Critical Values for IPS Panel Unit Root Tests: A Response Surface Analysis Rajaguru, Gulasekaran

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1 Bond Unversy Research Reposory Crcal Values for IPS Panel Un Roo ess: A Response Surface Analyss Rajaguru, Gulasearan Publshed: 0/0/00 Documen Verson: Peer revewed verson Ln o publcaon n Bond Unversy research reposory. Recommended caon(apa): Rajaguru, G. (00). Crcal Values for IPS Panel Un Roo ess: A Response Surface Analyss. he s Ausralan Conference of Economss, Adelade, Ausrala. General rghs Copyrgh and moral rghs for he publcaons made accessble n he publc poral are reaned by he auhors and/or oher copyrgh owners and s a condon of accessng publcaons ha users recognse and abde by he legal requremens assocaed wh hese rghs. For more nformaon, or f you beleve ha hs documen breaches copyrgh, please conac he Bond Unversy research reposory coordnaor. Download dae: 4 Apr 09

2 Crcal Values for IPS Panel Un Roo ess: A Response Surface Absrac Analyss By Gulasearan RAJAGURU Bond Busness School Bond Unversy Gold Coas Ausrala 49 Emal: rgulase@bond.edu.au hs paper presens he crcal values for he esng of un roos n heerogeneous panels. he paper develops an algorhm o generae he crcal values hrough Mone Carlo smulaons whch wll be compuaonally effcen as opposed o he radonal smulaon echnques used n he earler panel un roo sudes. he resuls from he smulaon expermens are used o consruc he response surface regressons n whch he crcal values depend on boh cross-seconal and me uns. he predcably of he response surface regressons are evaluaed hrough repored IPS crcal values. Key words: Panel un roo ess, response surface regressons, randomzaon, -bar.

3 . Inroducon he use of panel un roo ess has become very popular among appled economercans snce he developmen of panel un roo es procedures by Levn and Ln (99, 99). One of he advanages of hs procedure s ha he power of he es ncreases wh an ncrease n he number of panel seres compared o he well-nown low power of he sandard ADF un roo es agans near un roo alernaves. Increasngly, recen emprcal sudes use he es procedure nroduced by Im, Pesaran and Shn (00) (hereafer, IPS) whch can es he null hypohess of non-saonary n he presence of heerogeney across he panel. Mos of he emprcal leraure uses eher he crcal values repored by IPS whch are close o her sample szes or Mone Carlo expermens for her parcular sample szes. On he oher hand, oher researchers use sandardzed - bar es sascs (see secon ) o verfy he panel un roo properes of he daa. 4 he nferences based on IPS crcal values could be msleadng when he sample sze s approxmaed o he repored values. In hs paper, we propose an algorhm o oban he crcal values for non-sandardzed -bar sasc whou conducng an exensve smulaon for he ndvdual cross-secons. he crcal values obaned from he Augmened Dcey-Fuller (ADF) es for saonary has been exended o panel ess for saonary under models wh varous degrees of heerogeney by, for example Levn and Ln (99,99), Quah (994) and Im, Pesaran and Shn (00) (hereafer, IPS). he man dfference beween he panel un es procedures proposed by Quah (994), Levn and Ln (99) and IPS s ha whle he former consruc he es sasc under he alernave hypohess ha all componen seres n he panel are saonary, he laer (IPS) es he alernave ha a leas one of he ndvdual seres s saonary. hese panel un roo ess have been employed n varous sudes o verfy he valdy of he varous hypoheses and he economc heores. o ce a few, for example Purchasng Power Pary hypoheses (see, MacDonald (996), Oh (996), Wu (996), Franel and Ross (996), Coaley and Fueres (997, 000) Papel (997) and Flessg and Srauss (000)), panel un roo properes of nflaon rae (see Culver and Papel (997), Lee and Wu (00) and Holmes(00)), un roos n healh care expendure and GDP (see, McCosey and Selden (998)), Invesmen-Savng correlaon (see, Ho (00)), Mean reverson of neres raes (see, Wu and Chen (00)), Gbra s Law of Propornae effecs (see,goddard, Wlson, Blandon (00)). See Ho (00), Holmes (00), Wu and Chen (00), Goddard, Wlson and Blandon (00),Cushman, MacDonald and Samborsy (00), Chou and Chao (00), Holmes (00) and Srauss (000). 4 See, for example, Flessng and Srauss (000), Srazcch, Co and Lee (00) and Wu (000)

4 hese expermens are summarzed by means of response surface regressons n whch he crcal values depend on he sample sze (see MacKnnon (99)). he predcably of he response surface regressons are evaluaed by comparng he predced crcal values wh repored IPS crcal values. Boh n-sample and ou-of-sample predcably of he regressons are evaluaed hrough he errormercs such as roo mean squared error (RMSE), mean absolue error (MSE) and mean absolue percenage error (MAPE). Fnally, he paper repors he crcal values based on he esmaed response surface regresson for he IPS sample.. IPS Panel Un Roo es he heerogeneous panel daa model proposed by IPS s gven by y p ϕ y γ ε, =,,,, =,,,. () = = µ y he null and alernave hypoheses are H 0, H : s 0. Each equaon 0 : = < s esmaed separaely by OLS due o heerogeney and he es sascs are obaned as (sudenzed) averages of he es sascs for each equaon. he -bar sasc proposed by IPS s defned as he average of he ndvdual Dcey- ˆ Fuller τ sascs: = τ, where τ =. () ˆ = σ IPS repor he crcal values for he -bar sascs descrbed by () for he varous combnaons of and. he sandardzed -bar sasc proposed by IPS under he assumpon ha he crosssecons are ndependen s gven by

5 Γ = ( E( τ = 0) ) var( τ = 0). () he means E( τ = 0) and he varances var( τ = 0) are obaned by Mone Carlo smulaons and are abulaed n IPS. IPS conjecure ha he sandardzed -bar sasc Γ converges wealy o a sandard normal dsrbuon as and. Smulaon Expermens s y y. he underlyng daa generang process (DGP) consdered by IPS = ε, ε ~ (0,), =,,..., ; =,,...,, wh y 0. hey esmae - bar sascs based on (). he crcal values repored by IPS are compued va sochasc smulaon of 50,000 replcaons for he models wh ) a consan and ) a consan and a rend. In hs paper, we esmae he response surface funcon o approxmae he lower-al crcal values of percen, 5 percen and 0 percen for he models wh ) a consan and no rend and ) a consan and a rend. he smulaon echnque nroduced n hs paper s dfferen from he usual Mone Carlo expermens adoped by IPS. Insead of smulang he underlyng DGP and re-esmang he model () for he varous combnaons of across he, hs paper randomzes he -sasc across he replcaons obaned from a model wh a sngle cross-secon for a fxed sample sze,. We use M=00,000 replcaons for hs purpose. On he oher hand he smulaon expermen s conduced for he sample of = wh observaons only.. Algorhm y y he underlyng daa generang process n he smulaons s gven by = ε, ε ~ (0,), =,,,. In he frs sage, he underlyng DGP s 0 = generaed and he ADF regresson s fed for he smulaed daa of sze over M 4

6 replcaons. I should be noed ha he underlyng DGP s no generaed for he panel of sze as n he radonal approaches. he -sasc o es he null hypohess of = 0 n () s compued for a sngle cross-secon of sze over M replcaons. Le,..., M, be he correspondng esmaed -sasc for he frs crosssecon over M replcaons. Secondly, he -sascs over M replcaons for he remanng - cross-secons can be obaned by smply randomzng he frs M - sascs obaned from he cross-secon of sze (.e.,,..., M, ). ha s, he - sascs j, =,,... ; j =,,..., M are consruced by = [ ], where he replcaon j ndex s randomly drawn from a unform dsrbuon by a smple random samplng wh replacemen 5 (.e. ~ U[, M ] ). Here [] refers o he neger par of he gven argumen n. he cumulave averages over he cross-secons, m n = ; n =,,...,, consue he -bar sascs for he m-h replcaon. Fnally, he crcal values are obaned by exracng he s, 5h and 0 h qunles from he smulaed numercal dsrbuon. 6 I s observed ha durng he smulaons he proposed algorhm presens he same crcal values as he radonal Mone Carlo smulaon echnque. he proposed smulaon mechansm s abulaed n Appendx. Usng hs algorhm, one can oban he crcal values for n =,,... for he fxed sample sze hrough cumulave averages. However, he radonal smulaon approaches are able o provde he crcal values for fxed and. For a fxed, only M 5 he resuls obaned from smple random samplng wh replacemen (SRSWR) are conssen wh he smple random samplng whou replacemen (SRSWOR) as he number of replcaons M=00,000 are suffcenly large. 6 In hs exercse, we develop he response surface regresson for non-sandardzed es sascs as he sandardzed es sascs nvolves larger number of parameers ha has o be esmaed by sochasc smulaons. herefore, he samplng error of esmang non-sandardzed es sascs wll be smaller han ha of sandardzed ones. 5

7 expermens need o be conduced usng he proposed algorhm as opposed o he radonal approaches ha requre M expermens o oban he desred crcal values. he cos of compung he remanng (-)M relevan es sascs by randomzaon s sgnfcanly less han ha of he radonal one. he compuaonal me for he radonal approaches ncreases sgnfcanly as ncreases. I s expeced he new algorhm wll provde new nsghs for panel regresson sudes because s compuaonally effcen. 4. Response Surface Analyss In order o generalze he esmaors of he crcal values for any combnaon of cross-seconal un and he sample sze a a gven level of sgnfcance, we use he response surface regresson echnques proposed by MacKnnon (99, 996). Suppose α ha we are neresed n q (, ),.e., α quanle of he dsrbuon, where α = %, 5% and 0%. Response surfaces are esmaed for wo dfferen ess 7 : ) -bar sasc wh a consan ) -bar sasc wh a consan and a rend. In each case, hree response surfaces are esmaed based on he s, 5h and 0h quanles. Hence, a oal of sx response surface regressons are esmaed. We consder all combnaons of {,,,00} and {5,6,7,,00}. he number of observaons used n each response surface regresson s In conras o response surface regressons based on pure me seres sudes, n whch he regresson equaon s a funcon of sample sze, we consruc he response surfaces equaon whch s a funcon of and and he response surface equaon for he -bar es sasc: 7 See MacKnnon (996), MacKnnon, Haug and Mchels (999) for more deals abou response surface regresson echnques. 6

8 7 = ), ( q α (4) In he response surface equaons, he regressors are chosen o mnmze he roo mean squared error of he regresson. he regressors s and s capure he ndvdual me and cross-seconal effecs respecvely. I s observed ha for a fxed, he crcal value ), ( q α s an ncreasng funcon of and vce versa. he regressors and do no explan such effecs compleely. In order o capure such monooncy and o ensure he convergence of he response surface regressons for large and, we nroduce and as addonal explanaory varables. I s found durng he expermens ha he response surface equaon wh hese facors ouperforms he models whou hese facors. Furhermore, he response surface equaon s mproved by mulplyng hese facors by and. hese mulplcave erms hen ncorporae he effecs from he neracon of and. I s also observed ha he crcal values are more sensve o when s small han when s large. hese effecs are also capured hrough he neracon of and wh he facors and

9 . I s also observed ha he ncluson of such neracon facors for he hgher degree, for example, does no mprove he resuls. ============= ables and ============== he performance of he response surface regressons are evaluaed by boh whn-sample and ou-of-sample predcably of he crcal values. he response surface regressons are chosen o mnmze he roo mean squared error (RMSE) of he regressons. For he ou-of-sample predcons, we conduc Mone Carlo expermens for he combnaons of {,,,..., } and {00, 00, 400, 500}. hs consues 400 samples for each case. hs helps o evaluae he accuracy of he response surfaces for large. hree measuremens - roo mean squared errors (RMSE), mean absolue errors (MAE), mean absolue percenage errors (MAPE) are used o evaluae he performance of he esmaed response surface regressons for -bar es sascs. he resuls are repored n able. he predcably of he esmaed response surface equaon s also compared wh repored crcal values from he IPS sudy. I s also observed for he models wh a consan and a rend ha he repored crcal values for =5 n he IPS paper for 50,000 replcaons are que dfferen from he crcal values generaed (by Mone Carlo smulaon) n hs paper based on 00,000 replcaons. hese dscrepances could be due o he sgnfcan dfference n he number of replcaons. I s necessary o 8

10 have a large number of replcaons for he case of =5 because ndvdual Dcey Fuller regresson suffer from a lac of degrees of freedom for he models wh he consan and he rend because hree parameers wh a sample of 5 are esmaed. We have also verfed he accuracy of our crcal values by adopng 00,000 replcaons and he crcal values are same as for 00,000 replcaons. he error-mercs for he IPS sample by excludng =5 are also repored n able. Fnally, he crcal values for he IPS sample based on esmaed response surface funcons are repored n able 4. he esmaed response surface regressons are porrayed n appendx. ================= able and 4 ================= I s observed from able ha he response surface regressons provde smooh and accurae crcal values a decmal places and he average predcve error of hese regressons are less han half a percen n mos of he cases. he performance of he response surface regresson for he 0 percen crcal values s noably beer han ha of he response surface regressons for he 5 percen and percen crcal values. he performance of he models for he 5 percen crcal value s superor o he models wh he percen crcal values. In general, he esmaed models repored n ables and ouperform he oher compeve models based on hree crera: RMSE, MAE and 9

11 MAPE. 8 For he sae of brevy, he response surface regresson resuls for he oher compeve models are no repored. 5. Concluson he response surface regressons for he IPS crcal values should prove o be useful for appled economercans esng un roos n heerogeneous panels. he proposed algorhm o generae he crcal values provdes a new dmenson o panel sudes because s compuaonally effcen and powerful. he response surface regressons were developed based on he crcal values obaned from smulaon expermens and are funcons of he number of cross-secons and sample szes. he performance of hese regressons was compared wh repored IPS crcal values. he crcal values for he panel un roos of any combnaon of cross secons and sample szes can be calculaed n a spreadshee by subsung he panel dmensons n he response surface regressons whou conducng an exensve smulaon. References Chou, W.L., and C.C. Chao (00), Are Currency Devaluaons Effecve? A Panel Un Roo es, Economcs Leers, 7, 9-5. Cushman, D.O., R. MacDonald, and M. Samborsy (00), he Law of One Prce for ransonal Urane, Economcs Leers, 7, We have also consdered varous compeve models by allowng non-lneary such as where γ and δ could non-negers, n he response surface regressons. γ and δ, 0

12 Dcey, D.A. and W.A. Fuller (979), Dsrbuon of Esmaors of Auoregressve me Seres Wh a Un Roo, Journal of he Amercan Sascal Assocaon, 74, Engle, R.F., and C.W.J. Granger (987), Co-negraon and error correcon: represenaon, esmaon and esng, Economerca, 55, Flessng, A., and J. Srauss (00), Panel Un Roo ess of OECD Sochasc Convergence, Revew of Inernaonal Economcs, 9, 5-6. Goddard, J., J. Wlson, and P. Blandon (00), Panel ess of Gbra s Law for Japanese Manufacurng, Inernaonal Journal of Indusral Organzaon, 0, Ho,.W. (00), A Panel Co-negraon Approach o he Invesmen-Savng Correlaon, Emprcal Economcs, 7, Holmes, M.J. (00), Some ew Evdence on Exchange Raes, Capal Conrols and European Unon Fnancal Inegraon, Inernaonal Revew of Economcs and Fnance, 0, Holmes, M.J. (00), Panel Daa Evdence on Inflaon Convergence n he European Unon, Appled Economcs Leers, 9, Im, K.S., M.H. Pesaran, Y. Shn (00), esng for Un Roos n Heerogeneous Panels. Mmeo, Deparmen of Appled Economcs, Unversy of Cambrdge. Levn, A. and C.F. Ln (99), Un Roo ess n Panel Daa: Asympoc and Fne Sample Properes, unpublshed manuscrps, Unversy of Calforna, Sam Dego. MacKnnon, J.G. (994), Crcal values for Co-negraon ess, n Long-Economc Relaonshps: Readngs n Co-negraon, eds. R.F. Engle and C.W.J. Granger, Oxford: Oxford Unversy Press,

13 MacKnnon, J.G. (994), Approxmae Asympoc Dsrbuon Funcons for Un-roo and Co-negraon ess, Journal of Busness and Economc Sascs,, MacKnnon, J.G. (996), umercal Dsrbuon Funcons for Un-roo and Conegraon ess, Journal of Appled Economercs,, MacKnnon, J.G., A.A. Haug, and L. Mchels (999), umercal Dsrbuon Funcons Lelhood Rao ess for Co-negraon, Journal of Appled Economercs, forhcomng. McCosey, S.K., and.m. Selden (998), Healh care Expendure and GDP: Panel Daa Un Roo es Resuls, Journal of Healh Economcs, 7, Quah, D. (99), Inernaonal Paerns of Growh: I. Perssence n Cross-Counry Dspares, unpublshed manuscrps, London School of Economcs. Quah, D. (994), Explong Cross-Secon Varaons for Un Roo Inference n Dynamc Daa, Economcs Leers, 44, 9-9. Phllps, P.C.B., and P. Perron (988), esng for a Un Roo n me Seres Regresson, Bomera, 75, Srauss, J. (000), Is here a Permanen Componen n US Real GDP, Economcs Leers, 66, 7-4. Srazcch, M.C., C.Y. Co, and J. Lee (00) Are Shocs o Foregn Invesmen n Developng Counres Permanen or emporary? Evdence from Panel Un Roo ess, Economcs Leers, 70, Wu, J.L. (000), Mean Reverson of he Curren Accoun: Evdence from he Panel Un Roo es, Economcs Leers, 66, 5-.

14 Wu, J.L., and S.L. Chen (00), Mean Reverson of Ineres Raes n Eurocurrency Mare, Oxford Bullen of Economcs and Sascs, 6,

15 able : Response Surface Regressons for he -bar sascs: Consan bu no rend percen 5 percen 0 percen Coeffcen S.E Coeffcen S.E Coeffcen S.E R

16 able : Response Surface Regressons for he -bar sascs: Consan and rend percen 5 percen 0 percen Coeffcen S.E Coeffcen S.E Coeffcen S.E R

17 able : Predcably of Response Surface Regressons: -bar es sasc Consan bu no rend Consan and rend % 5% 0% % 5% 0% Whn RMSE Sample MAE MAPE 0.4% 0.9% 0.7% 0.% 0.5% 0.% Ou RMSE sample MAE MAPE 0.% 0.8% 0.7% 0.97% 0.44% 0.4% IPS RMSE Repored MAE values MAPE 0.% 0.4% 0.4% 0.6% 0.6% 0.0% IPS* RMSE Repored MAE values MAPE 0.% 0.5% 0.5% * Comparson wh IPS crcal values by excludng =5 case 6

18 able 4 Crcal Values of he -bar sasc based on Response Surface Regressons \ Panel A: DF regressons conanng only consans percen percen percen Panel A: DF regressons conanng consans and lnear rends percen percen

19 percen

20 Appendx : Smulaon Mechansm for Panel Un Roo es -sascs Replcaons Crossseconseconsecon Cross- Cross-.. Crosssecon = = = -bar sascs = = =.. = = = = = = j = = = = = j = = = = = = j j j j j j = j j = j j = j = = = = j = j j = M M M M M M = M M = M M = M M = M Smulaon oe: () s fxed. () Column (for cross-secon ) s obaned by sochasc smulaon of M replcaons based on equaon (). () Values n Columns hrough (.e., cross-secons hrough )are obaned by randomly drawng he values from column wh replacemen. = = =

21 Appendx : Response surface funcon -bar (%) bar (%) rend bar (5%) bar (5%) rend bar (0%) bar (0%) rend