Bootstrapping Unit Root Tests with Covariates 1

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1 Bootstrappig Uit Root Tests with Covariates 1 Yoosoo Chag Departmet of Ecoomics Rice Uiversity Robi C. Sickles Departmet of Ecoomics Rice Uiversity Woho Sog Departmet of Ecoomics Rice Uiversity Abstract We cosider the bootstrap method for the covariates augmeted Dickey- Fuller (CADF) uit root test suggested i Hase (1995) which uses related variables to improve the power of uivariate uit root tests. It is show that there are substatial power gais from icludig correlated covariates. The limit distributio of the CADF test, however, depeds o the uisace parameter that represets the correlatio betwee the equatio error ad the covariates. Hece, iferece based directly o the CADF test is ot possible. To provide a valid iferetial basis for the CADF test, we propose to use the bootstrap procedure to obtai critical values, ad establish the asymptotic validity of the bootstrap CADF test. Simulatios show that the bootstrap CADF test sigi catly improves the ite sample size performaces of the CADF test, especially whe the covariates are highly correlated with the error. Ideed, the bootstrap CADF test o ers drastic power gais over the covetioal ADF test. We apply our testig procedures to the exteded Nelso-Plosser data set for the post-1929 samples as well as postwar aual CPI-based real exchage rates for 14 OECD coutries. This versio: September 18, 21 JEL Classi catio: C12, C15, C22. Key words ad phrases: Uit root tests, covariates, bootstrap cosistecy. 1 We are grateful to Joo Park, Bill Brow, ad David Papell for helpful discussios ad commets. Correspodece address to: Robi Sickles, Departmet of Ecoomics - MS 22, Rice Uiversity, 61 Mai Street, Housto, TX , Tel: , Fax: , rsickles@rice.edu.

2 1. Itroductio Covetioal uivariate tests for the presece of uit roots i aggregate ecoomic time series have importat implicatios for the coduct of domestic macro ad iteratioal ecoomic policy. These tests ufortuately have bee plagued by reliace o relatively short time series with relatively low frequecies. Size distortios ad low power are wellkow problems with covetioal testig procedures (see, e.g., Stock 1991, ad Campbell ad Perro 1991, Domowitz ad El-Gamal, 21). Curret macroecoomic theory provides little i the way of guidace o how to icrease the power ad moderate size distortios other tha by icreasig the legth of the time series. Reliace o the sort of iformatio that was utilized i covetioal empirical macroecoomics before the Lucas critique took hold, amely the iformatio cotaied i the correlated errors of other overideti ed equatios i the structural system, has little apparet place i the curret uit root testig literature. Eve agreemet o the cadidate set of correlated series has little theoretical basis (Stock ad Watso, 1999). The rst wide-spread use of uivariate tests for the presece of uit roots was carried i the semial work of Nelso ad Plosser (1982) who foud that most U.S. macroecoomic time series could be characterized as a uivariate time series structure with a uit root. Subsequet empirical aalyses have largely co rmed their digs while the literature cotiues to ackowledge the low power of uit root tests ad a implicatio of this low power, that i ite samples it is almost impossible to discrimiate betwee a uit root process ad oe which is very close to it. Clearly the uit root hypothesis has importat implicatios for determiig the e ect of radom shocks o a ecoomic system ad the literature has ot bee silet o the may e orts to overcome the low power of covetioal uit root tests. Oe such cotributio was made by Hase (1995) who oted that covetioal uivariate uit root tests igore potetially useful iformatio from related time series ad that the iclusio of related statioary covariates i the regressio equatio may lead to a more precise estimate of the autoregressive coe ciet. He proposed to use the covariates augmeted Dickey-Fuller (CADF) uit root test rather tha covetioal uivariate uit root tests. He aalyzed the asymptotic local power fuctios for the CADF t-statistic ad discovered that eormous power gais could be achieved by the iclusio of appropriate covariates. His Mote Carlo study suggested that such gais were also possible i the ite sample power performaces of the CADF vis-a-vis covetioal ADF test. Hase showed that the limit distributio of the CADF test is depedet o the uisace parameter that characterizes the correlatio betwee the equatio error ad the covariates. Therefore, it is ot possible to perform valid statistical iferece directly usig the CADF test. To deal with this iferetial di culty, Hase (1995) suggested usig critical values based o a estimated uisace parameter. 2 His two-step procedure ca be a practical solutio for the implemetatio of the CADF test. Strictly speakig, however, the validity of the resultig test is questioable sice the variability i the uisace parameter estimate is igored. 2 Hase (1995) s Table 1 provides asymptotic critical values for the CADFt-statistic for values of the uisace parameter i steps of.1 via simulatios. For itermediate values of the uisace parameter, critical values are selected by iterpolatio. 2

3 I this paper, we applythe bootstrap method to the CADF test to deal with the uisace parameter depedecy ad to provide a valid basis for iferece based o the CADF test. I particular, we show the cosistecy of the bootstrap CADF test ad establish the asymptotic validity of the critical values from the bootstrap distributio of the test. The ite sample performaces of the bootstrap CADF test are ivestigated ad compared with those of the CADF test ad the usual ADF test through simulatios. The simulatios show that the CADF test based o the two-step procedure su ers serious size distortios, especially whe the covariates are highly correlated with the error, while our bootstrap CADF test sigi catly improves the ite sample size performaces of the CADF test. Moreover, the bootstrap CADF test o ers dramatic power gais over the covetioal ADF test. As a illustratio, we apply our covariate tests ad stadard uit root tests i a reexamiatio of the statioarity of U.S. domestic macroecoomic aggregates ad iteratioal rates of exchage. The former are aalyzed with the exteded Nelso-Plosser data set for the post-1929 period. We ivestigate whether the digs of uit roots i the Nelso-Plosser data set are reversed whe the more powerful covariate tests are used. The latter are examied usig postwar aual CPI-based real exchage rates for 14 OECD coutries, for which most previous studies failed to reject the ull hypothesis of a uit root. We d that our ew covariate test rejects the uit root hypothesis i all the series i the Nelso-Plosser data set for the period ad i most cases for the postwar real exchage rates. The paper is orgaized as follows. Sectio 2 itroduces the uit root test with covariates ad derives limit theories for the sample tests. Sectio 3 applies the bootstrap methodology to the sample tests cosidered i Sectio 2 ad establishes the asymptotic validity of the bootstrap test. Sectio 3 also provides a discussio of practical issues arisig from the implemetatio of the bootstrap methodology. I Sectio 4, we coduct simulatios to ivestigate the ite sample performaces of the bootstrap CADF test. Empirical applicatios are preseted i Sectio 5 while Sectio 6 cocludes. All mathematical proofs are provided i the Appedix. 2. Uit Root Tests with Covariates 2.1 Model ad Assumptios We cosider the time series (y t ) give by 4y t = y t 1 + u t (1) for t = 1;:::;, where 4 is the usual di erece operator. We let the regressio errors (u t ) i the model (1) to be serially correlated, ad also allow them to be related to other statioary covariates. To de e the data geeratig process for the errors (u t ) more explicitly, let (w t ) be a m-dimesioal statioary covariates. It is assumed that the errors (u t ) are give by a p-th order autoregressive (AR) process speci ed as (L)u t = (L) w t + " t (2) where L is the lag operator, (z) = 1 P p k=1 kz k ad (z) = P q k= r kz k. 3

4 We cosider the test of the uit root ull hypothesis = for (y t ) give as i (1), agaist the alterative of the statioarity <. The iitial value of y of (y t ) does ot a ect our subsequet aalysis so log as it is stochastically bouded, ad therefore we set it at zero for expositioal brevity. Uder the ull hypothesis of uit root, 4y t = u t ad we have from (2) that 4y t = y t 1 + px k 4y t k + k=1 qx k= r kw t k + " t (3) which is a autoregressio of 4y t augmeted by its lagged level y t 1 ad the leads ad lags of the m statioary covariates i w t. Ideed, the above regressio may be viewed as a further augmetatio of the usual ADF regressio, which is a autoregressio of 4y t augmeted by its lagged level y t 1 oly. Our test statistics for testig the uit root i (y t ), which are itroduced i the ext sectio, will be based o the least squares estimator for from this CADF regressio. For the subsequet aalysis, we also eed to be more explicit about the data geeratig process for the statioary variables (w t ) that are used as covariates. We assume that (w t ) is geerated by a AR(`) process as (L)w t+r+1 = t (4) where (z) = I m P`k=1 k z k. To de e explicitly the correlatio betwee the covariates (w t ) ad the series to be tested (y t ), we cosider together the iovatios ( t) ad (" t ) that geerate, respectively, the covariates (w t ) ad the regressio error (u t ), which i tur geerates (y t ). De e» t = (" t ; t) ad deote by j j the Euclidea orm: for a vector x = (x i ), jxj 2 = P i x 2 i ad for a matrix A = (a ij ), jaj = P i;j a 2 ij. We ow lay out assumptios eeded for the developmet of our asymptotic theory. Assumptio 2.1 We assume (a) Let (» t ) be a sequece of iid radom variables such that E» t =, E» t» t = > ad Ej» t j s < 1 for some s 4. (b) (z); det( (z)) 6= for all jzj 1. Here, we assume (» t ) to be a iid sequece, which is stroger tha eeded, to make the bootstrap procedure i the ext sectio meaigful. Assumptio 2.1 (a) states that the regressio error (" t ) i equatio (3) is serially ucorrelated ad idepedet of t+k for k 1. The coditio e ectively implies that the regressio error " t is orthogoal to the lagged di ereces of the depedet variable (4y t 1 ;:::;4y t p ) ad the leads ad lags of the statioary covariates (w t+r ;:::;w t q ), which is ecessary for the regressio (3) to be cosistetly estimated via usual least squares estimatio. Such orthogoalities ca be achieved by appropriately icreasig the orders p, q ad r i the lag polyomials (L) ad (L) as the sample size teds to i ity, as show i Saikkoe (1991). To discuss 4

5 it more explicitly, let v t = P q k= r kw t k + " t. The error v t becomes orthogoal to the lagged di ereces (4y t 1 ;:::;4y t p ) if we allow the order p i the lag polyomial (L) i (2) to icrease at a cotrolled rate as the sample size icreases. The orthogoality betwee the error " t i v t ad the leads ad lags of the statioary covariates (w t+r ;:::;w t q ) ca also be achieved if the orders q ad r i the lag polyomial (L) are su cietly large eough to white the error. Moreover, we may exted the aalysis to allow (L) to be a i ite order lag polyomial ad approximate it by a ite order AR with the order icreasig with the sample size, as doe by Berk (1974), Said ad Dickey (1984) ad more recetly by Chag ad Park (21). Our subsequet theory will also hold uder these schemes. As q ad r icrease, the error " t will become idepedet of the iovatio t at all leads ad lags, ad as p icreases " t becomes orthogoal to the lagged di ereces of the depedet variable. Hece, it is atural to thik of the error " t as a residual, i.e., k s ad k s are the coe ciets such that " t is orthogoal to all of the icluded regressors (4y t 1 ;:::;4y t p ;w t+r ;:::;w t q ). Uder Assumptio 2.1 (a), the followig ivariace priciple holds [s] 1 X p» t! d B(s) (5) for s 2 [; 1] as! 1. The limit process B = (B " ;B ) is a (1 + m)-dimesioal vector Browia motio with covariace matrix µ ¾ 2 = " ¾ " : (6) ¾ " The asymptotic behavior of (y t ) is determied by that of (u t ) as show i model (1), ad the latter is depedet upo the limitig behaviors of the statioary covariates (w t ) ad the iovatios (" t ) as idicated i the relatio (2). We may the derive the limit behavior of (u t ) usig the speci catio give i (2) from those of (" t ) ad (w t ) as follows: [s] 1 X p u t! d ¼(1) (1) ª(1)B (s) + B " (s) as! 1, where ¼(1) = 1= (1) ad ª(1) = (1) 1. This is derived i Lemma A.1 (b) i Appedix. The variace of the limit process give i the previous equatio is easily derived as ³ (1) ¾ 2 u = ¼(1) 2 ª(1) ª(1) (1) + ¾ 2 " + 2 (1) ª(1)¾ " (7) usig the parameters de ed i the precedig equatios. Let z t = (4y t 1 ;:::;4y t p ;w t+r;:::;w t q). We assume Assumptio 2.2 ¾ 2 u > ad Ez t z t >. The coditio ¾ 2 u > esures that the series (y t) is I(1) whe =, which is ecessary to be able to iterpret testig = as testig for a uit root i (y t ). The coditio Ez t z t > implies that the statioary regressors i z t are asymptotically liearly idepedet, which is 5

6 required alog with the coditio Assumptio 2.1 (a) for the cosistecy of the LS coe ciet estimates for (z t ) Covariates Augmeted Uit Root Tests To itroduce our test statistics more e ectively, we rst de e Now we have A = B = C = Ã X! Ã X! y t 1 " t y t 1 zt X 1 Ã z t zt X Ã X! Ã yt 1 2 X! y t 1 zt X 1 Ã z t zt X Ã X! Ã " 2 X! X 1 Ã! X t z t " t : " t zt z t zt ^ = A B 1 ³ ^¾ 2 = 1 C A 2 B 1 s(^ ) 2 = ^¾ 2 B 1! z t " t! z t y t 1 where ^ is the OLS estimator of from the covariates augmeted regressio (3), ^¾ 2 is the usual error variace estimator, ad s(^ ) is the estimated stadard error for ^. We also let px ^ (1) = 1 ^ k (8) where ^ k s are the OLS estimators of k s i the CADF regressio (3). The statistics that we will cosider i the paper are give by k=1 S = T = ^ ^ (1) (9) ^ s(^ ) : (1) Note that S is a test for the uit root based o the estimated uit root regressio coe ciet, ad T is the usual t-statistics for testig the uit root hypothesis from the CADF regressio (3). The test T is cosidered i Hase (1995). The limit theories for the tests S ad T are give i 3 As discussed below Assumptio 2.1, the etire statioary regressorsz t i the regressio (3) are orthogoal to the regressio error" t, i.e., Ez t" t = uder Assumptio 2.1 (a). 6

7 Theorem 2.3 Uder Assumptios 2.1 ad 2.2, we have as! 1, where ad P(s) = B " (s)=¾ ". S! d ¾ " T! d µ Q(s)dP(s) Q(s) 2 ds Q(s)dP(s) 1=2 Q(s) 2 ds Q(s) = (1) ª(1)B (s) + B " (s) The asymptotic distributios are preseted explicitly i terms of the Browia motios B " ad B via Q = (1) ª(1)B + B " ad P = B " =¾ ". I this way we ca easily relate the asymptotic distributios of the bootstrapped tests, which are developed i the ext sectio, to the limit distributios of the sample statistics give above. The ull asymptotic distributio for the CADF test T give i Theorem 2.3 is actually equivalet to the oe derived i Hase (1995, Theorem 3). To deal with the uisace parameter depedecy, the limit distributios there are, however, derived from the limit Browia motios (B " ;B v ) for the partial sum processes of (" t ;v t ) where v t = (L) w t +" t. Notice that B v = Q, where Q is de ed i Theorem 2.3 above. The limit distributios are the preseted i terms of the stadard Browia motios de ed from the Browia motios B " ad B v. To see this, let U ad W be idepedet stadard Browia motios. The we may express the limit Browia motio (B " ;B v ) = (B " ;Q) as µ B" Q = µ ¾" p 1 ½ 2 U + ¾ "v W=¾ v ¾ v W with ¾ 2 v = ¾ 2 u=¼(1) 2, ¾ v" = (1) ª(1)¾ " + ¾ 2 ", ad ½ 2 = ¾ 2 v"=(¾ 2 "¾ 2 v), where the uisace parameters are obtaied from (6) ad (7). We may further simplify B " as B " = ¾ " ( p 1 ½ 2 U + ½W): Let P = B " =¾ " = ( p 1 ½ 2 U + ½W) ad Q = ¾ v W. The, the ull asymptotics for T derived i Theorem 2.3 becomes WdW T! d ½ µ 1=2 + W 2 q 1 ½ 2 WdU µ 1=2 : W 2 Notice that the limit Browia motio U is idepedet of W, ad cosequetly the distributio of the stochastic itegral R 1 WdU is mixed ormal with the mixig variate R 1 W2. It 7 (11)

8 is ow easy to see that the limit distributio obtaied i the previous equatio is equivalet to the limit distributio derived i Hase (1995). The asymptotic distributios for both S ad T are ostadard ad deped upo the uisace parameters that characterize the correlatio betwee the covariates ad the regressio error. The limit distributios are therefore basically ukow. Cosequetly it is impossible to perform valid statistical iferece based directly o the CADF tests. As a feasible practical solutio, oe may simulate critical values for the tests for each value of the uisace parameter ad use its estimated value to obtai the most appropriate critical value available from the tabulated values. 4 This two-step procedure ca be a feasible practical solutio for the implemetatio of the CADF tests; however, the resultig tests will ot be valid i strict sese sice the variability i the uisace parameter estimate is ot properly take ito accout. The models with determiistic compoets ca be aalyzed similarly. Whe the time series (z t ) with a ozero mea is give by or with a liear time tred z t = ¹ + y t (12) z t = ¹ + ±t + y t (13) where (y t ) is geerated as i (1), we may test for the presece of the uit root i the process (y t ) from the CADF regressio (3) de ed with the tted values (y ¹ t ) or (y t ) obtaied from the prelimiary regressio (12) or (13). The limit theories for the CADF tests give i Theorem 2.3 exted easily to the models with ozero mea ad determiistic treds, ad are give similarly with the followig demeaed ad detreded Browia motios ad Q ¹ (s) = Q(s) Q (s) = Q(s) + (6s 4) Q(t)dt Q(t)dt (12s 6) tq(t)dt i the place of the Browia motio Q(s). I the ext sectio, we cosider bootstrappig the covariates augmeted tests S ad T to deal with the uisace parameter depedecy problem ad to provide a valid basis for iferece based o the covariates augmeted uit root tests. 3. Bootstrap Uit Root Tests with Covariates I this sectio, we cosider the bootstrap for the covariates augmeted uit root tests S ad T itroduced i the previous sectio. We establish the bootstrap cosistecy of the 4 Notig that the ull limit distributio of the CADFt-test depeds oly o the correlatio coe ciet ½ 2, Hase (1995, Table 1, p.1155) provides the asymptotic critical values for the CADFt-test for values of ½ 2 from.1 to 1 i steps of.1. The estimate for½ 2 is costructed as ^½ 2 = ^¾ 2 v"=^¾ 2 v^¾ 2 ", where ^¾ v",^¾ 2 v ad ^¾ 2 " are cosistet oparametric estimators of the correspodig parameters. 8

9 tests ad show the asymptotic validity of the tests. Throughout the paper, we use the usual otatio to sigify the bootstrap samples, ad use P ad E respectively to deote the probability ad expectatio coditioal o a realizatio of the origial sample. Various issues arisig i practical implemetatio of the bootstrap methodology are also addressed. To costruct the bootstrap CADF tests, we rst geerate the bootstrap samples for the m-dimesioal statioary covariates (w t ) ad the series (y t ) to be tested. We begi by costructig the tted residuals which will be used as the basis for geeratig the bootstrap samples. We rst let u t = 4y t ad t the regressio u t = px ~ k u t k + k=1 qx k= r ~ kw t k + ~" t (14) by the usual OLS regressio. It is importat to base the bootstrap samplig o regressio (14) with the uit root restrictio = imposed. The samples geerated by regressio (3) without the uit root restrictio do ot behave like uit root processes, ad this will reder the subsequet bootstrap procedures icosistet as show i Basawa et al. (1991). Next, we t the `-th order autoregressio of w t as w t+r+1 = ~ 1 w t+r + + ~ `w t+r `+1 + ~ t (15) by the usual OLS regressio. We may prefer, especially i small samples, to use the Yule- Walker method to estimate (15) sice it always yields a ivertible autoregressio, thereby esurig the statioarity of the process w t [see, e.g., Brockwell ad Davis (1991, Sectios 8.1 ad 8.2)]. As the sample size icreases, however, the problem of oivertibility i the OLS estimatio vaishes a.s., ad the two methods become equivalet. Our subsequet results are applicable also for the Yule-Walker method, sice it is asymptotically equivalet to the OLS method. We the geerate the (1+m)-dimesioal bootstrap samples» t = (" t; t ) by resamplig from the cetered tted residual vectors ( ~» t ) = (~" t ; ~ t) where (~" t ) ad (~ t) are the tted residuals from (14) ad (15). That is, obtai iid samples (» t ) from the empirical distributio of à ~» t 1! X ~»t : The bootstrap samples (» t ) costructed as such will satisfy E» t = ad E» t» t = ~, where ~ = (1=) P ~»t ~» t. 5 Next, we geerate the bootstrap samples for (w t ) recursively from ( t ) usig the tted autoregressio give by w t+r+1 = ~ 1 w t+r + + ~ `w t+r+1 ` + t (16) 5 Alteratively, we may resample" t ad t separately from the ~"t ad ~ t fort = 1;:::;. I this case, however, preservig the origial correlatio structure eeds more care. We basically eed to pre-white ~" t ad ~ t before resamplig, ad the re-color the resamples to recover the correlatio structure. More speci cally, we rst pre-white ~" t ad ~ t by pre-multiplyig ~ 1=2 to ~» t = (~" t;~ t ), fort = 1;:::;. Next, geerate» t = (" t; t ) by resamplig from the pre-whiteed ~" t ad ~ t ad subsequetly re-colorig them by pre-multiplyig ~ 1=2 to restore the origial depedece structure. 9

10 with appropriately chose `-iitial values of (w t ), where ~ k ; 1 k ` are the coe ciet estimates from the tted regressio (15). Iitializatio of (w t ) is uimportat for our subsequet theoretical developmet, though it may play a importat role i ite samples. 6 The we obtai (w t+r;:::;w t q) from the sequece (w t ), ad costruct the bootstrap samples (v t ) as v t = qx k= r ~ kw t k + " t (17) usig the LS estimates ~ k; r k q from the tted regressio (14). The we geerate (u t ) recursively from (v t ) usig the tted autoregressio give by u t = ~ 1 u t ~ p u t p + v t (18) with appropriately chose p-iitial values of (u t ), ad where ~ k ; 1 k p are the estimates for k s from the tted regressio (14). Fially, we geerate (y t ) from (u t ) with the ull restrictio = imposed. This is to esure the ostatioarity of the geerated bootstrap samples (y t ), which is claimed uder the ull hypothesis, ad to make the subsequet bootstrap tests valid. Thus we obtai (y t ) as tx y t = y t 1 + u t = y + u k (19) which also requires iitializatio y. A obvious choice would be to use the iitial value y of (y t ), ad geerate the bootstrap samples (y t ) coditioal o y. As discussed earlier, the choice of iitial value may a ect the ite sample performace of the bootstrap; however, the e ect of the iitial value becomes egligible asymptotically as log as it is stochastically bouded. If the mea or liear time tred is maitaied as i (12) or (13) ad the uit root test is performed usig the demeaed or detreded data, the e ect of the iitial value y of the bootstrap sample would disappear. We may therefore just set y = for the subsequet developmet of our theory i this sectio. To costruct the bootstrapped tests, we cosider the followig bootstrap versio of the covariates augmeted regressio (3), which was used to costruct the sample CADF tests S ad T i the previous sectio 4y t = y t 1 + px k 4y t k + k=1 k=1 qx k= r kw t k + " t: (2) We test for the uit root hypothesis = i (2) usig the bootstrap versios of the CADF tests, de ed i (23) ad (24) below, that are costructed aalogously as their sample couterparts S ad T de ed i (9) ad (1). 6 We may use the rst`-values of (w t) as the iitial values of (w t). The bootstrap samples (w t) geerated as such may ot be statioary processes. Alteratively, we may geerate a larger umber, say +M, of (w t) ad discard rstm-values of (w t). This will esure that (w t) become more statioary. I this case the iitializatio becomes uimportat, ad we may therefore simply choose zeros for the iitial values. 1

11 Similarly as before, we deote by ^ ad s(^ ) respectively the OLS estimator for ad the stadard error for ^ obtaied from the CADF regressio (2) based o the bootstrap samples. To de e them more explicitly, we let ad subsequetly de e A = B = z t = (4y t 1;:::;4y t p;w t+r;:::;w t q) Ã X y t 1" X t X y 2 t 1 Ã X y t 1z t y t 1z t! Ã X! Ã X z t z t z t z t! 1 Ã X! 1 Ã X ad the variace of the bootstrap sample (" t ), which is give by ~¾ 2 = 1! z t " t! z t y t 1 X (~" t ¹" ) 2 ; (21) where ¹" = 1 P ~" t. The we may write the OLS estimator of from the bootstrap CADF regressio (2) ad its estimated variace as ^ = A B 1 s(^ ) 2 = ~¾ 2 B 1 : We also de e, accordigly as ^ (1) itroduced i (8), ~ (1) = 1 where ~ k s are the estimates for k s from the tted regressio (14). Now we cosider the statistics px ~ k ; (22) k=1 S = T = ^ ~ (1) ^ s(^ ) (23) (24) correspodig to S ad T itroduced i (9) ad (1) of the previous sectio. For the costructio of the bootstrap statistics S ad T, it is possible to replace ~ (1) ad ~¾ 2 with ^ (1) ad ^¾ 2, the bootstrap couterparts to ^ (1) ad ^¾ 2. We ca compute ^ (1) ad ^¾ 2 from regressio (2) i the same way that their sample couterparts are computed from regressio (3). We may ideed show that such replacemets do ot a ect the limitig distributios of the statistics. For the theoretical aalysis i the paper, however, we oly cosider S ad T de ed i (23) ad (24) for the expositioal brevity. 11

12 To implemet the bootstrap CADF tests, we repeat the bootstrap samplig for the give origial sample ad obtai a ( ) ad b ( ) such that P fs a ( )g = P ft b ( )g = (25) for ay prescribed size level. The bootstrap CADF tests reject the ull hypothesis of a uit root if S a ( ); T b ( ): It will ow be show uder appropriate coditios that the tests are asymptotically valid, i.e., they have asymptotic size. We do ot aalyze i the paper the radomess associated with the bootstrap samplig i computig the bootstrap critical values a ( ) ad b ( ). We simply assume that eough umber of bootstrap iteratios are carried out to make it egligible. See Adrews ad Buchisky (1999) for a study o the umber of bootstrap iteratios to achieve the desired level of bootstrap samplig accuracy. We ow itroduce the otatio! d for bootstrap asymptotics. For a sequece of bootstrapped statistics (Z ), we write Z! d Z a.s. if the coditioal distributio of (Z ) weakly coverges to that of Z a.s. Here it is assumed that the limitig radom variable Z has distributio idepedet of the origial sample realizatio. We ow preset the limit theories for the bootstrap CADF tests S ad T. Theorem 3.3 Uder the ull hypothesis =, we have as! 1, S! d ¾ " T! d µ Q(s)dP(s) Q(s) 2 ds a.s. Q(s)dP(s) 1=2 a.s. Q(s) 2 ds uder Assumptios 2.1 ad 2.2, where Q(s) ad P(s) are de ed i Theorem 2.3. Theorem 3.3 shows that the bootstrap statistics S ad T have the same ull limitig distributios as the correspodig sample statistics S ad T. It implies, i particular, that the bootstrap CADF tests are asymptotically valid. To discuss the asymptotic validity of the tests usig bootstrap critical values, deote by S ad T the weak limits of S ad T respectively, ad de e a( ) ad b( ) to be the asymptotic critical values of the size tests based o S ad T, i.e., PfS a( )g = PfT b( )g = : 12

13 Sice the distributios of S ad T are absolutely cotiuous with respect to Lebesgue measure, we have from Theorem 3.3 P fs a( )g; P ft b( )g! a.s. (26) uder Assumptios 2.1 ad 2.2, ad the results i (26) imply (a ( );b ( ))! (a( );b( )) a.s.; where a ( ) ad b ( ) are the size bootstrap critical values de ed i (25). Cosequetly, we have uder Assumptios 2.1 ad 2.2 PfS a ( )g; PfT b ( )g! as! 1, which proves that the bootstrap CADF tests have size asymptotically. Our bootstrap theory here easily exteds to the tests for a uit root i models with determiistictreds, such as those itroduced i (12) or (13). It is straightforward to establish the bootstrap cosistecy for the CADF tests applied to the demeaed ad detreded time series, usig the results obtaied i this sectio. The bootstrap CADF tests are therefore valid ad applicable also for the models with determiistic treds. 4. Simulatios We perform a set of simulatios to ivestigate the ite sample performaces of the bootstrap CADF t-test. For the simulatios, we cosider (y t ) give by the uit root model (1) with the error (u t ) geerated by where the error term (v t ) is give by u t = 1 u t 1 + v t ; v t = w t + " t : I our simulatio study, the covariate (w t ) is assumed to follow a AR(1) process w t+1 = Áw t + t: The iovatios» t = (" t ; t) are i.i.d. N(; ); where Ã! 1 = : ¾" ¾ " 1 Uder this setup, we have the followig covariate augmeted ADF regressio: y t = y t y t 1 + w t + " t : (27) The correlatio betwee the iovatio v t = w t + " t ad the covariate w t depeds o two parameter values, the coe ciet o the covariate ad the ARcoe ciet Á of the covariate, 13

14 as ca be see clearly from the data geeratig process (DGP). Cosequetly, the relative merit of costructig a uit root test from the covariate augmeted regressio depeds o the parameters ad Á: We cotrol the degree of correlatio betwee the error v t ad the covariate w t through these parameters: The values of the parameters ad Á are allowed to vary amog f :8; :5; :5; :8g. Parameter 1 o the lagged di erece term does ot a ect the aforemetioed correlatio ad, hece, we set 1 = throughout the simulatios for coveiece. The cotemporaeous covariace, ¾ " ; is set at :5. For the test of the uit root hypothesis, we set = ad ivestigate ite sample sizes i relatio to correspodig omial test sizes. For ite sample powers, we cosider the cases, = :5; ad :1: The ite sample performaces of the bootstrap CADF test are compared with those of the sample CADF test computed from the regressio (27) ad also with those of the ADF test based o the usual ADF regressio. The usual ADF regressio does ot iclude the covariate w t as a regressor ad thus the regressio error e ectively becomes v t = w t + " t with the DGP cosidered i this simulatio setup. The e ective error v t is obviously serially correlated due to serial correlatio i w t. This ivalidates the use of the covetioal ADF test which uses the critical values from the Dickey-Fuller distributio. To make more meaigful comparisos with the results of the ADF test, we may icrease the umber of lagged di ereces to white the error v t. I this simulatio experimet we set the lag order for the ADF test at 3. 7 The regressio equatio for the CADF test cotais o AR lag terms ad icludes oly the curret value of the covariate i each regressio. All regressios iclude a tted itercept. The regressio equatio for covariate (15) is estimated usig AR(1) model. To implemet the CADF test, asymptotic criticalvalues, correspodig to each sample estimate of ½ 2 ; are take from Table 1 i Hase (1995). 8 Sample sizes of = 5; 1; ad 25 are examied for 1%; 5%; ad 1% omial size tests; we report the results for = 1 oly, sice those for = 5; ad 25 are qualitatively similar. Each replicatio discards the rst 1 observatios to elimiate start-up e ects. The reported results are based o 5, simulatio iteratios with the bootstrap critical values computed from 3, bootstrap repetitios. The ite sample sizes ad powers for the ADF, the CADF, ad the bootstrap CADF tests are reported, respectively, i Tables 1 ad 2. As ca be see clearly from Table 1, the bootstrap CADF test sigi catly improves the ite sample size performaces of the sample CADF test, especially whe the covariates are highly correlated with the error. More precisely, whe the parameter is large i absolute value ad the parameter Á is large with positive value, the size distortios of the CADF test are quite oticeable. The dowward size distortios of the CADF test for ( ;Á) = ( :8; :8) ad ( :5; :8) are very serious while the overrejectios for ( ;Á) = (:8; :8) ad (:5; :8) are ot egligible. This tedecy becomes more severe as the cotemporaeous covariace, ¾ " ; gets large i absolute value. For example, whe ¾ " = :8; actual 1% size of the CADF test for ( ;Á) = (:8; :8) goes up to 16% ad i the case of ( ;Á) = ( :8; :8) it 7 The ADF test with lag order smaller tha 3 teds to overreject the ull hypothesis. The lag order 3 seems to have the best overall size ad power performaces ad thus is chose for our simulatios. 8 Sample estimates of½ 2 are calculated usig the Parze kerel ad Adrews (1991) automatic badwidth estimator. 14

15 goes dow to as low as 1%. 9 I the case of the bootstrap CADF test, however, we do ot observe such size distortios. The bootstrap CADF test performs geerally very well for di eret choices of parameters. The size distortios of the ADF test are little or mild. The sigi cat improvemet i the ite sample sizes that the bootstrap CADF test o ers does ot come at the expese of ite sample powers. Ideed, the results i Table 2A ad 2B show that the bootstrap CADF test also o ers drastic power gais over the covetioal ADF test for all the 1%, 5%, ad 1% tests. I particular, the discrimiatory powers of the bootstrap CADF test are oticeably much higher tha those of the ADF test for the cases whe the parameter has positive values. For = :8; the powers of the bootstrap CADF test for = :1 are more tha three times as large as those of the ADF test. The discrepacies become four times as large whe = :5: The bootstrap CADF test performs much better tha the ADF test for low values of ^½ 2 whe is egative. Note that the powers of the bootstrap CADF test are comparable to those of the CADF test i most cases, ad eve better i some cases. 5. Empirical Applicatios We ext apply our testig procedures to a set of U.S. macroecoomic aggregates ad OECD acial time series. A umber of ecoometric studies have foud that stadard tests for a uit root, such as the ADF (Dickey ad Fuller, 1979, 1981) ad the Phillips ad Perro (1988) tests, have low power agaist statioary alteratives i the relatively small samples we cosider i this sectio (See, Dejog, Nakervis, Savi, ad Whitema 1992a, 1992b, amog others). This is especially true whe a series uder ivestigatio is a ear-itegrated process. Sice the low power of the uivariate uit root tests is the primary problem, it is importat to ivestigate whether or ot the ull hypothesis of a uit root is rejected by the more powerful covariate tests. Moreover, various uivariate uit root tests provide mixed results for a give time series whe they do ot cosistetly reject or accept the ull hypothesis. I this situatio makig a de itive coclusio about the (o)statioarity of the time series may be problematic. Hece, the use of more powerful tests may poit to sharper coclusios about the statioarity property of the particular time series. We cosider two data sets. The rst is the Nelso-Plosser data set exteded by Schotma ad Va Dijk (1991) to ed i The secod is aual CPI-based real exchage rates 1 for 14 OECD coutries from 1951 to The real exchage rate, r it ; for the i-th coutry is computed usig the U.S. dollar as umeraire currecy. Real exchage rates are aalyzed for Australia, Austria, Belgium, Caada, Frace, Filad, Germay, Italy, Japa, Luxemburg, Netherlads, Norway, Spai, ad the Uited Kigdom. The testig strategy is as follows. Note rst that the order of autoregressio has importat e ects o the size ad power performaces of the tests (see Ng ad Perro 1995, ad Schwert 1989, amog others). Secodly, ote also that a determiistic rule relatig lag legth p to sample size is iferior to a data-depedet rule that takes sample iformatio 9 The result for¾ " = :8 is ot reported i the paper. It is available upo request from the authors. 1 The real exchage rate is calculated asr it = log(e itp t=p it); wheree it;p t; adp it deote respectively omial spot exchage rate for thei-th coutry, the CPI for the U.S., ad the CPI for thei-th coutry. 15

16 ito accout (Ng ad Perro 1995). Based o these digs, we rst use the ADF test with di eret lag order selectio criteria such as AIC, BIC, ad STC (the sequetial test criterio 11 ), ad the Phillips ad Perro (PP) tests, Z t ad Z ; for di eret choices of trucatio lag parameter: If all of these tests cosistetly reject or accept the ull hypothesis of a uit root for a particular time series, we will use that variable as a possible cadidate for a covariate. If the variable to be used as a covariate is statioary, the we will use the level of it, ad if the variable is ostatioary, the we will take a rst di erece of it. Thus, oly statioary covariates will be utilized i our multivariate tests. To the time series whose test results are udetermied, we apply the CADF ad the bootstrap CADF tests with covariates selected by the above pre-tests which are ow kow to be either I() or I(1). Amog the cadidates for covariates we choose the oe which gives us the smallest ^½ 2 sice this covariate provides the most powerful test, as show i Sectio 4. Uless otherwise stated, all regressios iclude a costat ad a time tred. For the covariate tests, we do ot use future covariate values, i.e., we let r = i all the regressios. The AIC lag order selectio rule is used as a lag selectio criterio for the CADF test ad for the AR estimatio of covariates. For the CADF test, asymptotic critical values, correspodig to each sample estimate of ½ 2 ; are take from Table 1 i Hase (1995). For the bootstrap test we use critical values computed from 5, bootstrap iteratios. 5.1 The Exteded Nelso-Plosser Data Series The Nelso ad Plosser data set is oe of the most widely aalyzed macroecoomic aggregate time series data sets. Nelso ad Plosser (1982) studied the time series properties of 14 series ad foud that all of them, with the exceptio of the uemploymet rate series, were characterized by stochastic ostatioarity. I our empirical applicatio we use the exteded Nelso-Plosser data set of Schotma ad Va Dijk (1991). All variables are measured i logarithms. The estimated period is i cosideratio of a structural break i 1929 coicidig with the oset of the Great Depressio. We perform the ADF ad the PP tests for the 14 time series whose o-statioarity have bee questioed. Table 3 presets the results. With the ADF test we reject the ull hypothesis for 12 series at least at the 1% sigi cace level. With the PP test, the uit root hypothesis ca be rejected for 9 series at least at the 1% sigi cace level. Both tests fail to reject the uit root hypothesis for real wages. The ull hypothesis is rejected usig both tests for 4 series: real GNP, real per capita GNP, employmet, ad uemploymet rate 12. The stadard tests clearly do ot give de itive coclusios for 9 out of 14 time series (omial GNP, idustrial productio, GNP de ator, cosumer prices, wages, moey stock, velocity, iterest rate, ad commo stock prices). Pael (c) of Table 3 cotais 11 The value of lag legthpchose is determied by a test o the sigi cace of the estimated coe ciets ^ i. We actually used a fairly liberal procedure choosig a value ofpequal top if thetstatistic o ^ i was greater tha 1.6 i absolute value ad thetstatistic o ^ i forl>p was less tha 1.6 (with a maximum value forpof 8 for the Nelso-Plosser series, ad 6 for real exchage rates). This liberal procedure is justi ed i the sese that icludig too may extra regressors of lagged rst-di ereces does ot a ect the size of the test but oly decreases its power. Icludig too few lags may have a substatial e ect o the size of the test (Perro 1989, ad Ng ad Perro 1995). 12 I the case of the uemploymet rate series, the result is obtaied without a time tred i the regressio. 16

17 the test results of the remaiig 5 series for which the ull hypothesis ca be cosistetly accepted or rejected. 13 We will use these 5 series as covariates for the CADF test. Note that I(1) variables will be rst-di ereced before they are used as covariates. 14 To the 9 time series whose results are udetermied, we apply the covariate tests usig as covariates the series cosistetly foud to be either I() or I(1) by the above pre-tests. The results are give i Pael (a) of Table 3. For all cases the values of ^½ 2 are lower tha.1; thus we should expect, based o our simulatio results, more powerful test results with the CADF ad the bootstrap CADF tests tha with the ADF test. With these ew tests we ca reject the ull hypothesis of a uit root for 8 time series. The oly exceptio is for the series o idustrial productio. No-statioarity of the GNP de ator, wages, velocity, iterest rate, ad commo stock prices series are strogly rejected at the 1% sigi cace level. These results are fairly robust to the choice of maximum lag legth i the cases of omial GNP, idustrial productio, velocity, ad iterest rate, while for the other series the tests ted to reject the ull hypothesis for higher orders of lag legth. The results show that with the more powerful covariate tests there is much less ucertaity about the potetial statioarity of may commoly studied U.S. macroecoomic aggregates. We ca compare our digs with those of Perro (1989). For the post-1929 period, he could ot reject the ull hypothesis of a uit root for iterest rate. With the CADF ad the bootstrap CADF tests, however, we ca reject the uit root hypothesis i the iterest rate series at the 1% sigi cace level. We caot, however, reject the ull hypothesis for real wages ad idustrial productio. To check the possibility that this di erece may come from the e ect of the 1973 oil shock, we aalyze the two series for the period of Results are show i Pael (b) of Table 3. The ADF ad the PP tests give us mixed results, while the CADF ad the bootstrap CADF tests strogly reject the ull hypothesis at the 1% sigi cace level. Therefore, the CADF ad the bootstrap CADF tests reject the ostatioarity of all the series i the Nelso-Plosser data set for the post-1929 samples. Give that the uit root hypothesis ca be rejected for all the series, we ca assess the sigi cace of the other coe ciets sice the asymptotic distributios of their t-statistics are stadard ormal. I all cases except cosumer prices, iterest rate, ad commo stock prices, the time tred coe ciets are sigi cat at least at the 5% level. These results suggest that the uderlyig processes of these series are characterized maily by statioary uctuatios aroud a determiistic tred fuctio for the sample period. The cosumer prices, iterest rate ad commo stock prices series are aalyzed without a time tred ad the results idicate that these series are statioary aroud a costat. 13 We varied the maximum lag legth,q max; from 2 to 8 for the ADF test, ad similarly from 2 to 8 for the choice of trucatio lag parameter for the PP test. The results seem to be robust to the choice of maximum lag legths. 14 Stock ad Watso (1999) ote that curret theoretical literatures i macroecoomics provide either ituitio or guidace o which covariates are cadidates for our CADF ad bootstrap CADF tests other tha o the basis of statioarity. 17

18 5.2 Real Exchage Rates Aother situatio i which the covariate tests are especially useful is with certai types of pael data i which cross-sectioal correlatios betwee time series are preset. If this iformatio ca be properly modeled the it ca provide a e ciecy gai over uivariate methods as poited out by Hase (1995). To properly exploit these potetial e ciecy (ad power) gais, we aalyze aual CPI-based real exchage rates for 14 OECD coutries for the period of , the period icludig the Bretto Woods system ad exible exchage rate regime. We have a bit more ituitio here as to which covariates may be more importat sice covariace with close tradig parters has a geographical as well as a ecoomic ratioale. Movemets i real exchage rates are thought to be drive primarily by deviatios from purchasig power parity (PPP). Models of exchage rate determiatio are built o the assumptio that the PPP hypothesis holds. There are, however, co ictig empirical evideces. Recet studies by MacDoald (1996), Frakel ad Rose (1996), Oh (1996), Papell (1997), ad O Coell (1998), amog others, suggest that the issue is ot completely settled. I particular, whe cosiderig data for the recet exible rate experiece (1973- preset), may researchers have bee uable to reject the ull hypothesis of a uit root (see, e.g., Mark 199, ad Ediso ad Pauls 1993). Oe respose to this o-rejectio might be that the tests do ot ecompass a su cietly log time-spa to capture the mea reversio ecessary to reject the ull hypothesis. It is of iterest, therefore, if real exchage rates are mea revertig for loger spas of time tha the period of recet oat regime, usig low frequecy data. Oe advatage of usig relatively low frequecy aual or quarterly data is that it ca possibly icrease the power of statistical tests for radom walk behavior (Shiller ad Perro 1985). As pretests, the ADF ad the PP tests are performed for the 14 real exchage rate series. With the ADF test, we ca reject the uit root hypothesis for 9 coutries, while the ull hypothesis is rejected for 12 coutries with the PP test. We caot reject the ull hypothesis with both tests for the Netherlads ad Norway, ad both tests cosistetly reject the ull hypothesis of a uit root for 6 coutries. Table 4 shows the result. We therefore have 6 remaiig coutries whose statioarity properties are udetermied (Australia, Austria, Caada, Japa, Belgium, ad Germay). The results of the covariate tests for the 6 coutries are i Pael (a) of Table 4. For the chose covariates, the estimated ^½ 2 are quite low, ragig betwee.1 ad.42. This idicates that the powers of the CADF ad the bootstrap CADF tests should be cosiderably higher tha those of the ADF test (ad possibly higher tha those of the PP test). Results based o the CADF ad the bootstrap CADF tests ow idicate that we ca reject the ull hypothesis of a uit root for all the coutries, although the evidece for Belgium ad Germay is less compellig tha for Australia, Austria, Caada, Japa, ad the U.K. for which the ull hypothesis of a uit root is strogly rejected at the 1% sigi cace level (5% level i the case of Australia) For these two coutries, the CADF test caot reject the ull hypothesis. However, the critical values of the 1% test are 1:753 ad 1:952; respectively, ad their test statistics, t = 1:56 ad 1:811; respectively, are very close to the critical values. Thus the evidece for these two coutries is margial. 18

19 Our results based o the more powerful covariate tests are at variace with those based o stadard uivariate uit root tests. Frakel (1985) tested for a uit root i the real exchage rate of the Uited Kigdom usig the Dickey-Fuller test o aual data ad was uable to reject the radom walk hypothesis whe the sample period was limited to the postwar era ( ). Whitt (1992) tested for a uit root i the real exchage rates of the U.K., Japa, Australia, ad several other coutries usig the Dickey-Fuller test o aual data post World War II ad was uable to reject the ull hypothesis of a uit root for these coutries. Fly ad Boucher (1993), usig mothly data for the similar period, coducted tests for the PPP hypothesis ad cocluded that PPP does ot hold for Caada ad Japa. Now, with a expaded data set (through 1998) ad more powerful tests, we ca decisively reject the ull hypothesis of a uit root for Australia, Caada, Japa, ad the U.K.. These digs are supportive of the idea of Aizema (1984) o the PPP hypothesis that the doctrie of PPP should hold better betwee eighborig coutries, ad betwee coutries with larger potetial trade, because of the lower trasactio cost of trade i goods betwee such coutries Coclusio I this paper, we cosider the bootstrap procedure for the covariates augmeted Dickey- Fuller (CADF) uit root test which substatiallyimproves the power of uivariate uit root tests. Hase (1995) origially proposed the CADF test ad suggested a two-step procedure to overcome the uisace parameter depedecy problem. Here, we propose bootstrappig the CADF test i order to directly deal with the uisace parameter depedecy ad base ifereces o the bootstrapped critical values. We also establish the bootstrap cosistecy of the CADF test ad show that the bootstrap CADF test is asymptotically valid. The ite sample performaces of the bootstrap CADF test are ivestigated ad compared with those of the CADF test ad the usual ADF test through simulatios. The bootstrap CADF test sigi catly improves the ite sample size performaces of the CADF test, especially whe the covariates are highly correlated with the error. Ideed, the bootstrap CADF test o ers drastic power gais over the covetioal ADF test. As a illustratio, we apply the tests to the 14 macroecoomic time series i the exteded Nelso-Plosser data set for the post-1929 samples as well as postwar aual CPI-based real exchage rates for 14 OECD coutries. I cotrast to the results of previous studies usig the uivariate uit root tests, our empirical results show that the ull hypothesis of a uit root is rejected for all the series i the Nelso-Plosser data set ad for most of the coutries i the real exchage rates series. 7. Appedix Lemma A.1 Uder Assumptio 2.1, we have as! 1 16 I particular, Caada ad Japa are raked as the U.S. s top two tradig parters. 19

20 [s] 1 X (a) p w t! d ª(1)B (s) (b) [s] 1 X p u t! d ¼(1) (1) ª(1)B (s) + B " (s) where ª(1) = (1) 1 ad ¼(1) = 1= (1). Proof of Lemma A.1 To establish the stated results, we use the Beveridge-Nelso (BN) represetatios for the ite order lag polyomials (L); (L) ad (L) de ed i (2) ad (4) ad the limit theory from the ivariace priciple give i (5). Part (a) We begi by derivig the BN represetatio for (w t ) from (4). Let (1) = I m P`k=1 k. The we may easily get or (1)w t = t r 1 + `X k=1j=k `X j (w t k w t k+1 ) w t = ª(1) t r 1 + ( ¹w t 1 ¹w t ); (28) where ª(1) = (1) 1 ad ¹w t = P`k=1 ¹ k w t k+1, with ¹ k = ª(1) P`j=k j. Uder our coditio i Assumptio 2.1, f ¹w t g is well de ed both i a.s. ad L s sese [see Brockwell ad Davis (1991, Propositio 3.1.1)]. The we have tx w k = ª(1) k=1 tx k r 1 + ( ¹w ¹w t ): k=1 Note that ( ¹w t ) is stochastically of smaller order of magitude tha the sum P t k=1 k, ad hece will become egligible i the limit. The it follows directly from (5) [s] 1 X p w t = ª(1) 1 [s] X p t r 1 + p 1 ³ ¹w ¹w [s]! d ª(1)B (s) for s 2 [; 1], givig the stated result i part (a). Part (b) Let (1) = 1 P p k=1 k. Similarly, we derive the BN represetatio for u t from (2) as follows P 1 u t = (L) px p j=k w t + " j t + (u t k u t k+1 ) (1) (1) k=1 = ¼(1) (L) w t + " t + (¹ut 1 ¹u t ) where ¼(1) = 1= (1) ad ¹u t = P p k=1 ¹ ku t k+1, with ¹ k = ¼(1) P p j=k j. The process f¹u t g is also well de ed both i a.s. ad L r sese. We may also obtai the BN represetatio for (L) w t as follows (L) w t = (1) w t + q 1 X qx k=j=k+1 j (w t k 1 w t k ) + = (1) w t + ( ¹w + t 1 ¹w+ t ) + ( ¹w t ¹w t 1 ) 2 r 1 X rx k=j=k+1 j (w t+k+1 w t+k )