Heteroskedasticity. Heteroskedasticity means that the variance of the errors is not constant across observations.

Transcription

1 Heeroskedascy Heeroskedascy means ha he varance of he errors s no consan across observaons. In parclar he varance of he errors may be a fncon of explanaory varables. Thnk of food expendre for example. I may well be ha he dversy of ase for food s greaer for wealher people han for poor people. So yo may fnd a greaer varance of expendres a hgh ncome levels han a low ncome levels.

2 Heeroskedascy may arse n he conex of a random coeffcens model. Sppose for example ha a regressor mpacs on ndvdals n a dfferen way b a Y b a Y ε ε

3 Assme for smplcy ha and are ndependen. Assme ha and are ndependen of each oher. Then he error erm has he followng properes: Where s he varance of 0 E E E E ε ε ε σ σ ε ε ε Var Var Var ε σ ε ε ε

4 In boh scaer dagrams he average slope of he nderlyng relaonshp s he same exs realex Scaer Dagram HOMOSKEDASTICITY

5 00 00 exs realex Scaer Dagram RANDOM COEFFICIENTS MODEL HETEROSKEDASTICITY

6 Implcaons of Heeroskedascy Assmng all oher assmpons are n place, he assmpon garaneeng nbasedness of OLS s no volaed. Conseqenly OLS s nbased n hs model However he assmpons reqred o prove ha OLS s effcen are volaed. Hence OLS s no BLUE n hs conex We can devse an effcen esmaor by reweghng he daa appropraely o ake no accon of heeroskedascy

7 If here s heeroskedascy n or daa and we gnore hen he sandard errors of or esmaes wll be ncorrec However, f all he oher assmpons hold or esmaes wll sll be nbased. Snce he sandard errors are ncorrec nference may be msleadng

8 Correcng he Sandard errors for Heeroskedascy of nknown knd - The Ecker-Whe procedre If we sspec heeroskedascy b we do no know s precse form we can sll compe or sandard errors n sch a way ha he are robs o he presence of heeroskedascy Ths means ha hey wll be correc wheher we have heeroskedascy or no. The procedre s jsfed for large samples.

9 . replace exs 05*nvnormnform*rr 3*nvnormnform 4785 real changes made. regr exs rr, robs Y 0 v * Regresson wh robs sandard errors Nmber of obs 4785 F, Prob > F R-sqared Roo MSE Robs exs Coef. Sd. Err. P> [95% Conf. Inerval] rr _cons

10 . replace exs 00*nvnormnform*rr 3*nvnormnform 4785 real changes made. regr exs rr Y 0 v * Sorce SS df MS Nmber of obs F, Model Prob > F Resdal R-sqared Adj R-sqared 0.85 Toal Roo MSE exs Coef. Sd. Err. P> [95% Conf. Inerval] rr _cons

11 To see how we can do hs les go back o he dervaon of he varance for he esmaor of he slope coeffcen n he smple wo varable regresson model lecre 3 We had ha ] [ ] [ ] ˆ [ E E b b E N N N j j N j N

12 The problem arses becase E [ ] s no longer a consan σ. The varance of he resdal changes from observaon o observaon. Hence n general we can wre E [ ] σ We gave an example n he random coeffcens model how hs can arse. In ha case he varance depended on

13 The Varance of he slope coeffcen esmaed by OLS when here s heeroskedascy ] ] ˆ [ N N b b E σ

14 The Ecker-Whe formla To esmae hs varance we can replace he for each observaon by he sqared OLS resdal for ha observaon Ths we esmae he varance of he slope coeffcen by sng σ b a Y ˆ ˆ ˆ [ ] ^ ] ˆ ˆ NVar b Var N

15 Smmary of seps for esmang he varance of he slope coeffcens n a way ha s robs o he presence of Heeroskedascy Esmae regresson model by OLS. Oban resdals. Use resdals n formla of prevos page. A smlar procedre can be adaped for he mlple regresson model.

16 Seral Correlaon or Aocorrelaon We have assmed ha he errors across observaons are no correlaed: Assmpon 3 We now consder relaxng hs assmpon n a specfc conex: Wh daa aver me Sppose we have me seres daa: I.e. we observe Y, seqenally n reglar nervals over me. GDP, neres raes, Money Spply ec.. We se as a sbscrp o emphasze ha he observaonas are over me only.

17 Consder he regresson The model Y a b When we have seral correlaon he errors are correlaed over me. For example a large negave shock o GDP n one perod may sgnal a negave shock n he nex perod. One way o capre hs s o se an Aoregressve model for he resdals,.e. In hs formlaon he error hs perod depends on he error n he las perod and on an nnovaon v. v s assmed o sasfy all he classcal assmpons Assmpon o Assmpon 3. v

18 We consder he saonary aoregressve case only n whch he effec of a shock evenally des o. Ths wll happens f To see hs sbse o one perod back o ge And so on o ge Ths a shock ha occrs n perods back has an mpac of < < v v k k k k v v v v v n

19 Implcaons of seral correlaon Under seral correlaon of he saonary ype OLS s nbased f he oher assmpons are sll vald In parclar Assmpon OLS s no longer effcen Condons for he Gass Markov heorem are volaed. If we gnore he presence of seral correlaon and we esmae he model sng OLS, he varance of or esmaor wll be ncorrec and nference wll no be vald.

20 Esmang wh seral correlaon Defne a lag of a varable o be s pas vale. Ths - denoes he vale of one perod ago. The perod may be a year, or a monh or whaever s he nerval of samplng day or mne n some fnancal applcaons Wre: Sbrac he second from he frs o ge b a Y b a Y v b a a Y Y b b a a Y Y

21 Now sppose we knew Then we cold consrc he varables Y Y and Then he regresson wh hese ransformed varables sasfes he Assmpons -4. Ths, accordng o he Gass Markov heorem f we esmae b wh hese varables we wll ge an effcen esmaor. Ths procedre s called Generalsed Leas Sqares GLS. However we canno mplemen drecly becase we do no know

22 A wo sep procedre for esmang he regresson fncon when we have Aocorrelaon Sep : Regress Y on Y-, and -. The coeffcen of Y- wll be an esmae of Consrc Y ˆ Y ˆ and Sep. Rn he Regresson sng OLS o oban b: ˆ Y a * b ˆ Y v Ths procedre s called Feasble GLS

23 Smmary When we know GLS s BLUE When has o be esmaed n a frs sep hen feasble GLS s effcen n large samples only. In fac n small samples feasble GLS wll be generally based. However n pracce works well wh reasonably szed samples.

24 EAMPLE: Esmang he AR coeffcen n he error erm rho and ransformng he model o ake no accon of seral correlaon. regr lbp lpbr lpsmr lryae lag* Log Ber Prchases Monhly daa Sorce SS df MS Nmber of obs 50 one observaon los by laggng log ber prchases lbp Coef. Sd. Err. P> [95% Conf. Inerval] Log prce of ber lpbr Log prce of margarne lpsmr Log real ncome lryae One monh Lag of he above Lag of dependen varable: laglpbr laglpsmr laglryae Esmae of rho laglbp _cons

25 . regr lbprho lpbrrho lpsmrrho lryaerho Sorce SS df MS Nmber of obs F 3, Model Prob > F Resdal R-sqared Adj R-sqared Toal Roo MSE All varables now have been consrced as lbprho Coef. Sd. Err. P> [95% Conf. Inerval] lpbrrho lpsmrrho lryaerho _cons