1. Microeconometric Modeling
William Greene
Stern School of Business
New York University
New York NY USA
4.3 Mixed Models and Random Parameters

2. Concepts Models Random Effects Simulation Random Parameters
Maximum Simulated Likelihood
Cholesky Decomposition
Heterogeneity
Hierarchical Model
Conditional Means
Population Distribution
Nested Logit
Willingness to Pay (WTP)
Random Parameters and WTP
WTP Space
Endogeneity
Market Share Data
Random Parameters
RP Logit
Error Components Logit
Generalized Mixed Logit
Berry-Levinsohn-Pakes Model
Hybrid Choice
MIMIC Model

3. A Recast Random Effects Model

4. A Computable Log Likelihood

5. Simulation

6. Random Effects Model: Simulation
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| *** ( )
EDUC| *** ( )
HHNINC| ( )
|Means for random parameters
Constant| ** ( )
|Scale parameters for dists. of random parameters
Constant| ***
Implied  from these estimates is /( ) =

7. The Entire Parameter Vector is Random

8. Maximum Simulated Likelihood
True log likelihood
Simulated log likelihood

10. S M

11. MSSM

12. Modeling Parameter Heterogeneity

13. A Hierarchical Probit Model
Uit = 1i + 2iAgeit + 3iEducit + 4iIncomeit + it.
1i=1+11 Femalei + 12 Marriedi + u1i
2i=2+21 Femalei + 22 Marriedi + u2i
3i=3+31 Femalei + 32 Marriedi + u3i
4i=4+41 Femalei + 42 Marriedi + u4i
Yit = 1[Uit > 0]
All random variables normally distributed.

15. Estimating Individual Parameters
Model estimates = structural parameters, α, β, ρ, Δ, Σ, Γ
Objective, a model of individual specific parameters, βi
Can individual specific parameters be estimated?
Not quite – βi is a single realization of a random process; one random draw.
We estimate E[βi | all information about i]
(This is also true of Bayesian treatments, despite claims to the contrary.)

16. Estimating i

17. Conditional Estimate of i

18. Conditional Estimate of i

22. “Individual Coefficients”

23. The Random Parameters Logit Model
Multiple choice situations: Independent conditioned on the individual specific parameters

24. Random Parameters Model
Allow model parameters as well as constants to be random
Allow multiple observations with persistent effects
Allow a hierarchical structure for parameters – not completely random
Uitj = 1’xi1tj + 2i’xi2tj + i’zit + ijt
Random parameters in multinomial logit model
1 = nonrandom (fixed) parameters
2i = random parameters that may vary across individuals and across time
Maintain I.I.D. assumption for ijt (given )

25. Customers’ Choice of Energy Supplier
California, Stated Preference Survey
361 customers presented with 8-12 choice situations each
Supplier attributes:
Fixed price: cents per kWh
Length of contract
Local utility
Well-known company
Time-of-day rates (11¢ in day, 5¢ at night)
Seasonal rates (10¢ in summer, 8¢ in winter, 6¢ in spring/fall)
(TrainCalUtilitySurvey.lpj)

26. Population Distributions
Normal for:
Contract length
Local utility
Well-known company
Log-normal for:
Time-of-day rates
Seasonal rates
Price coefficient held fixed

27. Estimated Model Estimate Std error Price -.883 0.050
Contract mean
std dev
Local mean
std dev
Known mean
std dev
TOD mean*
std dev*
Seasonal mean*
std dev*
*Parameters of underlying normal.

28. Distribution of Brand Value
Standard deviation
=2.0¢
10% dislike local utility
2.5¢
Brand value of local utility

29. Random Parameter Distributions

30. Time of Day Rates (Customers do not like – lognormal coefficient
Time of Day Rates (Customers do not like – lognormal coefficient. Multiply variable by -1.)

31. Posterior Estimation of i
Estimate by simulation

32. Expected Preferences of Each Customer
Customer likes long-term contract, local utility, and non-fixed rates.
Local utility can retain and make profit from this customer by offering a long-term contract with time-of-day or seasonal rates.

33. Application: Shoe Brand Choice
Simulated Data: Stated Choice,
400 respondents,
8 choice situations, 3,200 observations
3 choice/attributes + NONE
Fashion = High / Low
Quality = High / Low
Price = 25/50/75,100 coded 1,2,3,4
Heterogeneity: Sex (Male=1), Age (<25, 25-39, 40+)
Underlying data generated by a 3 class latent class process (100, 200, 100 in classes)

34. Stated Choice Experiment: Unlabeled Alternatives, One Observation

35. Random Parameters Logit Model

38. WTP Application (Value of Time Saved)
Estimating Willingness to Pay for Increments to an Attribute in a Discrete Choice Model WTP = MU(attribute) / MU(Income)
Random

39. Extending the RP Model to WTP
Use the model to estimate conditional distributions for any function of parameters
Willingness to pay = -i,time / i,cost
Use simulation method

40. Simulation of WTP from i

42. WTP

43. A Generalized Mixed Logit Model

44. Generalized Multinomial Choice Model

45. Estimation in Willingness to Pay Space
Both parameters in the WTP calculation are random.

46. Estimated Model for WTP
Variable| Coefficient Standard Error b/St.Er. P[|Z|>z]
|Random parameters in utility functions
QUAL| *** renormalized
PRICE| (Fixed Parameter) renormalized
|Nonrandom parameters in utility functions
FASH| *** not rescaled
ASC4| *** not rescaled
|Heterogeneity in mean, Parameter:Variable
QUAL:AGE| interaction terms
QUA0:AGE|
PRIC:AGE|
PRI0:AGE|
|Diagonal values in Cholesky matrix, L.
NsQUAL| *** correlated parameters
CsPRICE| (Fixed Parameter) but coefficient is fixed
|Below diagonal values in L matrix. V = L*Lt
PRIC:QUA| (Fixed Parameter)......
|Heteroscedasticity in GMX scale factor
sdMALE| heteroscedasticity
|Variance parameter tau in GMX scale parameter
TauScale| *** overall scaling, tau
|Weighting parameter gamma in GMX model
GammaMXL| (Fixed Parameter)......
|Coefficient on PRICE in WTP space form
Beta0WTP| *** new price coefficient
S_b0_WTP| standard deviation
| Sample Mean Sample Std.Dev.
Sigma(i)| overall scaling
|Standard deviations of parameter distributions
sdQUAL| ***
sdPRICE| (Fixed Parameter)......
Estimated Model for WTP

56. Appendix: extensions

57. Modeling Variations Parameter specification Stochastic specification
“Nonrandom” – variance = 0
Correlation across parameters – random parts correlated
Fixed mean – not to be estimated. Free variance
Fixed range – mean estimated, triangular from 0 to 2
Hierarchical structure - ik = k + k’zi
Stochastic specification
Normal, uniform, triangular (tent) distributions
Strictly positive – lognormal parameters (e.g., on income)
Autoregressive: v(i,t,k) = u(i,t,k) + r(k)v(i,t-1,k) [this picks up time effects in multiple choice situations, e.g., fatigue.]

58. Model Extensions AR(1): wi,k,t = ρkwi,k,t-1 + vi,k,t
Dynamic effects in the model
Restricting sign – lognormal distribution:
Restricting Range and Sign: Using triangular distribution and range = 0 to 2.
Heteroscedasticity and heterogeneity

59. Continuous Random Variation in Preference Weights

60. Aggregate Data and Multinomial Choice: The Model of Berry, Levinsohn and Pakes

61. Resources
Automobile Prices in Market Equilibrium, S. Berry, J. Levinsohn, A. Pakes, Econometrica, 63, 4, 1995, (BLP)
A Practitioner’s Guide to Estimation of Random-Coefficients Logit Models of Demand, A. Nevo, Journal of Economics and Management Strategy, 9, 4, 2000,
A New Computational Algorithm for Random Coefficients Model with Aggregate-level Data, Jinyoung Lee, UCLA Economics, Dissertation, 2011
Elasticities of Market Shares and Social Health Insurance Choice in Germany: A Dynamic Panel Data Approach, M. Tamm et al., Health Economics, 16, 2007,

62. Theoretical Foundation
Consumer market for J differentiated brands of a good
j =1,…, Jt brands or types
i = 1,…, N consumers
t = i,…,T “markets” (like panel data)
Consumer i’s utility for brand j (in market t) depends on
p = price
x = observable attributes
f = unobserved attributes
w = unobserved heterogeneity across consumers
ε = idiosyncratic aspects of consumer preferences
Observed data consist of aggregate choices, prices and features of the brands.

63. BLP Automobile Market Jt t

64. Random Utility Model
Utility: Uijt=U(wi,pjt,xjt,fjt|), i = 1,…,(large)N, j=1,…,J
wi = individual heterogeneity; time (market) invariant. w has a continuous distribution across the population.
pjt, xjt, fjt, = price, observed attributes, unobserved features of brand j; all may vary through time (across markets)
Revealed Preference: Choice j provides maximum utility
Across the population, given market t, set of prices pt and features (Xt,ft), there is a set of values of wi that induces choice j, for each j=1,…,Jt; then, sj(pt,Xt,ft|) is the market share of brand j in market t.
There is an outside good that attracts a nonnegligible market share, j=0. Therefore,

65. Functional Form
(Assume one market for now so drop “’t.”) Uij=U(wi,pj,xj,fj|)= xj'β – αpj + fj + εij = δj + εij
Econsumers i[εij] = 0, δj is E[Utility].
Will assume logit form to make integration unnecessary. The expectation has a closed form.

66. Heterogeneity
Assumptions so far imply IIA. Cross price elasticities depend only on market shares.
Individual heterogeneity: Random parameters
Uij=U(wi,pj,xj,fj|i)= xj'βi – αpj + fj + εij
βik = βk + σkvik.
The mixed model only imposes IIA for a particular consumer, but not for the market as a whole.

67. Endogenous Prices: Demand side
Uij=U(wi,pj,xj,fj|)= xj'βi – αpj + fj + εij
fj is unobserved
Utility responds to the unobserved fj
Price pj is partly determined by features fj.
In a choice model based on observables, price is correlated with the unobservables that determine the observed choices.

68. Endogenous Price: Supply Side
There are a small number of competitors in this market
Price is determined by firms that maximize profits given the features of its products and its competitors.
mcj = g(observed cost characteristics c, unobserved cost characteristics h)
At equilibrium, for a profit maximizing firm that produces one product, sj + (pj-mcj)sj/pj = 0
Market share depends on unobserved cost characteristics as well as unobserved demand characteristics, and price is correlated with both.

69. Instrumental Variables (ξ and ω are our h and f.)

70. Econometrics: Essential Components

71. Econometrics

72. GMM Estimation Strategy - 1

73. GMM Estimation Strategy - 2

74. BLP Iteration

75. ABLP Iteration our ft.  is our (β,)
No superscript is our (M); superscript 0 is our (M-1).

76. Side Results

77. ABLP Iterative Estimator

78. BLP Design Data

79. Exogenous price and nonrandom parameters

80. IV Estimation

81. Full Model

82. Some Elasticities