Report

Discrete Choice Modeling William Greene Stern School of Business New York University Part 5 Panel Data Models Application: Health Care Panel Data German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods 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 HOSPITAL HSAT DOCVIS HOSPVIS PUBLIC ADDON HHNINC = = = = = = = = HHKIDS EDUC AGE MARRIED = = = = 1(Number of doctor visits > 0) 1(Number of hospital visits > 0) health satisfaction, coded 0 (low) - 10 (high) number of doctor visits in last three months number of hospital visits in last calendar year insured in public health insurance = 1; otherwise = 0 insured by add-on insurance = 1; otherswise = 0 household nominal monthly net income in German marks / 10000. (4 observations with income=0 were dropped) children under age 16 in the household = 1; otherwise = 0 years of schooling age in years marital status 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. Group Sizes Researchers rarely have any reason to treat the data as nonrandomly sampled. (This is good news.) 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. Do something about attrition bias. (Wooldridge, Inverse Probability Weighted M-Estimators for Sample Stratification and Attrition, Portuguese Economic Journal, 2002, 1: 117-139) Variable addition test using covariates of presence in the panel Nonconstructive – what to do next? Stringent assumptions about the process Model based on probability of being present in each wave of the panel We return to these in discussion of applications of ordered choice models Panel Data Models Benefits Modeling heterogeneity Rich specifications Modeling dynamic effects in individual behavior Costs More complex models and estimation procedures Statistical issues for identification and estimation Fixed and Random Effects Model: Feature of interest yit Probability distribution or conditional mean Observable covariates xit, zi Individual specific heterogeneity, ui Probability or mean, f(xit,zi,ui) Random effects: E[ui|xi1,…,xiT,zi] = 0 Fixed effects: E[ui|xi1,…,xiT,zi] = g(Xi,zi). The difference relates to how ui relates to the observable covariates. Household Income We begin by analyzing Income using linear regression. Fixed and Random Effects in Regression yit = ai + b’xit + eit How do we proceed for a binary choice model? Random effects: Two step FGLS. First step is OLS Fixed effects: OLS based on group mean differences yit* = ai + b’xit + eit yit = 1 if yit* > 0, 0 otherwise. Neither ols nor two step FGLS works (even approximately) if the model is nonlinear. Models are fit by maximum likelihood, not OLS or GLS New complications arise that are absent in the linear case. Pooled Linear Regression - Income ---------------------------------------------------------------------Ordinary least squares regression ............ LHS=HHNINC Mean = .35208 Standard deviation = .17691 Number of observs. = 27326 Model size Parameters = 2 Degrees of freedom = 27324 Residuals Sum of squares = 796.31864 Standard error of e = .17071 Fit R-squared = .06883 Adjusted R-squared = .06879 Model test F[ 1, 27324] (prob) = 2019.6(.0000) --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------Constant| .12609*** .00513 24.561 .0000 EDUC| .01996*** .00044 44.940 .0000 11.3206 --------+------------------------------------------------------------- Fixed Effects ---------------------------------------------------------------------Least Squares with Group Dummy Variables.......... Ordinary least squares regression ............ LHS=HHNINC Mean = .35208 Standard deviation = .17691 Number of observs. = 27326 Model size Parameters = 7294 Degrees of freedom = 20032 Residuals Sum of squares = 277.15841 Standard error of e = .11763 Fit R-squared = .67591 Adjusted R-squared = .55791 Model test F[***, 20032] (prob) = 5.7(.0000) --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------EDUC| .03664*** .00289 12.688 .0000 11.3206 --------+------------------------------------------------------------For the pooled model, R squared was .06883 and the estimated coefficient on EDUC was .01996. Random Effects ---------------------------------------------------------------------Random Effects Model: v(i,t) = e(i,t) + u(i) Estimates: Var[e] = .013836 Var[u] = .015308 Corr[v(i,t),v(i,s)] = .525254 Lagrange Multiplier Test vs. Model (3) =******* ( 1 degrees of freedom, prob. value = .000000) (High values of LM favor FEM/REM over CR model) Baltagi-Li form of LM Statistic = 4534.78 Sum of Squares 796.363710 R-squared .068775 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------EDUC| .02051*** .00069 29.576 .0000 11.3206 Constant| .11973*** .00808 14.820 .0000 --------+------------------------------------------------------------Note: ***, **, * = Significance at 1%, 5%, 10% level. ---------------------------------------------------------------------For the pooled model, the estimated coefficient on EDUC was .01996. Fixed vs. Random Effects Linear Models Fixed Effects Robust to both cases Use OLS Convenient Random Effects Inconsistent in FE case: effects correlated with X Use FGLS: No necessary distributional assumption Smaller number of parameters Inconvenient to compute Nonlinear Models Fixed Effects Usually inconsistent because of ‘IP’ problem Fit by full ML Extremely inconvenient Random Effects Inconsistent in FE case : effects correlated with X Use full ML: Distributional assumption Smaller number of parameters Always inconvenient to compute Binary Choice Model Model is Prob(yit = 1|xit) (zi is embedded in xit) In the presence of heterogeneity, Prob(yit = 1|xit,ui) = F(xit,ui) 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] Ignoring Unobserved Heterogeneity Assuming strict exogeneity; Cov(x it ,ui it ) 0 y it *=xit β ui it Prob[y it 1 | x it ] Prob[ui it -xitβ] Using the same model format: Prob[y it 1 | x it ] F xitβ / 1+u2 F( x it δ) This is the 'population averaged model.' Ignoring Heterogeneity in the RE Model Ignoring heterogeneity, we estimate δ not β. Partial effects are δ f( x it δ) not βf( x itβ) β is underestimated, but f( x itβ) is overestimated. Which way does it go? Maybe ignoring u is ok? Not if we want to compute probabilities or do statistical inference about β. Estimated standard errors will be too small. 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. Pooled vs. A 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 --------+------------------------------------------------------------- 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 --------+------------------------------------------------------------- Effect of Clustering Yit must be correlated with Yis across periods Pooled estimator ignores correlation Broadly, yit = E[yit|xit] + wit, E[yit|xit] = Prob(yit = 1|xit) wit is correlated across periods Assuming the marginal probability is the same, the pooled estimator is consistent. (We just saw that it might not be.) Ignoring the correlation across periods generally leads to underestimating standard errors. ‘Cluster’ Corrected Covariance Matrix C the number if clusters n c number of observations in cluster c H1 = negative inverse of second derivatives matrix gic = derivative of log density for observation C V H C 1 1 C nc c 1 i 1 gic 1 i 1 gic H nc Cluster Correction: Doctor ---------------------------------------------------------------------Binomial Probit Model Dependent variable DOCTOR Log likelihood function -17457.21899 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------| Conventional Standard Errors Constant| -.25597*** .05481 -4.670 .0000 AGE| .01469*** .00071 20.686 .0000 43.5257 EDUC| -.01523*** .00355 -4.289 .0000 11.3206 HHNINC| -.10914** .04569 -2.389 .0169 .35208 FEMALE| .35209*** .01598 22.027 .0000 .47877 --------+------------------------------------------------------------| Corrected Standard Errors Constant| -.25597*** .07744 -3.305 .0009 AGE| .01469*** .00098 15.065 .0000 43.5257 EDUC| -.01523*** .00504 -3.023 .0025 11.3206 HHNINC| -.10914* .05645 -1.933 .0532 .35208 FEMALE| .35209*** .02290 15.372 .0000 .47877 --------+------------------------------------------------------------- Modeling a Binary Outcome Did firm i produce a product or process innovation in year t ? yit : 1=Yes/0=No Observed N=1270 firms for T=5 years, 1984-1988 Observed covariates: xit = Industry, competitive pressures, size, productivity, etc. How to model? Binary outcome Correlation across time Heterogeneity across firms Application: Innovation A Random Effects Model U it xit ui , u i ~ N [0, u ] Ti = observations on individual i For each period, yit 1[U it 0] (given u i ) Joint probability for Ti observations is Prob( yi1 , yi 2 ,...) t 1 F ( yit , xit ui ) Ti For convenience, write u i = u vi , vi ~ N [0,1] T N log L | v1 ,...vN i i log t i 1 F ( yit , xit u vi ) It is not possible to maximize log L | v1 ,...vN because of the unobserved random effects. A Computable Log Likelihood The unobserved heterogeneity is averaged out Ti log L i 1 log t 1 F ( yit , xit u vi ) f vi dvi Maximize this function with respect to ,, u . N How to compute the integral? (1) Analytically? No, no formula exists. (2) Approximately, using Gauss-Hermite quadrature (3) Approximately using Monte Carlo simulation Quadrature – Butler and Moffitt This method is used in most commerical software since 1982 T logL i1 log t i 1 F(y it , x it u v i ) v i dv i -v 2 N 1 = i1 log g( v ) exp dv i 2 2 N (make a change of variable to w = v/ 2 = 1 i1 log g( 2w) exp -w2 dwi N The integral can be computed using Hermite quadrature. 1 N H log w hg( 2zh ) i1 h 1 The values of w h (weights) and zh (nodes) are found in published tables such as Abramovitz and Stegun (or on the web). H is by choice. Higher H produces greater accuracy (but takes longer). Quadrature Log Likelihood After all the substitutions, the function to be maximized: Not simple, but feasible. Ti N 1 H logL i1 log w F(y , x 2 z ) h it it u h t 1 h1 Simulation N N Ti logL i1 log t 1 F(y it , x it u v i ) v i dv i = i1 log g(v i ) This equals N i1 -v i2 exp dv i 2 2 1 log E[g( v i )] The expected value of the function of v i can be approximated by drawing R random draws v ir from the population N[0,1] and averaging the R functions of v ir . We maximize logL S i1 log N 1 R Ti F(y it , x it u v ir ) r 1 t 1 R Random Effects Model ---------------------------------------------------------------------Random Effects Binary Probit Model Dependent variable DOCTOR Log likelihood function -16290.72192 Random Effects Restricted log likelihood -17701.08500 Pooled Chi squared [ 1 d.f.] 2820.72616 Significance level .00000 McFadden Pseudo R-squared .0796766 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 using the Butler and Moffitt method 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 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. Fixed Effects Models Uit = i + ’xit + it For the linear model, i and (easily) estimated separately using least squares For most nonlinear models, it is not possible to condition out the fixed effects. (Mean deviations does not work.) Even when it is possible to estimate without i, in order to compute partial effects, predictions, or anything else interesting, some kind of estimate of i is still needed. Fixed Effects Models Estimate with dummy variable coefficients Uit = i + ’xit + it Can be done by “brute force” even for 10,000s of individuals log L i 1 N Ti t 1 log F ( yit , i xit ) F(.) = appropriate probability for the observed outcome Compute and i for i=1,…,N (may be large) Unconditional Estimation Maximize the whole log likelihood Difficult! Many (thousands) of parameters. Feasible – NLOGIT (2001) (‘Brute force’) (One approach is just to create the thousands of dummy variables – SAS.) Fixed Effects Health Model Groups in which yit is always = 0 or always = 1. Cannot compute αi. 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 β without having to estimate αi. Only feasible for the logit model. (Poisson and a few other continuous variable models. No other discrete choice models.) Binary Logit Conditional Probabiities ei xit Prob( yit 1| xit ) . i xit 1 e Ti Prob Yi1 yi1 , Yi 2 yi 2 , , YiTi yiTi yit t 1 Ti Ti exp yit xit exp yit xit β t 1 t 1 Ti Ti Ti exp d x exp d x β All different ways that t d it S i it it it it Si t 1 t 1 t dit can equal Si Denominator is summed over all the different combinations of Ti values of yit that sum to the same sum as the observed Tt=1i yit . If Si is this sum, T there are terms. May be a huge number. An algorithm by Krailo Si and Pike makes it simple. Example: Two Period Binary Logit e i xitβ Prob(y it 1 | xit ) . 1 e i xitβ Prob Yi1 y i1 , Yi2 y i2 , Prob Yi1 Prob Yi1 Prob Yi1 Prob Yi1 , YiTi y iTi y 0 , data it t 1 2 1, Yi2 0 y it 1 , data t 1 2 0, Yi2 1 y it 1 , data t 1 2 1, Yi2 1 y it 2 , data t 1 0, Yi2 0 2 Ti exp y x it it Ti t 1 y it , data . Ti t 1 exp d x tdit Si it it t 1 1. exp( x i1β) exp( x i1β) exp( x i2β) exp( x i2β) exp( x i1β) exp( x i2β) 1. Comments on Enumeration in the Logit Model " This can easily be generalized for any T. However the computations rise geometrically with T and are excessive for T > 10. See Greene (1993)." (Baltagi, Panel Data, 1st edition) "If T is large, getting ... can be cumbersome as one can guess from 3.5 with T = 3." (M.J.Lee, "Panel Data Econometrics") (Both unaware of Krailo and Pike...) 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 αi. Because the distribution of αi is unrestricted – in particular, E[αi] is not necessarily zero – it is hard to know what to plug in for αi. In addition, we cannot estimate average partial effects, as doing so would require finding E[Λ(xit + αi)], a task that apparently requires specifying a distribution for αi.” (Wooldridge, 2002) Binary Logit Estimation Estimate by maximizing conditional logL Estimate i by using the ‘known’ in the FOC for the unconditional logL ˆ x ) exp( i it ˆ ( y P ) 0, P t 1 it it it 1 exp(i ˆ xit ) Ti Solve for the N constants, one at a time treating as known. No solution when yit sums to 0 or Ti “Works” if E[i|Σiyit] = E[i]. Use the average of the estimates of i for E[i]. Works if the cases of Σiyit = 0 or Σiyit = T occur completely at random. Use this average to compute predictions and partial effects. Logit Constant Terms Step 1. Estimate β with Chamberlain's conditional estimator Step 2. Treating β as if it were known, estimate i from the first order condition 1 yi Ti Ti t 1 ˆ e i e xitβ ˆ 1 Ti ic it 1 t 1 1 c T i it i Ti 1 e i e xitβ Estimate i 1 / exp(i ) i log i c it t 1 c i it Ti ˆ) is treated as known data. c it exp( x it β Solve one equation in one unknown for each i. Note there is no solution if y i = 0 or 1. Iterating back and forth does not maximize logL. Fixed Effects Logit Health Model: Conditional vs. Unconditional Advantages and Disadvantages of the FE Model Advantages Allows correlation of effect and regressors Fairly straightforward to estimate Simple to interpret Disadvantages Model may not contain time invariant variables Not necessarily simple to estimate if very large samples (Stata just creates the thousands of dummy variables) The incidental parameters problem: Small T bias 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 αi) is consistent. Specific: Upward bias (experience with probit and logit) in estimators of . Exactly 100% when T = 2. Declines as T increases. Some Familiar Territory – 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. A Monte Carlo Study of the FE Probit Estimator Percentage Biases in Estimates of Coefficients, Standard Errors and Marginal Effects at the Implied Data Means Bias Correction Estimators Motivation: Undo the incidental parameters bias in the fixed effects probit model: Advantages (1) Maximize a penalized log likelihood function, or (2) Directly correct the estimator of β For (1) estimates αi so enables partial effects Estimator is consistent under some circumstances (Possibly) corrects in dynamic models Disadvantage No time invariant variables in the model Practical implementation Extension to other models? (Ordered probit model (maybe) – see JBES 2009) A Mundlak Correction for the FE Model “Correlated Random Effects” Fixed Effects Model : y*it i xit it ,i = 1,...,N; t = 1,...,Ti yit 1 if yit > 0, 0 otherwise. Mundlak (Wooldridge, Heckman, Chamberlain),... i xi ui (Projection, not necessarily conditional mean) where u is normally distributed with mean zero and standard deviation u and is uncorrelated with xi or (xi1 , xi 2 ,..., xiT ) Reduced form random effects model y*it xi xit it ui ,i = 1,...,N; t = 1,...,Ti yit 1 if yit > 0, 0 otherwise. Mundlak Correction A Variable Addition Test for FE vs. RE The Wald statistic of 45.27922 and the likelihood ratio statistic of 40.280 are both far larger than the critical chi squared with 5 degrees of freedom, 11.07. This suggests that for these data, the fixed effects model is the preferred framework. Fixed Effects Models Summary Incidental parameters problem if T < 10 (roughly) Inconvenience of computation Appealing specification Alternative semiparametric estimators? Theory not well developed for T > 2 Not informative for anything but slopes (e.g., predictions and marginal effects) Ignoring the heterogeneity definitely produces an inconsistent estimator (even with cluster correction!) A Hobson’s choice Mundlak correction is a useful common approach. 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) (The rabbit in the hat) 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 = y i0 x iδ + w i (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 mo del A Dynamic Model for Public Insurance Age Household Income Kids in the household Health Status Basic Model Add initial value, lagged value, group means Dynamic Common Effects Model