BRIDGE ESTIMATOR AS AN ALTERNATIVE TO DICKEY- PANTULA UNIT ROOT TEST

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1 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember BRIDGE ESTIMATOR AS AN ALTERNATIVE TO DICKEY- PANTULA UNIT ROOT TEST Hüseyi Güler Yeliz Yalҫi Çiğdem Koşar Absrac Ecoomic series may have ui roos. There are differe mehods o es ui roos i he lieraure. Rece sudies show he advaages of usig Bridge esimaor for choosig he lag legh of ui roo ess ad compuig es saisics. This mehod pealizes he parameers of he model wih a posiive shrikage parameer. For a suiable choice of he shrikage parameer he limiig disribuios of he Bridge esimaor ca have posiive probabiliy mass a 0 whe he rue value of he parameer is zero for saioary series. Therefore he Bridge esimaor provides a mehod ha ca be used for he model selecio ad esimaio simulaeously. I is kow i he lieraure ha Bridge esimaor does he model selecio while simulaeously disiguishig bewee saioary ad ui roo models for series wih a ui roo as well. I his sudy we propose Bridge esimaor o deermie he iegraio order of series wih more ha oe ui roo. I order o do his we apply he Bridge esimaor o Dickey-Paula model. We evaluae he size ad power of he Bridge esimaor wih a simulaio sudy ad compare our proposed esimaor wih he exise Dickey-Paula es. Key words Ui Roo Tes Oracle Propery Model Selecio Bridge Esimaor LASSO JEL Code C C5 C. Iroducio Saioariy is a very impora assumpio i may ecoomeric echiques. However i he pracice mos of he ecoomic ime series are osaioary ad igorig he osaioariy causes ureliable resuls. The augmeed Dickey-Fuller (ADF Dickey ad Fuller 979) ess are commoly applied o check wheher he series is saioary or o. Some researchers apply ADF es afer akig he differece of he series o ideify he order of iegraio. However ADF es assumes ha he series has sigle ui roo (Dickey ad Fuller 98). Accordig o Dickey ad Paula (987) applyig ADF es for differeced series causes some saisical problems. For his reaso i is suggesed o use Dickey-Paula ui roo es o deermie he order of iegraio of variables whe here is more ha oe ui roo. Recely Caer ad 5

2 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember Kigh (0) propose Bridge Esimaor o do model selecio while simulaeously deermie wheher he series is I or I0. They show ha Bridge esimaor ouperforms ADF ess i erms of size ad power. I his sudy we describe Bridge esimaor as a aleraive o Dickey-Paula ui roo es. We exed Caer ad Kigh s (0) approach o his model ad compare he size ad power of hese approaches i a simulaio sudy. Dickey-Paula Ui Roo Tes Dickey ad Paula (987) propose a proper sequece of saisical mehod o ivesigae series havig more ha oe ui roo. Le he ime series Y saisfy Y Y Y Y e () where e is a sequece of iid radom variables wih mea 0 ad variace. Le m m ad m deoe he roos of characerisic equaio m m m. Assume ha 0 m m m. Cosider he followig four hypoheses (a) H 0 m ( (b) H m m ( 0 Y is saioary - Y (c) H m m m ( (d) H m m m ( I ); has oe ui roo - Y Y I ); has wo ui roos - has hree ui roos - I I ). ); Tha is uder he hypohesis H d he d h differece of Y is esseially saioary (Dickey ad Paula 987). Now cosider a reparameerizaio of model () X Y Z W e () where Z Y Y W Z Z X W W are firs secod ad hird differeces of he sequece Y respecively. Dickey ad Paula (987) oe ha ) ad ha m )( m )( ) ( ( m ( m )( m ) ( m )( m ) ( m )( m ) mm m o wrie previous four hypoheses i erms of s as follows (a) H 0 ; (b) H 0 ; (c) H ; 54

3 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember (d) H Give observaios i is possible o regress X o Y Z ad W o ge ordiary leas squares (OLS) esimaes of ˆ ˆ ad ˆ ad he correspodig saisics () () (). However Dickey ad Paula argue ha i may o be appropriae o use sadard saisics. They show ha whe he model i () holds wih radom variables he saisic e a sequece of iid 0 for esig oe ui roo has differe asympoic disribuios depedig o he umber of ui roos prese. I addiio o his hey show ha a sequeial procedure based o he saisics i is o cosise. I order o solve his problem Dickey ad Paula (987) sugges a sequeial procedure based o pseudo saisics i ( p) where i ( p) is he i coefficie of B Y i he regressio of p Y B p Y saisic for he i o B Y B i Y B. Their sequeial procedure based o Dickey-Fuller saisics is as follows. Rejec he hypohesis H of hree ui roos ad go o sep if where ˆ ( ) ˆ was give by Fuller (976).. Rejec he hypohesis H of exacly wo ui roos ad go o sep if i addiio o you also fid ha ( ). ˆ ( ) ˆ. Rejec he hypohesis H of exacly oe ui roo i favor of he hypohesis H 0 of o ui roos if ( i ). ˆ i ( ) They show ha he sequeial procedure based o i ( ) is a cosise level- procedure. p Moreover he probabiliy ha he procedure chooses he rue umber of ui roos prese coverges o whe he series has a leas oe ui roo ad o whe he series is saioary. Dickey ad Paula (987) exed heir heoreical comparisos wih a umerical example ad simulaio sudy. They show ha heir mehod is superior o Dickey-Fuller saisics ad Hasza-Fuller F saisics i erms of size ad power. BRIDGE Esimaor o Deermie Iegraio Order of Series Le a liear regressio model wih observaios ad y X e () 55 p idepede variables

4 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember where y is he vecor of depede variable X p is he marix of idepede variables p is he vecor of ukow parameers ad uobservable error erms wih e 0 I 56 e is he vecor of ~. OLS miimizes sum of squared residuals y X y X ˆ mi ad ˆ ( X X ) X y. (4) Recely Bridge / Lasso ype esimaors are proposed o pealize model parameers while miimizig he sum of squared residuals as follows where ˆ mi y X y X 0 is he uig parameer ad 0 j j (5) is he shrikage parameer. Frak ad Friedma (99) ad Fu (998) call hese esimaors as Bridge esimaors. Tibshirai (0) defies he special case ( ) as Leas Absolue Shrikage (LASSO). Fu (998) use differe shrikage parameers ad compare Bridge OLS LASSO ad Ridge wih a simulaio sudy. Fu ad Kigh (000) esablish he limi law for LASSO. These esimaors shrik he esimaes of zero parameer(s) o zero wih probabiliy covergig o whe 0. This is called he oracle propery. Fa ad Li (00) show ha hese mehods simulaeously selec he model ad esimae he coefficies. Huag Horowiz ad Ma (008) exed Kigh ad Fu s (000) sudy o ivesigae he asympoic properies of Bridge esimaors for saioary series. Caer (009) cosider LS ad Geeralized Mehod of Momes (GMM) based Lasso esimaors respecively ad fids ha i small samples Lasso esimaor have smaller mea square error ad bias ha he oes chose by Akaike Iformaio Crierio (AIC) Schwarz Crierio (SC) ad sequeial esig procedures. I a rece paper Caer ad Kigh (0) fid he limiig disribuios of Bridge esimaors whe he series is I. Their proposed mehod is a pealized versio of ADF es. This mehod does he model selecio while simulaeously disiguishig bewee saioary ad ui roo models. By doig so hey elimiae he wo-sep procedure of model selecio (i.e. decidig he deermiisic compoes ad choosig he lag legh) ad he applyig a ui roo es. Noe ha Caer ad Kigh s (0) mehod is o a ui roo es sice here is o esig sep i heir procedure. Isead of ha hey propose Bridge esimaor o choose bewee ui roo ad saioary models o decide wheher he series is iegraed or o. Their simulaios show ha his mehod selec he opimal lag legh ad ui roo simulaeously ad makes a subsaial differece i erms of size ad power whe i is

5 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember compared wih he exisig ui roo ess wih lag selecio mehods. For he ADF model Caer ad Kigh (0) show ha i is possible o esimae he coefficie of lagged depede variable cosisely regardless of saioariy or osaioariy whe 0. I heir simulaios hey show ha he mehod performs well whe I his paper we exed Caer ad Kigh s (0) mehod o Dickey-Paula model. Sice Bridge esimaor performs well i model selecio we hope ha our approach leads o some gai i erms of size ad power over Dickey-Paula es especially whe he series coais more ha oe ui roo. I order o do his we pealize he parameers of () while miimizig he sum of squared residuals as follows ˆ mi X Y Z W j j (6) Noe ha (6) is a oliear opimizaio problem sice i ivolves he powers of model parameers. By followig Caer ad Kigh s (0) suggesio we use order o deermie he uig parameer Iformaio Crieria (MBIC) MBIC log X ˆ Y ˆ Z ˆ W 4. 4 i (6). I we use Caer ad Kigh s Modified Bayesia log S (7) where ˆ ˆ ad ˆ are Bridge esimaes ha correspod o a specific choice of ad S is he umber of ozero Bridge esimaes. The esimae of is obaied from ˆ arg mi MBIC. (8) For selecio of he values of we follow Caer ad Kigh (0) ad use he se However Bridge esimaors of he zero compoes may ake boh posiive ad egaive values. I order o solve his problem he compoes of he Bridge esimaor ha are close o zero are forced o be exacly zero. This is doe by comparig he Bridge esimaor wih a hard hresholdig parameer c 0. Caer ad Kigh (0) sugges o se he coefficie of he lagged depede variable o be zero whe is Bridge esimae is greaer ha. Similarly we se ˆ i 0 whe ˆ i c i where c i is he hard hresholdig c parameer for ˆ i ( i ). Caer ad Kigh (0) choose c ha correspods o 5% 57

6 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember wrog model selecio whe he rue daa geeraig process (DGP) is osaioary ad deoe his as c This is similar o creae he criical value of a es saisic by seig By followig heir approach we apply he same mehod o choose c i values for ˆi ( i ) i our simulaios. Oce Bridge esimaes of () are esimaed by (6) we ca check he ozero coefficies of ˆi ( i ). Similar o Dickey-Paula es we ca apply followig sequeial decisio crieria o deermie iegraio order of series. Rejec he hypohesis H of hree ui roos ad go o sep if ˆ 0.. Rejec he hypohesis H of exacly wo ui roos ad go o sep if i addiio o ˆ 0 you also fid ha ˆ 0.. Rejec he hypohesis H of exacly oe ui roo i favor of he hypohesis H 0 of o ui roos if ˆ 0 i ( i ). Followig hese seps we ca deermie he iegraio order of series. Please oe ha his is o a sequeial esimaio procedure as i Dickey-Paula es. Our mehod ivolves esimaio of he parameers simulaeously ad he sequeially checkig he ozero coefficies sarig from ˆ o ˆ. Tha is why we call his mehod as a sequeial decisio crieria. Oe ca also check he ozero coefficies simulaeously bu his migh lead o icoclusive decisios (for isace 0 bu 0 ). Our simulaios show ha his is a very ulikely case. However o avoid icoclusive decisios ha migh happe we sugges he sequeial process give above. Simulaio Experime I his secio we compare Dickey ad Paula s (987) sequeial procedure based o i ( p) wih he proposed Bridge esimaor. However as poied ou by Caer ad Kigh (0) Bridge is a model selecio/esimaio seup ad here is o esig procedure. I order o compare Bridge wih Dickey-Paula we ca use he correc model selecio perceages. The idea is o compare esimaed iegraio of orders whe he rue DGP coais ui roo(s) ad saioary. Sice we fix he wrog model selecio by usig exac criical values ad hard hresholdig parameers wrog ad correc model selecio perceages deoe he size ad size-adjused powers. Therefore we firs obai exac criical 58

7 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember values for Dickey-Paula es ad hard hresholdig parameers for Bridge esimaor. The we obai size-adjused powers o compare aforemeioed mehods. We use MATLAB o perform simulaios. We use he followig DGP o geerae daa Y Y Y Y e 00. (9) where 00 e are idepedely ad radomly geeraed from N 0 ad he iiial values Y 0 Y ad Y are se o zero. The parameers ad are compued from he riples of roos lised i he lef colum of Table. This allows us o corol he iegraio order of Y by varyig m m ad m. To obai exac criical values for Dickey-Paula es saisics we use roos m m m ( d ) m m m 0 ( d ) ad m m m 0 ( () d ) for () ad () respecively. The for each case we obai 5% empirical criical values wih 000 replicaios. These are ad for () () ad () respecively. These exac criical values are used isead of ˆ i he sequeial Dickey-Paula es. The same DGP is applied o our proposed Bridge esimaor o obai hard hresholdig parameers c i ( i ). We choose c c ad c so ha wrog model selecio perceages are se o 5% for m m m ( d ) m m m 0 ( d ) ad m m m 0 ( d ) respecively. These hard hresholdig parameers are c c 9. 4 ad c I order o obai size ad size-adjused powers we use he same DGP o geerae daa wih he roos lised i he lef colum of Table. The we fid he esimaed iegraio order of series wih Dickey-Paula es ad Bridge for replicaios. These are also repored i Table. Resuls show ha boh esimaors perform very well. There is o size disorio for m m m m m m 0 ad m m m 0. However Bridge esimaor has some size disorio for he se of roos ad I hese cases Bridge esimaor frequely cocludes he iegraio order as ad while he rue order is ad 0 respecively. This meas ha Bridge esimaor may lead o overdifferece i some cases. I is also observed ha he Dickey-Paula es uderdifferece he model more ha he Bridge esimaor does. Whe we compare he correc model selecio perceages for he remaiig se of roos we observe ha Bridge domiaes Dickey-Paula 59

8 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember es i 6 cases while Dickey-Paula es domiaes Bridge i 6 cases as well. I is observed ha Bridge ouperforms Dickey-Paula es especially whe he roos become closer o. Tab. Model selecio perceages for replicaios Seleced Model Roos d 50 d d d Noe The op lie o each se represes he Bridge esimaor; he boom lie represes he Dickey-Paula es. () idicaes he rue model ad bold cases idicaes he ouperformig mehod for he seleced se of roos.

9 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember Coclusio I his paper we suggesed Bridge esimaor for esig muliple ui roos. We compared our proposed esimaor wih he exisig Dickey-Paula es. Our simulaios showed ha oe of hese mehods is uiformly beer ha he oher. However resuls suggesed ha Dickey- Paula es uderdifferece he model more ha Bridge esimaor does. We also foud ha Bridge esimaor ouperforms Dickey-Paula es especially whe he roos become closer o oe. For he fuure research oe may fid he limiig disribuio of Bridge esimaor uder muliple ui roos. I is also possible o exed our model by icludig lags ad deermiisic compoes. We expec ha his migh icrease he performace of Bridge esimaor sice i is a model selecio procedure ad elimiaes he wo-sep aure of ui roo ess. Ackowledgme This research is suppored by Çukurova Uiversiy Research Fud (SDK ). Refereces Caer M. (009). Lasso-Type Gmm Esimaor. Eco. Theory Ecoomeric Theory 5(0) -. doi0.07/s Caer M. & Kigh K. (0). A aleraive o ui roo ess Bridge esimaors differeiae bewee osaioary versus saioary models ad selec opimal lag. Joural of Saisical Plaig ad Iferece 4(4) doi0.06/j.jspi Dickey D. A. & Fuller W. A. (979). Disribuio of he Esimaors for Auoregressive Time Series wih a Ui Roo. Joural of he America Saisical Associaio 74(66a) doi0.080/ Dickey D. A. & Fuller W. A. (98). Likelihood Raio Saisics for Auoregressive Time Series wih a Ui Roo. Ecoomerica 49(4) 057. doi0.07/957 Dickey D. A. & Paula S. G. (987). Deermiig he Order of Differecig i Auoregressive Processes. Joural of Busiess & Ecoomic Saisics 5(4) 455. doi0.07/9997 Fa J. & Li R. (00). Variable Selecio via Nococave Pealized Likelihood ad is Oracle Properies. Joural of he America Saisical Associaio 96(456) doi0.98/

10 The 0 h Ieraioal Days of Saisics ad Ecoomics Prague Sepember Frak L. E. & Friedma J. H. (99). A Saisical View of Some Chemomerics Regressio Tools. Techomerics 5() doi0.080/ Fu W. J. (998). Pealized Regressios The Bridge versus he Lasso. Joural of Compuaioal ad Graphical Saisics 7() 97. doi0.07/907 Fu W. & Kigh K. (000). Asympoics for lasso-ype esimaors. A. Sais. The Aals of Saisics 8(5) doi0.4/aos/ Fuller W. A. (976). Iroducio o saisical ime series. New York Wiley. Huag J. Horowiz J. L. & Ma S. (008). Asympoic properies of bridge esimaors i sparse high-dimesioal regressio models. A. Sais. The Aals of Saisics 6() doi0.4/ Tibshirai R. (0). Regressio shrikage ad selecio via he lasso A rerospecive. Joural of he Royal Saisical Sociey Series B (Saisical Mehodology) 7() 7-8. doi0./j x Coac Hüseyi GÜLER Çukurova Uiversiy Çukurova Uiversiy Faculy of Ecoomics ad Admiisraive Scieces Deparme of Ecoomerics Sarıçam Adaa / TURKEY hguler@cu.edu.r Yeliz YALÇIN Gazi Uiversiy Gazi Uiversiy Faculy of Ecoomics ad Admiisraive Scieces Deparme of Ecoomerics Beşevler Akara / TURKEY yyeliz@gazi.edu.r Çiğdem KOŞAR Çukurova Uiversiy Çukurova Uiversiy Faculy of Ecoomics ad Admiisraive Scieces Deparme of Ecoomerics Sarıçam Adaa / TURKEY ckosar@cu.edu.r 5

Inference of the Second Order Autoregressive. Model with Unit Roots

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