Chapter 7 - Modeling Issues
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1 Chapter 7 - Modelng Issues 7.1 Heterogenety 7. Comparng fxed and random effects estmators 7.3 Omtted varables Models of omtted varables Augmented regresson estmaton 7.4 Samplng, selectvty bas, attrton Incomplete and rotatng panels Unplanned nonresponse Non-gnorable mssng data
2 7.1 Heterogenety Also thnk of clusterng dfferent observatons from the same subject (observatonal unt) tend to be related to one another. Methods for handlng ths varables n common jontly dependent dstrbuton functons
3 Varables n Common These are latent (unobserved) varables May be fxed (mean structure) or random (covarance structure) May be organzed by the cross-secton or by tme If by cross-secton, may be subject orented (by ) or spatal May be nested
4 Jontly Dependent Dstrbuton Functons For the covarance structures, ths s a more general way to thnk about random varables that are common to subjects. Also ncludes addtonal structures not suggested by the common varable approach Example: For the error components model, we have Corr (y 1, y )=ρ where ρ >0. However, we need not requre postve correlatons for a general unform correlaton model.
5 Practcal Identfcaton wth Heterogenety may be dffcult Jones (1993) Example Subject 1 α 1 = Subject α = Subject 3 α3 = Tme 15 5
6 e Theoretcal Identfcaton wth Heterogenety may be Impossble Neyman Scott (1948) Example Example - Identfcaton of varance components Consder the fxed effects panel data model y t = α + ε t, =1,, n, t=1,, where Var ε t = σ and Cov (ε 1, ε ) = σ ρ. The ordnary least squares estmator of α s = (y 1 + y )/. Thus, the resduals are y y = (y 1 - y )/ and e = y y = (y - y 1 )/= - e 1. Thus, ρ cannot be estmated, despte havng n - n = n degrees of freedom avalable for estmatng the varance components. 1 = 1
7 Estmaton of regresson coeffcents wthout complete dentfcaton s possble If our man goal s to estmate or test hypotheses about the regresson coeffcents, then we do not requre knowledge of all aspects of the model. For example, consder the one-way fxed effects model y = α 1 + X β + ε. Apply the common transformaton matrx Q = I T -1 J to each equaton to get y * = Q y = QX β + Q ε = X * β + ε *, because Q 1 = 0. Use ols on the transformed data. For T =, σ R* = Q R Q = ρ ρ σ = 1 (1- ρ) Note that can estmate the quantty σ (1- ρ) yet cannot separate the terms σ and ρ.
8 7. Comparng fxed and random effects estmators Sometmes, the context of the problem does not make clear the choce of fxed or random effects estmators. It s of nterest to compare the fxed effects to the random effects estmators usng the data. In random effects models, we assume that { α } are ndependent of {ε t }. Thnk nstead of drawng {x t } at random and performng nference condtonal on {x t }. Interpret {α } to be the unobserved (tme-nvarant) characterstcs of a subject. Ths assumes that ndvdual effects { α } are ndependent of other ndvdual characterstcs {x t }, our strct condtonal exogenety assumpton SEC6.
9 A specal case Consder the error components model wth K = so that y t = α + β 0 + β 1 x t,1 + ε t, We can express (Fuller-Battese) the gls estmator as where x * t = x t b 1, EC n T = 1 t= 1 = n σ ε x 1 Tσ α + σ ( * * )( * * x ) t x yt y T ( * * x x ) = 1 t= 1 As σ α 0, we have that x t * x t,, y t * y t and b 1,EC b 1,OLS As σ α, we have that b 1,EC b 1,FE ε 1/ t y * t = y t y σ ε 1 Tσ α + σ ε 1/
10 A specal case Defne the so-called between groups estmator, b 1, B = n = 1 ( x x)( y y) n = 1 ( x x) Ths estmator can be motvated by averagng all observatons from a subject and then computng an ordnary least squares estmator usng the data. The followng decomposton due to Maddala (1971), b 1,EC = (1- ) b 1,FE + b 1,B where Var b1, EC = Var b1, B measures the relatve precson of the two estmators of β. { } n y x, = 1
11 A specal case To express the relatonshp between α and x, we consder E [α x ]. Specfcally, we assume that α = η + γ x,1, where {η } s..d. Thus, the model of correlated effects s y t = η + β 0 + β 1 x t,1 + γ x + ε.,1 t Surprsngly, one can show that the generalzed least squares estmator of β 1 s b 1,FE. Intutvely, by consderng devatons from the tme seres means, yt y, the fxed effects estmator sweeps out all tme-constant omtted varables. In ths sense, the fxed effects estmator s robust to ths type of model aberraton. Under the correlated effects model, the estmator b 1,FE s unbased, consstent and asymptotcally normal. the estmators b 1,HOM, b 1,B and b 1,EC are based and nconsstent.
12 A specal case To test whether or not to use the random or fxed estmator, we need only examne the null hypothess H 0 : γ = 0. Ths s customarly done usng the Hausman (1978) test statstc χ FE = ( b b ) Varb 1, EC 1, FE 1, EC Ths test statstc has an asymptotc (as n ) ch-square dstrbuton wth 1 degree of freedom. 1, FE - Varb Test statstc s large, go wth the fxed effects estmator Test statstc s small, go wth the random effects estmator
13 General Case Assume that E α = α and re-wrte the model as y = Z α + X β + ε *, where ε * = ε + Z (α - α) and Varε * = Z D Z + R = V. Ths re-wrtng s necessary to make the beta s under fxed and random effects formulatons comparable. Wth the approprate defntons, the extenson of Maddala s (1971) result s GLS ( I ) b FE b B b = + ( )( ) 1 where = Varb GLS Varb B The extenson of the Hausman test statstc s χ FE = 1 ( b b ) ( Var( b ) Var( b )) ( b b ) FE GLS FE GLS FE GLS
14 Case study: Income tax payments Consder the model wth Varable Intercepts but no Varable Slopes (Error Components) χ FE The test statstc s = 6.01 Wth K = 8 degrees of freedom, the p-value assocated wth ths test statstc s Pr ob( χ > 6.01). = For the model wth Varable Intercepts and two Varable Slopes χ FE The test statstc s = Wth K = 8 degrees of freedom, the p-value assocated wth ths test statstc s Prob( >13.68)= χ
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16 7.3 Omtted Varables I call these models of correlated effects. Secton 7. descrbed the Hausman/Mundlak model of tme-constant omtted varables. Chamberlan (198) an alternatve hypothess Omtted varables need not be tme-constant Hausman and Taylor (1981) another alternatve hypothess Some of the explanatory varables are not correlated wth α. To estmate these models, Arellano (1993) used an augmented regresson model. We wll also use ths approach. For a dfferent approach, Stata has programmed an nstrumental varable approach ntroduce by Hausman and Taylor as well as Amemya and Mac Curdy.
17 Unobserved Varables Models Introduced by Palta and Yao (1991) and coworkers. Let o =(z, x ) be observed varables and u be unobserved varables. Assumng multvarate normalty, we can express: [ ] ) E ( ) (, E 1 0 uo o o Σ Σ γ I X β Z α o α y + + = [ ] ( ) ) ( ) (, Var 1 0 γ I Σ Σ Σ Σ γ I R o α y + = uo uo u [ ] γ U X β Z α u o α y + + =,, E
18 Unobserved Varables Models The unobserved varables enter the lkelhood through: lnear condtonal expectatons (γ) correlaton between observed and unobserved varables (Σ uo ) The fxed effects estmator may be based, unlke the correlated effects model case. By examnng certan specal cases, we agan arrve at the Mundlak, Chamberlan and Hausman/Taylor alternatve hypotheses. Other alternatves are also of nterest. Specfcally, an extended Mundlak alternatve s: E * * * [ y α, o ] = β + z α + x β + z γ + x γ t 0 t t 1
19 Examples of Correlated Effects Models Assumng q = 1 and z t =1, ths s Chamberlan s alternatve. Chamberlan used the hypothess: * T Thus, E[ y t x ] = + β0 xtβ + t = x 1 tγt Assume an error components desgn for the x s. That s, Σ x,1 + Σ x, for s = t Cov( xs, xt ) = Σ x, for s t Assumng q = 1 and z t =1, ths s Mundlak s alternatve. That s E y x β + x β + x [ ] γ Further assume that the frst K-r x-varables are uncorrelated wth α. Ths s the Hausman/Taylor alternatve. t = * 0 H 1 : α = η t T + t = 1 x t γ t
20 Augmented Regresson Estmaton and Testng I advocate the augmented regresson approach that uses the model: E [y η o ] = Z η + X β + G γ. Random slopes η that do not affect the condtonal regresson functon. Thus, so that E [y o ] = X β + G γ. Choose G = G(X, Z ) to be a known functon of the observed effects. Choce of G depends on the alternatve model you consder. The test for omtted varables s thus H 0 : γ = 0. Defne b AR and γ AR to be the correspondng weghted least squares estmators.
21 Some Results The estmator b AR s unbased (even n the presence of omtted varables). The weghts correspondng to gls (W = V ) and ( ) G = Z Z R Z Z R X yelds b AR = b FE. Ths s an extenson of Mundlak s alternatve. The ch-square test for H 0 : γ = 0 s: χ ( Var γˆ AR ) γˆ AR 1 AR = γˆ AR
22 Determnants of Tax Lablty I examne a 4 % sample (58) of taxpayers from the Unversty of Mchgan/Ernst & Young Tax Data Base The panel conssts of tax years , 86, 87 Tax Lablty data, we use x t β = lnear functon of demographc and earnng characterstcs of a taxpayer z t α = α 1 + α LNTPI t + α 3 MR t y t = logarthmc tax lablty for the th taxpayer n the tth year
23 Emprcal Fts I present fts of four dfferent models Random effects ncludes varable ntercept plus two varable slopes +omtted varable correctons Random coeffcents Wth AR1 parameter effects +omtted varable correctons ( Extended Mundlak alternatve )
24 Results Secton 7. ndcated, wth only varable ntercepts, that the fxed effects estmator s preferable to the random effects estmator. For random effects, two addtonal varable slope terms were useful the random coeffcents model dd not yeld a postve defnte estmate of Var α, I used a thrd order factor analytc model New tests ndcate that both the fxed effects model wth 3 varable components and the extended Mundlak model are preferable to the random effects model wth 3 varable components Comparng fxed effects model wth 3 varable components and the extended Mundlak model, the AIC favors the former yet I advocate the latter (parsmony and so on).
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26 7.4 Samplng, selectvty bas and attrton Incomplete and rotatng panels Early longtudnal and panel data methods assumed balanced data, that s, T = T. Ths suggests technques from multvarate analyss. Data may not be avalable due to: Delayed entry Early ext Intermttent nonresponse If planned, then there s generally no dffculty. See the text for the algebrac transformaton needed. Planned ncomplete data s the norm n panel surveys of people.
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28 7.4. Unplanned nonresponse Types of panel survey nonresponse (source Verbeek and Njman, 1996) Intal nonresponse. A subject contacted cannot, or wll not, partcpate. Because of lmted nformaton, ths potental problem s often gnored n the analyss. Unt nonresponse. A subject contacted cannot, or wll not, partcpate even after repeated attempts (n subsequent waves) to nclude the subject. Wave nonresponse. A subject does not respond for one or more tme perods but does respond n the precedng and subsequent tmes (for example, the subject may be on vacaton). Attrton. A subject leaves the panel after partcpatng n at least one survey.
29 Mssng data models Let r j be an ndcator varable for the jth observaton, wth a one ndcatng that ths response s observed and a zero ndcatng that the response s mssng. Let r = (r 1,, r T ) and r = (r 1,, r n ). The nterest s n whether or not the responses nfluence the mssng data mechansm. Use y = (y 1,, y T ) to be the vector of all potentally observed responses for the th subject Let Y = (y 1,, y n ) to be the collecton of all potentally observed responses.
30 Rubn s (1976) Mssng data models Mssng completely at random (MCAR). The case where Y does not affect the dstrbuton of r. Specfcally, the mssng data are MCAR f f(r Y) = f(r), where f(.) s a generc probablty mass functon. Lttle (1995) - the adjectve covarate dependent s added when Y does not affect the dstrbuton of r, condtonal on the covarates. If the covarates are summarzed as {X, Z}, then the condton corresponds to the relaton f(r Y, X, Z) = f(r X, Z). Example: x=age, y=ncome. Mssngness could vary by ncome but s really due to age (young people don t respond)
31 General advce on mssng at random One opton s to treat the avalable data as f nonresponses were planned and use unbalanced estmaton technques. Another opton s to utlze only subjects wth a complete set of observatons by dscardng observatons from subjects wth mssng responses. A thrd opton s to mpute values for mssng responses. Lttle and Rubn note that each opton s generally easy to carry out and may be satsfactory wth small amounts of mssng data. However, the second and thrd optons may not be effcent. Further, each opton mplctly reles heavly on the MCAR assumpton.
32 Selecton Model Partton the Y vector nto observed and mssng components: Y = (Y obs, Y mss ). Selecton model s gven by f(r Y). Wth parameters θ and ψ, assume that the log lkelhood of the observed random varables s L(θ,ψ) = log f(r, Y obs,θ,ψ) = log f(y obs,θ) + log f(r Y obs,ψ). MCAR case f(r Y obs,ψ) = f(r ψ) does not depend on Y obs. Data mssng at random (MAR) f selecton mechansm model dstrbuton does not depend on Y mss but may depend on Y obs. That s, f(r Y) = f(r Y obs ). For both MAR and MCAR, the lkelhood may be maxmzed over the parameters separately n term. For nference about θ, the selecton model mechansm may be gnored. MAR and MCAR are referred to as the gnorable case.
33 Example Income tax payments Let y = tax lablty and x = ncome. The taxpayer s not selected (mssng) wth probablty ψ. The selecton mechansm s MCAR. The taxpayer s not selected f tax lablty < $100. The selecton mechansm depends on the observed and mssng response. The selecton mechansm cannot be gnored. The taxpayer s not selected f ncome < $0,000. The selecton mechansm s MCAR, covarate dependent. Assumng that the purpose of the analyss s to understand tax labltes condtonal on knowledge of ncome, stratfyng based on ncome does not serous bas the analyss. The probablty of a taxpayer beng selected decreases wth tax lablty. For example, suppose the probablty of beng selected s logt (-ψ y ). In ths case, the selecton mechansm depends on the observed and mssng response. The selecton mechansm cannot be gnored. The taxpayer s followed over T = perods. In the second perod, a taxpayer s not selected f the frst perod tax < $100. The selecton mechansm s MAR. That s, the selecton mechansm s based on an observed response.
34 Example - correcton for selecton bas Hstorcal heghts. y = the heght of men recruted to serve n the mltary. The sample s subject to censorng n that mnmum heght standards were mposed for admsson to the mltary. The selecton mechansm s non-gnorable because t depends on the ndvdual s heght. The jont dstrbuton for observables s f(r, Y obs, µ, σ) = f(y obs, µ, σ) f(r Y obs ) n = = 1 { f( y y > c) Prob( y > c) } ( Prob( y c) ) m Ths s easy to maxmze n µ and σ. If one gnored the censorng mechansms, then the log lkelhood s n = 1 1 log φ σ MLEs based on ths are dfferent, and based. y µ σ
35 Non-gnorable mssng data There are many models of mssng data mechansms - see Lttle and Rubn (1987). Heckman two-stage procedure Heckman (1976) developed for cross-sectonal data but also applcable to fxed effects panel data models. Thus, use y t = α + x t β + ε t. Further, assume that the samplng response mechansm s governed by the latent (unobserved) varable r t * r t * = w t γ+ η t. We observe r t 1 = 0 f r * t 0 otherwse
36 Assume {y t, r t } s multvarate normal to get E (y t r t * 0) = α + x t β + β λ λ(w t γ), ( a) where β λ = ρσ and λ( a) = φ. Φ( a) Heckman s two-step procedure Use the data {( r t, w t )} and a probt regresson model to estmate γ. Call ths estmator g H. Use the estmator g H to create a new explanatory varable, x t,k+1 = λ(w t g H ). Run a one-way fxed effects model usng the K explanatory varables x t as well as the addtonal explanatory varable x t,k+1. To test for selecton bas, test H 0 : β λ = 0. Estmates of β λ gve a correcton for the selecton bas.
37 Hausman and Wse procedure Use an error components model, y t = α + x t β + ε t. The samplng response mechansm s governed by the latent varable error components model r t * = x + w t γ + η t. The varances are: α σ 0 0 α σ αξ ε t 0 Var = ξ σ η t 0 f σ αξ = σ εη = 0, then the selecton process s ndependent of the observaton process. αξ σ σ ε 0 εη σ 0 ξ 0 σ σ εη 0 η
38 Hausman and Wse procedure Agan, assume jont normalty. Wth ths assumpton, one can check that: E( y t r σ T αξ σ εη σ = + + ξ ) x tβ g t g t g T + σ ξ σ η σ η Tσ ξ + σ η s= 1 s where g t = E (ξ + η t r ). Calculatng ths quantty s computatonally ntensve, requrng numercal ntegraton of multvarate normals. f σ αξ = σ εη = 0, then E (y t r ) = x t β.
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