Testing for Granger Non-causality in Heterogeneous Panels

Transcription

1 Testng for Granger on-causalty n Heterogeneous Panels Chrstophe Hurln y June 27 Abstract Ths paper proposes a very smple test of Granger (1969) non-causalty for heterogeneous panel data models. Our test statstc s based on the ndvdual Wald statstcs of Granger non causalty averaged across the groups. Frst, ths statstc s shown to converge sequentally to a standard normal dstrbuton. Second, for a xed T sample the sem-asymptotc dstrbuton of the average statstc s characterzed. A standardzed statstc based on an approxmaton of the moments of Wald statstcs s proposed. Monte Carlo experments show that our panel standardzed statstcs provde very good small sample propertes. Keywords : Granger non-causalty, Panel data, Wald Test. J.E.L Class caton : C23 LEO, Unversty of Orleans. emal: chrstophe.hurln@unv-orleans.fr. A substantal part of the work for ths paper was undertaken n the Department of Economcs of the Unversty Pars IX Dauphne, EURIsCO. y I am grateful for comments from partcpants of the econometrc semnars at Unversty Pars I, Pars X anterre, Marne-la-Vallée and Unversty of Geneva. 1

2 1 Introducton The am of ths paper s to propose a smple Granger (1969) non causalty test n heterogeneous panel data models wth xed coe cents. In the framework of a lnear autoregressve data generatng process, the extenson of standard causalty tests for panel data mples to test cross sectonal lnear restrctons on the coe cents of the model. As usually, the use of cross-sectonal nformaton may extend the nformaton set on the causalty from a gven varable to another. Indeed, n many economc problems t s hghly probable that f a causal relatonshp exsts for a country or an ndvdual, t exsts also for some other countres or ndvduals. In ths case, the causalty can be tested wth more e cency n a panel context wth T observatons. However, the use of the cross-sectonal nformaton mples to take nto account the heterogenety across ndvduals n the de nton of the causal relatonshp. As dscussed n Granger (23), the usual causalty test n panel asks f some varable, say X t causes another varable, say Y t ; everywhere n the panel [..]. Ths s rather a strong null hypothess. Then, we propose here a smple Granger non causalty test for heterogeneous panel data models. Ths test allows to take nto account both dmensons of the heterogenety n ths context: the heterogenety of the causal relatonshps and the heterogenety of the data generatng process (DGP ). Let us consder the standard mplcaton of the Granger causalty de nton 1. For each ndvdual; we say that the varable x s causng y f we are better able to predct y usng all avalable nformaton than f the nformaton apart from x had been used (Granger 1969). If x and y are observed on ndvduals, the ssue conssts n determnng the optmal nformaton set used to forecast y: Several solutons could be adopted. The most general s to test the causalty from the varable x observed on the th ndvdual to the varable y observed for the j th ndvdual, wth j = or j 6= : The second soluton, s more restrctve and s drectly derved from the tme seres analy- 1 The Granger causalty de nton s based on the two precepts that the cause preceded the e ect and the causal seres had nformaton about the e ect that was not contaned n any other seres accordng to the condtonal dstrbutons (Granger 23). The fact that the cause produces a superor forecasts of the e ect s just an mplcaton of these statements. However, t does provde sutable post sample tests as dscussed n Granger (198). 2

3 ss. It mples to test causal relatonshp for a gven ndvdual. The cross sectonal nformaton s then only used to mprove the spec caton of the model and the power of tests as n Holtz-Eakn, ewey and Rosen (1988). The baselne dea s to assume that there exsts a mnmal statstcal representaton, whch s common to x and y at least for a subgroup of ndvduals. In ths paper we use such a model. Then, causalty tests could be mplemented and consdered as a natural extenson of the standard tme seres tests n the cross sectonal dmenson. However, one of the man spec c stakes of panel data models s to specfy the heterogenety between cross-secton unts. In ths context, the heterogenety has two man dmensons. We propose to dstngush between the heterogenety of the DGP and the heterogenety of the causal relatonshps from x to y. Indeed, the DGP may be d erent from an ndvdual to another, whereas there exsts a causal relatonshp from x to y for all ndvduals. More generally, n a p order lnear vectoral autoregressve model, we de ne four knds of causal relatonshps. The rst, denoted Homogenous on Causalty (H C) hypothess, mples that there does not exst any ndvdual causalty relatonshps from x to y. The symmetrc case s the Homogenous Causalty (HC) hypothess, whch occurs when there exsts causalty relatonshps, and when the ndvdual predctors of y, obtaned condtonally to the past values of y and x are dentcal. The dynamcs of y s then totally dentcal for all the ndvduals of the sample. The two last cases correspond to heterogeneous process. Under the HEterogenous Causalty (HEC) hypothess, we assume that there exsts causalty relatonshps as n the HC case, but the dynamcs of y s heterogenous. The heterogenety does not a ect the causalty result. Fnally, under the HEterogenous on Causalty (HE C) hypothess, we assume that there exsts a subgroup of ndvduals for whch there s a causal relatonshp from x to y. Symmetrcally, there s at least one and at the most 1 non causal relatonshps n the model. That s why, n ths case, the heterogenety deals wth the causalty from x to y: To sum t up, n the HC hypothess, there does not exst any ndvdual causalty from x to y: On the contrary, n the HC and HEC cases, there s a causalty relatonshps for each ndvdual of the sample. In the HC case, the DGP s homogenous, 3

4 whereas t s not the case n the HEC hypothess. Fnally n the HEC hypothess; there s an heterogenety of the causalty relatonshps snce there s a subgroup of 1 unts for whch the varable x does not cause y: In ths paper, we propose a smple test of the Homogenous on Causalty (HC) hypothess. Under the null hypothess, there s no causal relatonshp for all the unts of the panel. However, we do not test ths hypothess aganst the HC hypothess as Holtz-Eakn, ewey and Rosen (1988). We specfy the alternatve as the HE C hypothess. There s two subgroups of unts: one wth causal relatonshps from x to y, but not necessarly wth the same DGP; and an another subgroup where there s no causal relatonshps from x to y: For that, our test s lead n an heterogenous panel data model wth xed coe cents. Under the null or the alternatve, the unconstraned parameters may be d erent from ndvdual to another. Then, whatever the result on the exstence of causal relatonshps, we assume that the dynamc of the ndvdual varables may be heterogeneous. As n the lterature devoted to panel unt root tests n heterogeneous panels, and partcularly n Im, Pesaran and Shn (23), we propose a statstc of test based on averagng standard ndvdual Wald statstcs of Granger non causalty tests. Under the assumpton of cross-secton ndependence (as used n rst generaton panel unt root tests), we provde d erent results. Frst, ths statstc s shown to converge sequentally n dstrbuton to a standard normal varate when the tme dmenson T tends to n nty, followed by the ndvdual dmenson : Second, for a xed T sample the sem-asymptotc dstrbuton of the average statstc s characterzed. In ths case, ndvdual Wald statstcs do not have a standard ch-squared dstrbuton. However, under very general settng, t s shown that ndvdual Wald statstcs are ndependently dstrbuted wth nte second order moments as soon as T > 5 + 2K; where K denotes the number of lnear restrctons. For a xed T, the Lyapunov central lmt theorem s su cent to get the dstrbuton of the standardzed average Wald statstc when tends to n nty. The two rst moments of ths normal sem-asymptotc dstrbuton correspond to emprcal mean of the correspondng theoretcal moments of the 4

5 ndvdual Wald statstcs. The ssue s then to propose an evaluaton of the two rst moments of standard Wald statstcs for small T sample. The rst soluton conssts n usng Monte Carlo or Bootstrap smulatons. The second soluton conssts n usng an approxmaton of these moments based on the exact moments of the rato of quadratc forms n normal varables derved from the Magnus (1986) theorem for a xed T sample, wth T > 5 + 2K. Gven these approxmatons, we propose a second standardzed average Wald statstc to test the HC hypothess n short T sample. The nte sample propertes of our test statstcs are examned usng Monte Carlo methods. The smulaton results clearly show that our panel based tests have very good propertes even n samples wth very small values of T and. The sze of our standardzed statstc based the sem-asymptotc moment s reasonably close to the nomnal sze for all values of T and. Besdes, the power of our panel test statstc substantally exceeds taht Granger non Causalty based on sngle tme seres n all experments and n partcular for very small values of T = 1, as soon as there are very few cross-secton unts n the panel ( = 5). Fnally, approxmated crtcal values are proposed for nte T and sample. The paper s organzed as follows. Secton 2 s devoted to the de nton of the Granger causalty test n heterogenous panel data models. Secton 3 sets out the asymptotc dstrbutons of the average Wald statstc. Secton 4 derves the semasymptotc dstrbuton for xed T sample and secton 5 proposes some results of Monte Carlo experments. Secton 6 extends the results to a xed sample and the last secton provdes some concludng remarks. 2 A non causalty test n heterogenous panel data models Let us consder two covarance statonary varables, denoted x and y; observed on T perods and on ndvduals. For each ndvdual = 1; ::; ; at tme t = 1; ::; T; we consder the followng lnear model: y ;t = + KX k=1 (k) y ;t k + 5 KX k=1 (k) x ;t k + " ;t (1)

6 wth K 2 and = (1) ; :::; (K) : For smplcty, ndvdual e ects are supposed to be xed. Intal condtons (y ; K ; :::; y ; ) and (x ; K ; :::; x ; ) of both ndvdual processes y ;t and x ;t are gven and observable. We assume that lag orders K are dentcal for all cross-secton unts of the panel and the panel s balanced. In a rst part, we allow for autoregressve parameters (k) (k) and regresson coe cents slopes to d er across groups. However, contrary to Wenhold (1996) and ar-rechert and Wenhold (21), parameters (k) and (k) are constant. It s mportant to note that our model s not a random coe cent model as n Swamy (197): t s a xed coe cents model wth xed ndvdual e ects. We make the followng assumptons. Assumpton (A 1 ) For each cross secton unt = 1; ::; ; ndvdual resduals " ;t ; 8t = 1; ::; T are ndependently and normally dstrbuted wth E (" ;t ) = and nte heterogeneous varances E " 2 ;t = 2 "; : Assumpton (A 2 ) Indvdual resduals " = (" ;1 ; ::; " ;T ) ; are ndependently dstrbuted across groups. Consequently E (" ;t " j;s ) = ; 8 6= j and 8 (t; s) : Assumpton (A 3 ) Both ndvdual varables x = (x ;1 ; :::; x ;T ) and y = (y ;1 ; :::; y ;T ) ; are covarance statonary wth E y;t 2 < 1 ; E x 2 ;t < 1; E (x ;t x j;z ) ; E (y ;t y j;z ) and E (y ;t x j;z ) are only functon of the d erence t E (y ;t ) are ndependent of t: z; whereas E (x ;t ) and Ths smple two varables model consttutes the basc framework to study the Granger causalty n a panel data context. As for tme seres, the standard causalty tests consst n testng lnear restrctons on vectors : However wth a panel data model, one must be very careful to the ssue of heterogenety between ndvduals. The rst source of heterogenety s standard and comes from the presence of ndvdual effects. The second source, whch s more crucal, s related to the heterogenety of the parameters. Ths knd of heterogenety drectly a ects the paradgm of the representatve agent and so, the conclusons about causalty relatonshps. It s well known that the estmates of autoregressve parameters get under the wrong hypothess = j ; 8 (; j) are based (see Pesaran and Smth 1995 for an AR(1) process). Then, f we mpose the homogenety of coe cents, the statstcs of causalty tests can lead to 6

7 a fallacous nference. Intutvely, the estmate b obtaned n an homogeneous model wll converge to a value close to the average of the true coe cents, and that f ths mean s tself close to zero, we rsk to accept at wrong the hypothess of no causalty. Beyond these statstcal stakes, t s evdent that an homogeneous spec caton of the relaton between the varables x and y does not allow to gve some nterpretaton of the relatons of causalty as soon as at least one ndvdual of the sample has an economc behavor d erent from that of the others. For example, let us assume that there exsts a relaton of causalty for a set of countres, for whch vectors are strctly dentcal. If we ntroduce nto the sample, a set of 1 countres for whch, on the contrary, there s no relaton of causalty, what are the conclusons? Whatever the value of the rato = 1 s, the test of the causalty hypothess s nonsenscal. Gven these observatons, we now propose to test the Homogenous on Causalty (H C) hypothess. Under the alternatve we allow that there exsts a subgroup of ndvduals wth no causalty relatons and a subgroup of ndvduals for whch the varable x Granger causes y: The null hypothess of HC s de ned as: wth = (1) H : = 8 = 1; :: (2) ; :::; (K) : Under the alternatve, we allow for to d er across groups. We also allow for some, but not all, of the ndvdual vectors to be equal to (non causalty assumpton). We assume that under H 1 ; there are 1 < ndvdual processes wth no causalty from x to y: Then, ths test s not a test of the non causalty assumpton aganst the causalty from x to y for all the ndvduals, as n Holtz-Eakn, ewey and Rosen (1988). It s more general, snce we can observe non causalty for some unts under the alternatve: H 1 : = 8 = 1; ::; 1 (3) 6= 8 = 1 + 1; 1 + 2; ::; where 1 s unknown but sats es the condton 1 = < 1: The fracton 1 = s necessarly nferor to one, snce f 1 = there s no causalty for all the ndvdual of the panel, and then we get the null hypothess HC: In the opposte case 1 = ; there 7

8 s causalty for all the ndvdual of the sample. The structure of ths test s smlar to the unt root test n heterogenous panels proposed by Im, Pesaran and Shn (23). In our context, f the null s accepted the varable x does not Granger cause the varable y for all the unts of the panel. On the contrary, let us assume that the HC s rejected and f 1 = ; we have seen that x Granger causes y for all the ndvduals of the panel : n ths case we get an homogenous result as far as causalty s concerned. The DGP may be not homogenous, but the causalty relatons are observed for all ndvduals. On the contrary, f 1 > ; then the causalty relatonshps s heterogeneous : the DGP and the causalty relatons are d erent accordng the ndvduals of the sample. In ths context, we propose to use the average of ndvdual Wald statstcs assocated to the test of the non causalty hypothess for unts = 1; ::;. De nton The average statstc W Hnc assocated to the null Homogenous on Causalty ( HC) hypothess s de ned as: W Hnc = 1 X W ;T (4) =1 where W ;T denotes the ndvdual Wald statstcs for the th cross secton unt assocated to the ndvdual test H : =. In order to express the general form of ths statstc, we stack the T perods observatons for the th ndvdual s characterstcs nto T elements columns as: y ;1 k x ;1 k " ;1 y (k) = x (k) = " = 6 : 7 4 (T;1) (T;1) (T;1) : 5 " ;T : : y ;T k : : x ;T k and we de ne two (T; K) matrces : Y = h y (1) : y (2) : ::: : y (K) X = h x (1) : x (2) : ::: : x (K) Let us denote Z the (T; 2K + 1) matrx Z = [e : Y : X ] ; where e denotes a (T; 1) unt vector, and = the vector of parameters of model. The HC hypothess test can be expressed as R = where R s a (K; 2K + 1) matrx wth R = [ : I K ] : 8

9 The Wald statstc W ;T assocated to the ndvdual test H : = s de ned for each = 1; ::; as: W ;T = b R h b 2 R Z Z 1 R 1 R b = b R hr (Z Z ) 1 R 1 R b b" b" = (T 2K 1) where b s the estmate of parameter get under the alternatve hypothess, b 2 the estmate of the varance of resduals. For a small T sample, the correspondng unbased estmator 2 may be expressed as b 2 = b" b" = (T 2K 1) : It s well known that ths Wald statstc can also be expressed as a rato of quadratc forms n normal varables correspondng to the true populaton of resduals, wth: e" W ;T = (T 2K 1) e" e" = 1; ::; (5) M e" where the (T; 1) vector e" = " = "; s dstrbuted accordng (; I T ) under assumpton A 1 : The matrx and M are postve sem de nte, symmetrc and dempotent (T; T ) matrx. = Z Z Z 1 R h R Z Z 1 R 1 R Z Z 1 Z (6) M = I T Z Z Z 1 Z (7) where I T s the dentty matrx of sze T: The matrx M corresponds to the standard projecton matrx of the lnear regresson analyss. The ssue s now to determne the dstrbuton of the average statstc W Hnc under the null hypothess of Homogenous on Causalty. For that, we rst consder the asymptotc case where T and tends to n nty, and n second part the case where T s xed. 3 Asymptotc dstrbuton We propose to derve the asymptotc dstrbuton of the average statstc W Hnc under the null hypothess of non causalty. For that, we consder the case of a sequental convergence when T tends to n nty and then tends to n nty. Ths sequental 2 It s also possble to use the standard formula of the Wald statstc by substtutng the term (T 2K 1) by T: However, several software (as Evews) use ths normalsaton. 9

10 convergence result can be deduced from the standard convergence result of the ndvdual Wald statstc W ;T n a large T sample. In a non dynamc model, the normalty assumpton n A 1 would be su cent to establsh the fact for all T; the Wald statstc has a ch-squared dstrbuton wth K degrees of freedom. But n our dynamc model, ths result can only be acheved asymptotcally. Let us consder the expresson (5). Gven that under A 1 the least squares estmate b s convergent, we know that plm " M " = (T 2K 1) = 2 ";. It mples that: e" plm M e" T 2K 1 = plm 1 " M " = 1 T 2K 1 T!1 T!1 2 "; Then, f the statstc W ;T has a lmtng dstrbuton, t s the same dstrbuton of the statstcs that results when the denomnator s replaced by ts lmtng value, that s to say 1. Thus, W ;T has the same lmtng dstrbuton as e" e" : Under assumpton A 1 ; the vector e" s dstrbuted across a (; I T ) : Snce s dempotent, the quadratc form e" e" s dstrbuted as a ch-squared wth a number of degrees of freedom equal to the rank of : The rank of the symmetrc dempotent matrx s equal to ts trace, that s to say K (cf. appendx??). Then, under the null hypothess of non causalty; each ndvdual Wald statstc converges to a ch-squared dstrbuton wth K degrees of freedom: W ;T d! T!1 2 (K) 8 = 1; ::; (8) In other words, when T tends to n nty, ndvdual statstcs fw ;T g =1 are dentcally dstrbuted. They are also ndependent snce under assumpton A 2 ; resdual " and " j for j 6= are ndependent. To sum t up: f T tends to n nty ndvdual Wald statstcs W ;T are ::d: wth E (W ;T ) = K and V (W ;T ) = 2K: Then, the dstrbuton of the average Wald statstc W Hnc when T! 1 rst and then! 1; can be deduced from a standard Lndberg-Levy central lmt theorem. Theorem 1 Under assumpton A 2 ; the ndvdual W ;T statstcs for = 1; ::; are dentcally and ndependently dstrbuted wth nte second order moments as T! 1, and therefore by Lndberg-Levy central lmt theorem under the H C null hypothess, the average statstc W Hnc HC Z Hnc = sequentally converges n dstrbuton. r 2K W Hnc K d! (; 1) (9) T;!1 1

11 wth W Hnc and then! 1: = (1=) P =1 W ;T, where T;! 1 denotes the fact that T! 1 rst For a large and T sample, f the realzaton of the standardzed statstc Z Hnc s superor to the normal correspondng crtcal value for a gven level of rsk, the homogeneous non causalty (H C) hypothess s rejected. Ths asymptotc result may be useful n some macro panels. However, t should be extended to the case where T and tend to n nty smultaneously. 4 Fxed T samples and sem-asymptotc dstrbutons Asymptotcally, ndvdual Wald statstcs W ;T for each = 1; ::; ; converge toward an dentcal ch-squared dstrbuton. However, ths convergence result can not be acheved for any tme dmenson T; even f we assume the normalty of resduals. The ssue s then to show that for a xed T dmenson, ndvdual Wald statstcs have nte second order moments even they do not have the same dstrbuton and they do not have a standard dstrbuton. Let us consder the expresson (5) of W ;T under assumpton A 1 : ths s a rato of two quadratc forms n a standard normal vector. Magnus (1986) gves general condtons whch nsure that the expectatons of a quadratc form n normal varables exsts. Let us consder the moments E [(x Ax=x Bx) s ], when x s normally dstrbuted vector ; 2 I T, A s a symmetrc (T; T ) matrx and B a postve sem de nte (T; T ) matrx of rank r 1: Let us denote Q a (T; T r) matrx of full column rank T r such that BQ = : If r T 1; Magnus (1986) s theorem dent es three condtons: () If AQ = ; then E [(x Ax=x Bx) s ] exsts for all s : () If AQ 6= and Q AQ =, then E [(x Ax=x Bx) s ] exsts for s < r and does not exst for s r: () If Q AQ 6=, then E [(x Ax=x Bx) s ] exsts for s < r=2 and does not exst for s r=2: 11

12 These general condtons are done n the case where matrces A and B are determnstc. In our case, the correspondng matrces M and are stochastc, even we assume that varables X are determnstc. However, gven a xed T sample, we propose here to apply these condtons to the correspondng realsatons denoted m and : Frst, n our case the rank of the symmetrc dempotent matrx m s equal to T 2K 1. Second, snce the matrx m s the projecton matrx assocated to the realzaton z of Z ; we have by constructon m z = ; where z of full column rank 2K + 1; snce T rank(m ) = 2K + 1 Then, for a gven realsaton by constructon, the product z s d erent from zero snce z = z z z 1 R h R z z 1 R 1 R 6= Besdes, the product z z s also d erent from zero, snce z z = R h R z z 1 R 1 R 6= Then, the Magnus theorem allows us to establsh that E e" e" = e" m e" s exsts as soon as s < rank(m ) =2: We assume that ths condton s also sats ed for W ;T : E [(W ;T ) s ] = (T 2K 1) s e" s E e" e" M e" exsts f s < T 2K 1 2 In partcular, gven the realzatons of and M, we can dentfy the condton on T whch assures that second order moments (s = 2) of W ;T exsts. Proposton 2 For a xed tme dmenson T 2, the second order moments of the ndvdual Wald statstc W ;T assocated to the test H ; : = ; exst f and only f: T > 5 + 2K (1) Hence for a small T, ndvdual Wald statstcs W ;T are not necessarly dentcally dstrbuted snce matrces and M are d erent from an ndvdual to another. Besdes, they do not have standard dstrbuton as n prevous secton. However, the condton whch nsures the exstence of second order moments are the same for all unts. The second order moments of W ;T exst as soon as T > 5 + 2K or equvalently T 6 + 2K. 12

13 For a xed T sample, the statstc of non causalty test W Hnc s the average of non dentcally dstrbuted varables W ;T ; but wth nte second order moments under the condton of proposton 2. Under assumpton A 2 ; resdual " and " j for j 6= are ndependent. Consequently, ndvdual Wald W ;T for = 1; ::; are also ndependent. Then, the dstrbuton of the non causalty test statstc W Hnc the Lyapunov central lmt theorem. can be derved accordng Theorem 3 Under assumpton A 2 ; f T > 5 + 2K the ndvdual W ;T statstcs 8 = 1; ::; are ndependently but not dentcally dstrbuted wth nte second order moments, and therefore by Lyapunov central lmt theorem under the H C null hypothess, the average statstc W b HC converges. If lm!1! X 1 2 X h 1 V ar (W ;T ) E jw ;T E (W ;T )j 3! 3 =1 =1 = the standardzed statstc Z Hnc p h wth W Hnc Z Hnc = W Hnc converges n dstrbuton: 1 P =1 E (W ;T ) q 1 P =1 V ar (W ;T ) d! (; 1) (11)!1 = (1=) P =1 W ;T, where E (W ;T ) and V ar (W ;T ) respectvely denote the mean and the varance of the statstc W ;T de ned by equaton (5). The decson of rule s the same as n the asymptotc case: f the realzaton of the standardzed statstc Z Hnc s superor to the normal correspondng crtcal value for a gven level of rsk, the homogeneous non causalty (H C) hypothess s rejected. For large T; the moments used n theorem (3) are expected to converge to E (W ;T ) = K and V ar (W ;T ) = 2K snce ndvdual statstcs W ;T converge n dstrbuton to a chsquared dstrbuton wth K degrees of freedom. Then, the statstc Z Hnc Z Hnc converges to and we nd the condtons of the theorem 1. However, these asymptotc moment values could lead to poor test results, when we have small values of T: The ssue s then to evaluate the mean and the varance of the Wald statstc W ;T ; whereas ths statstc does not have a standard dstrbuton for a xed T: 13

14 The ssue s now to compute the standardzed average statstc Z Hnc. There are two man approaches to compute the two rst moments of the ndvdual Wald statstcs W ;T. Frstly, these moments can be computed va stochastc smulaton (Monte Carlo or bootstrap) of the Wald under the null. In ths case, for each cross secton unt, t s necessary to estmate the parameters of the model (, and ) and the parameters of the DGP for varables x t. Then, the varable y s smulated under the null wth ::d: normal resdual " wth zero means and varance 2 (Monte Carlo) or wth re-sampled hstorcal resduals (bootstrap). For each smulaton of the processes y and x, the ndvdual Wald statstc W ;T s computed. Fnally, usng the replcatons of W ;T, the correspondng two rst moments are estmated for each cross-secton unt. We denote ez MC the correspondng standardzed average statstc. It s obvous that ths method can be tme consumng, especally f we consder very large panel. Secondly, we propose here an approxmaton of E (W ;T ) and V ar (W ;T ) based on the results of Magnus (1986) theorem. Let us consder the expresson of the Wald W ;T as a rato of two quadratc forms n a standard normal vector under assumpton A 1 : e" W ;T = (T 2K 1) e" e" (12) M e" where the (T; 1) vector e" = " = "; s dstrbuted accordng (; I T ) where matrces and M are dempotent and symmetrc (and consequently postve sem-de nte). For a gven T sample, let us denote respectvely and m, the realzatons of matrces and M : We propose here to apply the Magnus (1986) theorem to the quadratc forms n a standard normal vector de ned as: e" fw ;T = (T 2K 1) e" e" m e" where matrces and m are dempotent and symmetrc (and consequently postve sem-de nte). Theorem 4 (Magnus 1986) Let e" be a normal dstrbuted vector wth E (e" ) = and E e" e" = IT : Let P be an orthogonal (T; T ) matrx and a dagonal (T; T ) (13) matrx such that P m P = P P = I T (14) 14

15 Then, we have, provded the expectaton exsts for s = 1; 2; 3:: : 8 9 e" s E e" 1 X Z 1 < sy = e" = m e" (s 1)! s (v) : ts 1 j j [trace (R )] n j dt (15) ; v where the summaton s over all (s; 1) vectors v = (n 1 ; ::; n s ) whose elements n j are nonnegatve ntegers satsfyng P s j=1 jn j = s s (v) = s! 2 s sy [n j! (2j) n j ] j=1 and s a dagonal postve de nte (T; T ) matrx and R a symmetrc (T; T ) matrx 1 j=1 (16) gven by: = (I T + 2 t ) 1=2 R = P P (17) In our case, we are nterested by the two rst moments. For the rst order moment (s = 1), there s only one scalar v = n 1 whch s equal to one: Then, the quantty 1 (v) s equal to one. For the second order moment (s = 2), there are two vectors v = (n 1 ; n 2 ) whch are respectvely de ned by v 1 = (; 1) and v 2 = (2; ) : Consequently 2 (v 1 ) = 2 and 2 (v 2 ) = 1: Gven these results, we can compute the exact two correspondng moments of the statstc f W ;T as: E fw;t = (T 2K 1) fw;t 2 E = (T 2K 1) 2 2 Z 1 Z 1 t j j trace (R ) dt + j j trace (R ) dt (18) Z 1 t j j [trace (R )] 2 dt where matrces and R are de ned n theorem (4). Both quanttes j j and trace (R ) can be computed analytcally n our model gven the propertes of these matrces. Snce s ssued from the orthogonal decomposton of the dempotent matrx m ; wth rank(m ) = T 2K 1 (cf. appendx??), ths matrx s a zero except the rst block whch s equal to the T 2K 1 dentty matrx (correspondng to the characterstc roots of m whch are non null). Then, for a scalar t 2 R + ; the matrx = (I T + 2 t ) 1=2 can be parttoned as: D (t) (T 2K 1;T 2K 1) = (T;T ) (2K+1;T 2K 1) 15 (T 2K 1;2K+1) I 2K+1 (2K+1;2K+1) 1 C A (19)

16 where I p denotes the dentty matrx of sze p: The dagonal block D (t) s de ned as D (t) = (1 + 2t) 1 2 IT 2K 1 : Then, the determnant of can be expressed as: j j = (1 + 2t) ( T 2K 1 2 ) (2) Besdes, the trace of the matrx R can be computed as follows. Snce for any non sngular matrces B and C; the rank of BAC s equal to rank of A; we have here: rank (R ) = rank P P = rank P P snce the matrx s non sngular. sngularty of P, we get: Wth the same transformaton, gven the non rank (R ) = rank P P = rank ( ) Fnally, the rank of the realsaton s equal to K; the rank of. trace (R ) = K Gven these results, the two rst moments (equatons 18 and 19) of the statstc fw ;T based for a gven T sample on realzatons and m ; can be expressed as: E fw;t = (T 2K 1) K Z 1 fw;t 2 E = (T 2K 1) 2 2K + K 2 Then, we get the followng results. (1 + 2t) ( T 2K 1 2 ) dt Z 1 t (1 + 2t) ( T 2K 1 2 ) dt Proposton 5 For a xed T sample, where T sats es the condton of proposton (2), gven realzatons and m of matrces and M (equatons 6 and 7), the exact two rst moments of the ndvdual statstcs f W ;T ; for = 1; :::; ; de ned by equaton (13) are respectvely: E fw;t (T 2K 1) = K (T 2K 3) V ar fw;t = 2K (T 2K 1)2 (T K 3) (T 2K 3) 2 (T 2K 5) as soon as the tme dmenson T sats es T 6 + 2K. (21) (22) 16

17 The proof of ths proposton s done n appendx A. It s mportant to verfy that for large T sample, the moments of the ndvdual statstc f W ;T tend to the correspondng moments of the asymptotc dstrbuton of W ;T snce 8 = 1; :::; : lm E fw;t = K lm V ar fw;t = 2K T!1 T!1 Both moments correspond to the moments of a F (K; T 2K 1). Indeed, n ths dynamc model the F dstrbuton can be used as an approxmaton of the true dstrbuton of the statstc W ;T =K for a small T sample. Then, the use of the Magnus theorem gven the realzatons and m to approxmate the true moments of the Wald statstc s equvalent to assert that the true dstrbuton of W ;T can be approxmated by the F dstrbuton. We propose n ths paper to approxmate the two rst moments of the ndvdual Wald statstc W ;T by the two rst moments of the statstcs W f ;T based on the realzatons and m of the stochastc matrces and M (equatons 21 and 22). So, for T 6 + 2K; we assume that: 1 X =1 E (W ;T ) ' E fw;t (T 2K 1) = K (T 2K 3) (23) 1 X =1 V ar (W ;T ) ' V ar fw;t = 2K (T 2K 1)2 (T K 3) (T 2K 3) 2 (T 2K 5) (24) ez Hnc Gven these approxmatons, we compute an approxmated standardzed statstc for the average Wald average statstc W Hnc ez Hnc = p h W Hnc r V ar of the HC hypothess. E fw;t (25) fw;t For a large sample, under the Homogenous on Causalty (H C) hypothess, we assume that the statstc e Z Hnc average Wald statstc Z Hnc. follows the same dstrbuton as the standardzed 17

18 Proposton 6 Under assumptons A 1 and A 2 ; for a xed T dmenson wth T > 5 + 2K; the standardzed average statstc Z e Hnc s ez Hnc = wth W Hnc 2 K (T 2K 5) (T 2K 3) (T K 3) (T 2K 1) W Hnc = (1=) P =1 W ;T. converges n dstrbuton: d K! (; 1) (26)!1 Consequently, the testng procedure of the H C hypothess s very smple and works as follows. statstcs W ;T For each ndvdual of the panel, we compute the standard Wald assocated to the ndvdual hypothess H ; : = wth 2 R K Gven these realzatons, we get a realzaton of the average Wald statstc W Hnc : We compute the realzaton of the approxmated standardzed 3 statstc e Z Hnc the formula (26) or we compute the statstc e Z MC prevously descrbed. For a large sample, f the value of e Z Hnc accordng to based on the Monte Carlo procedure (or Z e MC ) s superor to the normal correspondng crtcal value for a gven level of rsk, the homogeneous non causalty (H C) hypothess s rejected. When the panel s unbalanced or when the lag order K s spec c to each cross-unt, the standardzed statstc Z e Hnc p h ez Hnc = = p " " W Hnc r 1 must be adapted as follows: 1 P =1 E fw;t 1 P =1 V ar fw;t W Hnc 1 X =1 X =1 # K (T 2K 1) (T 2K 3) # 1=2 2K (T 2K 1) 2 (T K 3) (T 2K 3) 2 (27) (T 2K 5) 3 If one uses the standard de nton of the Wald statstc wth the T normalzaton, t s necessary to adapt the formula (26) by substtutng the quantty T 2K 1 by T: More precsely, f the Wald ndvdual statstc W ;T s de ned as: W ;T = b h R R ZZ 1 R 1 R b = b" b" =T then the standardzed average Wald statstc Z e Hnc s de ned as: s ez Hnc (T 4) = 2 K (T + K 2) T T 2 W Hnc K 18

19 where T > 5 + 2K denotes the tme dmenson for the th cross-secton unt. 5 Monte Carlo smulaton results In ths secton, we propose some Monte Carlo experments to examne nte sample propertes of the alternatve panel-based non causalty tests. We consder three sets of Monte Carlo experments. The rst set focuses on the benchmark model 4 : y ;t = + y ;t k + x ;t k + " ;t (28) The parameters of the model are calbrated as follows. The auto-regressve parameters are drawn accordng to a unform dstrbuton on ] 1; 1[ n order to satsfy the statonarty assumpton A 3. The xed ndvdual e ects ; = 1; ::; are generated accordng to a (; 1). Indvdual resduals are drawn n normal dstrbuton wth zero means and heterogeneous varances 2 ";. The varance 2 "; are generated accordng to a unform dstrbuton on [:5; 1:5]. Under the null of HC; = for all. Under the alternatve, s d erent from for all,.e. 1 =. In ths case, parameters are generated accordng to a (; 1) at each smulaton. The second set of experments allows for heterogenety of the causalty relatonshps under the alternatve H 1 : = for = 1; ::; 1 and 6= for = 1 + 1; ::; : In these experments, we evaluate the emprcal power of our panel tests for varous values of the rato n 1 = 1 =. We consder a case n whch there s no causalty for one crosssecton unt out of two (n 1 = :5) and a case wth no causalty for nne cross-secton unts out of ten (n 1 = :9). The thrd set of experments focuses on a model wth K lags: y ;t = + P K k=1 (k) y ;t k + P K k=1 (k) x ;t k + " ;t (29) where the auto-regressve parameters (k) on ] K; K[ under the constrant that the roots of (z) = P K unt crcle. are drawn accordng to a unform dstrbuton k=1 (k) z k le outsde the The others parameters are calbrated as n the rst set of experments. We consder two cases, denoted A and B. In Monte Carlo experments of case A, we 4 We also carred out number of other experments wth other data generatng processess. The results are smlar to the ones reported n ths secton and are avalable from the author on request. 19

20 compute the sze and the power (n 1 = ) of our panel tests for a lag order K equal to 2. In case B, we assume that there s a ms-spec caton of the lag order. The underlyng data are generated by a model wth one lag (K = 1), but the ndvdual Wald statstcs (and the correspondng standardzed average panel statstcs) are computed from the smulated seres wth a regresson model wth two lags (K = 2). The second set of experments were carred out for = 6 (only for the case n 1 = :5), 1, 2, 5 and T = 1, 25, 5, 1. The other experments were carred out for = 1, 5, 1, 25, 5 and T = 1, 25, 5, 1. We used 1; replcatons to compute emprcal sze and power of the tests at the 5% nomnal sze. All the parameters values such as ; ; "; or are generated ndependently at each smulaton. All the experments are carred out usng the followng two statstcs: the Z Hnc based on the asymptotc moments (equaton 9) and the e Z Hnc based on the approxmaton of moments for a xed T sample (equaton 26). The results of the rst set of experments are summarzed n Table 1. As a benchmark, n the rst row of ths table we report the results for the Granger non-causalty test based on a Wald statstc and sngle tme seres ( = 1). For large T samples, the standardzed statstc Z Hnc based on the asymptotc moments K and 2K (whch are vald f T tends to n nty) has a correct sze. Our panel test s more powerful than tests based on sngle tmes seres even n panel wth very few cross-secton unts. For nstance, for a typcal panel of macroeconomc annual data (T = 5), the power of non causalty test rses from :71 wth a sngle tme seres test ( = 1) to :99 wth a panel test as soon as only ve crosssecton unts are ncluded ( = 5). However, for small values of T, the standardzed statstc Z Hnc s overszed and the extent of ths over-rejecton worsens as ncreases. Ths over-rejecton can be ntutvely understood as follows. The Wald statstc based on sngle tme seres s slghtly over-szed for small values of T. So, under the null, we can observe large values (superor to the ch-squared crtcal value) of the ndvdual Wald statstcs for some cross-secton unts. For a gven value of, these large values (that range from the ch-squared crtcal value to n nty) are not annhlated by the realsatons obtaned for other cross-secton unts snce the latter only range from to 2

21 the ch-squared crtcal value. Consequently, the cross-secton average (W Hnc statstc) tends to be larger than the correspondng normal crtcal value. The more ncreases, the more the probablty to obtan large values for some cross-secton unt ncreases. So, for small values of T, the Z Hnc and ths tendency s stronger as s allowed to ncrease. test tends to over-reject the null of non causalty On the contrary, the sze of the standardzed Z ehnc statstc based on the semasymptotc moments (de ned for xed values of T ) s reasonably close to the nomnal sze for all values of T and. The sem asymptotc standardzed Z ehnc statstc substantally augments the power of non-causalty tests appled to sngle tme seres even for very small values of. For example, when T = 1, the power of our panel test s equal to :73 even wth only ve cross-secton unts ( = 5). In ths case, the test based on tme seres ( = 1) has only a power of :43. The Z e Hnc statstc has a correct sze, and ts power rses monotoncally and quckly wth and T: For T = 1, when s larger than 1, the power of the Z e Hnc test s near to one. Ths mprovement n power can be ntutvely understood as follows. Indvdual statstcs are bounded from below (by zero) but may take arbtrarly large value. Hence, when averagng among ndvdual Wald statstcs, the abnormal realsatons (realsatons below the ch-squared crtcal value) are annhlated by the realsatons on the true sde (large). In power smulatons summarzed n Table 1, we assume that there s causalty for all the cross-secton unts of the panel. The ssue s now to determne the n uence of heterogenety of causalty relatonshps,.e. the relatve mportance to 1 wth respect to, on the power of our panel tests. The results for the second set of experments are summarzed n Table 2. For n 1 = :5 and n 1 = :9; we can verfy that the powers of the standardzed statstcs Z Hnc ;t and e Z Hnc are slghtly reduced compared to the case n 1 = (Table 1). But, even n the worse case studed (n whch there s causalty for only one cross-secton unt out of ten,.e. n 1 = :9), the powers of our panel tests reman reasonable even for very small values of T and. For nstance, wth T = 25 and = 1 ( 1 = 9), the power of the Z ehnc statstc s equal to :42. Wth twenty cross-secton unts (causalty for two cross-sectons unts f n 1 = :9), ts power ncreases 21

22 to :6. The results for the thrd set of experments are summarzed n Table 3. In case A, we consder a model wth two lags. The results are qute smlar to the results obtaned for the benchmark case wth one lag (Table 1): the powers of the panel average statstcs substantally exceed that of sngle tmes seres non-causalty test, the Z Hnc ;t over-szed and the e Z Hnc statstc s has a correct sze for all T and values. Smlar results (not reported) are obtaned when we consder heterogeneous lag orders K. In case B, we study the n uence of a m-spec caton of the lag-order. When the lag order s over estmated for all the cross-secton unts, the power of our panel test statstcs s reduced but remans reasonable. Wth T = 1, the power of the panel e Z Hnc :36 wth ve cross-secton unts to :87 wth twenty cross-secton unts 6 Fxed T and xed dstrbutons If and T are xed, the standardzed statstc e Z Hnc statstc rses from and the average statstc W Hnc not converge to standard dstrbutons under the H C hypothess. Two solutons are then possble: the rst conssts n usng the mean Wald statstc W Hnc do and to compute the exact sample crtcal values, denoted c () ; for the correspondng szes and T va stochastc smulatons. We propose the results of an example of such a smulaton n table??. As n Im, Pesaran and Shn (23), the second soluton conssts n usng the approxmated standardzed statstc e Z Hnc and to compute an approxmaton of the correspondng crtcal value for a xed. Indeed, we can show that: h Pr ez Hnc < ez () = Pr W Hnc < c () where ez () s the percent crtcal value of the dstrbuton of the standardzed statstc under the HC hypothess. The crtcal value c () of W Hnc r where E fw;t c () = ez () and V ar fw;t 1 var fw;t + E fw;t s de ned as: respectvely denote the mean and the varance of the ndvdual Wald statstc de ned by equatons (21) and (22). Gven the result of proposton (6), we know that the crtcal value ez () corresponds to the percent 22

23 crtcal value of the standard normal dstrbuton, denoted z f tends to n nty whatever the sze T: For a xed, the use of the normal crtcal value z to bult the correspondng crtcal value c () s not founded, but however we can propose an approxmaton ec () based on ths value. ec () = z r 1 var fw;t + E fw;t (3) or equvalently: ec () = z s (T 2K 1) (T 2K 3) 2K (T K 3) K (T 2K 1) + (T 2K 5) (T 2K 3) (31) In Table 4 the smulated 5% crtcal values c (:5) get from 5 replcatons of the benchmark model under H are reproduced. The approxmated 5% crtcal values ec (:5) are also reported. As we can observe, both crtcal values are very smlar and the same result can be obtaned for larger lag-order K. 23

24 7 Concluson In ths paper, we propose a smple Granger (1969) non-causalty test for heterogenous panel data models. Under the null hypothess of Homogeneous on Causalty (H C), there s no causal relatonshp for all the cross-secton unts of the panel. Under the alternatve, there are two subgroups of cross-secton unts: one wth causal relatonshps from x to y (but not necessarly wth the same DGP ) and another subgroup where there s no causal relatonshp from x to y: As n panel unt root test lterature, our test statstc s smply de ned as the cross-secton average of ndvdual Wald statstcs assocated to the standard Granger causalty tests based on sngle tme seres. Under a cross-secton ndependence assumpton, we show that ths average statstc converge to a standard normal dstrbuton when T and tend sequentally to n nty. For xed T sample; the sem-asymptotc dstrbuton s characterzed. In ths case, ndvdual Wald statstcs do not have a standard ch-squared dstrbuton. However, under very general settng, Wald statstcs are ndependently dstrbuted wth nte second order moments. For a xed T, the Lyapunov central lmt theorem s then su cent to get the dstrbuton of the standardzed average Wald statstc when tends to n nty. The two rst moments of ths normal sem-asymptotc dstrbuton correspond to the cross-secton averages of the correspondng theoretcal moments of the ndvdual Wald statstcs. The ssue s then to evaluate the two rst moments of standard Wald statstcs for small T samples. In ths paper we propose a general approxmaton of these moments and a correspondng standardzed average Wald statstc. One of the man advantages of our testng procedure s that t s very smple to mplement: the standardzed average Wald statstcs are smple to compute and have a standard normal asymptotc dstrbuton. Besdes, Monte Carlo smulatons show that our panel statstcs lead to substantally augment the power of the Granger noncausalty tests appled to sngle tme seres even for samples wth very small T and dmensons. Fnally, our test statstcs (based on cross secton average of ndvdual Wald statstcs) do not requre any partcular panel estmaton. 24

25 Our testng procedure has the same advantages, but also the same drawbacks as the approach used by Im, Pesaran and Shn (23) n the context of panel unt root tests. Frstly, the rejecton of the null of Homogeneous on Causalty does not provde any gudance as to the number or the dentty of the partcular panel members for whch the null of non causalty s rejected. Secondly, the asymptotc dstrbuton of our statstcs s establshed under the assumpton of cross-secton ndependence. As for panel unt root tests, t s now necessary to develop second generaton panel non causalty tests that allow for general or spec c cross-secton dependences. Ths s precsely our objectve for further researches. 25

26 Appendx A Exact moments of ndvdual Wald f W ;T The two noncentered moments of W f ;T are respectvely de ned as: Z 1 E fw;t = (T 2K 1) K (1 + 2t) ( T 2K 1 2 ) dt fw;t 2 E = (T 2K 1) 2 2K + K 2 Z 1 t (1 + 2t) ( T 2K 1 2 ) dt Let us denote for smplcty T e = (T 2K 1) =2: For the rst order moment, we get: Z 1 E fw;t = 2T e K (1 + 2t) et dt 2 3 = 2T e K 4 (1 + 2t) T e et + 1 Snce the quantty 2 et = 2 T e K 2 et 1 1 = T 2K 3 s strctly d erent from zero under the condton of proposton (2), we get E fw;t (T 2K 1) = K (T 2K 3) (32) For the second order moment, the de nton s: fw;t 2 E = 4 T e2 2K + K 2 Z 1 t (1 + 2t) et dt By ntegratng by parts, ths expresson can be transformed as: 82 3 fw;t 2 E = 4 T e2 2K + K 2 < 4 t (1 + 2t) T e : 2 et et + 1 Under under the condton of proposton (2) we have T e > 1; then: fw;t 2 E = 4 T e2 2K + K 2 Z 1 (1 + 2t) et dt 2 et 1 = 4 T e2 2K + K 2 2 et 1 = 4 T e2 2K + K 2 2 et (1 + 2t) T e et et 2 1 Z 1 9 = (1 + 2t) et dt ; 26

27 After smpl catons, we have : fw;t 2 T E = e2 2K + K 2 = (T 2K 1)2 2K + K 2 et 1 et 2 (T 2K 3) (T 2K 5) (33) Under the condton T > 5 + 2K; ths second order moment exsts as t was prevously establshed n proposton (2). Fnally, we can compute the second order centered moment, V ar fw;t as: V ar fw;t fw;t 2 = E 2 E fw;t = (T 2K 1)2 2K + K 2 (T 2K 3) (T 2K 5) K (T 2K 1) 2 (T 2K 3) After smpl catons, we have: V ar fw;t = 2 K (T 2K 1)2 (T K 3) (T 2K 3) 2 (T 2K 5) (34) 27

28 References [1] Granger, C.W.J Investgatng causal relatons by econometrc models and cross-spectral methods, Econometrca 37(3); [2] Granger, C.W.J Testng for causalty. Journal of Economc Dynamcs and Control 2, [3] Granger C.W.J. 23. Some aspects of causal relatonshps, Journal of Econometrcs 112, [4] Holtz-Eakn D., ewey W, Rosen H.S Estmatng vector autoregressons wth panel data. Econometrca 56, [5] Hsao, C. 23. Analyss of panel data, Cambrdge Unversty Press [6] Im, K.S., Pesaran, M.H., Shn, Y. 23. Testng for Unt Roots n Heterogenous Panels. Journal of Econometrcs, 54, [7] Magnus, J.R The exact moments of a rato of quadratc forms n normal varables, Annales d Econome et de Statstques, 4, [8] ar-rechert, U. Wenhold, D. 21, Causalty tests for cross-country panels: a look at FDI and economc growth n less developed countres, Oxford Bulletn of Economcs and Statstcs, 63, [9] Pesaran, H.M. Smth, R Estmatng long-run relatonshps from dynamc heterogenous panels, Journal of Econometrcs, 68, [1] Swamy, P.A E cent nference n a random coe cent regresson model, Econometrca, 38, [11] Wenhold, D Tests de causalté sur données de panel : une applcaton à l étude de la causalté entre l nvestssement et la crossance, Econome et Prévson, 126,

29 Table 1: Sze and Power of Panel Granger on-causalty Tests T = 1 T = 25 T = 5 T = 1 Test Sze Power Sze Power Sze Power Sze Power 1 Wald Z ch ez ch Z ch ez ch Z ch ez ch Z ch ez ch otes: Ths table reports the sze and power of the Wald statstc based on tme seres ( = 1), the panel standardzed statstc Z Hnc based on asymptotc moments de ned by (9) and the panel standardzed statstc Z e Hnc based on sem-asymptotc moments de ned by (26). The underlyng data are generated by y ;t = + y ;t k + x ;t k + " ;t, for = 1; ::; and t = 1, 99; ::; T: At each replcaton, the autoregressve parameters are drawn accordng to a unform dstrbuton on ] 1; 1[ and the xed ndvdual e ects are generated accordng to a (; 1). Indvdual resduals are ::d: ; 2 ";. The varance 2 "; are generated accordng to a unform dstrbuton on [:5; 1:5]. The sze ( = ; = 1; :; ) and the power of the tests are computed at the ve percent nomnal level. Under the alternatve (power smulatons), s d erent from for all,.e. 1 =. The parameters are generated accordng to a (; 1) : umber of replcatons s set to 1;. 29

30 Table 2: Power of Panel Granger on-causalty Tests: Experments wth Heterogenety n Causalty Relatonshps (n1>) Power of Panel HC Tests wth n 1 = :5 1 Test T = 1 T = 25 T = 5 T = Z ch ez ch Z ch ez ch Z ch ez ch Z ch ez ch Power of Panel HC Tests wth n 1 = :9 1 Test T = 1 T = 25 T = 5 T = 1 6 Z ch ez ch 1 9 Z ch ez ch Z ch ez ch Z ch ez ch otes: Ths table reports the power of the panel standardzed statstc Z Hnc based on asymptotc moments de ned by (9) and the panel standardzed statstc Z e Hnc based on sem-asymptotc moments de ned by (26). The underlyng data are generated by y ;t = + y ;t k + x ;t k + " ;t, for = 1; ::; and t = 1, 99; ::; T: At each replcaton, the auto-regressve parameters are drawn accordng to a unform dstrbuton on ] 1; 1[ and the xed ndvdual e ects are generated accordng to a (; 1). Indvdual resduals are ::d: ; 2 ";. The varance 2 "; are generated accordng to a unform dstrbuton on [:5; 1:5]. The power s computed at the ve percent nomnal level. We consder power smulatons wth heterogenety of causalty relatonshps. The parameters are equal to (non-causalty) for = 1; ::; 1 and d erent from (causalty) for = 1 +1; ::;. In ths case, are generated accordng to a (; 1) : The rato n 1 = 1 =; wth n 1 < 1; denotes the fracton of crosssecton unts for whch there s no causalty under the alternatve. 3