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Estimation of willingness to pay in preferencespace vs. WTP space
Arne Risa HoleUniversity of She¢ eld
Danish Choice Modelling Day - Odense 8th December 2011
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Background
Increased use of mixed logit to estimate willingness to pay inapplied economics
Standard approach: assume a distribution for the coe¢ cientsand derive WTP for an attribute as the ratio of the attributecoe¢ cient to an estimate of the marginal utility of money
Can lead to WTP distributions which are heavily skewed andthat may not even have de�ned moments
Train and Weeks (2005) suggested re-formulating the modelsuch that assumptions are made regarding the distribution ofWTP
Motivating example: health workers choice betweenhypothetical jobs
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Utility in preference space
The utility person n derives from choosing job j in choicesituation t is speci�ed as
eUnjt = αnwnjt + βnxnjt + εnjt/σn (1)
εnjt is assumed to be extreme value distributed with scale σn
Muliplying through by σn yields
Unjt = (σnαn)wnjt + (σnβn)xnjt + εnjt (2)
It is not possible to separately identify σn and αn/βn !standard practice to normalise σn to 1. Assumes εnjt ishomoscedastic (more on that later).
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Utility in willingness to pay space
Train and Weeks (2005) suggest rewriting equation (2) as
Unjt = αn [wnjt + γnxnjt ] + εnjt (3)
Uses the fact that the WTP for the attributes is given by
γn = βn/αn
The models are behaviourally equivalent but standardassumptions regarding the distributions of αn and βn in (2)can lead to unusual distributions for WTP
The WTP space approach avoids this problem by specifyingthe distributions for WTP directly
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
The Generalized Multinomial Logit Model
Let�s go back to the utility function in preference space
Unjt = (σnαn)wnjt + (σnβn)xnjt + εnjt
Fiebig et al. (2010) propose that instead of imposing theσn = 1 normalisation σn speci�ed as
exp(σ+ θzn + τε0n)
where ε0n � N(0, 1) and zn is a vector of characteristics ofperson n
Relaxes the assumption of homoscedastic errors
Since only relative scale di¤erences can be identi�ed σ is setto �τ2/2 which implies that E (σn) = 1 when θ = 0
This speci�cation is called GMNL-II by Fiebig et al.
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Hensher and Greene (2010) show that the GMNL model neststhe preference space and WTP space models
It is easy to see that the GMNL model reduces to thepreference space model when τ = θ = 0
We can also see that
Unjt = (σnαn)wnjt + (σnβn)xnjt + εnjt
= σn [αnwnjt + βnxnjt ] + εnjt
becomes the WTP space model when αn = 1
σn = exp(σ+ θzn + τε0n) with zn = 1 can then beinterpreted as the (log-normally distributed) wage parameter
βn can be interpreted as the WTP estimates
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
GMNL Stata command
Developed in collaboration with Yuanyuan Gu and StephanieKnox at the University of Technology, Sydney
Can be used to estimate models in WTP space and allvariants of the GMNL model described in Fiebig et al.
Postestimation commands for generating predictedprobabilities, individual-level parameter estimates etc.
The module, including an example dataset and a workingpaper is available at
http://www.shef.ac.uk/economics/people/hole/stata
Comments and suggestions are very welcome
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Example: Households�choice of electricity supplier
Subset of the data from Huber and Train (2000)
Residential electricity customers presented with a series ofexperiments with four alternative electricity suppliers
Price is either �xed or a variable rate that depends on thetime of day or the season
The following attributes are included in the experiment:
Price in cents per kWh if �xed price, 0 if TOD or seasonal ratesContract length in yearsWhether a local company (0-1 dummy)Whether a well-known company (0-1 dummy)TOD rates (0-1 dummy)Seasonal rates (0-1 dummy)
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
First 16 records in dataset
. use http://fmwww.bc.edu/repec/bocode/t/traindata.dta, clear
. list in 1/12, sepby(gid)
++| y price contract local wknown tod seasonal gid pid |||
1. | 0 7 5 0 1 0 0 1 1 |2. | 0 9 1 1 0 0 0 1 1 |3. | 0 0 0 0 0 0 1 1 1 |4. | 1 0 5 0 1 1 0 1 1 |
||5. | 0 7 0 0 1 0 0 2 1 |6. | 0 9 5 0 1 0 0 2 1 |7. | 1 0 1 1 0 1 0 2 1 |8. | 0 0 5 0 0 0 1 2 1 |
||9. | 0 9 5 0 0 0 0 3 1 |
10. | 0 7 1 0 1 0 0 3 1 |11. | 0 0 0 0 1 1 0 3 1 |12. | 1 0 0 1 0 0 1 3 1 |
++
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Model with �xed coe¢ cients (MNL)
. clogit y price contract local wknown tod seasonal, group(gid)
Iteration 0: log likelihood = 1379.3159(output omitted)Iteration 4: log likelihood = 1356.3867
Conditional (fixedeffects) logistic regression Number of obs = 4780LR chi2(6) = 600.47Prob > chi2 = 0.0000
Log likelihood = 1356.3867 Pseudo R2 = 0.1812
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
+price | .6354853 .0439523 14.46 0.000 .7216302 .5493403
contract | .13964 .0161887 8.63 0.000 .1713693 .1079107local | 1.430578 .0963826 14.84 0.000 1.241672 1.619485wknown | 1.054535 .086482 12.19 0.000 .8850338 1.224037
tod | 5.698954 .3494016 16.31 0.000 6.383769 5.01414seasonal | 5.899944 .35485 16.63 0.000 6.595437 5.204451
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Willingness to pay estimates
Based on the MNL results we can calculate the WTPestimates using the Stata command wtp:
. wtp price contract local wknown tod seasonal
contract local wknown tod seasonalwtp .21973759 2.2511589 1.6594175 8.9678781 9.2841551ll .27319536 1.8855365 1.3653397 9.2764201 9.6128415ul .16627982 2.6167813 1.9534953 8.6593361 8.9554687
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Model with �xed coe¢ cients in WTP space
. gen mprice = price
. gen const = 1
. constraint 1 [Mean]mprice = 1
. constraint 2 [tau]_cons = 0
. matrix start = 1,0,0,0,0,0,0,0
. gmnl y mprice contract local wknown tod seasonal, group(gid) id(pid) het(const)constraint(1 2) from(start, copy) nrep(1)
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Iteration 0: log likelihood = 6814.2393 (not concave)(output omitted)Iteration 14: log likelihood = 1356.3867
Generalized multinomial logit model Number of obs = 4780Wald chi2(5) = 5133.89
Log likelihood = 1356.3867 Prob > chi2 = 0.0000
( 1) [Mean]mprice = 1( 2) [tau]_cons = 0
(Std. Err. adjusted for clustering on pid)
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]+Mean |
mprice | 1 . . . . .contract | .2197376 .0272749 8.06 0.000 .2731953 .1662798
local | 2.251159 .1865454 12.07 0.000 1.885537 2.616781wknown | 1.659417 .1500424 11.06 0.000 1.36534 1.953495
tod | 8.967877 .1574222 56.97 0.000 9.276419 8.659336seasonal | 9.284155 .1677001 55.36 0.000 9.612841 8.955469
+Het |
const | .4533659 .0691633 6.56 0.000 .5889235 .3178082+
/tau | (omitted)The sign of the estimated standard deviations is irrelevant: interpret them asbeing positive
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
The price coe¢ cient is given by
. nlcom (price: exp([Het]const))
price: exp([Het]const)
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
+price | .6354856 .0439523 14.46 0.000 .7216305 .5493407
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Model with random coe¢ cients in WTP space
. matrix start = e(b),0.1,0.1,0.1,0.1,0.1
. gmnl y mprice, group(gid) id(pid) rand(contract local wknown tod seasonal)het(const) constraint(1) from(start, copy) nrep(100) gamma(0)
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Iteration 0: log likelihood = 1352.9782 (not concave)(output omitted)Iteration 11: log likelihood = 1122.3142
Generalized multinomial logit model Number of obs = 4780Wald chi2(5) = 2303.48
Log likelihood = 1122.3142 Prob > chi2 = 0.0000
( 1) [Mean]mprice = 1(Std. Err. adjusted for clustering on pid)
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
+Mean |
mprice | 1 . . . . .contract | .2535117 .0562202 4.51 0.000 .3637012 .1433221
local | 2.09998 .2201434 9.54 0.000 1.668507 2.531454wknown | 1.535489 .1582956 9.70 0.000 1.225235 1.845743
tod | 9.455161 .2898365 32.62 0.000 10.02323 8.887092seasonal | 9.379777 .2524339 37.16 0.000 9.874539 8.885016
+Het |
const | .0332425 .0856309 0.39 0.698 .201076 .1345909+SD |
contract | .4615008 .0593945 7.77 0.000 .5779119 .3450897local | 1.706872 .2254088 7.57 0.000 1.265079 2.148665wknown | 1.126835 .1862598 6.05 0.000 .7617721 1.491897
tod | 2.398136 .3486618 6.88 0.000 3.081501 1.714771seasonal | 2.205005 .329771 6.69 0.000 2.851344 1.558666
+/tau | .3172959 .1406606 2.26 0.024 .0416063 .5929856
The sign of the estimated standard deviations is irrelevant: interpret them asbeing positive
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Price is assumed to be log-normally distributed. The mean ofthe underlying normal distribution is
. nlcom (price_mean: [Het]const[tau]_cons^2/2)
price_mean: [Het]const[tau]_cons^2/2
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
+price_mean | .0835809 .0715274 1.17 0.243 .223772 .0566102
The "-[tau]_cons^2/2" term is due to the normalisation ofsigma in the GMNL model
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Example: Health workers choice of job
Joint work with Julie Riise Kolstad at the University of Bergen
Discrete choice experiment on the choice of health servicejobs among Tanzanian �nal-year students training to beClinical O¢ cers (COs)
The aim of the experiment was to elicit the students�preferences for di¤erent features of health service jobs
320 �nalists (around 60% of all CO �nalists in Tanzania in2007) from 10 randomly selected schools participated in theDCE
After excluding incomplete responses we were left with anestimation sample of 296 respondents
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
The attributes in the choice experiment were chosen followingextensive literature searches and early in-depth interviews
The attributes included the wage of the job, educationprospects and other characteristics related to the location ofthe job and the facilities of the workplace
We used a D-optimal design to construct the hypotheticalchoice situations
The result was a set of 32 choice situations that wererandomly divided into two blocks
Each respondent was presented with 16 choice situationswhere each of these represented the choice between twohypothetical jobs
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
An example choice set
Job AAvailability ofequipment &drugs:
Housing: Education opportunities/possibility of upgradingqualifications:
Workload: Infrastructure: Salary andallowances:
Location:
Sufficient No house isprovided.
Education offered after 6 yearsof service.
Normal: Nearly enough time tocomplete duties. One hour ofextra work per day.
The place has mobilecoverage, electricity andwater.
350,000TSH permonth
Regionalheadquarters
Job BAvailability ofequipment &drugs:
Housing: Education opportunities/possibility of upgradingqualifications:
Workload: Infrastructure: Salary andallowances:
Location:
Insufficient A decenthouse isprovided.
Education offered after 2 yearsof service.
Heavy; barely enough time tocomplete duties. Three hours ofextra work per day.
The place has unreliablemobile coverage, noelectricity or water.
500,000TSH permonth
A 3hour or morebus ride from thedistrictheadquarters
Considering your current situation, which of the two jobs would you choose?
Job A: Job B:
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Model speci�cation
We estimated six di¤erent choice models:
1: Logit2: ML with �xed wage coef.3: ML pref. space, all coefs. random4: ML pref. space, all coefs. random and some correlated5: ML WTP space, all coefs. random6: ML WTP space, all coefs. random and some correlated
Coe¢ cients for wage, education, infrastructure and equipmentare given log-normal distributions, all other coefs. arenormally distributed
We use 1000 Halton draws to estimate the ML models withindependent coefs. and 2500 draws for the models withcorrelated coefs.
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Key �ndings
Evidence of substantial amount of heterogeneity in preferencesfor wage, further education, equipment, infrastructure andworkload
Suggests that model 2 (�xed wage coef.) is too restrictive
Mean WTP estimates derived from pref. space models arevery high:
1 2 3 4 5 6
Education 2yrs 357 416 849 561 390 369Infrastructure 222 237 466 353 205 262
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
WTP for further education after 2 years
0.0
005
.001
.001
5.0
02.0
025
Den
sity
0 500 1000 1500WTP (1000 TSH per month)
Model 3 Model 4Model 5 Model 6
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
WTP for infrastructure
0.0
01.0
02.0
03.0
04.0
05D
ensit
y
0 200 400 600 800 1000WTP (1000 TSH per month)
Model 3 Model 4Model 5 Model 6
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Goodness of �t
1 2 3 4 5 6
LL -2424 -2335 -2267 -2226 �2278 -2228AIC 4872 4715 4580 4499 4601 4501BIC 4950 4857 4728 4647 4750 4650
Preference space models �t data better
But: bigger di¤erence between models allowing/not allowingfor preference heterogeneity in wages and correlated vs.uncorrelated coe¢ cients
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
Conclusions
Although it is common in practice to specify that thecoe¢ cient for the monetary attribute is �xed, this may beunrealistic
Allowing for preference heterogeneity is not straightforward asit can lead to implausible WTP estimates
In line with the literature from other �elds our evidencesuggest that models estimated in WTP space produce morerealistic WTP estimates
Preference space models �t data better but best �ttingmodels in the two regimes have similar GOF
Our results suggest that sensitivity testing is important
Arne Risa Hole University of She¢ eld Estimation of WTP in preference space vs. WTP space
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