Mapping the WTP Distribution from Individual Level Parameter Estimates

Matthew W. WindenUniversity of Wisconsin - Whitewater

SEA Conference – November 2012

Motivation Heterogeneity exists in respondents’ preferences, WTP, and

error variances within the population (Lanscar and Louviere 2008)

Traditional Models Used in Non-Market Valuation Impose Distributional Assumptions About Preference Heterogeneity in the Population (Train 2009, Revelt and Train 1999)

Top-Down Modeling Mixed Logit – Impose Continuous Distribution Latent Class Logit – Impose Discrete Distribution

Misspecification May Lead to Bias in Parameter, Marginal Price (MP), and Willingness-To-Pay (WTP) Estimates Leads to inefficient policy analysis and recommendations

SEA November 16th, 2012 2/12 Matthew Winden, UW - Whitewater

Individual Level Modeling As Solution Louviere et al. (2008) estimate individual level parameters

using conditional logit estimator (no welfare analysis) Convergence issue 1: Collinearity of attributes Convergence issue 2: Perfect Predictability Cognitive Burden (Number of Questions/Attributes)

Louviere et al. (2010) Best-Worst Scaling As Solution

Why Individual Level Models? “Bottom-Up Modeling Approach”

3/12 Matthew Winden, UW - Whitewater SEA November 16th, 2012

Top-Down Versus Bottom-Up Modeling

“Top-Down” “Bottom-Up”

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Assume

(, ) Estimate Derive

Estimate Derive

SEA November 16th, 2012

Contributions Objective 1: Use Monte-Carlo Simulation to Provide Evidence

of the Validity of Individual Level Estimation Techniques

Objective 2a: Estimate Traditional and Individual Level Models on a Stated Preference Dataset Eliminates Collinearity as a Convergence Problem

Objective 2b: Estimate Traditional and Individual Level Models on a Revealed Preference Dataset

Objective 3: Use Individual Level Estimates to Examine Potential Bias Resulting from Distributional Assumptions in Traditional Models

5/12 Matthew Winden, UW - Whitewater SEA November 16th, 2012

Traditional Mixed Logit

P(j|vi) = exp(Uji)/Σexp(Uji)

Utility of choice j for respondent i:

= αji + Βj + ΦjZji + ΘjiWji

where:

αji = alternative-specific constant

Βj = vector of fixed coefficients

Χi = fixed individual characteristics

Φj = vector of fixed coefficients

Θj = vector of varying coefficients

Zji & Wji = choice-varying attributes of choices6/12 Matthew Winden, UW - Whitewater SEA November 16th, 2012

Individual Level Simulation & Estimation Strategy 3 Datasets (A, B, C)

Known parameter, attribute, and error distributions 100 respondents, 100 choice scenarios Face 3 attributes (X1 & X2 - Uniform, X3 – Zero, Status Quo) Face 3 alternatives (Respondent Specific Error Term to Each Alternative) Have 3 individual specific betas for each of the three attributes

Simulation A Beta 1 = Normal, Beta 2 = Normal, Beta 3 = Normal

Simulation B Beta 1 = Normal, Beta 2 = Normal, Beta 3 = Uniform

Simulation C Beta 1 = Normal, Beta 2 = Normal, Beta 3 = Exponential

7/12 Matthew Winden, UW - Whitewater SEA November 16th, 2012

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Individual Level Model Simulation

Simulation ML LL ML+I LL ML X3 β ML+I X3 β True X3 β

A -1282.69 -1194.94 4.458 4.599 3.945

B -1597.3 -1302.81 2.701 4.584 3.826

C -1864.74 -1545.42 4.273 4.878 4.004

Matthew Winden, UW - Whitewater

LL Values Indicate Stronger Fit

Not Statistically Significantly Different

SEA November 16th, 2012

Table 34: Willingness-To-Pay Estimates ($/Gal)

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Traditional and Individual Model Comparisons

Mixed Logit 1(Normal β’s)

Mixed Logit 2(Log-Normal β’s)

Individual(Mapped β’s)

Attribute MP (S.E.)

MP (S.E.)

MP

ED 0.029 (0.002)

0.031 (0.003)

0.086

NR 0.022 (0.002)

0.024 (0.002)

0.065

HH 0.034 (0.003)

0.035 (0.003)

0.094

Adj. 0.365 0.335 0.515

LL -5372.6 -5623.7 -4099.23

0.00 0.00 0.00

Matthew Winden, UW - Whitewater SEA November 16th, 2012

Kolmogorov-Smirnov tests reject the null hypothesis of the equivalence of normal and log-normal distributions

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Individual Level Mapped Distribution

Matthew Winden, UW - Whitewater SEA November 16th, 2012

Conclusions? (So-Far) Result 1: Validity of Individual Estimation Demonstrated

through Simulation Kind Of...

Result 2: Individual Level Model Distributions, MPs, & WTPs Differ from Outcomes Using Traditional Models Role of Including or Excluding Individuals with Statistically

Significant (but possibly Lexicographic) Preferences on Estimates Role of Including or Excluding Individuals with Statistically

Insignificant values (Round to Zero?)

Result 3: Without knowing underlying distribution, may inadvertently choose incorrect mixing distribution based on LL Although in the SP Case, the results are not statistically different

11/12 Matthew Winden, UW - Whitewater SEA November 16th, 2012

Extensions E1: True (Full) Monte-Carlo Simulation For Individual Level

Specifcations Vary Over Types of Non-Traditional Distributions and Number of

Respondents, Choice Occasions, and Attributes

E2: Comparison using Revealed Preference Dataset Introduces Potential Collinearity as a Convergence Issue More Realistic Situation Under Which Heterogenity May Exist

E3: Significance Tests for Individual Level Models E4: Compare Results Against Latent Class Models E5: Scale Issues in Aggregation of Individual Respondents

12/12 Matthew Winden, UW - Whitewater SEA November 16th, 2012

Thank You!