Section 12 Time Series Regression with Non- Stationary Variables

Transcription

1 Secin Time Series Regressin wih Nn- Sainary Variables The TSMR assumpins include, criically, he assumpin ha he variables in a regressin are sainary. Bu many (ms?) ime-series variables are nnsainary. We nw urn echniques all quie recen fr esimaing relainships amng nnsainary variables. Sainariy Frmal definiin y y E y var cv, y s s The key pin f his definiin is ha all f he firs and secnd mmens f y are he same fr all Sainariy implies mean reversin: ha he variable revers ward a fixed mean afer any shck Kinds f nnsainariy Like ms rules, nnsainariy can be vilaed in several ways Nnsainariy due breaks Breaks in a series/mdel are he ime-series equivalen f a vilain f Assumpin #0. The relainship beween he variables (including lags) changes eiher abruply r gradually ver ime. Wih a knwn penial break pin (such as a change in plicy regime r a large shck ha culd change he srucure f he mdel): Can use Chw es based n dummy variables es fr sabiliy acrss he break pin. Inerac all variables f he mdel wih a sample dummy ha is zer befre he break and ne afer. Tes all ineracin erms (including he dummy iself) = 0 wih Chw F saisic. If breakpin is unknwn: Quand likelihd rai es finds he larges Chw-es F saisic, excluding (rimming) he firs and las 5% (r mre r less) f he sample as penial breakpins make sure ha each sub-sample is large enugh prvide reliable esimaes. ~ 4 ~

2 QLR es saisic des n have an F disribuin because i is he max f many F saisics. Deerminisic rends are cnsan increases in he mean f he series ver ime, hugh he variable may flucuae abve r belw is rend line randmly. y v v is sainary disurbance erm If he cnsan rae f change is in percenage erms, hen we culd mdel lny as being linearly relaed ime. This vilaes he sainariy assumpins because E y, independen f which is n Schasic rends allw he rend change frm perid perid be randm, wih given mean and variance. Randm walk is simples versin f schasic rend: y y v where v is whie nise. Randm walk is limiing case f sainary AR() prcess y y v as Slving recursively (cndiinal n given iniial value y 0 ), y y v, 0 y y v y v v, 0 y y v... v y v y v. 0 0 s 0 s 0 This vilaes sainariy assumpins because var y y0 var v v, which depends n, and uncndiinal 0 variance is infinie: var y var v v. 0 0 Ne cmparisn wih sainary AR(): y y v, 0 s v var y var v v 0 0 Randm walk wih drif allws fr nn-zer average change: y y v This als vilaes he cnsan-mean assumpin: y y0 v, y y v y v v, 0 y y0 v 0. ~ 5 ~

3 E y y0 y0, var y y. 0 v Bh cndiinal mean and cndiinal variance depend n Bh uncndiinal mean and uncndiinal variances are infinie Fr AR() wih nn-zer mean: y y v v E y, 0 v var. y v 0 Bh uncndiinal mean and variance are finie and independen f. Difference beween deerminisic and schasic rend Cnsider large negaive shck v in perid In deerminisic rend, he rend line remains unchanged. Because v is assumed sainary, is effec evenually disappears and he effec f he shck is emprary In schasic rend, he lwer y is he basis fr all fuure changes in y, s he effec f he shck is permanen. Which is mre apprpriae? N clear rule ha always applies Schasic rends are ppular righ nw, bu hey are cnrversial. Uni rs and inegrain in AR mdels Ne ha he randm-walk mdel is jus he AR() mdel wih =. In general, he sainariy f a variable depends n he parameers f is AR represenain: AR(p) is y y... y v, r L y v. p p (Can generalize allw v be any sainary prcess, n jus whie nise.) The sainariy f y depends n he rs (sluins) he equain L 0. (L) is a p-rder plynmial ha has p rs, which may be real r imaginary-cmplex numbers. AR() is firs-rder, s here is ne r: 0 0 ~ 6 ~ L L, L L L L, s / is he r f he

4 AR() plynmial. (Or / in he simpler AR() nain we used abve.) If he p rs f L 0 are all greaer han ne in abslue value (frmally, because he rs f a plynmial can be cmplex, we have say uside he uni circle f he cmplex plane ), hen y is sainary. By ur r crierin fr sainariy, he AR() is sainary if. This crrespnds he assumpin we presened earlier ha. If ne r mre rs f L 0 are equal ne and he hers are greaer han ne, hen we say ha he variable has a uni r. We call hese variables inegraed variables fr reasns we will clarify sn. Inegraed variables are jus barely nnsainary and have very ineresing prperies. (Variables wih rs less han ne in abslue value simply explde.) The randm-walk is he simples example f an inegraed prcess: y y v y y v L y L y v The r f L = 0 is L =, which is a uni r. Inegraed prcesses Cnsider he general AR(p) prcess y y y v, which we wrie in lag-perar nain as L y v. p p We ned abve ha he sainariy prperies f y are deermined by wheher he rs f (L) = 0 are uside he uni circle (sainary) r n i (nnsainary). (L) is an rder-p plynmial in he lag perar L L L p. We can facr (L) as p p L L L L L L, where p p,,, are he rs f (L). p We rule u allwing any f he rs be inside he uni circle because ha wuld imply explsive behavir f y, s we assume j., r ~ 7 ~

5 Suppse ha here are k p rs ha are equal ne (k uni rs) and p k rs ha are greaer han ne (uside he uni circle in he cmplex plane). We can hen wrie L L L L, where pk we number he rs s ha he firs p k are greaer han ne. Le L L L p kl. k Then L k k L y L L y L y v. Because (L) has all f is rs uside he uni circle, he series k y is sainary. We inrduce he erminlgy inegraed f rder k (r I(k)) describe a series ha has k uni rs and ha is sainary afer being differenced k imes. The erm inegraed shuld be hugh f as he inverse f differenced in much ha same way ha inegrals are he inverse f differeniain. The inegrain perar L accumulaes a series in he same way ha he difference perar L urns he series in changes. Inegraing he firs differences f a series recnsrucs he riginal series: L y L L y y If y is sainary, i is I(0). If he firs difference f y is sainary bu y is n, hen y is I(). Randm walks are I(). If he firs difference is nnsainary bu he secnd difference is sainary, hen y is I(), ec. In pracice, ms ecnmic ime series are I(0), I(), r ccasinally I(). Impacs f inegraed variables in a regressin If y has a uni r (is inegraed f rder > 0), hen he OLS esimaes f cefficiens f an auregressive prcess will be biased dwnward in small samples. Can es = 0 in an auregressin such as y y v wih usual ess Disribuins f saisics are n r clse nrmal Spurius regressin Nn-sainary ime series can appear be relaed wih hey are n. This is exacly he kind f prblem illusraed by he baseball aendance/bswana GDP example k ~ 8 ~

6 Shw he Granger-Newbld resuls/ables Dickey-Fuller ess fr uni rs Since he desirable prperies f OLS (and her) esimars depend n he sainariy f y and x, i wuld be useful have a es fr a uni r. The firs and simples es fr uni-r nnsainariy is he Dickey-Fuller es. I cmes in several varians depending n wheher we allw a nn-zer cnsan and/r a deerminisic rend Tesing he null ha y is randm walk wihu drif: DF es wih n cnsan r rend Cnsider he AR() prcess y y v The null hyphesis is ha y is I(), s H 0 : =. Under he null hyphesis, y fllws a randm walk wihu drif. Alernaive hyphesis is ne-sided: H : < and y is sainary AR() prcess We can jus run an OLS regressin f his equain and es = wih a cnveninal es because he disribuin f he saisic is n asympically nrmal under he null hyphesis ha y is I(). y y v y v, wih If we subrac y frm bh sides, we ge. If he null hyphesis is rue ( = r = 0) hen he dependen variable is nn-sainary and he cefficien n he righ is zer. We can es his hyphesis wih an OLS regressin, bu because he regressr is nnsainary (under he null), he saisic will n fllw he r asympically nrmal disribuin. Insead, i fllws he Dickey- Fuller disribuin, wih criical values sricer han hse f he nrmal. See Table. n p. 486 fr criical values. If he DF saisic is less han he (negaive) criical value a ur desired level f significance, hen we rejec he null hyphesis f nn-sainariy and cnclude ha he variable is sainary. Ne ha a ne-ailed es (lef-ailed) is apprpriae here because = shuld always be negaive. Oherwise, i wuld imply >, which is nn-sainary in a way ha cann be recified by differencing. The inuiin f he DF es relaes he mean-reversin prpery f sainary prcesses: y y v. ~ 9 ~

7 If < 0, hen when y is psiive (abve is zer mean) y will end be negaive, pulling y back ward is (zer) mean. If = 0, hen here is n endency fr he change in y be affeced by wheher y is currenly abve r belw he mean: here is n mean reversin and y is nnsainary. Tesing he null ha y is a randm walk wih drif: DF es wih cnsan bu n rend In his case, he null hyphesis is ha y fllws a randm walk wih drif Alernaive hyphesis is sainariy y y v y y v y v H 0 : 0 H : 0 Very similar DF es wihu a cnsan bu criical values are differen (See Table.) Tesing he null ha y is rend sainary : DF es wih cnsan and rend In his case, he null is ha he deviains f y frm a deerminisic rend are a randm walk Alernaive is ha hese deviains are sainary y y v y y v y v Ne ha under he alernaive hyphesis, y is nnsainary (due he deerminisic rend) unless = 0. Is v serially crrelaed? Prbably, and he prperies f he DF es saisic assume ha i is n. By adding sme lags f y n he RHS we can usually eliminae he serial crrelain f he errr. y y ay a y v is he mdel fr he Augmened p p Dickey-Fuller (ADF) es, which is similar bu has a differen disribuin ha depends n p. Saa des DF and ADF ess wih he dfuller cmmand, using he lags(#) pin add lagged differences. An alernaive he ADF es is use Newey-Wes HAC rbus sandard errrs in he riginal DF equain raher han adding lagged differences eliminae serial crrelain f e. This is he Phillips-Pern es: pperrn in Saa. Nnsainary vs. brderline sainary series Y Y u is a nnsainary randm walk Y 0.999Y u is a sainary AR() prcess ~ 0 ~

8 They are n very differen when T <. Shw graphs f hree series Can we hpe ha ur ADF es will discriminae beween nnsainary and brderline sainary series? Prbably n wihu lnger samples han we have. Since he null hyphesis is nnsainariy, a lw-pwer es will usually fail rejec nnsainariy and we will end cnclude ha sme highly persisen bu sainary series are nnsainary. Ne: The ADF es des n prve nnsainariy; i fails prve sainariy. DF-GLS es Anher useful es ha can have mre pwer is he DF-GLS es, which ess he null hyphesis ha he series is I() agains he alernaive f eiher I(0) r ha he series is sainary arund a deerminisic rend. Available fr dwnlad frm Saa as dfgls cmmand. DF-GLS es fr H 0 : y is I() vs. H : y is I(0) Quasi difference series: y, fr, z 7 y y, fr,3,, T. T, fr, x 7, fr,3,, T. T Regress z n x wih n cnsan (because x is essenially a cnsan): z 0z v. Calculae a derended (really demeaned here) y series as d y ˆ y 0. Apply he DF es he derended y d series wih crreced criical values (S&W Table 6. prvide criical values). DF-GLS es fr H 0 : y is I() vs. H : y is sainary arund deerminisic rend ~ ~

9 Cinegrain Quasi-difference series: y, fr, z 3.5 y y, fr,3,, T T, fr, x 3.5, fr,3,, T T, fr, x 3.5, fr,3,, T T Run rend regressin z x x v 0 d Calculae derended y as y ˆ ˆ y 0 d Perfrm DF es n y using criical values frm S&W s Table 6.. Sck and Wasn argue ha his es has cnsiderably mre pwer disinguish brderline sainary series frm nn-sainary series. I is pssible fr w inegraed series mve geher in a nnsainary way, fr example, s ha heir difference (r any her linear cmbinain) is sainary. Such series fllw a cmmn schasic rend. These series are said be cinegraed. Sainariy is like a rubber band pulling a series back he fixed mean. Cinegrain is like a rubber band pulling he w series back (a fixed relainship wih) each her, even hugh bh series are n pulled back a fixed mean. If y and x are bh inegraed, we cann rely n OLS sandard errrs r saisics. By differencing, we can avid spurius regressins: If y x e hen y x e. Ne he absence f a cnsan erm in he differenced equain: he cnsan cancels u. If a cnsan were be in he differenced equain, ha wuld crrespnd a linear rend in he levels equain. e is sainary as lng as e is I(0) r I() The differenced equain has n hisry. Is e sainary r nnsainary? Suppse ha e is I(). ~ ~

10 This means ha he difference e y x is n meanrevering and here is n lng-run endency fr y say in he fixed relainship wih x. N cinegrain beween y and x. e is I(0). Bygnes are bygnes: if y is high (relaive x ) due a large psiive e, hen here is n endency fr y cme back x afer. Esimain f differenced equain is apprpriae. Nw suppse ha e is I(0). Tha means ha he levels f y and x end say clse he relainship given by he equain. Suppse ha here is a large psiive e ha pus y abve is lngrun equilibrium level in relain x. Wih sainary e, we expec he level f y reurn he lngrun relainship wih x ver ime: sainariy f e implies ha crr(e, e + s ) 0 as s. Thus, fuure values f y shuld end be smaller (less psiive r mre negaive) han hse prediced by x in rder clse he gap. In erms f he errr erms, a large psiive e shuld be fllwed by negaive e values reurn e zer if e is sainary. In he cinegraed case This is he siuain where y and x are cinegraed. This is n refleced in he differenced equain, which says ha bygnes are bygnes and fuure values f y are nly relaed he fuure x values here is n endency eliminae he gap ha pened up a. If we esimae he regressin in differenced frm we are missing he hisry f knwing hw y will be pulled back in is lng-run relainship wih x. If we esimae in levels, ur es saisics are unreliable because he variables (hugh n he errr erm) are nnsainary. The apprpriae mdel fr he cinegraed case is he errr-crrecin mdel f Hendry and Sargan. ECM cnsiss f w equains: Lng-run (cinegraing) equain: y x e, where (fr he rue values f and ) e is I(0) ~ 3 ~

11 Shr-run (ECM) adjusmen equain: y y x y y x x v p p 0 q q Ne he presence f he errr-crrecin erm wih cefficien in he ECM equain. This erm reflecs he disance ha y is frm is lng-run relainship x. If < 0, hen y abve is lng-run level will cause y be negaive (her facrs held cnsan), pulling y back ward is lng-run relainship wih x. There is n cnsan erm in he differenced regressin (hugh many peple include ne) because he cnsan erm in he x, y relainship cancels u in he differencing prcess. Because bh y and x are I(), heir firs differences are I(0). Because hey are cinegraed wih cinegraing vecr,, he difference in he errrcrrecin erm is als I(0). This erm wuld n be sainary if hey weren cinegraed and he ECM regressin wuld be invalid. Esimain f cinegraed mdels The ECM equain can be esimaed by OLS wihu undue difficuly because all he variables are sainary. The cinegraing regressin can be esimaed super-cnsisenly by OLS (alhugh he esimaes will be nn-nrmal and he sandard errrs will be invalid). HGL sugges esimaing bh equains geher by nnlinear leas squares Sck and Wasn recmmend an alernaive dynamic OLS esimar fr he cinegrain equain: p y x jx j v jp This can be esimaed by OLS and he HAC-rbus sandard errrs are valid fr. Dn include he erms in he errr-crrecin erm in he ECM regressin, which remains y ˆ ˆ x. Nrmally, we wuld have crrec he sandard errrs f he ECM fr he fac ha he errr-crrecin variable is calculaed based n esimaed raher han knwn wih cerainy. Hwever, because he esimars f are super-cnsisen in he cinegrain case, hey cnverge asympically faser he rue han ~ 4 ~

12 he and esimaes and can be reaed as if hey were rue parameer values insead f esimaes. Mulivariae cinegrain The cncep f cinegrain exends muliple variables Wih mre han w variables, here can be mre han ne cinegraing relainship (vecr) Ineres raes n bnds issued by Oregn, Washingn, Idah migh be relaed by r O = r W = r I. Tw equal signs means w cinegraing relainships. Vecr errr-crrecin mdels (VECM) allw fr he esimain f errrcrrecin regressins wih muliple cinegraing vecrs. We will sudy hese sn. Saa des his using he vec cmmand. Tesing fr cinegrain The earlies es fr cinegrain is Engle and Granger s exensin f he ADF es: Esimae he cinegraing regressin by OLS. Tes he residuals wih an ADF es, using revised criical values as in S&W s Table 6.. Oher, mre ppular ess include he Jhansen-Juselius es, which generalizes easily muliple variables and muliple cinegraing relainships. Vecr auregressin VAR was develped in he macrecnmics lieraure as an aemp characerize he jin ime-series f a se (vecr) f variables wihu making he resricive (and perhaps false) assumpins ha wuld allw he idenificain f srucural dynamic mdels. VAR can be hugh f as a reduced-frm represenain f he jin evluin f he se f variables. Hwever, in rder use he VAR fr cndiinal frecasing, we have make assumpins abu he causal srucure f he variables in he mdel. The need fr idenifying resricins ges pushed frm he esimain he inerpreain phase f he mdel. Tw-variable VAR(p) fr x and y (which shuld be sainary, s hey migh be differences): y y y x x v x y y x x v y y 0 y yp p y yp p x x 0 x xp p x xp p Ne absence f curren values f he variables n he RHS f each equain. This reflecs uncerainy abu wheher he crrelain beween y and x is because x causes y r because y causes x. ~ 5 ~

13 Crrelain beween y and x will mean ha he w errr erms are crrelaed wih ne anher, hwever. (This is assumed n happen in he simplified HGL example, bu in pracice hey are always crrelaed.) y x This means ha we can hink f v as a pure shck y and v as a pure shck x: ne f hem will have be respnding he her in rder fr hem be crrelaed. Esimae by OLS SUR is idenical because regressrs are same in each equain Hw many variables? Mre adds p cefficiens each equain Generally keep he sysem small (6 variables is large) Hw many lags? Can use he AIC r Schwarz crierin n he sysem as a whle evaluae: ln de ˆ ln T SC p u k kp where k is he number f T, variables/equains and p is he number f lags. The deerminan is f he esimaed cvariance marix f he errrs, calculaed as he sample variances and cvariances f he residuals. Wha can we use VARs fr? Granger causaliy ess The seup is naural fr bidirecinal (r mulidirecinal) Granger causaliy ess. Frecasing VAR is a simple generalizain f predicing a single variable based n is wn lags: we are predicing a vecr f variables based n lags f all variables. We can d frecasing wihu any assumpins abu he underlying srucural equains f he mdel. N idenificain issues fr frecasing. T d muli-perid frecass, we jus plug in he prediced values fr fuure years and generae lnger-erm frecass recursively. Idenificain in VARs: Impulse-respnse funcins and variance decmpsiins In rder use VARs fr simulain f shcks, we need be able idenify he shcks. Is v x a pure shck x wih n effec n y? Is v y a pure shck y wih n effec n x? Bh cann generally be rue if he w v erms are crrelaed. Tw pssible inerpreains (idenifying resricins) v x is a pure x shck; sme par f v y is a respnse v x and he remainder f v y is a pure y shck. x is firs and y respnds x in he curren perid. y des n affec x. ~ 6 ~

14 v y is a pure y shck; sme par f v x is a respnse v y and he remainder f v x is a pure x shck. Oppsie assumpin abu cnempraneus causaliy. If we dn make ne f hese assumpins, hen shcks are n idenified and we can d simulains. (Can sill frecas and d Granger causaliy, hugh.) If we make ne r he her idenifying resricin, hen we can cnduc simulains f he effecs f shcks x r y. Suppse ha we assume ha x affecs y cnempraneusly, bu n he her way arund. Shck f ne uni (we fen use ne sandard deviain insead) v x causes a ne-uni increase in x and a change in y ha depends n he cvariance beween v x and v y, which we can esimae. In +, he changes x and y will affec x + and y + accrding he cefficiens x, y, x, and y. (We assume ha all v erms are zer afer.) Then in +, he changes x, y, x +, and y + will affec he values in +. This prcess feeds frward indefiniely. x s y s The sequence and fr s = 0,,, is called he x x v v impulse-respnse funcin wih respec a shck x. We can analyze a ne-uni shck y in he same basic way, excep ha y x by assumpin v has n effec n v r x. This gives he IRF wih respec a shck y. Ne ha he IRF will vary depending n ur chice f idenifying cndiin. If we assume ha y affecs x bu n vice versa (raher han he her way arund), hen we ge a differen IRF. The idenificain and IRF calculain is similar fr mre han w variables. Wih k >, he ms cmmn idenificain scheme is idenificain by rdering assumpin. We pick ne variable ha can affec all he hers cnempraneusly, bu is n immediaely affeced by any hers. Then we pick he secnd variable ha is affeced nly by he firs in he immediae perid bu can affec all bu he firs. This amuns an rdering where variables can have a cnempraneus effec nly n variable belw hem in he lis. (Of curse, all variable in he mdel are assumed affec all hers wih a ne-perid lag.) The her cmmn upu frm a VAR is he variance decmpsiin. This asks he same quesin abu hw he varius shcks affec he varius variables, bu frm he her direcin. ~ 7 ~

15 Hw much f he variance in y + s is due shcks x, shcks y and shcks her variables? The variance decmpsiin breaks dwn he variance f y + s in he shares aribued each f he shcks. We wn alk abu he frmulas used calculae hese. The Enders ex n he reading lis prvides mre deails. Vecr errr-crrecin mdels Wha if x and y are I() variables ha are cinegraed? We can d a w-variable (vecr) errr-crrecin mdel Wih w variables, here can be nly ne cinegraing relainship linking he lng-run pahs f he w variables geher. Wih m variables, here can be m cinegraing relainships, bu we wn wrry abu his. We can use OLS esimae he cinegraing regressin y 0 x e and calculae he residuals e ˆ, which are I(0) if x and y are cinegraed. We can hen esimae a VAR in he differences f x and y, using he residuals as cinegrain erms n he righ-hand side f each equain. Fr example: y y ˆ y0 ye yy... ypy p yx... ypx p v x x eˆ y... y x... x v x 0 x x xp p x xp p Time-varying vlailiy: Auregressive cndiinal heerskedasiciy (ARCH) mdels Financial ecnmiss have ned ha vlailiy in asse prices seems be aucrrelaed: if here are highly vlaile reurns n ne day, hen i is likely ha reurns will have high vlailiy n subsequen days as well. This is called vlailiy clusering. Des high vlailiy (large errr variance) end persis ver ime? Heerskedasiciy in a ime-series cnex means ha he errr variance depends n : One pssibiliy wuld be mdel as a deerminisic funcin f (a rend?) r f her ime-dependen variables We can als mdel ime-dependen errr variance as a randm variable ARCH mdels errr variance as an AR r MA prcess: This is a paricular paern f heerskedasiciy where here are psiive r negaive shcks he errr variance each perid and where shcks end persis ~ 8 ~

16 Engle mdeled his by making he variance f he errr erm depend n he square f recen errr erms: y y xe, e ~ N 0,, e e 0 p p This errr srucure is he ARCH(p) mdel.. Ne ha he cndiinal heerskedasiciy here is really mving average raher han auregressive because here are n lagged erms. (HGL leave u he lagged y and x erms lk nly a a single sainary variable) A nw-mre-cmmn generalizain is he GARCH(p, q) mdel: e e 0 p p q q ARCH and GARCH mdels (and a variey f her varians) are esimaed by ML. In Saa, he arch cmmand esimaes bh ARCH and GARCH mdels, as well as her varians. ~ 9 ~