ECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity

Transcription

1 ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur

2 . Breusch Paga test Ths test ca be appled whe the replcated data s ot avalable but oly sgle observatos are avalable. Whe t s suspected that the varace s some fucto (but ot ecessarly multplcatve) of more tha oe explaatory varable, the Breusch Paga test ca be used. Assumg that uder the alteratve hypothess s expressble as ' * * ˆ = hz ( γ) = h( γ + Zγ ) where h s some uspecfed fucto ad s depedet of, ' *' Z = (, Z ) = (, Z, Z,..., Z ) 3 p s the vector of observable explaatory varables wth frst elemet uty ad ukow coeffcets related to varables. These Z s may also clude some X s also. β γ = ( γ, γ ) = ( γ, γ,..., γ ) * s a vector of wth frst elemet beg the tercept term. The heterogeety s defed by these p p Specfcally, assume that = γ+ γz γ pzp. The ull hypothess H : = =... = 0 ca be expressed as H0 γ γ3 γ p : = =... = = 0. If H 0 s accepted, t mples that Z, Z,..., Z do ot have ay effect o ad we get = γ. 3 p

3 The test procedure s as follows:. Igorg heteroskedastcty, apply OLS to y = β + β X + + β X + ε... k k ad obta resdual e = y Xb 3 b= X X X y ( ' ) '.. Costruct the varables g e = = e = e SS res where SS res s the resdual sum of squares based o e s. 3. Ru regresso of g o ad get resdual sum of squares * Z, Z,..., Zp SS res. 4. For testg, calculate the test statstc Q = g SS = * res whch s asymptotcally dstrbuted as χ 5. The decso rule s to reject H 0 f Q> χ α ( m ). dstrbuto wth (p - ) degrees of freedom. Ths test s very smple to perform. A farly geeral form s assumed for heteroskedastcty, so t s a very geeral test. Ths s a asymptotc test. Ths test s qute powerful the presece of heteroskedastcty.

4 4 3. Goldfeld Quadt test Ths test s based o the assumpto that s postvely related to X j,.e., oe of the explaatory varable explas the heteroskedastcty the model. Let j th explaatory varable explas the heteroskedastcty, 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 mddle. c c So each part cotas observatos provded > k. 3. Ru two separate regresso the two parts usg OLS ad obta the resdual sum of squares SS ad. 4. The test statstc s SSres F0 = SSres whch follows a F - dstrbuto,.e., c, c F k k whe H 0 true. c c 5. The decso rule s to reject H 0 wheever F0 > F α k, k. res SS res

5 5 Ths test s smple test but t s based o the assumpto that oly oe of the explaatory varable helps s determg the heteroskedastcty. Ths test s a exact fte sample test. Oe dffculty ths test s that the choce of c s ot obvous. If large value of c s chose, the t reduces the c c degrees of freedom k ad the codto > k may be volated. O the other had, f smaller value of c s chose, the the test may fal to reveal the heteroskedastcty. The basc objectve of orderg of observatos ad deleto of observatos the mddle part may ot reveal the heteroskedastcty effect. Sce the frst ad last values of gve the maxmum dscreto, so deleto of smaller value may ot gve the proper dea of heteroskedastcty. Keepg those two pots vew, the workg choce of c s suggested as c =. 3 X j Moreover, the choce of s also dffcult. Sce X, so f all mportat varables are cluded the model, the t may be dffcult to decde that whch of the varable s fluecg the heteroskedastcty. j

6 6 4. Glesjer test Ths test s based o the assumpto that s flueced by oe varable Z,.e., there s oly oe varable whch s fluecg the heteroskedastcty. Ths varable could be ether oe of the explaatory varable or t ca be chose from some extraeous sources also. The test procedure s as follows:. Use OLS ad obta the resdual vector e o the bass of avalable study ad explaatory varables.. Choose Z ad apply OLS to e = δ + δ Z + v h 0 where v s the assocated dsturbace term. 3. Test H0 : δ = 0 usg t-rato test statstc. 4. Coduct the test for h =±, ±. So the test procedure s repeated four tmes. I practce, oe ca choose ay value of h. For smplcty, we choose h =. The test has oly asymptotc justfcato ad the four choces of h gve geerally satsfactory results. Ths test sheds lght o the ature of heteroskedastcty.

7 7 5. Spearma s rak correlato test It d deotes the dfferece the raks assged to two dfferet characterstcs of the th object or pheomeo ad s the umber of objects or pheomeo raked, the the Spearma s rak correlato coeffcet s defed as d = r = 6 ; r. ( ) Ths ca be used for testg the hypothess about the heteroskedastcty. Cosder the model y = β + β X + ε 0.. Ru the regresso of y o X ad obta the resduals e.. Cosder e. 3. Rak both e ad X ( or yˆ ) a ascedg (or descedg) order. 4. Compute rak correlato coeffcet r based o e ad X ( or yˆ ). 5. Assumg that the populato rak correlato coeffcet s zero ad > 8, use the test statstc t 0 r = r whch follows a t-dstrbuto wth ( - ) degrees of freedom. 6. The decso rule s to reject the ull hypothess of heteroskedastcty wheever t0 t α ( ). 7. If there are more tha oe explaatory varables, the rak correlato coeffcet ca be computed betwee e ad each of the explaatory varables separately ad ca be tested usg t 0.

8 Estmato uder heteroskedastcty Cosder the model y = Xβ + ε 8 wth k explaatory varables ad assume that E( ε ) = V ( ε ) =Ω=. 0 0 The OLSE s b= X X X y ( ' ) '. Its estmato error s ad b β = ( X ' X) X ' ε Eb X X X E ( β) = ( ' ) ' ( ε) = 0. Thus OLSE remas ubased eve uder heteroskedastcty. The covarace matrx 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 same as covetoal expresso. So OLSE s ot effcet uder heteroskedastcty as compared uder homoskedastcty

9 Now we check f Ee ( ) = The resdual vector s e = y Xb = Hε e = = or ot where e s the th resdual. [ 0,0,...,0,,0,...0] ' Hεε ' H e e e where s a x vector wth all elemets zero except the th elemet whch s uty ad H = I X( X ' X) X '. The 9 e = '.' ee = ' Hεε ' H ( ) = ' ( εε ') = ' Ω E e HE H H H 0 0 h h h h H = = h h 0 h h 0 0 Ee ( ) = h, h,..., h h. 0 0 h

10 0 Thus Ee ( ) ad so becomes a based estmator of the presece of heteroskedastcty. I the presece of heteroskedastcty, use the geeralzed least squares estmato. The geeralzed least squares estmator (GLSE) of β s e ˆ ( X ' X) X ' y. β = Ω Ω Its estmato error s obtaed as ˆ = ( X ' Ω X) X ' Ω ( X + ) β β ε ˆ = ( X ' Ω X) X ' Ω. β β ε Thus E ˆ β β = X Ω X X Ω E ε = ( ) ( ' ) ' ( ) 0 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 ( ' ).

11 Example Cosder a smple lear regresso model y = β0 + β x + ε, =,,...,. The varaces of OLSE ad GLSE of Var( b) = = ( x x) = ( x x) β are Var( ˆ β ) = = ( x x) ( x x) ( x x) = = x x = Square of the correlato coeffcet betwee ( 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).

12 The geeralzed least squares estmato assumes that specfed. Based o ths assumpto, the possbltes of followg two cases arse: Ω Ω s completely specfed or s ot completely specfed Ω s kow,.e., the ature of heteroskedastcty s completely We cosder both the cases as follows: Case : ' s are prespecfed Suppose,,..., are completely kow the model y = β+ βx β X + ε. k k,.e., y X Xk ε = β + β βk +. Now deflate the model by Let * ε ε =, the E * * ( ε ) = 0, Var( ε ) = =. Now OLS ca be appled to ths model ad usual tools for drawg statstcal fereces ca be used. Note that whe the model s deflated, the tercept term s lost as a software output. β / s tself a varable. Ths pot has to be take care

13 Case : Ω may ot be completely specfed 3 Let or but,,..., X λ j X λ are partally kow ad suppose = j s ot avalable. Cosder the model y β β X β X ε = k k + X λ j ad deflate t by as y β X Xk ε = + β βk +. λ λ λ λ λ X X X X X j j j j j Now apply OLS to ths trasformed model ad use the usual statstcal tools for drawg fereces. A cauto s to be kept s md whle dog so. Ths s llustrated the followg example wth oe explaatory varable model. Cosder the model y = β + β + ε 0 x. Deflate t by x, so we get y β ε 0 = + β +. x x x β0 β β0 β β β0 Note that the roles of ad orgal ad deflated models are terchaged. I orgal model, s tercept term ad s slope parameter whereas deflated model, becomes the tercept term ad becomes the slope parameter. So essetally, oe ca use OLS but eed to be careful detfyg the tercept term ad slope parameter, partcularly the software output.