Nonlinear Meta-Regression for Benefit Transfer · 2018. 12. 13. · Intro Adding-up NL-MRM Model...

Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Nonlinear Meta-Regression for Benefit Transfer

Klaus Moeltner

Presented at the 2018 ACES Meetings,Washington, DC Area, Dec. 3-6, 2018

This research was partially funded under U.S. EPA contract number EP-C-13-039. The findings, conclusions,

and views expressed in this paper are those of the author and do not necessarily represent those of the U.S.

EPA. No agency endorsement should be inferred.

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Intro

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Intro

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Intro

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Intro

log (wtp) = A + βlog(y) + γ0q0 + γ∆ (q1 − q0) , or

wtp = yβexp (A + γ0q0 + γ∆ (q1 − q0))

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Intro

Sensitivity to scope?

∂wtp

∂q1|q0 = γ∆ ∗ wtp,

γ∆ > 0?

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Intro

Sensitivity to scope?

∂wtp

∂q1|q0 = γ∆ ∗ wtp,

γ∆ > 0?

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Intro

Adding-up property?

wtp (q2 − q0)?= wtp (q1 − q0) + wtp (q2 − q1)

yβexp (A) exp (γ0q0 + γ∆ (q2 − q0)) 6=yβexp (A) ∗(exp (γ0q0 + γ∆ (q1 − q0)) + exp (γ0q1 + γ∆ (q2 − q1))) ,

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Intro

Adding-up property?

wtp (q2 − q0)?= wtp (q1 − q0) + wtp (q2 − q1)

yβexp (A) exp (γ0q0 + γ∆ (q2 − q0)) 6=yβexp (A) ∗(exp (γ0q0 + γ∆ (q1 − q0)) + exp (γ0q1 + γ∆ (q2 − q1))) ,

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Intro

Adding-up property?

wtp (q2 − q0)?= wtp (q1 − q0) + wtp (q2 − q1)

yβexp (A) exp (γ0q0 + γ∆ (q2 − q0)) 6=yβexp (A) ∗(exp (γ0q0 + γ∆ (q1 − q0)) + exp (γ0q1 + γ∆ (q2 − q1))) ,

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

• Adding-Up violation is a real “practical headache”

• Many EPA policies proceed in incrementsover time and/or space

• Example: Chesapeake Bay watershed implementation plan toreduce TMDLs

• Multiple phases across sectors and jurisdictions

• Example: Steam Electric Power Plants rule

• EPA needs to know partial benefits from partial cleanup ANDbenefits “still on the table”

• First raised in the peer-reviewed literature byNewbold et al. (2018)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

• Adding-Up violation is a real “practical headache”

• Many EPA policies proceed in incrementsover time and/or space

• Example: Chesapeake Bay watershed implementation plan toreduce TMDLs

• Multiple phases across sectors and jurisdictions

• Example: Steam Electric Power Plants rule

• EPA needs to know partial benefits from partial cleanup ANDbenefits “still on the table”

• First raised in the peer-reviewed literature byNewbold et al. (2018)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

• Adding-Up violation is a real “practical headache”

• Many EPA policies proceed in incrementsover time and/or space

• Example: Chesapeake Bay watershed implementation plan toreduce TMDLs

• Multiple phases across sectors and jurisdictions

• Example: Steam Electric Power Plants rule

• EPA needs to know partial benefits from partial cleanup ANDbenefits “still on the table”

• First raised in the peer-reviewed literature byNewbold et al. (2018)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

• Adding-Up violation is a real “practical headache”

• Many EPA policies proceed in incrementsover time and/or space

• Example: Chesapeake Bay watershed implementation plan toreduce TMDLs

• Multiple phases across sectors and jurisdictions

• Example: Steam Electric Power Plants rule

• EPA needs to know partial benefits from partial cleanup ANDbenefits “still on the table”

• First raised in the peer-reviewed literature byNewbold et al. (2018)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

• Adding-Up violation is a real “practical headache”

• Many EPA policies proceed in incrementsover time and/or space

• Example: Chesapeake Bay watershed implementation plan toreduce TMDLs

• Multiple phases across sectors and jurisdictions

• Example: Steam Electric Power Plants rule

• EPA needs to know partial benefits from partial cleanup ANDbenefits “still on the table”

• First raised in the peer-reviewed literature byNewbold et al. (2018)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

• Adding-Up violation is a real “practical headache”

• Many EPA policies proceed in incrementsover time and/or space

• Example: Chesapeake Bay watershed implementation plan toreduce TMDLs

• Multiple phases across sectors and jurisdictions

• Example: Steam Electric Power Plants rule

• EPA needs to know partial benefits from partial cleanup ANDbenefits “still on the table”

• First raised in the peer-reviewed literature byNewbold et al. (2018)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

• Adding-Up violation is a real “practical headache”

• Many EPA policies proceed in incrementsover time and/or space

• Example: Chesapeake Bay watershed implementation plan toreduce TMDLs

• Multiple phases across sectors and jurisdictions

• Example: Steam Electric Power Plants rule

• EPA needs to know partial benefits from partial cleanup ANDbenefits “still on the table”

• First raised in the peer-reviewed literature byNewbold et al. (2018)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

Example: from “boatable” (q=2.5) to “fishable” (q=5.1) to“swimmable” (q=7.0)

γ̂0 = −0.124

γ̂∆ = 0.2095

q0 = 2.5, q1 = 5.1, q2 = 7.0

wtp (q1 − q0) + wtp (q2 − q1)

wtp (q2 − q0)= 1.09

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

Example: from “boatable” (q=2.5) to “fishable” (q=5.1) to“swimmable” (q=7.0)

γ̂0 = −0.124

γ̂∆ = 0.2095

q0 = 2.5, q1 = 5.1, q2 = 7.0

wtp (q1 − q0) + wtp (q2 − q1)

wtp (q2 − q0)= 1.09

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

Example: from “boatable” (q=2.5) to “fishable” (q=5.1) to“swimmable” (q=7.0)

γ̂0 = −0.124

γ̂∆ = 0.2095

q0 = 2.5, q1 = 5.1, q2 = 7.0

wtp (q1 − q0) + wtp (q2 − q1)

wtp (q2 − q0)= 1.09

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

Example: smaller steps around “fishable”

γ̂0 = −0.124

γ̂∆ = 0.2095

q0 = 5.0, q1 = 5.1, q2 = 5.2

wtp (q1 − q0) + wtp (q2 − q1)

wtp (q2 − q0)= 1.95

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

Example: smaller steps around “fishable”

γ̂0 = −0.124

γ̂∆ = 0.2095

q0 = 5.0, q1 = 5.1, q2 = 5.2

wtp (q1 − q0) + wtp (q2 − q1)

wtp (q2 − q0)= 1.95

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Adding-up

Example: smaller steps around “fishable”

γ̂0 = −0.124

γ̂∆ = 0.2095

q0 = 5.0, q1 = 5.1, q2 = 5.2

wtp (q1 − q0) + wtp (q2 − q1)

wtp (q2 − q0)= 1.95

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

NL-MRM

wtp =

∫ q1

q0

yβexp (A) exp (γq)dq =

yβexp (A)

(1

γ(exp (γq1)− exp (γq0))

)log (wtp) = A + βlog (y) + log

(1

γ(exp (γq1)− exp (γq0))

)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

NL-MRM

yis = x′isβ + log(γ−1 (exp (γq1,is)− exp (γq0,is))

)+ us + εis

us ∼ n(0, σ2

u

),

εis ∼ n(0, σ2

εωis

), where ωis ∼ ig

(ν2 ,

ν2

)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Model Comparison

Meta-Data:

• 140 obs., 51 studies

• wtp for water quality change, 100-point scale

• 22 explanatory variables

• used by EPA to inform steam electric rule revision (ongoing)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Model Comparison

Meta-Data:

• 140 obs., 51 studies

• wtp for water quality change, 100-point scale

• 22 explanatory variables

• used by EPA to inform steam electric rule revision (ongoing)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Model Comparison

Meta-Data:

• 140 obs., 51 studies

• wtp for water quality change, 100-point scale

• 22 explanatory variables

• used by EPA to inform steam electric rule revision (ongoing)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Model Comparison

Meta-Data:

• 140 obs., 51 studies

• wtp for water quality change, 100-point scale

• 22 explanatory variables

• used by EPA to inform steam electric rule revision (ongoing)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Model Comparison

Meta-Data:

• 140 obs., 51 studies

• wtp for water quality change, 100-point scale

• 22 explanatory variables

• used by EPA to inform steam electric rule revision (ongoing)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Model Comparison

Model comparison via Bayes Factors

NL-MRM vs: log-Bayes Factor

MRM1linlin -24.285MRM1linlog -20.958MRM1loglog -23.742

MRM2 -6.560MRM2Slin -9.053MRM2Slog -23.870

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

BT context:

• riperian ecosystem

• can be borrow from remaining (lake) data?

• Bayesian Stochastic Search Variable Selection

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

BT context:

• riperian ecosystem

• can be borrow from remaining (lake) data?

• Bayesian Stochastic Search Variable Selection

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

BT context:

• riperian ecosystem

• can be borrow from remaining (lake) data?

• Bayesian Stochastic Search Variable Selection

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

BT context:

• riperian ecosystem

• can be borrow from remaining (lake) data?

• Bayesian Stochastic Search Variable Selection

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

River data (n=96)

Lake data (n=44)

WTP

WTP

Interactionterms

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

ys = X1sβx + Msβm + Zsδ+

log(

(γ + δγds)−1 (exp ((γ + δγds)q1,s)− exp ((γ + δγds)q0,s)))

+ isus + εs

p (δj ) = p0 ∗ n(0, c2t2

)+ (1− p0) n

(0, t2

),

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

ys = X1sβx + Msβm + Zsδ+

log(

(γ + δγds)−1 (exp ((γ + δγds)q1,s)− exp ((γ + δγds)q0,s)))

+ isus + εs

p (δj ) = p0 ∗ n(0, c2t2

)+ (1− p0) n

(0, t2

),

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

Scenario 1: WQI 25-51, 51-70 (boatable, fishable, swimmable)

BT results, SSVS, total WTP ($)

NL-MRM MRM1linlinscenario (WQI) low mean high low mean high

25 to 51 0.00 50.99 156.53 0.72 32.33 85.5951 to 70 0.00 26.33 80.10 0.94 24.08 63.6525 to 70 0.00 77.19 235.89 0.91 41.12 108.12

adding-up error 0.17% 37.18%

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

Scenario 1: WQI 25-51, 51-70 (boatable, fishable, swimmable)

BT results, SSVS, total WTP ($)

NL-MRM MRM1linlinscenario (WQI) low mean high low mean high

25 to 51 0.00 50.99 156.53 0.72 32.33 85.5951 to 70 0.00 26.33 80.10 0.94 24.08 63.6525 to 70 0.00 77.19 235.89 0.91 41.12 108.12

adding-up error 0.17% 37.18%

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

Scenario 2: WQI 73-73.5, 73.5-74(option D of Steam Electric rule)

BT results, SSVS, total WTP ($)

NL-MRM MRM1linlinscenario (WQI) low mean high low mean high

73 to 73.5 0.00 0.57 1.74 0.46 16.26 43.8973.5 to 74 0.00 0.57 1.73 0.47 16.21 43.7973 to 74 0.00 1.14 3.47 0.28 16.36 43.98

adding-up error 0.18% 98.40%

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

Scenario 2: WQI 73-73.5, 73.5-74(option D of Steam Electric rule)

BT results, SSVS, total WTP ($)

NL-MRM MRM1linlinscenario (WQI) low mean high low mean high

73 to 73.5 0.00 0.57 1.74 0.46 16.26 43.8973.5 to 74 0.00 0.57 1.73 0.47 16.21 43.7973 to 74 0.00 1.14 3.47 0.28 16.36 43.98

adding-up error 0.18% 98.40%

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

• Option D of SER: $ 1.14/HH

• 84.5 million HHs affected

• Benefits: $96 million (2007 dollars)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

• Option D of SER: $ 1.14/HH

• 84.5 million HHs affected

• Benefits: $96 million (2007 dollars)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Empirical Example

• Option D of SER: $ 1.14/HH

• 84.5 million HHs affected

• Benefits: $96 million (2007 dollars)

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Conclusion

• Existing MRMs for quality change theoretically flawed

• Adding-Up: serious problem of high practical relevance

• NL-MRM provides compromise of theoretical consistency andreasonable empirical fit

• Bayesian methods allow for rigorous model comparison,full exploitation of meta-data for BT application

THANK YOU!

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Conclusion

• Existing MRMs for quality change theoretically flawed

• Adding-Up: serious problem of high practical relevance

• NL-MRM provides compromise of theoretical consistency andreasonable empirical fit

• Bayesian methods allow for rigorous model comparison,full exploitation of meta-data for BT application

THANK YOU!

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Conclusion

• Existing MRMs for quality change theoretically flawed

• Adding-Up: serious problem of high practical relevance

• NL-MRM provides compromise of theoretical consistency andreasonable empirical fit

• Bayesian methods allow for rigorous model comparison,full exploitation of meta-data for BT application

THANK YOU!

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Conclusion

• Existing MRMs for quality change theoretically flawed

• Adding-Up: serious problem of high practical relevance

• NL-MRM provides compromise of theoretical consistency andreasonable empirical fit

• Bayesian methods allow for rigorous model comparison,full exploitation of meta-data for BT application

THANK YOU!

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Conclusion

• Existing MRMs for quality change theoretically flawed

• Adding-Up: serious problem of high practical relevance

• NL-MRM provides compromise of theoretical consistency andreasonable empirical fit

• Bayesian methods allow for rigorous model comparison,full exploitation of meta-data for BT application

THANK YOU!

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Intro Adding-up NL-MRM Model Comparison Empirical Example Conclusion

Conclusion

• Existing MRMs for quality change theoretically flawed

• Adding-Up: serious problem of high practical relevance

• NL-MRM provides compromise of theoretical consistency andreasonable empirical fit

• Bayesian methods allow for rigorous model comparison,full exploitation of meta-data for BT application

THANK YOU!

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