panel data seminar - · pdf fileit+1 as an additional set of covariates ... (crest-insee)...

Panel Data SeminarDiscrete Response Models

Romain Aeberhardt Laurent Davezies

Crest-Insee

11 April 2008

Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April 2008 1 / 29

Overview and Strategies

Contents

1 Overview and Strategies

2 Simple Approaches and their DrawbacksLinear Probability ModelFixed effects : the Incidental Parameters ProblemRandom Effects : the assumptions are too strong

3 Classical RemediesConditional Logit : removing the Fixed EffectsChamberlain’s and Mundlak’s Approaches : relaxing the RandomEffects assumption

4 ExtensionsDynamic frameworkSemi-Parametric approach



Introduction

Panel data characterized by an outcome of the form :yit = F (xitβ + αi + uit)

Main advantage of panel data : possibility to take into account theunobserved heterogeneity αi

Main difficulty with panel data : dealing with unobservedheterogeneity, in particular : relationship between αi and xit



Important reminder

The usual denomination of “Fixed Effects” and “Random Effects” ismisleading

Fixed Effects “means” no assumption concerning the dependencebetween αi and xit

Random Effects “means” in general an independence assumptionbetween αi and xit (although it can be relaxed)



Simple strategies

Linear Probability Model

Good for a quick startBut bad properties (worse than in cross section)

Probit / Logit with Fixed Effects as dummies

Conceptually simpleBut ML estimators are consistent only when N →∞ and T →∞(incidental parameters problem)

Simple Random Effects Probit

Computationaly quite easy (already implemented)But one strong assumption of no correlation between unobservedheterogeneity and covariatesSo one misses the point of using panel data



Classical Remedies

Conditional Logit

In the spirit of the Within or FD transformationsNo assumptions required on the correlation between unobservedheterogeneity and covariatesBut the identification hinges on the functional form (logit)

Chamberlain’s and Mundlak’s Approaches

Based on the RE framework, computationaly easyRelaxes the no correlation assumptionAllows only for a restricted relation between unobserved heterogeneityand covariates



Extensions

Dynamic framework

Relaxes the strict exogeneity assumptionIn particular, allows for the presence of the lagged dependent variableamong the covariatesQuestion of state dependence vs. unobserved heterogeneityRaises a new issue : the initial conditions problem

Semi-parametric models



Main Reference for this class

Econometric Analysis of Cross Section and Panel Data, J.M.Wooldridge


Simple Approaches and their Drawbacks

Contents






Simple Approaches and their Drawbacks Linear Probability Model

Linear Probability Model : good for a quick start

Main advantage : allows to use all the simple and well known methodsdevelopped for linear models (FE, RE, Chamberlain’s approach, ...)

Same problems as in the cross section case (predicted values outsidethe unit interval, heteroskedasticity)

Even less appealing : it implies −xiβ ≤ αi ≤ 1− xiβ


Simple Approaches and their Drawbacks Fixed effects : the Incidental Parameters Problem

First idea : using dummies for fixed effects

Interest : no assumption on the correlation structure between αi andxit

A priori simple : just add dummies in the equation and use standardestimation procedures

Danger : MLE estimators are asymptotically unbiased and consistentonly if N →∞ and T →∞

Intuition : in the ML framework the number of regressors is fixed, andhere it increases with NFixed effects are biased and poorly estimated when T is smallIt contaminates the rest of the coefficients through the MLE procedureDifference with the linear case : the estimation of β did not depend onthe αi (Frish-Waugh)

This is called the “incidental parameters problem”


Simple Approaches and their Drawbacks Fixed effects : the Incidental Parameters Problem

Chamberlain’s illustration of the incidental parametersproblem

Very simple framework : ML estimation of a logit model with twoindependent time periods, fixed effects and one explanatory variable xit s.t.∀i , xi1 = 0 and xi2 = 1

P(yit = 1|x , α) =eαi+xitβ

1 + eαi+xitβ

if yi1 = 0 and yi2 = 0 then αi = −∞if yi1 = 1 and yi2 = 1 then αi = +∞if yi1 + yi2 = 1 then αi = −β/2

and β = 2 log(n2/n1)P−→ 2β

with n1 = #{i |yi1 = 1, yi2 = 0} and n2 = #{i |yi1 = 0, yi1 = 1}


Simple Approaches and their Drawbacks Random Effects : the assumptions are too strong

RE : simple procedure but strong assumptions

Basic assumptions :

P(yit = 1|xit , αi ) = Φ(xitβ + αi )yi1, yi2, . . . , yiT independent conditional on (xi , αi )

Density of (yi1, . . . , yiT ) conditional on (xi , αi ) :

f (yi1, . . . , yiT |xi , αi , β)

=T∏

t=1

f (yit |xit , αi , β)

=T∏

t=1

Φ(xitβ + αi )yit [1− Φ(xitβ + αi )]1−yit


Simple Approaches and their Drawbacks Random Effects : the assumptions are too strong

RE : simple procedure but strong assumptions

One needs to integrate out αi , which requires an additionalassumption :

αi |xi ∼ N (0, σ2α)

The conditional density becomes

f (yi1, . . . , yiT |xi , β, σα) =

∫ +∞

−∞[

T∏t=1

f (yit |xit , α, β)]1

σαϕ

(α

σα

)dα

This is already implemented or easy to implement in standardsoftwares

The independance assumption of αi and xi is very strong

One misses the point of using panel data

But this procedure will be the basis for more complicated approaches


Classical Remedies

Contents






Classical Remedies Conditional Logit : removing the Fixed Effects

Conditional Logit : make the αi vanish

In the spirit of the linear FE model

Requires no assumption on αi

yi1, . . . , yiT independent conditional on (xi , αi )

The distribution of (yi1, . . . , yiT ) conditional on

xi , αi and ni =T∑

t=1

yit

does not depend on αi




Example with T = 2, the result is based on

P(yi1 = 1, yi2 = 0|αi , xi )

P(yi1 = 0, yi2 = 1|αi , xi )= eβ(xi1−xi2)

and then

P(yi1 = 0, yi2 = 1|yi1 + yi2 = 1, αi , xi ) =1

1 + eβ(xi1−xi2)

independent of αi and hence,

P(yi1 = 0, yi2 = 1|yi1 + yi2 = 1, xi ) =1

1 + eβ(xi1−xi2)




Conditional log likelihood for observation i is

clli (β) = 1{ni=1}(wi log Λ[(xi2 − xi1)β]

+ (1− wi ) log(1− Λ[(xi2 − xi1)β]))

Same properties as the “usual” likelihood

The identification uses only the individuals who change state

Only drawback : the identification hinges on the functional form(logit) and there is no similar strategy with probit for example

There is still a conditional independance assumption for the yit : i.e.no serial correlation in the uit , and no state dependence.


Classical RemediesChamberlain’s and Mundlak’s Approaches : relaxing the

Random Effects assumption

Back to the RE

Relaxing the crucial RE assumption : αi |xi ∼ N (0, σ2α) by specifying a

special form of dependence

Mundlak (1978) : αi |xi ∼ N (ψ + xiξ, σ2a)

Chamberlain (1980), more general form : instead of xi , he uses thevector of all explanatory variables across all time periods xi

We can use standard RE probit software by just adding all the xi toall time periods (Chamberlain), or only the xi (Mundlak)

Restrictive in the sense that it specifies a distribution of αi w.r.t. xi

Still strong assumptions on the distribution tails for αi

At least allows for some correlation

Can be extended, for instance by specifying the distribution of thehigher moments of αi |xi


Classical RemediesChamberlain’s and Mundlak’s Approaches : relaxing the

Random Effects assumption

Strict exogeneity

All the previous procedures hinge on the strict exogeneity of xit

conditional on αi :

xit independent of uit′ at all time periods t ′

Very difficult to correct for endogeneity in nonlinear models

But an easy test can be implemented :

Let wit be a subset of xit which potentially fail the strict exogeneityassumptionInclude wit+1 as an additional set of covariatesUnder the null hypothesis of strict exogeneity,the coefficients on wit+1 should be statistically insignificant


Extensions

Contents






Extensions Dynamic framework

State dependence vs. unobserved heterogeneity

Dynamic framework :

P(yit = 1|yit−1, . . . , yi0, xi , αi ) = G (xitδ + ρyit−1 + αi )

xit are supposed to be strictly exogenous, but yit−1 appears on theRHS so we lose the strict exogeneity (yit−1 depends on uit−1)

Extensions of the previous approaches

Conditional logit cf Chamberlain (1985, 1993), Magnac (2000), HonoreKyriazidou (1997)Extension of the RE framework but raises the initial conditions problem



Conditional Logit in a dynamic framework

You need at least 4 observations per individual

Intuition : in order to make the αi vanish, you need to consider thetwo sets of events :A = {yi0 = d0, yi1 = 0, yi2 = 1, yi3 = d3}andB = {yi0 = d0, yi1 = 1, yi2 = 0, yi3 = d3}With no other covariates, see Chamberlain (1985), Magnac (2000)

Extensions with strictly exogenous covariates, see Honore andKyriazidou (2000)



Back to RE framework, the initial conditions problem

Form of the joint density of the observations ranging from 0 to T for anindividual i :

f (yi0, yi1, . . . , yiT |αi , xi , β) =T∏

t=1

f (yit |yit−1, xit , αi , β)f (yi0|xi0, αi )

Goal : integrating out αi in order to obtain :

f (yi0, yi1, . . . , yiT |xi , β) =

∫ T∏t=1

f (yit |yit−1, xit , αi , β)f (yi0|xi , αi )g(αi |xi )dαi

Initial conditions problem : specifying f (yi0|xi , αi )



Initial conditions problem : Heckman’s approach

Specify f (yi0|xi , αi ) and then specify a density for αi given xi

For instance, assume that yi0 follows a probit model with successprobability Φ(η + xiπ + γαi )

Then integrate out αi by specifying for instance αi |xi ∼ N (mi , σ2i )

Problem : it is very difficult to specify the density of yi0 given (xi , αi )

Problem : because the ”true” density of yi0 given (xi , αi ) is not knownand is supposed to depend on yi−1, estimators are biased whenT < +∞



Initial conditions problem : Wooldridge’s approach

Instead of working on the full density

f (yi0, yi1, . . . , yiT |αi , xi , β)

Wooldridge prefers to work on the conditional density

f (yi1, . . . , yiT |yi0, αi , xi , β)

Advantage : remaining agnostic on the density of yi0 given (xi , αi )Then specify a density for αi given (yi0, xi )and keep conditioning on yi0 in addition to xi

f (yi1, . . . , yiT |yi0, xi , θ) =

∫ +∞

−∞f (yi1, . . . , yiT |yi0, xit , α, β)h(α|yi0, xi , γ)dα

For example, with h(α|yi0, xi , γ) ∼ N (ψ + ξ0yi0 + xiξ, σ2a)

yit = 1{ψ+xitδ+ρyit−1+ξ0yi0+xiξ+ai+eit>0}

We can use standard RE probit software by just adding yi0 and xi toall time periods


Extensions Semi-Parametric approach

Reminder on Manski’s approach in cross section (1988)

Model yi = 1{xiβ+εi>0}

Until now, the conditional density f (ε|xi ) was specified

Can we relax this assumption ?

E(ε|X ) = 0 is not enough to identify β (Manski, 1988)med(ε|X ) will allow to identify β/ ‖β‖ under one more technicalassumption concerning X : there must be one continuous variable Xk ,s.t. the density of Xk |X−k is positive everywhere a.s.

β0 = arg maxβ

E((2Y − 1)1{X ′β>0})

βMS ∈ arg maxβ

n∑i=1

Yi1{X ′β≥0} + (1− Yi )1{X ′β<0}



Reminder on Manski’s approach in cross section (1988)

βMSP→ β0

n1/3(βMS − β0

)L→ D

See Kim and Pollard (1990) for the exact definition of D



Extensions to panel data

See Honore and Kyriazidou (1997) :

Extension to dynamic panel data with exogenous covariates

P(yi0 = 1|xi , αi ) = p0(xi , αi )

P(yit = 1|xi , αi , yi0, . . . , yit−1) = F (xitβ + γyit−1 + αi )

with T = 4, β and γ may be estimated by maximizing w.r.t. b an g

n∑i=1

1{xi2−xi3=0}(yi2 − yi1)sgn((xi2 − xi1)b + g(yi3 − yi0))


panel data seminar - · pdf fileit+1 as an additional set of covariates ... (crest-insee)...

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