Stationarity and Unit Root tests

Transcription

1 Saioari ad Ui Roo ess

2 Saioari ad Ui Roo ess. Saioar ad Nosaioar Series. Sprios Regressio 3. Ui Roo ad Nosaioari 4. Ui Roo ess Dicke-Fller es Agmeed Dicke-Fller es KPSS es Phillips-Perro Tes 5. Resolvig Nosaioari

3 Saioar ad Nosaioar Series A give ime series is saioar whe mea ad variace are cosa or idepede of ime. E( ) cosa mea Var ( ) cosa variace cov(, ) cov(, ime idepede covariace s s) s Time series is o-saioar if he mea ad variace is o cosa or is chagig over ime. Ma ecoomic variables sch as GDP, GDP compoes, iflaio, exchage raes, sock prices, labor force evolve over ime. I is impora o check wheher hese series are saioar or o saioar before a ecoomeric esimaio becase esimaio sig o-saioar variables ma geerae a sprios relaioship: repored o have relaioship whe here is o relaioship. 3

4 Nosaioar Series The series do o have a cosa mea I ca be observed ha he firs series does o displa wha is kow as a mea reversio behavior: i waders p ad dow radoml wih o edec o rer o a pariclar poi. he secod series o average grows each period b a drif. 4

5 Nosaioar Series Alhogh he series has a cosa mea, i does o have a cosa variace. The variace of he daa ca be imagied sig he viral bads o eiher side of he mea of he daa which will cover almos all observaios. The series does o displa secod order homogeei becase whe we slide he frame alog he horizoal axis, we observe ha he firs frame differs sbsaiall from he secod ad he hird. 5

6 Sprios Regressio Cosider he followig regressio: Y X If Y ad X are osaioar, he leas sqares esimaio of his regressio ca ield resls which are compleel wrog. For isace, eve if he re vale of β is, OLS ca ield a esimae which is ver differe from zero. Saisical ess (sig he -sa) ma idicae ha β is o zero. Frhermore, if β =, he he R shold be zero. I fac he R will ofe be qie large. OLS geeraes sprios regressio if variables ivolved are o saioar. If oe or all variables are osaioar he all he sal regressio resls migh be misleadig ad icorrec. This is he so-called sprios regressio problem. 6

7 Ui Roo ad Nosaioari Cosider a sochasic aoregressive AR() series: I is a i-roo process if ρ =. The he previos process becomes a radom walk: If If, he is saioar., he is osaioar. To see how is osaioar whe ρ =, we assme ha ~ N(, ). This meas ha E ( ) ad Var ( ). 7

8 Ui Roo ad Nosaioari We rer o he aoregressive AR() series: A ime = we ge: () A ime = we ge: () A ime = we ge: A ime = 3 we ge: A ime = we ge: 8 () () (3) 3 3 (3) Hece, we fod ha:... i i i (4)

9 Ui Roo ad Time Depede Variace Wha are he firs wo momes (mea ad variace) of he process? If ρ = holds, he (4) becomes: Ths, we have becase we have assmed here is o aocorrelaio amog he residals, ad The variace is : 9... i i.. ) ( ) (... ) ( ) (... E E E E E E.. ) ( ) (... ) ( ) (... Var Var Var Var Var Var ). (, ~ N Ths he variace of error erm icreases wih ime. This makes his series o saioar.

10 Ui Roo ess Dicke-Fller (DF) es The objecive is o es he presece of a i roo vs. he aleraive of a saioar model. Ths he goal is o es wheher ρ = holds. I he coex of a AR () model, if we sbrac from each side of he process he erm we obai where or similarl we ca have: (A) if we se: θ = ρ

11 Ui Roo ess Dicke-Fller (DF) es We ca es for a i roo (i.e. o-saioari of a process) i erms of model (A) b examiig he ll hpohesis H: θ = agais he aleraive H: θ < If H: θ = is valid, he ρ = ρ =, which meas ha he series is a radom walk process, which meas ha he ime series is osaioar. If H: θ < is valid, he ρ < ρ <, which meas ha he series is saioar. To es he ll hpohesis ha H: θ =, i is possible o se he sadard -saisic, b wih differe criical vales calclaed sig he Dicke- Fller disribio. We also se wo addiioal model specificaios o es for i roo: a radom walk model wih drif a radom walk model wih drif arod a sochasic red

12 Ui Roo ess Dicke-Fller (DF) es We are also ieresed i esig for a i roo (H: θ = ) i erms of a AR () process wih a cosa: a (B) Ma ecoomic ime series are redig. I is impora o disigish bewee wo impora cases: () A saioar process wih a deermiisic red: Shocks have rasior effecs. () A process wih a sochasic red or a i roo: Shocks have permae effecs. Therefore we es for a i roo i erms of a AR() process wih a red: a a (C)

13 Ui Roo ess Agmeed Dicke-Fller (ADF) es The Dicke-Fller es is exeded o a AR(p) process: a a a... a p p where he mber of agmeig lags (p) is deermied b miimizig he Schwarz Baesia iformaio crierio or miimizig he Akaike iformaio crierio or lags are dropped il he las lag is saisicall sigifica. The ll hpohesis of he Agmeed Dicke-Fller es is H: θ = ad he aleraive hpohesis is H: θ < As i he DF es, we ca se for he i roo es he -saisic associaed wih he Ordiar leas sqares esimae of he coefficie θ. Agai, he - saisic does o follow a sadard -disribio. 3

14 Ui Roo ess Agmeed Dicke-Fller (ADF) es How do we appl he ADF es i Grel? Selec he ime series of ieres, he choose Variable, Ui roo ess, ad Agmeed Dicke fller es (see picre). 4

15 Ui Roo ess Agmeed Dicke-Fller (ADF) es How do we appl he ADF es i Grel? The followig box will appear. Firs, we deermie he mber of agmeig lags (p) [] The, we specif how he es procedre will aomaicall selec he opimal lag order of he AR process. This ca be doe b miimizig he Schwarz Baesia iformaio crierio (BIC) or miimizig he Akaike iformaio crierio (AIC) or sig he -saisic [] Click o he hree radom walk model specificaios [3], ad he press OK. [] [] [3] 5

16 Ui Roo ess Agmeed Dicke-Fller (ADF) es The es resls for each radom walk specificaio will appear. Grel repors he esimae of he coefficie θ, he correspodig -saisic ad he p-vale. The ll hpohesis H: θ = (here is o-saioari) is sppored becase he p- vale is larger ha.5 for all hree specificaios. Ths, he series is osaioar. p-vale of he -saisic sa 6

17 Ui Roo ess A Reversed Tes: KPSS Someimes i is coveie o have saioari as he ll hpohesis. A i roo es, ha examies he ll hpohesis of saioari verss he aleraive hpohesis of he presece of a i roo, is he KPSS (Kwiakowski, Phillips, Schmid, ad Shi) es. We esimae he model b leas sqares : We se he es saisic: a a LM S where S i, for,,..., T are parial sms of errors, ad is a i heeroskedasici ad aocorrelaio correced (HAC) esimaor (Newe- Wes) of he variace of. 7

18 Ui Roo ess A Reversed Tes: KPSS How do we appl he KPSS es i Grel? Selec he ime series of ieres, he choose Variable, Ui roo ess, ad KPSS es (see picre). 8

19 Ui Roo ess A Reversed Tes: KPSS How do we appl he KPSS es i Grel? The followig box will appear. Firs, we deermie he mber of lags (p) for he compaio of he HAC esimaor [] Secod, click o he red selecio i order o iclde a red i or specificaio [] The, press OK. [] [] 9

20 Ui Roo ess A Reversed Tes: KPSS How do we appl he KPSS es i Grel? Grel repors he LM es resl, he criical vales for levels of saisical sigificace %, 5%, ad %, ad he p-vale of he es. Sice he LM es saisic vale (.743) is larger ha he criical vale a level 5% (.48), we rejec he ll hpohesis of saioari. Ths, he ime series is osaioar. LM es saisic

21 Ui Roo ess The Phillips-Perro es The Phillips-Perro (PP) i roo ess differ from he ADF ess mail i how he deal wih serial correlaio ad heeroskedasici i he errors. Similar o ADF es, he esimae he model: a where we ma exclde he cosa or iclde a red erm. The hpohesis o be esed: H: ρ - = agais H: ρ - < The PP es saisic have he same disribio as he ADF -saisic. Oe advaage of he PP ess over he ADF ess is ha he PP ess are robs o geeral forms of heeroskedasici i he error erm.

22 Ui Roo ess The Phillips-Perro es The PP es correcs for a serial correlaio ad heeroskedasici i he errors of he es regressio b direcl modifig he es saisic ρ=. The modified saisic, deoed Z, is give b. error s Z The erms ad are cosise esimaes of he variace parameers, ) ( k,, q j q j j i j i i j, where k is he mber of he parameers of he regressio, q is he mber of lags o se i calclaig (HAC esimaor), is he variace of he error erms, ad.. error s

23 Ui Roo ess The Phillips-Perro es Phillips ad Perro s Z es saisic ca be viewed as Dicke Fller saisic ha has bee made robs o serial correlaio b sig he HAC (heeroskedasici- ad aocorrelaio-cosise covariace marix) esimaor. Whe j>, is a esimaor of he covariace bewee wo error erms j periods apar. Whe he covariaces are zero i.e. here is o aocorrelaio bewee error erms, he Hece he secod erm i he formla of disappears ad we ge: Sice, he Z es saisic becomes: 3 j,,. j, q j q j. Z error s Z Therefore, whe here is o aocorrelaio, he Z saisic is he ADF saisic.

24 Ui Roo ess The Phillips-Perro es How do we appl he Phillips-Perro es i Grel? Selec he ime series of ieres, he choose Variable, Ui roo ess, ad Phillips-Perro es. The followig picre will appear. Selec he series o es for i roo [] Choose wheher o iclde a cosa, a red, or boh []. [] [] 4

25 Ui Roo ess The Phillips-Perro es How do we appl he Phillips-Perro es i Grel? Grel repors he es resl, ad he correspodig p-vale. Z Sice he p-vale of he es saisic (i.e.,.753) is larger ha.5, here is srog sppor of he ll hpohesis of a i roo. Ths, he ime series is o-saioar. Z Z es saisic 5

26 Resolvig Nosaioari The mai mehod for idcig saioari is o differece he daa. For isace if is osaioar, he we calclae he firs differece of he series: Whe a variable coais a i roo, i is said o be I() (iegraed of order oe) ad eeds o be differeced oce o become saioar. Whe a variable does o coai a i roo, i is said o be I() (iegraed of order zero) ad we do o have o differece he series (we se i as i is). Whe a variable coais wo i roos, i is said o be I() ad eeds o be differeced wice o idce saioari. For isace if is I(), he we calclae he secod differece of he daa. Usall we calclae he differece of he differeced daa: 6

27 Resolvig Nosaioari Whe sig a i roo es, he daa is firs esed o deermie if i coais a i roo, i.e. i is I() ad o I() Therefore, he ll ad aleraive hpoheses of he i roo es are cosidered o be H H : : If i is o I(), i cold be I(), I() or have a higher order of i roos. I his case he i roo es eeds o be codced o he differeced variable o deermie if i is I() or I(). (I is ver rare o fid I(3) or higher orders). Therefore, we will es ~ ~ I I H H : : ~ ~ I I ~ ~ I I 7

28 Resolvig Nosaioari Example: we esed wheher he series of US/EURO exchage raes have a i roo (i.e., he series is I()) b implemeig he Agmeed Dicke Fller es. We fod ha he daa have a i roo; herefore he series is I(). So we have o differece he daa o idce saioari. 8

29 Resolvig Nosaioari We ca calclae he firs differece of he daa b selecig Add, ad he firs differece (see picre). The ew ime series will have a ame sarig wih d_. 9

30 Resolvig Nosaioari The we will es wheher he differeced daa have a i roo (i.e., ha he series is I()). We appl he ADF es o he ew series. The picre preses he es resls. The p-vales of he hree specificaios are almos zero (mch smaller ha.5). Ths he differeced daa do o have a i roo. Ths, US/EURO exchage raes are I(). 3