Research in Applied Econometrics Chapter 2. Contingent Valuation Econometrics

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Research in Applied Econometrics

Chapter 2. Contingent Valuation Econometrics

Pr. Philippe Polomé, Université Lumière Lyon 2

M1 APE Analyse des Politiques Économiques M1 RISE Gouvernance des Risques Environnementaux

2020 – 2021

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Research in Applied Econometrics Chapter 2. Contingent Valuation Econometrics Principle of Economic Value

Outline

Principle of Economic Value

Principles of Contingent Valuation A Summary of Logit & Probit Models

A Summary of Maximum Likelihood Estimation Single-bounded CV Estimation

Application

Computing Welfare Measures Double-bounded CV Estimation

Application : Exxon-Valdez

Questionnaire Issues and Biases

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Jules Dupuit (1804 – 1866)

French civil/bridge engineer & economist

Had to choose among many demands for building new bridges.

If a fee (péage) made it possible to pay for the bridge building & exploitation, investment will (in the long run) be prof- itable/reasonnable.

However, Dupuit argues that some of the individuals who cross the bridge would be willing to pay more than the fee.

Thus, from the viewpoint of a public operator, one should consider the maximum amount that individuals are willing to pay.

The difference between this amount and the (paid) fee is the consumer surplus.

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Economic Impact of the Bridge

I

1º approximation : fee

×

number of passages

I Price×quantities

I

Dupuit : among those who cross, some are willing to pay more than the fee

I They have asurplus

I We should therefore sum these to the fee to find the economic value of the bridge

I

Whether the fee is actually paid or not

does not change

the economic value of the bridge

I It changes only the division of the value between the bridge manager and users (who gets what part)

I However, the absence of a fee means it actually becomes zero and thus the bridge is (likely) used by more people

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Graphical Representation of the Bridge Example

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Economic Value

I

Therefore the Willingness to pay (WTP) is the proper measure of economic value in a public decision framework

I Based on individual utility I NOT a price or a cost

I non-economists often confuse them I NOT financial or accounting

I May be purely immaterial

I No actual (or future) transaction is required to define a value

I

Willingness to accept compensation is also valid in case of loss

I

They are a standard notion of value in applied economics

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Issues in Non-market Values

I

Commensurability

I Is the environment really amenable to a single quality index measure ? I e.g. what is air “quality”

? what pollutants ? How do you combine them ? I

Measurement, actually

getting proper data

I

Substitutability: can people actually compare coffee and whales ?

I

Sustainability : is natural

capital actually convertible

to financial capital ?

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Nonmarket Values

I

WTP are limited by the individual budget

I =⇒ in this sense, they represent a capacity to pay I There is an interpretation in terms of public finance : the

budget that a collectivity could levy to finance the environnemental corresponding to the WTP

I =⇒ Other things equal, with the utility function, a rich person’s WTP will be higher than a poor’s

I So that the rich person’s “opinion” will weight more in the collectivity budget

I

WTP and compensations are expressed in money

I They are thus comparable between individuals and can be added

I Usually not the case w/ non-economic notions of value

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Understanding the sources of economic value : a typology

Source Example in a forest context

Direct Consumptive Use (private goods)

Hunting and gathering products Wooden products / Cultivation Direct Recreational Use

(public goods)

Hunting and gathering practices Hiking / Nature watching

Indirect/functional Use

Water: Filter / Flood protection Air: Filter / Fixing carbon Soil: Erosion / Desertification Landscape

Option Use: Preserve future / 3rd party use Quasi-use: Value of information

Non-use

“Patrimonial”: Existence & Heritage

“Moral”: Role of humanity wrt nature, Non-human rights

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French Guidelines (Valeurs tutélaires) for Transport

¹

I

Context of road infrastructure, mainly

I

Value of Statistical Life VSL (VVS) : 3 M€ 2010

I Value of a Year of Life VYL (VAV) : 115 000 € 2010 I Value of a seriously injured : 15 % of VSL, 450 000 € 2010 I Value of a lightly injured : 2 % of VSL, soit 60 000 € 2010 I

Value of carbon

I Value 2013 : 32 € 2010/tCO2 I Value 2030 : 100 € 2010/tCO2 I

Value of time depends on

I Motive (professional, holiday...) I Distance (urban, <20km, 20-80km, ...) I Mode

I

Multiples values in transport sector : Environment, noise...

1Commissariat général à la stratégie et à la prospective, L’évaluation socioéconomique des investissements publics , www.strategie.gouv.fr, sept.

2013

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Database of valuation studies : www.evri.ca

4000+ records

Benefit Transfert

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Research in Applied Econometrics Chapter 2. Contingent Valuation Econometrics Principles of Contingent Valuation

Outline

Principle of Economic Value

Principles of Contingent Valuation

A Summary of Logit & Probit Models

A Summary of Maximum Likelihood Estimation Single-bounded CV Estimation

Application

Computing Welfare Measures Double-bounded CV Estimation

Application : Exxon-Valdez

Questionnaire Issues and Biases

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Purpose

I Operationalize

the theoretical notions of values

I

In practice : impossible to measure every individual’s benefit

I Resort to statistical techniques

I Representative samplesw/ control variables to enable inference to the population

I Individual benefits areneveridentified I Econometric techniques are thus essential I

We’ll need the packages

I DCchoice

I Ecdat, stats (should be there already)

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Classification of Valuation Techniques

I

Based on

stated

preferences

I Contingent Valuation (“Évaluation” Contingente) I Choice experiments / Contingent choices

I

Based on

revealed

preferences

I Transport cost – estimating demand for transport I Principle : one must travel to enjoy a site I Hedonic prices – estimating demand for housing

I Principle : house prices depend on their environment

I

Based on (infered) prices

I Principle : at the marginal buyer, price is the WTP I

Others, not based on preferences

I Land compensation

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Stated Preferences Techniques

I

A sample of people is

surveyed directly

on their preferences about a public project

I To infer a measure of individual statistical value I at the population level

I

Interviews can be anything : telephone, postal mail, e-mail, website

I Preferably face-to-face, but it’s more expensive I or combinaisons

I

The sample depends on the objective

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Questionnaire Structure : wide - precise - wide

I

Opening questions

I Possible filters to select certain respondents

I

General Q on the environment to bring to the particular case of interest

I While making the respondent think about it I We want informed, thought about, answers I

Evaluation Q

I

Debriefing

I Why did the respondent answer what s.he did ? I Did s.he not believe the scenario ?

I

Collect data on specific potential explanatory variables

I e.g. if survey on a lake quality, what use has the respondent of the lake ?

I Tourism, recreational (boat, fish...)

I

Socio-econ data

I Primarily for inference to population

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Contingent Valuation

I

One potential environmental change

z0→z1

is described

I Together with its statedcost

I The context of such cost is important : taxes, fees, prices...

I

A single question is asked : for or against said change

I The question is sometimes repeated w/ another cost

I At this stage, model only the 1st Q

I This is called the dichotomous format (yes/no), next slide, there are others

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The Dichotomous Format

I

This is the most popular, least controversial “elicitation”

format

I Assumed least prone to untruthful answers I While not too demanding for the respondent

I

Consider an environmental change from

z0

to

z1

I To simplify, consider only an improvement I Respondents are proposed a “bid”b

I They answer yes or no

I They may also state that they don’t know or refuse to answer - but we don’t consider that in this course

I This is similar to “posted price” market context I There is a good (the environmental improvement) I The situation is a bit like asking whether to buy it I Respondents are routinely in such situation I Except, not in a public good context I Further, “buying” cannot really be that I So we need a context, but we discuss that later

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The Dichotomous Format

I

Formalizing, let the Indirect Utility (= utility as a function of prices, environment and income)

v

(

z,y

) +

I

It represents individual’s preferences

I from the point of view of the researcher

I The error term reflect multiple influences the researcher does not know about

I These influences are modeled as a random variable, but that does not mean people act randomly

I This is called a Random Utility Model (RUM) I If the answer is “Yes”, then it must be that

v(z1,y−b) +1≥v(z0,y) +0

I and thus, thatWTP>b

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How does the survey let us find the WTP ?

I

4 to 6 (different) bids are proposed to different respondents

I Each respondent ever sees only one bid

I Consider the proportion of “Yes” for each of these bids

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From proportions to WTP distribution

I

Assume that

I For a bid of 0, the proportion of Yes is 100%

I For some high bid the proportion might be zero

I Respondents have a single shape of their utility function I but differ according to observable dataX and unobservables

I

“connect-the-dots” as an estimate of the WTP distribution

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WTP distribution

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To sum up

I

In a survey, we ask people whether they are WTP b for some environmental improvement

I 4 to 6 bids b are used in the survey, every respondent is presented only one b

I thus we have 4 to 6 proportions of “yes”

I When the survey is properly done, we find that as b%the proportion of “yes”&

I

We saw that people answer yes if b<WTP

I We want to find the Pr{Yes|(b,X)}

I That is, the probability that a respondent answers Yes given a bid b and personnal characteristics X

I Estimating this probability for every bid is the same as estimating the distribution of the WTP

I From the distribution, we findE(WTP|(b,X))

I

To estimate, we need

I a discrete choice model : logit or probit

I and an estimation technique : Maximum Likelihood

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Research in Applied Econometrics Chapter 2. Contingent Valuation Econometrics A Summary of Logit & Probit Models

Outline

Principle of Economic Value Principles of Contingent Valuation

A Summary of Logit & Probit Models

A Summary of Maximum Likelihood Estimation Single-bounded CV Estimation

Application

Computing Welfare Measures Double-bounded CV Estimation

Application : Exxon-Valdez

Questionnaire Issues and Biases

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WTP distribution

I

Going back to the IUF, we had the answer is “Yes” if the individual is WTP at least

b

to have

z1

instead of

z0

, that is

v

(

z1,y−b

) +

1 ≥v

(

z0,y

) +

0

I

Include also individual characteristics

X

, so we write

v

(z

1,y−b,X

) +

1 ≥v

(z

0,y,X

) +

0

Assume

V

(z

j,y,X) =αj

+

θyj

+

δzj j

= 0, 1

I θis the marginal utility of income

I In principle, we would like it to decrease with income, but we simplify

I incomey differs in the two states of the environmentz0andz1

I αj=Pk=K

k=1 γkjxk is short for all the individual characteristics I x1could be a constant so that we could have an intercept I we assume that the coefficientsγkj could differ whenj= 0 or

j= 1

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WTP distribution

I

In other words,

Pr

{Yes|b,X}

= Pr

{v

(z

1,y−b,X

) +

1 ≥v

(z

0,y,X) +0}

= Pr

{₀−1 ≤v

(

z1,y−b,X

)

−v

(

z0,y,X

)}

= Pr

{₀−₁ ≤α₁−α₀−θb

+

δ

(

z1−z0

)}

I

Since

α_j

=

^P^k=K_k=1 γ_kjxk

is short for the individual characteristics

I Thenα₁−α₀=Pk=K

k=1 xk(γ_k1−γ_k₀)

I So the vectorsγ1andγ0 arenot identifiedseparately I only the difference is

I Writeγk1−γk0=γk

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Identification issues on z

I

Write

0−1

as

and

α1−α0

as

α

=

^P^k=K_k=1 γkxk

Pr

{Yes|b,X}

= Pr

{≤α−θb

+

δ

(

z1−z0

)}

I

As mentioned, it would be difficult to reduce an environmental change to a single index change

z0→z1

I But from an econometric point of view, the problem is that the change is the same for all individuals

I so coefficientδcannot be estimated : it is not identified I The model has 2 “intercepts”γ1(from theαindex) and

δ(z1−z0)

I We simply aggregate them into a single interceptγ0so that the model is identified

I But we can recover neitherγ1norδ I So now :α=γ₀+Pk=K

k=2 γ_kxk

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WTP distribution

I

The model is now

Pr

{Yes|b,X}

= Pr

{≤α−θb}

I

To estimate the coefficients of the model, there is a number of ways

I a classical one is to assume a specific mathematical form (a specification) for Pr{≤α−θb}

I and apply Maximum Likelihood I

Call this form

G

(α

−θb

)

I G takes values between zero and one : 0≤G(α−θb)≤1 I since it is a probability

I Therefore,G must be anon linearfunction of its argument α−θb

I

If the distribution of

is :

I Logistic, then the specification is the Logit model I Normal, then this is the Probit model

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Logit & Probit

I Logit

model,

G

is the distribution function (cumulative density) of the standard

logistic

r.v. :

G

(α

−θb) = exp (α−θb

)

/

[1 + exp (α

−θb)] = Λ (α−θb) I Probit

model,

G

is the distribution function of the standard

normal

r.v., of which the density is noted

φ

(.) :

G

(α

−θb

) =

Z α−θb

−∞

φ

(

t

)

dt

= Φ (α

−θb

)

with

φ

(α

−θb

) = (2π)

^−1/2

exp

−

(α

−θb

)

²/2

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Logit vs. Probit

I

The logistic and normal distributions are similar

I

Logistic makes computations and analysis easier

I & allows for simplifications in more advanced models

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How do we compute the E(WTP) ?

I

The WTP is s.t.

α0

+

θy

+

δz0

+

0

=

α1

+

θ

(

y−WTP

) +

δz1

+

1

I Solving and rewriting as above : I WTP=α+

θ , withα=γ0+Pk=K

k=2 γkxk and=1−0

I So that e.g.E(WTP|X) =α/θ

I However, we have written the model as Pr{≤α−θb} I so the parameters that will be estimated are ˆαand−θˆ I thusE(WTP\|X) =−ˆα/ˆθ

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Log-logistic / Log-normal

I

If we assume :

I ∼log−normal

Pr{Yes|b}= Φ (α−θln (b)) I ∼log−logistic

Pr{Yes|b}= Λ (α−θln (b))

I

These are still the Probit and Logit models, respectively,

I But the ln of the bid is used instead of the bid itself

I And that is still compatible with RUM I It guarantees thatWTP≥0

I We will compute the welfare measures in this case in the application below

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Marginal effect a continuous regressor x

_j

I

The effect of a marginal change in

xj

I on theresponseproba Pr{Yes|X}=G(α−θb) I withα=γ0+Pk=K

k=2 γkxk

I is given by the partial derivative

∂G

γ₀+Pk=K

k=2 γ_kxk −θb

∂xj

=g(α−θb)γj

whereg() is the derivative ofG(), that is the density function I

This is called the

marginal effect

of

xj

I it depends on the values taken byallthe regressors, not justxj

I In linear regressions, the marginal effect is constant, it is the coefficient of the regressor

I Since the density functiong >0, the sign of the marginal effect ofxj is the sign ofγj

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Marginal effect a continuous regressor x

_j

I

The marginal effect is a non-linear combination of regressors

I

It can be calculated at “interesting” points of

X

I e.g. ¯X, the sample average :

∂G

γ0+Pk=K

k=2 γkx¯k−θ¯b

∂xj

= ∂G(¯x)

∂xj

I However, that does not mean much for discrete regressors e.g.

gender or the bid

I Or it can be calculated for eachi in the sample

∂G

γ0+Pk=K

k=2 γkxki−θbi

∂xji

I and then we can compute an average of the “individual”

marginal effect ∂G(.)

∂xj

I In general, these are not the same : ∂G(¯x)

∂xj

6= ∂G(.)

∂xj

I Often, the average marginal effect is more interesting

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Marginal effect of a discrete regressor x

_j

I

Effect of a change in

x2

discrete

I fromx20 tox21 (often, from 0 to 1)

I on the response proba Pr{Yes|X}=G(α−θb) I The discrete change ∆ in Pr{Yes|X}_i is

G γ0+

k=K

X

k=3

γkxki−θbi+γ2x21

!

−G γ0+

k=K

X

k=3

γkxki−θbi+γ2x20

!

I

Such discrete effect differs from individual to individual

I Even the sign could in principle differ

I

In R, such effects are not calculated automatically

I The above formula must be calculated explicitly

(38)

Research in Applied Econometrics Chapter 2. Contingent Valuation Econometrics A Summary of Maximum Likelihood Estimation

Outline

Principle of Economic Value Principles of Contingent Valuation A Summary of Logit & Probit Models

A Summary of Maximum Likelihood Estimation

Single-bounded CV Estimation

Application

Computing Welfare Measures Double-bounded CV Estimation

Application : Exxon-Valdez

Questionnaire Issues and Biases

(39)

Density

I f

(y

|θ) : probability density function,pdf, of a random

variable

y

I conditioned on a set of parametersθ

I It represents mathematically thedata generating processof each obs. of a sample of data

I

The joint density of n

independent and identically distributed

(iid) obs.

I = the product of the individuali densities : f(y1, ...,yn|θ) =

n

Y

i=1

f(yi|θ) =L(θ|y) I

This joint density is the

likelihood function

I a function of the unknown parameter vectorθ I y is used to indicate the collection of sample data

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Likelihood Function

I

Intuitively, this is much the same as a joint probability

I Consider 2 (6-sided) dices

I What is the probability of rolling a 36 ?

I The likelihood function is the idea of the probability of the sample

I Except that points have probability mass zero

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Conditional Likelihood

I

Generalize the likelihood function to allow the density to depend on conditioning variables :

f

(

yi|xi, θ)

I Take the classical LRMyi=xiβ+i

I Supposeis normally distributed with mean 0 and variance σ²:∼n 0, σ²

I Then,yi ∼n xiβ, σ²

I Thusyi arenot iid: they have different means

I But they are independant, so that (yi−xiβ)/σ∼n(0,1) I thusL(θ|y,X) =

Π

if(yi|xi, θ) = Π

i

√ 1

2πσ²exp (

−(yi−xiβ)² σ²

)

(42)

Conditional log-Likelihood

Usually simpler to work with the

log

: ln

L

(θ|

y

) =

n

X

i=1

ln

f

(

yi|θ)

thus ln

L

(θ|

y,X

) =

X

ln

f

(y

i|xi, θ) =−

1 2

n

X

i=1

h

ln

σ²

+ ln (2π) + (y

i −xiβ)²/σ²ⁱ

where X is the

n × K

matrix of data with ith row equal to

xi

(43)

Maximum Likelihood Estimation Principle

I

We see that with a discrete rv

I f(yi|θ) is the probability of observingyi conditionnally on θ I The likelihood function is then the probability of observing the

sampleY conditionnally on θ

I Assume that the sample that we have observed is the most likely

I What value ofθ makes the observed sample most likely ? I Answer : The value ofθthat maximizes the likelihood function

I since then the observed sample will have maximum probability

I

When

y

is a continuous rv, instead of a discrete one,

I we cannot say anymore thatf(yi|θ) is the probability of

observingyi conditionnally onθ, I but we retain the same principle.

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Maximum Likelihood Estimation Principle

I

The value ˆ

θ_ML

of the parameter vector

θ

that maximizes

L

(θ|

data

) is the

maximum likelihood estimatesθ

ˆ

I The value vector ˆθML that maximizesL(θ|data) is the same as the one that maximizes lnL(θ|data)

I

The necessary condition for maximizing ln

L

(θ|

data) is

∂

ln

L

(θ|

data

)

/∂θ

= 0

I This is called thelikelihood equations

I There are as many as elements inθ

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ML Properties

I Conditionnally on correct distributional assumptions I and under regularity conditions

I ML has very good properties

I In a sense, because the information supplied to the estimator is very good : not only the sample but also the full distribution I Consistent but biased

I Asymptotically normal

I Asymptotically efficient(Achieves the Cramer–Rao lower bound for consistent estimators)

I

ML does not have an explicit solution (no formula !)

I But yieldnumericalestimates ˆβ_MV

I

What if the distributional assumptions are not correct ?

I e.g. if you used Probit : what if is not normal ?

I Sometimes the properties remain, that is a complex topic I if the distributions are close (e.g. logistic intead of normal), it

is likely that the properties are close I We usually focus more on endogeneity ofX

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Maximum Likelihood Estimation of Logit & Probit

I

Likelihood Function for the dichotomous case is

Y

i

[Pr

{willing|β, σ,Xi}^yⁱ

] [1

−

Pr

{willing|β, σ,Xi}]^(1−yⁱ⁾ I

The

sizes

of the estimated coefficients

are not

comparable

between probit and logit

I Approximately, multiply probit coef by 1.5 yields the logit coef (rule of thumb !)

I The marginal effects should be approximately the same

(47)

Measures of the quality of fit

I

The

correctly predicted percentage I may be appealing

I ∀i compute the fit proba thatyi takes value 1, G Xiβˆ I If≥.5 we “predict”yi = 1 and zero otherwise

I Compute the % of correct predictions

I

Problem : it is possible to see high % of correctly predicted while the model is not much useful

I e.g. in a sample of 200, 180 obs. ofyi = 0 for which 150 are predicted zero and 20 obs. ofyi= 1 all predicted zero

I The model is clearly poor

I But we still have 75% correct predictions I A flat prediction of 0 has 90% correct predictions

I

A better measure is a 2

×

2 table as in the next slide

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Goodness of Fit : Predictive Table

Observed

yi

= 1

yi

= 0 Total

Predict

ˆ

yi

= 1

350

122 472 ˆ

yi

= 0 78

203

281 Total 428 325 753

(49)

Goodness of Fit : Pseudo R-square

I Pseudo−R²

= 1

−

ln

L_UR/

ln

L₀

I lnLUR log-likelihood of the estimated model

I lnL0log-likelihood of a model with only the intercept I i.e. forcing allβ= 0 except for the intercept

I

Similar to an

R²

for OLS regression

I sinceR²= 1−SSRUR/SSR0

I

There exists other measures of the quality of fit

I but the fit is not what maximum likelihood is seeking

I contrarily to LS

I I stress more the statistical significance of regressors

(50)

Research in Applied Econometrics Chapter 2. Contingent Valuation Econometrics Single-bounded CV Estimation

Outline

Principle of Economic Value Principles of Contingent Valuation A Summary of Logit & Probit Models

A Summary of Maximum Likelihood Estimation

Single-bounded CV Estimation

Application

Computing Welfare Measures Double-bounded CV Estimation

Application : Exxon-Valdez

Questionnaire Issues and Biases

(51)

Estimation

I

“Parametric” estimation (such as we have seen, with coefficients to estimate)

I GLM package (core distribution) I DCchoice package

I

Example with the

NaturalPark

data of the

Ecdat

package (Croissant 2014)

I Here we follow RAE2020.R

(52)

Application

Outline

Principle of Economic Value Principles of Contingent Valuation A Summary of Logit & Probit Models

A Summary of Maximum Likelihood Estimation

Application

Computing Welfare Measures Double-bounded CV Estimation

Application : Exxon-Valdez

Questionnaire Issues and Biases

(53)

NaturalPark data

I

From the

Ecdat

package (Croissant 2014)

I consists of WTP and other socio-demographic variables I of 312 individuals regarding the preservation of the Alentejo

natural park in Portugal

I

In

help

type NaturalPark, execute

summary(NaturalPark) I 7 variables

I bid1 is the first bid I min 6, max 48

I bidh & bidl are 2nd bids that we do not look into for the moment

I answers is a factor {YY, YN, NY, NN} of answers to 2 bids I We’ll use only the 1st letter, Y for Yes

I Socio-demographics are Age, Sex, Income

I Certainly, there should be more, this is a simplified real-world case

(54)

Application

Data Transformation

I

Rename

NP <- NaturalPark

for simplicity

I

Extract the answer to the 1st bid from “answers”

I NP$R1 <- ifelse(NP$answers == "yy" | NP$answers ==

"yn", 1, 0)

I What does that do ?

I A call to the logical function ifelse( ) takes the form ifelse(condition, true, false)

I It returns true if condition is true, and false otherwise.

I The vertical bar | is an “or” operator in a logical expression.

I The prefix NP$ is a command that makes it possible to access each variable contained in NP directly.

I

Convert bid1 to log (log in English is ln in French)

I NP$Lbid1 <- log(NP$bid1)

I summary(NP)

reveals that things have gone smoothly

(55)

Estimation using glm

I glm

is a classical package for many models of type

G

(X

β) I Its use is much likelm

I But you have to specify the link functionG using option family =

I This is fairly flexible, but a bit complicated

I summary

works on

glm I As several usual commands I

Output is in part similar to

lm

I Coef values next to var names with their significance I This is interpreted in a way similar tolm

I Negative (signif.) coef implies a negative impact on Pr{Yes|b}

I Note “sexfemale” indicates that the Pr{Yes|b}is smaller for a woman other things equal (cf. discussion on marginal effect) I Also gives the lnL value at optimum

(56)

Iterations and Convergence

Non-linear models do not have explicit solutions Solved numerically by algorithms

I

Newton-type in plot

I Iterates until condition I

Risk of local max

I poor start point(s)

(57)

Estimation using DCchoice

I DCchoice

is designed for such data

I But not for other contexts

I Format for single-bounded issbchoice(formula, data, dist =

"log-logistic")

I

the default dist is log-logistic

I This is in fact logistic

I But the bid variable is interpreted in log I formula followsResponse ~ X | bid

I | bid is mandatory

I

the output is much more directly relevant for valuation purposes

I bidalways shown last

I log(bid)if log-logistic or log-normal were selected I but you must supply the log of the bid

I Measures of mean WTP, we will see later

(58)

Application

Goodness of fit

I

table(predict(fitted model, type = "probability")>.5,NP$R1)

I This is a contingency table that counts the number of

predicted Yes

I predicted prob>.5 (returns TRUE or FALSE) I against the actual Yes/No

I per individual, so with each individual’sXi (individual predictions)

SB.NP.DC.logit 0 1 FALSE (predicted 0) 85 38

TRUE (predicted 1) 56 133

(59)

Plotting

I sbchoice

produces an object that can be

plotted directly I directplotof the object is the fitted probability of Yes for the

bid range

I probably conditionnaly on average age, income, sex – the package isn’t explicit

I

Using a predict method helps

I observe what is outside the range I compare logit & probit fitted curves

I In particular : logit has slightly fatter tails, inducing a higher WTP

I To use predict

I Creates a matrix of new data

I Chooses the proper type, here we want a proba, we call it

“response”

(60)

Application

Logit vs. Probit predict

I

As can be seen, for the same data

I Logit has slightly fatter tails than probit

(61)

Essay #2

Write an R-code that does the following.

Take the dataset you selected from Essay #1.

1.

Using the

ifelse

command from line 186 of the script, convert the dependant variable you selected in Essay 1 in a

dichotomous variable thta is 1 for each observation above the mean and 0 otherwise. This can be interpreted as a situation in which only an indicator of the dependant variable is available and not the full numeric variable.

2.

Using the same regressors as in Essay #1 or different ones, replicate the analysis that has been done for the NaturalPark dataset.

3.

Discuss the result, including significance, in the R code in

comments immediately following each command. Note that

(62)

Computing Welfare Measures

Outline

Principle of Economic Value Principles of Contingent Valuation A Summary of Logit & Probit Models

A Summary of Maximum Likelihood Estimation

Application

Computing Welfare Measures Double-bounded CV Estimation

Application : Exxon-Valdez

Questionnaire Issues and Biases

(63)

I

Recall we assumed

V

(

zj,y

) =

α_j

+

θyj

+

δzj j

= 0, 1

I

Then WTP is s.t.

α0

+

θy

+

δz0

+

0

=

α1

+

θ

(y

−WTP) +δz1

+

1

I Solving and rewriting as aboveWTP=α+ θ , with α=α1−α0and=1−0

I So that WTP is a random variable and we can compute its e.g.E(WTP) =α/θ

I However, we had written the model as Pr{≤α−θb} I so the parameters that will be estimated are ˆαand−θˆ I thusE\(WTP) =−ˆα/θˆ

I Rememberα=γ₀+Pk=K

k=2 γ_kxk is individual :α_i I thenE(WTP) is also individual

I So that we have to think about how we aggregate individual expected WTP over the sample

(64)

Other measures of welfare

I E

(

WTP

) is the most obvious measure

I However, the expectation

is strongly influenced by the tail of the

distributionG(.)

I While we do not actually have data to fit it

I Since there are not many bids

I And it does not feel very serious

to ask a very high bid

(65)

Truncated expectations

I E

(

WTP

) =

Z ∞

0

Pr

{Yes|b}db

I Historically, the highest bid has been used to truncate E(WTP)

Z b max

0

Pr{Yes|b}db

I However, that is not a proper expectation since the support of Pr{Yes|b}does not stop atb max

I An alternative uses the truncated distribution : Z b max

0

Pr{Yes|b}/(1−Pr (Yes|b max))db that is, the actual distribution of the WTP when we do not allow WTP > b max

(66)

Median WTP

I

Finally the median has been suggested as a more robust measure :

bmedian

s.t.

Pr

{Yes|bmedian}

=

¹/2

I i.e. the bid s.t. 50%

would answer Yes

(67)

Shape of the WTP function

I

The shape of the WTP function depends on the shape of the indirect utility function

V

(.)

I For some shapes, some values ofθlead to impossibilities

Distribution Expected Median

Normal

^α_θ ^α_θ

Logistic

^α_θ ^α_θ

Log-normal exp

^α_θ

exp

_2θ¹2

exp

^α_θ

Log-logistic exp

^α_θ

Γ

1

−¹_θ

Γ

1 +

¹_θ

exp

^α_θ I

Again, since

αi

=

γ0

+

^P^k=K_k=2 γkxki

, each of these forms are

individual

I So the question arises whether to compute a sample mean or a sample median

I DCchoice appears to compute a sample mean I But is not explicit

(68)

How do we choose a welfare measure ?

I

3 welfare measures : untruncated, properly truncated, median

I Of course we would not select an infinite one

I The smallest estimates to avoid criticism ?

I Do these measures differ significantly from each other ? I We will see that later

I

4 well-known distributions

I (log-)normal, (log-)logistic : they do not differ substantially I There are others

I There are also other estimators e.g. non-parametric I The estimate of the best-fit model ?

I

2 aggregation rules : sample mean or sample median

I researchers usually take the first, but the median is often less extreme

I

DCchoice does not provide any guidance

(69)

Computing the welfare measure

I

DCchoice computes automatically

I

With glm, use the above formulas

I

Much as I like DCchoice, I must note that for the data we use (NaturalPark)

I The mean WTP does not coincide with the median I for the symmetrical distributions (normal & logistic) I That is a problem I should write the authors

(70)

Confidence Intervals

I

In the end, WTP, under any of its forms, is an estimate

I As such it has a confidence interval

I Much as for ˆβ in a linear regression, you should always report the CI

I At a minimum to give an idea of the variance

I and to show whether it is significantly different from zero

I

There are 2 main methods

I Krinsky & Robb I Bootstrap

I

These methods are much broader than valuation

I They are useful in all types of research in applied econometrics

(71)

Constructing Confidence Intervals : Krinsky and Robb

I

Start with the estimated vector of coefficients

I

By the properties of ML, it is distributed (multivariate) normally

I Its matrix of variance-covariance has been estimated in the ML estimation process

I It is stored in the ML object

I

Draw D times from a multivariate normal distribution with

I mean = the vector of estimated coefficients

I the estimated variance-covariance matrix of these estimated coefficients

I

So, we have D vectors of estimated coefficients

I If D is large, the average of these D vectors is just our original vector of coef.

(72)

Constructing Confidence Intervals : Krinsky and Robb

I

Compute the welfare measure for each such replicated coefficient vector

I Thus we have D estimated welfare measures I some large, some small : an empirical distribution

I order them from smallest to largest I the 5% most extreme are deemed random I the 95% most central are deemed reasonnable I and contitute the 95% confidence interval

I

The lower and upper bounds of the 95% confidence interval

I corresponds to the 0.025 and 0.975 percentiles of the

measures, respectively

(73)

Krinsky and Robb : Implementation in DCchoice

I

Function

krCI(.)

I constructs CI for the 4 different WTPs

I estimated by functionssbchoice(.)ordbchoice(.)

I

call as

krCI(obj, nsim = 1000, CI = 0.95)

I objis an object of either the “sbchoice” or “dbchoice” class, I nsimis the number of draws from the multidimensional normal

I influences machine time

I CIis the percentile of the confidence intervals to be estimated I returns an object “krCI”

I table of the simulated confidence intervals I vectors containing the simulated WTPs

I

Is there a package that does Krinsky & Robb for glm objects ?

(74)

Constructing Confidence Intervals : Bootstrap

I

Similar to Krinsky & Robb

I except in the way the new estimated coefficients are obtained I Essentially, instead of simulating new estimates

I We simulate new data

I and then calculate new estimates

(75)

Mediocrity principle

I

Consider that our sample is

mediocre

in the population

I This means : it does not have anything exceptional

I

Then, if we could draw a new sample from that population,

I we would surely obtain a fairly mediocre sample

I that is, fairlysimilarto the original one

(76)

Bootstrap principle

I

It’s not possible to draw a new sample

I

But imagine that using the original sample, we draw one obs,

I and we “put it back” in the sample (“replace”)

I then we draw again

I repeat until we have the same number n of obs as in the original sample

I call thisa bootstrap sample

I Each original obs. appears 0 to n times

I

By the

mediocrity principle

I the bootstrap sample is fairly close to a real new sample I or at least, a “not unlikely” sample

I Estimate a new vector of coefficients I Repeat D times

(77)

Bootstrap : Implementation in DCchoice

I

Function

bootCI()

carries out the bootstrapping

I and returns the bootstrap confidence intervals I callbootCI(obj, nboot = 1000, CI = 0.95)

I

Longer than K&R since each sample must be generated

I and then compute new estimates

I while K&R only simulates new estimates I

In the end, the results are similar

I

Note : another mean would be the

resample

cmd

I Applicable toglm

I But I don’t develop here

(78)

Differences of welfare measures

I

Sometimes we want to know whether a welfare measure is significantly different from another

I In other words : is their difference significantly different from zero

I In terms of CI : does the CI of their difference include zero ? I

To compute that : Bootstrap similarly

I Krinsky and Robb is also possible but

I If the welfare measures are independant, their difference has variance-covariance that is the sum of each

variance-covariance

I If they are not independant, then it’s difficult

(79)

Outline

Principle of Economic Value Principles of Contingent Valuation A Summary of Logit & Probit Models

A Summary of Maximum Likelihood Estimation Single-bounded CV Estimation

Application

Computing Welfare Measures

Double-bounded CV Estimation

Application : Exxon-Valdez

Questionnaire Issues and Biases

(80)

Research in Applied Econometrics Chapter 2. Contingent Valuation Econometrics Double-bounded CV Estimation

Double-bounded CV Estimation

I

To increase the amount of information collected by the survey,

I The valuation question is asked a 2nd time

I If answer is Yes (No), then ask with a higher (lower) bid

I

Called

double-bounded

dichotomous choice

I Or dichotomous choice with follow-up I

More precisely, the phrasing could be

I If ∆Z cost you b€, would you be WTP it ?

I If answered Yes : would you be WTPb^U€ ?b^U>b I If answered No : would you be WTPb^L€ ?b^L<b

(81)

Double-bounded CV Estimation

I

There are 4 outcomes for each respondent

I YY, YN, NY, NN

I YY indicates that WTP>b^U I YN that b<WTP<b^U I NY thatb^L<WTP<b I NN that WTP<b^L

I

Thus the answers are intervals

I Probit & Logit do not suffice I Many use ML

I But the likelihood function is different

(82)

Estimation with DBDC data

I

To develop the likelihood function

I it is necessary to express probabilities first I

Write

P^YY

as the probability to answer Yes, Yes

I P^YY = PrbÛ <WTP = 1−G bÛ I P^YN = Prb<WTP<bÛ =G bÛ

−G(b) I P^NY = Pr

b^L<WTP<b =G(b)−G b^L I P^NN= PrWTP<b^L =G b^L

I

For a sample of

n

obs. ln

L

=

N

X

n=1

hdn^YYPn^YY

+

dn^YNPn^YN

+

dn^NYPn^NY

+

dn^NNPn^NN

i

where

n

indexes individuals and

dn^XX

indicates whether

n

answered

XX

(dich. variable)

(83)

Estimation with DBDC data

I

There is no direct command corresponding to such likelihood

I It must be programmed

I This is called “Full Information Maximum Likelihood” FIML I We don’t do this

I It is pre-programmed in DCchoice

I For the same basic choices of distribution as for SBDC data

I

Endogeneity issue

I The 2nd bid is not exogenous

I Since it depends on the previous answer

I Thus it contains unobserved characteristics of the individuals I Such unobservables also determine the 2nd choice

I This is in principle addressed by FIML I

A more general model is Bivariate probit

I allowing the 2 answers to have less than perfect correlation I not covered by DCchoice

(84)

Estimation with DBDC data : Std normal cdf

I P^YY

= Pr

ⁿb^U <WTP^o

= 1

−

Φ

−α

+

βb^U I P^YN = Pr

b<WTP<b^U = Φ −α+βb^U

−Φ (−α+βb) I P^NY = Pr

b^L<WTP<b = Φ (−α+βb)−Φ −α+βb^L I P^NN

= Pr

ⁿWTP <b^L^o

= Φ

−α

+

βb^L

I

So : we estimate the same coefficients

α

and

β

as in SBDC

I But with more data, so that it is more efficient

I Assuming people answer in the same way to both valuation questions

I

So : the computation of the welfare measures is the same

(85)

Outline

Principle of Economic Value Principles of Contingent Valuation A Summary of Logit & Probit Models

A Summary of Maximum Likelihood Estimation Single-bounded CV Estimation

Application

Computing Welfare Measures

Double-bounded CV Estimation

Application : Exxon-Valdez

Questionnaire Issues and Biases

(86)

Application : Exxon-Valdez

Context

I

1989, about 35 000 tons of crude oil at sea

I

Ended up extending on about 26 000

km²

at sea

I and soil 1 600 km of coastline I For comparison :

I Deepwater Horizon (2010)500-600 000 m3 (rank 3) I Exxon-Valdez (1989)40-100 000 m3 (rank 21)

I

Most of the damage was in Prince William Sound and the Gulf of Alaska up to the Kodiak Islands

I

Several types of damages

I Professional fishing (minimal) I Tourism (possibly a benefit) I Environmental heritage loss

I Punitive damages (supposed to be incentive) I

Valuation survey

(87)

(88)

Exxon-Valdez Questionnaire

(89)

(90)

Exxon-Valdez Questionnaire

(91)

(92)

Exxon-Valdez Questionnaire

(93)

(94)

Exxon-Valdez Questionnaire

(95)

(96)

Exxon-Valdez Questionnaire

(97)

(98)

Exxon-Valdez Questionnaire

(99)

(100)

Application : Exxon-Valdez

Exxon-Valdez Questionnaire

I

Avoid Willingness to accept Q :

I For assumed strategic behaviour I

The basis Q is

I Compensation for the loss of an environmental heritage during 10 years ?

I Scenario : after 10 years environmental damage will be fully recovered

I

Convert such “WTA” into a WTP to avoid that the catastrophy happens again for 10 years

I Scenario : in 10 years time, similar cat. will be impossible due to double-hull

I

The scenario need not be true

I Since we are investigating human preferences for things that may never happen

I However, it must appear credible to respondents

(101)

(102)

Exxon-Valdez Questionnaire : valuation scenario

(103)

(104)

Exxon-Valdez Questionnaire : follow-up valuation Q

(105)

(106)

Exxon-Valdez Questionnaire

(107)

(108)

Exxon-Valdez Questionnaire

(109)

(110)

Exxon-Valdez Questionnaire

(111)

(112)

Exxon-Valdez Questionnaire

(113)

Reading the Exxon-Valdez Data

I data(“CarsonDB”)

I it is only a frequency table for the DBDC survey I Thus without theX data

T1 TU TL yy yn ny nn

1 10 30 5 119 59 8 78

2 30 60 10 69 69 31 98

3 60 120 30 54 75 25 101

4 120 250 60 35 53 30 139

I

So there are 6 distinct bids

I 5, 10, 30, 60, 120, 250

I There is always a large proportion of nn : part of protest I The proportion of yy decreases with b

I The proportion of yn & of ny is roughly constant with b I with about 2 to 3 more yn than ny