Deparmen of Economcs Unversy of Torono ECO408F M.A. Economercs Lecure Noes on Heeroskedascy
Heeroskedascy o Ths lecure nvolves lookng a modfcaons we need o make o deal wh he regresson model when some of he assumpons ha are needed for he OLS o be BLUE are volaed o In parcular he lecure wll focus on he presence of heeroskedascy (varance no consan) Heeroskedascy (have homoskedascy of resduals f he varance s consan) o Arses when varance of resdual s no consan o e.g., f look a a cross-secon of frms n an ndusry error erms wh large frms mgh have larger varance han errors assocaed wh smaller frms o INSERT e.g. GRAPH locaed n margn o e.g., sales of larger frms mgh be more volale han hose of smaller frms o e.g., prce of a smaller company s sock mgh be more volale han ha of a blue chp company. o Ths s a problem ha usually occurs n a cross-seconal daa o In OLS ( ) Ε Homoskedasc ( ) Ε Heeroskedascy o Wh heeroskedascy, OLS places more wegh on he observaons wh large error varances han hose wh smaller varances Correcng for Heeroskedascy o Correc for heeroskedascy by ransformng he model so ha has consan varance and apply OLS o he ransformed model o Ths procedure s called weghed leas squares (WLS) o Ths esmaor s also par of he famly of esmaon procedures referred o as Generalzed Leas Squares (GLS) o Example : Suppose have model α ( ) (.e., s he nercep or consan erm)
o Defne The followng varables: o And esmae α o Wh OLS, noe ha he resduals n hs ransformed model are homoskedasc ( ) o So he resduals n he ransformed model sasfy he assumpons of he OLS model, so OLS on he ransformed model s BLUE o Example : Suppose ( ) C ance of resduals s drecly relaed o one of he regressors n he regresson Need o ransform model so ha ( ) s consan Wegh he regresson wh so ransform he model so ha: Transformed regresson:
Noe ha ( ) C C ance of ransformed model s consan, so OLS on ransformed model s BLUE Tesng for Heeroskedascy A. Goldfeld Quand Tes: Tes sasc s derved under he assumpon ha under Η, he sample could be poroned no wo subses of observaons where he error varance s dfferen for each subse, bu s consan whn a subse The alernave hypohess s Η :... ; he null hypohess s Τ homoskedascy Under Η, he observaons can be ordered accordng o ncreasng varances To mplemen es:. Assumng B. Breusch-Pagan Tes: Η s rue, order he observaons accordng o ncreasng error varance.. Om cenral observaons.. Run wo separae regressons. One usng he frs ( T r / observaons ) and he oher usng he las ( T r) / observaons. 4. Compue he es sasc λ s / s, where s and s are resdual sums of squares from he frs and second regressons, respecvely. Under he null hypohess of homoskedascy, he es sasc λ has an F dsrbuon T r k T r k wh, degrees of freedom. 5. Compare λ wh crcal value and accep or rejec. Ths s a small sample es no an asympoc one ance s a funcon, no necessarly mulplcave of more han one explanaory varable. Under Η : h Z α h α Z α Z, Z α ( α,..., α ) s 4
C. Whe Tes: ( Z Z ) Z,..., Under Η : α 0 and he resduals have a normal dsrbuon, can be shown ha 0 ( TSS ESS), TSS oal sum of squares and ESS error sum of squares, from he regresson eˆ ˆ Z α v s, where eˆ y x ˆ and T eˆ ˆ OLS T Wll be asympocally dsrbued as a χ s If don assume normaly hen can ake he R from he regresson of e on Z mulpled by he sample sze as a es sasc, whch wll have he ˆ asympoc dsrbuon d TR χ s Wh Goldfeld-Quand or Breusch-Pagan es have o make an assumpon abou he form he heeroskedascy. The Whe es s an asympoc es ha does no requre one o specfy he form or he varables causng he heeroskedascy. Example: Suppose he model s Y x e X (, X, X ) Whe es requres runnng an auxlary regresson on he regressors and her squares and cross producs. So n our example:, x, x, x, x, x x So run he regresson of ê on, x, x, x, x, x x Under Η d TR χ, q s # of varables n auxlary regresson 0 q k( k ) Wh q Ths es has become popular n recen years because of he fac you don have o specfy he form of he heeroskedascy. Anoher hng ha has emerged s ha now many people esmae he model wh OLS and use a suggeson Whe made o correc he errors. (Done n many sofware packages, lke STATA) 5