A Brief Guide to Recognizing and Coping With Failures of the Classical Regression Assumptions

Transcription

1 A Bef Gude to Recognzng and Copng Wth Falues of the Classcal Regesson Assumptons Model: Y 1 k X 1 X fxed n epeated samples IID 0, I. Specfcaton Poblems A. Unnecessay explanatoy vaables 1. OLS s no longe BLU o effcent but t becomes so fo ; t s stll unbased and consstent and the nfeence machney s all stll vald.. You can ecognze that such an explanatoy vaable s unnecessay usng a t test. 3. But don t dop the ntecept, even f t s nsgnfcant! B. Omtted explanatoy vaables 1. OLS s based and nconsstent (and the nfeence machney s nvald) f you wongly omt an explanatoy vaable whch s coelated (has sample vaaton whch s lnealy elated to) that of any explanatoy vaable ncluded n the model. If the sample vaaton of an omtted explanatoy vaable s not elated to that of any ncluded explanatoy vaable, then ts omsson causes no poblem except, of couse, fo the fact that the model wll not ft the data as well.. If data on the omtted vaable s avalable, ty ncludng t.

2 3. If you estmate a lnea model when the elatonshp s nonlnea, you have n essence omtted an explanatoy vaable. Check fo nonlnea elatonshps by lookng at scattedagams. If, on that o some othe bass, you suspect that the elatonshp may be nonlnea, ty ncludng an (X ) n the model. II. ~ gaussan A. Volaton of ths assumpton mples that s not effcent, but t s stll BLU, unbased, and consstent. The nfeence machney wll be vald only fo lage. B. Testng fo volaton of ths assumpton: 1. Plot hstogam of fttng eos. Does t look gaussan?. Hee s a smple lage-sample test fo gaussan eos. Compute m, m, and m 3 4 the nd, 3d, and 4th sample moments (espectvely) of the fttng eos: m k 1 1 e k k, 3, 4 6 m 3 m 3 4 m 4 m 3 Unde the null hypothess that the eos ae gaussan, the test statstc 1 gven above s dstbuted (), fo lage. C. Copng wth the esults: Thnk about tansfomatons of the dependent vaable (and pehaps the explanatoy vaables as well) whch wll make the gaussanty assumpton moe 1 A. Spanos, Statstcal Foundatons of Econometc Modellng, Cambdge Unvesty Pess, Cambdge, 1981.

3 ealstc. The most commonly-ted choce s the logathm, because so many of the economc vaates whch ae not nomally dstbuted ae lognomally dstbuted. III. E( ) = 0 Ensue ths by ncludng an ntecept n the model specfcaton! IV. Homoskedastcty E( ) = A. Volaton mples: 1. OLS s no longe BLU o effcent (even fo lage ), but t s stll unbased and consstent.. The nfeence machney s nvald. B. Testng fo heteoskedastcty: 1. If you ae wllng to assume that takes on one value (e.g., u ) ove one pat of the sample and anothe value (e.g., H o: u = H A: u ) ove the est, you can test by unnng the egesson sepaately ove each pat of the sample, theeby obtanng two estmates of the model ae satsfed, then : s and s. If all of the othe assumptons of u n u u k s u n u k ndependent of n k s n k

4 so that, unde H, o s u s F n u k, n k allowng you to do the test.. (Smplfed Beusch/Pagan) Altenatvely, f you ae wllng to assume that vaes wth (f at all) because the vaance of s a functon of some vaable Z {whee Z s an obseved vaable whch mght (o mght not) be one the explanatoy vaables n the egesson} you could poceed as follows: a. Apply OLS to the model, obtanng esduals e... e and 1 ˆ 1 1 e b. Calculate ẽ e ˆ fo 1 c. Then apply OLS to the egesson ẽ o 1 Z

5 and compute R and SSE fo ths last equaton. Then, fo lage, 1 (SSE) R 1 R 1 0,1 unde the null hypothess of homoskedastcty. C. Copng wth heteoskedastcty 1. If, fo example, / 9, meely multply all the data fo the pat of the u sample by 3. If you ae usng s u /s to estmate the ato u /, ths wll elmnate the heteoskedastcty poblem s s suffcently lage.. Thnk about tansfomng the data so as to educe (o elmnate) the heteoskedastcty. Fo example, wth tme sees data, you mght consde usng gowth ates nstead of changes, o vce-vesa. O you mght choose to model pe capta spendng (o constant dolla spendng) nstead of actual spendng wth ethe tme sees o coss-sectonal data. The Beusch/Pagan test can be qute nfomatve n ths egad f t eects homoskedastcty n favo of an altenatve hypothess that the vaance of depends on populaton fo obsevaton, but does not eect homoskedastcty when othe easonable choces fo Z ae made, that suggests that pe capta tansfomatons ae called fo.

6 V. on-autocoelaton E( ) = 0 fo {elevant only fo tme sees data} A. Volaton mples: 1. OLS s no longe BLU o effcent (even fo lage ), but t s stll unbased and consstent so long as the explanatoy vaables ae fxed.. The nfeence machney s nvald. Typcally, the nfeental dstotons caused by unteated autocoelaton ae fa moe sevee than those caused by heteoskedastcty. 3. If you model uses lagged values of Y as explanatoy vaables, autocoelaton causes to become based and nconsstent. B Testng fo autocoelaton n the egesson eos The most common test s due to Dubn and Watson, but I wll descbe a smple test whch I pefe. Apply OLS to the followng egesson equaton: e t e t 1 t ols and use the estmated t ato fo to test H o : 0 H A : 0 Ths test s vald (fo lage ) f the autocoelaton s of the fom t t 1 t whee t IID 0, (called AR(1) ) so that = 0 coesponds to nonautocoelaton.

7 C. Copng wth autocoelaton {I am wtng ths out fo the bvaate egesson model; smla esults hold fo the multple egesson model.} 1. Suppose that t t 1 t whee t IID 0, and we somehow knew the value of. Then X t t and 1 X t 1 t 1 mply that 1 (1 ) X t1 X t 1 t t 1 1 (1 ) X t X t 1 t (1 ) X t t * * so that a egesson of on X t wll have the nce eo tem, t. Thus, the OLS estmato of and 1/(1 - ) tmes the OLS estmato of the ntecept n ths equaton would be (nealy) BLU and effcent and the nfeence machney would be vald. ols. Of couse, we don t know but we do have an estmato fo t,, whch s ols consstent fo f the autocoelaton s AR(1). Consequently, f we compute and use t to calculate

8 Ỹ t ˆ ols 1 X t X t ˆ ols X t 1 then and estmated by applyng OLS to Ỹ t X t t * ols (and usng the esultng to estmate and the esultng /(1 - ) to estmate ) wll be apoxmately BLU and effcent fo lage and the nfeence machney wll be vald (fo suffcently lage ) so long as the fom of the autocoelaton n the ognal equaton was n fact AR(1). 3. Altenatvely, note that the equaton n secton V.C.1 maked wth the * also mples that 1 (1 ) X t X t 1 t so that one could poductvely vew the AR(1) autocoelaton n the ognal egesson s eos as the egesson technology s way of tellng us to put some lagged vaables n the model specfcaton. If we do, the eo tem s nonautocoelated!

9 VI. X fxed n epeated samplng A. Volaton mples: 1. OLS s no longe BLU o effcent o unbased (even fo lage ).. OLS s stll consstent (and the nfeence machney s at least vald fo lage ) so long as the explanatoy vaables ae uncoelated wth the eo tem n the egesson equaton. 3. Howeve, any explanatoy vaables whch wee n fact ontly detemned along wth the dependent vaable (as pat of the soluton of a set of smultaneous equatons) wll n fact be coelated wth the eo tem. {E.g., suppose that Y s t the equlbum quantty sold n peod t and X s the equlbum maket pce n t peod t. Then and X t ae ontly detemned by a pa of smultaneous equatons the supply and demand functons.} In such cases, OLS yelds based and nconsstent paamete estmates and (of couse) the nfeence machney s nvald. B. Copng wth smultanety Specalzed econometc methods have been developed fo obtanng consstently estmatng the paametes n systems of smultaneous equatons and fo obtanng nfeences whch ae at least vald n lage samples n such stuatons. The most popula such method s a vaaton on OLS called SLS; futhe dscusson of t s beyond the scope of ths couse, howeve.