1. Discrete Choice Modeling William Greene Stern School of Business IFS at UCL February 11-13, 2004

2. Discrete Choice Modeling  Econometric Methodology Binary Choice Models Multinomial Choice  Model Building Specification Estimation Analysis Applications  NLOGIT Software

3. Our Agenda 1. Methodology 2. Discrete Choice Models 3. Binary Choice Models 4. Panel Data Models for Binary Choice 5. Introduction to NLOGIT 6. Discrete Choice Settings 7. The Multinomial Logit Model 8. Heteroscedasticity in Utility Functions 9. Nested Logit Modeling 10. Latent Class Models 11. Mixed Logit Models and Simulation Based Estimation 12. Revealed and Stated Preference Data Sets

4. Part 1 Methodology

5. Measurement as Observation Population Measurement Theory Characteristics Behavior Patterns

6. Individual Behavioral Modeling  Assumptions about behavior Common elements across individuals Unique elements  Prediction Population aggregates Individual behavior

7. Modeling Choice  Activity as choices  Preferences  Behavioral axioms  Choice as utility maximization

8. Inference Population Measurement Econometrics Characteristics Behavior Patterns Choices

9. Econometric Frameworks  Nonparametric  Parametric Classical (Sampling Theory) Bayesian

10. Likelihood Based Inference Methods Behavioral Theory Statistical Theory Observed Measurement Likelihood Function The likelihood function embodies the theoretical description of the population. Characteristics of the population are inferred from the characteristics of the likelihood function. (Bayesian and Classical)

11. Modeling Discrete Choice  Theoretical foundations  Econometric methodology Models Statistical bases Econometric methods  Estimation with econometric software  Applications

12. Part 2 Basics of Discrete Choice Modeling

13. Modeling Consumer Choice: Continuous Measurement What do we measure? What is revealed by the data? What is the underlying model? What are the empirical tools? Example: Travel expenditure based on price and income Expenditure Income Low price High price

14. Discrete Choice  Observed outcomes Inherently discrete: number of occurrences (e.g., family size; considered separately) Implicitly continuous: the observed data are discrete by construction (e.g., revealed preferences; our main subject)  Implications For model building For analysis and prediction of behavior

15. Two Fundamental Building Blocks  Underlying Behavioral Theory: Random utility model The link between underlying behavior and observed data  Empirical Tool: Stochastic, parametric model for binary choice A p latform for models of discrete choice

16. Random Utility A Theoretical Proposition About Behavior  Consumer making a choice among several alternatives  Example, brand choice (car, food)  Choice setting for a consumer: Notation Consumer i, i = 1, …, N Choice setting t, t = 1, …, T i (may be one) Choice set j, j = 1,…, J i (may be fixed)

17. Behavioral Assumptions  Preferences are transitive and complete wrt choice situations  Utility is defined over alternatives: U ijt  Utility maximization assumption If U i1t > U i2t, consumer chooses alternative 1, not alternative 2.  Revealed preference (duality) If the consumer chooses alternative 1 and not alternative 2, then U i1t > U i2t.

18. Random Utility Functions U itj =  j +  i ’x itj +  i ’z it +  ijt  j = Choice specific constant x itj = Attributes of choice presented to person  i = Person specific taste weights z it = Characteristics of the person  i = Weights on person specific characteristics  ijt = Unobserved random component of utility Mean: E[  ijt ] = 0, Var[  ijt ] = 1

19. Part 3 Modeling Binary Choice

20. A Model for Binary Choice  Yes or No decision (Buy/Not buy)  Example, choose to fly or not to fly to a destination when there are alternatives.  Model: Net utility of flying U fly =  +  1Cost +  2Time +  Income +  Choose to fly if net utility is positive  Data: X = [1,cost,terminal time] Z = [income] y = 1 if choose fly, U fly > 0, 0 if not.

21. What Can Be Learned from the Data? (A Sample of Consumers, i = 1,…,N) Are the attributes “relevant?” Predicting behavior - Individual - Aggregate Analyze changes in behavior when attributes change

22. Application  210 Commuters Between Sydney and Melbourne  Available modes = Air, Train, Bus, Car  Observed: Choice Attributes: Cost, terminal time, other Characteristics: Household income  First application: Fly or other

23. Binary Choice Data Choose Air Gen.Cost Term Time Income 1.0000 86.000 25.000 70.000.00000 67.000 69.000 60.000.00000 77.000 64.000 20.000.00000 69.000 69.000 15.000.00000 77.000 64.000 30.000.00000 71.000 64.000 26.000.00000 58.000 64.000 35.000.00000 71.000 69.000 12.000.00000 100.00 64.000 70.000 1.0000 158.00 30.000 50.000 1.0000 136.00 45.000 40.000 1.0000 103.00 30.000 70.000.00000 77.000 69.000 10.000 1.0000 197.00 45.000 26.000.00000 129.00 64.000 50.000.00000 123.00 64.000 70.000

24. An Econometric Model  Choose to fly iff U FLY > 0 U fly =  +  1Cost +  2Time +  Income +  U fly > 0   > -(  +  1Cost +  2Time +  Income)  Probability model: For any person observed by the analyst, Prob(fly) = Prob[  > -(  +  1Cost +  2Time +  Income)]  Note the relationship between the unobserved  and the outcome

25.  +  1Cost +  2TTime +  Income

26. Econometrics  How to estimate ,  1,  2,  ? It’s not regression The technique of maximum likelihood Prob[y=1] = Prob[  > -(  +  1Cost +  2Time +  Income)] Prob[y=0] = 1 - Prob[y=1]  Requires a model for the probability

27. Completing the Model: F(  )  The distribution Normal: PROBIT, natural for behavior Logistic: LOGIT, allows “thicker tails” Gompertz: EXTREME VALUE, asymmetric, underlies the basic logit model for multiple choice  Does it matter? Yes, large difference in estimates Not much, quantities of interest are more stable.

29. Estimated Binary Choice Models LOGIT PROBIT EXTREME VALUE Variable Estimate t-ratio Estimate t-ratio Estimate t-ratio Constant 1.78458 1.40591 0.438772 0.702406 1.45189 1.34775 GC 0.0214688 3.15342 0.012563 3.41314 0.0177719 3.14153 TTME -0.098467 -5.9612 -0.0477826 -6.65089 -0.0868632 -5.91658 HINC 0.0223234 2.16781 0.0144224 2.51264 0.0176815 2.02876 Log-L -80.9658 -84.0917 -76.5422 Log-L(0) -123.757 -123.757 -123.757

30.  +  1Cost +  2Time +  (Income+1) Effect on predicted probability of an increase in income (  is positive)

31. Marginal Effects in Probability Models  Prob[Outcome] = some F(  +  1Cost…)  “Partial effect” =  F(  +  1Cost…) /  ”x” (derivative) Partial effects are derivatives Result varies with model  Logit:  F(  +  1Cost…) /  x = Prob * (1-Prob) *   Probit:  F(  +  1Cost…) /  x = Normal density  Scaling usually erases model differences

32. The Delta Method

33. Marginal Effects for Binary Choice  Logit  Probit

34. Estimated Marginal Effects Estimate t-ratio Estimate t-ratio Estimate t-ratio GC.003721 3.267.003954 3.466.003393 3.354 TTME-.017065-5.042-.015039-5.754-.016582-4.871 HINC.003869 2.193.004539 2.532.033753 2.064 Logit Probit Extreme Value

35. Marginal Effect for a Dummy Variable  Prob[y i = 1|x i,d i ] = F(  ’x i +  d i ) =conditional mean  Marginal effect of d Prob[y i = 1|x i,d i =1]=Prob[y i = 1|x i,d i =0]  Logit:

36. (Marginal) Effect – Dummy Variable  HighIncm = 1(Income > 50) +-------------------------------------------+ | Partial derivatives of probabilities with | | respect to the vector of characteristics. | | They are computed at the means of the Xs. | | Observations used are All Obs. | +-------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Characteristics in numerator of Prob[Y = 1] Constant.4750039483.23727762 2.002.0453 GC.3598131572E-02.11354298E-02 3.169.0015 102.64762 TTME -.1759234212E-01.34866343E-02 -5.046.0000 61.009524 Marginal effect for dummy variable is P|1 - P|0. HIGHINCM.8565367181E-01.99346656E-01.862.3886.18571429 (Autodetected)

37. Computing Effects  Compute at the data means? Simple Inference is well defined  Average individual effects More appropriate? Asymptotic standard errors. (Not done correctly in the literature – terms are correlated!)

38. Elasticities  Elasticity =  How to compute standard errors? Delta method Bootstrap  Bootstrap the individual elasticities? (Will neglect variation in parameter estimates.)  Bootstrap model estimation?

39. Estimated Income Elasticity for Air Choice Model +------------------------------------------+ | Results of bootstrap estimation of model.| | Model has been reestimated 25 times. | | Statistics shown below are centered | | around the original estimate based on | | the original full sample of observations.| | Result is ETA =.71183 | | bootstrap samples have 840 observations.| | Estimate RtMnSqDev Skewness Kurtosis | |.712.266 -.779 2.258 | | Minimum =.125 Maximum = 1.135 | +------------------------------------------+ Mean Income = 34.55, Mean P =.2716, Estimated ME =.004539, Estimated Elasticity=0.5774.

40. Odds Ratio – Logit Model Only  Effect Measure? “Effect of a unit change in the odds ratio.”

41. Inference for Odds Ratios  Logit coefficient = , estimate = b  Coefficient = exp(  ), estimate = exp(b)  Standard error = exp(b) times se(b)  t ratio is the same

42. How Well Does the Model Fit?  There is no R squared  “Fit measures” computed from log L “pseudo R squared = 1 – logL0/logL Others… - these do not measure fit.  Direct assessment of the effectiveness of the model at predicting the outcome

43. Fit Measures for Binary Choice  Likelihood Ratio Index Bounded by 0 and 1 Rises when the model is expanded  Cramer (and others)

44. Fit Measures for the Logit Model +----------------------------------------+ | Fit Measures for Binomial Choice Model | | Probit model for variable MODE | +----------------------------------------+ | Proportions P0=.723810 P1=.276190 | | N = 210 N0= 152 N1= 58 | | LogL = -84.09172 LogL0 = -123.7570 | | Estrella = 1-(L/L0)^(-2L0/n) =.36583 | +----------------------------------------+ | Efron | McFadden | Ben./Lerman | |.45620 |.32051 |.75897 | | Cramer | Veall/Zim. | Rsqrd_ML | |.40834 |.50682 |.31461 | +----------------------------------------+ | Information Akaike I.C. Schwarz I.C. | | Criteria.83897 189.57187 | +----------------------------------------+

45. Predicting the Outcome  Predicted probabilities P = F(a + b1Cost + b2Time + cIncome)  Predicting outcomes Predict y=1 if P is large Use 0.5 for “large” (more likely than not)  Count successes and failures

46. Individual Predictions from a Logit Model Observation Observed Y Predicted Y Residual x(i)b Pr[Y=1] 81.00000.00000.0000 -3.3944.0325 85.00000.00000.0000 -2.1901.1006 89 1.0000.00000 1.0000 -2.6766.0644 93 1.0000 1.0000.0000.8113.6924 97 1.0000 1.0000.0000 2.6845.9361 101 1.0000 1.0000.0000 2.4457.9202 105 1.0000.00000 1.0000 -3.2204.0384 109 1.0000 1.0000.0000.0311.5078 113.00000.00000.0000 -2.1704.1024 117.00000.00000.0000 -3.3729.0332 445.00000 1.0000 -1.0000.0295.5074 Note two types of errors and two types of successes.

47. Predictions in Binary Choice Predict y = 1 if P > P* Success depends on the assumed P*

48. ROC Curve  Plot %Y=1 correctly predicted vs. %y=1 incorrectly predicted  45 0 is no fit. Curvature implies fit.  Area under the curve compares models

50. Aggregate Predictions Frequencies of actual & predicted outcomes Predicted outcome has maximum probability. Threshold value for predicting Y=1 =.5000 Predicted ------ ---------- + ----- Actual 0 1 | Total ------ ---------- + ----- 0 151 1 | 152 1 20 38 | 58 ------ ---------- + ----- Total 171 39 | 210

51. Analyzing Predictions Frequencies of actual & predicted outcomes Predicted outcome has maximum probability. Threshold value for predicting Y=1 is P*.5000. (This table can be computed with any P*.) Predicted ------ -------------------- + ----- Actual 0 1 | Total ------ ----------------------+------- 0 N(a0,p0) N(a0,p1) | N(a0) 1 N(a1,p0) N(a1,p1) | N(a1) ------ ----------------------+ ----- Total N(p0) N(p1) | N

52. Analyzing Predictions - Success  Sensitivity = % actual 1s correctly predicted = 100N(a1,p1)/N(a1) % [100(38/58)=65.5%]  Specificity = % actual 0s correctly predicted = 100N(a0,p0)/N(a0) % [100(151/152)=99.3%]  Positive predictive value = % predicted 1s that were actual 1s = 100N(a1,p1)/N(p1) % [100(38/39)=97.4%]  Negative predictive value = % predicted 0s that were actual 0s = 100N(a0,p0)/N(p0) % [100(151/171)=88.3%]  Correct prediction = %actual 1s and 0s correctly predicted = 100[N(a1,p1)+N(a0,p0)]/N [100(151+38)/210=90.0%]

53. Analyzing Predictions - Failures  False positive for true negative = %actual 0s predicted as 1s = 100N(a0,p1)/N(a0) % [100(1/152)=0.668%]  False negative for true positive = %actual 1s predicted as 0s = 100N(a1,p0)/N(a1) % [100(20/258)=34.5%]  False positive for predicted positive = % predicted 1s that were actual 0s = 100N(a0,p1)/N(p1) % [100(1/39)=2/56%]  False negative for predicted negative = % predicted 0s that were actual 1s = 100N(a1,p0)/N(p0) % [100(20/171)=11.7%]  False predictions = %actual 1s and 0s incorrectly predicted = 100[N(a0,p1)+N(a1,p0)]/N [100(1+20)/210=10.0%]

54. Aggregate Prediction is a Useful Way to Assess the Importance of a Variable Frequencies of actual & predicted outcomes. Predicted outcome has maximum probability. Threshold value for predicting Y=1 =.5000 Predicted ------ ---------- + ----- Actual 0 1 | Total ------ ---------- + ----- 0 145 7 | 152 1 48 10 | 58 ------ ---------- + ----- Total 193 17 | 210 Predicted ------ ---------- + ----- Actual 0 1 | Total ------ ---------- + ----- 0 151 1 | 152 1 20 38 | 58 ------ ---------- + ----- Total 171 39 | 210 Model fit without TTMEModel fit with TTME

55. Simulating the Model to Examine Changes in Market Shares Suppose TTME increased by 25% for everyone. Before increase After increase Predicted ------ ---------- + ----- Actual 0 1 | Total ------ ---------- + ----- 0 151 1 | 152 1 20 38 | 58 ------ ---------- + ----- Total 171 39 | 210 Predicted ------ ---------- + ----- Actual 0 1 | Total ------ ---------- + ----- 0 152 0 | 152 1 29 29 | 58 ------ ---------- + ----- Total 181 29 | 210 The model predicts 10 fewer people would fly NOTE: The same model used for both sets of predictions.

56. Scaling U itj =  j +  i ’x itj +  i ’z it +  ijt  ijt = Unobserved random component of utility Mean: E[  ijt ] = 0, Var[  ijt ] = 1  Why assume variance = 1?  What if there are subgroups with different variances? Cost of ignoring the between group variation? Specifically modeling  More general heterogeneity across people Cost of the homogeneity assumption Modeling issues

57. Choice Between Two Alternatives  By way of example: Automobile type Choices (1) SUV or (2) Sedan, J i = 2 One choice situation: T i = 1 Attribute: x ij = price, perhaps others Characteristic: z i = income No variation in taste parameters,  i =   What do revealed choices tell us?

58.  Modeling the Binary Choice U i,suv =  suv +  P suv +  suv Income +  i,suv U i,sed =  sed +  P sed +  sed Income +  i,sed  Chooses SUV: U i,suv > U i,sed U i,suv - U i,sed > 0  (  SUV -  SED ) +  (P SUV -P SED ) + (  SUV-  sed)Income +  i,suv -  i,sed > 0   i > -[  +  (P SUV -P SED ) +  Income]

59. Probability Model for Choice Between Two Alternatives  i > -[  +  (P SUV -P SED ) +  Income]

60. Individual vs. Grouped Data  Proportions and Frequencies  Likelihood is the same  Yj i may be 1s and 0s, proportions, or frequencies for the two outcomes.

61. Weighting and Choice Based Sampling  Weighted log likelihood for all data types  Endogenous weights for individual data “Biased” sampling – “Choice Based”

62. Choice Based Sample SamplePopulationWeight Air27.62%14%0.5068 Ground72.38%86%1.1882

63. Choice Based Sampling Correction  Maximize Weighted Log Likelihood  Covariance Matrix Adjustment V = H -1 G H -1 (all three weighted) H = Hessian G = Outer products of gradients “Robust” covariance matrix (?) (Above without weights. What is it robust to?)

64. Effect of Choice Based Sampling Unweighted +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Constant 1.784582594 1.2693459 1.406.1598 GC.2146879786E-01.68080941E-02 3.153.0016 TTME -.9846704221E-01.16518003E-01 -5.961.0000 HINC.2232338915E-01.10297671E-01 2.168.0302 +---------------------------------------------+ | Weighting variable CBWT | | Corrected for Choice Based Sampling | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Constant 1.014022236 1.1786164.860.3896 GC.2177810754E-01.63743831E-02 3.417.0006 TTME -.7434280587E-01.17721665E-01 -4.195.0000 HINC.2471679844E-01.95483369E-02 2.589.0096

65. Hypothesis Testing – Neyman/Pearson  Comparisons of Likelihood Functions Likelihood Ratio Tests Lagrange Multiplier Tests  Distance Measures: Wald Statistics (All to be demonstrated in the lab)

66. Heteroscedasticity in Binary Choice Models  Random utility: Y i = 1 iff  ’x i +  i > 0  Resemblance to regression: How to accommodate heterogeneity in the random unobserved effects across individuals?  Heteroscedasticity – different scaling Parameterize: Var[  i ] = exp(  ’z i ) Reformulate probabilities  Probit:  Partial effects are now very complicated

67. Application: Credit Data  Counts of major derogatory reports)  “Deadbeat” = 1 if MAJORDRG > 0  Mean depends on AGE, INCOME, OWNRENT, SELFEMPLOYED  Variance depends on AVGEXP, DEPENDT (average monthly expenditure, number of dependents)  Probit model with heteroscedasticity

68. Probit with Heteroscedasticity +---------------------------------------------+ | Binomial Probit Model | | Dependent variable DEADBEAT | | Number of observations 1319 | | Log likelihood function -639.3388 | | Restricted log likelihood -653.3217 | | Chi-squared 27.96596 | | Degrees of freedom 6 | | Significance level.9535906E-04 | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Index function for probability Constant -1.272312665.13598690 -9.356.0000 AGE.1126209389E-01.40404726E-02 2.787.0053 33.213103 INCOME.5286782288E-01.20239074E-01 2.612.0090 3.3653760 OWNRENT -.2049230056.88518106E-01 -2.315.0206.44048522 SELFEMPL.1143040149.13825044.827.4084.68991660E-01 Variance function AVGEXP -.4768665802E-03.12613317E-03 -3.781.0002 185.05707 DEPNDT.6880605703E-02.42546206E-01.162.8715.99393480 +-------------------------------------------+ | 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] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Index function for probability Constant -.3768739381.54283831E-01 -6.943.0000 AGE.3335964337E-02.12357954E-02 2.699.0069 33.213103 INCOME.1566006938E-01.65292318E-02 2.398.0165 3.3653760 OWNRENT -.6070059841E-01.24667682E-01 -2.461.0139.44048522 SELFEMPL.3385819023E-01.41052591E-01.825.4095.68991660E-01 Variance function AVGEXP -.1133874143E-03.31868469E-04 -3.558.0004 185.05707 DEPNDT.1636042704E-02.10080807E-01.162.8711.99393480

69. Part 4 Panel Data Models for Binary Choice

70. Panel Data and Binary Choice Models 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 + v i )  Same outcome mechanism: Y it = [U it > 0]

71. Fixed and Random Effects Models  Fixed Effects Robust to both specifications Inconvenient to compute (many parameters) Incidental parameters problem  Random Effects Inconsistent if correlated with X Small number of parameters Easier to compute  Computation – available estimators

72. Fixed Effects  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 “Estimating Econometric Models with Fixed Effects” at www.stern.nyu.edu/~wgreene

73. Random Effects  U it =  +  ’x it + (  it +  v v i )  Logit model (can be generalized)  Joint probability for individual i | v i =  Unobserved component v i must be eliminated  Maximize wrt ,  and  v  How to do the integration? Analytic integration – quadrature; most familiar software Simulation

74. Estimation by Simulation is the sum of the logs of E[Pr(y1,y2,…|v i )]. Can be estimated by sampling vi and averaging. (Use random numbers.)

75. Random Effects is Equivalent to a Random Constant Term  U it =  +  ’x it + (  it +  v v i ) = (  +   v i ) +  ’x it +  it =  i +  ’x it +  it   i is random with mean  and variance  View the simulation as sampling over  i Why not make all the coefficients random?

76. A Sampling Experiment  CLOGIT data using GC, TTME, INVT and HINC  Standardized data: each X it * is (X it – Mean(X))/S x  Constructed utilities U it = 0 + 1  GC it * + 1  TTME it * + 1  INVT it * + (Random number it + HINC i *)  Treat 4 observations in each group as a panel with T = 4.  (We will examine a “live” panel data set in the lab.)

77. Estimated Fixed Effects Model +---------------------------------------------+ | FIXED EFFECTS Logit Model | | Maximum Likelihood Estimates | | Dependent variable Z | | Weighting variable None | | Number of observations 840 | | Iterations completed 5 | | Log likelihood function -342.1919 | | Sample is 4 pds and 210 individuals. | | Bypassed 51 groups with inestimable a(i). | | LOGIT (Logistic) probability model | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Index function for probability GC.6708935970E-02.18621919E-01.360.7186 112.29560 TTME.3648053834E-01.57989428E-02 6.291.0000 34.779874 INVT.3338438006E-02.25104319E-02 1.330.1836 492.25314 INVC.6795479927E-02.19477804E-01.349.7272 48.448113 Partial derivatives of E[y] = F[*] with respect to the characteristics. Computed at the means of the Xs. Estimated E[y|means,mean alphai]=.501 Estimated scale factor for dE/dx=.250 GC.1677222976E-02.46555287E-02.360.7186 112.29560 TTME.9120074679E-02.14482840E-02 6.297.0000 34.779874 INVT.8346040194E-03.62727700E-03 1.331.1833 492.25314 INVC.1698858823E-02.48687627E-02.349.7271 48.448113 WHY?

78. Estimated Random Effects Model (1) +---------------------------------------------+ | Logit Model for Panel Data | | Maximum Likelihood Estimates | | Dependent variable Z | | Weighting variable None | | Number of observations 840 | | Iterations completed 15 | | Log likelihood function -494.6084 | | Hosmer-Lemeshow chi-squared = 15.81181 | | P-value=.04515 with deg.fr. = 8 | | Random Effects Logit Model for Panel Data | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Characteristics in numerator of Prob[Y = 1] Constant -2.074416165.20930847 -9.911.0000 GC.9739427161E-02.53423005E-02 1.823.0683 TTME.8353847679E-02.30194645E-02 2.767.0057 INVT.1252315669E-03.69864222E-03.179.8577 INVC -.1215241461E-02.55156025E-02 -.220.8256 RndmEfct.9492940742E-01.18841088.504.6144 -.58755677E-07

79. Estimated Random Effects Model (2) +---------------------------------------------+ | Random Coefficients Logit Model | | Maximum Likelihood Estimates | | Dependent variable Z | | Weighting variable None | | Number of observations 840 | | Iterations completed 14 | | Log likelihood function -494.5136 | | Restricted log likelihood -496.1793 | | Chi-squared 3.331300 | | Degrees of freedom 1 | | Significance level.6797315E-01 | | Sample is 4 pds and 210 individuals. | | LOGIT (Logistic) probability model | | Simulation based on 100 random draws | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Nonrandom parameters GC.1928882840E-01.40879229E-02 4.718.0000 110.87976 TTME.2364065236E-01.24280249E-02 9.737.0000 34.589286 INVT.5332059842E-03.54092102E-03.986.3243 486.16548 INVC -.6668386903E-02.41649216E-02 -1.601.1094 47.760714 Means for random parameters Constant -2.942970074.15967241 -18.431.0000 Scale parameters for dists. of random parameters Constant.5338591567.56357583E-01 9.473.0000 Conditional Mean at Sample Point.4886 Scale Factor for Marginal Effects.2499 GC.4819681744E-02.10205421E-02 4.723.0000 110.87976 TTME.5907067980E-02.59571899E-03 9.916.0000 34.589286 INVT.1332316870E-03.13504534E-03.987.3239 486.16548 INVC -.1666223679E-02.10411841E-02 -1.600.1095 47.760714

80. Commands for Panel Data Models  Model: LOGIT ; Lhs = … ; Rhs = … ; Pds = number of periods  Common effect Fixed effects; FEM $ or ; Fixed $ Random; Random Effects $ Simulation; RPM ; Fcn=One(N) $  Use with Probit, Logit (and many others)