11

Mixture Models of Mixture Models of Choice under RiskChoice under Risk

Peter G. MOFFATTPeter G. MOFFATTUniversity of East Anglia, University of East Anglia, Norwich, United Kingdom.Norwich, United Kingdom.

(with Anna CONTE and John D. HEY, both LUISS, Rome)

22

Estimating preference Estimating preference functionals for choice under functionals for choice under

risk from the choice behaviour risk from the choice behaviour of individuals.of individuals.

Data: Hey (2001, Data: Hey (2001, Exp. Ec.Exp. Ec.).). 53 subjects53 subjects a sequence of 500 tasksa sequence of 500 tasks spread over 5 daysspread over 5 days they choose between risky they choose between risky

prospects of the following kind:prospects of the following kind:

33

€ 100

Random Lottery Incentive system used (one chosen lottery selected at random and played for real).

Lottery Lottery pp Lottery Lottery qq

€ 0

€50

€ 100

€50

€ 0

44

All 500 problems involved All 500 problems involved three three of of the four outcomes £0, £50, £100 the four outcomes £0, £50, £100

and £150.and £150. Let us denote them by Let us denote them by

xxii ( (i= 1, 2, 3, 4i= 1, 2, 3, 4) )

and the corresponding utility values byand the corresponding utility values by

uuii ( (i = 1, 2, 3, 4i = 1, 2, 3, 4)).. The probabilities of the four outcomes The probabilities of the four outcomes

in these two lotteries in pairwise-in these two lotteries in pairwise-choice problem choice problem tt ( (t = 1, … , 500t = 1, … , 500))

are pare p1t1t p p2t2t p p3t3t p p4t4t

and and qq1t1t q q2t2t q q3t3t q q4t4t

55

Subjects Subjects compute:compute:

where the where the P’s P’s and and Q’s Q’s are not are not necessarily the correct probabilities, necessarily the correct probabilities, but they are derived from the correct but they are derived from the correct probabilities in the following manner:probabilities in the following manner:

1 1 2 2 3 3 4 4t t t tP u P u P u P u

1 1 2 2 3 3 4 4t t t tQ u Q u Q u Q u

(p-lottery)

(q-lottery)

66

Here the function Here the function w(.) w(.) is a is a probability weighting function.probability weighting function.

1 2 3 4

2 2 3 4 3 4

3 3 4 4

4 4

1t t t t

t t t t t t

t t t t

t t

P w p p p

P w p p p w p p

P w p p w p

P w p

1 2 3 4

2 2 3 4 3 4

3 3 4 4

4 4

1t t t t

t t t t t t

t t t t

t t

Q w q q q

Q w q q q w q q

Q w q q w q

Q w q

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THE MIXTURE ASSUMPTIONTHE MIXTURE ASSUMPTION EITHER the subject is EU ( ), EITHER the subject is EU ( ), OR the subject is Rank Dependent EU (RDEU).OR the subject is Rank Dependent EU (RDEU).

RDEU: Two versions:RDEU: Two versions:– Power weighting function: Power weighting function:

0w p p – Quiggin weighting function: Quiggin weighting function:

1/ 0.2791

pw p

p p

w p p

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Subjects choose the lottery Subjects choose the lottery pp ( (qq) if and ) if and

only ifonly if

where where

1 1 2 2 3 3 4 4 0t t t t tD u D u D u D u

1,2,3,4jt jt jtD P Q j

here here t is a is a FechnerianFechnerian error term, with error term, with

represents extent ofrepresents extent of 2~ 0,t N . .

computational error by subjects.computational error by subjects.

99

Assume CRRA utility:Assume CRRA utility:

r r represents risk attitude. represents risk attitude.

0150

rx

u x r

Observed binary dependent variable:Observed binary dependent variable:

1

1it

it

y if subject i chooses p in problem t

y if subject i chooses q in problem t

1010

DISTRIBUTIONAL DISTRIBUTIONAL ASSUMPTIONSASSUMPTIONS

Risk attitude (EU maximisers):Risk attitude (EU maximisers):

2ln ~ ,r N

Risk attitude and rank-dependent Risk attitude and rank-dependent parameter (RDEU maximisers):parameter (RDEU maximisers):

2

2min

ln( )~ ,

ln

r SN

M S S

1111

Tremble parameterTremble parameter

Probability ω that subject loses concentration and chooses randomly.

A proportion p of the population is RDEU; a proportion 1-p is EU.

Mixing proportionMixing proportion

1212

Likelihood contribution for a given subject:Likelihood contribution for a given subject:

min

500

2 2 3 3 41

500

2 2 3 3 41

, , , , , , , , ,

(1 ) 1 / / 2 ; ,

1 / / 2 , ; , , , ,

t t t tt

t t t tt

L M S p

p y d u d u d f r dr

p y D u D u D g r M S d dr

Sample log-likelihood:53

1i

i

LogL LogL

LogL maximised using maximum

simulated likelihood.

1313

Results: Maximised LogLResults: Maximised LogL

Weighting Weighting functionfunction

EU onlyEU only RDEU RDEU onlyonly

Mixture Mixture ModelModel

QuigginQuiggin -7210.27-7210.27 -6860.18-6860.18 -6716.49-6716.49

PowerPower -7210.27-7210.27 -6845.15-6845.15 -6773.37-6773.37

1414

(Marginal) Log-likelihood function (Marginal) Log-likelihood function (Quiggin)(Quiggin)

1515

CRRA utility. Quiggin weighting function (n = 53; T = 500)mixture model

EU only RDEU only EU-type RDEU-type

θ (EU) Θ (RDEU)

-1.15599(0.07217)

-0.95338(0.02332)

-0.76438(0.09492)

-0.95425(0.01996)

δ (EU) Δ (RDEU)

0.55348 (0.03040)

0.54652(0.01755)

0.32431(0.06369)

0.53947(0.01500)

M --0.49629(0.02043)

--0.55465(0.03041)

S -0.24676

(0.01965)-

0.24031(0.01820)

-0.25563

(0.10011)-

0.33793(0.08296)

ω0.01375

(0.00173)0.01534

(0.00184)0.01139

(0.00157)

σ0.04823

(0.00089)0.04134

(0.00089)0.07438

(0.00297)0.03398

(0.00081)

p - - 0.80266 (0.05623)

Log-likelihood -7210.27887 -6860.18750 -6716.49907

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Computation of posterior probabilities:Computation of posterior probabilities:

min

1 500

500

2 2 3 3 41

1 500

500

2 2 3 3 41

( | )

(1 ) 1 / / 2 ; ,

( | )

1 / / 2 , ; , , , ,

t t t tt

t t t tt

P subject is EU y y

p y d u d u d f r dr

L

P subject is RDEU y y

p y D u D u D g r M S d dr

L

1717

Posterior Probabilities (Quiggin)Posterior Probabilities (Quiggin)

1818

Distribution of RD Distribution of RD parameterparameter

1919

95% bounds for weighting 95% bounds for weighting function (Quiggin)function (Quiggin)

2020

Summary (1)Summary (1)

We have taken into account We have taken into account heterogeneity in behaviour between heterogeneity in behaviour between individuals and within individuals:individuals and within individuals:

2121

by ‘heterogeneity between by ‘heterogeneity between individuals’ we mean that people are individuals’ we mean that people are different, not only in terms of which different, not only in terms of which type of preference functional that type of preference functional that they have, but also in terms of their they have, but also in terms of their parameters for these functionals.parameters for these functionals.

This means that trying to estimate a This means that trying to estimate a ‘representative agent’ preference ‘representative agent’ preference functional to represent the functional to represent the preference functional of all the preference functional of all the individuals may well lead to biased individuals may well lead to biased estimates.estimates.

Summary (2)Summary (2)

2222

by ‘heterogeneity within by ‘heterogeneity within individuals’ we mean that individuals’ we mean that behaviour can differ for a given behaviour can differ for a given individual solving the same individual solving the same choice problem.choice problem.

Summary (3)Summary (3)

2323

solutions to both these problems, solutions to both these problems, concentrating particularly on using a concentrating particularly on using a mixture model and introducing mixture model and introducing unobserved heterogeneity terms to unobserved heterogeneity terms to capture the heterogeneity of capture the heterogeneity of preference functionals across preference functionals across individuals.individuals.

an econometric approach that deals an econometric approach that deals with technical difficulties associated with technical difficulties associated with these issues.with these issues.

We have We have proposed:proposed:

2424

80% of the population are RDEU 80% of the population are RDEU with their own (Quiggin) with their own (Quiggin) weighting function; 20% are EU.weighting function; 20% are EU.

Mixture model tells us (with high Mixture model tells us (with high probability) which subjects are of probability) which subjects are of which type.which type.

Results:Results:

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Mixture Models of Mixture Models of Choice under RiskChoice under Risk

Peter MOFFATTPeter MOFFATT

University of East Anglia, University of East Anglia, NorwichNorwich(with Anna CONTE and John D. HEY, both LUISS, Rome)

Thank you