Empirical Methods for Microeconomic Applications University of Lugano, Switzerland May 27-31, 2013 William Greene Department of Economics Stern School of Business 1C. Extensions of Binary Choice Models Agenda for 1C • • • • • • Endogenous RHS Variables Sample Selection Dynamic Binary Choice Model Bivariate Binary Choice Simultaneous Equations Ordered Choices • • Ordered Choice Model Application to BHPS Endogeneity Endogenous RHS Variable • U* = β’x + θh + ε y = 1[U* > 0] E[ε|h] ≠ 0 (h is endogenous) • • • Case 1: h is continuous Case 2: h is binary, e.g., a treatment effect Approaches • • Parametric: Maximum Likelihood Semiparametric (not developed here): GMM Various approaches for case 2 Endogenous Continuous Variable U* = β’x + θh + ε = ρ. y = 1[U* > 0] Correlation This is the source of the endogeneity h = α’z +u E[ε|h] ≠ 0 Cov[u, ε] ≠ 0 Additional Assumptions: (u,ε) ~ N[(0,0),(σu2, ρσu, 1)] z = a valid set of exogenous variables, uncorrelated with (u,ε) Endogenous Income in Health Income responds to Age, Age2, Educ, Married, Kids, Gender 0 = Not Healthy 1 = Healthy Healthy = 0 or 1 Age, Married, Kids, Gender, Income Determinants of Income (observed and unobserved) also determine health satisfaction. Estimation by ML (Control Function) Probit fit of y to x and h will not consistently estimate (,) because of the correlation between h and induced by the correlation of u and . Using the bivariate normality, x h ( / )u u Prob( y 1| x, h) 2 1 Insert ui = (hi - αz )/u and include f(h|z ) to form logL logL= hi - αz i x h i i u (2 y 1) log i 2 1 N i=1 log 1 hi - αz i u u Two Approaches to ML (1) Full information ML. Maximize the full log likelihood with respect to (,, u , , ) (The built in Stata routine IVPROBIT does this. It is not an instrumental variable estimator; it is a FIML estimator.) (2) Two step limited information ML. (Control Function) (a) Use OLS to estimate and u with a and s. (b) Compute vˆi = uˆi /s = (hi az i ) / s x h vˆ i i ˆ ˆ x h vˆ log (c) log i i i i 2 1 The second step is to fit a probit model for y to (x,h,vˆ) then solve back for (,,) from (,,) and from the previously estimated a and s. Use the delta method to compute standard errors. FIML Estimates ---------------------------------------------------------------------Probit with Endogenous RHS Variable Dependent variable HEALTHY Log likelihood function -6464.60772 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------|Coefficients in Probit Equation for HEALTHY Constant| 1.21760*** .06359 19.149 .0000 AGE| -.02426*** .00081 -29.864 .0000 43.5257 MARRIED| -.02599 .02329 -1.116 .2644 .75862 HHKIDS| .06932*** .01890 3.668 .0002 .40273 FEMALE| -.14180*** .01583 -8.959 .0000 .47877 INCOME| .53778*** .14473 3.716 .0002 .35208 |Coefficients in Linear Regression for INCOME Constant| -.36099*** .01704 -21.180 .0000 AGE| .02159*** .00083 26.062 .0000 43.5257 AGESQ| -.00025*** .944134D-05 -26.569 .0000 2022.86 EDUC| .02064*** .00039 52.729 .0000 11.3206 MARRIED| .07783*** .00259 30.080 .0000 .75862 HHKIDS| -.03564*** .00232 -15.332 .0000 .40273 FEMALE| .00413** .00203 2.033 .0420 .47877 |Standard Deviation of Regression Disturbances Sigma(w)| .16445*** .00026 644.874 .0000 |Correlation Between Probit and Regression Disturbances Rho(e,w)| -.02630 .02499 -1.052 .2926 --------+------------------------------------------------------------- Partial Effects: Scaled Coefficients Conditional Mean E[ y | x, h] (x h) h z u z u v where v ~ N[0,1] E[y|x,z,v] =[x (z u v)] Partial Effects. Assume z = x (just for convenience) E[y|x,z,v] [x (z u v)]( ) x E[y|x,z ] E[y|x,z,v] Ev ( ) [x (z u v)](v)dv x x The integral does not have a closed form, but it can easily be simulated : R E[y|x,z ] 1 Est. ( ) [x (z u vr )] x R r 1 For variables only in x, omit k . For variables only in z, omit k . Endogenous Binary Variable U* = β’x + θh + ε Correlation = ρ. This is the source of the endogeneity y = 1[U* > 0] h* = α’z +u h = 1[h* > 0] E[ε|h*] ≠ 0 Cov[u, ε] ≠ 0 Additional Assumptions: (u,ε) ~ N[(0,0),(σu2, ρσu, 1)] z = a valid set of exogenous variables, uncorrelated with (u,ε) Endogenous Binary Variable P(Y = y,H = h) = P(Y = y|H =h) x P(H=h) This is a simple bivariate probit model. Not a simultaneous equations model - the estimator is FIML, not any kind of least squares. Doctor = F(age,age2,income,female,Public) Public = F(age,educ,income,married,kids,female) FIML Estimates ---------------------------------------------------------------------FIML Estimates of Bivariate Probit Model Dependent variable DOCPUB Log likelihood function -25671.43905 Estimation based on N = 27326, K = 14 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------|Index equation for DOCTOR Constant| .59049*** .14473 4.080 .0000 AGE| -.05740*** .00601 -9.559 .0000 43.5257 AGESQ| .00082*** .681660D-04 12.100 .0000 2022.86 INCOME| .08883* .05094 1.744 .0812 .35208 FEMALE| .34583*** .01629 21.225 .0000 .47877 PUBLIC| .43533*** .07357 5.917 .0000 .88571 |Index equation for PUBLIC Constant| 3.55054*** .07446 47.681 .0000 AGE| .00067 .00115 .581 .5612 43.5257 EDUC| -.16839*** .00416 -40.499 .0000 11.3206 INCOME| -.98656*** .05171 -19.077 .0000 .35208 MARRIED| -.00985 .02922 -.337 .7361 .75862 HHKIDS| -.08095*** .02510 -3.225 .0013 .40273 FEMALE| .12139*** .02231 5.442 .0000 .47877 |Disturbance correlation RHO(1,2)| -.17280*** .04074 -4.241 .0000 --------+------------------------------------------------------------- Partial Effects Conditional Mean E[ y | x, h] (x h) E[ y | x, z ] Eh E[ y | x, h] Prob(h 0 | z )E[ y | x, h 0] Prob( h 1| z )E[ y | x, h 1] (z ) (x) (z ) (x ) Partial Effects Direct Effects E[ y | x, z ] (z )(x) (z )(x ) x Indirect Effects E[ y | x, z ] z (z ) (x) (z ) (x ) (z ) (x ) (x) Identification Issues • • • • Exclusions are not needed for estimation Identification is, in principle, by “functional form” Researchers usually have a variable in the treatment equation that is not in the main probit equation “to improve identification” A fully simultaneous model • • • y1 = f(x1,y2), y2 = f(x2,y1) Not identified even with exclusion restrictions (Model is “incoherent”) Selection A Sample Selection Model U* = β’x + ε Correlation = ρ. y = 1[U* > 0] This is the source of the “selectivity: h* = α’z +u h = 1[h* > 0] E[ε|h] ≠ 0 Cov[u, ε] ≠ 0 (y,x) are observed only when h = 1 Additional Assumptions: (u,ε) ~ N[(0,0),(σu2, ρσu, 1)] z = a valid set of exogenous variables, uncorrelated with (u,ε) Application: Doctor,Public 3 Groups of observations: (Public=0), (Doctor=0|Public=1), (Doctor=1|Public=1) Sample Selection Doctor = F(age,age2,income,female,Public=1) Public = F(age,educ,income,married,kids,female) Sample Selection Model: Estimation f(y1,y 2 ) = Prob[y1 = 1| y 2 =1] * Prob[y 2 =1] (y1 =1,y 2 =1) = Prob[y1 = 0 | y 2 =1] * Prob[y 2 =1] (y1 = 0,y 2 =1) = Prob[y 2 = 0] (y 2 = 0) Terms in the log likelihood : (y1 =1,y 2 =1) Φ2 (β1 x i1,β2 x i2 ,ρ) (Bivariate normal) (y1 = 0,y 2 =1) Φ2 (-β1 x i1,β2 x i2 ,-ρ) (Bivariate normal) (y 2 = 0) Φ(-β2 x i2 ) (Univariate normal) Estimation is by full information maximum likelihood. There is no "lambda" variable. ML Estimates ---------------------------------------------------------------------FIML Estimates of Bivariate Probit Model Dependent variable DOCPUB Log likelihood function -23581.80697 Estimation based on N = 27326, K = 13 Selection model based on PUBLIC Means for vars. 1- 5 are after selection. --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------|Index equation for DOCTOR Constant| 1.09027*** .13112 8.315 .0000 AGE| -.06030*** .00633 -9.532 .0000 43.6996 AGESQ| .00086*** .718153D-04 11.967 .0000 2041.87 INCOME| .07820 .05779 1.353 .1760 .33976 FEMALE| .34357*** .01756 19.561 .0000 .49329 |Index equation for PUBLIC Constant| 3.54736*** .07456 47.580 .0000 AGE| .00080 .00116 .690 .4899 43.5257 EDUC| -.16832*** .00416 -40.490 .0000 11.3206 INCOME| -.98747*** .05162 -19.128 .0000 .35208 MARRIED| -.01508 .02934 -.514 .6072 .75862 HHKIDS| -.07777*** .02514 -3.093 .0020 .40273 FEMALE| .12154*** .02231 5.447 .0000 .47877 |Disturbance correlation RHO(1,2)| -.19303*** .06763 -2.854 .0043 --------+------------------------------------------------------------- Estimation Issues • This is a sample selection model applied to a nonlinear model • • • • There is no lambda Estimated by FIML, not two step least squares Estimator is a type of BIVARIATE PROBIT MODEL The model is identified without exclusions (again) A Dynamic Model Dynamic Models y it 1[x it y i,t 1 it ui > 0] Two similar 'effects' Unobserved heterogeneity State dependence = state 'persistence' Pr(y it 1 | y i,t 1 ,..., y i0 , x it ,u] F[ x it y i,t 1 ui ] How to estimate , , marginal effects, F(.), etc? (1) Deal with the latent common effect (2) Handle the lagged effects: This encounters the initial conditions problem. Dynamic Probit Model: A Standard Approach (1) Conditioned on all effects, joint probability P(y i1 , y i2 ,..., y iT | y i0 , x i ,ui ) t 1 F( x it β y i,t 1 ui , y it ) T (2) Unconditional density; integrate out the common effect P(y i1 , y i2 ,..., y iT | y i0 , x i ) P(y i1 , y i2 ,..., y iT | y i0 , x i ,ui )h(ui | y i0 , x i )dui (3) Density for heterogeneity h(ui | y i0 , x i ) N[ y i0 x iδ, u2 ], x i = [x i1 ,x i2 ,...,x iT ], so ui = yi0 x iδ + wi (contains every period of x it ) (4) Reduced form P(y i1 , y i2 ,..., y iT | y i0 , x i ) T t 1 F( x it β y i,t 1 y i0 x iδ u w i , y it )h(w i )dw i This is a random effects model Simplified Dynamic Model Projecting ui on all observations expands the model enormously. (3) Projection of heterogeneity only on group means h(ui | y i0 , x i ) N[ y i0 x iδ, u2 ] so ui = y i0 x iδ + w i (4) Reduced form P(y i1 , y i2 ,..., y iT | y i0 , x i ) T t 1 F( x it β y i,t 1 y i0 x iδ u w i , y it )h(w i )dw i Mundlak style correction with the initial value in the equation. This is (again) a random effects model A Dynamic Model for Public Insurance Dynamic Common Effects Model Bivariate Model Gross Relation Between Two Binary Variables Cross Tabulation Suggests Presence or Absence of a Bivariate Relationship +-----------------------------------------------------------------+ |Cross Tabulation | |Row variable is DOCTOR (Out of range 0-49: 0) | |Number of Rows = 2 (DOCTOR = 0 to 1) | |Col variable is HOSPITAL (Out of range 0-49: 0) | |Number of Cols = 2 (HOSPITAL = 0 to 1) | +-----------------------------------------------------------------+ | HOSPITAL | +--------+--------------+------+ | | DOCTOR| 0 1| Total| | +--------+--------------+------+ | | 0| 9715 420| 10135| | | 1| 15216 1975| 17191| | +--------+--------------+------+ | | Total| 24931 2395| 27326| | +-----------------------------------------------------------------+ Tetrachoric Correlation A correlation measure for two binary variables Can be defined implicitly y1 * = μ1 + ε1, y1 = 1(y1* > 0) y 2 * = μ2 + ε 2 ,y 2 = 1(y 2 * > 0) 0 1 ρ ε1 ~ N , ε 0 ρ 1 2 ρ is the tetrachoric correlation between y1 and y 2 Log Likelihood Function for Tetrachoric Correlation logL = i=1logΦ2 (2yi1 -1)μ1,(2y i2 -1)μ2 ,(2y i1 -1)(2y i2 -1)ρ n = i=1logΦ2 qi1μ1,qi2μ2 ,qi1qi2ρ n Note : qi1 = (2y i1 -1) = -1 if y i1 = 0 and +1 if y i1 = 1. Φ2 = Bivariate normal CDF - must be computed using quadrature Maximized with respect to μ1,μ2 and ρ. Estimation +---------------------------------------------+ | FIML Estimates of Bivariate Probit Model | | Maximum Likelihood Estimates | | Dependent variable DOCHOS | | Weighting variable None | | Number of observations 27326 | | Log likelihood function -25898.27 | | Number of parameters 3 | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Index equation for DOCTOR Constant .32949128 .00773326 42.607 .0000 Index equation for HOSPITAL Constant -1.35539755 .01074410 -126.153 .0000 Tetrachoric Correlation between DOCTOR and HOSPITAL RHO(1,2) .31105965 .01357302 22.918 .0000 A Bivariate Probit Model • • • Two Equation Probit Model No bivariate logit – there is no reasonable bivariate counterpart Why fit the two equation model? • • Analogy to SUR model: Efficient Make tetrachoric correlation conditional on covariates – i.e., residual correlation Bivariate Probit Model y1 * = β1x1 + ε1, y1 = 1(y1* > 0) y 2 * = β2 x 2 + ε 2 ,y 2 = 1(y 2 * > 0) 0 1 ρ ε1 ~ N , ε 0 ρ 1 2 The variables in x 2 and x 2 may be the same or different. There is no need for each equation to have its 'own variable.' ρ is the conditional tetrachoric correlation between y1 and y 2 . (The equations can be fit one at a time. Use FIML for (1) efficiency and (2) to get the estimate of ρ.) Estimation of the Bivariate Probit Model (2yi1 -1)β1 xi1, n logL = i=1logΦ2 (2yi2 -1)β2 x i2 , (2yi1 -1)(2y i2 -1)ρ = i=1logΦ2 qi1β1 x i1,qi2β2 x i2 ,qi1qi2ρ n Note : qi1 = (2yi1 -1) = -1 if y i1 = 0 and +1 if y i1 = 1. Φ2 = Bivariate normal CDF - must be computed using quadrature Maximized with respect to β1,β2 and ρ. Parameter Estimates ---------------------------------------------------------------------FIML Estimates of Bivariate Probit Model for DOCTOR and HOSPITAL Dependent variable DOCHOS Log likelihood function -25323.63074 Estimation based on N = 27326, K = 12 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------|Index equation for DOCTOR Constant| -.20664*** .05832 -3.543 .0004 AGE| .01402*** .00074 18.948 .0000 43.5257 FEMALE| .32453*** .01733 18.722 .0000 .47877 EDUC| -.01438*** .00342 -4.209 .0000 11.3206 MARRIED| .00224 .01856 .121 .9040 .75862 WORKING| -.08356*** .01891 -4.419 .0000 .67705 |Index equation for HOSPITAL Constant| -1.62738*** .05430 -29.972 .0000 AGE| .00509*** .00100 5.075 .0000 43.5257 FEMALE| .12143*** .02153 5.641 .0000 .47877 HHNINC| -.03147 .05452 -.577 .5638 .35208 HHKIDS| -.00505 .02387 -.212 .8323 .40273 |Disturbance correlation (Conditional tetrachoric correlation) RHO(1,2)| .29611*** .01393 21.253 .0000 ---------------------------------------------------------------------| Tetrachoric Correlation between DOCTOR and HOSPITAL RHO(1,2)| .31106 .01357 22.918 .0000 --------+------------------------------------------------------------- Marginal Effects • What are the marginal effects • • • Possible margins? • • • • Effect of what on what? Two equation model, what is the conditional mean? Derivatives of joint probability = Φ2(β1’xi1, β2’xi2,ρ) Partials of E[yij|xij] =Φ(βj’xij) (Univariate probability) Partials of E[yi1|xi1,xi2,yi2=1] = P(yi1,yi2=1)/Prob[yi2=1] Note marginal effects involve both sets of regressors. If there are common variables, there are two effects in the derivative that are added. Bivariate Probit Conditional Means Prob[yi1 = 1,y i2 = 1] = Φ2 (β1xi1,β2 xi2 ,ρ) This is not a conditional mean. For a generic x that might appear in either index function, Prob[y i1 = 1,y i2 = 1] = gi1β1 + gi2β2 x i β x - ρβx βx - ρβ x 2 i2 1 i1 2 i2 ,gi2 = φ(β2 x i2 )Φ 1 i1 gi1 = φ(β1 x i1 )Φ 2 2 1- ρ 1- ρ The term in β1 is 0 if x i does not appear in x i1 and likewise for β2 . Φ (βx ,β x ,ρ) E[yi1 | x i1, x i2 ,yi2 = 1] = Prob[y i1 = 1| x i1, x i2 ,y i2 = 1] = 2 1 i1 2 i2 Φ(β2 x i2 ) E[yi1 | x i1, x i2 ,yi2 = 1] Φ (β x ,β x ,ρ)φ(β x ) 1 = gi1β1 + gi2β2 - 2 1 i1 2 i2 2 2 i2 β2 x i Φ(β2 x i2 ) [Φ(β2 x i2 )] gi1 gi2 Φ2 (β1 xi1,β2 xi2 ,ρ)φ(β2 xi2 ) = β + 1 β2 2 Φ( β x ) Φ( β x ) [Φ( β x )] 2 i2 2 i2 2 i2 Direct Effects Derivatives of E[y1|x1,x2,y2=1] wrt x1 +-------------------------------------------+ | 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] = .819898 | | Observations used for means are All Obs. | | These are the direct marginal effects. | +-------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ AGE .00382760 .00022088 17.329 .0000 43.5256898 FEMALE .08857260 .00519658 17.044 .0000 .47877479 EDUC -.00392413 .00093911 -4.179 .0000 11.3206310 MARRIED .00061108 .00506488 .121 .9040 .75861817 WORKING -.02280671 .00518908 -4.395 .0000 .67704750 HHNINC .000000 ......(Fixed Parameter)....... .35208362 HHKIDS .000000 ......(Fixed Parameter)....... .40273000 Indirect Effects Derivatives of E[y1|x1,x2,y2=1] wrt x2 +-------------------------------------------+ | 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] = .819898 | | Observations used for means are All Obs. | | These are the indirect marginal effects. | +-------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ AGE -.00035034 .697563D-04 -5.022 .0000 43.5256898 FEMALE -.00835397 .00150062 -5.567 .0000 .47877479 EDUC .000000 ......(Fixed Parameter)....... 11.3206310 MARRIED .000000 ......(Fixed Parameter)....... .75861817 WORKING .000000 ......(Fixed Parameter)....... .67704750 HHNINC .00216510 .00374879 .578 .5636 .35208362 HHKIDS .00034768 .00164160 .212 .8323 .40273000 Marginal Effects: Total Effects Sum of Two Derivative Vectors +-------------------------------------------+ | 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] = .819898 | | 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| +---------+--------------+----------------+--------+---------+----------+ AGE .00347726 .00022941 15.157 .0000 43.5256898 FEMALE .08021863 .00535648 14.976 .0000 .47877479 EDUC -.00392413 .00093911 -4.179 .0000 11.3206310 MARRIED .00061108 .00506488 .121 .9040 .75861817 WORKING -.02280671 .00518908 -4.395 .0000 .67704750 HHNINC .00216510 .00374879 .578 .5636 .35208362 HHKIDS .00034768 .00164160 .212 .8323 .40273000 Marginal Effects: Dummy Variables Using Differences of Probabilities +-----------------------------------------------------------+ | Analysis of dummy variables in the model. The effects are | | computed using E[y1|y2=1,d=1] - E[y1|y2=1,d=0] where d is | | the variable. Variances use the delta method. The effect | | accounts for all appearances of the variable in the model.| +-----------------------------------------------------------+ |Variable Effect Standard error t ratio (deriv) | +-----------------------------------------------------------+ FEMALE .079694 .005290 15.065 (.080219) MARRIED .000611 .005070 .121 (.000511) WORKING -.022485 .005044 -4.457 (-.022807) HHKIDS .000348 .001641 .212 (.000348) Computed using difference of probabilities Computed using scaled coefficients Simultaneous Equations A Simultaneous Equations Model Simultaneous Equations Model y1 * = β1x1 + γ1y 2 + ε1, y1 = 1(y1* > 0) y 2 * = β2 x 2 + γ 2 y1 + ε 2 ,y 2 = 1(y 2 * > 0) 0 1 ρ ε1 ~ N , ε 0 ρ 1 2 This model is not identified. (Not estimable. The computer can compute 'estimates' but they have no meaning.) bivariate probit;lhs=doctor,hospital ;rh1=one,age,educ,married,female,hospital ;rh2=one,age,educ,married,female,doctor$ Error 809: Fully simultaneous BVP model is not identified Fully Simultaneous ‘Model’ (Obtained by bypassing internal control) ---------------------------------------------------------------------FIML Estimates of Bivariate Probit Model Dependent variable DOCHOS Log likelihood function -20318.69455 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------|Index equation for DOCTOR Constant| -.46741*** .06726 -6.949 .0000 AGE| .01124*** .00084 13.353 .0000 43.5257 FEMALE| .27070*** .01961 13.807 .0000 .47877 EDUC| -.00025 .00376 -.067 .9463 11.3206 MARRIED| -.00212 .02114 -.100 .9201 .75862 WORKING| -.00362 .02212 -.164 .8701 .67705 HOSPITAL| 2.04295*** .30031 6.803 .0000 .08765 |Index equation for HOSPITAL Constant| -1.58437*** .08367 -18.936 .0000 AGE| -.01115*** .00165 -6.755 .0000 43.5257 FEMALE| -.26881*** .03966 -6.778 .0000 .47877 HHNINC| .00421 .08006 .053 .9581 .35208 HHKIDS| -.00050 .03559 -.014 .9888 .40273 DOCTOR| 2.04479*** .09133 22.389 .0000 .62911 |Disturbance correlation RHO(1,2)| -.99996*** .00048 ******** .0000 --------+------------------------------------------------------------- A Latent Simultaneous Equations Model Simultaneous Equations Model in the latent variables y1 * = β1 x1 + γ1y 2* + ε1, y1 = 1(y1* > 0) y 2 * = β2 x 2 + γ 2 y1* + ε 2 , y 2 = 1(y 2 * > 0) 0 1 ρ ε1 ε ~ N 0 , ρ 1 2 Note the underlying (latent) structural variables in each equation, not the observed binary variables. This model is identified. It is hard to interpret. It can be consistently estimated by two step methods. (Analyzed in Amemiya (1979) and Maddala (1983).) A Recursive Simultaneous Equations Model Recursive Simultaneous Equations Model y1 * = β1x1 + ε1, y1 = 1(y1* > 0) y 2 * = β2 x 2 + γ 2 y1 + ε 2 ,y 2 = 1(y 2 * > 0) 0 1 ρ ε1 ~ N , ε 0 ρ 1 2 This model is identified. It can be consistently and efficiently estimated by full information maximum likelihood. Treated as a bivariate probit model, ignoring the simultaneity. Ordered Choices Ordered Discrete Outcomes • • E.g.: Taste test, credit rating, course grade, preference scale Underlying random preferences: • • • • • Existence of an underlying continuous preference scale Mapping to observed choices Strength of preferences is reflected in the discrete outcome Censoring and discrete measurement The nature of ordered data Bond Ratings Health Satisfaction (HSAT) Self administered survey: Health Satisfaction (0 – 10) Continuous Preference Scale Modeling Ordered Choices • Random Utility (allowing a panel data setting) Uit = + ’xit + it = ait + • • it Observe outcome j if utility is in region j Probability of outcome = probability of cell Pr[Yit=j] = Prob[Yit < j] - Prob[Yit < j-1] = F(j – ait) - F(j-1 – ait) Ordered Probability Model y* βx , we assume x contains a constant term y 0 if y* 0 y = 1 if 0 < y* 1 y = 2 if 1 < y* 2 y = 3 if 2 < y* 3 ... y = J if J-1 < y* J In general : y = j if j-1 < y* j , j = 0,1,...,J -1 , o 0, J , j-1 j , j = 1,...,J-1 Combined Outcomes for Health Satisfaction Probabilities for Ordered Choices Prob[y=j]=Prob[ j-1 y* j ] = Prob[ j-1 βx j ] = Prob[βx j ] Prob[βx j1 ] = Prob[ j βx ] Prob[ j1 βx ] = F[ j βx ] F[ j1 βx] where F[] is the CDF of . Probabilities for Ordered Choices μ1 =1.1479 μ2 =2.5478 μ3 =3.0564 Coefficients What are the coefficients in the ordered probit model? There is no conditional mean function. Prob[y=j|x ] [f( j1 β'x) f( j β'x)] k x k Magnitude depends on the scale factor and the coefficient. Sign depends on the densities at the two points! What does it mean that a coefficient is "significant?" Effects of 8 More Years of Education An Ordered Probability Model for Health Satisfaction +---------------------------------------------+ | Ordered Probability Model | | Dependent variable HSAT | | Number of observations 27326 | | Underlying probabilities based on Normal | | Cell frequencies for outcomes | | Y Count Freq Y Count Freq Y Count Freq | | 0 447 .016 1 255 .009 2 642 .023 | | 3 1173 .042 4 1390 .050 5 4233 .154 | | 6 2530 .092 7 4231 .154 8 6172 .225 | | 9 3061 .112 10 3192 .116 | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Index function for probability Constant 2.61335825 .04658496 56.099 .0000 FEMALE -.05840486 .01259442 -4.637 .0000 .47877479 EDUC .03390552 .00284332 11.925 .0000 11.3206310 AGE -.01997327 .00059487 -33.576 .0000 43.5256898 HHNINC .25914964 .03631951 7.135 .0000 .35208362 HHKIDS .06314906 .01350176 4.677 .0000 .40273000 Threshold parameters for index Mu(1) .19352076 .01002714 19.300 .0000 Mu(2) .49955053 .01087525 45.935 .0000 Mu(3) .83593441 .00990420 84.402 .0000 Mu(4) 1.10524187 .00908506 121.655 .0000 Mu(5) 1.66256620 .00801113 207.532 .0000 Mu(6) 1.92729096 .00774122 248.965 .0000 Mu(7) 2.33879408 .00777041 300.987 .0000 Mu(8) 2.99432165 .00851090 351.822 .0000 Mu(9) 3.45366015 .01017554 339.408 .0000 Ordered Probit Partial Effects Partial effects at means of the data Average Partial Effect of HHNINC Predictions of the Model:Kids +----------------------------------------------+ |Variable Mean Std.Dev. Minimum Maximum | +----------------------------------------------+ |Stratum is KIDS = 0.000. Nobs.= 2782.000 | +--------+-------------------------------------+ |P0 | .059586 .028182 .009561 .125545 | |P1 | .268398 .063415 .106526 .374712 | |P2 | .489603 .024370 .419003 .515906 | |P3 | .101163 .030157 .052589 .181065 | |P4 | .081250 .041250 .028152 .237842 | +----------------------------------------------+ |Stratum is KIDS = 1.000. Nobs.= 1701.000 | +--------+-------------------------------------+ |P0 | .036392 .013926 .010954 .105794 | |P1 | .217619 .039662 .115439 .354036 | |P2 | .509830 .009048 .443130 .515906 | |P3 | .125049 .019454 .061673 .176725 | |P4 | .111111 .030413 .035368 .222307 | +----------------------------------------------+ |All 4483 observations in current sample | +--------+-------------------------------------+ |P0 | .050786 .026325 .009561 .125545 | |P1 | .249130 .060821 .106526 .374712 | |P2 | .497278 .022269 .419003 .515906 | |P3 | .110226 .029021 .052589 .181065 | |P4 | .092580 .040207 .028152 .237842 | +----------------------------------------------+ This is a restricted model with outcomes collapsed into 5 cells. Fit Measures • • • • There is no single “dependent variable” to explain. There is no sum of squares or other measure of “variation” to explain. Predictions of the model relate to a set of J+1 probabilities, not a single variable. How to explain fit? • • • Based on the underlying regression Based on the likelihood function Based on prediction of the outcome variable Log Likelihood Based Fit Measures Fit Measure Based on Counting Predictions This model always predicts the same cell. A Somewhat Better Fit Different Normalizations • NLOGIT • • • • Y = 0,1,…,J, U* = α + β’x + ε One overall constant term, α J-1 “thresholds;” μ-1 = -∞, μ0 = 0, μ1,… μJ-1, μJ = + ∞ Stata • • • Y = 1,…,J+1, U* = β’x + ε No overall constant, α=0 J “cutpoints;” μ0 = -∞, μ1,… μJ, μJ+1 = + ∞ α̂ μˆ j αˆ μˆ j - αˆ Generalizing the Ordered Probit with Heterogeneous Thresholds Index = βxi Threshold parameters Standard model : μ-1 = -, μ0 = 0, μ j > μ j-1 > 0, μJ = + Preference scale and thresholds are homogeneous A generalized model (Pudney and Shields, JAE, 2000) μij = α j + γ j zi Note the identification problem. If zik is also in xi (same variable) then μij - βxi = α j + γzik - βzik +... E.g., ( j + 1Age + 2Married) - ( + 1Age 2Sex ) ( j + 2Married) - ( + (1 1 ) Age 2Sex ) No longer clear if the variable is in x or z (or both) Differential Item Functioning People in this country are optimistic – they report this value as ‘very good.’ People in this country are pessimistic – they report this same value as ‘fair’ Panel Data • Fixed Effects • • • The usual incidental parameters problem Partitioning Prob(yit > j|xit) produces estimable binomial logit models. (Find a way to combine multiple estimates of the same β. Random Effects • Standard application Incidental Parameters Problem Table 9.1 Monte Carlo Analysis of the Bias of the MLE in Fixed Effects Discrete Choice Models (Means of empirical sampling distributions, N = 1,000 individuals, R = 200 replications) Random Effects Dynamic Ordered Probit Model Model for Self Assessed Health • British Household Panel Survey (BHPS) • • • • • Waves 1-8, 1991-1998 Self assessed health on 0,1,2,3,4 scale Sociological and demographic covariates Dynamics – inertia in reporting of top scale Dynamic ordered probit model • • Balanced panel – analyze dynamics Unbalanced panel – examine attrition Dynamic Ordered Probit Model Latent Regression - Random Utility h *it = xit + H i ,t 1 + i + it xit = relevant covariates and control variables It would not be appropriate to include hi,t-1 itself in the model as this is a label, not a measure H i ,t 1 = 0/1 indicators of reported health status in previous period Hi ,t 1 ( j ) = 1[Individual i reported h it j in previous period], j=0,...,4 Ordered Choice Observation Mechanism h it = j if j 1 < h*it j , j = 0,1,2,3,4 Ordered Probit Model - it ~ N[0,1] Random Effects with Mundlak Correction and Initial Conditions i = 0 1H i ,1 + 2 xi + u i , u i ~ N[0,2 ] Testing for Attrition Bias Three dummy variables added to full model with unbalanced panel suggest presence of attrition effects. Attrition Model with IP Weights Assumes (1) Prob(attrition|all data) = Prob(attrition|selected variables) (ignorability) (2) Attrition is an ‘absorbing state.’ No reentry. Obviously not true for the GSOEP data above. Can deal with point (2) by isolating a subsample of those present at wave 1 and the monotonically shrinking subsample as the waves progress. Inverse Probability Weighting Panel is based on those present at WAVE 1, N1 individuals Attrition is an absorbing state. No reentry, so N1 N2 ... N8. Sample is restricted at each wave to individuals who were present at the previous wave. d it = 1[Individual is present at wave t]. d i1 = 1 i, dit 0 d i ,t 1 0. xi1 covariates observed for all i at entry that relate to likelihood of being present at subsequent waves. (health problems, disability, psychological well being, self employment, unemployment, maternity leave, student, caring for family member, ...) Probit model for dit 1[xi1 wit ], t = 2,...,8. ˆ it fitted probability. t Assuming attrition decisions are independent, Pˆit s 1 ˆ is ˆ d it Inverse probability weight W it P̂it Weighted log likelihood logLW i 1 t 1 log Lit (No common effects.) N 8 Estimated Partial Effects by Model Partial Effect for a Category These are 4 dummy variables for state in the previous period. Using first differences, the 0.234 estimated for SAHEX means transition from EXCELLENT in the previous period to GOOD in the previous period, where GOOD is the omitted category. Likewise for the other 3 previous state variables. The margin from ‘POOR’ to ‘GOOD’ was not interesting in the paper. The better margin would have been from EXCELLENT to POOR, which would have (EX,POOR) change from (1,0) to (0,1).