Over-Rejections By Weighted Symmetric Unit Root Tests In The Presence Of Multiple Structural Breaks

Transcription

1 Over-Rejecions By Weied Symmeric Uni Roo ess In e Presence Of Muliple Srucural Breaks akasi Masuki Deparmen of Economics, Osaka Gakuin Universiy, masuki@uc.osaka-u.ac.jp Keywords: Uni roo; Weied symmeric esimaor; Muliple breaks; Over-rejecions. EXENDED ABSRAC Panula e al. 994 proposed modified Dickey Fuller DF ype uni roo ess usin a weied symmeric esimaor ereafer, ess. In eneral, ese ess are considered o ave more power an e correspondin DF ess wen ere is no srucural break in a ime series. For e case of e presence of oneime srucural cane in e sample, Leybourne and Newbold and Cook 4 invesiaed e beavior of e ess. Leybourne and Newbold concluded a e weied symmeric version of e DF -ype es will properly rejec e uni roo null ypoesis a some specified sinificance level wen e rue daa-enerain process is an I process. On e oer and, Cook s Mone Carlo sudy suesed a e ess mi suffer from low power under e saionariy alernaive ypoesis were a series is eneraed by an I process. However, ese consideraions reardin e weied symmeric uni roo ess are limied o cases avin one or no break. us, wen a ime series as muliple wo or more breaks, no evidence exiss o ascerain e poenial effecs of ese breaks on e ypoesis es by e ess. a is, weer e ess remain robus o various possible combinaions of parameers expressin locaions and maniudes of breaks needs o be deermined. Given ese facs, o consider muliple srucural canes, is sudy inroduces muliple sifs ino e daa-enerain model of a series and invesiaes e beavior of weied symmeric uni roo ess under is model. samples, e es saisics beave in various combinaions of locaions of wo breaks wi increasin break sizes. e obained resuls from is sudy reveal a e ess mi be biased oward over-rejecions of e uni roo null ypoesis because of e presence of muliple srucural breaks. e limiin disribuions of e es saisics are derived wen muliple srucural breaks occur in e sample period. en, for e case of double breaks, e beavior of e es saisics is examined for some exreme cases of e nuisance parameers a express locaions and sizes of e breaks. A Mone Carlo experimen is conduced in order o examine e manner in wic in finie 864

2 . INRODUCION Perron 989 and Leybourne e al. 998 ave indicaed a e sandard Dickey Fuller DF ess commonly used in preliminary invesiaions o observe e non-saionariy of a series can ave very low power and serious size disorion wen e rue daa-enerain process as a srucural break. Many relaed sudies ave been conduced; for example, Monanes and Reyes 998, Leybourne and Newbold, Lee, and Sen. However, Panula e al. 994 proposed modified DF ype uni roo ess usin a weied symmeric esimaor ereafer, ess. In e absence of a break, ese are enerally considered o be more powerful an e correspondin DF ess Panula e al. 994, Park and Fuller 995, Paerson and Heravi. For ese ess, in e presence of a srucural cane in e sample, Leybourne and Newbold invesiaed e asympoic properies of e weied symmeric version of e DF -ype es under e uni roo null ypoesis. ey concluded a e spurious rejecion of e uni roo ypoesis will no be presen in e -es even wen a break occurs in e level or slope of e rend funcion of a series. Cook 4 sudied e small-sample performance of e and recursive mean-adjused uni roo ess in a saionary series avin one mean sif. Cook s Mone Carlo resuls sues a ese modified ess can suffer from low power under e saionariy alernaive ypoesis. However, ese consideraions reardin e weied symmeric uni roo ess are limied o cases avin one or no break. us, wen a ime series as muliple wo or more breaks, no evidence exiss o ascerain e poenial effecs of ese breaks on e ypoesis es by e ess. a is, weer e ess remain robus o various possible combinaions of parameers expressin locaions and maniudes of breaks needs o be deermined. Given ese facs, o consider muliple srucural canes, we inroduce muliple mean sifs ino e daa-enerain model of a series y as follows: y = α β d z d z = z / H k = DU = ρ ε, ρ =, =, Κ, were denoes e order of a break =, Κ, H, k denoes e size of e break, ε is an independenly and idenically disribued innovaion process wi a zero mean and a finie variance σ, and e sample size of e series is. DU denoes e break in e level of e rend funcion, were DU = for > and oerwise, were is e fracion of e break, wic is defined as B / for all B is e dae of e break, and < < Λ < H <. < H < Le e denoe e residual in e reression of y on an inercep and ime rend for =, Κ, ; en, e weied symmeric esimaor is obained by minimizin W = w e e = Q ρ = w e ρe ρ were w is e wei variable, wic is iven as. e weied symmeric esimaor of ρ is obained as ˆρ = e e e e. e saisics = = = based on is esimaor are defined as = ˆ σ e = ˆ ρ ˆ ρ e = were ˆ σ = ˆ ρ. Q W / In e followin secions, we will repor only e resul of e saisic because e beavior of e ˆ ρ saisic is quie similar o a of e saisic, and refer o e es uilizin e saisic as e weied symmeric -es. In e nex secion, we will derive e limiin disribuion of e es saisic and explore is asympoic beavior in e case of double breaks for cerain exreme cases wi respec o e daes and sizes of e breaks. o examine e smallsample properies of e es, some Mone Carlo experimens are also conduced in Secion. e conclusion is described in Secion 4.. LIMIING BEHAVIOR OF HE ES We sow e limiin disribuion of e weied symmeric -es saisic wen muliple mean sifs exis in e ime series. eorem. For model, σ c σ W r dr c l / 865

3 { W r dw r W r dr } σ c l l s c l ] were represens weak converence, W r is a derended sandard Wiener process defined as W r W r a dr, were W r is a sandard Wiener process, a s W s ds and d 6 s W s ds. c, l, and s are iven by c = k 6 c c H H = > { k k 6 6 = k = k H H = > l = σ k W l } k k { = σ k W W W 6 r / dw r σ k W r dr { σw } l = s = k were Σ is e summaion from = o H. is eorem sows a e limiin disribuion of e saisic depends on e nuisance parameers a indicae e locaions and sizes of e breaks. In urn, we discuss e asympoic beavior of e es saisic for double breaks H =. ree exreme cases of e fracions of breaks are considered:,,,,,, and,,, were bo e break fracions lie a e beinnin of e sample, bo lie a e end of e sample, and e fracions of e firs and e second breaks lie a e beinnin and e end of e sample, respecively. For simpliciy, in e followin proposiion, we se σ = wiou loss of eneraliy. } Proposiion. For e case of double breaks H =, e limiin disribuion of e weied symmeric -es saisic usin ρˆ and σˆ wi σ = in e ree cases of, is as,, / A k k k k as,, { W } { A k k k k W } as,, / { A k W k W } / were A = W r dw r W r dr W and B = k k W r dr. Reardin e DF -es, Leybourne and Newbold ave sown a as e break fracion of a break approaces zero, e limiin disribuion of e saisic conains e erm of e squared break size parameer k in e numeraor Equaion, and is erm causes e spurious rejecion of e uni roo null ypoesis. In Proposiion, as,, and,, e limiin disribuion of e -saisic conains e deerminisic erm k k in e numeraor of is equaion. erefore, we can infer a is erm can cause overrejecions of e uni roo null in e -es if is value is neaive and is absolue value is lare enou o dominae socasic erms in e limiin disribuion. However, Leybourne and Newbold also proved a in e case of a sinle break, e deerminisic erm correspondin o k k in Proposiion does no appear in e limiin disribuion of e -es Equaion 5. Weer or no e deerminisic erm of break size parameers is included in e limiin disribuion sems from e difference in e number of breaks considered in e daa-enerain model one or muliple breaks. is implies a muliple breaks in e level can ave varyin impacs on e -es compared o a sinle break in e level.. MONE CARLO SUDY o invesiae e small-sample properies of e weied symmeric uni roo -es, we implemen a Mone Carlo simulaion. e series, y, is eneraed by model wi ρ =, H =, and α = β =, and ε is i. i. d. N,. e sample size is and ere are 5 replicaions. e 866

4 empirical size of e -es a e nominal level 5% is compued in e reion of < < < a. inervals. e sizes of wo breaks, k, k, assume some possible combinaions of.5,.,.5, and.. able sows a wen one of e wo maniudes of breaks as a neaive value, frequen rejecions of e null occur a e near daes of wo breaks in e es. Moreover, bo breaks become sinifican wen eir absolue values k, k =.,. and.,. are lare. is resul is consisen wi e findin of Proposiion ; i can erefore be concluded a reardless of e sample size of e series, e over-rejecion penomenon occurs in e -es in e presence of muliple mean sifs. able. e empirical size of e weied symmeric -es a e nominal 5% level 4. CONCLUSIONS We ave invesiaed e beavior of e uni roo -es based on a weied symmeric esimaor wen e rue daa-enerain process is ineraed of order one wi muliple srucural breaks in e level. e limiin disribuion of e es saisic was derived under e uni roo null ypoesis, inorin e breaks in e process. As sown by Leybourne and Newbold, in esin e uni roo null ypoesis in e presence of one break, e weied symmeric -es ouperforms e correspondin DF -es in erms of e absence of e bias oward spurious rejecions of e null ypoesis. However, we ave revealed a e problem of over-rejecions of e null can occur even in e -es wen wo muliple mean sifs exis in e ime series. 5. ACKNOWLEDGEMENS is paper owes muc o elpful commens of Misuru Nakaawa Osaka Ciy Universiy on my presenaion a e 4 Fall Meein of e Japanese Economic Associaion Okayama. e researc for e paper was suppored in par by e Minisry of Educaion, Culure, Spors, Science and ecnoloy MEX, Gran-in-Aid for Encouraemen of Youn Scieniss #47 Japan. 867

5 6. FOONOES. Lumsdaine and Papell 997 suesed a uni roo es a permis wo breaks a unknown daes under e alernaive ypoesis, and Lee and Srazicic proposed a minimum Larane Muliplier LM uni roo es a allows for wo endoenous breaks under bo e null and alernaive ypoeses.. We ave also analyzed ree oer cases: e / cases were d = k D, were D = for > and oerwise muliple slope canes, d = k DU, and d = k D, were e summaion Σ is from = o H. Consequenly, i as been found a e -es is asympoically invarian o e exisence of any ype of muliple breaks.. For e case of more an wo breaks, e correspondin deerminisic par, wic may become e cause of e misjudmen on e ypoesis es, is presen in e limiin disribuion. Journal of Business & Economic Saisics,, Park, H.J. and W.A. Fuller 995, Alernaive esimaors and uni roo ess for e auoreressive process, Journal of ime Series Analysis, 6, Paerson, K. and S. Heravi, Weied symmeric ess for a uni roo: response funcions, power, es dependence and es conflic, Applied Economics, 5, Perron, P. 989, e Grea cras, e oil price sock, and e uni roo ypoesis, Economerica, 57, 6-4. Sen, A., Beaviour of Dickey Fuller F- ess under e rend-break saionary alernaive, Saisics and Probabiliy Leers, 55, REFERENCES Cook, S. 4, Finie-sample properies of modified uni roo ess in e presence srucural cane, Applied Maemaics and Compuaion, 49, Lee, J., On e end-poin issue in uni roo ess in e presence of a srucural break, Economics Leers, 68, 7-. Lee, J. and M.C. Srazicic, Minimum Larane muliplier uni roo es wi wo srucural breaks, e Review of Economics and Saisics, 85, Leybourne, S.J.,.C. Mills, and P. Newbold 998, Spurious rejecion by Dickey Fuller ess in e presence of a break under e null, Journal of Economerics, 87, 9-. Leybourne, S.J. and P. Newbold, Beaviour of e sandard and symmeric Dickey Fullerype ess wen ere is a break under e null ypoesis, Economeric Journal,, -5. Lumsdaine, R.L and D.H. Papell 997, Muliple rend breaks and e uni-roo ypoesis, e Review of Economics and Saisics, 79, -8. Monanes, A. and M. Reyes 998, Effec of a sif in e rend funcion on Dickey Fuller uni roo ess, Economeric eory, 4, Panula, S.G., G.Gonzalez-Farias and W.A. Fuller 994, A comparison of uni-roo es crieria, 868