Failure of Assumptions

Transcription

1 of 9 Falre of Assptons Revew... Basc Model - 3 was to wrte t: paraeters; observatons or or U Y Y U Estatng - there are several was to wrte t ot: Y U Assptons - fall nto three categores: regressors, error ters 3 & 4, or both. E,..., - error ncorrelated to conteporar regressors. and E/ onsnglar - cobned wth asspton, these are the dentfcaton or reglart condtons.e., we can fnd n theor or n practce & wth soe techncal stat stff sa [ ], UU E 3. E - error ters are nrelated to each other 4. E - hoosedastct error ters have the sae varance 3 & 4 sa, Assptons on regressors and relatonshp between regressors and error ters.e., assptons & are reqred for to be consstent Assptons on error ters.e., 3 & 4 are anl st to spl calclatons for Var Best Estate - As long as the for assptons hold, s or best estate gven the data set.e., has the lowest varance; ths s tre even f we had addtonal nfo sch as 3 Basc Proofs - alost all proofs n econoetrcs rel on st two thngs: Saple averages converges to poplaton ean Saple average over sqare root of s norall dstrbted central lt theore

2 Heterosedastct - Var s not constant so E E whch we sed to splf the calclatons to fnd Var ; ore realstc becase we woldnt epect a bg fr to have the sae varaton as a sall fr or bg state verss sall state or rch person verss poor person; snce we cant splf, we have E E Whte Heterosedastct Consstent Covarance Estator - ote :, s not good estate for, bt s OK for ote : stll have asspton 3 no correlaton between and ote 3: ts safer to se the WHCCE; t rato and Wald Test are vald even when sng WHCCE; stll need to chec for hoosedastct before sng F test thogh Hoosedastct - prel statstcal asspton Detectng Heterosedastct - E so the varance s not constant; we dont need to now what s or ts dstrbton to detect that ts not constant Inforal Wa - Rn regresson Save resdals û Sqare the Plot aganst each regressor Loo for patterns Lagrange Mltpler Test - foral wa Do nforal wa and let z be coln vector of regressors that are correlated wth note that z Lnear Fnctonal For - asse z α α z α We want to test H : α,..., α vs. H a : soe α dont care whch one; ths s st the F-test thats reported when we rn a regresson More General - nder H, were actall testng f h where h s an fncton becase nder H, h s a constant α ; sng lnear fnctonal for s fne for detectng heterosedastct Generalzed Least Sqares - we cold se the WHCCE entoned earler to adst for heterosedastct or we cold get fanc; heres the theor: We start wth the basc odel: wth E z z Dvde both sdes b : Ths elnates the heterosedastct and preserves the other assptons Proof: Var z E z E z constant! GLS Estator - Y also called Weghted Least Sqares Estator GLS Loos good z Loos sspect s ncreasng wth of 9

3 In Practce - sonds good, bt we dont now We cold asse s that s, theres soe constant varance n all the error ters and the var b soe scalar ltple of that varance, bt ths s prett rs becase we cold have s, well ae a ore general asspton: z α α z α z Well rn an OLS regresson on z α αz α z and then let be the predctons fro that odel: z α α z α z Then we rn the feasble generalzed least sqares: whch wll be the sae as GLS for large saples Proble - no garantee that > a., z or so we cheat: [ ] whatever nber o decde s sall enogh; there shold onl be a few observatons that ths s an sse for; f there are an, the fnctonal for for a be wrong Correlated Error Ters - we assed E Te Seres - sall get error ters correlated seqental, hence seral correlaton: ρ ρ γ Cross Secton - and - doesnt ean anthng; sall called spatal, networ, or clster correlaton e.g., frs net to each other; fal ebers; grops of frends/slar nterests etwor Model - sppose M grops: G, G,..., G M are sets contanng nde of observatons n each grop; each grop can have a dfferent nber of observatons Stata - G s represented b a sngle varable wth vales:,,,,,, 3, 3, etc. denotng whch clster each observaton belongs to Basc Idea - no correlaton n error ters between grops, bt error ters wthn grop are correlated at a constant rate... E ρ f, G Proble - E UU I ; an dagonal s stll assng no heterosedastct; proble s off dagonal ters; soe are le the shold be; others are ρ E E E, E, E E, V E UU E, E, E Estate - can stll se saple varance: s K 3 of 9

4 Estate ρ - two cases:, G ρ Dfferent for Each Grop - ρ dvded b # pars n G; G need each grop to have a large saple; ths s what Stata ses, G, G, G ρ Sae for Each Grop - ρ Total # pars In both cases, we se and ρ to estate V Choles Decoposton - V ; hard to do n Stata Transfor Data - Y U GLS Estator - V V GLS Y Potental Proble - stll need to be consstent; eaple where ts not: h w h f f α αw f α h h f OLS Te Seres - an was for error ters to be correlated: a ρ b... f and f are correlated, then h f and are correlated ρ ρ ρ ρ ρ34 can be an prevos error ter c Proble - cold lead to RHS regressor correlated wth whch volates dentfcaton condton so s not consstent e.g., f s serall correlated, ts probabl correlated to - Detectng -. spposed s consstent; estate. Rn regresson based on how o thn error ters are correlated: a. ρ... chec t-rato for ρ b. ρ ρ... chec the F-test for all paraeters ontl se F-test for an ltple lag proble le ths Drbn-Watson Statstc - sae as case a above.e., frst order seral correlaton; ver hgh or ver low vales ndcate correlaton present; proble wth ths statstc s that we dont now the dstrbton so we dont now the dea vale Stata - generate lagged varables: generate lag [_n-] Fng - f ρ s are sgnfcant odf varables: a. ρ ρ b. ρ ρ ρ ρ 4 of 9

5 Heterosedastct & Correlated Error Ters - two probles; eact solton depends on tpe of correlaton; asse ρ and E z that that heterosedastct of cases heterosedastct of ; z s coln vector of regressors that are correlated wth see heterosedastct secton; steps:. Regress on and get OLS. Regress on and get ρ and ρ 3. Regress on z and get ρ ρ ρ 4. Modf varables and rern regresson: z z z Mltcollneart - RHS regressors are etreel correlated Pre Mltcollneart - have redndant regressor.e., ts a lnear cobnaton of the other regressors; s not nvertble Stata - wll atoatcall drop the proble regressor and tell o ear Mltcollneart - s nvertble bt have at least one egenvale close to zero shold all be > Proble - wll be nstable cold swtch sgn f we reove a regressor; hpothess tests and nterpretaton of cant be trsted; no probles for forecastng thogh Case - probabl have too an pro varables for the sae thng Detectng - t-ratos are sall so regressors see nsgnfcant; no nqe rle or procedre to detect near ltcollneart As Method - regress each regressor on the other regressors; f R >.95 we shold be concerned; t-rato tells whch repressors are correlated Soltons -. Increase saple sze a st have a bad saple. Consder droppng proble varable... cold lead to probles wth econoc theor; safe thng to do s re-rn regresson and ae sre paraeters of ncorrelated varables dont change... eaple: Asse we want to rn: We rn α α α η and get R.98 and 4,..., are sgnfcant; and 3 arent correlated to Drop and rn, where α ; effect of wll be "pced p" b the new paraeters; for those regressors that werent correlated wth wed epect the paraeter not to change ch: α becase α snce and were not hghl correlated Bas - or dependng on how o loo at t cold be based f α s large; dont now f ts to hgh or too low becase we dont now 5 of 9

6 Measreent Error - ver coon Addtve - st one tpe of error easest to deal wth Tre Model -... note, were leavng off the nde of the observatons to eep the notaton sple.e., not wrtng Rando Error - v and.e., ; we asse the error s not delberate.e., ts rando so we have E v, E, E v and E... actall we can go farther and asse none of the regressors s correlated wth an of the error ters Observed Model - solve the error eqatons for and and sbsttte nto the odel: v v Observed Error - η v E η E v E v... even when we asse E, E v, and E, we have E η E....e., we cold, whch can be re-wrtten: [ ] Proble - [ ] [ ] have regressors correlated wth the error ter so a not be consstent Effect on - Y... s a postve defnte atr so the onl wa s nbased s f, well actall loo at the s cancel; we dvde b so the ter converges as saple sze ncreases [ ] η E η E η η E η E E η E η Case - f there s no easreent error n the regressors.e.,, then s stll consstent.e., easreent error n dependent varable doesn t atter Case - onl a sngle regressor has easreent error e.g.,... all paraeter estates are affected so s not consstent Drecton of Bas - E > so E has opposte sgn of, so all are based toward the orgn Case 3 - two or ore regressors have easreent error; stll have all based, bt cant deterne drecton of bas Mltplcatve - v and Rando Error - E v, E, E v and E ; well also asse the errors are ndependent of ther correspondng varables Observed Model - start wth tre odel: ± to left sdes: 6 of 9

7 ± on rght: Move all ters to end: [ ] Observed Error - η Sb v and : η v Gather ters: η v Proble - E η E E v E E Use ndependence: E η E E E v E E E E Case - f there s no easreent error n the regressors.e.,, then E η E E E v so s stll consstent sae reslt as addtve error... bt now we have heterosedastct Var depends on Var Case - onl a sngle regressor has easreent error e.g.,, then E η E E E assed E Sbsttte : E η E E e e E Dfference fro addtve s so st le before all paraeter estates are > affected and s not consstent based toward the orgn... bt now we have heterosedastct Var depends on Var Mltplcatve Error n ln Model - ln ln ln Addtve Error - v and becoe ln ln ln v and ln ln ln so ltplcatve error n ln odel becoes sae as addtve error elnates heterosedastct proble Otted Varable Bas Tre Model - Observed Model - f we leave ot : [ ] Proble - error ter η cold be correlated wth regressors.e., not consstent: E η E E E... we now E b asspton, bt n general E Best Case - none of the regressors are correlated so ths snt a proble et Best - onl one regressor, sa, s correlated wth ; all paraeter estates are based and the drecton depends on E Solton - cold be ssng becase theres no data; eas f s to fnd a pro varable 7 of 9

8 Pro Varable Pro - z s pro varable for f E,,, z E,,.e., z doesnt contan addtonal nforaton on ; wed rather se f we had t, bt z wll wor Asspton - α α α α z Plg that nto odel: α α α z α Gather ters: α α α α z Proble - cold bas all coeffcents depends on correspondng α Good Pro - want α.e., want to be correlated to z onl and not an of the other regressors; n ths case sgnfcance test on α s "good enogh".e., roghl the sae as test on Mltple Proes - α α α α z α z Model becoes: α α α α z α z ow do ont test on paraeters for z and z to test f s sgnfcant Proble - snce z and z are hopefll hghl correlated to, there probabl correlated to each other so we cold have near ltcollneart Redndant Regressors o effect on.e., stll consstent Probles - lose effcenc, cold have near ltcollneart, ore lel to have regressor correlated to error ter Restrcted Regresson If there are restrctons on the paraeters, we can enforce the n the regresson b:. Rn nrestrcted regresson and copte û. Solve the for as an paraeters as there are restrctons 3. Sbsttte these nto the orgnal odel 4. Collect ters; ters that do not have a paraeter are oved to the left hand sde 5. Rn restrcted regresson and copte / 6. ew F-test: F, 4 / 4 Eaple - Model: Restrctons: Step : solve for : 3 8 of 9

9 Sbsttte that nto : Solve for : 3 Step : solve these nto orgnal odel: Step 3: collect ters: Developng the Restrctons - trans-log cost fncton reqres C KP, KP KC P, P, P ln P ln P r λ ln P λ ln P ln P λ ln P δ ln P δ ln P KP, KP ln K ln P ln K ln P r ln Q λ ln K λ ln K ln P ln P λ ln K λ ln K ln P λ ln K ln P λ ln P ln P λ ln K λ ln K ln P ln P δ δ ln Q ln K δ ln Q ln P δ ln Q ln K δ ln Q ln P lnc KP, KP ln K lnc P, P lnc P lnc δ λ λ Restrcton: Loo at ters that cancel lnc P, P ln P ln P r λ ln P λ ln P ln P λ ln P δ δ ln P δ ln P lnc KP, KP ln K ln P ln K ln P r ln Q λ ln K λ ln K ln P λ ln P λ ln K λ ln K ln P λ ln K ln P λ ln P ln P λ ln K λ ln K P ln ln P δ δ ln Q ln K δ ln Q ln P δ ln Q ln K δ ln Q ln P λ That eans ln K ln K λln K λ ln K ln P λ ln K λ ln K ln P λ ln K ln P λ ln K λ ln K ln P δ ln K δ ln Q ln K ln K Collect ln K ters: ln K ters: λ λ λ ln K ln P ters: λ Collect Collect λ 5 restrctons Collect ln K ln P ters: λ λ Collect ln Q ln K ters: δ δ 9 of 9