A COMPARISON OF SEVERAL TESTS FOR EQUALITY OF COEFFICIENTS IN QUADRATIC REGRESSION MODELS UNDER HETEROSCEDASTICITY
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1 Colloquum Bomtrcum COMPISON OF SEVEL ESS FO EQULIY OF COEFFICIENS IN QUDIC EGESSION MODELS UNDE HEEOSCEDSICIY Małgorzata Szczpa Dorota Domagała Dpartmt of ppld Mathmatcs ad Computr Scc Uvrsty of Lf Sccs Głęboa Lubl Polad -mals: malgorzata.szczpa@up.lubl.pl dorota.domagala@up.lubl.pl Summary hs papr rvws ad compars svral tsts of qualty btw th vctors of coffcts th two polyomal (quadratc) rgrsso modls. h mprcal sgfcac lvls of th xamd tsts ar calculatd by mas of Mot Carlo smulatos. h tst ras ar prstd basd o th loss fucto. Kywords ad phrass: htroscdastcty rgrsso modls Mot Carlo smulatos Classfcato MS 00: 6F03 6J99 65C05. Itroducto Wh a rgrsso s usd to rprst a rlatoshp btw varabls th rsarchr could as whthr th sam rlatoshp holds for two groups of xprmtal uts. h aswr ca b obtad by tstg th qualty of rgrsso coffcts. W cosdr th two rgrsso modls
2 0 MŁGOZ SZCZEPNIK DOO DOMGŁ whr y s vctor y ε (.) s matrx wth ra s vctor of rgrsso coffcts. W assum that rror trms ε 0 ~ ad ε N I s dpdt of ε. h ull hypothss to b tstd s H 0 : agast th altratv H 0 :. Chow (960) proposd th followg statstc to tst th qualty of rgrsso coffcts F : (.) whr > ad y y s th rsdual sum of squars th modl cotag obsrvatos of both modls (.) ad y y ar th rsdual sums of squars calculatd sparatly for ach of th modls (.). h vctors ar th last squars stmators of rspctvly. Chow showd that f (wh homoscdastcty holds) ad udr H 0 statstc (.) s dstrbutd F( + ). Howvr th assumpto about homoscdastcty s ot oft fulflld ad thr s htroscdastcty btw th modls whch mas that. umbr of tsts wr dvsd to compar two htroscdastc rgrsso modls for xampl th paprs by Corly ad Masfld (988) Hoda ad Ohta (986) hursby (99) Wrahad (987). h comparsos of chos tsts wth th assumpto of lar rgrsso wr coductd by hursby (99) surum ad Shfl (985) ad Szczpa ad Wsołowsa-Jaczar (006).. sts udr htroscdastcty Kadyala ad Gupta (978) proposd a tst basd o W statstcs whr
3 COMPISON OF SEVEL ESS FO EQULIY OF COEFFICIENS S S W (.) ad S ad th crtcal valu s F(α; + ). hs tst wll b rfrrd to as th Chow tst (follow by hursby 99). Hoda ad Ohta (986) modfd tst statstc (.) as d d W whr. d S h crtcal valu s χ (α; ). Wrahad s (987) tst s basd o p-valu 0 d f B p v whr B s th bta fucto. h radom varabl has th bta dstrbuto wth paramtrs ad r F f r whr r ad r F s th cumulatv dstrbuto fucto of th F dstrbuto wth r dgrs of frdom. 3. Mot Carlo dsg h Mot Carlo xprmts ar basd o th rgrsso quatos: x c x b a y x c b x a y (3.)
4 MŁGOZ SZCZEPNIK DOO DOMGŁ whr w assum that ~ N0 ad ~ N0. o xam th mprcal sgfcac lvl w ta all th smulatos: a b c. For th valus ar usd to chc tsts udr th xtrm ad modrat htroscdastcty ad also udr homoscdastcty. h combatos of pars wr ta to cosdrato amly (00) (00) (3030) (5050) (00) (050) (030) (050) ad (3050). h valus of x ad x ar calculatd as th qudstat ad crasg umbrs from [09] trval. Wh w obta x x. I vry xprmt for appotd ad ( ) th valus of y accordg to (3.) wr gratd. h Mrs wstr psudoradom umbr grator (Matsumoto ad Nshmura 998) was usd th smulatos. h smulatos ad calculatos wr programmd Pascal laguag. lso th softwar lbrary LPCK (Lar lgbra PCKag 03) was vry hlpful to carry out th calculatos. For ach of 7 xprmts th * mprcal sgfcac lvl was calculatd as th quott of th umbr of rctd ull (tru) hypothss to th umbr of smulatos (000000). h omal sgfcac lvl s part of th rsults s prstd abls -4. abl. Chow tst. Emprcal sgfcac lvl x 00 σ σ ( ) (00) (00) (5050) (00) (050) (030) (3050)
5 COMPISON OF SEVEL ESS FO EQULIY OF COEFFICIENS 3 abl. Chow tst. Emprcal sgfcac lvl x 00 σ σ ( ) (00) (00) (5050) (00) (050) (030) (3050) h followg coclusos ca b draw rlato to Chow tst: xprmts wth = : mprcal sgfcac lvls ar always qual or largr tha h rsults ar clos to th omal (0.05) sgfcac lvl wh modrat htroscdastcty (σ = ) ad homoscdastcty occur. h mprcal lvls for xtrm htroscdastcty ar mardly dffrt from 0.05 * xprmts wth < : th valus ar dffrt from th omal sgfcac lvl wh σ. Morovr th mprcal lvls of sgfcac ar always lss tha th omal wh σ < σ. O th othr had α s largr tha 0.05 wh σ > σ. h coclusos about Chow tst ar as follows: th rsults ar always largr tha or qual to 0.05 ad ar also lss dsprsd compard wth Chow tst wh < xprmts wth = : th sam α valus ar obtad as Chow tst bcaus x = x xprmts wth < : th tst prforms wll stuatos whr σ = h largr htroscdastcty th largr α (xcpt for th cas = 0 =50).
6 4 MŁGOZ SZCZEPNIK DOO DOMGŁ abl 3. Hoda -Ohta tst. Emprcal sgfcac lvl x 00 σ σ ( ) (00) (00) (5050) (00) (050) (030) (3050) abl 4. Wrahad tst. Emprcal sgfcac lvl x 00 σ σ ( ) (00) (00) (5050) (00) (050) (030) (3050) h coclusos about Hoda-Ohta tst: th rsults ar always largr tha or qual to 0.05 xprmts wth = : α valus ar slghtly largr compard to Chow ad Chow tsts
7 COMPISON OF SEVEL ESS FO EQULIY OF COEFFICIENS 5 xprmts wth < : th mprcal sgfcac lvls ar slghtly largr compard to Chow tst xcpt for th stuatos wh σ = 5 0 ad th cas = 0 =50. h coclusos about Wrahad tst: th rsults ar always lss tha 0.05 xprmts wth = : α valus ar clos to th omal wh th assumd htroscdastcty s mor xtrmal: σ = xprmts wth < : th bst rsults ar obtad for σ = h comparso of th tsts o sum up ad compar rsults of th xamd tst th loss fucto proposd hursby (99) s usd: * 00 5 LS whr dots umbr of xprmts. h valus of LS ar prstd abls 5 ad 6. abl 5. h loss valus ad ras of tsts all combatos of sampl szs (all xprmts) = 7 = = 36 st LS ra LS ra Chow Chow Hoda-Ohta Wrahad
8 6 MŁGOZ SZCZEPNIK DOO DOMGŁ abl 6. h loss valus ad ras of tsts wth rspct to th dgr of htroscdastcty modrat htroscdastcty ad homoscdastcty σ = = 36 < σ = 5 0 = 0 < σ = = 0 st LS ra LS ra LS ra Chow Chow Hoda-Ohta Wrahad Coclusos W hav cosdrd th mprcal sgfcac lvls of four tsts. s show abl 5 th rsults of all 7 xprmts mply that Hoda-Ohta tst s th bst. Howvr wh = Chow ad Chow tsts sm to b good. I th xprmts wth modrat htroscdastcty ad homoscdastcty (abl 6) aga Hoda-Ohta tst sms to b th bst. Wrahad tst s th bst th cas of htroscdastcty of 5 ad 0 wh <. For th xprmts whr < ad th varac of th frst modl s 0. or 0. th Chow tst ad th Hoda-Ohta tst appar to b good chocs. h obtad ras of tsts ar smlar to rsults Szczpa ad Wsołowsa-Jaczar (006). h dffrcs cocr xtrm htroscdastcty. Howvr t should b otd that th assumptos about th modls wr ot th sam ad th x ad x (3.) wr chos dffrt way at th prst xprmts. Examg th mprcal powrs of ths tsts wll b th xt stp of our studs.
9 COMPISON OF SEVEL ESS FO EQULIY OF COEFFICIENS 7 frcs Chow G. C. (960). sts of Equalty btw Sts of Coffcts wo Lar grssos. Ecoomtrca Corly M. D. Masfld E.. (988). pproxmat st for Comparg Htroscdastc grsso Modls. Joural of th mrca Statstcal ssocato 83 No Hoda Y. Ohta K. (986). Modfd Wald tsts tsts of qualty btw sts of coffcts two lar rgrssos udr htroscdastcty. h Machstr School of Ecoomc ad Socal Studs Kadyala K.. Gupta S. (978). sts for poolg cross-sctoal data th prsc of htrosdastcty Bullt of th Isttut of Mathmatcal Statstcs Matsumoto M. Nshmura. (998). Mrs wstr: 63-dmsoally qudstrbutd uform psudoradom umbr grator. CM ras. o Modlg ad Computr Smulato 8 No Szczpa M. Wsołowsa-Jaczar M. (006). Porówa tstów do sprawdzaa dtyczośc dwóch modl rgrs w przypadu htroscdastyczośc. Colloquum Bomtrycz hursby J. G. (99). comparso of svral xact ad approxmat tsts for structural shft udr htroscdastcty. Joural of Ecoomtrcs surum H. Shfl N. (985). Som tsts for th costacy of rgrsso udr htroscdastcty. Joural of Ecoomtrcs Wrahad S. (987). stg rgrsso qualty wth uqual varacs. Ecoomtrca 55 No LPCK Lar lgbra PCKag. (03).
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