The Classical Model. Heteroskedasticity and Correlations Across Errors

Transcription

1 The Classcal Model Heteroskedastcty ad Correlatos Across Errors

2 Heteroskedastcty Recall Assumpto 5 of the CLRM: that all errors have the same varace. That s, Var(ε ) = σ for all = 1,,, Heteroskedastcty s a volato of ths assumpto. It occurs f dfferet observatos errors have dfferet varaces. For example, Var(ε ) = σ I ths case, we say the errors are heteroskedastc. Because heteroskedastcty volates a assumpto of the CLRM, we kow that least squares s ot BLUE whe the errors are heteroskedastc. Heteroskedastcty occurs most ofte cross-sectoal data. These are data where observatos are all for the same tme perod (e.g., a partcular moth, day, or year) but are from dfferet ettes (e.g., people, frms, provces, coutres, etc.)

3 Pure Heteroskedastcty There are two basc types of heteroskedastcty (pure & mpure) Pure Heteroskedastcty arses f the model s correctly specfed, but the errors are heteroskedastc, e.g., the DGP s: Y = β 0 + β 1 X 1 + ε where Var(ε ) = σ There are may ways to specfy the heteroskedastc varace σ. A very smple specfcato s dscrete heteroskedastcty, where the errors are draw from oe of two dstrbutos, a wde dstrbuto (wth Large varace σ L ) or a arrow dstrbuto (wth Small varace σ S ) draw a pcture. A commo specfcato s to assume that the error varace s proportoal to a the square of varable Z (that may or may ot be oe of the depedet varables). I ths case, Var(ε ) = σ = σ Z ad each observato s error s draw from ts ow dstrbuto wth mea zero ad varace σ Z. (draw some pctures) A example: suppose we re modelg household expedtures o lesure actvtes (moves, skg, vacatos, etc.). At low levels of household come, there wll be less household-to-household varato lesure spedg ( $ terms) tha at hgh levels of household come. That s, the error varace wll be proportoal to household come (Z). Ths s because poor people have less room ther budget for such varace.

4 Ieffcecy Why s OLS effcet whe we have pure heteroskedastcty? It s because there s aother lear estmator that uses the data better, ad ca delver a lowervarace estmated coeffcet Eg, what f some observatos had zero-varace o ther errors, but others had postve varace A lear estmator that delvers a lower-varace coeffcet s to ru OLS o olythose observatos wth zero-varace. Trash all the rest of the data

5 Impure Heteroskedastcty mpure heteroskedastcty ca arse f the model s ms-specfed (e.g., due to a omtted varable) ad the specfcato error duces heteroskedastcty. For example, suppose the DGP s Y = β 0 + β 1 X 1 + β X + ε but we estmate Y = β 0 + β 1 X 1 + ε * where ε * = β X + ε ad where ε s a classcal error term. The f X tself has a heteroskedastc compoet (e.g., the value of X comes from ether a wde or arrow dstrbuto), the omttg t from the model makes the ms-specfed error term ε * behave heteroskedastcally. Of course the soluto here s smple: do t omt X from the model!

6 Cosequeces of Heteroskedastcty We kow that heteroskedastcty volates Assumpto 5 of the CLRM, ad hece OLS s ot BLUE. What more ca we say? OLS estmates rema ubased We oly eed Assumptos 1-3 to show that the OLS estmator s ubased, hece a volato of Assumpto 5 has o effect o ths property Varace of the OLS estmator s flated Cosder the case whe the varace of the error rses wth X: the varace of the estmated slope coeffcet s bgger (draw pcture). Stadard formulae for stadard errors of OLS estmates are wrog. They do t take accout of the extra samplg varato (# above)

7 ( X X ) OLS Varace s Wrog ( ) ( ) Let Y X ad E X X ad let be heteroskedastc E = α + β + ε, ε = 0 ε : ε = σ ε ˆ = 1 β = β + ( X X ) = 1 E ˆ β = β ( X X ) ε ˆ ˆ ˆ = 1 1 V β = E ( β E β ) = E = E ( X X ) ε ( X X ) ( X X = 1 ) = 1 = 1 1 = E ( X1 X )( X1 X ) ε1ε 1 + ( X1 X )( X X ) ε1ε ( X 1 X )( X X ) ε 1ε + ( X X )( X X ) εε ( X X ) = 1 ( X ) ( ) ( ) X E ε X X 1 = 1 = 1 E ( X X ) ( ) ε = 1 ( X X ) ( X X ) ( X X ) = = = = 1 = 1 = 1 σ

8 Testg for Heteroskedastcty There are may formal tests avalable I addto to formal tests always look at resdual plots!! Look for specfcato errors too, sce apparet heteroskedastcty may just be due to a omtted varable, for stace. Of the may formal tests avalable, the most useful s the Whte Test. It s qute geeral, ad desged to test for heteroskedastcty of a ukow form (e.g., whe we do t kow that the error varace s proportoal to some Z) The Whte test s qute commo, ad you ca do t EVews wth a couple of clcks (we ll see ths a momet). The ext slde dscusses how the test works

9 Three Steps of the Whte test 1. Estmate the regresso model of terest (call ths equato 1) & collect the resduals, e. Suppose equato 1 s: Y = β 0 + β 1 X 1 + β X + β 3 X 3 + ε. Square the resduals. Regress e o all the depedet varables from equato 1, ther squares, ad cross products (call ths equato ). So our equato s: e = α + α X + α X + α X + α X 1 + α X + α X α 7 X 1 X + α 8 X 1 X 3 +α 9 X X 3 + u 3. Test the overall sgfcace of equato. The test statstc s R, where s the sample sze ad R s the proporto of varato explaed equato. Uder the ull of o heteroskedastcty, ths test statstc has a Ch-square(k * ) dstrbuto asymptotcally, where k * s the umber of slope coeffcets equato. Crtcal values of the Ch-square dstrbuto are the text (table B-8). f the test statstc exceeds the crtcal value, reject the ull. I Evews, you frst ru the regresso, the, uder Vew, select Resdual Dagostcs, select Heteroskedastcty Tests, select Whte

10 What to do f errors are heteroskedastc If you fd evdece of heteroskedastcty whether through a formal test by lookg at resdual plots you have several optos 1. Use OLS to estmate the regresso ad fx the stadard errors A. We kow OLS s ubased, t s just that the usual formula for the stadard errors s wrog (ad hece tests ca be msleadg) B. We ca get cosstet estmates of the stadard errors (as the sample sze goes to fty, a cosstet estmator gets arbtrarly close to the true value a probablstc sese) called Whte s Heteroskedastcty- Cosstet stadard errors C. Whe specfyg the regresso EVews, clck the OPTIONS tab, check the Coeffcet Covarace Matrx box, ad the Whte butto D. Most of the tme, ths approach s suffcet. Try Weghted Least Squares (WLS) f you kow the source of the heteroskedastcty ad wat a more effcet estmator 3. Try re-defg the varables aga, f you thk you uderstad the source of the problem (takg log of depedet varable ofte helps)

11 The Park Test If we suspect we kow the source of the heteroskedastcty, e.g., f we suspect Var(ε ) = σ = σ Z the we ca do better. There are three steps: 1. Estmate the equato of terest: Y = β 0 + β 1 X 1 + β X + ε ad collect the resduals e. Estmate the auxlary regresso: l(e ) = α 0 + α 1 lz + u where the error term u satsfes the assumptos of the CLRM. (why logs?). 3. Test the statstcal sgfcace of lz the auxlary regresso wth a t-test,.e., test the ull H 0 : α 1 = 0, H 1 : α 1 0 Problem wth ths approach: we eed to kow Z!

12 Weghted Least Squares We kow that OLS s ot BLUE whe errors are heteroskedastc Suppose we wat to estmate the regresso: Y = β 0 + β 1 X 1 + β X + ε ad we kow (or suspect) that Var(ε ) = σ = σ Z. We could rewrte the model as: We could rewrte the model as: Y = β 0 + β 1 X 1 + β X + Z u where Var(u ) = σ. The BLUE of ths model s called Weghted Least Squares (WLS). It s just least squares o the trasformed model: Y /Z = β 0 /Z + β 1 X 1 /Z + β X /Z + u where we dvde everythg through by Z. Notce that the trasformed model has a homoskedastc error, ad hece OLS s BLUE the trasformed model. You ca do all ths EVews usg the Weght opto (but ote EVews defes the weght as 1/Z )

13 Redefg Varables Sometmes the best alteratve s to go back to the drawg board ad redefe the varables a way that s cosstet wth ecoomc theory & commo sese ad that makes the errors homoskedastc. Usg a logarthmc depedet varable may help homoskedastcty the sem-log model meas the error varace s a costat proporto of the depedet varable. Other trasformatos may help, too, e.g., deflatg by a scale varable

14 Redefg Varables A Example Aother example: suppose we re estmatg the regresso: EXP = β 0 + β 1 GDP + β POP + β 3 CIG + ε where EXP s medcal expedture provce GDP s GDP provce POP s populato provce CIG s the umber of cgarettes sold provce The large provces (Quebec ad Otaro) wll have much larger error varace tha small provces (e.g., PEI), just because the scale of ther medcal expedtures (ad everythg else) wll be so much bgger. We could take logs, but eve better, estmate: EXP /POP = α 0 + α 1 GDP /POP + α 3 CIG /POP + u whch puts expedtures, GDP, ad cgarette sales per-capta terms. Ths way, each provce cotrbutes about the same amout of formato, ad should stablze the error varaces too.

15 Seral Correlato Seral correlato occurs whe oe observato s error term (ε ) s correlated wth aother observato s error term (ε j ): Corr(ε, ε j ) 0 We say the errors are serally correlated Ths usually happes because there s a mportat relatoshp (ecoomc or otherwse) betwee the observatos. Examples: Tme seres data (whe observatos are measuremets of the same varables at dfferet pots tme) Cluster samplg (whe observatos are measuremets of the same varables o related subjects, e.g., more tha oe member of the same famly, more tha oe frm operatg the same market, etc.) Example: Suppose you are modelg calore cosumpto wth data o a radom sample of famles, oe observato for each famly member. Because famles eat together, radom shocks to calore cosumpto (.e., errors) are lkely to be correlated wth famles. Seral correlato volates Assumpto 4 of the CLRM. So we kow that least squares s ot BLUE whe errors are serally correlated.

16 Pure Seral Correlato There are two basc types of seral correlato (pure & mpure) Pure Seral Correlato arses f the model s correctly specfed, but the errors are serally correlated, e.g., the DGP s: Y t = β 0 + β 1 X 1t + ε t where ε t = ρε t-1 + u t ad u t s a classcal error term (.e., t satsfes the assumptos of the CLRM) Note: we used the subscrpt t (stead of ) to deote the observato umber. Ths s stadard for models of tme seres data (t refers to a tme perod), whch s where seral correlato arses most frequetly. Note also: ths kd of seral correlato s called frst-order autocorrelato (or frst-order autoregresso, or AR(1) for short), ad ρ s called the autocorrelato coeffcet ths kd of seral correlato s very commo tme-seres settgs

17 Impure Seral Correlato Impure seral correlato arses f the model s ms-specfed (e.g., due to a omtted varable) ad the specfcato error duces seral correlato. For example, suppose the DGP s Y t = β 0 + β 1 X 1t + β X t + ε t but we estmate Y t = β 0 + β 1 X 1t + ε * t where ε * t = β X t + ε t ad suppose X t = γx t-1 + u t ad where ε t ad u t are classcal error terms. Because of the specfcato error (omttg X from the model), the error term the ms-specfed model s: ε * t = β X t + ε t = β (γx t-1 + u t ) + ε t = γε * t-1+ β u t + ε t -γε t-1 ad s therefore correlated wth the error term of observato t-1 (that s, ε * t-1) Ths omtted varables problem does ot cause bas because the omtted varable s ot correlated wth the cluded regressor. But, t does cause effcecy because the omtted varable s correlated wth tself over tme.

18 Some examples We saw the example of frst-order autocorrelato already: ε t = ρε t-1 + u t ths requres -1 < ρ < 1 (why? what f ρ = 0?) ρ < 0 s a example of egatve seral correlato, where ε t ad ε t-1 ted to have opposte sgs. Ths case s dffcult to terpret & pretty rare ecoomc data. (Draw some pctures) ρ > 0 s a example of postve seral correlato. I ths case, ε t adε t-1 ted to have the same sg. Ths s very commo ecoomc data (draw some pctures). Happes frequetly tme seres data f macroecoomc shocks take tme to work ther way through the ecoomy Example: modelg the prce of orages. Orages ca oly be grow warm clmates. They are trasported by truck before beg sold to cosumers. Thus ther prce s flueced by the prce of gasole (ad hece of ol). A uexpected shock to the supply of ol (say, due to the vaso of a ol producg coutry ) leads to a crease the prce of ol that may last several years, ad shows up as a seres of postve shocks to the prce of orages. We ca also have hgher order autocorrelato, e.g., ε t = ρ 1 ε t-1 + ρ ε t- + u t (secod-order autocorrelato) Autocorrelato eed ot be betwee adjacet perods, e.g., wth quarterly data we mght have ε t = ρε t-4 + u t (seasoal autocorrelato, where today s error s correlated wth the error 1 year ago, say due to seasoal demad)

19 Cosequeces of Seral Correlato We kow that seral correlato volates Assumpto 4 of the CLRM, ad hece OLS s ot BLUE. What more ca we say? 1. OLS estmates rema ubased We oly eed Assumptos 1-3 to show that the OLS estmator s ubased, hece a volato of Assumpto 4 has o effect o ths property. The OLS estmator s o loger the best (mmum varace) lear ubased estmator Seral correlato mples that errors are partly predctable. For example, wth postve seral correlato, the a postve error today mples tomorrow s error s lkely to be postve also. The OLS estmator gores ths formato; more effcet estmators are avalable that do ot. 3. Stadard formulae for stadard errors of OLS estmates are wrog. Stadard formulae for OLS stadard errors assume that errors are ot serally correlated have a look at how we derved these lecture 1 (we eeded to use assumpto 4 of the CLRM). Sce our t-test statstc depeds o these stadard errors, we should be careful about dog t-tests the presece of seral correlato.

20 Testg for Seral Correlato There are a umber of formal tests avalable I addto to formal tests always look at resdual plots!! The most commo test s the Durb-Watso (DW) d Test Ths s a test for frst-order autocorrelato oly Some caveats: the model eeds to have a tercept term, ad ca t have a lagged depedet varable (.e., Y t-1 as oe of the depedet varables) If we wrte the error term as ε t = ρε t-1 + u t DW tests the ull hypothess H 0 : ρ 0 (o postve autocorrelato) or H 0 : ρ = 0 (o autocorrelato) agast the approprate alteratve. Ths test s so commo that almost every software package automatcally calculates the value of the d statstc wheever you estmate a regresso but they almost ever report p-values for reasos we ll see a momet

21 The Durb-Watso dtest The test statstc s based o the least squares resduals e 1, e,, e T (where T s the sample sze). The test statstc s: T ( e ) t= t e t 1 d = T Cosder the extremes: e t= 1 t f ρ = 1, the we expect e t = e t-1. Whe ths s true, d 0. f ρ = -1, the we expect e t = - e t-1 ad e t - e t-1 = -e t-1. Whe ths s true, d 4. f ρ = 0, the we expect d. Hece values of the test statstc far from dcate that seral correlato s lkely preset. Ufortuately, the dstrbuto theory for d s a lttle fuy for some values of d, the test s coclusve For a gve sgfcace level, there are two crtcal values: 0 < d L < d U < For a oe-sded test, H 0 : ρ 0, H 1 : ρ > 0 Reject H 0 f d < d L Do ot reject H 0 f d > d U Test s coclusve f d L d d U (see the text for decso rules for a two-sded test) Do a example ad show how to plot the resduals

22 What to do f errors are serally correlated If you fd evdece of seral correlato whether through a formal test or just by lookg at resdual plots you have several optos avalable to you 1. Use OLS to estmate the regresso ad fx the stadard errors A. We kow OLS s ubased, t s just that the usual formula for the stadard errors s wrog (ad hece tests ca be msleadg) B. We ca get cosstet estmates of the stadard errors (as the sample sze goes to fty, a cosstet estmator gets arbtrarly close to the true value a probablstc sese) called Newey-West stadard errors C. Whe specfyg the regresso EVews, clck the OPTIONS tab, check the coeffcet covarace matrx box, ad the HAC Newey- West butto D. Most of the tme, ths approach s suffcet. Try Geeralzed Least Squares (GLS) f you wat a more effcet estmator

23 Geeralzed Least Squares We kow that OLS s ot BLUE whe errors are serally correlated Rather, the BLUE s a geeralzato of OLS called Geeralzed Least Squares (GLS) Suppose we wat to estmate the regresso: Y t = β 0 + β 1 X 1t + ε t but we suspect that ε t s serally correlated. I partcular, suppose we thk (say, based o a DW test) that ε t = ρε t-1 + u t The we could wrte the model as: Y t = β 0 + β 1 X 1t + ρε t-1 + u t GLS s a method of estmatg β 0,β 1 ad ρ (ad t s the BLUE of β 0 β 1 ) There are several dfferet ways of calculatg the GLS estmator (the text dscusses two) -- the mechacs are beyod the scope of ths course (we have computers for that!) The smplest: EVews, just add AR(1) as a depedet varable! (or AR(4) for the seasoal model we saw before, or AR(1) AR() for the secod-order autoregressve model, etc.).

24 No-Sphercal Errors If errors are ucorrelated across observatos ad have detcal varaces, we say they are sphercal. If errors are correlated across observatos or have varaces that dffer across observatos, we say that they are o- sphercal. Whe you have osphercalerrors, OLS gves the wrog varaces, but you ca get correct oes: Whte or HAC- Newey-West (or clustered) Whe you have osphercalerrors, OLS s ot effcet. Its varace s ot the smallest possble amog lear estmators. But, WLS s effcet f you kow the form of the heteroskedastcty, ad GLS s effcet f you kow the form of the correlato across observatos.