Recall MLR 5 Homskedasticity error u has the same variance given any values of the explanatory variables Var(u x1,...,xk) = 2 or E(UU ) = 2 I

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1 Chapter 8 Heterosedastcty Recall MLR 5 Homsedastcty error u has the same varace gve ay values of the eplaatory varables Varu,..., = or EUU = I Suppose other GM assumptos hold but have heterosedastcty. Varu =

2 Why mght ecoomc data have heterosedastcty?. Sewess of oe of the s. Outlers 3. Bary y 4. Error learg models suppose have data o hours of typg practce ad typg errors for a buch of dfferet people. Hgher varace at lower hours, but as people lear, varace errors falls. 5. X s come, Y s somethg le savgs as come grows, have more scope for choce about what to do wth come 6. Specfcato errors fuctoal form 7. Clustered data data o dvduals, some varables are state level averages

3 What problems does ths volato cause? Do ths a two varable case: OLS Estmator u u X y If we tae the epected value of ths, as log as we have eogeety, OLS estmator s ubased R s fe eve wth heterosedastcty

4 However, f have heterosedastcty, usual estmator for var s based. Recall var Dervato u Tae the varace of ths: var = 0 + SST var u = SST varu = SST where SST Or Var = σ Uder homosedastcty, we get σ Var = = or X X - multvarate case Recall that σ = / = u However, we do t ow the true u Istead we have u = y β 0 β Recall that smply replacg u wth u leads to a based estmator of σ see Mar s otes Istead we use σ = So the usual estmator for the varace of the estmated coeffcets s varβ σ = = I multvarate case varβ j = σ X X - u

5 But uder heterosedastcty, ths s based. Stll the case that Var = σ But assumg equal varace σ = σ wll result based stadard errors Why do we care f the estmator for ths s based? If the estmator for ths s based fereces wll be wrog LM, F, t stats wll be wrog.

6 8.3 Testg for Heterosedastcty: y = β o + β β +u A. BREUSCH-PAGAN TEST ths s the Koeer verso there are several BP tests, but Wooldrdge calls ths verso the BP test What s Heterosedastcty? Ho = varu,,.... = Alteratve ot detcal Matr form EUU = I draw ths out So f Ho s true ad u has zero codtoal mea varu X = Eu X so Ho = Eu,,.... = Eu = Use a LM test procedure for ths Steps:. Ru restrcted regresso regular OLS model wth the restrcto of homosedastcty u ~ Number of restrctos s. Ru aullary regresso u~ 0... error 3. LM = * R from above regresso 4. Compare to B. WHITE TEST test wth weaer assumptos that BP test: Istead of Ho = varu,,.... = Tests that Corru,,, j = 0 turs out ths tests for all forms of heterosedastcty that could valdate OLS stadard errors Aga, aother LM test procedure Steps:. Ru restrcted regresso regular OLS model u ~ Number of restrctos s

7 . Ru aullary regresso error u ~ 3. LM = * R from above regresso Chews up lots of degrees of freedom... C. Oe more LM test procedure Specal case of Whte test:. Costruct ftted values of y y... 0 If square these ftted values, get a partcular fucto of all squares ad cross-products of s. Estmate error y y u 0 3. LM = * R from above regresso ~

8 Solutos to Heterosedastcty. Geerate Robust stadard errors t, F, LM statstcs that are vald presece of heterosedastcty of uow form.. Use Weghted Least Squares more geerally, GLS wth a arbtrary covarace matr less commoly used geeral FGLS 3. Use Weghted Least Squares more geerally, GLS wth a specfc covarace matr less commoly used

9 Soluto #. Heterosedastc Robust Iferece after OLS estmato Secto 8. Recall wth oly oe X: y = β o + β +u Other GM assumptos hold but have heterosedastcty. Varu = var = SST where SST Note that f =, the we have usual form Use a estmator for σ = = u SST What f have heterosedastcty? The eed a cosstet estmate of var

10 Whte shows ca use the followg:. Regress y o Predcted errors u y 0 u. var put hat over var SST 3. Ths estmator s cosstet see otes Wooldrdge o ths proof. Coverges probablty to SST Eteso to multple regresso framewor: Huber-Whte or Whte or Huber or Robust stadard errors: r j u var j = SSR s hat over var j where r j s th resdual from regressg j o all other ad SSR s sum of squared resduals from j o all other s STATA: reg yvar var var, robust These stadard errors ad the assocated t-stats are oly vald as sample sze gets large. IF homosedastcty holds AND sample sze s small, may ot wat to add robust opto. But most of tme wll estmate everythg wth robust stadard errors. What about testg restrctos? WALD statstc Recall that we taled about F tests. There s a heterosedastcty robust verso of F that s a Wald statstc. LM tests see boo

11 Soluto # Weghted Least Squares If ow the form of the varace, ca use a dfferet estmator: WLS. WLS s more effcet tha OLS IF ow the form of the varace. If do t ow the form of the varace, s ot ecessarly more effcet. But f have STRONG heterosedastcty, WLS ca be more effcet. I practce, mostly use OLS wth robust stadard errors. But good to see how WLS wors a specal case of GLS. Suppose have heterosedastcty s ow up to a multplcatve costat GENERALIZED LEAST SQUARES GLS Here also called WEIGHTED least squares essetally, mmze sum of squared errors, where weght each observato by /h gve more weght to obs wth lower varace. Fgurg out the weghts s somewhat arbtrary uless for some reaso KNOW the form of heterosed. Oe case where do ow form of Heterosed whe have averages of dvdual level data stead of estmatg dvduals, have frm or state or coutry averages. Suppose that dvdual s errors are ucorrelated. Varace decreases wth sze of group. I ths case, h = /m where m s umber of members the group frm or state or coutry. Homewor has you wor through ths more detal.

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