3/3/2014. CDS M Phil Econometrics. Heteroskedasticity is a problem where the error terms do not have a constant variance.

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1 3/3/4 a Plla N OS Volao of Assmpos Assmpo of Sphercal Dsrbaces Var T T I Var O Cov, j, j,..., Therefore he reqreme for sphercal dsrbaces s ad j I O homoskedascy No aocorrelao Heeroskedascy: Defo Heeroscedascy Heeroskedascy s a problem where he error erms do o have a cosa varace. CDS Phl coomercs 3 Tha s, hey may have a larger varace whe vales of some X or he Y s hemselves are large or small. 4 xample of Heeroskedascy fy x. x x x 3.. y x β + β x x 5 Heeroskedascy: Defo Ths ofe gves he plos of he resdals by he depede varable or approprae depede varables a characersc fa or fel shape Seres 6

2 3/3/4 CDS Phl coomercs Resdal Aalyss for Resdal Aalyss for qal Varace qal Varace No-cosa varace Cosa varace x x Y x x Y resdals resdals 7 8 Heeroskedascy Heeroskedascy: Defo : Defo 9 Heeroskedascy Heeroskedascy Wh osphercal errors e.g. heeroskedascy ad/or aocorrelao o loger apples. I Var Ω T O O ω ω ω O Ω Σ Heeroskedascy Heeroskedascy Aocorrelao Aocorrelao T O O ρ ρ ρ ρ ρ ρ Σ Ω Heeroskedascy Heeroskedascy Gve or model, y Xβ + where X s a o-sochasc marx wh fll colm rak ad Σ Ω The OS esmaor of β s X XX ˆ + β β β β ˆ So OS s sll based

3 3/3/4 The varace marx s Var ˆ β {ˆβ βˆβ β } Therefore ay ferece based o s XX wll be correc. CDS Phl coomercs Heeroskedascy [ XXXX ] XX XX XXXX XX X XX ΩXXX XΩXXX s may be a based esmaor of 3 Heeroskedascy: Cases I may be cased by: odel msspecfcao - omed varable or mproper fcoal form. earg behavors across me Chages daa colleco or defos. Olers or breakdow model. Freqely observed cross secoal daa ses where demographcs are volved poplao, GNP, ec. 4 Heeroskedascy: Implcaos Heeroskedascy: Implcaos co. The regresso βs are based /cosse. B hey are o loger he bes esmaor. They are o BU o mmm varace - hece o effce. So cofdece ervals are vald. Wrog ferece 5 Types of Heeroskedascy There are a mber of ypes of heeroskedascy. Addve lplcave ARCH Aoregressve codoal heeroskedasc - a me seres problem. 6 Tesg for Heeroskedascy A mber of formal ess : Ramsey RST es Park es Glejser es Goldfeld-Qad es Bresch-Paga es Whe es 7 sseally wa o es H : Var x, x,, x k s, eqvale o H : x, x,, x k s If assme he relaoshp bewee ad x j s lear, ca es H as a lear resrco So, for d + d x + + d k x k + v hs meas esg»h : d d d k Tesg for Heeroskedascy 8 3

4 3/3/4 The Bresch-Paga Tes smae he resdals from he OS regresso Ge z û / ha s he resdals sqared dvded by Regress z o all of he xs. û / The Bresch-Paga Tes ca have 3 ess:. Θ ½ RSS, where RSS regresso sm of sqares from regressg z o all of he xs ; Θ χ k df. ca have 3 ess: 9 The Bresch-Paga Tes. The F sasc s js he repored F sasc for overall sgfcace of he regresso, The Bresch-Paga Tes 3. The Bresch-Paga-Godfrey sasc s R, F [R /k] / [ R / k ], whch s dsrbed F k, k whch s dsrbed as χ k- The Bresch-Paga Tes : A xample The Bresch-Paga Tes : A xample Cosmpo $ Icome $ I Saa Sascs: ear models ad relaed Regresso dagoscs Specfcao ess, ec. 3 χ 3.84, α 5% χ 6.63, α % F, 8 4., α 5% F, , α % 4 4

5 3/3/4 The Whe Tes: Whe s Geeralzed Heeroskedascy es The Bresch-Paga es wll deec ay lear forms of heeroskedascy The Whe es allows for oleares by sg sqares ad crossprodcs of all he xs sg a F or o es wheher all he x j, x j, ad x j x k are joly sgfca ca ge o be weldy 5 The Whe Tes: Whe s Geeralzed Heeroskedascy es The es proceeds as follows: Sep : smae he orgal eqao by leas sqares ad oba he resdals Sep : Regress he sqared resdals o a cosa, all he regressors, he regressors sqared ad her cross-prodcs eracos. For example, wh wo explaaory varables x x x x x x where x represes he cosa erm [ ] 3 3 x3 6 The Whe Tes: Whe s Geeralzed Heeroskedascy es The Whe Tes: Whe s Geeralzed Heeroskedascy es: A xample Sep 3: The es sasc s ~ R χ k H : Cosa varace If R > χ he we have a sse wh heeroskedascy. 7 8 The Whe Tes: Whe s Geeralzed Heeroskedascy es: A xample The Whe Tes: A xample Geerae varables Saa NR 5 x χ dsrbo wh 5 df.75, α 5% 9 Coclso? R < χ 3 homoskedascy 5

6 3/3/4 Alerave form of he Whe es Cosder ha he fed vales from OS, ŷ, are a fco of all he xs Ths, ŷ wll be a fco of he sqares ad crossprodcs ad ŷ ad ŷ ca proxy for all of he x j, x j, ad x j x k ; so Regress he resdals sqared o ŷ ad ŷ ad se he R o form a F or sasc Remedes for Heeroskedascy Ths depeds o he form heeroskedascy akes. Idrec: Re-specfy he model; Use heeroscedasc-cosse Ss Drec: GS WS adjs he varace-covarace marx 3 3 Heeroskedasc Cosse Ss OS esmae: based ad cosse. B Varˆ β Ths ca be re-wre as Varˆ β XX where.e., we eed o esmae all he ' s - whch s mpossble. XX XΩXXX XDag XX X Dag Dag,,..., Ω O 33 Heeroskedasc Cosse Ss Whe 98 arges ha all we really eed s a esmae of X ΩX Uder very geeral codos, ca be show ha X ΩX xx e xx Therefore he adjsed varace s es.asym.varˆ β XX exxxx 34 Heeroskedasc Cosse Ss A cosse esmae of he varace, he sqare roo ca be sed as a sadard error for ferece Typcally kow as robs sadard errors. Somemes he esmaed varace s correced for degrees of freedom by mlplyg by / k. As s all he same, hogh. 35 Heeroskedasc Cosse Ss: Robs Ss Impora o remember: Robs sadard errors oly have asympoc jsfcao wh small sample szes sascs formed wh robs sadard errors wll o have a dsrbo close o he, ad fereces wll o be correc I Saa, ear regresso: S/Robs: selec robs defal 36 6

7 3/3/4 Geeralzed eas Sqares I s always possble o esmae robs sadard errors for OS esmaes, B f we kow somehg of he specfc form of he heeroskedascy, we ca oba more effce esmaes ha OS The basc dea s o rasform he model o oe ha has homoskedasc errors called geeralzed leas sqares Geeralzed eas Sqares Geeralzed eas Sqares Gve marx. a posve defe Ay posve defe marx ca be expressed he form: PP, where P s osglar: Ω PP, so ha P Ω P I ad P P Ω Ω Now premlply he model y Xβ + by P o ge y * X * β + * Where y * P y ; X * P X ; * P 39 4 Geeralzed eas Sqares Geeralzed eas Sqares Gve P Ω P I ad y * X * β + * Where y * P y ; X * P X ; * P Now * * P P P ΩP P ΩP I : Homoscedasc OS assmpos sasfed Ω y * X * β+ * OS esmae of βs: b X * X * X * y * X Ω X X Ω y A BU of βwh Varb X * X * X Ω X b s he Geeralzed eas Sqares GS or Ake esmaor of β A based esmae of s: ˆ eω e Where e y Xb k 4 4 7

8 3/3/4 Geeralzed eas Sqares Geeralzed eas Sqares: Weghed eas sqares If s ormally dsrbed, so s * Ths b s a esmaor So has m var he class of all based esmaors. GS s a weghed leas sqares WS procedre where each sqared resdal s weghed by he verse of Var x Weghed eas Sqares Weghed eas Sqares e heeroskedascy be modeled as Var x s hx, where hx h o be specfed. Now / h x, becase h s oly a fco of x, ad Var / h x s, becase we kow Var x s h For example, A commo specfcao: var o oe of he regressors or s sqare: x k hx x k ; h xk The weghed rasformed S regresso model: y x x βk + β + β xk xk xk xk So dvde he whole eqao by h ad we have a model wh homoskedasc error 45 / x k, ad Var / x k s Homoscedasc WS mmzes he weghed sm of sqares weghed by /h 46 Feasble GS Feasble GS ore ypcal s he case where we do kow he form of he heeroskedascy I hs case, eed o esmae hx Ths s he case of FGS R he orgal OS model, save he resdals, û, sqare hem Regress û o all of he depede varables ad ge he fed vales, ê Do WS sg /ê as he wegh

9 3/3/4 FGS: Saa FGS: Saa 49 5 FGS: Saa FGS: Saa Also Dowload wls, sg 5 5 Resdals FGS: Saa Prce Rs. Resdals Adversg Rs s Oher wegh ypes abse ad loge ad sqared fed vales xb. 54 9

10 3/3/4 FGS: Saa FGS: Saa Oher wegh ypes abse ad loge ad sqared fed vales xb. Compare wh he FGS doe by seps Aocorrelao Aocorrelao: Defo Aocorrelao s correlao of he errors resdals over me Here, resdals show a cyclc paer, o radom. Cyclcal paers are a sg of posve aocorrelao Resdals Tme Resdal Plo CDS Phl coomercs 57 Tme Volaes he regresso assmpo ha resdals are radom ad depede 58 Aocorrelao: Defo Aocorrelao: Defo The assmpo volaed s j Ths he Pearso s r bewee he resdals from OS ad he same resdals lagged o perod s o-zero. Types of Aocorrelao Aoregressve AR processes ovg Average A processes 59 6

11 3/3/4 Aocorrelao: Defo Aocorrelao: Defo Aoregressve processes ARp The resdals are relaed o her precedg vales. ρ + ε Ths s classc s order aocorrelao: AR process Aoregressve processes co. I d order aocorrelao he resdals are relaed o her - vales as well AR: ρ ρ + ε + arger order processes may occr as well: ARp ρ ρ... ρ + ε p p 6 6 Aocorrelao: Defo Aocorrelao: Defo ovg Average Processes Aq The error erm s a fco of some radom error ad a poro of he prevos radom error. A process ε θε Hgher order processes for Aq also exs. ε θ ε θ ε... θ ε q q The error erm s a fco of some radom error ad some poros of he prevos radom errors Aocorrelao: Defo Aocorrelao: Defo xed processes ARAp,q θ ε ρ θ ε + ρ... θ ε ρ q The error erm s a complex fco of boh aoregressve {ARp} ad movg average {Aq} processes. q p p + ε AR processes represe shocks o sysems ha have log-erm memory. A processes are qck shocks o sysems, b have oly shor erm memory

12 3/3/4 Aocorrelao: Implcaos Aocorrelao: Cases Coeffce esmaes are based, b he esmaes are o BU The varaces are ofe grealy deresmaed based small Hece hypohess ess are excepoally sspec. Specfcao error Omed varable Wrog fcoal form agged effecs Daa Trasformaos Ierpolao of mssg daa dfferecg d The Drb-Waso sasc s sed o es for aocorrelao ˆ ˆ Aocorrelao: Tess H : resdals are o correlaed H : posve aocorrelao s prese ˆ The possble rage s d 4 d shold be close o f H s re d < posve aocorrelao, d > egave aocorrelao 69 Tesg for +ve Aocorrelao H : posve aocorrelao does o exs H : posve aocorrelao s prese Calclae he Drb-Waso es sasc d Usg Saa or SPSS Fd he vales d ad d U from he D-W able for sample sze, ad mber of depede varables, k Decso rle: rejec H f d < d Rejec H Icoclsve Do o rejec H d d U 7 Tesg for +ve Aocorrelao H : posve aocorrelao does o exs H : posve aocorrelao s prese Drb-Waso d Tes: Decso Rles Nll Hypohess Decso If No + aocorrelao Rejec < d < d Decso rle: rejec H f d < d or 4 d < d < 4 No + aocorrelao No Decso d d d U No - aocorrelao Rejec 4 d < d < 4 No - aocorrelao No Decso 4 d U d 4 d No +/- aocorrelao Do o rejec d U < d < 4 d d U d Do o rejec H d 4 d U 4 d 4 7 Tesg for +ve Aocorrelao 7

13 3/3/4 Tesg for +ve Aocorrelao Tesg for +ve Aocorrelao Sppose we have he followg me seres daa: Sales 6 4 Is here aocorrelao? y x R Tme coed 73 xample wh 5: Drb-Waso Calclaos Sm of Sqared Dfferece of Resdals Sm of Sqared Resdals Drb-Waso Sasc.494 d T ˆ ˆ T ˆ Sales y x R Tme Tesg for +ve Aocorrelao Aocorrelao: Tess co. Here, 5 ad k : oe depede varable Usg he Drb-Waso able, d.9 ad d U.45 d.494 < d.9, Therefore he gve lear model s o he approprae model o forecas sales Decso: rejec H sce d.494 < d sgfca +ve aocorrelao exss Drb-Waso d co. Noe ha he d s symmerc abo., so ha egave aocorrelao wll be dcaed by a d >.. Use he same dsaces above. as pper ad lower bods. Rejec H Icoclsve Do o rejec H d.9 d U.45 CDS Phl coomercs Aocorrelao: Tess co. Aocorrelao: Remedes Drb s h Cao se DW d f here s a lagged edogeos varable he model d h T TS y S y- s he esmaed varace of he Y - erm h has a sadard ormal dsrbo Geeralzed eas Sqares Frs dfferece mehod Take s dffereces of yor Xs ad Y Regress Y o X Assmes ha ρ + Geeralzed dffereces Reqres ha ρbe kow

14 3/3/4 Aocorrelao: Remedes Cochra-Orc mehod smae model sg OS ad oba he resdals,. Usg he resdals r he followg regresso. û ˆû + ρ v Aocorrelao: Remedes Cochra-Orc mehod co. 3 sg he ρ obaed, perform he regresso o he geeralzed dffereces Y ˆY ρ B ˆ ρ + BX ˆX ρ + ˆ ρ 4 Sbse he vales of B ad B o he orgal regresso o oba ew esmaes of he resdals. 5 Rer o sep ad repea l ρ o loger chages