Introduction Intuition Models Structures Extremum

Introduction to Identification

Abi Adams

HT 2017

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Introduction Intuition Models Structures Extremum

Outline for Weeks 1-3

I Lectures 1-2: The Language of Identification

II Lecture 3: Casual vs Structural Identification

III Lectures 4-6: Topics in Simultaneity

i (Linear) Supply & Demand

ii Social Interactions

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Introduction Intuition Models Structures Extremum

Outline for Today

I Intuitive Introduction to Identification

II Models

III Structures

IV Alternative Definitions

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Introduction Intuition Models Structures Extremum

Why is this Theory Based Empirical Analysis?

I Think of economic processes as generating data

I What can be known about the underlying process given thedata it yields?

I What is the nature of the knowledge that can be obtained?Point or set identification?

I What minimal restrictions do we require for knowledge ofparticular features of a process?

I How does the extent and nature of measurement affect theanswers to these questions?

I What are the implications for the design of surveys,experiments etc?

I Need to think about this prior to estimation!

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Warm Up

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Introduction Intuition Models Structures Extremum

The Basics

I Let θ be parameters, vectors, and/or functions that we wantto learn about (and hopefully estimate!)

I Identification: what can be learned about θ fromobservable data?

I If we knew the population that data are drawn from, whatwould be knowable about θ

I Logically precedes estimation, inference, and testing

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The BasicsI Typically, what we learn from the data are distributions, e.g.

F (Y |X )

I Crude definition: θ is identified if it is uniquely determinedgiven what we know can be learned about F

I Might seem a bit circular — the starting point is one ofassuming features of F (Y |X ) are identified (i.e. can belearned from the data) and then using those features todetermine whether θ is identified

I Assuming that some features of F (Y |X ) are identified isjustified by deeper information regarding the underlyingData Generating Process (DGP)

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Starting PointI E.g. If DGP is of n→∞ i.i.d observations {y , x}, then

F (Y ,X ) can be learned from the dataI Assume F (Y ,X ) is identified, and use this as starting point

for identifying θ

I E.g. For each observation:I Choose a value of X from its support

I Conditional on that value of X , randomly draw anobservation of Y , independent from other draws of Y

I E.g. X is the temperature you run an experiment at and Yis the outcome of the experiment

I Assume F (Y |X ) can be learned from the data but only atthe values of X we choose

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Introduction Intuition Models Structures Extremum

Starting Point

I E.g. Revealed preference approachI Often takes a finite data approach in modern Afriat-Varian

incarnation — do not know the demand function

I Not always the case: Samuelson (1938, 1948) andHouthakker (1950), for example, assumed observability ofthe demand system as their starting point

I Let φ denote the features of the data that we assume areknowable to begin with.

I Identification depends on what features we are willingto assume are known or knowable about the DGP

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Introduction Intuition Models Structures Extremum

An Aside: Big Data

I In many ways, ‘big data’ is about identification

I Varian (2014): “In this period of ‘big data’, it seems strangeto focus on sampling uncertainty, which tends to be smallwith large datasets, while completely ignoring modeluncertainty, which may be quite large”

I With big data, the observed sample is so large that it cantreated as if it were the population.

I Identification deals precisely with what can be learnedabout the relationships among variables given thepopulation.

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Quick ExampleI Linear model for scalar Y and scalar X

Y = Xθ + e (1)

with E(Xe) = 0I Assume that can learn the first two moments of (Y ,X )

from the data

E(XY ) = θE(X 2) + E(Xe) (2)

I For identification of θ require E(X 2) 6= 0 — in this case, θ isuniquely determined from the data

θ =E(XY )

E(X 2)(3)

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Introduction Intuition Models Structures Extremum

Quick ExampleI Simple IV model

Y = α + Xβ + e, E(e|Z = z) = c (4)

I Assume that can learn E(Y |Z ) and E(X |Z ) at, at least twopoints in the support of Z at which

E(X |Z = z1) 6= E(X |Z = z2) (5)

I Then,

E(Y |Z = z1) = α + βE(X |Z = z1) + c (6)E(Y |Z = z2) = α + βE(X |Z = z2) + c (7)

I and θ is uniquely determined from the data

β =E(Y |Z = z1)− E(Y |Z = z2)

E(X |Z = z1)− E(X |Z = z2)(8)

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Introduction Intuition Models Structures Extremum

Historical Example: Demand Curves

I Textbook example of an identification problem ineconomics first recognised by Phillip Wright (1915)

I What are the features of demand curves?

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Historical Example: Demand CurvesI Moore’s (1914) “new type” of demand curve

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Introduction Intuition Models Structures Extremum

Historical Example: Demand Curves

I Moore’s (1914) “new type” of demand curve

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Introduction Intuition Models Structures Extremum

Historical Example: Demand CurvesI Is this a demand curve?

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Introduction Intuition Models Structures Extremum

Historical Example: Demand CurvesI Is this a demand curve? e.g. Working (1927)

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Introduction Intuition Models Structures Extremum

Introduction: Equilibrium Market ModelI Q is the amount buyers will buy at price P. Z is a list of

market characteristics.

Q = γdP + Z ′βd + Ud (9)

I P is the minimum price at which sellers deliver an amountQ

P = γsQ + Z ′βs + Us (10)I In each period Q and P are such that the market clears

and these equations are simultaneously satisfied.

I (Ud ,Us) and Z are mean independent, andE([UdUs]|Z = z) = (0,0)

I Can the data be informative about the unknownparameters?

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Introduction: Equilibrium Market ModelI Need to solve the simultaneous equations

Q = γdP + Z ′βd + Ud

P = γsQ + Z ′βs + Us(11)

I Write as:

[Q P]

[1 −γs−γd 1

]= Z ′[βdβs] + [Ud Us] (12)

I Require 1− γdγs 6= 0 for matrix to be nonsingular

[Q P] =(Z ′[βdβs] + [Ud Us]

) 11− γdγs

[1 γsγd 1

](13)

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Introduction Intuition Models Structures Extremum

Introduction: Equilibrium Market Model

I This yields the ‘reduced form’ (more later!):

Q = Z ′ (βd + βsγd )

1− γdγs+

Ud + γdUs

1− γdγs

P = Z ′ (βs + βdγs)

1− γdγs+

Us + γsUd

1− γdγs

(14)

I Given mean independence

E(Q|Z = z) = z ′ (βd + βsγd )

1− γdγs= z ′δd

E(P|Z = z) = z ′ (βs + βdγs)

1− γdγs= z ′δs

(15)

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Introduction Intuition Models Structures Extremum

Introduction: Equilibrium Market Model

I If Z has K elements, we have enough information to know2K parameters

I However, there are 2K + 2 unknown parameters!

I (γd , βd , γs, βs) is not identified without further restrictions...

I One cannot separately identify demand and supply curveson the basis of equilibrium observations alone withoutfurther restrictions

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Introduction Intuition Models Structures Extremum

Introduction: Equilibrium Market ModelI We’re going to need two restrictions....

I Landmark: Wright “the Tariff on Animal and Vegetable Oils’1928, develops IV

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Introduction Intuition Models Structures Extremum

Outline for Today

I Intuitive Introduction to Identification

II Models

III Structures

IV Alternative Definitions

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Introduction Intuition Models Structures Extremum

Formal Ingredients

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Introduction Intuition Models Structures Extremum

The Economic ModelI Economic models will typically specify e.g. the economic

agents involved, their objective functions, information,interactions &c

I Key distinctionsI Which variables are observable and which are

unobservable?

I Which are determined outside (exogenous) and inside(endogenous) the model?

Observable UnobservableExogenous X εEndogenous Y Y

I Will also have a list of primitive functions and distributions,H, F

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Introduction Intuition Models Structures Extremum

The Economic Model

I Models describe a relationship between the primitivefunctions and distributions, and the observable andunobserved variables

I Represent these relationships by a vector function v :

h(Y ,Y,X , ε,H,F) = 0 (16)

I These relationships can be used to derive the jointdistribution of the observable variables (Y ,X ) as a functionof model primitives

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Introduction Intuition Models Structures Extremum

Example: Utility Maximisation

I Consumer consumes two goods to maximise utility(dependent on demographic characteristics, w , andunobserved tastes, ε) subject to budget constraint

maxy1,y2

U (y1, y2,w , ε) s.t. py1 + y2 ≤ I (17)

I Assume that utility is strictly increasing, strictly concave, &twice differentiable

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Introduction Intuition Models Structures Extremum

Example: Utility Maximisation

I Optimisation plus two-good assumption implies that y1satisfies:

Uy1 (y1, I − py1,w , ε)− pUy2 (y1, I − py1,w , ε) = 0 (18)

I This determines the joint distribution of the vector ofobservables, (Y ,p, I,w)

h(Y ,X , ε,U) = h(Y ,p, I,w , ε,U)

= Uy1 (Y , I − pY ,w , ε)− pUy2 (Y , I − pY ,w , ε)= 0

(19)

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Introduction Intuition Models Structures Extremum

Example: Utility MaximisationI Under our assumptions, the value of Y that satisfies this

equations for given (p, I,w , ε) is unique

I The reduced form model is then

Y = m(p, I,w , ε) (20)

I This maps the observable and unobservable explanatoryvariables into the observable exogenous variables withoutspecifying behavioural and equilibrium conditions fromwhich the mapping is derived

I To answer many questions, the reduced form issufficient.... however, for many, knowledge of the primitivemodel functions is what we are after

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Introduction Intuition Models Structures Extremum

Outline for Today

I Intuitive Introduction to Identification

II Models

III Structures

IV Alternative Definitions

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Introduction Intuition Models Structures Extremum

StructuresI We have seen that outcomes, Y , can be thought of as

determined by a set of structural equations:

h(Y ,X , ε) = 0 (21)

I There is a probability distribution FεX (or Fε|X )

I The pair S ≡ {h,FεX} is called a structure. Each Sgenerates a distribution FYX (or FY |X )

I The observable distribution FYX corresponds to the truevalue S0 of S

I The model, Γ, comprises the restrictions on admissiblestructures

I LetMΓ be the set of admissible structures defined by amodel, Γ

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Introduction Intuition Models Structures Extremum

Example: Utility Maximisation

I In the consumer case, elements of MΓ are the pairs

S ={

Uy1 (Y , I − pY ,w , ε)− pUy2 (Y , I − pY ,w , ε) ,Fε,p,I,w}

(22)such that for all (w , ε):

I U(·, ·,w , ε) is strictly increasing, strictly concave, & twicedifferentiable

I Fε,p,I,w is a distribution function

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Introduction Intuition Models Structures Extremum

Observational Equivalence

I Key point: There may be many structures that imply aparticular distribution of the data

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

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Introduction Intuition Models Structures Extremum

Identification

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

φ = F S′

φ

I The model Γ is uniformly identifying if it identifies allS ∈MΓ

I i.e. there is a one-to-one correspondence between thestructures admitted by a uniformly identifying model and theprobability distributions they imply

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Introduction Intuition Models Structures Extremum

Restrictiveness & Just Identification

I If M1 ⊂ M0, then M0 admits structures that are not admittedby M1, then M1 is more restrictive

I Suppose M1 is a uniformly identifying model and that nomodel M0 with M1 ⊂ M0 is uniformly identifying

I M0 is a just identifying model

I Such models lose their uniform identifying power if any ofits restrictions are relaxed

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Introduction Intuition Models Structures Extremum

Misspecification

I A structure S is satisfied if F S′

φ = FφI Observed data satisfies the restrictions laid down by the

structure

I A model is misspecified if there is no structure S ∈ MΓ

that is satisfied

I No specification of the model which could give rise toobserved data

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Introduction Intuition Models Structures Extremum

Identification of Structural Features

I There is often interest only in some features of a model,e.g. sign of a coefficient

I A model might identify a structural feature when it is notuniformly identifying

I Even though a model admits observationally equivalentstructures, a feature is invariant within any set ofobservationally equivalent structures

I E.g. Level of utility versus marginal rates of substitution

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Introduction Intuition Models Structures Extremum

Identification of Structural Features

I Consider a feature of a structure θ(S) and a model Γ

I Γ point identifies θ(S0) if θ(S) is constant across allstructures admitted by Γ that are observationally equivalentto S0

I Γ set identifies θ(S0) to within Θ0 if for all admissiblestructures that are observationally equivalent to S0,θ(S) ∈ Θ0

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Introduction Intuition Models Structures Extremum

Identification of Structural Features

I Chesher (2006): if there exists some function G(FYX ) suchthat for all S′ ∈MΓ,

θ(S′) = a→ G(F S′

YX ) = a (23)

then Γ uniformly point identifies θ

I Overidentification: more than one distinct function G

I Analogue estimation: G(

F̂YX

)

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Introduction Intuition Models Structures Extremum

Quick ExampleI Structural equations: true value X̃ but observe X

Y = α + βX̃ + U X = X̃ + W

Y = α + βX̃ + (U − βW )(24)

I Model Restriction: E (U − βW |Z = z) = c

I Knowable Data: Conditional expectation identified at, atleast two values in the support of Z at which:

E(X |Z = z1) 6= E(X |Z = z2) (25)

I Then

β =E(Y |Z = z1)− E(Y |Z = z2)

E(X |Z = z1)− E(X |Z = z2)= G(FYXZ ) (26)

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Introduction Intuition Models Structures Extremum

Outline for Today

I Intuitive Introduction to Identification

II Models

III Structures

IV Extremum Based Identification

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Introduction Intuition Models Structures Extremum

Other Terms You Might Hear

I Many different terms and lots of different notation in thisfield

I Other alluded to ‘types’ of identification fit into theframework above — depends on what data assume haveavailable

I Wright-Cowles Ordinary Identification

I Likelihood Based Identification

I Extremum based Identification

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Introduction Intuition Models Structures Extremum

Extremum Identification

I Extremum estimators: defined by maximisation of anobjective function, e.g. least squares, maximum likelihood,GMM

θ̂ = arg maxθ∈Θ

Qn(θ) (27)

I θ identified if it is the only parameter vector that uniquelymaximises the probability limit of the extremum estimatorobjective function

I Implicitly assume that only information available is thevalue of the objective function

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Introduction Intuition Models Structures Extremum

Extremum Identification

I Extremum estimators: defined by maximisation of anobjective function, e.g. least squares, maximum likelihood,GMM

θ̂ = arg maxθ∈Θ

Qn(θ) (28)

I θ identified if it is the only parameter vector that uniquelymaximises the probability limit of the extremum estimatorobjective function

I Implicitly assume that only information available is thevalue of the objective function

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Introduction Intuition Models Structures Extremum

Extremum Identification

I Example: i.i.d data Wi = {yi ,Xi} with

θ̂ = arg maxθ∈Θ

1n

n∑i=1

m(Wi , θ) (29)

I Probability limit : E [m(Wi , θ)] = M(θ) = φ

I Structure: restriction that M(θ′) = maxθ∈Θ M(θ)

I Structures θ′ and θ′′ are observationally equivalent if

M(θ′) = M(θ′′) = max θ ∈ ΘM(θ) (30)

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Introduction Intuition Models Structures Extremum

Extremum IdentificationI θ0 is identified if and only if M(θ) has a unique argmax in Θ

I Tells you whether a structure is identified given a particularestimator

I However, doesn’t tell you whether a structure could havebeen identified using other information that might beavailable

I Identification failure might be due to poor choice ofobjective function

I E.g. Scalar Wi , θ = E(W ), and estimator

θ̂ = arg maxθ∈Θ

1n

n∑i=1

(Wi − |θ|)2 (31)

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