Presentation on theme: "Discrete Choice Modeling"— Presentation transcript:
1Discrete Choice Modeling William GreeneStern School of BusinessNew York UniversityLab Sessions
2Bivariate Extensions of the Probit Model Lab Session 3Bivariate Extensions of the Probit Model
3Bivariate Probit Model Two equation modelGeneral usage ofLHS = the set of dependent variablesRH1 = one set of independent variablesRH2 = a second set of variablesEconomical use of namelists is useful hereNamelist ; x1=one,age,female,educ,married,working $Namelist ; x2=one,age,female,hhninc,hhkids $BivariateProbit ;lhs=doctor,hospital;rh1=x1;rh2=x2;marginal effects $
4Heteroscedasticity in the Bivariate Probit Model General form of heteroscedasticity in LIMDEP/NLOGIT: Exponentialσi = σ exp(γ’zi) so that σi > 0γ = 0 returns the homoscedastic case σi = σEasy to specifyNamelist ; x1=one,age,female,educ,married,working ; z1 = … $Namelist ; x2=one,age,female,hhninc,hhkids ; z2 = … $BivariateProbit ;lhs=doctor,hospital;rh1=x1 ; hf1 = z1;rh2=x2 ; hf2 = z2$
5Heteroscedasticity in Marginal Effects Univariate case:If the variables are the same in x and z, these terms are added.Sign and magnitude are ambiguousVastly more complicated for the bivariate probit case. NLOGIT handles it internally.
7Marginal Effects: Total Effects | Partial derivatives of E[y1|y2=1] with || respect to the vector of characteristics. || They are computed at the means of the Xs. || Effect shown is total of 4 parts above. || Estimate of E[y1|y2=1] = || Observations used for means are All Obs. || Total effects reported = direct+indirect. ||Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|Constant (Fixed Parameter)AGEFEMALEEDUCMARRIEDWORKINGHHNINCHHKIDS
10Imposing Fixed Value and Equality Constraints Used throughout NLOGIT in all models,model parameters appear as a long list:β1 β2 β3 β4 α1 α2 α3 α4 σ and so on.M parameters in total.Use ; RST = list of symbols for the model parameters, in the right orderThis may be used for nonlinear models. Not in REGRESS. Use ;CLS:… for linear modelsUse the same name for equal parametersUse specific numbers to fix the values
11BivariateProbit ; lhs=doctor,hospital ; rh1=one,age,female,educ,married,working; rh2=one,age,female,hhninc,hhkids; rst = beta1,beta2,beta3,be,bm,bw,beta1,beta2,beta3,bi,bk, $Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X|Index equation for DOCTORConstant| ***AGE| ***FEMALE| ***EDUC| ***MARRIED| ***WORKING||Index equation for HOSPITALHHNINC| ***HHKIDS| **|Disturbance correlationRHO(1,2)| (Fixed Parameter)......
12Multivariate Probit MPROBIT ; LHS = y1,y2,…,yM ; Eq1 = RHS for equation 1; Eq2 = RHS for equation 2…; EqM = RHS for equation M $Parameters are the slope vectors followed by thelower triangle of the correlation matrix
13Estimated Multivariate Probit | Multivariate Probit Model: 3 equations. || Number of observations || Log likelihood function || Number of parameters || Replications for simulated probs. = ||Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|Index function for IP84ConstantFDIUMSPIndex function for IP85FDIUMSPIndex function for IP86FDIUMSPCorrelation coefficientsR(01,02)R(01,03)R(02,03)