Limited Dependent Variables and Panel Data. Tibor Hanappi
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1 Lmted Dependent Varables and Panel Data Tbor Hanapp
2 Lmted Dependent Varables Dscrete: Varables that can take onl a countable number of values Censored/Truncated: Data ponts n some specfc range cannot be observed Focus on dscrete dependent varables! 6/30/2010 2
3 Outlook Revew: Cross-sectonal models Panel data: Fxed effects Panel data: Random effects Dscusson: FE and RE Implementaton n STATA 6/30/2010 3
4 Dscrete Dependent Varables Estmatng probabltes of observng a partcular outcome (e.g. bnar outcomes Applcatons: Labour market outcomes: Emploment, self selecton, labour suppl Consumer demand: Purchase decsons, nvestments Programme partcpaton: Health, educaton, nsurance schemes Other: Retrement decsons, transportaton mode choce 6/30/2010 4
5 Dscrete Dependent Varables Bnar outcomes: t t t 0,1 1f * t 0 otherwse x u t t 0 Outcome for ndvdual at tme pont t Cross-sectonal analss: drop subscrpt t 6/30/2010 5
6 Cross-Sectonal Analss Interested n: Pr( 1x Pr( 1x Pr( * 1 Pr( u Pr( u 0 x Pr( x u 0 x x Probablt that the outcome s postve for ndvdual 6/30/2010 6
7 Cross-Sectonal Analss In the bnar case t follows that: Possble approach: Lnear probablt model (LPM: Usual panel data methods could be appled But: Estmated probabltes are not restrcted to the unt nterval 6/30/ Pr( 1 Pr( 1 0 Pr( 0 ( x x x x E e x e x x E 1 Pr( (
8 Cross-Sectonal Analss Non-lnear models: Pr( 1x Pr( u x 1 Pr( u x Pr( u x F( x The last lne holds onl as long as the dstrbuton of u s smmetrc around zero Functon F(. restrcts the outcomes to be wthn the unt nterval 6/30/2010 8
9 Cross-Sectonal Analss Intutve llustraton: Pr( u x I u - x u du x f ( u f ( u du Integral over an ndcator functon showng whether the outcome s postve gven the values of the error term Other wa to see ths: Integrate over all those values of u for whch the outcome s postve Gves the probablt that the error term s such that a postve outcome occurs (gven what s known about x 6/30/2010 9
10 Cross-Sectonal Analss Specfng a dstrbuton for u: Bnar Logt: F( c ( c exp( c 1 exp( c Bnar Probt: F( c ( c c 1 exp z 2 dz Logt leads to closed form outcome probabltes Probt s computatonall more ntense; but offers a more general treatment 6/30/
11 Cross-Sectonal Analss Maxmum Lkelhood: Outcome probabltes are ndependent of each other Lkelhood functon takes the followng form L( N 1 Pr( Outcome probabltes are F( and F( respectvel Where F(. s ether (. or 1X Pr( 0 X Maxmzaton of log L(β wth respect to β gves MLE Probt: Outcome probabltes have to be approxmated numercall (e.g. through smulaton x (. x 6/30/
12 Panel Data: Fxed Effects For panel data ut Outcome probabltes n the fxed effects model Pr( t 1x t Logt: Vew μ and β as unknown parameters to be estmated b maxmzng log L(β,μ However: As, for a fxed T, μ ncrease wth N N μ cannot be consstentl estmated for a fxed T t Pr( * t 0 Pr( t t x t F( x Incdental parameter problem! 6/30/
13 Panel Data: Fxed Effects Cannot get rd of μ through wthn-transformaton (as n lnear models Possble soluton: Fnd a mnmum suffcent statstc for μ T For the logt model s a mnmum suffcent statstc for μ t 1 t B defnton, condtonng L(β on the mnmum suffcent statstc gves a condtonal lkelhood functon that does not depend on μ (Chamberlan, 1980 L C ( N 1 Pr T 1,..., T t 1 t, X 6/30/
14 Panel Data: Fxed Effects Logt model wth FE: condtonng on the mnmum suffcent statstc for μ elds outcome probabltes that are ndependent of μ Maxmzng the condtonal lkelhood gves consstent β estmates (whle retanng closed-form outcome probabltes However: Onl observatons for ndvduals who swtched status can be used n estmaton Dependent varable takes the value 1 f t swtches from 0 to 1 and 0 f t swtches from 1 to 0 In ths case the dfferences x t x t 1 are used as ndependent varables 6/30/
15 Panel Data: Random Effects 2 In the Probt model wth RE ut t wth ~ IIN(0, 2 and t ~ IIN(0, ndependent of each other and the xt 2 Now: E( u t u s for t s and the jont lkelhood of ( 1,..., T nvolves a T-dmensonal ntegral L 1,..., T X... f ( u 1, u2,..., ut du 1du... dut N (, Pr 2 1 Maxmzaton wth respect to β and gets to be nfeasble f T s large Numercal approxmaton nvolves smulaton as well as nonsmulaton procedures 6/30/
16 Panel Data: Prelmnar Summar Possble combnatons of models wth effects specfcatons: Logt Fxed Effects Yes No Probt Random Effects Yes Yes Fxed effects specfcaton not possble n the Probt framework (condtonal lkelhood approach does not lead to smplfcatons Logt model wth random effects s feasble, but (mabe not ver useful Potental advantage of Logt: Closed-form outcome probabltes (lost n case of RE! Dsadvantage of Logt: Onl outcome swtches can be used for estmaton 6/30/
17 Panel Data: Dscusson Possble nterpretatons of fxed ndvdual effects n models wth dscrete dependent varables: Indvdual-specfc unobserved effects on outcome that are not pcked up b x t Labour market context: E.g. ablt Other contexts: E.g. tme-nvarant ndvdual preferences Logt: To test for FE use a varant of the Hausman-test based on the dfference between the condtonal MLE and the usual MLE wthout FE 6/30/
18 Panel Data: Dscusson Possble nterpretatons of random ndvdual effects: Idea: N ndvduals are randoml drawn from a large populaton; N s large; FE would mpl large losses n degrees of freedom E.g. household panel studes wth representatve samples Indvdual effects vewed as random and estmates vald for the populaton from whch the sample was drawn However: Populaton does not consst of an nfnt of ndvduals Alternatve vew: Populaton as an nfnt of decsons (Haavelmo, 1944 behavoural nterpretaton Probt: To test for RE use LR-prncple to evaluate the lkelhood for the pooled regresson and for the RE estmator 6/30/
19 Implementaton n STATA xtlogt Uses RE as default, FE optonal Automatcall omts groups of observatons wthout wthn-group varaton However: Condtonal lkelhood approach has to be mplemented through data transformaton (swtches, dfferences xtprobt Onl RE possble Uses mean and varance adaptve Gauss-Hermte quadrature as ntegraton method Other varants of Gauss-Hermte quadrature can be defned Apparentl: Other smulaton methods not avalable (Accept-Reject, smoothed Accept-Reject, GHK smulator Estmaton of RE Probt models sometmes not possble because maxmum of the lkelhood functon dffcult to fnd 6/30/
20 Defnton Suffcent statstc: If T(X s a suffcent statstc for S, then an nference about S should depend on the sample X onl through the value of T(X. That s f x and are two sample ponts such that T(x=T( then nference about S should be the same whether X=x or X= s observed. (Casella and Berger, Statstcal Inference, 2002, p.272 6/30/
21 Lterature Baltag, Bad, 2001, Econometrc Analss of Panel Data, 2 nd Edton, John Wle & Sons. Casella, George and Roger L. Berger, 2002, Statstcal Inference, Duxbur. Chamberlan, G., 1980, Analss of Covarance wth Qualtatve Data, Revew of Economc Studes, Vol.47, pp Greene, Wllam, 2008, Dscrete Choce Modellng, n The Handbook of Econometrcs: Vol. 2, Appled Econometrcs, Part 4.2., ed. T. Mlls and K. Patterson, Palgrave, London. Haavelmo, Trgve, 1944, The Probablt Approach n Econometrcs, Econometrca, Vol.12 Supplement, pp Tran, Kenneth, 2009, Dscrete Choce Methods wth Smulaton, 2 nd Edton, Cambrdge Unverst Press. 6/30/
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