Institut für Halle Institute for Economic Research Wirtschaftsforschung Halle

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1 Insttut fü Halle Insttute fo Economc Reseach Wtschaftsfoschung Halle A Smple Repesentaton of the Bea-Jaque-Lee Test fo Pobt Models Joachm Wlde Dezembe 2007 No. 13 IWH-Dskussonspapee IWH-Dscusson Papes

2 A Smple Repesentaton of the Bea-Jaque-Lee Test fo Pobt Models Joachm Wlde Dezembe 2007 No. 13

3 IWH Autho: D Joachm Wlde Chef Econometcan Emal: Joachm.Wlde@wh-halle.de Tel.: The esponsblty fo dscusson papes les solely wth the ndvdual authos. The vews expessed heen do not necessaly epesent those of the IWH. The papes epesent pelmnay wok and ae cculated to encouage dscusson wth the autho. Ctaton of the dscusson papes should account fo the povsonal chaacte; a evsed veson may be avalable dectly fom the autho. Comments and suggestons on the methods and esults pesented ae welcome. IWH-Dscusson-Papes ae ndexed n RePEc-Econpapes. Heausgebe: INSTITUT FÜR WIRTSCHAFTSFORSCHUNG HALLE IWH Pof. D. Ulch Blum Päsdent, D. Hubet Gabsch Foschungsdekto Das IWH st Mtgled de Lebnz-Gemenschaft Hausanschft: Klene Mäkestaße 8, Halle Saale Postanschft: Postfach , Halle Saale Telefon: Telefax: Intenetadesse: 2 IWH-Dskussonspapee 13/2007

4 IWH A Smple Repesentaton of the Bea-Jaque-Lee Test fo Pobt Models Abstact The nfeence n pobt models eles on the assumpton of nomalty. Howeve, tests of ths assumpton ae not mplemented n standad econometc softwae. Theefoe, the pape pesents a smple epesentaton of the Bea-Jaque-Lee test, that does not eque any matx algeba. Futhemoe, the epesentaton s used to compae the Bea-Jaque- Lee test wth the RESET-type test poposed by Papke and Woolddge Keywods: pobt model, Lagange multple test, nomalty assumpton, atfcal egesson JEL Classfcaton: C25 Zusammenfassung De statstsche Infeenz n Pobtmodellen beuht maßgeblch auf de Gültgket de Nomalvetelungsannahme fü de Stögößen. Dennoch snd Tests auf Nomalvetelung n velen ökonometschen Paketen ncht mplementet und weden dahe oft auch ncht duchgefüht. Das Pape päsentet deshalb ene enfache Dastellung des Bea- Jaque-Lee Tests, de de Nomalvetelung gegen de veglechswese allgemene Altenatve de Peason-Vetelungen testet. De Dastellung beuht auf ene Hlfsegesson, we se von Davdson und MacKnnon 1984 fü enen Test gegen Heteoskedaste vogeschlagen wude. Se kann mt jede Statstk- ode Ökonometesoftwae, de de Schätzung enes Pobtmodells emöglcht, poblemlos beechnet weden. Daübe hnaus zegt de Dastellung, dass de Test auf Nomalvetelung asymptotsch dentsch st mt enem von Papke und Woolddge 1996 vogeschlagenen RESET Test, de de Lneatät des Ewatungswetes de latenten Vaablen übepüft. Folglch kann ohne zusätzlche Infomatonen ncht zwschen de funktonalen Fom des Ewatungswetes und dem Vetelungstyp de Stögößen untescheden weden. Sofen alledngs de Lneatät des Ewatungswetes untestellt weden kann, st de künstlche Regesson des Bea-Jaque-Lee Tests nfomatve, da se Hnwese daauf gbt, ob de Ablehnung de Nullhypothese duch Schefe ode ene von de Nomalvetelung abwechende Kutoss veusacht wude. Schlagwöte: Pobtmodell, Lagange Multple Test, Nomalvetelungsannahme JEL-Codes: C25 IWH-Dskussonspapee 13/2007 3

5 IWH A Smple Repesentaton of the Bea-Jaque-Lee Test fo Pobt Models 1 Intoducton Pobt models ae often appled wthout testng the nomalty assumpton. Ths s athe poblematc because the standad maxmum lkelhood estmato of the pobt model s mostly based fo nonnomal dstubances cf. Geene 2003, p One eason mght be that sutable tests ae not mplemented n standad econometc softwae. Theefoe, the pape pesents a smple epesentaton of the test of Bea, Jaque, and Lee It only eques the estmaton of a pobt model and some standad tansfomatons. Bea, Jaque, and Lee test the assumpton of nomalty aganst a athe geneal altenatve, wheeas othe authos assume moe specfc dstbutons e.g. Slva The epesentaton s based on an atfcal egesson whch was suggested by Davdson and MacKnnon 1984 fo a smla test on homoscedastc eos. The dependent vaable s the so-called standadzed esdual of the pobt estmaton cf. Woolddge 2002, p An altenatve atfcal egesson s based on a egessand of ones. Howeve, the small sample popetes ae athe poo cf. Davdson and MacKnnon 1993, p Hence, the pape does not consde ths altenatve. The atfcal egesson of the Bea-Jaque-Lee test can be tansfomed nto the atfcal egesson of the RESET-type test poposed by Papke and Woolddge Although two egessos ae numecally changed by the tansfomaton, the explaned sums of squaes of both egessons and thus the test statstcs ae dentcal. Theefoe, wthout addtonal nfomaton the dstbutonal fom of the dstubances s not dstngushable fom the functonal fom of the mean of the latent vaable. Secton 2 summazes the deas of Bea, Jaque and Lee n a slghtly dffeent notaton whch makes t easy to pesent the esults n Secton 3. In Secton 3 the smple epesentaton of the test wll be deved and ntepeted. Secton 4 concludes wth a compason of the two tests. 4 I thank an anonyomous efeee fo athe useful comments. The usual dsclame apples. IWH-Dskussonspapee 13/2007

6 IWH The Bea-Jaque-Lee BJL test BJL consde the usual pobt model: y = β x + u, u d N 0, σ, = 1,..., N * 2 * 1, y > 0 y = 0, othewse, 1 x beng a K 1-vecto of exogenous vaables, β a vecto of unknown paametes, y * a latent tendency, y an obseved ndcato, and u a stochastc eo tem, whch s stochastcally ndependent of the exogenous vaables. They deve a Lagangemultple test of the null hypothess of nomally dstbuted eo tems. Unde the altenatve hypothess BJL assume the Peason famly of dstbutons. Implyng a mean of zeo, the densty functon fu of ths famly s chaactezed by the followng dffeental equaton BJL 1984, p. 565: 1 ln f u c u = u c c u + c u The nomal dstbuton s a specal case of 2 wth c 0 =σ 2 and c 1 =c 2 =0. Hence, the followng hypotheses ae tested: H 0 : c 1 =c 2 =0 aganst H 1 : c 1 0 and/ o c 2 0. "c 1 0" mples a skew dstbuton, "c 2 0" a kutoss dffeent fom 3 gven the dstbuton s symmetc cf. Johnson, Kotz and Balakshnan 1994, p. 22. The vaance σ 2 s nomalzed to 1 as usual BJL 1984, p Theefoe, the vecto θ of the emanng unknown paametes s equal to β, c 1, c 2. The Peason famly of dstbutons contans many common dstbutons as specal cases, fo nstance the t-dstbuton o the gamma dstbuton Johnson, Kotz and Balakshnan 1994, pp. 18 and 21. Nevetheless, the altenatve hypothess s not as geneal as that of the well establshed Jaque-Bea test n the lnea egesson model BJL 1984, p Howeve, t s the most geneal altenatve hypothess n the lteatue. Usng the nfomaton matx equalty the Lagange multple test statstc can be denoted as IWH-Dskussonspapee 13/2007 5

7 IWH ln L y, θ 1 ln L y, θ LM = I H θ 0, θ θ ln L y, θ ln L y, θ IH θ 0 = E y, θ θ 3 lnl beng the log-lkelhood functon unde H 1, =, y y 1... y N θ 0 0 = β, and β the pobt maxmum lkelhood estmato of β. 2 2 Defne X = x 1/ 3 x 1 1/ 4 x 3 β β + β x x, x θ = β 0 0, ϕ = ϕ β, Φ = Φ β, X, ϕ, Φ the analogous functons afte substtutng β by β, ϕ and Φ beng the densty functon and the dstbuton functon of a standad nomal dstbuton. Gven the ndependence of the sample unts the test statstc 3 s obtaned as cf. bd., pp. 570, 571 y 2 ϕ y 1 N N N Φ ϕ Φ ϕ LM = X X X X = 1 Φ 1 Φ = 1 Φ 1 Φ = 1 Φ 1 Φ. 4 It s asymptotcally χ 2 2 dstbuted. 6 IWH-Dskussonspapee 13/2007

8 IWH 3 A smple epesentaton of the test The statstc 4 s equal to the explaned sum of squaes of an atfcal egesson. Defne as vecto of the standadzed esduals of a pobt estmaton,.e. y Φ = 1,..., N, =, =1,..., N, Φ 1 Φ and 5 R M 6 1 R Φ Φ 1 ϕ R =, R = X ', = 1,..., N. N Usng 5 and 6 the statstc 4 can be wtten as follows: 1 LM = R R R R. 7 Defnng the atfcal egesson = Rδ + ε, 8 and lettng δ be the odnay least squaes estmato of δ 7 s equvalent to LM = Rδ Rδ, 9.e. the LM statstc s the explaned sum of squaes of 8. Based on 7-9 LM statstc and atfcal egesson can be ntepeted as follows: The test ejects the hypothess of nomally dstbuted eos f the vaables on the ght hand sde of 8 "explan" the standadzed esduals of the pobt model. The poducts of those vaables wth the dependent vaable ae aleady equal to the gadent of the loglkelhood functon unde H 1 cf. 3 and 4. The fst K components of ths vecto ae equal agan to the gadent of the log-lkelhood functon of a standad pobt model cf. Geene 2003, p Hence, the fst K egessos of the atfcal egesson ae othogonal to the dependent vaable. Futhemoe, n case the null hypothess s vald θ s close to the unestcted maxmum lkelhood estmate. Theefoe, the devates wth espect to c 1 and c 2 ae close to zeo,.e. the last two egessos of the atfcal egesson ae almost othogonal to the dependent vaable. Thus, no egesso explans the dependent vaable, and the LM statstc becomes small. Howeve, n case of skewness o a kutoss unequal to 3 θ mght be athe dffeent fom the unestcted maxmum lkelhood estmate. Then, the IWH-Dskussonspapee 13/2007 7

9 IWH last two egessos ae not othogonal to the vecto of the standadzed esduals, and the LM statstc becomes lage. Concenng a sample unt egesson and defnng f = Φ ϕ 1 Φ 8 s y Φ = δ 1f x1 + K + δkfxk Φ 1 Φ δk+ 1f β x 1 + δk+ 2f β x 3 + β x + ε Usng 10 all vaables of the atfcal egesson can be calculated easly wthout matx algeba. Meely standad foecasts of a pobt estmaton ae needed. Aftewads, the LM statstc s computed va 9 and compaed wth the ctcal value of a χ 2 2 dstbuton. The LIMDEP code of the complete pocedue can be found n the appendx. 8 IWH-Dskussonspapee 13/2007

10 IWH 4 Compason wth a RESET-type test * Papke and Woolddge 1996, p. 624 test the hypothess that * functon of x aganst the altenatve that E y x s a lnea E y x s a nonlnea functon of x. Usng the notaton above, the atfcal egesson s 1 Φ 2 3 y Φ = δ % 1f x1 + + δ % Kf xk + δ% K+ 1f β x + δ% K+ 2f β x + ε% Φ K. 11 The test statstc s N R u 2, R u 2 beng the constant-unadjusted coeffcent of detemnaton of 11. Snce 1/N' conveges to 1 Engle 1984, p. 818, N R u 2 s asymptotcally equvalent to the explaned sum of squaes of 11. It s easly seen that 10 can be tansfomed nto 11 gven that x ncludes a constant. Theefoe, the JBL test aganst a Peason dstbuton of the esduals and the RESETtype test of Papke and Woolddge aganst a nonlnea functon of the condtonal mean appoxmated by a quadatc and cubc tem ae asymptotcally dentcal. Addtonal nfomaton s needed to clafy the eason fo a ejecton of the null hypothess. Howeve, gven that the mean s a lnea functon of x, egesson 10 contans moe nfomaton than egesson 11. Fo t ndcates whethe the null s ejected because of skewness o because of a non-standad kutoss. IWH-Dskussonspapee 13/2007 9

11 IWH Appendx: LIMDEP code?= Replace va2,, vak by you explanatoy vaables, and nset the dependent dummy vaable namelst ;x_pob=one, va2,, vak $ ceate ;y= $ pobt ;lhs=y; hs=x_pob $ matx ;bx=x_pob*b $ ceate ;bex=bx $ ceate ;bexph=phbex ;ya=y-bexph ;ya=bexph*1-bexph ;yat=1/ya^0.5 ;y_at=ya*yat ;xa=n01bex ;xa=xa*yat ;x_at1=xa $?= Complement the lnes, and eplace va2,, vak by you explanatoy vaables ceate ;x_at2=xa*va2 ; ;x_atk=xa*vak $ ceate ;x_ats=-1/3*xa*bex^2-1 ;x_atk=0.25*xa*bex*3+bex^2$?= Complete the lst of egessos fo the atfcal egesson namelst ;x_ateg=x_at1,x_at2,...,x_atk,x_ats,x_atk$ egess ;lhs=y_at; hs=x_ateg $ matx calculate ;y_hat=x_ateg*b$ ;lst; LM=y_hat'y_hat; p_value=1-chlm,2$ 10 IWH-Dskussonspapee 13/2007

12 IWH Refeences Bea, A. K.; Jaque, C. M.; Lee, L. F. 1984: Testng the nomalty assumpton n lmted dependent vaable models. Intenatonal Economc Revew 25, pp Davdson, R.; MacKnnon, J. G. 1984: Convenent specfcaton tests fo logt and pobt models. Jounal of Econometcs 25, pp Davdson, R; MacKnnon, J. G. 1993: Estmaton and nfeence n econometcs. Oxfod Unvesty Pess: New Yok. Engle, R. F. 1984: Wald, Lkelhood Rato, and Lagange Multple tests n econometcs, n: Z. Glches, M. Intlgato eds, Handbook of econometcs, Vol. 2. Noth Holland, Amstedam, pp Geene, W. 2003: Econometc analyss 5 th ed. Pentce Hall: Uppe Saddle Rve. Johnson, N. L.; Kotz, S.; Balakshnan, N. 1994: Contnuous unvaate dstbutons Vol. 1 2 nd ed. Wley: New Yok. Papke, L. E.; Woolddge, J. M. 1996: Econometc methods fo factonal esponse vaables wth an applcaton to 401 k plan patcpaton ates. Jounal of Appled Econometcs 11, pp Slva, J. M. C. 2001: A scoe test fo non-nested hypotheses wth applcatons to dscete data models. Jounal of Appled Econometcs 16, pp Woolddge, J. M. 2002: Econometc analyss of coss secton and panel data. MIT Pess: Cambdge MA. IWH-Dskussonspapee 13/