Recap WTP Normalisations Special Reg

Introduction to Identification

Abi Adams

HT 2017

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Recap WTP Normalisations Special Reg

Outline for Today

Aim: Understand basic concepts so that we can move on toapply them in a range of applied settings in future lectures

I Recap

II Normalisations (in the context of discrete choice)

III Special Regressors

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Recap WTP Normalisations Special Reg

Recap

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Recap

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Recap: Observational Equivalence

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Recap: Point Identification

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Recap: Structural Features

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Recap: Uniform Identification

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Recap: Uniform Identification

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Recap: Terminology

I Observational Equivalence 1S′ and S′′ such that F S′

YX = F S′′

YX are observationallyequivalent

I Observational Equivalence 2S′ and S′′ such that F S′

φ = F S′′

φ are observationallyequivalent given the features of the data that are knowable,φ

I The model Γ identifies S0 if there is no S′ ∈MΓ such thatF S0

φ = F S′

φ

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Proving Identification

I There are a number of ways that one might demonstrateidentification

I The most common way is to prove identification byconstruction: given a structure, one is able to write aclosed form expression for θ as a function of φ

I However, not necessary that a closed form expressionexists for a structure to be identified

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Example: Identifying WTP

I Lewbel, Linton, McFadden (2011): want to recover thedistribution of people’s willingness to pay (WTP), W ?,FW?(w).

I Dataset used by Hanemann et al. (1991) to elicit the WTPfor protecting wetland habitats and wildlife in California’sSan Joaquin Valley

I For each person in the sample, researchers draw a price Pfrom a known distribution function and ask if they would bewilling to pay $P or more to preserve the wetland

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Example: Identifying WTPI Binary choice: D denotes an individual’s response

D = I (W ? > P) (1)

I Given random assignment of prices, P is distributedindependently of W ?

E(D|P = p) = Pr (W ? > P|P = p)

= Pr (W ? > P)

= 1− Pr (W ? < P)

= 1− FW?(p)

(2)

I Here, identification is proved by construction: FW? isuniquely determined by the function E(D|P = p), which isassumed to be known given φ

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Example: Identifying WTP

I Note that given the experimental design, the function FW?

might not be identified everywhere

I In the motivating experiment, P could take on one of 14values between $25 and $375 — can identify thedistribution function only at w? = p at these particularvalues

I To identify the entire distribution function FW? , would wantto design an experiment so that P could take on any valuethat W ? could equal — p should be drawn from acontinuous distribution with support at least as large as therange of possible values of W ?

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Proving Identification: Extremum

I Another common method is to prove that θ0 is the uniquesolution to some maximisation problem defined given S

I E.g. Show that the likelihood is globally concave, thenmaximum likelihood will have a unique maximising value

I Establish identification by showing that the uniquemaximiser in the population equals the true θ0

I Note trend to attempt to show this with complicatedstructural models by graphing marginal likelihood functionat the estimated parameter vector — don’t do this forpresentations!

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Discrete Choice

I The WTP Example provides some insight into specialregressor methods, often used in discrete choice modelswhen we want to be flexible about the distribution ofunobserved preference heterogeneity

I Before starting, a brief recap on discrete choice to allow usto discuss the role of normalisations

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Binary Choice

I Imagine a consumer choosing whether to consume agood/enter into treatment/start working

I Choose the action if:

α + βXi > εi (3)

I The probability that they choose

Pr(Yi = 1|X = xi) = Pr(α + βxi > εi |X = xi)

= Pr(α + βxi > εi)

= Fε(α + βxi)

(4)

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Binary Choice

I In most of the models you will have encountered thus far,you proceed by putting a functional formal assumption onthe distribution of the errors

I Example: Probit: εi ∼ N(µ, σ2)

I However, without further restrictions, {α, β} are notidentified

Pr(Yi = 1|X = xi) = Fε(α + βxi)

= Φ

(α + βxi − µ

σ

) (5)

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Binary Choice: LocationI Different combinations of {µ, α} are observationally

equivalent

Pr(Yi = 1|X = xi) = Φ

(α + βxi − µ

σ

)= Φ

((α + κ) + βxi − (µ+ κ)

σ

)= Φ

(α̃ + βxi − µ̃

σ

) (6)

I Standard: restrict µ = 0 — the location normalisation

Pr(Yi = 1|X = xi) = Φ

(α + βxi

σ

)(7)

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Binary Choice: ScaleI Different combinations of {σ, α, β} are observationally

equivalent

Pr(Yi = 1|X = xi) = Φ

(α + βxi

σ

)= Φ

(κα + κβxi

κσ

)= Φ

(α̃ + β̃xi

σ̃

) (8)

I Standard: restrict σ = 1 — the scale normalisation

Pr(Yi = 1|X = xi) = Φ (α + βxi) (9)

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Normalisations

I In parametric models, common to impose these restrictionson the distribution of the error term as we have just seen

I For example, in the Probit model above, assume that ε hasa standard normal distribution

I However, note that we could have imposed the locationand scale restrictions on {α, β} rather than {µ, σ}

I For example, α = 0 and βk = 1, allowing ε to have anarbitrary mean and variance

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Normalisations

I Normalisations common in semi- and nonparametricmodels

I Example: structure is a linear index model

E(Y |X ) = g(α + Xβ) (10)

I Features of interest: θ = {g, β, α}

I Normalisations/restrictions typically imposed on parametervectors in semiparametric models

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Normalisations

I For any nonzero constant κ, define θ̃ = {g̃, β̃, α̃} withβ̃ = β/κ, α̃ = α/κ and g̃(z) = g(κz)

I Then θ̃ is observationally equivalent to θ

I All elements β̃ in the identified set have β̃ proportional to β— identified up to a scale

I Require a scale normalisation, usually βk = 1 or β′β = 1

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Normalisations

I For any nonzero constant κ, define θ̃ = {g̃, β, α̃} withα̃ = α + κ and g̃(z) = g(z − κ)

I Then θ̃ is observationally equivalent to θ

I Require a location normalisation, usually α = 0 — excludea constant

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Normalisations

I What makes something a normalisation rather than arestriction?

I Calling a restriction a normalisation implies that is does notrestrict or limit behaviour — ‘without loss of generality’

I Thus, whether a restriction can be thought of in this waydepends in part on how we will use and interpret the model

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Normalisations

I If one is simply interested in, e.g. marginal effects, then thescale normalisation is indeed without loss of generality

∂Pr(Yi = 1)

∂X=β

σφ

(α + βX − µ

σ

)(11)

I If however want to imbue coefficients with meaning, onemight need to be careful!

I Caution: direct comparison of discrete choice coefficientsacross different samples/specifications

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Normalisations: Outside OptionsI ‘Outside option’ normalisations are also common in

discrete choice models

I Let utility from choice Y = y for y = 0,1

αy + βyX + εy (12)

I Utility maximisation means that choose good 1 if:

α1 + β1X + ε1 > α0 + β0X + ε0

(α1 − α0) + (β1 − β0)X + (ε1 − ε0) > 0α + βX + ε > 0

(13)

I Interpret α + βX as the utility from Y = 1 if assume thenormalisation that the utility of the outside option is zero

α0 + β0X + ε0 = 0 (14)

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Normalisations: Outside Options

I In static discrete choice models, this is usually a freenormalisation, without loss of generality

I However, this might not be the case in dynamic discretechoice models

I Assuming that the outside option has the same utility inevery period imposes real restrictions on preferences andhence on behaviour — be careful!

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Relaxing Assumptions on ε

I The parametric assumptions placed on the distribution ofunobserved errors are essentially arbitrary and can havevery restrictive behavioural implications

I To introduce some more common concepts in theliterature, explore the use of ‘special regressors’ in discretechoice and their role in identification

I Intuitively, variation in special regressors allow one to traceout the distribution of unobservables

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Special Regressor MethodsI Let’s pick up on the example from the beginning of the

lecture, casting in the standard notation used in theliterature:

D = I (W ? > P)

= I (W ? − P > 0)

= I (W ? + V > 0)

(15)

I Let H(v) = E(D|V = v) and suppose V is continuouslydistributed (V is the special regressor!)

H(v) = Pr (W ? + V > 0)

= Pr (W ? > −V )

= 1− FW?(−v)

(16)

I If the support of V contains the support of −W ?, then theentire distribution function FW? would be identified

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Special Regressor Methods

I Want to identify, e.g., the average willingness to pay; thespecial regressor allows one to do this

E(W ?) =

∫ wu

wl

wfw?(w) dw

=

∫ wu

wl

w∂Fw?(w)

∂wdw

=

∫ wu

wl

w∂ [1− H(−w)]

∂wdw

(17)

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Special Regressor Methods

I Key assumptions on the special regressor:I Independence (or conditional independence)

I Additive

I Continuity

I Large support

I Pop up a lot even if not always identified as specialregressors!

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Special Regressor Methods

I Large support important for identification of certainfeatures that rely on knowledge of the tails of a distribution

E(W ?) =

∫ wu

wl

w∂ [1− H(−w)]

∂wdw (18)

I If supp(V) bounded to a ≤ V ≤ b, then Fw?(w) onlyidentified for −b ≤W ? ≤ −a

I In this case, E(W ?) is not even set identified

I No bounds on E(W ?) because Fw? could have massarbitrarily far below −b or above −a

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Special Regressor: Random Coefficients

I Random coefficients often used to allow for moresophisticated unobserved preference heterogeneityspecifications

Y = I(V εv + X εx > 0) (19)

where εv and εx are random coefficients

I Assume εv > 0 and let ε = εx/εv — a scale normalisation

Y = I(V + X ε > 0) (20)

I Is the distribution of ε identified from the data?

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Special Regressor: Random CoefficientsI Assume that V is a special regressor, distributed

independently of X

Y = I(V + X ε > 0)

= I(V + U > 0)(21)

I Using the same argument as before FU|X is identified byvariation in the special regressor

E(Y |X = x ,V = v) = Pr(v + U > 0|X = x)

= 1− FU|X (−v)(22)

I So the distribution of ε is identified!

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Conclusion

I These two lectures have introduced the concept ofobservational equivalence and introduced its role inproving identification of structural features

I Next week we will consider the connection between this‘structural’ approach to identification and a ‘causal’approach to identification

I We will apply these results to consider identification insimple equilibrium settings

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