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Choice with multiple alternatives Specification of the deterministic - - PowerPoint PPT Presentation
Choice with multiple alternatives Specification of the deterministic - - PowerPoint PPT Presentation
Choice with multiple alternatives Specification of the deterministic part Michel Bierlaire Introduction to choice models Nonlinear specifications: heteroscedasticity Heteroscedasticity Logit is homoscedastic in i.i.d. across both i and
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Heteroscedasticity
Logit is homoscedastic
◮ εin i.i.d. across both i and n. ◮ In particular, they all have the same variance.
Motivation
◮ People may have different level of knowledge (e.g. taxi drivers) ◮ Different sources of data
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Heteroscedasticity
Data
◮ G groups in the population. ◮ Each individual n belongs to exactly one group g. ◮ Characterized by indicators:
δng = 1 if n belongs to g,
- therwise
and
g δng = 1, for all n.
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Heteroscedasticity
Assumption: variance of error terms is different across groups
Consider individual n1 belonging to group 1, and individual n2 belonging to group 2. Uin1 = Vin1 + εin1 Uin2 = Vin2 + εin2 and Var(εin1) = Var(εin2)
Modeling
Without loss of generality: Var(εin1) = α2
2 Var(εin2)
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Heteroscedasticity
Modeling: scale parameters
Uin1 = Vin1 + εin1 = Vin1 + ε′
in1
α2Uin2 = α2Vin2 + α2εin2 = α2Vin2 + ε′
in2
Variance
Var(ε′
in2)
= Var(α2εin2)
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Heteroscedasticity
Modeling: scale parameters
Uin1 = Vin1 + εin1 = Vin1 + ε′
in1
α2Uin2 = α2Vin2 + α2εin2 = α2Vin2 + ε′
in2
Variance
Var(ε′
in2)
= Var(α2εin2) = α2
2 Var(α2εin2)
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Heteroscedasticity
Modeling: scale parameters
Uin1 = Vin1 + εin1 = Vin1 + ε′
in1
α2Uin2 = α2Vin2 + α2εin2 = α2Vin2 + ε′
in2
Variance
Var(ε′
in2)
= Var(α2εin2) = α2
2 Var(α2εin2)
= Var(εin1)
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Heteroscedasticity
Modeling: scale parameters
Uin1 = Vin1 + εin1 = Vin1 + ε′
in1
α2Uin2 = α2Vin2 + α2εin2 = α2Vin2 + ε′
in2
Variance
Var(ε′
in2)
= Var(α2εin2) = α2
2 Var(α2εin2)
= Var(εin1) = Var(ε′
in1)
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Heteroscedasticity
Modeling: scale parameters
Uin1 = Vin1 + εin1 = Vin1 + ε′
in1
α2Uin2 = α2Vin2 + α2εin2 = α2Vin2 + ε′
in2
Variance
Var(ε′
in2)
= Var(α2εin2) = α2
2 Var(α2εin2)
= Var(εin1) = Var(ε′
in1)
ε′
in1 and ε′ in2 can be assumed i.i.d.
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Heteroscedasticity
Modeling: utility function
µnVin + εin where µn =
G
- g=1
δngαg and αg, g = 1, . . . , G are unknown parameters to be estimated from data.
Remarks
◮ Even if Vin = j βjxjin is linear-in-parameters, µnVin = j µnβjxjin is not. ◮ Normalization: one αg must be normalized.