Double Autocorrelation in Two Way Error Component Models

Transcription

1 Open Journal of Statstcs o:046/ojs00 Publshe Onlne October 0 ( 85 Double Autocorrelaton n wo Way Error omponent Moels Abstract Jean Marceln osson rou Eugene Kouass Kern O Kymn Department of Economcs Unversty of ocoy Abjan ote- vore Resource Economcs West Vrgna Unversty Morgantown USA Dvson of Fnance an Economcs West Vrgna Unversty Morgantown USA E-mal: kernkymn@malwvueu Receve September 5 0; revse October 6 0; revse October 0 0 n ths paper we exten the works by [-5] accountng for autocorrelaton both n the tme specfc effect as well as the remaner error term Several transformatons are propose to crcumvent the ouble autocorrelaton problem n some specfc cases Estmaton proceures are then erve Keywors: wo Way Ranom Effect Moel Double Autocorrelaton F ntroucton Followng the works of [6] the regresson moel wth error components or varance components has become a popular metho for ealng wth panel ata A summary of the man features of the moel together wth a scusson of some applcatons s avalable n [7-0] among others However relatvely lttle s known about the two way error component moels n the presence of ouble autocorrelaton e autocorrelaton n the tme specfc effect an n the remaner error term as well hs paper extens the works by [-5] on the one-way ranom effect moel n the presence of seral autocorrelaton an by [] on the sngle autocorrelaton two-way approach t nvestgates some potental transformatons to crcumvent the ouble autocorrelaton ssue along wth some estmaton proceures n partcular we erve several transformatons when the two sturbances follow varous structures: from autoregressve an movng-average processes of orer to a general case of ouble seral correlaton We euce several estmators as well as ther asymptotc propertes an prove a F verson he remaner of ths paper s organze as follows: Secton consers smple transformatons on the presence of relatvely manageable ouble autocorrelaton structure n Secton general transformatons are consere when the ouble autocorrelaton s more complex estmators are erve n Secton 4 Asymptotc propertes of the estmators are consere n Secton 5 Secton 6 proves a F counterpart approach Fnally some conclung remarks appear n Secton 7 Smple ransformatons o crcumvent the ouble autocorrelaton ssue we frst nee to transform the moel base on the varance-covarance matrx he general regresson moel consere s yt 0 xt ut ; t 0 s the ntercept an s a k vector of slope coeffcents x t s a k row vector of explanatory varables whch are uncorrelate wth the usual two-way error components sturbances ut t (see [7]) n matrx form we wrte y X u t When the Errors Follow AR() Structures f the tme specfc term follows an AR() structure t t t wth t D0 an the remaner error term also follow an AR() structure t t et wth et D0 e we can efne two transformaton matrces of mensons respectvely an an opyrght 0 ScRes

2 86 J M ROU E AL an snce we have an e e e e () () the transforme errors an follow two fferent MA() processes of parameters an respectvely hus by applyng the approprate transformaton matrces the autoregressve error structure can be change nto a movng-average one he only cost s the loss of the ntal an frst pseuo-fferences whch has no serous consequence for a long tme menson As a result we focus on the MA() error structure When the Errors Follow MA() Structures Here the tme specfc term t follows an MA() structure t t t wth t D 0 whle the remaner error term t also follows an MA () structure t et et wth et D0 e For convergence purpose an assumng normalty the ntal values are efne an 0 0 e 0 0 he varance-covarance matrx of the three components error term s gven by e () an are postve efnte matrces of orer an s efne by xoepltz x x0 0 he exact nverse of such matrces suggeste by [] an [] oes not nvolve the parameters an Followng [] let P be the Pesaran orthogonal matrx whose t-th row s gven by tπ tπ tπ Lt sn sn sn P P ag tπ t cos P P D an tπ D a g t wth t cos Pre-multplyng the moel by Pyels the fol- of u P u lowng varance-covarance matrx D (4) e P General ransformatons We are now n the context of a general case of ouble autocorrelaton ssue an lea to a sutable error covarance matrx smlar to Equaton (4) an ts nverse Frst ransformaton Let P enote the matrx such that PP Such a matrx oes exst for an s a postve-efnte matrx ransformaton of the ntal moel y X u by yels P y P y X u (5) an the varance-covarance of the transforme errors s E u u P P P P hs transformaton has remove the autocorrelaton n the tme-specfc effect t Unfortunately by ong so t has nfecte the s t a n worsene the ntal correlaton n the reman er sturbances An atonal treatment s therefore neee Secon ransformaton (6) We now conser an orthogonal matrx P an a - agonal matrx D such that PPPP D (agonalzaton of P P ) hus appl yng a secon transformaton P yels y P y X u (7) he unerlyng varance-covarance matrx of the errors s E u u P P P P opyrght 0 ScRes

3 J M ROU E AL 87 or P P P P P P (8a) Eu u D PP PP P P P P D f an (8b) Here because of the choce of matrces P an P we en up wth snce P s an orthogonal ma- Generally speakng an D just n ee to have trx zero off-agonal elements e to be agonal matrces he ouble autocorrelaton structure s thus absorbe an one can easly accommoate wth the non-sphercal form of by means of an accurate nverson process omputng the nverse he nverse of s obtane usng the proceure e- one velope by [] After a bt of algebra gets E K E L J S D E J L D J K D L D D S S S 4 S S an S ag s s wth s t t (see the Appenx) 4 Estmaton We begn wth the efnton of the estmator followe by ts nterpretaton an weghte average property (9) 4 he Estmator Proposton : he estmator s X X X y (0) (Straghtforwar) 4 nterpretaton n classcal two-way regresson moels [] prove an nterpretaton of the estmator whch s appealng n vew of the sources of varaton n sample ata n the straght lne of ther work the estmator may be vewe as obtane by poolng three uncorrelate estmators: the covarance estmator (or wthn estmator) the between-nvual estmator an the wthn-nvual estmator hey are the same as those suggeste by [] except for the last one whch was labele between-tme estmator We have ) he covarance estmator A E K AX X X Ay ; ) he between-nvual estmator X A X X A y A J S an ) h e wthn-nvual estmator A E L X AX X A y t s mportant to note that these estmators are obtane from some transformatons of the regresson Equaton (7) e y P y X u he covarance estmator s obtane when Equaton (7) s pre-multple by M E K A; the transformaton annhlates the nvual- an tme-effects as well as the column of ones n the matrx of explanatory varables t s equvalent to the wthn estmator n the classcal two-way error component moel (see [-7]) he between-nvual estmator transformaton of Equaton (7) by the matrx comes from the opyrght 0 ScRes

4 88 J M ROU E AL M hs s equvalent to averagng n- each tme pero vual equatons for he wthn-nvual estmator s erve when Equaton (7) s transforme by M E L A he presence of the empotent matrx E ncates that ths transformaton wpes out the constant term as well as the tme specfc error term t Howe ver the nvual effect remans 4 as a Weghte Average Estmator As n [4] the estmator s a weghte average of the three estmators efne above Proposton : F F F () wth F X X X A X F X X X A X an F X X X A X F F From Equaton (0) t comes that wth X y X X X y X A y X A y an are re- X A y y efnton the estmators spectvely such that an herefore X A y X X A X A y X A X X A y X A X X y X AX X AX X AX Or X X X AX hus X AX X AX X X X AX X X X A X X X X A X F F F wth F F an F efne accorng to Equaton ( ) We shoul also note that the three estmators an are uncorrelate n fact A A EJ KDS 0 an A A E K DL 0 E whle A A 0 because K 0 snce JE 0 As a result cov cov cov 0 () Moreover followng [] the fact that rankmrank MrankM () gves evence on the use of all avalable nformaton from the sample he estmators an together use up the entre set of nformaton to bul the estmator wth no loss at all 5 Asymptotc Propertes Uner regular assumptons the an the three pseuo estmators of the coeffcent vector say an are all consstent an asymptotcally equvalent t s a result smlar to the one obtane n the classcal two-way error component moel (see [5]) 5 Assumptons We assume that the xt s are weakly non-stochastc e o not repeat n repeate samples We also state that the followng matrces exst an are postve efnte: opyrght 0 ScRes

5 J M ROU E AL 89 a X A X X E K X plm plm for the frst transformaton; b X J S X X A X plm plm for the secon transformaton; an plm X A X plm X E L X c for the thr transformaton Furthermore n the straght lne of [] we also assume that X Au X E K u a plm plm 0 for the frst transformaton; X A u b plm X J S u plm 0 for the secon transformaton; X Au X E L u c plm plm 0 for the thr transformaton n aton lm D so that the varance-components quantty D enotes by remans nfnte as he lmts an probabltes are taken as an All along ths secton followng [] we conser the usual assumptons re- garng the error vector u as state n [6] an [7] whch ensures the asymptotc normalty 5 Asym ptotc Prop erty of the ovarance Estmator Proposton : he covarance estmator Snce s consstent X E K X X E K u Hence X E K X plm plm X E K u plm 0 Makng use of assumptons (a) an (a) we establsh the consstency of the covarance estmator plm Proposton 4: he covarance estmator has an asymptotc normal strbuton gven by a X E K X plm Uner the M-transformaton we have E E K u E K E u 0 (4) Moreover ts varance s gven by V E K u E K an ts nverse s equal to E D whle assumpton (a) states the absence of correlaton between regressors an stur- bances uner the M transformaton We have an X E K u X 0 plm E K X (5) X E K X X E K u from whch we euce that X E K X 0 plm hus the asymptotc normalty of the covarance estmator mmeately follows X E a K X plm opyrght 0 ScRes

6 90 J M ROU E AL 5 Asymptotc Property of the etween-me Estmator Proposton 5: he between tme estmator Snce s consstent X AX X Au Hence accorng to assumptons (b) an (b) J X S X plm plm J X S u plm 0 Makng use of assumptons (b) an (b) we establsh the consstency of the between tme estmator plm Proposton 6: he between-tme estmator has an asymptotc normal strbuton gven by X J S X a plm (6) Uner the M -transformaton we get E u E u he varance of ths error term s wrtten as V u S 0 ts nverse s S Agan assumpton (b) states the absence of correlaton between regressors an sturbances uner the M transformaton We get J X S u n aton we have J X S X 0 plm J X S X from whch we euce that J X S u J X S X 0 plm hus the asymptotc normalty of the between-tme estmator mmeately follows X J S X a plm 54 Asymptotc Property of the Wthn-nvual Estmator Proposton 7: he wthn nvual estm ator s a consstent estmator Snce Hence X AX X Au plm plm X E L X X E L u plm 0 opyrght 0 ScRes

7 J M ROU E AL 9 Makng use of assumptons (c) an (c) we establsh the consstency of the covarance estmator plm has an asymp- Proposton 8: he wthn nvual estmator totc normal strbuton gven by h a X E L X plm (7) Uner the M -transformaton we obtan E E L u E L E u e varance of E L u s obtane as V E L u E L 0 he nverse of ths matrx s E L Assumpton (c) states the absence of correlaton between regressors an sturbances uner the M transformaton We have an X E L u X E L X 0 plm from whch we euce that X E L X X E L u X E L X 0 plm hus the asymptotc normalty of the wthn nvual estmator mmeately follows a X E L X plm 55 Asymptotc Property of the Estmator Proposton 9: he estmator s asymptotcally equvalent to the covarance estmator an therefore a X E K X plm From Equaton (0) we get (8) X X X u On the one han we have w X X X E K X X E L X X J S X here D as herefore from assumpton (a) we fn that X E L X 0 when Lke wse X J S X assumpton (a) leas us to 0 when Hence plm X X plm X E K X On the other han we can wrte X u X E K u X E L u X J S u Uner the M an M transformatons we get X E L u plm X J S u plm 0 opyrght 0 ScRes

8 9 J M ROU E AL leang to plm X u plm X E K u As a result e plm X E K X plm X E K u plm plm plm has the same lmtng strbu- hs shows the asymptotc equvalence of the two estmators an We then euce that Fnally ton as a X E KX plm hus the estmator suggeste uner the ouble autocorrelaton error structure has the esre asymptotc propertes 6 F Estmaton n practce the varance-covarance ts etermnaton herefore a F approach s requre he metho matrx s unknown as well as all the parameters nvolve n use conssts n removng the tme specfc effect to obtan a one-way error component moel only t carres the seral correlaton (see [8] an []) hs metho has been rectly apple to AR() an MA() processes n separate subsectons 6 Feasble Double AR() Moel We assume that t t e t t t t et D0 e t D 0 he wthn error term s u E u E E (9) he assocate varance-covarance matrx s E uu E E (0) Snce t follows an AR() process of parameter we efne the matrx as the famlar [9] transformaton matrx wth parameter hs matrx s su ch that e he resultng estmator s gven by W X X X y () y y an X X he covarance matrx of u u trck s usng [0] e E J e E E an J E J () Followng [] another estmator can be erve We label ths estmator the wthn-type estmator an s gven by () W X X X y wth y y an X X n orer to get the estmates of numerous parameters nvolve n the moel we frst nee an estmate of the correlaton coef- fcent he autocorrelaton functon of the error term u s gven by h h E ut u t h for h 0 t We euce from t that 0 (4) t then leas opyrght 0 ScRes

9 J M ROU E AL 9 to a convergent estmator of h h (see [7]) e 0 ut u th wth t th u efne as the OLS resuals o f the wthn equaton y X u Hence we get an (5a) (5b) Furthermore the QU estmate of e s also avalable as u E E u e (6) u beng the OLS estmate of u As a consequence we get an e (7a) 0 We now nee to fn an he autocovarance functon of the ntal error term u s gven h h for h 0 t t comes that h h h (7b) (8) We mmeately euce a convergent estmator of the secon correlaton coeffcent e 0 0 h u ut th wth h th u t (9) e- notng the OLS resuals of y X u he varances e s estmate by e n aton to the estmators mentone n Secton 4 other estmators su ch as the wthn estmator W an the wthn-type estmator W can all be performe as well Actually the knowlege of the AR() parameters an enttles us to bul the matrces nvolve n the etermnato n of say matrces P D K L S an S 6 Feasble Double MA() Moel (0) We now state that t e t e t wth an et D 0 e Agan evatons from nvual means lea to the moel y X u wth u t t he varance-covarance matrx of u Equaton (0) wth now s stll gven by oepltz r 0 0 () Here we set enotng the correlaton correcton matrx as ef ne by [8] n ther orthogonalzng algorthm We then transform the wthn moel by he n ew er ror term u ha s the followng covarance matrx E J E E () ecause of the movng average nature of the process lnear estmaton of the correlaton parameter s not easly obtanable nstea r proves useful he autocorrelaton functon of the wthn error term u t s gven by wth h E ut u t h h () for h 0 t h enotng the autocovarance functon of t As a conseque nce j j an 0 0 for some (4) opyrght 0 ScRes

10 94 J M ROU E AL h h ut u th s the emprcal th autocovarance functon an u t s are the OLS resuals of the wthn equaton We also get for some j r (5) j We then apply the [8] matrx to the ata (for nstance to the wthn transforme epenent vector y ) Moreover wll be apple to the vector of constants to get estm ates of the ts W e have n the straght lne of [8] the followng steps: y Step : ompute y an y t y t r y g t g t t g for t r g t for t g t Step : ompute y y knowng that he estmates of the s are t obtane as an r g t t fo r t g t We then obtan the estmate of as t t he autocovarance functon h Eutu th h h of the ntal composte error term u an ts emprcal counterpart h u t u h th th ( u t beng the OLS resuals of the ntal two-way moel) permt the estmaton of an r an 0 (6a) r r ( 6b) he wthn estmator W an the wthn-type one W are now obtanable However the estmator can be estmate prove the MA() parameters an are known especally uner the contons 4r 0 an 4 r 0 n other wors the estmates r an r shoul b ot h le nse th e open nterval as a pre-requste to a rect estmaton of an 7 Fnal Remarks hs paper has consere a complex but realstc correlaton structure n the two-way error component moel: the ouble autocorrelaton case t ealt wth some parsmonous moels especally the AR() an MA() ones as well as the general framework hrough a precse formula of the varance-covarance matrx of the errors we erve the estmator an relate asymptotc propertes An nvestgaton of the F s also consere n the paper 8 References [] S Revankar Error omponent Moels wth Seral orrelate me Effects Journal of the nan Statstcal Assocaton Vol pp 7-60 [] H altag an Q L A ransformaton that wll rcumvent the Problem of Autocorrelaton n an Error omponent Moel Journal of Econometrc Vol 48 o 99 pp 85-9 o:006/ (9) [] H altag an Q L Precton n the One-Way Er- orrelaton Journal ror omponent Moel wth Seral of Forecastng Vol o 6 99 pp o:000/for [4] H altag an Q L Estmatng Error omponent Moels wth General MA(q) Dsturbances Econometrc heory Vol 0 o 994 pp o:007/s x [5] J W Galbrath an V Zne-Walsh ransformng the Error omponent Moel for Estmaton wth general ARMA Dsturbances Journal of Econometrcs Vol 66 o pp o:006/ (94)06-6 [6] P alestra an M erlove Poolng ross-secton an me-seres Data n the Estmaton of a Dynamc Moel: he Deman for atural Gas Econometrca Vol 4 o 966 pp o:007/90977 [7] H altag Econometrc Analyss of Panel Data r Eton John Wley an Sons ew York 008 [8] Hsao Analyss of Panel Data ambrge Unversty Press ambrge 00 [9] G S Maala Lmte Depenent an Qualtatve Varables n Econometrcs ambrge Unversty Press opyrght 0 ScRes

11 J M ROU E AL 95 ambrge 98 [0] G S Maala he Econometrcs of Panel Data Vols an Ewar Elgar Publshng heltenham 98 [] M H Pesaran Exact Maxmum Lkelhoo Estmaton of a Regresson Equaton wth a Frst Orer Movng Average Errors he Revew of Economc Stues Vol 40 o 4 97 pp [] P A V Swamy an S S Arora he Exact Fnte Sample Propertes of the Estmators of oeffcents n the Error omponents Regresson Mo els Econometrca Vol 40 o 97 pp 6-75 o:007/ [] M erlove A ote on Error omponents Moels Econometrca Vol 9 o 97 pp 8-96 o:007/95 [4] G S Maala he Use of Varance omponents Moels n Poolng ross Secton an me Seres Data Econometrca Vol 9 o 97 pp 4-58 o:007/949 [5] Amemya he Estmaton of the Varances n a Varance-omponents Moel nternatonal Economc Revew Vol o 97 pp - o:007/5549 [6] H hel Prncples of Econometrcs John Wley an Sons ew York 97 [7] D Wallace an A Hussan he Use of Error omponents Moels n ombnng ross-secton an me Seres Data Econometrca Vol 7 o 969 pp 55-7 o:007/90905 [8] A Maury he Use of me Seres Processes to Moel the Error Structure of Earnngs n a Longtunal Data Analyss Journal of Econometrcs Vol 8 o 98 pp 8-4 o:006/ (8) [9] S J Pras an Wnsten ren Estmators an Seral orrelaton Unpublshe owles ommsson Dscusson Paper: Stat o 8 hcago 954 [0] W A Fuller an G E attese Estmaton of Lnear Moels wth ross-error Structure Journal of Econometrcs Vol o 974 pp o:006/ (74)9000-x [] J Wansbeek an A Kapteyn A Smple Way to obtan the Spectral Decomposton of Varance omponents Moels for alance Data ommuncatons n Statstcs Vol o 8 98 pp 05- opyrght 0 ScRes

12 96 J M ROU E AL Appenx: omputng the nverse of We establshe that D w th D ag t an ag ettng S G D we can re- wrte the varance covarance matrx as G J G J J y the means of an upate formula we euce an expresson of the nverse of G G J J G J J G We nee to obtan G an the nverse of the brack- ete expresson On th e one han G D D Let H enote the matrx At th s step the nverse of D D H s requre Let a be a orthogonal matrx hen H D herefore wth H ag t s worth mentonng that D a 0 for an a are fferent columns of the same ag onal matrx t s therefore obvous that H has alreay been agonalze As a consequence the nverse of H s gven by H ag H a a A A ag Snce 0 an we have a a a a a E herefore H E A t then follows that G D E A G D E D S n whch S ags s t wth s t t t On the other han the matrx J G J to be etermne We get J G J E D S or has opyrght 0 ScRes

13 J M ROU E AL 97 hus Hence J G J E D S J G J D E S JG J S D D an J G J a b S D a D an b Snce b a b a a b we euce JG J We are now ntereste n the expresson We have G JJ G J J G G JJG J JG b G JJ G G J J G a a b From the efntons of the matrces G an J we can wrte an so that G J E D S G JJ G G J S 4 E D D S S an lastly t then comes that G J J G S S 4 G JJG J JG 4 E D D 4 S S a a a 4 b 4 a a b n other wors SS S S G J J G J J G E D D a 4 a 4 Fnally the nverse of wth a a b S S can be erve as E D D D J 4 a J S S 4 S a b An alternatve expresson for s avalable Settng L D D D K D L an we get E D L L D D a J S S a b S E K E L D J S opyrght 0 ScRes

14 98 J M ROU E AL e S S S a b 4 4 S S S S S S Hence we fnally get E K E L J S D opyrght 0 ScRes