Testing for seasonal unit roots in heterogeneous panels

Transcription

1 Testng for seasonal unt roots n heterogeneous panels Jesus Otero * Facultad de Economía Unversdad del Rosaro, Colomba Jeremy Smth Department of Economcs Unversty of arwck Monca Gulett Aston Busness School Unversty of Aston January 004 ABSTRACT Ths paper uses the approach of Im, Pesaran Shn (003) to propose seasonal unt root tests for dynamc heterogeneous panels based on the means of the ndvduals HEGY test statstcs. The stardsed t-bar F-bar statstcs are smply averages of the HEGY tests across groups. These statstcs converge to stard normal varates. Keywords: Heterogeneous dynamc panels; Monte Carlo; seasonal unt roots; JEL classfcaton: C; C; C; C3 Contact Address: Jesus Otero Facultad de Economía Unversdad del Rosaro Calle 4 # 4-69 Bogota, Colomba Phone: (+) Ext. 66 E-mal: jotero@urosaro.edu.co *Ths paper was started whle the frst author was a vstng research fellow n the Department of Economcs at the Unversty of arwck. The frst author would lke to thank the Unversdad del Rosaro for ts fnancal assstance.

2 . Introducton Im, Pesaran Shn (003) (IPS) proposed a test for the presence of unt roots n panels, that combnes nformaton from the tme-seres dmenson wth that from the cross-secton dmenson, such that fewer tme observatons are requred for the test to have power. Many economc tme seres contan mportant seasonal components a varety of tests have been proposed to test for seasonal unt roots see Osborn Ghysels (00) for a revew of these tests. Of these tests the one proposed by Hylleberg, Engle, Granger Yoo (990) (HEGY) has proved to be the most popular. In ths paper, we look at usng the approach of IPS to nvestgate the performance of the HEGY test n dynamc heterogeneous panels. Based on Monte Carlo smulatons we fnd that the stardsed averaged test statstcs from the HEGY auxlary regresson follow a stard normal dstrbuton even for a relatvely small number of data ponts. The plan of the paper s as follows. Secton brefly revews the IPS approach to unt root testng n panels sets up the model used to develop the HEGY panel seasonal unt root tests. Secton 3 presents the Monte Carlo results.. IPS unt root test basc framework IPS presented a method to test for the presence of unt roots n dynamc heterogeneous panels. They consder a sample of N cross secton unts observed over T tme perods. The IPS test averages the (Augmented) Dckey-Fuller statstc obtaned across the N cross-sectonal unts of the panel (denoted as NT N tt N tbar, where t T s the ADF test for the th cross-sectonal unt). show that a sutable

3 stardsaton of the tbar NT statstc, denoted as Z tbar, follows a stard normal dstrbuton. Generalsng the HEGY test for seasonal unt roots, to a panel n whch there s sample of N cross sectons (ndustres, countres) observed over T tme perods: ( L) y y y y y,,, N, t,, T () 4t t t t 3 3t 4 3t t s where t D, t s st j D st n season s, ( L) s a 0 otherwse th p ordered polynomal n the lag operator, L, y t y t y t y t y t 3, t ~ N(0, ) yt yt yt yt yt3, y3 t yt yt y4t 4yt yt yt 4. HEGY test for the exstence of a unt root by testng H 0 : 0 aganst H: 0, for the exstence of a seasonal unt root by testng H0 : 0 aganst H: 0 smultaneously testng H0 : aganst H: 3 0, 4 0. A null hypothess of a seasonal unt root s only rejected when both the t-test for the jont F-test for 3 4 are rejected. Subsequently, Ghysels et. al. (994) suggest usng a test of H0 : aganst H: 0, 3 0, 4 0. In a panel context, the null hypothess to test the presence of a unt root, for example, becomes H 0 : 0 aganst H 0 : 0 for,, N, 0, for N, N,, N. Ths allows some, but not all, of the ndvdual seres to have a unt root, but assumes that a non-zero fracton of the processes are statonary. 3. Monte Carlo smulaton results In ths secton we undertake Monte Carlo smulaton to examne the fnte sample propertes of the HEGY-IPS test. Smulatons are undertaken under the null hypothess, n equaton (): 3

4 y y y,,, N, t,, T () t t4 t j 4 t t j p where t ~ N(0, ) ~ U[0.,.], are generated ndependently of t are fxed for all replcatons, where N (,,0,,,40) T (0,3,40,60,00). The HEGY statstcs from estmatng equaton () for the th group are gven by the t-ratos on j, j, the F-tests of the jont sgnfcance of, 3,, 3 4. Denote the estmated t-rato as jt t, t jt ˆ j 0 se( ˆ ) j j, the F-test as F jt, F ( ˆ ) ˆ ˆ ( ˆ jt Rj RV j R )/,,3 j Rj j j where R, R the, ˆ ˆ ˆ ˆ ˆ 3 4 estmated varance-covarance matrx from equaton (), s wrtten n parttoned form Vˆ Cˆ Cˆ as: Vˆ ˆ ˆ ˆ C V C Cˆ Cˆ Vˆ, where, for example, V ˆ s the estmated (4 4) varance-covarance matrx for the coeffcents on, C ˆ s the estmated varance-covarance matrx between the terms. For a fxed T defne the average statstcs: 4

5 N tbar t j, j NT jt N N F bar F, j,3. j NT jt N Followng IPS, consder the stardsed statstcs: N N t jbarnt E t jt( p,0 0) N t (0,),, jbar N j N Var t jt ( p,0 0) N N N F jbarnt E F jt ( p,0 0) N N(0,), j,3 Fbar j N Var F jt ( p,0 0) N EF (,0 0) jt p where E tjt ( p,0 0) Var t (,0 0) jt p Var F jt ( p,0 0) are the mean varance of t ( F jt jt ) n the HEGY model, when Table reports the values of Et jt ( p,0 0) Var t jt ( p,0 0), j, E F jt ( p,0 0) Var F (,0 0) jt p, j,3, for dfferent values of T p, for dfferent combnatons of determnstc components n the HEGY model. These results are based upon 0,000 replcatons. Through smulatons t appears that n the HEGY model (when p 0 ), the second moment of t jt exsts only for T 6 (when there s a constant trend) for T 0 (when there are seasonal dummy varables). In addton, for the second

6 moment of F jt to exst requres at least T 0 for all combnatons of the determnstc components. e now consder three Monte Carlo experments to examne the sze power (at the % sgnfcance level) of the HEGY-IPS test, usng,000 replcatons. Table reports the sze of the tests when there s no seral correlaton the model ncludes a constant a constant trend as determnstc components. The tests for both the tbar tbar are approxmately correctly szed. However, both the Fbar tests are slghtly over-szed especally for smaller N T. Ths table also reports the power of the HEGY-IPS test, when the data s generated as y 0.9 y,,, N, t,, T. t t4 t In a second set of experments, we allow for the presence of heterogeneous AR() seral correlaton n t, such that, t t t,, N, t,, T where t ~ N(0, ), ~ U[0.,0.4] s generated ndependently of t. Table 3 reports the sze of the HEGY-IPS test for p=0,,,3,4, when there s only a constant n the HEGY model. The table demonstrates the mportance of not underestmatng the order of the lag length, wth the emprcal sze for p=0, substantally dfferent from the nomnal %, wth tbar markedly under-szed, but all of the other tests becomng ncreasngly over-szed as T ncreases. There are lttle costs n terms of sze to over-specfyng the lag length, wth the emprcal sze of Fbar actually mprovng. However, the power of all of the tests falls wth an over-specfed lag length. In a thrd set of experments, we allow for the presence of heterogeneous MA() seral correlaton n t, such that, 6

7 t t t,, N, t,, T where t ~ N(0, ), ~ U[ 0.4, 0.] s generated ndependently of t. Table 4 reports the sze of the HEGY-IPS test for p=0,,,3,4. In ths case there are severe sze dstortons for p=0, wth tbar massvely over-szed. The other tests are also over-szed ths becomes ncreasngly so as both T N ncrease. Increasng p mproves the sze of these tests, but even for p=3 there s consstent evdence that tbar, Fbar are all stll margnally over-szed. References Ghysels, E., Lee, H. S. Noh, J. (994), Testng for unt roots n seasonal tme seres some theoretcal extensons a Monte Carlo nvestgaton, Journal of Econometrcs, 6, Ghysels, E. Osborn, D. R. (00), The econometrcs analyss of seasonal tme seres, Cambrdge Unversty Press. Hylleberg, S., Engle, R. F., Granger, C.. J. Yoo, B. S. (990), Seasonal ntegraton contegraton, Journal of Econometrcs, 69, -. Im, K. S., Pesaran, M. H. Shn, Y. (003), Testng for unt roots n heterogeneous panels, Journal of Econometrcs,, 3-4. Maddala, G. S. u, S. (999), A comparatve study of unt root tests wth panel data a new smple test, Oxford Bulletn of Economcs Statstcs, 6, 63-6.

8 Table : Mean varance correcton for t bar t bar tbar tbar p T=0 T=3 T=40 T=60 T=00 T=0 T=3 T=40 T=60 T=00 Constant, seasonal dummes, trend 0 varance mean varance mean varance mean varance mean varance Constant, seasonal dummes, no trend 0 varance mean varance mean varance mean varance mean varance Constant, no seasonal dummes, tend 0 varance mean varance mean varance mean varance mean varance Constant, no seasonal dummes, no trend 0 varance mean varance mean varance mean varance mean varance No constant, no seasonal dummes, no trend 0 varance mean varance mean varance mean varance mean varance

9 Table (cont d): Mean varance correcton for Fbar Fbar p T=0 T=3 T=40 T=60 T=00 T=0 T=3 T=40 T=60 T=00 Constant, seasonal dummes, trend 0 varance mean varance mean varance mean varance mean varance Constant, seasonal dummes, no trend 0 varance mean varance mean varance mean varance mean varance Constant, no seasonal dummes, trend 0 varance mean varance mean varance mean varance mean varance Constant, no seasonal dummes, trend 0 varance mean varance mean varance mean varance mean varance No constant, no seasonal dummes, no trend 0 varance mean varance mean varance mean varance mean varance

10 Table : Sze power of the HEGY-IPS test: No seral correlaton N T=0 T=3 T=40 T=60 T=00 tbar tbar Fbar tbar t bar F bar F 3 bar tbar t bar SIZE Constant, no seasonal dummes, no trend Constant, no seasonal dummes, trend POER Constant, no seasonal dummes, no trend Constant, no seasonal dummes, trend NOTE: For power the DGP s wrtten as yt yt t, where 0.9, ( ), ~ N(0,), t ~ N(0, ) ~ U[0.,.]. F bar F 3 bar tbar t bar F bar F 3 bar tbar t bar F bar F 3 bar 0

11 Table 3: Sze of the HEGY-IPS test: AR() errors ~ U[0.,0.4], constant, no seasonal dummes, no trend N p T=0 T=3 T=40 T=60 T=00 tbar tbar Fbar tbar t bar F bar F 3 bar tbar t bar F bar F 3 bar tbar t bar F bar F 3 bar tbar t bar F bar F 3 bar

12 Table 4: Sze of the HEGY-IPS test: MA() errors ~ U[ 0.4, 0.], constant, no seasonal dummes, no trend N p T=0 T=3 T=40 T=60 T=00 tbar tbar Fbar tbar t bar F bar F 3 bar tbar t bar F bar tbar t bar F bar F 3 bar tbar t bar F bar F 3 bar