Basics of heteroskedasticity

Transcription

1 Sect 8 Heterskedastcty ascs f heterskedastcty We have assumed up t w ( ur SR ad MR assumpts) that the varace f the errr term was cstat acrss bservats Ths s urealstc may r mst ecmetrc applcats, especally crss sects Heterskedastcty ccurs whe dfferet bservats have dfferet errr varace Effects f heterskedastcty We wll see that OLS estmatrs are ubased ad csstet the presece f heterskedastcty, but they are t effcet ad the estmated stadard errrs are csstet, s test statstcs usg the stadard errr are t vald Detect f heterskedastcty At a vsual level, we ca lk fr heterskedastcty by eamg the plt f resduals agast predcted values r dvdual eplaatry varables t see f the spread f resduals seems t deped these varables There are tests that frmalze these vsual descrpts, regressg the squared resduals predcted values r eplaatry varables Dealg wth heterskedastcty: Tw chces Use effcet OLS estmatr but use rbust stadard errrs that allw fr the presece f heterskedastcty Ths s the easest ad mst cmm slut Use weghted least squares (WLS) t calculate effcet estmatrs, cdtal crrect kwledge f the patter f heterskedastcty Ths s the better slut f we kw the patter, whch we usually d t Effects f heterskedastcty Smple regress (multple s smlar) mdel wth heterskedastcty: y e, 1 e E e var, 0, cv e, e 0, j j Cvarace matr f errr vectr s a dagal matr, but t a scalar matr ~ 75 ~

2 We derved earler that the OLS slpe estmatr culd be wrtte as b we, e wth w 1 OLS s ubased uder heterskedastcty: Eb E we 1 1 we e Ths uses the assumpt that the values are fed t allw the epectat f e t g sde the prduct Varace f OLS uder heterskedastcty s t the usual frmula var b var we 1 w var e wwj cv e, ej 1 1 j1 j 1 1 w 1 If s cstat, the we ca take t ut f the umeratr summat ad the umeratr summat devats f cacels e f the dematr summats, leavg the usual frmula: If the varace s t cstat, we ca t d ths ad the rdary varace estmatr s crrect OLS s effcet wth heterskedastcty 1 ~ 76 ~

3 We d t prve ths, but the Gauss-Markv Therem requres hmskedastcty, s the OLS estmatr s lger LUE Detectg heterskedastcty The eye-ball test s a smple but casual way t lk fr heterskedastcty Plt the resduals (r the squared resduals) agast the eplaatry varables r the predcted values f the depedet varable If there s a apparet patter, the there s heterskedastcty f the type that the varace s related t r β The reusch-paga test s a frmal way t test whether the errr varace depeds aythg bservable Suppse that var e E e h 1 z, Sz, S, where the z varables may be the same r dfferet frm the varables the regress, ad h may be ay kd f fuct T test ths, we regress the squared resduals the z varables ad test the hypthess that α α S are all zer: eˆ z z v 1, S, S Several pssble tests f H 0 : 3 S 0: Lagrage multpler test s R ~ S 1 Reject hmskedastcty f test statstc > crtcal value Ths s asympttc test The Whte test s a test that s smlar t the reusch-paga test, usg as the z varables All f the varables the rgal equat The squares f all f the varables Optally, the crss-prducts f the varables Ths leads t lts f varables f K s large Dummes ca t be cluded as squares The Gldfeld-Quadt test s sutable fr samples whch the data ca be dvded t tw grups ad wth varace dfferg ly betwee the grups Suppse that the grups are A ad wth varaces A ad Ru separate regresss fr the tw sub-samples A ad ad calculate the estmated errr varaces frm the resduals ˆ A/ A F ~ F A K, K ˆ / If the ull hypthess s that A, the the rat f the estmated varaces s the F statstc ad we ca d a e-taled r tw-taled test ~ 77 ~

4 Must be careful wth the tw-taled F test, thugh, because F tables ly reprt the rght-had tal area ad crtcal values Make A the sample wth the larger varace s that all f the crtcal area s the rght The e-taled test wth alteratve hypthess A s just the rdary F test wth the usual crtcal value Fr the tw-taled test, a 5% crtcal value becmes a 10% crtcal value because f the pssblty that the varace f A s smaller tha the varace f Heterskedastcty-csstet stadard errrs The frst, ad mst cmm, strategy fr dealg wth the pssblty f heterskedastcty s heterskedastcty-csstet stadard errrs (r rbust errrs) develped by Whte We use OLS (effcet but) csstet estmatrs, ad calculate a alteratve ( rbust ) stadard errr that allws fr the pssblty f heterskedastcty Frm abve, var b 1 1 Lgcal estmatr fr varace s ˆ e / K 1 var b rbust / 1 uder heterskedastcty Ths cmpares wth the hmskedastcty-ly estmatr f var b hmskedastc ˆ / 1 1 e K I matr terms, the cvarace matr f the ceffcet vectr s 1 1 var b XX ˆ XX, wth rbust ˆ ˆ e K Stata calculates the Whte heterskedastcty-csstet stadard errrs wth the pt rbust mst regress cmmads May ecmetrcas argue that e shuld pretty much always use rbust stadard errrs because e ever ca cut hmskedastcty ~ 78 ~ 1

5 Weghted least squares If e wats t crrect fr heterskedastcty by usg a fully effcet estmatr rather tha acceptg effcet OLS ad crrectg the stadard errrs, the apprprate estmatr s weght least squares, whch s a applcat f the mre geeral ccept f geeralzed least squares The GLS estmatr apples t the least-squares mdel whe the cvarace matr f e s a geeral (symmetrc, pstve defte) matr Ω rather tha I 1 ˆ 1 1 GLS X X X y The mst tutve apprach t GLS s t fd the Chlesky rt matr P such that PP s equal t Ω -1 Ths gves us Py PX Pe, fr whch the OLS estmatr s trasfrmed 1 1 XPPX XPPy b PX PX PX Py ˆ GLS X X X y Thus we ca use the usual OLS prcedure the trasfrmed mdel t get the effcet GLS estmatr Ths estmatr s smetmes called feasble GLS because t requres that we kw Ω, whch we usually d t Feasble GLS s whe we use a estmatr fr Ω rather tha the actual value Fr the case f heterskedastcty, , ad the crrespdg F matr s P Multplyg the X, y, ad e matrces by P trasfrms each bservat by dvdg, y, ad e by ~ 79 ~

6 Thus, WLS cssts f OLS the trasfrmed mdel * y *, j * e y,, j, e te that the cstat term s lger just es, because each 1 s dvded by a dfferet * ecause var e, var e 1 Thus dvdg each bservat by smethg prprtal t the errr stadard devat fr the bservat cverts the mdel t a hmskedastc e var e HGL eample: If we dvde by the the varace f the trasfrmed mdel s fr all bservats Ths s why we call t weghted least squares: we weght each bservat by the recprcal f ts stadard devat, gvg greatest weght t the bservats fr whch the errr varace s smallest If patter f heterskedastcty s ukw, we may estmate a mdel such as l eˆ z z v ad use ftted values as estmates f the errr 1, S, S varace f each bservat Ths feasble GLS s csstet ad asympttcally effcet as lg as we have the patter f heterskedastcty specfed crrectly ~ 80 ~