behavor of a cross-sectonal data over tme the ncreasng use of anel data, one new research area s examnng the roertes of non-statonary tme-seres data n

Transcription

1 On the Estmaton Inference of a Contegrated Regresson n Panel Data Chhwa Kao Mn-Hsen Chang Syracuse Unversty Frst Draft: May, 995 Frst Revsed Draft: January 997 Ths Draft: February 7, 997 Abstract In ths aer, we study the asymtotc dstrbutons for least-squares (OLS), fully moded (FM), dynamc OLS (DOLS) estmators n contegrated regresson models n anel data. We show that the OLS, FM, DOLS estmators are all asymtotcally normally dstrbuted. However, the asymtotc dstrbuton of the OLS estmator s shown to have a non-zero mean. Monte Carlo results examne the samlng behavor of the roosed estmators show that () the OLS estmator has a non-neglgble bas n nte samles, () the FM estmator does not mrove over the OLS estmator n general, () the DOLS out-erforms both the OLS FM estmators. Key Words: Panel Data, OLS Estmator; FM Estmator, DOLS Estmator, Heterogeneous Panels. Introducton Evaluatng the statstcal roertes of data along the tme dmenson has roven to be very derent from analyss of the cross-secton dmenson. As economsts have ganed access to better data wth more observatons across tme, understng these roertes has grown ncreasngly mortant. An area of artcular concern n tme seres econometrcs has been the use of non-statonary data. Wth the desre to study the We are thankful to three anonymous referees Peter Phlls for ontng out several techncal errors n an earler verson rovdng comments that led to mrovements of ths aer. We also thank Peter Pedron for hs helful comments Bangtan Chen for hs research assstance on an earler draft of ths aer. An electronc verson of the aer n ostscrt format can be retreved from htt://web.syr.edu/~cdkao. Address corresondence to: Chhwa Kao, Center for Polcy Research, 46 Eggers, Syracuse Unversty, Syracuse, Y 44; e-mal: cdkao@maxwell.syr.edu.

2 behavor of a cross-sectonal data over tme the ncreasng use of anel data, one new research area s examnng the roertes of non-statonary tme-seres data n anel form. It s an ntrgung queston to ask: how exactly does ths hybrd style of data combne the statstcal elements of tradtonal cross-sectonal analyss tme-seres analyss? In artcular, what s the correct way to analyze non-statonarty, the surous regresson roblem, contegraton n anel data? Gven the mmense nterest n testng for unt roots contegraton n tme-seres data, not much attenton has been ad to testng the unt roots n anel data. The only theoretcal studes as far as we know n ths area are Bretung Meyer (994), Quah (994), Levn Ln (99), Im, Pesaran, Shn (995), Maddala Wu (996). Bretung Meyer (994) derved the asymtotc normalty of the Dckey- Fuller test statstc for anel data wth a large cross-secton dmenson a small tme-seres dmenson. Quah (994) studed a unt root test for anel data that have smultaneously extensve cross-secton tme-seres varaton. He showed that the asymtotc dstrbuton for the roosed test s a mxture of the stard normal Dckey-Fuller-Phlls asymtotcs. Levn Ln (99) derved the asymtotc dstrbutons for unt roots on anel data showed that the ower of these tests ncreases dramatcally as the cross-secton dmenson ncreases. Im et al. (995) crtqued the Levn Ln anel unt root statstcs roosed alternatves. Maddala Wu (996) rovded a comarson of the tests of Im et al. (995) Levn Ln (99). They suggested a new test based on the Fsher test. However, to ths date, lttle s known about contegraton tests estmaton wth regresson models n anel data. Excetons are Kao (996), McCoskey Kao (996), Pedron (995, 996). In the rst half of Kao (996), he studed a surous regresson n anel data. Asymtotc roertes of the least-squares (OLS) estmator other conventonal statstcs were examned. Kao (996) showed that the OLS estmator s consstent for ts true value; but the t-statstc dverges so that nferences about the regresson coecent, ; are wrong wth robablty that goes to one as! T!:Furthermore, Kao (996) examned the Dckey-Fuller (DF) the augmented Dckey-Fuller (ADF) tests to test the null hyothess of no contegraton n anel data. McCoskey Kao (996) roosed further tests for the null hyothess of contegraton n anel data. Pedron (995) derved asymtotc dstrbutons for resdual based tests of contegraton for both homogenous heterogenous anels. Pedron (996) roosed a fully moded estmator for heterogenous anels. On the other h, Park Ogak (99) derved asymtotc dstrbutons for contegraton coecent estmators related t-statstcs for anel data usng CCR transformatons. Although they used an SUR aroach rather than dmenson asymtotcs, many of the ssues they dealt wth are smlar. In ths aer, we study the lmtng dstrbutons for the ordnary least squares (OLS), fully moded (FM), Dynamc OLS (DOLS) estmators n anel contegrated regresson models.

3 Secton ntroduces the model assumtons. Secton develos the asymtotc theory for OLS, FM DOLS estmators. Secton 4 gves the lmtng dstrbutons of FM DOLS estmators for heterogeneous anels. Secton 5 develos the lmtng dstrbutons of the Wald statstcs. Secton 6 resents some Monte Carlo results to evaluate the nte samle roertes of the OLS, FM, dynamc OLS estmators. Secton 7 summarzes the ndngs. All roofs are n the Aendx. R R Aword on notaton. We wrte the ntegral W (s)ds as W when there s no ambguty over lmts. 0 We dene to be any matrx such that n 0 : We use kak to denote tr A Ao 0 ; jaj to denote the determnant of A, d! to denote convergence n dstrbuton,! to denote convergence n robablty, [x] to denote the largest nteger x, I(0) I() to sgnfy a tme seres that s ntegrated of order zero one, resectvely, BM () to denote Brownan moton wth covarance matrx. The Model Assumtons Consder the followng xed eect anel regresson: y t + x 0 t + u t; ; :::; ; t ; :::T; () where fy t g are ; s a k vector of the sloe arameters, f g are the ntercets, fu t g are the statonary dsturbance terms. We assume that fx t g are k ntegrated rocesses of order one for all ; where x t x t + t : Under these seccatons, () descrbes a system of contegrated regressons,.e., y t s contegrated wth x t : The ntalzaton of ths system s y 0 x 0 0for all. Assumton fy t; x t g are ndeendent across. Assumton The cross-secton dmenson s a monotonc functon of the tme-seres dmenson,.e., (T ), so that the law of large numbers (Theorem 6., Bllngsley, 986,. 8) the central lmt theorem (Theorem 7., Bllngsley, 986,. 69) for trangular arrays can be aled. ext, we characterze the nnovaton vector w t u t ; t 0 0. We assume that w t s a lnear rocess that satses the followng assumton. Assumton (e.g., Phlls, 995) (a) w t (L) t P j0 j t j ; P j0 ja k j k < ; j()j 60for some a>:

4 (b) t s..d. wth zero mean, varance matrx ; nte fourth order cumulants. Assumton mles that (e.g., Phlls Solo, 99) the artal sum rocess T P [Tr] t w t satses the followng multvarate nvarance rncle: where X [Tr] d w t! B (r) BM () as T!; () T t B The long-run covarance matrx of fw t g s gven by 4 B u B X j 5 : E w j w 0 0 () () 0 where X j u u u 5 ; E w j w u u u E w 0 w u u u 5 () 5 (4) are arttoned conformably wth w t : Assumton 4 s non-sngular,.e., fx t g are not contegrated. Dene u: u u u: (5) Then, B can be rewrtten as B 4 B u B 5 4 u: u V W 5 ; 4

5 where 4 V W 5 BM (I) s a stardzed Brownan moton. Dene the one-sded long-run covarance wth + X j0 E w j w u u u 5 : Remark Here we assume that anels are homogeneous,.e., the varances are constant across the crosssecton unts. We wll relax ths assumton n Secton 4 to allow for derent varances for derent. OLS, Fully Moded, Dynamc OLS Estmators Let us rst study the lmtng dstrbuton of the OLS estmator for equaton (). The OLS estmator of s It follows that b OLS X t TX X (x t x )(x t x ) 0 TX t T bols h P P T h P T (x t t x )(x t x ) 0 h P h P T T [ T ] T ; (x t x )(y t y ) P T T (x t t : (6) x ) u t where x T P T t x t; y T P T t y t; T T P T t (x t x ) u t ; T T P T t (x t x )(x t x ) 0, P T P T, T T : Before gong nto the next theorem, we need to consder some relmnary results. All lmts n (a) (d) n Lemma are taken as T!:Also, all lmts n (e) (f ) n Lemma Theorems 4 are taken as! T!: Lemma If Assumtons (a) T d! (b) T d! (c) E [ ] u + u ; 4 hold, then R fw dv u: + R fw f W 0 ; R fw dw 0 u + u ; 5

6 (d) E [ ] 6 ; (e) T! u + u ; (f) T! 6 ; where f W W R W : Remark u s due to the endogenety of the regressor x t ; u s due to the seral correlaton. Thus, we have establshed the followng theorem: Theorem If Assumtons 4 hold, then (a) T bols! u +6 u; (b) T bols T d! 0;6 u: ; where T X T t TX (x t x t )(x t x ) 0 X fw dw 0 u + u : Remark We notce that T! u +6 u: Remark 4 The normalty of the OLS estmator comes naturally. When summng across, the nonstard asymtotc dstrbuton due to unt root n the tme dmenson s smoothed out. However, t s mortant to note that the OLS estmator s asymtotcally based. The asymtotc bas s b OLS T T u +6 u T whch decreases as T ncreases. Remark 5 u: can be seen as the long-run sgnal-to-nose rato. 0 Remark 6 If w t u t ; 0 t are..d., then whch was examned by Kao Chen (995). T! u 6

7 Chen, McCoskey, Kao (996) nvestgated the nte samle roretes of the OLS estmator n (6), the t-statstc, the bas-corrected OLS estmator, the bas-corrected t-statstc. They found that the bas-corrected OLS estmator does not mrove over the OLS estmator n general. The results of Chen, McCoskey, Kao (996) suggest that alternatves, such as the FM estmator or DOLS estmator (e.g., Sakkonen, 99; Stock Watson, 99) may be more romsng n contegrated anel regressons. Thus, we begn our study by examnng the lmtng dstrbuton of FM estmator, b FM : Followng Pedron (996), we begn our study by examnng the lmtng dstrbuton of the FM estmator, b FM. In contrast to Pedron (996), we ntally consder the case where s common across members of the anel n order to focus on the role that the sgnal to nose rato, u:, can lay n the asymtotc dstrbuton of an FM estmator. The FM estmator s constructed by makng correctons for endogenety seral correlaton to the OLS estmator b OLS n (6). Let b u b are consstent estmates of u : Dene ote that 4 u+ t t whch has the long-run covarance matrx u + t u t u t ; bu + t u t b u b t; y + t y t u t; by + t y t b u b t : 5 4 u 0 Ik 4 u: u t t where Ik s a k k dentty matrx. The endogenety correcton s acheved by modfyng the varable y t n () wth the transformaton by + t y t b u b t The seral correlaton correcton term has the form 5 ; 5 ; + x 0 t + u t b u b t: b + u bu b b u b b b u ; 7 b b u A

8 where u b b are kernel estmates of u : Therefore, the FM estmator s X TX X! TX b FM (x t x )(x t x ) 0 (x t x ) by t + T b + u t t ow, we state the lmtng dstrbuton of b FM : : (7) Theorem If Assumtons 4 hold, then T bfm d! 0;6 u: : Remark 7 ote that Pedron (996) allowed the drfts for the ntegrated regressors n hs contegrated system. Ths aer only consders the regresson n whch ntegrated regressors do not have drfts. Also we roose the FM estmators for multle regresson. ext, we roose a DOLS estmator, b D ; whch uses the ast future values of 4x t as addtonal regressors. We then show that the lmtng dstrbuton of b D s the same as the FM estmator, b FM : But rst, we need the followng addtonal assumton: Assumton 5 The rocess fu t g can be rojected ontof t g to get where u t X j X j c j t+j + v t ; (8) kc j k < ; fv t g s statonary wth zero mean, fv t g f t g are uncorrelated not only contemoraneously but also n all lags leads. Remark 8 Assumton 5 can be guaranteed by followng the condtons n Sakkonen (99,. ). Remark 9 In ractce, the leads lags may be truncated whle retanng Assumton 5 aroxmately, so that u t q X j q c j t+j + v t : Ths s because fc j g are assumed tobe absolutely summable,.e., P j kc jk < : We also need to requre that q q tend to nnty wth T at a sutable rate,.e., Assumton 6 q T! 0; q T! 0; T X jjj>q or q kc j k!0: (9) 8

9 We then substtute (8) nto () to have y t + x 0 t + q X j q c j t+j + v t : Therefore, we obtan the DOLS of ; b D ; by runnng the followng regresson: y t + x 0 t + q X j q c j 4x t+j + v t : (0) ext, we show that b D has the same lmtng dstrbuton b FM as n Theorem. Theorem If Assumtons 6 hold, then T bd d! 0;6 u: : 4 Heterogeneous Panels The aer so far assumes that the anel data are homogeneous. The substantal heterogenety exhbted by actual data n the cross-sectonal dmenson severely restrcts the ractcal alcablty of such estmators. Also, the estmators n Sectons are not easly extended to cases of broader cross-sectonal heterogenety snce the varances bases are seced n terms of the asymtotc covarance arameters that are assumed to be shared cross sectonally. Recently, Pedron (996) roosed an FM estmator for heterogeneous anels. Pedron (996) roosed the followng anel FM estmator (usng hs notatons):! X TX b T bl b TX L (x t x ) u t Tb ; () bl t (x t x )! X where b L s the lower trangular decomoston of a consstent estmator of the asymtotc covarance matrx t ; where u t s gven by u t u t bl bl 4 x t the seral correlaton adjustment arameter b s gven by b b + b 0 bl b + 0 : bl Pedron (996) then derved the followng result (hs Prooston.): T b T! (0;v); 9

10 where v 8 < : x y 0 6 else In ths secton, we roose an alternatve reresentaton of the anel FM estmator for heterogeneous anels. Agan, n contract to Pedron (996), ths secton only consders the regresson that ntegrated regressors do not have drfts. Also we roose an FM estmator for multle regresson. Before we dscuss the FM estmator we need the followng assumtons: Assumton 7 We assume the anels are heterogeneous,.e., ; : are vared for derent : We also assume the nvarance rncle n (), (8) n Assumton 5, (9) n Assumton 6 stll hold. Let x t b x t ; () u t b u: bu + t ; () y t b u: by + t ; (4) where b b u: are consstent estmators of u: ; resectvely. Assumton 8 b s not sngular for all. where Then, we dene the FM estmator for heterogeneous anels as b FM X b FM can be wrtten as T b h h FM P TX X (x t x )(x t x )0 t b + u bu b b u b b b u :! TX (x t x ) y t T b + u t P T T t (x t x )(x t x h P )0 P 4T h P T [ 4T ] T ; b b u A ; (5) P T T t (x t x ) u t T b + u 0

11 where x T P T t x t ; T T P T t (x t x ) u t T b + u ; 4T T P T t (x t x )(x t x )0, T P P T, 4T 4T : It s clear that from Lemma that T d! fw dv ; 4T d! fw f W 0; It follows that 4T! 6 I k; T! 6 I k: T b FM d! (0; 6I k) : Hence, we have establshed the followng theorem: Theorem 4 If Assumtons 7 8 hold, then T b FM d! (0; 6I k) : The DOLS estmator for heterogeneous anels, b D; can be obtaned by runnng the followng regresson: yt + x 0 t + q X j q c j 4x t+j + v t : It s straghtforward to show that b D also has the same lmtng dstrbuton as b FM: Theorem 5 If Assumtons 7 8 hold, then T b D d! (0; 6I k) : Remark 0 Theorems 4 5 show that the lmtng dstrbutons of b FM b D are free of nusance arameters. 5 Hyothess Testng We now consder a lnear hyothess that nvolves the elements of the coecent vector : We show that hyothess tests constructed usng the FM DOLS estmators have asymtotc ch-squared dstrbutons. The null hyothess has the form: H 0 : R r; (6)

12 where r s a m known vector R s a known m k matrx descrbng the restrctons. A natural test statstc of the Wald test usng FM b or D b for homogeneous anels s W Rb 6 T FM r 0h Rb b u: R Rb 0 FM r : (7) For the heterogeneous anels, a natural statstc usng bfm or bd to test the null hyothess s W Rb 0h 6 T FM r RR Rb 0 FM r : (8) It s clear that W W converge n dstrbuton to a ch-squared rom varable wth k degrees of freedom, k ; as! T!under the null hyothess. Hence, we establsh the followng theorem: Theorem 6 If Assumtons 8 hold, then under the null hyothess (6), (a) W d! k ; (b) W d! k : Remark Because the FM the DOLS estmators have the same asymtotc dstrbuton, t s easy to verfy that the Wald statstcs based on the FM estmator share the same lmtng dstrbutons wth those based on the DOLS estmator. 6 Monte Carlo Smulatons To comare the erformance of OLS, FM, DOLS estmators, we conducted Monte Carlo exerments based on the desgn smlar to Phlls Hansen (990) Phlls Loretan (99). The data generatng rocess (DGP) was y t + x t + u t x t x t + t for ; :::; ; t ; :::T; where u t t wth u t 0 u t t t A d 0 A : 0:4 0:6 0 A 5 ; u t t 5 A : A

13 We generated from a unform dstrbuton, U [0; 0]; set. We allowed to vary consdered values of f0:8; 0:4; 0:0; 0:8g for f 0:8; 0:4; 0:4g for : Rom numbers for (u t ; t ) were generated by the GAUSS rocedure RDS. At each relcaton, we generated (T + 000) length of rom numbers then slt t nto seres so that each seres had the same mean varance. The rst ; 000 observatons were dscarded for each seres. fu t g f t g were constructed wth u : Once the estmates of w t ; bw t were estmated, we used to estmate : was estmated by ( X b T TX t b T bw t bw 0 t + T X TX t lx $ l T X bw t bw 0 t (9) t+ bwt bw 0 t + bw t bw 0 t ) ; (0) where $ l saweght functon or a kernel: Usng Phlls Durlauf (986) the law of large numbers for trangular arrays, b b can be shown to be consstent for : The estmate of the long-run covarance matrx n (0) was obtaned by usng the rocedure KEREL n COIT :0 wth a Bartlett wndow of lag length ve. Results wth other kernels, such as Parzen QS kernels, are not reorted, because no essental derences were found for most cases. ext, we recorded the results from our Monte Carlo exerments that examned the nte-samle roertes of the OLS estmator, b OLS ; the FM estmator, b FM ; the DOLS estmator, b D : The smulatons were erformed by a Sun SarcServer ; 000. GAUSS :: COIT :0 were used to erform the smulatons. The results we reort are based on 0; 000 relcatons are summarzed n Tables. The FM estmator was obtaned by usng a Bartlett wndow of lag length ve as n (0). Four lags two leads were used for the DOLS estmator. Table reorts the Monte Carlo means stard devatons (n arentheses) of bfm ; bd bols ; for samle szes T ( )(0;40; 60) : The bases of the OLS estmator, b OLS ; decrease at a rate of T. For examle, wth 0:8 0:8;the bas at T 0s 0:0 at T 40s 0:04. Also, the bases ncrease n (f > 0) decrease n : In general, the FM estmator, b FM ; resents the same degree of dculty wth bas as does the OLS estmator, b OLS : For examle, whle the FM estmator, b FM ; reduces the bas substantally outerforms b OLS when > 0 < 0; the ooste s true when > 0 > 0; the ooste s true. Lkewse, when 0:8; b FM s less based than b OLS for values of 0:8: Yet, for values of 0:4; the bas n b OLS s less than the bas n b FM : There seems to be lttle to choose between b OLS b FM when

14 < 0: Ths s robably due to the falure of the semarametrc correcton rocedure n the resence of a negatve seral correlaton of the errors,.e., a negatve MAvalue, < 0: Fnally, for the cases where 0:0,b FM outerforms b OLS when < 0: On the other h, b FM s more based than b OLS when > 0: In contrast, the results n Table show that the DOLS, D b ; s dstnctly sueror to the OLS FM estmators for all cases n terms of the mean bases. Clearly, the DOLS out-erformed both the OLS FM estmators. Whle the lmtng theory deends on the assumton that the cross-secton tme-seres dmensons are comarable n magntude, the actual anel data have a wde varety of cross-secton tme-seres dmensons. It s mortant to know the eects of the varatons n anel dmensons. Table consders 0 derent settngs for T, each rangng from 0 to 0. Frst, we notce that the cross-secton dmenson has sgncant eect on the bases of OLS b ; FM b ; D b when s ncreased from to 0. However, when s ncreased from 0 to 40 on, there s lttle eect on the bases of OLS b ; FM b ; D b : The results n Table also conrm the suerorty of the DOLS. 7 Concluson Ths aer derves lmtng dstrbutons for the OLS, FM, DOLS estmators n a contegrated regresson shows they are asymtotcally normal. We also nvestgated the nte samle roretes of the OLS, FM, DOLS estmators. Our ndngs are summarzed as follows:. The OLS estmator has a non-neglgble bas n nte samles.. The FM estmator does not mrove over the OLS estmator n general.. The DOLS estmator may be more romsng than OLS or FM estmators n estmatng the contegrated anel regressons. Aendx A Proof of Lemma Proof. (a) (b) are taken from the lterature,.e., 4

15 where P T T T (x t t x )(x t x ) 0 d R! fww f 0 ; fw W T T P T t (x t x ) u t d! ; R fw dv u: + W ; R fw dw 0 u + u () where 4 u u + u : The row, column dagonal element of R f W f W 0 element s W W j s R W W R W, the row, column j o-dagonal W j : Usng E W W 6 Var W W 45 ; the corresondng generalzaton s E fww f 0 6 I k; where IK s a k k dentty matrx. It then follows that E [ ] 6 Ik () 6 () establshng (d): To rove (c); we use the fact that E 0 hr fw dv E hr fw dw 0 I k 5

16 resultng n E [ ] u + u : Fnally, T X T T! X T! u + u as requred for (e) (f ) by the law of large numbers for trangular arrays. 6 (4) B Proof of Var h R gw dv u: 6 u: Proof. ote that Var fw dv u: 0 E fw dv u: fw dv u: 0 fw dv fw dv u: E R usng E fw R 0 dv fw dv 6 I k: u: 6 I k 6 u: C Proof of Theorem Proof. (a) s mmedately evdent from Lemma. Recall that h T bols P h P h P T [ T ] T : ote that the sequence n P T T (x hp t t x )(x t x ) 0 n T T ( T ) P R fw dw 0 R fw dw 0 n T u + u o o u + u n R o fw dw 0 u + u R fw dw 0 u + u 6

17 s a trangular array sequence; thus, a central lmt theorem of a trangular array s needed. Recall that the varances are assumed to be the same across ;whch mles that the Lndeberg condton wll hold. We can readly see that the T wll converge to a normal varable wth an arorate normalzaton that T wll converge to 6 n robablty byalaw of large number for a trangular array. Frst, we note from Lemma that It follows that X T T fw dw 0 d u + u! fw dv u: : fw dw 0 u + u Var fw dv u: 6 u: from Aendx B. Usng the Slutsky theorem, we obtan! E fw dv u: 0 [ T ] T d! 0;6 u: : Hence, rovng (b), where T X TX T t Therefore, we establsh Theorem. T bols : d! 0;6 u: T (x t x )(x t x ) 0 X ; (5) fw dw 0 u + u : (6) D Proof of Theorem The FM estmator of can be rewrtten as follows b FM hp P T hp (x t t x )(x t x ) 0 P T (x t t x ) by t + T b + u hp P T hp + (x t t x )(x t x ) 0 P T (x t t x ) bu + t T b + u : (7) 7

18 Frst, we note that T bfm P where T T T t Let w t + where u + t h h P P T [ T ] h(x t x ) bu + t b + u 0 t 0 P T P T (x t t x )(x t x ) 0 h P T P T T t h(x t x ) bu b + t + u T ; (8) we have X [Tr] w t + T t + 4 B+ u Let B d! P ; T T. 4 B+ u B 4 u: I u + T T From Lemma the consstency of b u b + T 5 BM + as T!; (9) 0 I d! TX t 5 (x t x ) bu + t : 4 B u B we note that eb db u + ++ u ; 5 : where + u u u u: u A It follows n terms of stard Wener rocesses that + d T! fw dv u: ++ u : 8

19 ow let Clearly, from Lemma we know that T + T b + u : E [ T ]0 Var[ T ] 6 u: : A smlar argument to the roof of Theorem yelds as requred. T bfm d! 0;6 u: E Proof of Theorem Frst we wrte (0) n vector from: y e + x + q C + v x + D + v (say), where y s a T vector of y t ; e astunt vector, q s the T (q + q ) matrx of observatons on the q + q regressors 4x t q ; ;4x t+q ; x savector of T k of x t ; C s a (q + q ) vector of c j ; v s a T vector of v t ; s a T (q + q +)matrx, (e; q ); D s a (q + q +)vector of arameters. Let Q I ( 0 ) 0 : It follows that We then wrte where 5T P bd X T bd P [ 6T ] T (x 0 Q x ) X x 0 Q x (x 0 Q v ) P P 6T h 5T ; 5T ; 5T T (x0 Q v ) ; 6T 9 P P 5T : T (x0 Q v ) 6T ; 6T T (x 0 Q x ) :

20 Observe that from Sakkonen (99) 6T X X X X 6T T (x0 Q x ) x 0 T W T x + o () T TX q 0 (x t x )(x t x ) + o (); tq + 5T X X X X 5T T (x0 Q v ) x 0 T W T v + o () T TX q tq + (x t x ) v t + o (); T TX q tq + (x t x ) v t d! eb db u + ; where W T I T T ee 0 : Usng arguments smlar to those n Theorem, we establsh the followng: as requred. T bd d! 0;6 u: References [] Bllngsley, P. (986), Probablty Measure, John Wley, ew York. [] Bretung, J., Meyer, W. (994), Testng for Unt Roots n Panel Data: Are Wages on Derent Barganng Levels Contegrated? Aled Economcs, 6,

21 [] Chen, B., McCoskey, S., Kao, C. (996), Estmaton Inference of a Contegrated Regresson n Panel Data: A Monte Carlo Study, Manuscrt, Center for Polcy Research, Syracuse Unversty. [4] Im, K., Pesaran, H., Shn, Y. (995), Testng for Unt Roots n Heterogeneous Panels, Manuscrt, Unversty of Cambrdge. [5] Kao, C. (996), Surous Regresson Resdual-Based Tests for Contegraton n Panel Data when the Cross-Secton Tme-Seres Dmensons are Comarable, Manuscrt, Center for Polcy Research, Syracuse Unversty. [6] Kao, C., Chen, B. (995), On the Estmaton Inference for Contegraton n Panel Data when the Cross-Secton Tme-Seres Dmensons are Comarable, Manuscrt, Center for Polcy Research, Syracuse Unversty. [7] Levn, A., Ln, C.-F. (99), Unt Root Tests n Panel Data: ew Results, Dscusson Paer, Deartment of Economcs, UC-San Dego. [8] Maddala, G. S., Wu, S. (996), A Comaratve Study of Unt Root Tests wth Panel Data a ew Smle Test: Evdence From Smulatons the Bootstra, Manuscrt, Deartment of Economcs, Oho State Unversty.. [9] McCoskey, S., Kao, C. (996), Resduals-Based Tests of the ull of Contegraton n Panel Data, Manuscrt, Center for Polcy Research, Syracuse Unversty. [0] Park, J., Ogak, M. (99), Seemngly Unrelated Canoncal Contegratng Regressons, The Rochester Center for Economc Research, Workng Paer o. 80. [] Pedron, P. (995), Panel Contegraton: Asymtotcs Fnte Samle Proertes of Pooled Tme Seres Tests wth an Alcaton to the PPP Hyothess, Indana Unversty Workng Paers n Economcs, o [] Pedron, P. (996), Fully Moded OLS for Heterogeneous Contegrated Panels the Case of Purchasng Power Party, Indana Unversty Workng Paers n Economcs, o [] Phlls, P. C. B. (995), Fully Moded Least Squares Vector Autoregresson, Econometrca, 6, [4] Phlls, P. C. B., Durlauf, S.. (986), Multle Tme Seres Regresson wth Integrated Processes, Revew of Economc Studes, 5,

22 [5] Phlls, P. C. B., Hansen, B. E. (990), Statstcal Inference n Instrumental Varables Regresson wth I() Processes, Revew of Economc Studes, 57, [6] Phlls, P. C. B., Loretan, M. (99), Estmatng Long-Run Economc Equlbra, Revew of Economc Studes, 58, [7] Phlls, P. C. B., Solo, V. (99), Asymtotcs for lnear rocesses, Annals of Statstcs, 0, [8] Quah, D. (994), Exlotng Cross Secton Varaton for Unt Root Inference n Dynamc Data, Economcs Letters, 44, 9-9. [9] Sakkonen, P. (99), Asymtotcally Ecent Estmaton of Contegratng Regressons, Econometrc Theory, 58, -. [0] Stock, J., Watson, M. (99), A Smle Estmator of Contegratng Vectors n Hgher Order Integrated Systems, Econometrca, 6,

23 Table : Means Bases Stard Devatons of OLS, FM, DOLS Estmators 0:8 0:4 0:8 b OLS b FM b D b OLS b FM b D b OLS b FM b D 0:8 T (.049) (.047) (.040) (.0) (.05) (.0) (.0) (.06) (.009) T (.09) (.07) (.0) (.0) (.0) (.0) (.004) (.006) (.00) T (.00) (.009) (.007) (.007) (.007) (.006) (.00) (.00) (.00) 0:4 T (.08) (.06) (.07) (.00) (.09) (.0) (.0) (.08) (.0) T (.04) (.009) (.07) (.0) (.0) (.009) (.005) (.006) (.004) T (.007) (.005) (.005) (.006) (.006) (.005) (.00) (.00) (.00) 0:0 T (.07) (.05) (.07) (.06) (.0) (.06) (.06) (.09) (.07) T (.009) (.005) (.005) (.06) (.0) (.06) (.00) (.007) (.005) T (.005) (.00) (.00) (.005) (.008) (.008) (.00) (.004) (.00) 0:8.000 T (.06) (.0) (.008) (.07) (.05) (.04) (.04) (.07) (.0) T (.006) (.00) (.00) (.006) (.005) (.004) (.0) (.009) (.009) T (.00) (.00) (.00) (.00) (.00) (.00) (.007) (.005) (.005) ote: (a) T. (b) The lag length 5 of the Bartlett wndows s used for the FM estmator. (c) The 4 lags leads are used for the DOLS estmator.

24 Table : Means Bases Stard Devatons of OLS, FM, DOLS Estmators for Derent T (,T) OLS b FM b D b (,0) (.84) (.89) (.97) (,40) (.09) (.09) (.06) (,60) (.06) (.06) (.064) (,0) (.0) (.0) (.09) (0,0) (.00) (.09) (.0) (0,40) (.06) (.05) (.04) (0,60) (.00) (.009) (.009) (0,0) (.005) (.005) (.005) (40,0) (.0) (.0) (.0) (40,40) (.0) (.0) (.009) (40,60) (.007) (.007) (.007) (40,0) (.004) (.00) (.00) (60,0) (.07) (.07) (.08) (60,40) (.009) (.009) (.008) (60,60) (.006) (.006) (.005) (60,0) (.00) (.00) (.00) (0,0) (.0) (.0) (.0) (0,40) (.006) (.006) (.006) (0,60) (.004) (.004) (.004) (0,0) (.00) (.00) (.00) ote: (a) The lag length 5 of the Bartlett wndows s used for the FM estmator. (b) The 4 lags leads are used for the DOLS estmator. 4