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Part 16: Nonlinear Effects [ 1/103] Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

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Econometric Analysis of Panel Data 16. Nonlinear Effects Models and Models for Binary Choice

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Part 16: Nonlinear Effects [ 3/103] The Panel Probit Model

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Part 16: Nonlinear Effects [ 4/103] FIML See Greene, W., Convenient Estimators for the Panel Probit Model: Further Results, Empirical Economics, 29, 1, Jan. 2004, pp

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Part 16: Nonlinear Effects [ 5/103] GMM

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Part 16: Nonlinear Effects [ 6/103] GMM Estimation-1

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Part 16: Nonlinear Effects [ 7/103] GMM Estimation-2

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Part 16: Nonlinear Effects [ 8/103] GEE Estimation

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Part 16: Nonlinear Effects [ 9/103] Fractional Response

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Part 16: Nonlinear Effects [ 10/103] Fractional Response Model

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Part 16: Nonlinear Effects [ 11/103] Fractional Response Model

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Part 16: Nonlinear Effects [ 12/103]

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Part 16: Nonlinear Effects [ 13/103]

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Part 16: Nonlinear Effects [ 14/103]

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Part 16: Nonlinear Effects [ 15/103]

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Part 16: Nonlinear Effects [ 16/103]

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Part 16: Nonlinear Effects [ 17/103]

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Part 16: Nonlinear Effects [ 18/103]

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Part 16: Nonlinear Effects [ 19/103]

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Part 16: Nonlinear Effects [ 20/103]

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Part 16: Nonlinear Effects [ 21/103] 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 = 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 / (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

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Part 16: Nonlinear Effects [ 22/103] Unbalanced Panels Group Sizes 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.)

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Part 16: Nonlinear Effects [ 23/103] 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, 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: ) Stringent assumptions about the process Model based on probability of being present in each wave of the panel

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Part 16: Nonlinear Effects [ 24/103] Panel Data Binary Choice Models Random Utility Model for Binary Choice U it = + x it + it + Person i specific effect Fixed effects using dummy variables U it = i + x it + it Random effects using omitted heterogeneity U it = + x it + it + u i Same outcome mechanism: Y it = 1[U it > 0]

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Part 16: Nonlinear Effects [ 25/103] Pooled Model

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Part 16: Nonlinear Effects [ 26/103] Ignoring Unobserved Heterogeneity

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Part 16: Nonlinear Effects [ 27/103] Ignoring Heterogeneity in the RE Model

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Part 16: Nonlinear Effects [ 28/103] 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.

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Part 16: Nonlinear Effects [ 29/103] 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| AGE|.01532*** EDUC| *** HHNINC| ** Unbalanced panel has 7293 individuals Constant| AGE|.02232*** EDUC| *** HHNINC| Rho|.44990***

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Part 16: Nonlinear Effects [ 30/103] 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*** EDUC| *** HHNINC| ** |Based on the panel data estimator AGE|.00620*** EDUC| *** HHNINC|

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Part 16: Nonlinear Effects [ 31/103] Effect of Clustering Y it must be correlated with Y is across periods Pooled estimator ignores correlation Broadly, y it = E[y it |x it ] + w it, E[y it |x it ] = Prob(y it = 1|x it ) w it 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.

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Part 16: Nonlinear Effects [ 32/103] Cluster Corrected Covariance Matrix

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Part 16: Nonlinear Effects [ 33/103] Cluster Correction: Doctor Binomial Probit Model Dependent variable DOCTOR Log likelihood function Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X | Conventional Standard Errors Constant| *** AGE|.01469*** EDUC| *** HHNINC| ** FEMALE|.35209*** | Corrected Standard Errors Constant| *** AGE|.01469*** EDUC| *** HHNINC| * FEMALE|.35209***

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Part 16: Nonlinear Effects [ 34/103] Random Effects

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Part 16: Nonlinear Effects [ 35/103] Quadrature – Butler and Moffitt (1982)

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Part 16: Nonlinear Effects [ 36/103] Quadrature Log Likelihood

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Part 16: Nonlinear Effects [ 37/103] Simulation Based Estimator

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Part 16: Nonlinear Effects [ 38/103] Random Effects Model: Quadrature Random Effects Binary Probit Model Dependent variable DOCTOR Log likelihood function Random Effects Restricted log likelihood Pooled Chi squared [ 1 d.f.] 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| AGE|.02232*** EDUC| *** HHNINC| Rho|.44990*** |Pooled Estimates Constant| AGE|.01532*** EDUC| *** HHNINC| **

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Part 16: Nonlinear Effects [ 39/103] Random Parameter Model Random Coefficients Probit Model Dependent variable DOCTOR (Quadrature Based) Log likelihood function ( ) Restricted log likelihood Chi squared [ 1 d.f.] Simulation based on 50 Halton draws Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] |Nonrandom parameters AGE|.02226*** (.02232) EDUC| *** ( ) HHNINC| (.00660) |Means for random parameters Constant| ** ( ) |Scale parameters for dists. of random parameters Constant|.90453*** Using quadrature, a = Implied from these estimates is /( ) = compared to using quadrature.

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Part 16: Nonlinear Effects [ 40/103] A Dynamic Model

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Part 16: Nonlinear Effects [ 41/103] Dynamic Probit Model: A Standard Approach

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Part 16: Nonlinear Effects [ 42/103] Simplified Dynamic Model

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Part 16: Nonlinear Effects [ 43/103] A Dynamic Model for Public Insurance Age Household Income Kids in the household Health Status Add initial value, lagged value, group means

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Part 16: Nonlinear Effects [ 44/103] Dynamic Common Effects Model

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Part 16: Nonlinear Effects [ 45/103] Fixed Effects

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Part 16: Nonlinear Effects [ 46/103] Fixed Effects Models Estimate with dummy variable coefficients U it = i +x it + it Can be done by brute force for 10,000s of individuals F(.) = appropriate probability for the observed outcome Compute and i for i=1,…,N (may be large) See FixedEffects.pdf in course materials.

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Part 16: Nonlinear Effects [ 47/103] Unconditional Estimation Maximize the whole log likelihood Difficult! Many (thousands) of parameters. Feasible – NLOGIT (2001) (Brute force)

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Part 16: Nonlinear Effects [ 48/103] Fixed Effects Health Model Groups in which y it is always = 0 or always = 1. Cannot compute α i.

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Part 16: Nonlinear Effects [ 49/103] Conditional Estimation Principle: f(y i1,y i2,… | 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 α i. Only feasible for the logit model. (Poisson and a few other continuous variable models. No other discrete choice models.)

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Part 16: Nonlinear Effects [ 50/103] Binary Logit Conditional Probabiities

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Part 16: Nonlinear Effects [ 51/103] Example: Two Period Binary Logit

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Part 16: Nonlinear Effects [ 52/103] Estimating Partial Effects The fixed effects logit estimator of immediately gives us the effect of each element of x i on the log-odds ratio… Unfortunately, we cannot estimate the partial effects… unless we plug in a value for α i. Because the distribution of α i is unrestricted – in particular, E[α i ] is not necessarily zero – it is hard to know what to plug in for α i. In addition, we cannot estimate average partial effects, as doing so would require finding E[Λ(x it + α i )], a task that apparently requires specifying a distribution for α i. (Wooldridge, 2002)

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Part 16: Nonlinear Effects [ 53/103] Logit Constant Terms

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Part 16: Nonlinear Effects [ 54/103] Fixed Effects Logit Health Model: Conditional vs. Unconditional

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Part 16: Nonlinear Effects [ 55/103] 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

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Part 16: Nonlinear Effects [ 56/103] 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 α i ) is consistent. Specific: Upward bias (experience with probit and logit) in estimators of

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Part 16: Nonlinear Effects [ 57/103] 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.

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Part 16: Nonlinear Effects [ 58/103] Fixed Effects Attention mostly focused on index function models; f(y it |x it ) = some function of x itβ+α i. Incidental parameters problems Bias of estimator of β is O(1/T) How do we estimate α i ? How can we compute interesting partial effects? Models Linear model: No problem Poisson (nonlinear model): No problem 1 or 2 other models: No problem Other nonlinear models: The literature speaks in generalities The probit and logit models have been analyzed at length Almost nothing is known about any other model save for Greenes ( ) limited Monte Carlo studies (frontier, tobit, truncation, ordered probit, probit, logit)

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Part 16: Nonlinear Effects [ 59/103] Bias Correction Estimators Motivation: Undo the incidental parameters bias in the fixed effects probit model: (1) Maximize a penalized log likelihood function, or (2) Directly correct the estimator of β Advantages For (1) estimates α i 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)

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Part 16: Nonlinear Effects [ 60/103] Bias Reduction Parametric (probit and logit) models with fixed effects (We examine non- and semiparametric methods at the end of the course.) Recent references: All about probit and logit models. [1] Carro, J., Estimating dynamic panel data discrete choice models with fixed effects, JE, 140, 2007, pp [2] Val, F., Fixed Effects estimation of structural parameters and marginal effects in panel probit models, JE, 2010 [3] Hahn, J. and G. Kuersteiner, Bias reduction for dynamic nonlinear panel models with fixed effects, UCLA, 2003 [4] Hahn, J. and W. Newey, Jackknife and Analytical Bias reduction for nonlinear panel models, Econometrica, See, also, bibliographies and work of T. Woutersen, B. Honoré and E. Kyriazidou.

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Part 16: Nonlinear Effects [ 61/103] Bias Reduction – 1: Hahn All rely on a large T approximation to the bias when T is (very small) All analyze the equivalent of the brute force, unconditional estimator. Hahn/Kuersteiner and Hahn/Newey plim b MLE = β + B(T) where B(T) is O(1/T) Derive an expression for B(T) The bias corrected estimator is obtained by subtraction No further analysis is obtained to estimate fixed effects or partial effects.

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Part 16: Nonlinear Effects [ 62/103] Bias Reduction – 2: Val Plim b MLE = β + B(T) Find, D(T) a large sample approximation such that Plim b MLE +D(T) = β + F(T) where F(T) is O(1/T 2 ) Finds a counterpart approximation to the marginal effects.

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Part 16: Nonlinear Effects [ 63/103] Bias Reduction – 3: Carro Change the log likelihood. Maximum Modified Likelihood Estimator = MMLE Maximize MMLE such that the solution to MMLE is b MMLE plim b MMLE = β +G(T) where G(T) is O(1/T 2 ). Also obtains a solution for α i (unlike the others). a i,MMLE = f(b MMLE ) (A problem? When y it is always the same, there is no solution for a i.)

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Part 16: Nonlinear Effects [ 64/103] Bias Reduction? Approximations rely on large T Work moderately well when T is as low as 8 or 10. Completely miss the mark when T=2, 3,4 Nothing is known about any other models.

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Part 16: Nonlinear Effects [ 65/103] A Mundlak Correction for the FE Model

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Part 16: Nonlinear Effects [ 66/103] Mundlak Correction

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Part 16: Nonlinear Effects [ 67/103] A Variable Addition Test for FE vs. RE The Wald statistic of and the likelihood ratio statistic of are both far larger than the critical chi squared with 5 degrees of freedom, This suggests that for these data, the fixed effects model is the preferred framework.

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Part 16: Nonlinear Effects [ 68/103] 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 Hobsons choice Mundlak correction is a useful common approach.

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Part 16: Nonlinear Effects [ 69/103] Conditional vs. Unconditional Dep. Var. = Healthy Note, this estimator is not consistent – Incidental Parameters Problem

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Part 16: Nonlinear Effects [ 70/103] Escaping the FE Assumptions

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Part 16: Nonlinear Effects [ 71/103] Modeling a Binary Outcome Did firm i produce a product or process innovation in year t ? y it : 1=Yes/0=No Observed N=1270 firms for T=5 years, Observed covariates: x it = Industry, competitive pressures, size, productivity, etc. How to model? Binary outcome Correlation across time Heterogeneity across firms

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Part 16: Nonlinear Effects [ 72/103] Application

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Part 16: Nonlinear Effects [ 73/103]

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Part 16: Nonlinear Effects [ 74/103] Estimates of a Fixed Effects Probit Model FIXED EFFECTS Probit Model Dependent variable IP Log likelihood function Estimation based on N = 6350, K = 953 Inf.Cr.AIC = AIC/N =.958 Model estimated: Apr 16, 2013, 10:00:53 Unbalanced panel has 1270 individuals Skipped 552 groups with inestimable ai PROBIT (normal) probability model | Standard Prob. 95% Confidence IP| Coefficient Error z |z|>Z* Interval |Index function for probability EMPLP|.12108D D D-03 LOGSALES| IMUM| FDIUM| ** Note: nnnnn.D-xx or D+xx => multiply by 10 to -xx or +xx. Note: ***, **, * ==> Significance at 1%, 5%, 10% level

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Part 16: Nonlinear Effects [ 75/103] | Probit Regression Start Values for IP | |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| EMPLP D LOGSALES IMUM FDIUM Constant | Random Effects Binary Probit Model | EMPLP D LOGSALES IMUM FDIUM Constant Rho | FIXED EFFECTS Probit Model | EMPLP D LOGSALES IMUM FDIUM Pooled, Fixed Effects and Random Effects Probit

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Part 16: Nonlinear Effects [ 76/103] Fixed Effects Advantages Allows correlation of effect and regressors Fairly straightforward to estimate Simple to interpret Disadvantages Not necessarily simple to estimate if very large samples (Stata just creates the thousands of dummy variables) The incidental parameters problem: Small T bias. No time invariant variables

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Part 16: Nonlinear Effects [ 77/103] Incidental Parameters Problems: Conventional Wisdom General: Biased in samples with fixed T except in special cases such as linear or Poisson regression Specific: Upward bias (experience with probit and logit) in estimators of

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Part 16: Nonlinear Effects [ 78/103] What We KNOW - Analytic Newey and Hahn: MLE converges in probability to a vector of constants. (Variance diminishes with increase in N). Abrevaya and Hsiao: Logit estimator converges to 2 when T = 2. Han, Schmidt, Greene: Probit estimator converges to 2 when T = 2.

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Part 16: Nonlinear Effects [ 79/103] What We THINK We Know – Monte Carlo Heckman: Bias in probit estimator is small if T 8 Bias in probit estimator is toward 0 in some cases Katz (et al – numerous others), Greene Bias in probit and logit estimators is large Upward bias persists even as T 20

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Part 16: Nonlinear Effects [ 80/103] Heckmans Monte Carlo Study

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Part 16: Nonlinear Effects [ 81/103] Some Familiar Territory – A Monte Carlo Study of the FE Estimator: Probit vs. Logit (Greene, The Econometrics Journal, 7, 2004, pp ) 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.

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Part 16: Nonlinear Effects [ 82/103] A Monte Carlo Study of the FE Probit Estimator Percentage Biases in Estimates of Coefficients and Marginal Effects at the Implied Data Means

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Part 16: Nonlinear Effects [ 83/103] Dynamic Models

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Part 16: Nonlinear Effects [ 84/103] Application – Doctor Visits Riphahn, Million Wambach, JAE, 2003 German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods Variables in the file are 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. This is a large data set. 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). Note, the variable NUMOBS below tells how many observations there are for each person. This variable is repeated in each row of the data for the person. DOCTOR = 1(Number of doctor 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 / (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 EDUC = years of education

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Part 16: Nonlinear Effects [ 85/103] Application: Innovations Bertschek and Lechner, J of Econometrics, 1998

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Part 16: Nonlinear Effects [ 86/103] Application Stewart, JAE, 2007 British Household Panel Survey ( ) 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.

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Part 16: Nonlinear Effects [ 87/103] Application: Direct Approach

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Part 16: Nonlinear Effects [ 88/103] 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( y i,0, y i,1,...) = Prob(y i,0 ) × Prob(y i,1 | y i,0 ) × ×Prob(y i,T | y i,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.

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Part 16: Nonlinear Effects [ 89/103] A Dynamic RE Probit

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Part 16: Nonlinear Effects [ 90/103] Problems with Dynamic RE Probit Assumes y i,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. Doesnt allow time invariant variables in the model. The normality assumption in the projection.

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Part 16: Nonlinear Effects [ 91/103] Heckmans Solution

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Part 16: Nonlinear Effects [ 92/103] Dynamic Probit Model: A Simplified Approach (Wooldridge, 2005)

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Part 16: Nonlinear Effects [ 93/103] Distributional Problem Normal distributions assumed throughout Normal distribution for the unique component, ε i,t Normal distribution assumed for the heterogeneity, u i Sensitive to the distribution? Alternative: Discrete distribution for u i. Heckman and Singer style, latent class model. Conventional estimation methods. Why is the model not sensitive to normality for ε i,t but it is sensitive to normality for u i ?

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Part 16: Nonlinear Effects [ 94/103] A Multinomial Logit Common Effects Model How to handle unobserved effects in other nonlinear models? Single index models such as probit, Poisson, tobit, etc. that are functions of an x it 'β can be modified to be functions of x it 'β + c i. Other models – not at all obvious. Rarely found in the literature. Dealing with fixed and random effects? Dynamics makes things much worse.

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Part 16: Nonlinear Effects [ 95/103] A Multinomial Logit Model

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Part 16: Nonlinear Effects [ 96/103] A Heterogeneous Multinomial Logit Model

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Part 16: Nonlinear Effects [ 97/103] Common Effects Multinomial Logit

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Part 16: Nonlinear Effects [ 98/103] Simulation Based Estimation

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Part 16: Nonlinear Effects [ 99/103] Application Shoe Brand Choice S imulated Data: Stated Choice, N=400 respondents, T=8 choice situations, 3,200 observations 3 choice/attributes + NONE J=4 Fashion = High / Low Quality = High / Low Price = 25/50/75,100 coded 1,2,3,4; and Price 2 H eterogeneity: Sex, Age (<25, 25-39, 40+) U nderlying data generated by a 3 class latent class process (100, 200, 100 in classes) T hanks to (Latent Gold)www.statisticalinnovations.com

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Part 16: Nonlinear Effects [ 100/103] Application

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Part 16: Nonlinear Effects [ 101/103] No Common Effects | Start values obtained using MNL model | | Log likelihood function | |Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| FASH | QUAL | PRICE | PRICESQ | ASC4 | B1_MAL1 | B1_YNG1 | B1_OLD1 | B2_MAL2 | B2_YNG2 | B2_OLD2 | B3_MAL3 | B3_YNG3 | B3_OLD3 |

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Part 16: Nonlinear Effects [ 102/103] Random Effects MNL Model | Error Components (Random Effects) model | Restricted logL = | Log likelihood function | Chi squared(3) = (Crit.Val.=7.81) |Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Nonrandom parameters in utility functions FASH | QUAL | PRICE | PRICESQ | ASC4 | B1_MAL1 | B1_YNG1 | B1_OLD1 | B2_MAL2 | B2_YNG2 | B2_OLD2 | B3_MAL3 | B3_YNG3 | B3_OLD3 | Standard deviations of latent random effects SigmaE01| SigmaE02| SigmaE03|

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Part 16: Nonlinear Effects [ 103/103] Implementations of RE Models Linear, Probit, Logit, Poisson, 1 or 2 other models: SAS, about 10 others Linear, Probit, Logit, 4 or 5 others: MLWin (Using Bayesian MCMC methods) Linear, Probit, Logit, Poisson, 4 or 5 others: Stata (using quadrature, Proc = GLAMM) Linear, Probit, Logit, Poisson, MNL, Tobit, about 50 others: LIMDEP/NLOGIT (using maximum simulated likelihood)

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