Chapter 17 Handout: Autocorrelation (Serial Correlation)
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- Giles Pearson
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1 Chapr 7 Handou: Auocorrlaion (Srial Corrlaion Prviw Rviw o Rgrssion Modl o Sandard Ordinary Las Squars Prmiss o Esimaion Procdurs Embddd wihin h Ordinary Las Squars (OLS Esimaion Procdur o Covarianc and Indpndnc Wha Is Auocorrlaion (Srial Corrlaion? Consquncs of Auocorrlaion o h Mahmaics o Our Suspicions o Confirming our Suspicions Accouning for Auocorrlaion: An Eampl Jusifying h Gnralizd Las Squars (GLS Esimaion Procdur Robus Sandard Errors: An Alrnaiv Approach Rviw: Rgrssion Modl y = Cons + + y = Dpndn variabl = Eplanaory variabl = Error rm =,,, = Sampl siz h rror rm is a random variabl ha rprsns random influncs: Man[ = 0 Rviw: Sandard Ordinary Las Squars (OLS Rgrssion Prmiss Error rm Equal Varianc Prmis: h varianc of h rror rm s probabiliy disribuion for ach obsrvaion is h sam; all h variancs qual Var[: Error rm/error rm Indpndnc Prmis: h rror rms ar indpndn: Cov[ i, j = 0. Eplanaory Variabl/Error rm Indpndnc Prmis: h planaory variabls, h s, and h rror rms, h s, ar no corrlad. Rviw: Ordinary Las Squars (OLS Esimaion Procdur Svral simaion procdurs ar mbddd wih h ordinary las squars simaion procdur. For our purposs, h hr mos imporan ar a procdur o sima h valu of h cofficin, : varianc of h rror rm s probabiliy disribuion, Var[ varianc of h cofficin sima s probabiliy disribuion, Var[b Summary: Whn h sandard OLS rgrssion prmiss ar saisfid: Each of h simaion procdurs mbddd wihin h ordinary las squars (OLS simaion procdur is ; h ordinary las squars (OLS simaion procdur for h cofficin valu is h. Criical Poin: Whn h ordinary las squars (OLS simaion procdur prforms is calculaions i implicily assums ha h hr sandard ordinary las squars (OLS prmiss ar saisfid. Rviw: Covarianc and Indpndnc wo variabls ar indpndn whnvr h valu of on variabl dos no hlp us prdic h valu of h ohr. If wo variabls ar indpndn, hir covarianc quals 0. Cov[, y y y y y y y N N N N y y N
2 Scar Diagrams Dviaions from Mans Dviaions From Mans 0 Nasdaq Dviaions From Mans 0 Nasdaq Prcipiaion Dow Jons Covarianc =.9 0 Covarianc = Var[ + y = Var[ + y = Var[ + Cov[, y + Var[y Indpndn: W can ignor h. Wha is Auocorrlaion? Auocorrlaion ofn appars whn analyzing im sris daa. Auocorrlaion iss whnvr h valu of h obsrvaion s rror rm allows us o prdic h valu of obsrvaion s rror rm. Modl of Auocorrlaion: = v s ar indpndn Lab 7., h Grk lr rho, has bn usd o rprsn auocorrlaion. Whn quals 0, no auocorrlaion is prsn. On h ohr hand, whn dos no qual 0, auocorrlaion is prsn. = 0 0 = v dpnds on No auocorrlaion Auocorrlaion prsn = 0 =.9 Whn auocorrlaion is prsn If > 0 h rror rm/rror rm < 0 > 0 indpndnc prmis is. ypically, ypically,
3 Consquncs of Auocorrlaion: Ar h Ordinary Las Squars Esimaion Procdurs Sill Unbiasd? Qusion: How dos h prsnc of auocorrlaion affc h ordinary las squars (OLS simaion procdurs for h valu of h cofficin, : varianc of h rror rm s probabiliy disribuion, Var[ varianc of h cofficin sima s probabiliy disribuion, Var[b Qusion: Mor spcifically, ar hs simaion procdurs sill unbiasd in h prsnc of auocorrlaion? Esimaion Procdur for h Valu of h Cofficin Rviw: Arihmic of Mans Man of a consan plus a variabl: Man[c + = c + Man[ Man of a consan ims a variabl: Man[c = c Man[ Man of h sum of wo variabls: Man[ + y = Man[ + Man[y ( ( ( Man[ b Man[ ( Applying Man[c + = c + Man[ ( ( ( Man[ ( Rwriing h fracion as a produc Man [( (( ( ( ( Applying Man[c = cman[ Man [(( ( ( ( Applying Man[ + y = Man[ + Man[y Man[( Man[( Man[( Applying Man[c = cman[ ( Man[ ( Man[ ( Man[ ( [ ( [ Sinc Man[ = Man[ = Man[ = 0 Qusion: Hav w rlid on h rror rm/rror m indpndnc varianc prmis o show ha h ordinary las squars (OLS simaion procdur for h cofficin valu is unbiasd (Man[b =? Qusion: In h prsnc of auocorrlaion, should w pc h ordinary las squars (OLS simaion procdur for h cofficin valu sill o b unbiasd?
4 4 Esimaion Procdur for h Varianc of h Cofficin Esima s Probabiliy Disribuion Qusion: In h prsnc of auocorrlaion, is h ordinary las squars (OLS simaion procdur for h varianc of h cofficin sima s probabiliy disribuion unbiasd? Rviw h sragy w usd o sima h varianc of h cofficin sima s probabiliy disribuion: Sp : Esima h varianc of h rror rm s probabiliy disribuion from h availabl informaion; ha is, informaion from h firs quiz SSR EsVar[ = Dgrs of Frdom Sp : Apply h rlaionship bwn h variancs of cofficin sima s and rror rm s probabiliy disribuions Var[ b Var[ ( Focus on Sp : Qusion: Is h quaion, EsVar[ b EsVar[ b ( EsVar[ ( EsVar[, valid whn auocorrlaion is prsn? Rviw: Arihmic of Variancs and h Logic Usd o Driv h Equaion Varianc of a consan ims a variabl: Var[c = c Var[ Varianc of h sum of a consan and a variabl: Var[c + = Var[ Varianc of h sum of wo indpndn variabls: Var[ + y = Var[ + Var[y [ ( ( ( Var[ b Var ( [ Applying Var[c + = Var[ ( ( ( Var ( Rwriing h fracion as a produc Var [( (( ( ( ( Applying Var[c = c Var[ Var [(( ( ( [( Error rm/error rm Indpndnc Prmis: Var[ + y = Var[ + Var[y Var[( Var[( Var[( ( [ [ Applying Var[c = c Var[ ( Var[ ( Var[ ( Var[ [( [ Error rm qual varianc prmis: Var[ = Var[ = Var[ = Var[
5 5 ( Var[ ( Var[ ( Var[ [( [ Facoring ou h Var[ [( [ ( Var[ Var[ ( Simplifying Gnralizing: Var[ b Var[ ( Whn auocorrlaion is prsn, howvr, h prmis ha all rror rms ar indpndn is violad. How dos his affc h logic?. Var [(( ( ( [( Error rm/error rm Indpndnc Prmis: Var[ + y = Var[ + Var[y Var[( Var[( Var[( [( [. Var[ Var[ b ( How dos his affc our sragy o sima h varianc of h cofficin sima s probabiliy disribuion? Sp : Esima h varianc of h rror rm s probabiliy disribuion from h availabl informaion; ha is, informaion from h firs quiz SSR EsVar[ = Dgrs of Frdom Sp : Apply h rlaionship bwn h variancs of cofficin sima s and rror rm s probabiliy disribuions Var[ b Var[ ( EsVar[ b EsVar[ ( Qusion: Hav w rlid on h rror rm/rror m indpndnc varianc prmis o show ha Var[? Var[ b ( Qusion: In h prsnc of auocorrlaion, can w pc h OLS simaion procdur for h varianc of h cofficin sima s probabiliy disribuion sill o b unbiasd?
6 6 Suspicions Afr rviwing our drivaions of Man[b and Var[b,w suspc ha whn auocorrlaion is prsn h ordinary las squars OLS simaion procdur for h cofficin valu will b unbiasd. varianc of h cofficin sima s probabiliy disribuion may b flawd. W can us our simulaion o confirm our suspicions. Bu, firs rviw h modl of auocorrlaion. Modl of Auocorrlaion: = v s ar indpndn = 0 0 = v dpnds on No auocorrlaion Auocorrlaion prsn Auocorrlaion Simulaion Lab 7. Sampl Siz = 0 Is simaion Is simaion procdur procdur for h for h varianc of h cofficin s valu of h cofficin sima s unbiasd? probabiliy disribuion unbiasd? Acual Esima of Varianc of Esima of h varianc cofficin cofficin simad cofficin for cofficin sima s valu valu valus probabiliy disribuion Man (Avrag of Varianc of Avrag of Acual h Esimad h Esimad Cof Esimad Variancs, Esim Valu Valus, b, from Valus, b, from EsVar[b, from Rho Proc of All Rpiions All Rpiions All Rpiions 0 OLS.0.6 OLS.0 Summary: Is h simaion procdur: Sd Prmiss Auo an unbiasd simaion procdur for h OLS OLS o cofficin valu? o varianc of h cofficin sima s probabiliy disribuion?
7 7 Accouning for Auocorrlaion: An Eampl Sp : Apply h Ordinary Las Squars (OLS Esimaion Procdur o Esima h modl s paramrs wih h ordinary las squars (OLS simaion procdur. Sp : Considr h Possibiliy of Auocorrlaion o Ask whhr hr is rason o suspc ha auocorrlaion may b prsn. o Us h ordinary las squars (OLS rgrssion rsuls o g a sns of whhr auocorrlaion is a problm by amining h rsiduals. o Us h Lagrang Muliplir approach by simaing an arificial rgrssion o s for h prsnc of auocorrlaion. o Esima h valu of h paramr in h auocorrlaion modl. Sp : Apply h Gnralizd Las Squars (GLS Esimaion Procdur o Apply h modl of auocorrlaion and algbraically manipula h original modl o driv a nw, wakd modl in which h rror rms do no suffr from auocorrlaion. o Us h ordinary las squars (OLS simaion procdur o sima h paramrs of h wakd modl. An Eampl: Disposabl Incom and Consumr Durabls ConsDur Consumpion of durabls in monh (billions of 005 dollars Inc Disposabl incom in monh (billions of 005 dollars radiional Kynsian hory posulas ha highr lvls of disposabl incom incras h consumpion of consumr durabls: Modl: ConsDur = Cons + I Inc + hory: I > 0. Highr disposabl incom incrass consumpion of durabls. Sp : Apply h Ordinary Las Squars (OLS Esimaion Procdur NB: W ar using monhly daa. Consumpion and Disposabl Incom Ordinary Las Squars (OLS Dpndn Variabl: ConsDur Eplanaory Variabl(s: Esima SE -Saisic Prob Inc Cons Numbr of Obsrvaions 7 Esimad Equaion: ConsDur = Inrpraion: W sima ha a $ incras in ral disposabl incom incrass h ral consumpion of durabl goods by $. Criical Rsul: h Inc cofficin sima quals. his vidnc, h sign of h cofficin sima, suggss ha highr disposabl incom h consumpion of consumr durabls hrby h hory. H 0 : I = 0 Highr disposabl incom dos no affc consumpion of durabls H : I > 0 Highr disposabl incom incrass consumpion of durabls Using h ails probabiliy: Prob[Rsuls IF H 0 ru =.
8 8 Sp : Considr h Possibiliy of Auocorrlaion. Is hr rason b bliv ha auocorrlaion may b prsn? Ky Obsrvaion: Businss cycls nd o las for many monhs. Srong conomic priod Wak conomic priod Consumr confidnc In h las monh, consumr was high las monh; confidnc was low; consumrs spn frly, consumrs spn frly, consum, las monh. consum, las monh. ypically, consumr confidnc ypically, consumr confidnc will coninu o b will coninu o b his monh; consumrs will his monh; consumrs will spnd frly, spnd frly, consum, his monh consum, his monh Suspicion: h rror rms ar : auocorrlaion. Us h ordinary las squars (OLS rgrssion rsuls o g a sns of whhr auocorrlaion is a problm by amining h rsiduals. W can hink of h rsiduals as h simad rror rms: h rror rm, h s, h rsidual, h Rs s, ar unobsrvabl ar obsrvabl y = Cons + + Rs = y Esy = y ( Cons + Rs = y (b Cons + b sinc Esy = b Cons + b I is convnin o hink of h rsiduals o b simas of h rror rms. Consumpion and Disposabl Incom Do hs graphs suggs h prsnc of auocorrlaion?
9 9 Us h Lagrang Muliplir (LM chniqu o s for Auocorrlaion W can moiva h Lagrang muliplir (LM s for auocorrlaion wih a lil algbra: Original Modl: y = Cons + + s ar unobsrvabl Auocorrlaion Modl: = v s ar indpndn Ordinary Las Squars Esima: Esy = b Cons + b Rsiduals: Rs = y Esy Rs s ar obsrvabl Rs = y Esy Subsiuing for y y = Cons + + = Cons + + Esy Subsiuing for = = Cons + + Esy Subsiuing for Esy Esy = b Cons + b = Cons + + (b Cons + b Rarranging rms = ( Cons b Cons + ( b + Canno obsrv us Rs insad = ( Cons b Cons + ( b + Rs NB: Sinc h v s ar indpndn; w nd no worry abou auocorrlaion hr. Ging Sard in EViws Run h rgrssion. In h Equaion window, click Viw Click Rsidual Diagnosics Click Srial Corrlaion LM s Chang h numbr of Lags o includ from o. Consumpion and Disposabl Incom Lagrang Muliplir (LR Dpndn Variabl: Rsid Eplanaory Variabl(s: Esima SE -Saisic Prob Inc Cons Rsid( Numbr of Obsrvaions 7 Prsampl missing valu laggd rsiduals s o zro. Criical Rsul: h Rsid(- cofficin sima quals. h sign of h cofficin sima suggss ha an incras in las priod s rsidual his priod s rsidual. his vidnc suggss ha auocorrlaion. H 0 : = 0 No auocorrlaion prsn H : > 0 Auocorrlaion prsn Using h ails probabiliy: Prob[Rsuls IF H 0 ru =.
10 0 Esima h auocorrlaion paramr,. Auocorrlaion Modl: = v s ar indpndn Ging Sard in EViws Run h original rgrssion. EViws auomaically modifis Rsid vry im a rgrssion is run. o avoid clashs gnra wo nw variabls: Rsidual = Rsid RsidualLag = Rsidual( Now, spcify Rsidual as h dpndn variabl and RsidualLag as h planaory variabl; do no forg dl h consan. Consumpion and Disposabl Incom Ordinary Las Squars (OLS Dpndn Variabl: Rsidual Eplanaory Variabl(s: Esima SE -Saisic Prob RsidualLag Numbr of Obsrvaions 7 Esimad Equaion: Rsidual =.0890RsidualLag Esima of = Es =.89 Sp : Apply h Gnralizd Las Squars (GLS Esimaion Procdur Apply h modl of auocorrlaion and algbraically manipula h original modl o driv a nw, wakd modl in which h rror rms do no suffr from auocorrlaion: Original modl: y = Cons + + Auocorrlaion modl: = v s ar indpndn Original modl for priod : y = Cons + + Original modl for priod : y = Cons + + y = Cons + + Muliplying by Rwri h quaions for y and by y : y = Cons + + y = Cons + + Subracing y y = Cons Cons + + Facoring ou y y = Cons Cons + ( + Subsiuing for y y = Cons Cons + ( + Simplifying y y = ( Cons Cons + ( v s ar ; in h nw modl, auocorrlaion prsn.
11 Us h ordinary las squars (OLS simaion procdur o sima h paramrs of h wakd modl. wakd Modl: y y = ( Cons Cons + ( Rplac wih Es y Esy = ( Cons Es Cons + ( Es Adjy = ( Cons Cons + Adj Nw dpndn variabl: Adjy = y Esy AdjConsDur = ConsDur.89ConsDur Nw planaory variabl: Adj = Es AdjInc = Inc.89Inc Consumpion and Disposabl Incom Ordinary Las Squars (OLS Dpndn Variabl: AdjConsDur Eplanaory Variabl(s: Esima SE -Saisic Prob AdjInc Cons Numbr of Obsrvaions 7 Comparing h Ordinary Las Squars (OLS and Gnralizd Las Squars (GLS Esimas Cofficin Sandard ails Esima Error -Saisic Probabiliy Ordinary Las Squars (OLS <.000 Gnralizd Las Squars (GLS Lab 7.4 Jusifying h Gnralizd Las Squars (GLS Esimaion Procdur Auocorrlaion Simulaion Sampl Siz = 0 Man (Avrag of Varianc of Avrag of Acual h Esimad h Esimad Cof Esimad Variancs, Esim Valu Valus, b, from Valus, b, from EsVar[b, from Rho Proc of All Rpiions All Rpiions All Rpiions 0 OLS OLS GLS.0 Summary: Whn auocorrlaion is prsn: Can h ordinary las squars (OLS simaion procdur for h varianc of h cofficin sima s probabiliy disribuion b rusd? Which simaion procdur for h cofficin valu is br, h ordinary las squars (OLS or h gnralizd las squars (GLS? Is h ordinary las squars (OLS simaion procdur for h cofficin valu h bs linar unbiasd simaion procdur (BLUE?
12 Robus Sandard Errors: An Alrnaiv Approach wo issus mrg wih h ordinary las squars (OLS simaion procdur whn auocorrlaion is prsn: h sandard rror calculaions mad by h ordinary las squars (OLS simaion procdur ar flawd. Whil h ordinary las squars (OLS for h cofficin valu is unbiasd, i is no h bs linar unbiasd simaion procdur (BLUE. Robus sandard rrors addrss h firs issu and ar paricularly appropria whn h sampl siz is larg. Ging Sard in EViws Run h ordinary las squars (OLS rgrssion. In h quaion window, click Esima and Opions In h Cofficin covarianc mari bo slc HAC (Nwy-Ws from h Esimaion dfaul drop down lis. Click OK. Consumpion and Disposabl Incom HAC (Nwy-Ws sandard rrors Dpndn Variabl: ConsDur Eplanaory Variabl(s: Esima SE -Saisic Prob Inc Cons HAC (Nwy-Ws sandard rrors Numbr of Obsrvaions 7 Sandard rrors basd on h rror rm/rror rm indpndnc prmis Ordinary Las Squars (OLS Dpndn Variabl: ConsDur Eplanaory Variabl(s: Esima SE -Saisic Prob Inc Cons Numbr of Obsrvaions 7
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