frequentl used for estimating the parameters of bivariate probit model is Maximum Likelihood Estimation (MLE) Furthermore, the test statistic that usuall be used to evaluate significance of model parameters is Maximum Likelihood Ratio est hus, this paper will focus to discuss further about estimation and test statistic in bivariate probit model ( r c) Corresponding Author: Dr Vita Ratnasari, Doctorate Program in Statistics, Facult of Mathematics and Natural Science, Sepuluh Nopember Institute of echnolog (IS), Surabaa, Indonesia 78

2 Ratnasari et al, 0 MEHODS he probit model r categor is built from a latent regression in the same man-ner as the binomial probit model We begin with i β xi ε i, where x i is a vector of predictor variable for the i th observation and β is the unknown parameter As usual, is unobserved variable, that follow as: (Greene, 008) 0 if if 0 r if r r he probabilit for each observed respon has r categor, ie: P( 0) P( ) ( β x) ( β x ) 0 0 P( ) P( ) ( β x) ( β x ) P( r ) P( ) ( β x) ( β x ) r r r r Bivariate probit model ( r c) is a probit model which involves two response variables, ie β x ε and 0 if if r β x 0 if r r Whereas, the second has c categor, that is 0 if if Variables 3 c 0 if c c ε he first variable has r categor that is (, ) that satisf normal bivariate distribution can be written as (, ) ~ N( μ, ) For example, there is p predictor variables x, x,, xp, with E( ) β x and E( ) β x So, it can be written Bivariate normal densit function (, ) is: β x μ β x and 79

3 J Basic Appl Sci Res, (3): 78-88, 0 β x β x f (, ) exp β x β x he probabilit of bivariate normal densit function follows: (, ) with thresholds and is as P(, ) f (, ) d d Suppose so E( ), if so β x, and E( ) if β x herefore, bivariate normal densit function (, ) is as follows: j k ji ki P(, ) P(, ) (, ) d d where (,, ) exp Z Z Z Z, j 0,,,, r and k 0,,,, c, 0 0 β x, β x, β x,, ( r ) ( r) β x and 0 β x, β x, β x,, ( c) ( c) β x, β 0 p and β 0 x x x x p Value 0,,, ( r ) is threshold on the first response variable with ( r ) categor, and value 0,,, ( c ) is threshold on the second response variable with ( c ) categor Both response variables are formed in contingenc table ( r c) as in able able will have ( r c) categories, ie Y00, Y0, Y0,, Y( r ) ( c ) which value 0 or Y is an event that happens on the first response variable of j categor and the second response variable of k categor, where Y 00 r c Y j k 0 0 p Y 00 is an event that happens in the area of 0 and 0 Y 0 is an event that happens in the area of 0 and 0 Y 0 is an event that happens in the area of 0 and 0 Y is an event that happens in the area of 0 and 0 80

4 Ratnasari et al, 0 Y is an event that happens in the area of ( r )( c ) and r c he event on able follows Multinomial distribution, ie ( Y, Y, Y,, Y ) ~ M (; P, P, P,, P ) 0 0 ( r)( c) 0 0 ( r)( c) he densit function of Multinomial distribution, is as follows: r c ( r)( c) ( r)( c) j0 k0 P( Y, Y, Y,, Y ) P able : able of Frequenc Contingenc and Probabilit of wo Response Variables ( r c) Y Y 0 ( c ) otal 0 Y00 ; P 00 Y0 ; P 0 Y0( c) ; P0( c ) P0 Y0 ; P 0 Y ; P Y( c ) ; P( c ) P Y ; P Y ; P Y ; P ( r ) ( r )0 ( r )0 ( r) ( r ) ( r)( c) ( r)( c ) P( r ) otal P 0 P P ( c ) P In details, probabilit value on able is as follows 0 0 P (,, ) d d P (,, ) d d 0 (, ) (, ) (, ) 0 P (,, ) d d (, ) (, ) P (,, ) d d k j P (,, ) d d ( k ) ( j) (, ) (, ) (, ) (, ) 8

5 J Basic Appl Sci Res, (3): 78-88, 0 (, ) (, ) (, ) (, ) j k ( j) k j ( k ) ( j) ( k) P (,, ) d d ( r )( c ) ( c) ( r ) ( ) ( ) (, ) ( r) ( c) ( r) ( c) RESULS AND DISCUSSION he following is the discussion of parameter estimation and test statistic of bivariate binar probit model ( r c) Parameter Estimation able shows that the event on each respondent will have Multinomial distribution, ie ( Y, Y, Y,, Y ) ~ M (; P, P, P,, P ) Based on (Greene, 008) 0 0 ( r)( c) 0 0 ( r)( c) and (Gujarati, 003), the parameters on probit model could be estimated b using MLE method he initial step to gain parameter estimation b MLE method is b taking n sampel randoml, ie Y, Y, Y, Y,, Y X, X,, X, where i,,, n 00i 0i 0i i ( r)( c) i i i pi Function of the likelihood can be written as follows or n β 00i 00i, i 0i, 0i 0 i,,, ( r)( c) i ( r)( c) i L P Y Y Y Y L i n r c β P i j0 k0 Y Function ln of the likelihood is available on the equation () n r c ln L β ln P () i j0 k 0 Probabilit on the equation () contains parameter derived to its parameter β and β, ie β β β hen, ln L β is n r c ln L ln P and β β i j0 k 0 n r c P i j0 k0 P β () 8

6 Ratnasari et al, 0 n r c ln L ln P β β i j0 k0 where: n r c P i j0 k0 P β (3) P (, ) (, ) (, ) (, ) j k ( j) k j ( k) ( j) ( k) P (, ) (, ) (, ) (, ) ji k i ( j)i k i j i ( k) i ( j) i ( k )i β β β β β (4) P (, ) (, ) (, ) (, ) (5) β β β β β ji k i ( j)i k i j i ( k) i ( j) i ( k )i Equations (4) and (5) is obtained b substituting: (, ) ji k i ji k i β β (, ) ( j)i k i β β (, ) d d ( j)i k i (, ) i i j i ( k ) i j i ( k ) i β β (, ) ( j) i ( k )i β β (, ) d d i i (, ) d d ( j) i ( k ) i i i (, ) d d i i he estimates obtained b Equations () and (3) are not close form, then one of numerical approaches that can be used to find the estimates is Newton-Raphson method hrough the process of Newton-Raphson iteration, the maximum likelihood estimator can be obtained for β, where m β is the parameter estimation at iteration m Newton Raphson iteration process is the need of the vector g( β ) and the Hessian matrix Vector g( β ) is the first derivative of the function ln likelihood to its parameters Hessian Matrix elements H( β ) are the second derivatives of their parameter Vector component g( β ) which sies [( p ) ] is as follows ln L( β) β g( β) ln L( β) β [( p) ] he vector components g( β ) could be calculated in equation () and (3) Hessian matrix sies is [( p ) ( p )], ie: 83

7 J Basic Appl Sci Res, (3): 78-88, 0 ln L( β) ln L( β) β β β β H( β) ln L( β) ln L( β) β β β β [( p) ( p)] In details, matrix components of Hessian could be found on equations (6), (7), and (8) he equation (6) is derived from equation () toβ, ie β ln L P P c r n rci ββ k 0 j0 i β P β P ββ (6) where: β P P P β he value of P β is calculated from equation (4) and P (, ) (, ) (, ) (, ) β β β β β β β β β β ji k i ( j)i k i j i ( k ) i ( j) i ( k)i Equation (7) is obtained b deriving equation (3) to β, ie β ln L P c r n ββ k 0 j0 i β P β P ββ P (7) where: β P P P β Whereas, P β is obtained from (5) and P (, ) (, ) (, ) (, ) β β β β β β β β β β ji k i ( j)i k i j i ( k) i ( j) i ( k)i Equation (8) is found from equation () to β, ie β ln L P c r n ββ k 0 j0 i β P β P ββ P (8) where: 84

8 Ratnasari et al, 0 β P P P β hen, P β is derived from equation (4) and P (, ) (, ) (, ) (, ) β β β β β β β β β β ji k i ( j)i k i j i ( k) i ( j) i ( k)i Parameter Significance esting o test the goodness of fit of the model, the parameters are evaluated It is intended to determine whether the predictor variables contained in the model have a significant effect or not esting of the model parameters is carried out either at simultaneousl and individual he method used to obtain the test statistic is MLR Simultaneousl hpothesis is a hpothesis that state whether the variable x, x,, x p has a significant effect on the response variable P he hpotheses are: H : 0 0 p p H : at least has one 0, where s, and t,,, p st he parameter under population is { 0,,, p, 0,,, p}, whereas the parameter under H 0 is { 0, 0} est Statistic is obtained b making ratios of L( ˆ ) and L( ˆ ), L( ˆ ) L( ˆ ) H 0 is rejected if L( ˆ ) 0, where 0 L( ˆ 0 hus, G ln is obtained and according to (Agresti, ) 00), G follows distribution, ie L( ˆ ) G ln ln ln L( ˆ ) ln L( ˆ ) L( ˆ ) In details, L( ˆ ) is: L( ˆ ) (, ) (, ) (, ) (, ) (, ) ( j, k ) ( ( j), k ) ( j, ( k) ) ( ( j), ( k ) ) or ( ( r) ) ( ( c) ) ( ( r), ( c) ) ( r )( c) 85

9 J Basic Appl Sci Res, (3): 78-88, 0 ˆ ˆ ˆ ˆ ˆ ˆ L( ˆ ) ( β x, β x) ( β x, β x) ( β x, β x ) 0 ( βˆ ˆ ˆ ˆ x, 0 β x) ( 0 β x, 0 β x ) ( ˆ ˆ ˆ ˆ ˆ ˆ j β x, k β x) ( ( j) β x, k β x) ( j β x, ( k) β x ) ˆ ˆ ( ( j) β x, ( k) β x) ( r )( c) ˆ ˆ ˆ ˆ ( ( r) β x) ( ( c) β x) ( ( r) β x, ( c) β x ) he values of ˆβ and ˆβ could be obtained b using equations () and (3) Moreover, the value of L( ˆ ) is as follows: ˆ ˆ ˆ ˆ ˆ ˆ L( ˆ ) (, ) (, ) (, ) 0 ( ˆ ˆ ˆ ˆ 0, 0 0 ) ( 0 0, 0 0 ) ( ˆ ˆ ˆ ˆ ˆ ˆ j 0, k 0 ) ( ( j) 0, k 0 ) ( j 0, ( k) 0 ) ( ˆ ˆ ( j) 0, ( k ) 0 ) ( ˆ ˆ ˆ ˆ ( r) 0 ) ( ( c) 0 ) ( ( r) 0, ( c) 0 ) ( r )( c) hen, the values ˆ 0 and ˆ 0 could be found b using equations (9) and (0) Derivative of ln likelihood to 0 is P (9) ln L n r c 0 i j0 k0 P 0 hus, the result of P 0 is P (, ) (, ) (, ) (, ) ji k i ( j)i k i j i ( k ) i ( j) i ( k ) i where, ji ji 0 ( k) i ( k) i 0 and ( j) i ( j)i 0, k i k i 0 While, the first derivative of ln( β ) to 0 is 86

10 Ratnasari et al, 0 P (0) ln L n r c 0 i j0 k0 P 0 where the derivative of probabilit Pi and P 0i to 0 P (, ) (, ) (, ) (, ) ji k i ( j)i k i j i ( k ) i ( j) i ( k ) i is If G,v then H 0 is rejected, where degree of freedom (v) is the number of model parameters under population subtracted b the number of model parameters under H 0, with a large population After conducting the test of parameter significance simultaneousl, the next step is the partial testing In this testing, it is desirable to know the contribution of each predictor variable he hpothesis of individual testing on bivariate binar probit model is H 0 : st 0 H : st 0, where s, and t 0,,,, p Set of the parameters () under the null hpothesis is,, ; 0,,, ;, where: s t s t p s s t t ˆ ˆ ˆ ˆ ˆ ˆ 0 β x 0 β x β x 0 β x 0 β x 0 β x 00 0 L( ˆ ) (, ) (, ) (, ) 0 ( βˆ ˆ ˆ ˆ x, 0 β x ) ( 0 β x, 0 β x ) ( ˆ ˆ ˆ ˆ ˆ ˆ j β x, k β x ) ( ( j) β x, k β x ) ( j β x, ( k ) β x ) ˆ ˆ ( ( j) β x, ( k ) β x ) ˆ ˆ ˆ ˆ ( ( r) β x ) ( ( c) β x ) ( ( r) β x, ( c) β x ) ( r )( c) and x {, x, x,, x, x,, x } ( t) ( t) p β ˆ {,,,,,, } and 0 ( t) ( t) p β ˆ {,,, } or 0 p β ˆ {,,, } and 0 p β ˆ {,,,,,, } 0 ( t) ( t) p he partial test statistic was done b using MLR method as in simultaneous testing, so test ˆ rs statistic t was gained, that is t For large sample, t follows Normal standard SE( ˆ rs ) distribution, ie t ~ N (0,) If t, then the null hpothesis is rejected Z 87

11 J Basic Appl Sci Res, (3): 78-88, 0 Conclusion his paper alread discussed about theoretical part of bivariate probit model, particularl about estimation method and test statistic he results showed that estimation of model parameters b using MLE ielded not closed form solution hen, these estimated parameters could be obtained b Newton-Raphson iteration Furthermore, two test statistics to validate significance of model parameters could be constructed b MLR method based on asmptotic properties of these estimators, ie G for overall test and t for partial test REFERENCES Agresti, A (00), Categorical Data Analsis, John Wile & Sons, Inc, Hoboken, New Jerse Aitchison, J & Silve, SD (957) he Generaliation of Probit Analsis to the Case of Multiple Responses Biometrika 44(), 3-40 Boes, S & Winkelmann, R (005), Ordered Response Models, echnical Report, Socioeconomic Institute, Universit of Zurich Bokosi, FK (007) Household Povert Dinamics in Malawi: A Bivariate Probit Analsis Journal of Applied Sciences 7(), 58-6 Greene, WH (008), Econometrics Analsis, Fourth Edition, Prentice Hall, Englewood Cliffs, New Jerse Gujarati, DN(003) Basic Econometric Fourth Edition McGraw Hill, New York Jackman, S (000), Models for Ordered Outcomes, echnical Report, Stanford Universit McKelve, RD & Zavoina, W (975) A Statistical Model for the Analsis of Ordinal Level Dependent Variables Journal of Mathematical Sociolog 4, 03-0 Moumder, P, Raheem, N, alberth, J, & Berrens, RP (008) Investigating intended evacuation from wildfires in the wildland urban interface: Application of a bivariate probit model Forest Polic and Economics 0, Ronning, G & Kukuk, M (996) Efficient Estimation of Ordered Probit Models Journal of the American Statistical Association 9(435), 0-9 Snapinn, SM & Small, RD (986) est of Significance Using Regression Models for Ordered Categorical Data Biometrics 4, Yamamoto, & Shankar,VN (004) Bivariate ordered-response probit model of driver s and passenger s injur severities in collisions with fixed objects Accident Analsis & Prevention 36,