Panel Data Binary Choice Models Microeconometric Modeling

1/62: Topic 2.3 Panel Data Binary Choice Models Microeconometric Modeling William Greene Stern School of Business New York University New York NY USA 2.3 Panel Data Models for Binary Choice

2/62: Topic 2.3 Panel Data Binary Choice Models Concepts Models Unbalanced Panel Attrition Bias Inverse Probability Weight Heterogeneity Population Averaged Model Clustering Pooled Model Quadrature Maximum Simulated Likelihood Conditional Estimator Incidental Parameters Problem Partial Effects Bias Correction Mundlak Specification Variable Addition Test Random Effects Progit Fixed Effects Probit Fixed Effects Logit Dynamic Probit Mundlak Formulation Correlated Random Effects Model

3/62: Topic 2.3 Panel Data Binary Choice Models

4/62: Topic 2.3 Panel Data Binary Choice Models Application: Health Care Panel Data German Health Care Usage Data Data downloaded from Journal of Applied Econometrics Archive. This is an unbalanced panel with 7,293 individuals. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary choice. There are altogether 27,326 observations. The number of observations ranges from 1 to 7. (Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987). Variables in the file are DOCTOR = 1(Number of doctor visits > 0) HOSPITAL = 1(Number of hospital visits > 0) HSAT = health satisfaction, coded 0 (low) - 10 (high) DOCVIS = number of doctor visits in last three months HOSPVIS = number of hospital visits in last calendar year PUBLIC = insured in public health insurance = 1; otherwise = 0 ADDON = insured by add-on insurance = 1; otherswise = 0 HHNINC = household nominal monthly net income in German marks / 10000. (4 observations with income=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC = years of schooling AGE = age in years MARRIED = marital status

5/62: Topic 2.3 Panel Data Binary Choice Models Unbalanced Panels Most theoretical results are for balanced panels. Most real world panels are unbalanced. Often the gaps are caused by attrition. The major question is whether the gaps are missing completely at random. If not, the observation mechanism is endogenous, and at least some methods will produce questionable results. Researchers rarely have any reason to treat the data as nonrandomly sampled. (This is good news.) Group Sizes

6/62: Topic 2.3 Panel Data Binary Choice Models Unbalanced Panels and Attrition Bias Test for attrition bias. (Verbeek and Nijman, Testing for Selectivity Bias in Panel Data Models, International Economic Review, 1992, 33, 681-703. Variable addition test using covariates of presence in the panel Nonconstructive what to do next? Do something about attrition bias. (Wooldridge, Inverse Probability Weighted M-Estimators for Sample Stratification and Attrition, Portuguese Economic Journal, 2002, 1: 117-139) Stringent assumptions about the process Model based on probability of being present in each wave of the panel

7/62: Topic 2.3 Panel Data Binary Choice Models

8/62: Topic 2.3 Panel Data Binary Choice Models Inverse Probability Weighting Panel is based on those present at WAVE 1, N individuals Attrition is an absorbing state. No reentry, so N Sample is restricted at each wave to individuals who were present at the pre vious wave. d = 1[Individual is present at wave t]. d = 1 i, d 0 d 0. covariates observed for all i at entry that relate to likelihood of being present at subseque nt waves. (health problems, disability, psychological well being, self employment, unemployment, maternity leave, student, caring for family member, ...) Probit model for d 1[ = it x1 ], t = 2,...,T. + i it w 1 N ... N . 1 2 T it = = + 1 , 1 i it i t = ix 1 = it fitted probability. is s t = Assuming attrition decisions are independent, P it = 1 d P = it Inverse probability weight W it it i N T = Weighted log likelihood logL log (No commo n effects.) L W it = = 1 1 t

9/62: Topic 2.3 Panel Data Binary Choice Models

10/62: Topic 2.3 Panel Data Binary Choice Models Panel Data Binary Choice Models Random Utility Model for Binary Choice Uit = + xit + it + Person i specific effect Fixed effects using dummy variables Uit = i + xit + it Random effects using omitted heterogeneity Uit = + xit + it + ui Same outcome mechanism: Yit = 1[Uit > 0]

11/62: Topic 2.3 Panel Data Binary Choice Models Ignoring Unobserved Heterogeneity + = Assuming strict exogeneity; Cov( y *= u Prob[y Using the same model format: x ,u ) 0 it i it it + 1| x ] + x it i it it = = + it Prob[u - x ] it it i ) ( it it 2 u = = = Prob[y 1| x ] F x / 1+ F( x ) it it This is the 'population averaged model.'

12/62: Topic 2.3 Panel Data Binary Choice Models

13/62: Topic 2.3 Panel Data Binary Choice Models Ignoring Heterogeneity in the RE Model Ignoring heterogeneity, we estimate not . Partial effects are f( is underestimated, but f( Which way does it go? Maybe ignoring u is ok? Not if we want to compute probabilities or do statistical inference about errors will be too small. it it x ) not f( x x ) it ) is overestimated. Estimated standard . Scale factor is too large is too small

14/62: Topic 2.3 Panel Data Binary Choice Models = Age 43.527 P ( .37176 = + = .01625Age).01625 .006128 Age ( ) P Age = 1 x 1 RE ( ) ( .53689 + .02338Age) 1 .45298 .02338 = 1 .45298 = .00648 Population average coefficient is off by 43% but the partial effect is off by 6%

15/62: Topic 2.3 Panel Data Binary Choice Models Ignoring Heterogeneity (Broadly) Presence will generally make parameter estimates look smaller than they would otherwise. Ignoring heterogeneity will definitely distort standard errors. Partial effects based on the parametric model may not be affected very much. Is the pooled estimator robust? Less so than in the linear model case.

16/62: Topic 2.3 Panel Data Binary Choice Models Pooled vs. RE Panel Estimator ---------------------------------------------------------------------- Binomial Probit Model Dependent variable DOCTOR --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- Constant| .02159 .05307 .407 .6842 AGE| .01532*** .00071 21.695 .0000 43.5257 EDUC| -.02793*** .00348 -8.023 .0000 11.3206 HHNINC| -.10204** .04544 -2.246 .0247 .35208 --------+------------------------------------------------------------- Unbalanced panel has 7293 individuals --------+------------------------------------------------------------- Constant| -.11819 .09280 -1.273 .2028 AGE| .02232*** .00123 18.145 .0000 43.5257 EDUC| -.03307*** .00627 -5.276 .0000 11.3206 HHNINC| .00660 .06587 .100 .9202 .35208 Rho| .44990*** .01020 44.101 .0000 --------+-------------------------------------------------------------

17/62: Topic 2.3 Panel Data Binary Choice Models Partial Effects ---------------------------------------------------------------------- Partial derivatives of E[y] = F[*] with respect to the vector of characteristics They are computed at the means of the Xs Observations used for means are All Obs. --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Elasticity --------+------------------------------------------------------------- |Pooled AGE| .00578*** .00027 21.720 .0000 .39801 EDUC| -.01053*** .00131 -8.024 .0000 -.18870 HHNINC| -.03847** .01713 -2.246 .0247 -.02144 --------+------------------------------------------------------------- |Based on the panel data estimator AGE| .00620*** .00034 18.375 .0000 .42181 EDUC| -.00918*** .00174 -5.282 .0000 -.16256 HHNINC| .00183 .01829 .100 .9202 .00101 --------+-------------------------------------------------------------

18/62: Topic 2.3 Panel Data Binary Choice Models Random Effects + = Assuming strict exogeneity; Cov( y *= + u Prob[y 1| ] = x x ,u ) 0 it i it it + + = x it i it it + - - Prob[u x ] it it i it ) ( it it 2 u = = = + Prob[y 1| x ] F + x / 1+ F( x ) it it 0 This is the 'population averaged model.' Prob[y 1| = x it i ,u] Prob[y 1| ,u] This is the structural model it = = x - - + Prob[ F + x - v ] it it u i ( ) it = x v it it i u i

19/62: Topic 2.3 Panel Data Binary Choice Models Quadrature Butler and Moffitt (1982) Th is method is used in most commerical software since 198 2 ( ) v = T N + x + logL l og F(y , v ) dv i it it u i i i = i 1 = t 1 2 1 2 -v 2 2 u u ~ N[0, = where v ~ N[0,1] ] N = log g( v ) exp dv i i = i 1 v (make a change of variable to w = v/ 2 i 1 u i 1 ( ) N 2 = l og g( 2w) exp -w dw i i = The integral can be 1 computed using Hermite quadr ature. N H log w g( 2z ) h h = i 1 = h 1 The values tab les such as A of w (weights) and z (node s) are found i n published h bramovitz and Stegun (or on the web). H is by choice. Higher H produces greater accuracy (but takes longer). h

20/62: Topic 2.3 Panel Data Binary Choice Models Quadrature Log Likelihood After all the substitutions, the function to be maximized: Not simple, but 1 logL log feasib l e. ( ) N H T = + + w F(y , x 2 z ) i h it it u h = i 1 = = h 1 t 1 1 N H T = + + lo g w F(y , x z ) i h it it h = i 1 = = h 1 t 1

21/62: Topic 2.3 Panel Data Binary Choice Models Simulation Based Estimator ( ) v T N = + x + logL log F(y , v ) dv i it it u i i i = i 1 = t 1 2 -v 1 2 N = log g(v ) ex p dv i i i 2 = i 1 N T his equals log E[ g( v ) ] i = i 1 The expected value of the functio by dr aw ing R random draws v from the population N[0,1] and averaging the R functions of v . We maxi 1 logL log R n of v can be a ppro xima te d i i r m ze i ir iT t 1 N R = + x + F(y , v ) S it it u ir = i 1 = r 1 =

22/62: Topic 2.3 Panel Data Binary Choice Models Random Effects Model: Quadrature ---------------------------------------------------------------------- Random Effects Binary Probit Model Dependent variable DOCTOR Log likelihood function -16290.72192 Restricted log likelihood -17701.08500 Chi squared [ 1 d.f.] 2820.72616 Estimation based on N = 27326, K = 5 Unbalanced panel has 7293 individuals --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- Constant| -.11819 .09280 -1.273 .2028 AGE| .02232*** .00123 18.145 .0000 43.5257 EDUC| -.03307*** .00627 -5.276 .0000 11.3206 HHNINC| .00660 .06587 .100 .9202 .35208 Rho| .44990*** .01020 44.101 .0000 --------+------------------------------------------------------------- |Pooled Estimates Constant| .02159 .05307 .407 .6842 AGE| .01532*** .00071 21.695 .0000 43.5257 EDUC| -.02793*** .00348 -8.023 .0000 11.3206 HHNINC| -.10204** .04544 -2.246 .0247 .35208 --------+------------------------------------------------------------- Random Effects Pooled

23/62: Topic 2.3 Panel Data Binary Choice Models Random Parameter Model ---------------------------------------------------------------------- Random Coefficients Probit Model Dependent variable DOCTOR (Quadrature Based) Log likelihood function -16296.68110 (-16290.72192) Restricted log likelihood -17701.08500 Chi squared [ 1 d.f.] 2808.80780 Simulation based on 50 Halton draws --------+------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] --------+------------------------------------------------- |Nonrandom parameters AGE| .02226*** .00081 27.365 .0000 ( .02232) EDUC| -.03285*** .00391 -8.407 .0000 (-.03307) HHNINC| .00673 .05105 .132 .8952 ( .00660) |Means for random parameters Constant| -.11873** .05950 -1.995 .0460 (-.11819) |Scale parameters for dists. of random parameters Constant| .90453*** .01128 80.180 .0000 --------+------------------------------------------------------------- Using quadrature, a = -.11819. Implied from these estimates is .904542/(1+.904532) = .449998 compared to .44990 using quadrature.

24/62: Topic 2.3 Panel Data Binary Choice Models A Dynamic Model it = + + + it y Two similar 'effects' Unobserved heterogeneity State dependence = state 'persistence' Pr(y 1| y ,...,y ,x ,u] 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. 1[ x y u > 0] it i,t 1 i it = = + + F[ x y u] it i,t 1 i0 it i,t 1 i

25/62: Topic 2.3 Panel Data Binary Choice Models Dynamic Probit Model: A Standard Approach (1) Conditioned on all effects, joint probability T it = + + P(y ,y ,...,y |y , x ,u ) F( x y u ,y ) i1 i2 iT i0 i i i,t 1 i it = t 1 (2) Unconditional density; integrate out the common effect (3) Density for heterogeneity h(u |y , ) N[ y u = y x (4) Reduced form P(y ,y ,...,y |y , ) i x = P(y ,y ,...,y |y , x ) P(y ,y ,...,y |y , x ,u )h(u |y , x )du i1 i2 iT i0 i i1 i2 iT i0 i i i i0 i i i 2 u = + + + x x , ], w (contains every x = [ x x , ,..., x ], so period of i i0 i i0 i i1 i2 iT i + + x ) i i0 u i it = i1 i2 iT i0 T it i + + + + + F( x y y x w ,y )h(w )dw i,t 1 i0 u i it i i = t 1 This is a random effects model

26/62: Topic 2.3 Panel Data Binary Choice Models Simplified Dynamic Model Projecting u on all observations expands the model enormously. (3) Projection of heterogeneity only on group means h(u | y , ) N[ y u = y + w (4) Re = t 1 Mundlak style correction with the initial value in the equation. This is (again) a random effects mo i i 2 u = + + + x x x , ] so i i0 i i0 i + i i0 i duced form P(y ,y ,...,y | y , x ) i1 i2 iT i0 i T it i + + + + + F( x y y x w ,y )h(w )dw i,t 1 i0 u i it i i = del

27/62: Topic 2.3 Panel Data Binary Choice Models A Dynamic Model for Public Insurance Age Household Income Kids in the household Health Status Add initial value, lagged value, group means

28/62: Topic 2.3 Panel Data Binary Choice Models Dynamic Common Effects Model

29/62: Topic 2.3 Panel Data Binary Choice Models Application Stewart, JAE, 2007 British Household Panel Survey (1991-1996) 3060 households retained (balanced) out of 4739 total. Unemployment indicator (0.1) Data features Panel data unobservable heterogeneity State persistence: Someone unemployed at t-1 is more than 20 times as likely to be unemployed at t as someone employed at t-1.

30/62: Topic 2.3 Panel Data Binary Choice Models Application: Direct Approach

31/62: Topic 2.3 Panel Data Binary Choice Models GHK Simulation/Estimation The presence of the autocorrelation and state dependence in the model invalidate the simple maximum likelihood procedures we examined earlier. The appropriate likelihood function is constructed by formulating the probabilities as Prob( yi,0, yi,1, . . .) = Prob(yi,0) Prob(yi,1 | yi,0) Prob(yi,T| yi,T-1) . This still involves a T = 7 order normal integration, which is approximated in the study using a simulator similar to the GHK simulator.

32/62: Topic 2.3 Panel Data Binary Choice Models Problems with Dynamic RE Probit Assumes yi,0 and the effects are uncorrelated Assumes the initial conditions are exogenous OK if the process and the observation begin at the same time, not if different. Doesn t allow time invariant variables in the model. The normality assumption in the projection.

33/62: Topic 2.3 Panel Data Binary Choice Models Distributional Problem Normal distributions assumed throughout Normal distribution for the unique component, i,t Normal distribution assumed for the heterogeneity, ui Sensitive to the distribution? Alternative: Discrete distribution for ui. Heckman and Singer style, latent class model. Conventional estimation methods. Why is the model not sensitive to normality for i,t but it is sensitive to normality for ui?

34/62: Topic 2.3 Panel Data Binary Choice Models Fixed Effects Modeling Advantages: Allows correlation of covariates and heterogeneity Disadvantages: Complications of computing all those dummy variable coefficients solved problem (Greene, 2004) No time invariant variables not solvable in the FE context Incidental Parameters problem persistent small T bias, does not go away Strategies Unconditional estimation Conditional estimation Rasch/Chamberlain Hybrid conditional/unconditional estimation Bias corrrections Mundlak estimator

35/62: Topic 2.3 Panel Data Binary Choice Models Fixed Effects Models Estimate with dummy variable coefficients Uit = i+ xit+ it Can be done by brute force for 10,000s of individuals N T = + x log log ( , ) L F y i = it i it = 1 1 i t F(.) = appropriate probability for the observed outcome Compute and i for i=1, ,N (may be large) See FixedEffects.pdf in course materials.

36/62: Topic 2.3 Panel Data Binary Choice Models Unconditional Estimation Maximize the whole log likelihood Difficult! Many (thousands) of parameters. Feasible NLOGIT (2004) ( Brute force )

37/62: Topic 2.3 Panel Data Binary Choice Models Fixed Effects Health Model Groups in which yit is always = 0 or always = 1. Cannot compute i.

38/62: Topic 2.3 Panel Data Binary Choice Models Conditional Estimation Principle: f(yi1,yi2, | some statistic) is free of the fixed effects for some models. Maximize the conditional log likelihood, given the statistic. Can estimate without having to estimate i. Only feasible for the logit model. (Poisson and a few other continuous variable models. No other discrete choice models.)

39/62: Topic 2.3 Panel Data Binary Choice Models Binary Logit Conditional Probabilities it + x e + i = = x Prob( 1| ) . y it it it + x 1 e i T i = = = Prob , , , Y y Y y Y y y 1 1 2 2 i i i i iT iT it i i = 1 t T T i i it it x x exp exp y y it it = = 1 1 t t = = . T T i i it i x x exp exp d d it T S can equal S i All ways that different it it t = t i d S t i i = = 1 1 t t d t i Denominator is summed over all the different combinations of T values of y that sum to the same sum as the observed T i T t=1 . If S is this sum, y i it i it there are terms. May be a huge number. An algorithm by Krailo S i and Pike makes it simple.

40/62: Topic 2.3 Panel Data Binary Choice Models Example: Two Period Binary Logit it + x e + i = = Prob(y 1| ) . x it it it + x 1 e i T i it exp y x it T i t 1 = = = = = Prob Y y , Y y , , Y y y ,data . i1 i1 i2 i2 iT iT it T i i i t 1 = it exp d x it = t it d S i t 1 = 2 = = = = Prob Y 0, Y 0 y 0,data 1. i1 i2 it t 1 = 2 i1 exp( i1 x x ) = = = = Prob Y 1, Y 0 y 1,data i1 i 2 it i2 + exp( ) exp( i2 x x ) = t 1 2 exp( i1 x ) = = = = Prob Y 0, Y 1 y 1,data i1 i2 it i2 + exp( ) exp( x ) = t 1 2 = = = = Prob Y 1, Y 1 y 2,data 1. i1 i2 it t 1 =

41/62: Topic 2.3 Panel Data Binary Choice Models Example: Seven Period Binary Logit y X Prob[ = (1,0,0,0,1,1,1)| exp( 1 exp( There are 35 different sequences of y (permutations) that sum to 4. ]= 1 + i + + x x ) exp( 1 exp( + ) ... 1 7 x i + i + + ) 1 exp( + x x ) ) 1 2 7 i i i it * it p= For example, y might b e (1,1,1,1,0,0,0). Etc. | 1 7 t x exp y = 1 it it 7 t X Prob[y=(1,0,0,0,1,1,1)| , y =4] = it = 1 i 35 7 t * it p x exp y = 1 | it = 1 p

42/62: Topic 2.3 Panel Data Binary Choice Models

43/62: Topic 2.3 Panel Data Binary Choice Models With T = 50, the number of permutations of sequences of y ranging from sum = 0 to sum = 50 ranges from 1 for 0 and 50, to 2.3 x 10 for 15 or 35 up to a maximum of 1.3 x 10 for sum =25. These are the numbers of terms that must be summed for a model with T = 50. In the application below, the sum ranges from 15 to 35. 12 14

44/62: Topic 2.3 Panel Data Binary Choice Models The sample is 200 individuals each observed 50 times.

45/62: Topic 2.3 Panel Data Binary Choice Models The data are generated from a probit process with b1 = b2 = .5. But, it is fit as a logit model. The coefficients obey the familiar relationship, 1.6*probit.

46/62: Topic 2.3 Panel Data Binary Choice Models Estimating Partial Effects The fixed effects logit estimator of immediately gives us the effect of each element of xi on the log-odds ratio Unfortunately, we cannot estimate the partial effects unless we plug in a value for i. Because the distribution of i is unrestricted in particular, E[ i] is not necessarily zero it is hard to know what to plug in for i. In addition, we cannot estimate average partial effects, as doing so would require finding E[ (xit + i)], a task that apparently requires specifying a distribution for i. (Wooldridge, 2010)

47/62: Topic 2.3 Panel Data Binary Choice Models Fixed Effects Logit Health Model: Conditional vs. Unconditional

48/62: Topic 2.3 Panel Data Binary Choice Models Incidental Parameters Problems: Conventional Wisdom General: The unconditional MLE is biased in samples with fixed T except in special cases such as linear or Poisson regression (even when the FEM is the right model). The conditional estimator (that bypasses estimation of i) is consistent. Specific: Upward bias (experience with probit and logit) in estimators of

49/62: Topic 2.3 Panel Data Binary Choice Models

50/62: Topic 2.3 Panel Data Binary Choice Models A Monte Carlo Study of the FE Estimator: Probit vs. Logit Estimates of Coefficients and Marginal Effects at the Implied Data Means Results are scaled so the desired quantity being estimated ( , , marginal effects) all equal 1.0 in the population.