Transcript Valuing an Environmental Amenity

Independence of
Irrelevant Alternatives
The conditional logit model has
imbedded in it a property
which, if violated in reality,
makes the use of the model
inappropriate.
IIA: the relative probability of
choosing between two
alternatives is independent of
the other alternatives in the
choice set.
Interpretation of Coefficients
in RUMs
 P r(choosingsite k )
  P r( j ) * P r(k )  time cos t
timecostj
 ln P r(choosingsite k )
  P r( j )  time cos t
timecostj
 ln[P r(k ) / P r( j )]
  time cos t
[ timecostk - timecostj ]
Note that effect of change in attribute of site j on
the probability of choosing site k is independent
of all other alternatives.
Evaluating some marginal effects…
Example: what’s the effect of an increase in the
time costs of accessing site 1 by one hour?
Change in
Elasticity
Probability
Choice=Site1
-26.153
-4.67
Choice=Site2
.572
6.168
Choice=Site3
.572
5.506
Choice=Site4
6.068
The constant.572
elasticity over all other sites
with
a change in the time cost of site
1 is a
Choice=Site5
8.411
result of the .572
IIA property.
Testing for Violations
Hausman and McFadden have
developed a test for violations of the
IIA restriction:
Based on comparison of results
when the model is estimated
a) in unrestricted form and
b) with one alternative excluded.
Using the St. Lucia model, evidence
that we have violation of IIA.
Using Nested RUMs
One potential solution to violations of
the IIA restriction is to nest or group
alternatives, such that substitution
within a group is fundamentally
different from substitution across
groups.
IMPORTANT:
Independence of irrelevant
alternatives will be a property that is
forced to hold within any given
“nest” but not across nests.
Setting up a Possible
Nested RUM for the
St Lucia Problem
St Lucia Choice Problem
Car Mode
Bus Mode
Walking
S1 S2 S3 S4 S5 S1 S2 S3 S4 S5 S1 S2 S3 S4 S5
S1, …, S5 are the 5 different sites
Some Notation
To illustrate, let’s subscript the “top
level nest” with k and the “bottom
level nest” with j.
There are Jk bottom level alternatives
associated with top level alternative k.
k=1 car mode
k=2 bus mode
k=3 walk mode
J1 = 5, there are 5 sites available to the
individual when he drives
J2 = 5, there are 5 sites available to the
individual when he takes the bus
J3 = 5, there are 5 sites available to the
individual when he walks
There are all sorts of
ways to nest decisions.
An example for another type of
problem:
Choice among hiking sites
Forested Sites
Site F1 Site F2 Site F3 Site F4
Shoreline Sites
Site S1 Site S2
Site S3
The Nesting Structure is
a Strategic Modeling
Decision
• Sometimes the order of the “tree
structure” is obvious –
• When it is not, the results can
be sensitive to ordering.
The Nesting Structure is a
Strategic Modeling Decision
Sometimes the ordering of nests is obvious:
Choice among hiking sites
Forested Sites
Shoreline Sites
Site F1 Site F2 Site F3 Site F4
Site S1 Site S2
Site S3
But sometimes it is not - such as our case.
The results can be sensitive to ordering of nests.
The Logic of the Nested
Logit is the Same
The utility accruing to individual i
if he chooses mode=k, site=j is:
Ui ( j, k )  Vi ( j, k )  ijk
Where the Vi(j,k) is a function of
the attributes of the jth site and
of the kth mode.
 ijk is assumed to be distributed
as generalized extreme value.
An individual’s contribution to the
likelihood function is his
probability of selecting the
alternative that he is observed to
select:
 k 1
V ( j, k ) 
V ( h, k ) 
exp(
)  exp(
)
 k  h 1
k 
Pr( j, k ) 
m
Jm
K

V ( h, m ) 
)

 exp( 
m 1  h 1
m

Jk
m is a scale parameter that can vary
for different nests. It reflects the
degree of substitutability across
nests. If ’s = 1 then collapses to
simple, unnested RUM.
A Modification of the Utility
Function
To illustrate assume a simple
modification to the form of V(j,k):
V ( j, k )  1cijk  2tijk  3bjk  4sik
Travel cost, time cost, and beach size
vary over all sites,
sik is a variable that has a significance
for mode choice –
sik = 1 for k=car and i owns a car;
sik = 0 otherwise
Partitioning the Probability for
Ease of Interpretation
Pr(j,k) = Pr(j|k) * Pr(k)
 1k cijk   2 k tijk   3k b jk )
exp(
k
Pri ( j | k )  J
 1k cihk   2 k tihk   3k bhk )
exp(

k
h 1
k
P ri ( k ) 
exp( 3sik   k I ik )
K
 exp( s
m 1
3 im
  k I im )
Im is known as the inclusive value and
m is the inclusive value parameter
 1mcihm  2mtihm  3mbhm
I im  ln[ exp(
)]
Jm
h 1
m
Results from A Nested Specification
BTIMECB
BTIMEW
BCOST
BSIZ
BPIGCB
BPIGW
BCAR
Coeff.
1.8804
-0.82423
-0.44
0.736724
1.61992
2.60767
4.56708
Std.Err. t-ratio
1.03115 1.82359
0.261362 -3.15359
0.134826 -3.26349
0.197813 3.72434
0.617403 2.62377
0.799694 3.26084
1.12399 4.06327
P-value
0.068213
0.001613
0.001101
0.000196
0.008696
0.001111
4.84E-05
BTIMECB=coefficient on time cost for car,bus
BTIMEW = coefficient on time cost for walking
BPIGCB = coefficient on Pigeon Pt site for car,bus
BPIGW = coefficient on Pigeon Pt site for walking
Estimates of Inclusive Value Parameter, m
CAR
BUS
WALK
Coeff.
-0.50531
0.526598
1.45374
Std.Err. t-ratio
0.748416 -0.67517
0.476543 1.10504
0.868406 1.67403
P-value
0.499566
0.269143
0.094124
’s are significantly different from 1, not 0.
Welfare Measurement
Using the Nested RUM
If valuing a price change at one or
more sites:
1
(  1cihm
  2 mtihm   3bhm   4 sim )  m
ln{ [ exp
] }
m
CV  m 1 h 1
1
K
Jm
0
(  1cihm
  2 mtihm   3bhm   4 sm )  m
ln{ [ exp
] }
m
 m 1 h 1
1
K
Jm
Estimate for loss due to $5 parking charge is now
$1.25 per person per choice occasion.
Welfare Measurement
Using the Nested RUM
If valuing elimination of one or
more sites:
K
CV 
Jm
ln{ [ exp
m 1 h  2
K

Jm
(  1cihm   2 mtihm   3bhm   4 sim )  m
] }
m
1
ln{ [ exp
m 1 h 1
(  1cihm   2 mtihm   3bhm   4 sm )  m
] }
m
1
WTP for Loss of Sites:
Site Eliminated
Site 1
Site 2
Site 3
Site 4
Site 5
Sites 4 + 5
WTP/person/occasion
$
2.48
$
0.48
$
0.70
$
1.41
$
5.44
$
10.18
Note:
Eliminating both sites 4 and 5 simultaneously
produces more of a loss than the sum of the
two individually, because fewer
substitutes to switch to.
WTP per Choice
Occasion…
All of the measures we’ve gotten
to this point are
WTP per individual
per choice occasion
What does that mean?
Per choice occasion really means
“per trip” in this model.
But trips are an endogenous decision!
It is possible that if quality changes or
sites disappear, individuals may change
the number of trips taken.
So, how do we calculate
the total gains/losses?
Problem:
We have a WTP/trip for the change.
We might be able to calculate the
change in number of trips due to the
exogenous change (quality or site loss)
But can not logically solve for
total change in WTP from this information.
Alternative Solutions in the
Literature
• Kuhn-Tucker model(Phaneuf, Herriges, and Kling )
Completely consistent and utility theoretic,
but very difficult to do.
• Repeat Logit Model –
(Morey)
Depends on identifying “choice occasions”
when individual chooses not to take a trip.
• “Linked Trip and Site Choice
Model –
Ad hoc linking of RUM and
demand function for trips model
Combine RUM and
Single Site Model
The link between these two
models is the inclusive value
calculated for the initial and
subsequent situations.
Inclusive Value (initial situation):
0
(  1cihm
 2 mtihm  3bhm   4 sim ) m
I i  ln{[ exp
] }
m
m 1 h 1
0
K
Jm
Inclusive Value (afterchange):
1
(  1cihm
  2 mtihm  3bhm   4 sim ) m
I  ln{[ exp
] }
m
m 1 h 1
K
1
i
Jm
The Linked Model
1. Estimates the RUM model on site choice.
2. Calculate for each individual, the inclusive
value, Ii0, (which is an index of utility)
for the recreational choice occasion.
3. Estimate a demand for recreational trips
model, irrespective of site chosen.
This takes the form:
zi  f ( I , si )  i
0
i
Where zi = total trips by i
Ii0 is i’s inclusive value
si are socio-demographic variables of i
Points to Note
• The inclusive value captures all
the information about costs and
quality from the set of
alternatives each individual
faces
• The model must be estimated
using one of the truncation
approaches – e.g. the truncated
or endogenously stratified
Poisson Model.
• To do this, you must have total
trip data for the individuals you
intercepted.
Results of Poisson
Model
Coeff.
Std.Err.
t-ratio
Constant
1.11626 0.227897
4.8981
Inclusive Value
0.19337 0.053111
3.6411
Very Low Income
-0.53877 0.246578 -2.18499
Low Income
0.06586 0.119763 0.54989
Average Income
0.44973 0.099848 4.50415
High income
0.05873 0.097249 0.60393
Number of Children 0.12865 0.021924 5.86819
P-value
9.68E-07
0.000272
0.028889
0.582397
6.66E-06
0.545892
4.41E-09
Note: Income variables are dummy variables
for categories, leaving out the very high income
category. Interpretation is effect relative to very high
income group.
Some Useful References
General:
Haab, T, and K. McConnell, 2002, Valuing
Environmental and Natural Resources: The
Econometrics of Non-Market Valuation, Edward
Elgar.
Herriges, Kling, and Phaneuf. 1999 In Valuing
Recreation and the Environment: Revealed
Preferences Methods in Theory and Practice
Greene, William, LIMDEP 7.0,
Choice Sets:
Ben-Akiva and Lerman. 1985. Discrete Choice
Analysis.
Haab and Kicks. 1997. Journal of Environmental
Economics and Management.
Parsons and Hauber. 1998. Land Economics.
Peters, Adamowizc, and Boxall. 1995. Water Resources
Research
Hicks and Strand. 2000. Land Economics
Parson, Plantinga and Boyle. 2000. Land Economics
Nonlinear Income:
Herriges and Kling. 1997. Review of Economics and
Statistics
Nesting Structures:
Kling and Thompson. 1996. American Journal of
Agricultural Economics
Linking Discrete and Continuous Models:
Phaneuf, Herriges, and Kling. 2000. Review of
Economics and Statistics
Morey, Rowe and Wateson. 1993. American Journal of
Agricultural Economics
Bockstael, Hanemann, and Kling. 1987. Water
Resources Research
Mixed Logit (Random Parameters Logit):
Train. 1999. In Valuing Recreation and the
Environment: Revealed Preferences Methods in
Theory and Practice