DIFFERENCING AND UNIT ROOT TESTS

Transcription

1 DIFFERENCING AND UNIT ROOT TESTS In he Box-Jenkins approach o analyzing ime series, a key quesion is wheher o difference he daa, i.e., o replace he raw daa {x } by he differenced series {x x }. Experience indicaes ha 1 mos economic ime series end o wander and are no saionary, bu ha differencing ofen yields a saionary resul. A key example, which ofen provides a fairly good descripion of acual daa, is he random walk, x = x +ε, where {ε } is whie noise, assumed here o be independen, each having he 1 same disribuion (e.g., normal,, ec.). The random walk is said o have a uni roo. To undersand wha his means, le s recall he condiion for saionariy of an AR (p ) model. In Chaper 3, par II, we said ha he AR (p ) series x =α x +α x αpx p +ε will be saionary if he larges roo θ of he equaion (in he complex variable z ) p z =αz +αz p 1 2 p 2 +αp 1z +α p (1) saisfies θ < 1. So saionariy is relaed o he locaion of he roos of Equaion (1). We can hink of he random walk as an AR (1) process, x =αx 1 +ε wih α=1. Bu since i has α=1, he random walk is no saionary. Indeed, for an AR (1) o be saionary, i is necessary ha all roos of he equaion z =α have "absolue value" less han 1. Since he roo of he equaion z =α is jus α, we see we see ha he AR (1) is saionary if and only if 1 <α<1. For he random walk, we have a uni roo, ha is, a roo equal o one. The firs difference of a random walk is saionary, however, since x x 1 =ε, a whie noise process. In general, we say ha a ime series {x }isinegraed of order 1, denoed by I (1), if {x }isno saionary bu he firs difference {x x } is saionary and inverible. If {x }isi (1), i is considered 1 imporan o difference he daa, primarily because we can hen use all of he mehodologies developed for saionary ime series o build a model, or o oherwise analyze, he differenced series. This, in urn, improves our undersanding (e.g., provides beer forecass) of he original series, {x }. For example, in he Box-Jenkins ARIMA (p,1,q) model, he differenced series is modeled as a saionary ARMA (p, q ) process. In pracice, hen, we need o decide wheher o build a saionary model for he raw daa or for

2 -2- he differenced daa. More generally, here is he quesion of how many imes we need o difference he daa. In he ARIMA (p, d, q ) model, he d h difference is a saionary ARMA (p, q ). The series is inegraed of order d, denoed by I (d ), where d is an ineger wih d 1, if he series and all is differences up o he d 1 s are nonsaionary, bu he d h difference is saionary. A series is said o be inegraed of order zero, denoed by I (0), if he series is boh saionary and inverible. (The imporance of inveribiliy will be discussed laer). If he series {x }isi (d ) wih d 1, hen he differenced series {x x } 1 is I (d 1). For an example of an I (2) process, consider he AR (2) series x = 2x x +ε. This process is n o saionary. Equaion (1) becomes z = 2z 1, ha is, z 2z + 1 = 0. Facoring his gives (z 1) (z 1) = 0, so he equaion has wo uni roos. Since he larges roo (i.e., one) does no have " absolue value" less han one, he process is no saionary. I can be shown ha he firs difference is no saionary eiher. The second difference is x x [x x ] = x 2x + x, which is equal o ε by he definiion of our AR (2) process. Since he second difference is whie noise, {x }isanarima (0, 2, 0). Since he second difference is saionary, {x }isi(2). In general, for any ARIMA process which is inegraed of order d, Equaion (1) will have exacly d uni roos. In pracice, however, he only ineger values of d which seem o occur frequenly are 0 and 1. So here, we will limi our discussion o he quesion of wheher or no o difference he daa one ime. If we fail o ake a difference when he process is nonsaionary, regressions on ime will ofen yield a spuriously significan linear rend, and our forecas inervals will be much oo narrow (opimis- ic) a long lead imes. For an example of he firs phenomenon, recall ha for he Deflaed Dow Jones series, we go a -saisic for he slope of 5.27 (creaing he illusion of a very srong indicaion of rend), bu he mean of he firs differences was no significanly differen from zero. For an example of he second phenomenon, le s compare a random walk wih a saionary AR (1) model. For a random walk, he variance of he h -sep forecas error is

3 -3- var [x x ] = var [ε +ε +... (ε +ε +... )] n +h n n+h n+h 1 n n 1 = var [ε n +h ε n+1] = h var [ε ], which goes o as h increases. The widh of he forecas inervals will be proporional o h, indica- ing ha our uncerainy abou he fuure value of he series grows wihou bound as he lead ime is increased. On he oher hand, for he saionary AR (1) process x =αx +ε wih 1 <α<1, he bes linear h -sep forecas is f n, h h n 1 =α x, which goes o zero as h increases. The variance of he forecas h error is var [xn +h α x n ], which ends o var [x ], a finie consan. So as he lead ime h is increased, he widh of he h -sep predicion inervals grows wihou bound for a random walk, bu remains bounded for a saionary AR (1). Clearly, hen, if our series were really a random walk, bu we failed o difference i and modeled i insead as a saionary AR (1), hen our predicion inervals would give us much more faih in our abiliy o predic a long lead imes han is acually warraned. I is also undesirable o ake a difference when he process is saionary. Problems arise here because he difference of a saionary series is no inverible, i.e., canno be represened as an AR ( ). For example, if x =.9x +ε, so ha {x } is really a saionary AR (1), hen he firs difference {z }is 1 he non-inverible ARMA (1, 1) process z =.9z +ε ε, which has more parameers han he origi nal process. (Recall ha an ARMA (p, q ) is inverible if he larges roo θ of he equaion q z + bz q 1 + b q = 0 saisfies θ < 1, where b 1,...,bq are he MA parameers.) Because of he non-inveribiliy of {z }, is parameers will be difficul o esimae, and i will be difficul o consruc a forecas of z. Consequenly, aking an unnecessary difference (i.e., overdifferencing ) will end o +h degrade he qualiy of forecass. Ideally, hen, wha we would like is a way o decide wheher he series is saionary, or inegraed of order 1. A mehod in widespread use oday is o declare he series nonsaionary if he sample auo- correlaions decay slowly. If his paern is observed, hen he series is differenced and he auocorrelaha he differenced series is saionary. This mehod is somewha ad hoc, however. Wha is really ions of he differenced series are examined o make sure ha hey decay rapidly, hereby indicaing needed is a more objecive way of deciding beween he wo hypoheses, I (0) and I (1), wihou making

4 -4- any furher assumpions. Unforunaely, each of hese hypoheses covers a vas range of possibiliies, and any classical approach o discriminae beween hem seems doomed o failure unless we limi he scope of he hypoheses. The Dickey-Fuller Tes of Random Walk Vs. Saionary AR(1) A es involving much more narrowly-specified null and alernaive hypoheses was proposed by Dickey and Fuller in In is mos basic form, he Dickey-Fuller es compares he null hypohesis H : x = x +ε, 0 1 i.e., ha he series is a random walk wihou drif, agains he alernaive hypohesis H : x = c +ρx +ε, 1 1 where c and ρ are consans wih ρ < 1. According o H, he process is a saionary AR (1) wi 1 h mean =c /(1 ρ). To see his, noe ha, under H, we can wrie 1 x =(1 ρ) +ρx +ε 1, so ha x =ρ(x ) +ε. 1 Noe ha by making he random walk he null hypohesis, Dickey and Fuller are expressing a prefer- ence for differencing he daa unless a srong case can be made ha he raw series is saionary. This is consisen wih he convenional wisdom ha, mos of he ime, he daa do require differencing. A Type I error corresponds o deciding he process is saionary when i is acually a random walk. In his case, we will fail o recognize ha he daa should be differenced, and will build a saionary model for our nonsaionary series. A Type II error corresponds o deciding he process is a random walk when i is acually saionary. Here, we will be inclined o difference he daa, even hough differencing is no desirable. We should menion wo addiional imporan differences beween he AR (1) and he random walk. Whereas he innovaion ε has a emporary (exponenially decaying) effec on he AR (1), i has permanen effec on he random walk. Whereas he expeced lengh of ime beween crossings of is a

5 -5- finie for he AR (1) (so he AR (1) flucuaes around is mean of ), he expeced lengh of ime beween crossings of any paricular level is inf inie for he random walk (so he random walk has a endency o wander in a non-sysemaic fashion from any given saring poin). The Dickey-Fuller es is easy o perform. Given daa x,...,x, we run an ordinary linear 1 n r 2 n egression of he observaions (x,...,x ) of he "dependen variable" {x }, agains he observaions (x,...,x ) of he "independen variable" {x }, ogeher wih a consan erm. Under boh H and 1 n H 1, he daa x obey he linear regression model x = c +ρx +ε, 1 a 0 nd H corresponds o ρ=1, c = 0. Denoe he -saisic for he leas squares esimae ρˆ by τ =(ρˆ 1)/s, ρˆ where s is he esimaed sandard error for ρˆ. Noe ha τ is easy o calculae, since ρˆ and s can b ρˆ ρˆ e obained direcly from he oupu of he sandard compuer regression packages. For he Deflaed Dow daa, regressing x,...,x on x,...,x, we obain he following regression oupu: Residual Sandard Error = , Muliple R-Square = N = 546, F-saisic = on 1 and 544 df, p-value = 0 coef sd.err.sa p.value Inercep X The R-Square saisic is.9907, indicaing a very srong linear relaionship beween {x } and { x 1 ρˆ }. The esimaed slope is ρˆ =.9963, and s = We calculae τ =(ρˆ 1)/s = ( )/.0041 =.9024 ρˆ. Noe ha we do NOT use he saisic ( ) from he oupu, since his was compued relaive o a null value of zero, insead of 1.

6 -6- The saisic τ can be used o es H versus H. The perceniles of τ under H are given in he aached able. The null hypohesis is rejeced if τ is less han he abled value. The abulaions fo r finie n were based on simulaion, assuming ε disribuion (n = ) are valid as long as he ε are iid Gaussian. The abled values for he asympoic are iid wih finie variance. (No Gaussian assumpion is needed here.) I should be noed ha τ does no have a disribuion in finie samples, and does no have a sandard normal disribuion asympoically. In fac, he asympoic disribuion is longer-ailed han he sandard normal. For example, he asympoic.01 percenage poin of τ is a 3.43, insead of for a sandard normal. Thus, use of he sandard normal able would resul in an excess of spuri- ous declaraions of saionariy. For he Deflaed Dow daa, we obained τ =.9024, which is no significan according o he Table. So we are no able o rejec he random walk hypohesis. As usual in saisical hypohesis es- ing, his does no mean ha we should conclude ha he series is a random walk. In fac, from our ear- lier analysis we have srong saisical evidence ha he series is no a random walk, since he lag-1 auocorrelaion for he firs differences is highly significan. All we can conclude from he Dickey-Fuller es is ha here is no srong evidence o suppor he hypohesis H ha he series is a saionary AR (1). This is he ype of alernaive ha he es was designed o deec. The quesion of wheher he firs difference has any auocorrelaion is anoher issue alogeher, and he es was no designed o deec his ype of failure of he random walk hypohesis. In any case, he resuls of he es indicae ha i ing he ACF of he raw daa, bu he Dickey-Fuller es provides a more objecive basis for making his would be a good idea o difference he daa. We could have come o his same conclusion by examindecision. As an illusraion of he long ails in he τ disribuion, consider he random walk daa (n = 547) 1 which was used in he las handou for comparison wih he Dow and Deflaed Dow series. For his random walk daa se, we obain he following regression oupu.

7 -7- Residual Sandard Error = , Muliple R-Square = N = 546, F-saisic = on 1 and 544 df, p-value = 0 coef sd.err.sa p.value Inercep X We herefore ge τ =( )/.0057 = If τ had a sandard normal disribuion, we would obain a p -value of.063 (one-sided), indicaing some evidence in favor of he alernaive hypohesis (ha he series is a saionary AR (1)). Of course, we know ha his series was in fac a random walk, and so i is somewha disressing ha we are almos being led o commi a Type I error. Bu when we use he rue disribuion of τ under he null hypohesis (see able) we find ha he acual significance level is subsanially greaer han.10, alhough he able is no precise enough o allow us o find he exac p -value. Of course, he null and alernaive hypoheses H and H described above are oo narrow o be 0 1 very useful in a wide variey of siuaions. Ofen, we will wan o consider differencing he daa because we hope he difference may be saionary, bu we do no wan o commi ourselves o he assumpion ha he series is eiher a random walk or a saionary AR (1). Forunaely, alhough we will no describe he deails here, here is a similar es known as he Augmened Dickey-Fuller es, which allows us o es an ARIMA (p,1,0) null hypohesis versus an ARIMA (p +1,0,0) alernaive, where p 0 is known. If p = 1, for example, he null hypohesis would be ha he series is nonsaionary, bu is firs difference is a saionary AR (1); he alernaive hypohesis would be ha he series is a saion- ary AR (2). In rerospec, i seems ha he Deflaed Dow series is beer described by he above null hypohesis han by he one which was acually esed, i.e., he random walk. Bu i is never a good idea o change a saisical hypohesis afer looking a he daa; i can desroy he validiy of he es. Furh- ermore, he use of he random walk as a null hypohesis for financial ime series seems wise as a gen- eral rule. Difference Saionariy Vs. Trend Saionariy In he ordinary Dickey-Fuller (τ ) es, he series is assumed o be free of deerminisic rend, under boh he null and alernaive hypoheses. Many acual series do have rend, however, and i is of

8 -8- ineres o sudy he naure of his rend. Perhaps he mos imporan issue is he way in which he rend is combined wih he random aspecs of he series. In he case of a random walk wih drif x = c + x +ε where {ε } is zero mean whie noise, here is a mixure of deerminisic and sochasic 1 rend, he process has a uni roo, and he forecas inervals grow wihou bound as he lead ime increases. Differencing {x } yields a saionary series, so {x } is said o be difference saionary. (This is he same as I (1)). Anoher way o combine rend and randomness is o sar wih a deerminisic linear rend and bury i in whie noise: x =α +α +ε. This is a sandard linear regression (rend-line) model, which 0 1 can be analyzed wihou using ime series mehods. If he parameers (α, α, var [ε ]) are known, he 0 1 n he forecas of x is simply f =α +α (n +h ). If he ε are normally disribued, a forecas inerval n n +h n, h 0 1 +h n, h α/2 for x is given for large h by f ± z var ε. The widh of his forecas inerval does no end o infiniy as he lead ime increases. More generally, any series x =α 0 +α 1 + y formed by adding a deerminisic linear rend o a saionary, inverible, zero mean "noise" series {y } is said o be rend saionary. Trend saionary series do no conain a uni roo. The widh of heir forecas inervals for large h is 2 z var y, which does no end o infiniy. Trend saionary series are α/2 no difference saionary, since i can be shown ha he difference of {y } is no inverible. Since he rend saionary series obeys a regression model wih auocorrelaed errors, we can use generalized leas squares (a popular linear regression echnique) o esimae he rend and assess is saisical significance. Here, we show how o es a specific form of difference saionariy agains a specific form of rend saionariy, using a varian (τ ) of he Dickey-Fuller es. The null hypohesis is τ H 0 : x = c + x 1 +ε, a random walk wih drif (which is difference saionary), versus H : x =α +α + y ; y =ρy +ε

9 -9- Under H,{x } is rend saionary, and he "noise" erm is AR (1). If we pu ρ=0, hen we ge he 1 rend-line model. I can be shown ha under H 1,{x } can be expressed as x =β +β +ρx +ε, (2) where β and β are consans. (Specifically, β =α (1 ρ) +ρα and β =α (1 ρ).) If we pu ρ=1 0 hen Equaion (2) reduces o , x =α +x +ε, 1 1 i.e., a random walk wih drif. Thus, we wan o es he null hypohesis ha ρ=1 versus he alernaive ha ρ<1 in Equaion (2). To perform he es, we run an ordinary linear regression of he "dependen variable" {x } agains he explanaory variables ime ( ) and {x }, ogeher wih a consan erm. The observaions on {x } 1 are (x,...,x ), he observaions on are (2,...,n), and he observaions on {x } are 2 n 1 ( x 1,...,x n 1 ). The es saisic is he sandardized esimae of ρ in Equaion (2), τ= τ ρˆ 1. s ρˆ Alhough his may appear o be he same as he ordinary Dickey-Fuller saisic τ, i is acuall y differen because of he presence of ime as an explanaory variable. The perceniles of τ under he null τ hypohesis (ρ=1) are given in he aached able. The null hypohesis is rejeced if τ is less han he τ τ abled value. The perceniles of τ are considerably less han he corresponding perceniles of τ, indicaing he effecs of including ime as an explanaory variable. For example, he asympoic.01 percenage poin of τ is a 3.96 for τ, compared wih 3.43 for τ. τ τ The log10 Dow daa seems o conain a rend, bu wha is he naure of his rend? Would i be more appropriae o model his daa as a random walk wih drif, or as a rend line plus saionary AR (1) errors? In our original analysis of his daa, we firs ried an ordinary rend-line model, and found a highly significan rend. We hen quesioned he validiy of his finding, since he Durbin-Wason saisic showed srong error auocorrelaion. We could have pursued he use of a rend saionary model ( i.e., linear rend plus auocorrelaed errors) for his series, by re-esimaing he rend line using general-

10 -10- ized leas squares. This sill would no have answered he quesion as o wheher such a model is more appropriae han a random walk wih drif, however. To address his quesion, we now run he τ es. τ The regression described above yielded Residual Sandard Error = , Muliple R-Square = N = 546, F-saisic = on 2 and 543 df, p-value = 0 coef sd.err.sa p.value Inercep Time x.lag where Time denoes (2,...,547), x.lag is (x,...,x ) and he dependen variable is ( x 2,...,x 547 ρˆ ). The esimaed coefficien of x.lag is ρˆ =.9923, and s = We calculae τ=(ρˆ 1)/s = ( )/.0054 = τ ρˆ Since his is no less han he abled value of 3.42, we do no rejec he null hypohesis of random walk wih drif a level.05. In fac, examinaion of he able reveals ha our observed τ is no small a τ all, wih a p -value around.9, indicaing ha here is virually no evidence in favor of rend saionariy for his series. This does no mean ha he log10 Dow daa is acually a random walk wih drif. (Indeed, we previously found srong evidence ha he differences of his daa are no uncorrelaed, even hough hey seem o have a nonzero expecaion.) I jus means ha we canno rejec he random walk wih drif hypohesis in favor of rend saionariy.