Transcript Part 24: Stated Choice [1/47] Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business.

Part 24: Stated Choice [1/47]
Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
Part 24: Stated Choice [2/68]
Econometric Analysis of Panel Data
24A.
Multinomial
Choice
Extensions
Part 24: Stated Choice [3/68]
Rank Data and Best/Worst
Part 24: Stated Choice [4/47]
Part 24: Stated Choice [5/47]
Rank Data and Exploded Logit
Alt 1 is the best overall
Alt 3 is the best among
remaining alts 2,3,4,5
Alt 5 is the best among
remaining alts 2,4,5
Alt 2 is the best among
remaining alts 2,4
Alt 4 is the worst.
Part 24: Stated Choice [6/47]
Exploded Logit
U[j] = jth favorite alternative among 5 alternatives
U[1] = the choice made if the individual indicates only the favorite
Prob{j = [1],[2],[3],[4],[5]} = Prob{[1]|choice set = [1]...[5]} 
Prob{[2]|choice set = [2]...[5]} 
Prob{[3]|choice set = [3]...[5]} 
Prob{[4]|choice set = [4],[5]} 
1
Part 24: Stated Choice [7/47]
Exploded Logit
U[j] = jth favorite alternative among 5 alternatives
U[1] = the choice made if the individual indicates only the favorite
Individual ranked the alternatives 1,3,5,2,4
Prob{This set of ranks}
exp(x3 )
exp(x1 )
=


 j 1,2,3,4,5 exp(x j )  j 2,3,4,5 exp(x j )

exp(x5 )
j  2,4,5
exp(x j )


exp(x 2 )
j  2,4
exp(x j )
1
Part 24: Stated Choice [8/47]
Best Worst



Individual simultaneously ranks best and
worst alternatives.
Prob(alt j) = best = exp[U(j)] / mexp[U(m)]
Prob(alt k) = worst = exp[-U(k)] / mexp[-U(m)]
Part 24: Stated Choice [9/47]
Part 24: Stated Choice [10/47]
Choices
Part 24: Stated Choice [11/47]
Best
Part 24: Stated Choice [12/47]
Worst
Part 24: Stated Choice [13/47]
Part 24: Stated Choice [14/47]
Uses the result that if U(i,j) is the lowest utility, -U(i,j) is the highest.
Part 24: Stated Choice [15/47]
Uses the result that if U(i,j) is the lowest utility, -U(i,j) is the highest.
Part 24: Stated Choice [16/47]
Nested Logit Approach.
Part 24: Stated Choice [17/47]
Nested Logit Approach – Different Scaling for Worst
8 choices are two blocks of 4.
Best in one brance, worst in the
second branch
Part 24: Stated Choice [18/47]
Part 24: Stated Choice [19/47]
Part 24: Stated Choice [20/47]
Part 24: Stated Choice [21/47]
Model Extensions
βi = β + Δzi + Γwi


AR(1): wi,k,t = ρkwi,k,t-1 + vi,k,t
Dynamic effects in the model
Restricting sign – lognormal distribution:
βi,k = exp(μk + δk zi + γk wi )


Restricting Range and Sign: Using triangular
distribution and range = 0 to 2.
Heteroscedasticity and heterogeneity
σk,i = σk exp(θhi )
Part 24: Stated Choice [22/47]
Error Components Logit Modeling


Alternative approach to building cross
choice correlation
Common ‘effects.’ Wi is a ‘random individual
effect.’
U(brand1)i = β1Fashion1,i +β2Quality1,i +β3Price1,i + ε Brand1,i + σ Wi
U(brand2)i = β1Fashion2,i +β2Quality 2,i +β3Price2,i + ε Brand2,i + σ Wi
U(brand3)i = β1Fashion3,i +β2Quality 3,i +β3Price3,i + εBrand3,i + σ Wi
U(None)
= β4
+ εNo Brand,i
Part 24: Stated Choice [23/47]
Implied Covariance Matrix
Nested Logit Formulation
Var[ε] = π 2 / 6 =1.6449
Var[W] =1
 εBrand1 + σW  1.6449 + σ 2
σ2
σ2
0 


ε

2
2
2
+
σW
σ
1.6449
+
σ
σ
0
Brand2

 = Var ε + σW  =  σ 2
2
2
σ
1.6449 + σ
0 
Brand3




ε
0
0
0
1.6449
NONE

 

2
2
Cross Brand Correlation = σ / [1.6449 + σ ]
Part 24: Stated Choice [24/47]
Error Components Logit Model
----------------------------------------------------------Error Components (Random Effects) model
Dependent variable
CHOICE
Log likelihood function
-4158.45044
Estimation based on N =
3200, K =
5
Response data are given as ind. choices
Replications for simulated probs. = 50
Halton sequences used for simulations
ECM model with panel has
400 groups
Fixed number of obsrvs./group=
8
Number of obs.= 3200, skipped
0 obs
--------+-------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
--------+-------------------------------------------------|Nonrandom parameters in utility functions
FASH|
1.47913***
.06971
21.218
.0000
QUAL|
1.01385***
.06580
15.409
.0000
PRICE|
-11.8052***
.86019
-13.724
.0000
ASC4|
.03363
.07441
.452
.6513
SigmaE01|
.09585***
.02529
3.791
.0002
--------+--------------------------------------------------
Random Effects Logit Model
Appearance of Latent Random
Effects in Utilities
Alternative
E01
+-------------+---+
| BRAND1
| * |
+-------------+---+
| BRAND2
| * |
+-------------+---+
| BRAND3
| * |
+-------------+---+
| NONE
|
|
+-------------+---+
Correlation = {0.09592 / [1.6449 + 0.09592]}1/2 = 0.0954
Part 24: Stated Choice [25/68]
Hybrid Choice Models
Part 24: Stated Choice [26/47]
What is a hybrid choice model?

Incorporates latent variables in choice model

Extends development of discrete choice
model to incorporate other aspects of
preference structure of the chooser
Develops endogeneity of the preference
structure.

Part 24: Stated Choice [27/47]
Endogeneity

"Recent Progress on Endogeneity in Choice Modeling" with Jordan Louviere &
Kenneth Train & Moshe Ben-Akiva & Chandra Bhat & David Brownstone & Trudy
Cameron & Richard Carson & J. Deshazo & Denzil Fiebig & William Greene & David
Hensher & Donald Waldman, 2005. Marketing Letters Springer, vol. 16(3), pages
255-265, December.

Narrow view:

Broader view:
U(i,j) = b’x(i,j) + (i,j), x(i,j) correlated with (i,j)
(Berry, Levinsohn, Pakes, brand choice for cars, endogenous price attribute.)
Implications for estimators that assume it is.




Sounds like heterogeneity.
Preference structure: RUM vs. RRM
Heterogeneity in choice strategy – e.g., omitted attribute models
Heterogeneity in taste parameters: location and scaling
Heterogeneity in functional form: Possibly nonlinear utility functions
Part 24: Stated Choice [28/47]
Heterogeneity

Narrow view: Random variation in marginal
utilities and scale




RPM, LCM
Scaling model
Generalized Mixed model
Broader view: Heterogeneity in preference
weights



RPM and LCM with exogenous variables
Scaling models with exogenous variables in
variances
Looks like hierarchical models
Part 24: Stated Choice [29/47]
Heterogeneity and the MNL Model
P[choice j | i] =
exp(α j + β'x ij )

J(i)
exp(α
+
β
'
x
)
j
ij
j=1
Part 24: Stated Choice [30/47]
Observable Heterogeneity in
Preference Weights
Hierarchical model - Interaction terms
Uij = α j + βi x ij + γj zi + ε ij
βi = β + Φhi
Parameter heterogeneity is observable.
Each parameter βi,k = βk + φkhi
Prob[choice j | i] =
exp(α j + βi x ij + γj zi )



exp(α
+
β
x
+
γ
zi )
j
i
ij
j=1
Ji
Part 24: Stated Choice [31/47]
‘Quantifiable’ Heterogeneity in Scaling
Uij = α j + β'x ij + γj z i + ε ij
Var[ε ij ] = σ exp(δj w i ), σ = π / 6
2
j
2
1
wi = observable
characteristics:
age, sex, income, etc.
2
Part 24: Stated Choice [32/47]
Unobserved Heterogeneity in Scaling
HEV formulation: U ij  xij  (1 / i )ij
Generalized model with  = 1 and  = [0].
Produces a scaled multinomial logit model with  i  i 
exp(i xij )
Prob(choicei = j) =
, i  1,..., N , j  1,..., J i
Ji
 j 1 exp(i xij )
The random variation in the scaling is
i  exp(2 / 2  wi )
The variation across individuals may also be observed, so that
i  exp(2 / 2  wi  z i )
Part 24: Stated Choice [33/47]
A helpful way to view hybrid choice models



Adding attitude variables to the choice model
In some formulations, it makes them look like
mixed parameter models
“Interactions” is a less useful way to interpret
Part 24: Stated Choice [34/47]
Observable Heterogeneity in Utility Levels
Uij = α j + β'xij + γj zi + εij
Prob[choice j | i] =
exp(α j + β'xij + γjzi )

J(i)
j=1
exp(α j + β'xij + γzi )
Choice, e.g., among brands of cars
xitj = attributes: price, features
zit = observable characteristics: age, sex,
income
Part 24: Stated Choice [35/47]
Unbservable heterogeneity in utility levels
and other preference indicators
zi  bw i  i
Multinomial Choice Model
Uij = α j + β'x ij + γj zi + ε it
Prob[choice j | i] =
exp(α j + β'x ij + γj zi )

Jt (i)
j=1
exp(α j + β'x ij + γzi )
Indicators (Measurement) Model(s)
Outcomes yim = f m (zi ,vim )
Part 24: Stated Choice [36/47]
Part 24: Stated Choice [37/47]
Part 24: Stated Choice [38/47]
Part 24: Stated Choice [39/47]
Observed Latent
Observed
x

z* 
z1*  1h1  u1
y
z2*  2h 2  u2
z3*  3h 3  u3
y1  g1 ( z1* , 1 )
y2  g 2 ( z1* , 1 )
y3  g3 ( z1* , z2* , 1 )
y4  g 4 ( z2* ,  2 )
y5  g5 ( z2* ,  2 )
y6  g 6 ( z3* , 3 )
y7  g 7 ( z3* , 3 )
Part 24: Stated Choice [40/47]
MIMIC Model
Multiple Causes and Multiple Indicators
X
βx+w  z*
z*
 y1

 y2
 ...

 yM
  1
 
   2
  ...
 
  M
Y

 1


 z*    2

 ...



M






Part 24: Stated Choice [41/47]
should be  kl
Note. Alternative i,
Individual j.
xik
Part 24: Stated Choice [42/47]
U ij =

k


k


k


k


k xik  
l
k xik  
k
k xik  
k
k xik  

 kl xik  jl  ij
k

 kl  jl xik  ij
l


   kl  jl  xik  ij
 l

  x
*
kj
ik
 ij
k
*



 k kj  xik  ij
k
This is a mixed logit model. The interesting extension
is the source of the individual heterogeneity in the
random parameters.
Part 24: Stated Choice [43/47]
“Integrated Model”
Incorporate attitude measures in preference
structure
Part 24: Stated Choice [44/47]
Part 24: Stated Choice [45/47]
Part 24: Stated Choice [46/47]


Hybrid choice
Equations of the MIMIC Model
Part 24: Stated Choice [47/47]
Identification Problems



Identification of latent variable models with
cross sections
How to distinguish between different latent
variable models. How many latent variables are
there? More than 0. Less than or equal to the
number of indicators.
Parametric point identification