Panel Data Seminar. Discrete Response Models. Crest-Insee. 11 April 2008

Transcription

1 Panel Data Seminar Discrete Response Models Romain Aeberhardt Laurent Davezies Crest-Insee 11 April 2008 Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

2 Contents Overview and Strategies 1 Overview and Strategies 2 Simple Approaches and their Drawbacks Linear Probability Model Fixed effects : the Incidental Parameters Problem Random Effects : the assumptions are too strong 3 Classical Remedies Conditional Logit : removing the Fixed Effects Chamberlain s and Mundlak s Approaches : relaxing the Random Effects assumption 4 Extensions Dynamic framework Semi-Parametric approach Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

3 Introduction Overview and Strategies Panel data characterized by an outcome of the form : y it = F (x it β + α i + u it ) Main advantage of panel data : possibility to take into account the unobserved heterogeneity α i Main difficulty with panel data : dealing with unobserved heterogeneity, in particular : relationship between α i and x it Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

4 Overview and Strategies Important reminder The usual denomination of Fixed Effects and Random Effects is misleading Fixed Effects means no assumption concerning the dependence between α i and x it Random Effects means in general an independence assumption between α i and x it (although it can be relaxed) Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

5 Overview and Strategies Simple strategies Linear Probability Model Good for a quick start But bad properties (worse than in cross section) Probit / Logit with Fixed Effects as dummies Conceptually simple But 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 unobserved heterogeneity and covariates So one misses the point of using panel data Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

6 Overview and Strategies Classical Remedies Conditional Logit In the spirit of the Within or FD transformations No assumptions required on the correlation between unobserved heterogeneity and covariates But the identification hinges on the functional form (logit) Chamberlain s and Mundlak s Approaches Based on the RE framework, computationaly easy Relaxes the no correlation assumption Allows only for a restricted relation between unobserved heterogeneity and covariates Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

7 Extensions Overview and Strategies Dynamic framework Relaxes the strict exogeneity assumption In particular, allows for the presence of the lagged dependent variable among the covariates Question of state dependence vs. unobserved heterogeneity Raises a new issue : the initial conditions problem Semi-parametric models Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

8 Overview and Strategies Main Reference for this class Econometric Analysis of Cross Section and Panel Data, J.M. Wooldridge Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

9 Simple Approaches and their Drawbacks Contents 1 Overview and Strategies 2 Simple Approaches and their Drawbacks Linear Probability Model Fixed effects : the Incidental Parameters Problem Random Effects : the assumptions are too strong 3 Classical Remedies Conditional Logit : removing the Fixed Effects Chamberlain s and Mundlak s Approaches : relaxing the Random Effects assumption 4 Extensions Dynamic framework Semi-Parametric approach Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

10 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 methods developped for linear models (FE, RE, Chamberlain s approach,...) Same problems as in the cross section case (predicted values outside the unit interval, heteroskedasticity) Even less appealing : it implies x i β α i 1 x i β Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

11 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 α i and x it A priori simple : just add dummies in the equation and use standard estimation procedures Danger : MLE estimators are asymptotically unbiased and consistent only if N and T Intuition : in the ML framework the number of regressors is fixed, and here it increases with N Fixed effects are biased and poorly estimated when T is small It contaminates the rest of the coefficients through the MLE procedure Difference with the linear case : the estimation of β did not depend on the α i (Frish-Waugh) This is called the incidental parameters problem Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

12 Simple Approaches and their Drawbacks Fixed effects : the Incidental Parameters Problem Chamberlain s illustration of the incidental parameters problem Very simple framework : ML estimation of a logit model with two independent time periods, fixed effects and one explanatory variable x it s.t. i, x i1 = 0 and x i2 = 1 P(y it = 1 x, α) = eα i +x it β 1 + e α i +x it β if y i1 = 0 and y i2 = 0 then ˆα i = if y i1 = 1 and y i2 = 1 then ˆα i = + if y i1 + y i2 = 1 then ˆα i = ˆβ/2 P and ˆβ = 2 log(ñ 2 /ñ 1 ) 2β with ñ 1 = #{i y i1 = 1, y i2 = 0} and ñ 2 = #{i y i1 = 0, y i1 = 1} Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

13 Simple Approaches and their Drawbacks Random Effects : the assumptions are too strong RE : simple procedure but strong assumptions Basic assumptions : P(y it = 1 x it, α i ) = Φ(x it β + α i ) y i1, y i2,..., y it independent conditional on (x i, α i ) Density of (y i1,..., y it ) conditional on (x i, α i ) : f (y i1,..., y it x i, α i, β) T = f (y it x it, α i, β) = t=1 T Φ(x it β + α i ) y it [1 Φ(x it β + α i )] 1 y it t=1 Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

14 Simple Approaches and their Drawbacks Random Effects : the assumptions are too strong RE : simple procedure but strong assumptions One needs to integrate out α i, which requires an additional assumption : α i x i N (0, σ 2 α) The conditional density becomes f (y i1,..., y it x i, β, σ α ) = + T [ t=1 f (y it x it, α, β)] 1 ( ) α ϕ dα σ α σ α This is already implemented or easy to implement in standard softwares The independance assumption of α i and x i is very strong One misses the point of using panel data But this procedure will be the basis for more complicated approaches Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

15 Contents Classical Remedies 1 Overview and Strategies 2 Simple Approaches and their Drawbacks Linear Probability Model Fixed effects : the Incidental Parameters Problem Random Effects : the assumptions are too strong 3 Classical Remedies Conditional Logit : removing the Fixed Effects Chamberlain s and Mundlak s Approaches : relaxing the Random Effects assumption 4 Extensions Dynamic framework Semi-Parametric approach Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

16 Classical Remedies Conditional Logit : make the α i vanish Conditional Logit : removing the Fixed Effects In the spirit of the linear FE model Requires no assumption on α i y i1,..., y it independent conditional on (x i, α i ) The distribution of (y i1,..., y it ) conditional on does not depend on α i x i, α i and n i = T t=1 y it Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

17 Classical Remedies Conditional Logit : make the α i vanish Conditional Logit : removing the Fixed Effects Example with T = 2, the result is based on and then P(y i1 = 1, y i2 = 0 α i, x i ) P(y i1 = 0, y i2 = 1 α i, x i ) = eβ(x i1 x i2 ) P(y i1 = 0, y i2 = 1 y i1 + y i2 = 1, α i, x i ) = independent of α i and hence, P(y i1 = 0, y i2 = 1 y i1 + y i2 = 1, x i ) = e β(x i1 x i2 ) e β(x i1 x i2 ) Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

18 Classical Remedies Conditional Logit : make the α i vanish Conditional Logit : removing the Fixed Effects Conditional log likelihood for observation i is cll i (β) = 1 {ni =1}(w i log Λ[(x i2 x i1 )β] Same properties as the usual likelihood + (1 w i ) log(1 Λ[(x i2 x i1 )β])) 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 y it : i.e. no serial correlation in the u it, and no state dependence. Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

19 Back to the RE Classical Remedies Chamberlain s and Mundlak s Approaches : relaxing the Random Effects assumption Relaxing the crucial RE assumption : α i x i N (0, σ 2 α) by specifying a special form of dependence Mundlak (1978) : α i x i N (ψ + x i ξ, σ 2 a) Chamberlain (1980), more general form : instead of x i, he uses the vector of all explanatory variables across all time periods x i We can use standard RE probit software by just adding all the x i to all time periods (Chamberlain), or only the x i (Mundlak) Restrictive in the sense that it specifies a distribution of α i w.r.t. x i Still strong assumptions on the distribution tails for α i At least allows for some correlation Can be extended, for instance by specifying the distribution of the higher moments of α i x i Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

20 Strict exogeneity Classical Remedies Chamberlain s and Mundlak s Approaches : relaxing the Random Effects assumption All the previous procedures hinge on the strict exogeneity of x it conditional on α i : x it independent of u it at all time periods t Very difficult to correct for endogeneity in nonlinear models But an easy test can be implemented : Let w it be a subset of x it which potentially fail the strict exogeneity assumption Include w it+1 as an additional set of covariates Under the null hypothesis of strict exogeneity, the coefficients on w it+1 should be statistically insignificant Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

21 Contents Extensions 1 Overview and Strategies 2 Simple Approaches and their Drawbacks Linear Probability Model Fixed effects : the Incidental Parameters Problem Random Effects : the assumptions are too strong 3 Classical Remedies Conditional Logit : removing the Fixed Effects Chamberlain s and Mundlak s Approaches : relaxing the Random Effects assumption 4 Extensions Dynamic framework Semi-Parametric approach Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

22 Extensions Dynamic framework State dependence vs. unobserved heterogeneity Dynamic framework : P(y it = 1 y it 1,..., y i0, x i, α i ) = G(x it δ + ρy it 1 + α i ) x it are supposed to be strictly exogenous, but y it 1 appears on the RHS so we lose the strict exogeneity (y it 1 depends on u it 1 ) Extensions of the previous approaches Conditional logit cf Chamberlain (1985, 1993), Magnac (2000), Honoré Kyriazidou (1997) Extension of the RE framework but raises the initial conditions problem Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

23 Extensions Dynamic framework Conditional Logit in a dynamic framework You need at least 4 observations per individual Intuition : in order to make the α i vanish, you need to consider the two sets of events : A = {y i0 = d 0, y i1 = 0, y i2 = 1, y i3 = d 3 } and B = {y i0 = d 0, y i1 = 1, y i2 = 0, y i3 = d 3 } With no other covariates, see Chamberlain (1985), Magnac (2000) Extensions with strictly exogenous covariates, see Honoré and Kyriazidou (2000) Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

24 Extensions Dynamic framework Back to RE framework, the initial conditions problem Form of the joint density of the observations ranging from 0 to T for an individual i : f (y i0, y i1,..., y it α i, x i, β) = T f (y it y it 1, x it, α i, β)f (y i0 x i0, α i ) t=1 Goal : integrating out α i in order to obtain : f (y i0, y i1,..., y it x i, β) = T t=1 f (y it y it 1, x it, α i, β)f (y i0 x i, α i )g(α i x i )dα i Initial conditions problem : specifying f (y i0 x i, α i ) Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

25 Extensions Dynamic framework Initial conditions problem : Heckman s approach Specify f (y i0 x i, α i ) and then specify a density for α i given x i For instance, assume that y i0 follows a probit model with success probability Φ(η + x i π + γα i ) Then integrate out α i by specifying for instance α i x i N (m i, σ 2 i ) Problem : it is very difficult to specify the density of y i0 given (x i, α i ) Problem : because the true density of y i0 given (x i, α i ) is not known and is supposed to depend on y i 1, estimators are biased when T < + Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

26 Extensions Dynamic framework Initial conditions problem : Wooldridge s approach Instead of working on the full density f (y i0, y i1,..., y it α i, x i, β) Wooldridge prefers to work on the conditional density f (y i1,..., y it y i0, α i, x i, β) Advantage : remaining agnostic on the density of y i0 given (x i, α i ) Then specify a density for α i given (y i0, x i ) and keep conditioning on y i0 in addition to x i f (y i1,..., y it y i0, x i, θ) = + f (y i1,..., y it y i0, x it, α, β)h(α y i0, x i, γ)dα For example, with h(α y i0, x i, γ) N (ψ + ξ 0 y i0 + x i ξ, σ 2 a) y it = 1 {ψ+xit δ+ρy it 1 +ξ 0 y i0 +x i ξ+a i +e it >0} We can use standard RE probit software by just adding y i0 and x i to all time periods Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

27 Extensions Semi-Parametric approach Reminder on Manski s approach in cross section (1988) Model y i = 1 {xi β+ε i >0} Until now, the conditional density f (ε x i ) 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 technical assumption concerning X : there must be one continuous variable X k, s.t. the density of X k X k is positive everywhere a.s. ˆβ MS arg max β β 0 = arg max E((2Y 1)1 {X β β>0}) n Y i 1 {X β 0} + (1 Y i )1 {X β<0} i=1 Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

28 Extensions Semi-Parametric approach Reminder on Manski s approach in cross section (1988) ˆβ MS P β0 n 1/3 ( ˆβMS β 0 ) L D See Kim and Pollard (1990) for the exact definition of D Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29

29 Extensions to panel data Extensions Semi-Parametric approach See Honoré and Kyriazidou (1997) : Extension to dynamic panel data with exogenous covariates P(y i0 = 1 x i, α i ) = p 0 (x i, α i ) P(y it = 1 x i, α i, y i0,..., y it 1 ) = F (x it β + γy it 1 + α i ) with T = 4, β and γ may be estimated by maximizing w.r.t. b an g n 1 {xi2 x i3 =0}(y i2 y i1 )sgn((x i2 x i1 )b + g(y i3 y i0 )) i=1 Aeberhardt and Davezies (Crest-Insee) Panel Data Seminar 11 April / 29