Post on 10-Mar-2018
Recap WTP Normalisations Special Reg
Introduction to Identification
Abi Adams
HT 2017
Abi Adams
TBEA
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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 ?
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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!
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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)
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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)
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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)
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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)
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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)
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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!
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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)
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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!
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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?
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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!
Abi Adams
TBEA
Recap WTP Normalisations Special Reg
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
Abi Adams
TBEA