Chapter 8 Heteroskedasticity

Transcription

1 Chapter 8 Heteroskedastct I the ultple regresso odel Xβ + ε, t s assued that e, V ( ε) I, Var( ε ), Cov( εε ), j,,, j I ths case, the dagoal eleets of covarace atrx of ε are sae dcatg that the varace of each ε s sae ad off-dagoal eleets of covarace atrx of ε are zero dcatg that all dsturbaces are parwse ucorrelated Ths propert of costac of varace s tered as hooskedastct ad dsturbaces are called as hooskedastc dsturbaces I a stuatos, ths assupto a ot be plausble ad the varaces a ot rea sae The dsturbaces whose varaces are ot costat across the observatos are called heteroskedastc dsturbace ad ths propert s tered as heteroskedastct I ths case ( ε),,,, Var ad dsturbaces are parwse ucorrelated The covarace atrx of dsturbaces s V ( ε) dag(,,, ) Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur

2 Graphcall, the followg pctures depct hooskedastct ad heteroskedastct Hooskedastct Heteroskedastct (Var() creases wth x) Heteroskedastct (Var() decreases wth x) Exaples : Suppose a sple lear regresso odel, x deote the coe ad deotes the expedture o food It s observed that as the coe creases, the varato expedture o food creases because the choce ad varetes food crease, geeral, upto certa extet So the varace of observatos o wll ot rea costat as coe chages The assupto of hooscedastct ples that the cosupto patter of food wll rea sae rrespectve of the coe of the perso Ths a ot geerall be a correct assupto real stuatos Rather the cosupto patter chages ad hece the varace of ad so the varaces of dsturbaces wll ot rea costat I geeral, t wll be creasg as coe creases Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur

3 I aother exaple, suppose a sple lear regresso odel, x deotes the uber of hours of practce for tpg ad deotes the uber of tpg errors per page It s expected that the uber of tpg stakes per page decreases as the perso practces ore The hooskedastc dsturbaces assupto ples that the uber of errors per page wll rea sae rrespectve of the uber of hours of tpg practce whch a ot be true s practce Possble reasos for heteroskedastct: There are varous reasos due to whch the heteroskedastct s troduced the data Soe of the are as follows: The ature of pheoeo uder stud a have a creasg or decreasg tred For exaple, the varato cosupto patter o food creases as coe creases, slarl the uber of tpg stakes decreases as the uber of hours of tpg practce creases Soetes the observatos are the for of averages ad ths troduces the heteroskedastct the odel For exaple, t s easer to collect data o the expedture o clothes for the whole fal rather tha o a partcular fal eber Suppose a sple lear regresso odel j β + β xj + εj,,,,, j,,, j deotes the expedture o cloth for the th j fal havg j ebers ad x j deotes the age of the th perso th j fal It s dffcult to record data for dvdual fal eber but t s easer to get data for the whole fal So j ' s are kow collectvel The stead of per eber expedture, we fd the data o average expedture for each fal eber as j j j j ad the odel becoes + + β β x ε It we assue E ( εj ), Var( εj ), the E( ε ) Var( ε ) j Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 3

4 whch dcates that the resultat varace of dsturbaces does ot rea costat but depeds o the uber of ebers a fal costat ol whe all j ' s are sae j So heteroskedastct eters the data The varace wll rea 3 Soetes the theoretcal cosderatos troduces the heteroskedastct the data For exaple, suppose the sple lear odel β + β x + ε,,,,, deotes the eld of rce ad x deotes the quatt of fertlzer a agrcultural experet It s observed that whe the quatt of fertlzer creases, the eld creases I fact, tall the eld creases whe quatt of fertlzer creases Graduall, the rate of crease slows dow ad f fertlzer s creased further, the crop burs So otce that β chages wth dfferet levels of fertlzer I such cases, whe β chages, a possble wa s to express t as a rado varable wth costat ea β ad costat varace θ lke wth β β + v,,,, E v Var v E v ( ), ( ) θ, ( ε ) So the coplete odel becoes β + β x + ε β β+ v β + βx+ ( ε+ xv ) β + β x + w where w ε + xv s lke a ew rado error copoet So Ew ( ) Var w ( ) E( w ) E ε + xe v + xe εv + x θ + + x θ ( ) ( ) ( ) So varace depeds o ad thus heteroskedastct s troduced the odel Note that we assue hooskedastc dsturbaces for the odel β + β x + ε, β β + v but fall ed up wth heteroskedastc dsturbaces Ths s due to theoretcal cosderatos Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 4

5 4 The skewess the dstrbuto of oe or ore explaator varables the odel also causes heteroskedastct the odel 5 The correct data trasforatos ad correct fuctoal for of the odel ca also gve rse to the heteroskedastct proble Tests for heteroskedastct The presece of heteroskedastct affects the estato ad test of hpothess The heteroskedastct ca eter to the data due to varous reasos The tests for heteroskedastct assue a specfc ature of heteroskedastct Varous tests are avalable lterature for testg the presece of heteroskedastct, eg, Bartlett test Breusch Paga test 3 Goldfeld Quadt test 4 Glesjer test 5 Test based o Speara s rak correlato coeffcet 6 Whte test 7 Rase test 8 Harve Phllps test 9 Szroeter test Peak test (oparaetrc) test We dscuss the frst fve tests Bartlett s test It s a test for testg the ull hpothess H : Ths hpothess s tered as the hpothess of hooskedastct Ths test ca be used ol whe replcated data s avalable Sce the odel β X + β X + + β X + ε E ε Var ε k k, ( ), ( ),,,,, ol oe observato s avalable to fd overcoe f replcated data s avalable So cosder the odel of the for X β + ε * *, so the usual tests ca ot be appled Ths proble ca be Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 5

6 where * s a vector, X s k atrx, β s k vector ad ε s vector So replcated data * s ow avalable for ever X β + ε cossts of * * X β + ε cossts of * * X β + ε cossts of * * * the followg wa: observatos observatos observatos All the dvdual odel ca be wrtte as * * X ε * * X ε β + * * X ε or * Xβ + ε* where * s a vector of order, X s k vector Applcato of OLS to ths odel elds ˆ ( ' ) β X X X ' * ad obta the resdual vector e X β * * ˆ Based o ths, obta s s * * k e ' e ( ks ) ( k) Now appl the Bartlett s test as atrx, β s k vector ad ε * s s χ ( k) log C s whch has asptotc χ dstrbuto wth ( ) degrees of freedo where C + 3( ) k ( k) Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 6

7 Aother varat of Bartlett s test Aother varat of Bartlett s test s based o lkelhood rato test statstc If there are depedet oral rado saples where there are observatos the th saple The the lkelhood rato test statstc for testg where H s s u s / s j s ( ) j,,,, ;,,, j s To obta a ubased test ad a odfcato of - l u whch s a closer approxato to χ uder H, Bartlett test replaces b ( ) ad dvde b a scalar costat Ths leads to the statstc M ˆ ( ) log ˆ ( ) log + 3( ) whch has a χ dstrbuto wth ( ) degrees of freedo uder H ad ˆ ( ) ˆ j j ˆ ( ) I experetal sceces, t s easer to get replcated data ad ths test ca be easl appled I real lfe applcatos, t s dffcult to get replcated data ad ths test a ot be appled Ths dffcult s overcoe Breusch Paga test Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 7

8 Breusch Paga test Ths test ca be appled whe the replcated data s ot avalable but ol sgle observatos are avalable Whe t s suspected that the varace s soe fucto (but ot ecessarl ultplcatve) of ore tha oe explaator varable, the Breusch Paga test ca be used Assug that uder the alteratve hpothess s expressble as hz ( ˆ γ) h( γ + Zγ ) ' * * where h s soe uspecfed fucto ad s depedet of, Z (, Z )(, Z, Z,, Z ) ' *' 3 p s the vector of observable explaator varables wth frst eleet ut ad γ ( γ, γ ) ( γ, γ,, γ ) s a * p vector of ukow coeffcets related to β wth frst eleet beg the tercept ter The heterogect s defed b these p varables These Z ' s a also clude soe X ' s also Specfcall, assue that γ+ γz + + γ pzp The ull hpothess H : ca be expressed as : H γ γ3 γ p If H s accepted, t ples that Z, Z3,, Z p do ot have a effect o The test procedure s as follows: Igorg heteroskedastct, appl OLS to β β X β X ε k k + ad obta resdual e Xb b X X XY ( ' ) ' ad we get γ Costruct the varables g e e SS e res Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 8

9 where SS s the resdual su of squares based o e ' s res 3 Ru regresso of g o Z, Z,, Z p ad get resdual su of squares 4 For testg, calculate the test statstc Q g SS * res whch s asptotcall dstrbuted as * SS res χ dstrbuto wth ( p ) degrees of freedo 5 The decso rule s to reject H f Q> χ α ( ) Ths test s ver sple to perfor A farl geeral for s assued for heteroskedastct, so t s a ver geeral test Ths s a asptotc test Ths test s qute powerful the presece of heteroskedastct 3 Goldfeld Quadt test Ths test s based o the assupto that s postvel related to X, j e, oe of the explaator varable explas the heteroskedastct the odel Let th j explaator varable explas the heteroskedastct, so X j or X j The test procedure s as follows: Rak the observatos accordg to the decreasg order of X j Splt the observatos to two equal parts leavg c observatos the ddle c c So each part cotas observatos provded > k 3 Ru two separate regresso the two parts usg OLS ad obta the resdual su of squares SS res ad SS res SSres 4 The test statstc s F SS res c c whch follows a F dstrbuto, e, F k, k whe H true Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 9

10 c c 5 The decso rule s to reject H wheever F > F α k, k Ths test s sple test but t s based o the assupto that oe of the explaator varable helps s deterg the heteroskedastct Ths test s a exact fte saple test Oe dffcult ths test s that the choce of c s ot obvous If large value of c s chose, the t c c reduces the degrees of freedo k ad the codto > k a be volated O the other had, f saller value of c s chose, the the test a fal to reveal the heteroskedastct The basc objectve of orderg of observatos ad deleto of observatos the ddle part a ot reveal the heteroskedastct effect Sce the frst ad last values of gve the axu dscreto, so deleto of saller value a ot gve the proper dea of heteroskedastct Keepg those two pots vew, the workg choce of c s suggested as Moreover, the choce of c 3 X j s also dffcult Sce Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur, so f all portat varables are cluded the odel, the t a be dffcult to decde that whch of the varable s fluecg the heteroskedastct 4 Glesjer test: Ths test s based o the assupto that s flueced b oe varable Z, e, there s ol oe varable whch s fluecg the heteroskedastct Ths varable could be ether oe of the explaator varable or t ca be chose fro soe extraeous sources also The test procedure s as follows: Use OLS ad obta the resdual vector e o the bass of avalable stud ad explaator varables Choose Z ad appl OLS to e δ + δ Z + v h where v s the assocated dsturbace ter 3 Test H : δ usg t -rato test statstc 4 Coduct the test for h ±, ± So the test procedure s repeated four tes I practce, oe ca choose a value of h For splct, we choose h The test has ol asptotc justfcato ad the four choces of h gve geerall satsfactor results Ths test sheds lght o the ature of heteroskedastct X j

11 5 Speara s rak correlato test It d deotes the dfferece the raks assged to two dfferet characterstcs of the th object or pheoeo ad s the uber of objects or pheoeo raked, the the Speara s rak correlato coeffcet s defed as d r 6 ; r ( ) Ths ca be used for testg the hpothess about the heteroskedastct Cosder the odel β + β X + ε Ru the regresso of o X ad obta the resduals e Cosder e 3 Rak both e ad X (or ˆ ) a ascedg (or descedg) order 4 Copute rak correlato coeffcet r based o e ad X (or ˆ ) 5 Assug that the populato rak correlato coeffcet s zero ad > 8, use the test statstc t r r whch follows a t -dstrbuto wth ( ) degrees of freedo 6 The decso rule s to reject the ull hpothess of heteroskedastct wheever t t ( ) If there are ore tha oe explaator varables, the rak correlato coeffcet ca be coputed betwee e ad each of the explaator varables separatel ad ca be tested usg t α Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur

12 Estato uder heteroskedastct Cosder the odel Xβ + ε wth k explaator varables ad assue that E( ε ) V ( ε ) Ω The OLSE s b X X X ( ' ) ' Its estato error s ad b β ( X ' X) X ' ε Eb X X X E ( β) ( ' ) ' ( ε) Thus OLSE reas ubased eve uder heteroskedastct The covarace atrx of b s Vb ( ) Eb ( β)( b β)' ( X ' X) X ' E( εε ') X( X ' X) ( X ' X) X ' ΩX( X ' X) whch s ot the sae as covetoal expresso So OLSE s ot effcet uder heteroskedastct as copared uder hookedastct Now we check f The resdual vector s e Xb Hε e Ee ( ) or ot where e s the th resdual [,,,,,,] ' Hεε ' H e e e Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur

13 where s a vector wth all eleets zero except the th eleet whch s ut ad H I X( X ' X) X ' The e '' ee ' Hεε ' H E e HE H H H ( ) ' ( εε ') ' Ω h h h h H h h h h h Ee ( ) h, h,, h h Thus Ee ( ) ad so e becoes a based estator of the presece of heteroskedastct I the presece of heteroskedastct, use the geeralzed least squares estato The geeralzed least squares estator (GLSE) of β s ˆ ( X ' X) X ' β Ω Ω Its estato error s obtaed as ˆ ( X ' Ω X) X ' Ω ( X + ) ˆ ( X ' X) X ' ) β β ε β β Ω Ω ε Thus E ˆ β β X Ω X X Ω E ε V( ˆ β) E( ˆ β β)( ˆ β β) ( ) ( ' ) ' ( ) ( X ' Ω X) X ' Ω E( εε ) Ω X( X ' Ω X) ( X ' Ω X) X ' Ω ΩΩ X( X ' Ω X) X Ω X ( ' ) Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 3

14 Exaple: Cosder a sple lear regresso odel β + β x + ε,,,, The varaces of OLSE ad GLSE of β are Var( b) Var( ˆ β ) ( x x) ( x x) ( x x) ( x x) ( x x) x x Square of the correlato coeffcet betweee ( x x) ad Var( ˆ β ) Var( b) So effcet of OLSE ad GLSE depeds upo the correlato coeffcet betwee ( x x) ad ( x ) x The geeralzed least squares estato assues that Ω s kow, e, the ature of heteroskedastct s copletel specfed Based o ths assupto, the possbltes of followg two cases arse: Ω s copletel specfed or Ω s ot copletel specfed We cosder both the cases as follows: Case : ' s are pre-specfed: Suppose,,, are copletel kow the odel β + β X + + β X + ε k k Now deflate the odel b, e, Let * X X ε β + β + + β + k k ε * *, E ε Var ε ε the drawg statstcal fereces ca be used ( ), ( ) Now OLS ca be appled to ths odel ad usual tools for Note that whe the odel s deflated, the tercept ter s lost as β / s tself a varable Ths pot has to be take care a software output Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 4

15 Case : Ω a ot be copletel specfed Let,,, are partall kow ad suppose X λ j or but X λ j s ot avalable Cosder the odel β β X β X ε k k + ad deflate t b X λ j as β X X ε X X X X X k β β λ λ λ k λ λ j j j j j Now appl OLS to ths trasfored odel ad use the usual statstcal tools for drawg fereces A cauto s to be kept s d whle dog so Ths s llustrated the followg exaple wth oe explaator varable odel Cosder the odel Deflate t b β + β x + ε x, so we get β ε x x x + β + Note that the roles of β ad β orgal ad deflated odels are terchaged I orgal odel, β s tercept ter ad β s slope paraeter whereas deflated odel, β becoes the tercept ter ad β becoes the slope paraeter So essetall, oe ca use OLS but eed to be careful detfg the tercept ter ad slope paraeter, partcularl the software output Ecooetrcs Chapter 8 Heteroskedastct Shalabh, IIT Kapur 5