Bootstrap Unit Root Tests

Transcription

1 Bootstrap Uit Root Tests Joo Y. Park Departmet of Ecoomics Rice Uiversity ad School of Ecoomics Seoul Natioal Uiversity Abstract We cosider the bootstrap uit root tests based o fiite order autoregressive itegrated models drive by iid iovatios, with or without determiistic time treds. A geeral methodology is developed to approximate asymptotic distributios for the models drive by itegrated time series, ad used to obtai asymptotic expasios for the Dickey-Fuller uit root tests. The secod-order terms i their expasios are of stochastic orders O p /4 ) ad O p /2 ), ad ivolve fuctioals of Browia motios ad ormal radom variates. The asymptotic expasios for the bootstrap tests are also derived ad compared with those of the Dickey-Fuller tests. We show i particular that the bootstrap offers asymptotic refiemets for the Dickey-Fuller tests, i.e., it corrects their secod-order errors. More precisely, it is show that the critical values obtaied by the bootstrap resamplig are correct up to the secod-order terms, ad the errors i rejectio probabilities are of order o /2 ) if the tests are based upo the bootstrap critical values. Through simulatios, we ivestigate how effective is the bootstrap correctio i small samples. First Draft: February, 999 This versio: October, 22 Key words ad phrases: bootstrap, uit root test, asymptotic expasio. I am grateful to Coeditor, Joel Horowitz, ad three aoymous referees for may costructive suggestios, which have greatly improved the expostio of the paper. I also wish to thak Bill Brow ad Yoosoo Chag for helpful discussios ad commets. Lars Muus suggested the research reported here a decade ago, while he was my colleague at Corell. Earlier versios of this paper were preseted at World Cogress of the Ecoometric Society, Seattle, 2 ad Texas Ecoometrics Camp, Taglewood, 999.

2 . Itroductio It is ow well perceived that the bootstrap, if applied appropriately, helps to compute the critical values of asymptotic tests more accurately i fiite samples, ad that the tests based o the bootstrap critical values geerally have actual fiite sample rejectio probabilities closer to their asymptotic omial values. See, e.g., Hall 992) ad Horowitz 2). The bootstrap uit root tests, i.e., the uit root tests relyig o the bootstrap critical values, seem particularly attractive i this respect. For most of the commoly used uit root tests, the discrepacies i the actual ad omial rejectio probabilities are kow to be large ad ofte too large for the tests to be ay reliable. It has ideed bee observed by various authors icludig Ferretti ad Romo 996) ad Nakervis ad Savi 996) that the bootstrap tests have actual rejectio probabilities that are much closer to their omial values, compared to the asymptotic tests, i the uit root models. The mai purpose of this paper is to provide a theory for the asymptotic refiemet of bootstrap uit root tests. Bootstrap theories for uit root models have previously bee studied by, amog others, Basawa et al. 99a, 99b), Datta 996), Park 22) ad Chag ad Park 22). However, they have all bee restricted to the cosistecy ad icosistecy) of the bootstrap estimators ad statistics from uit root models. Noe of them cosiders the asymptotic refiemet of bootstrap. I this paper, we develop asymptotic expasios that are applicable for a wide class of uit root tests ad their bootstrap versios, ad provide a framework withi which we ivestigate the bootstrap asymptotic refiemet of various uit root tests. Our asymptotic expasios are obtaied by aalyzig the Skorohod embeddig, i.e., the embeddig of the partial sum process ito a Browia motio defied o a exteded probability space. I the paper, we cosider more specifically the Dickey-Fuller uit root tests for the fiite order autoregressive uit root models drive by iid errors, possibly with costat ad liear time tred. It ca be clearly see, however, that our methodology may also be used to aalyze may other uit root tests as well. For the Dickey-Fuller uit root tests, the expasios have as the leadig term the fuctioals of Browia motio represetig their asymptotic distributios. This is as expected. The secod-order terms i the expasios are, however, quite differet from the stadard Edgeworth-type expasios for the statioary models. They are represeted by fuctioals of Browia motios ad ormal radom variates, which are of stochastic orders O p /4 ) ad O p /2 ). The secod-order expasio terms ivolve various ukow model parameters. The expasios are obtaied for the tests i models with determiistic treds, as well as for the tests i purely stochastic models. They have similar characteristics. We show that the limitig distributios of the bootstrap statistics have expasios that are aalogous to the origial statistics. The bootstrap statistics have the same leadig expasio terms. This is well expected, sice the statistics that we cosider are asymptotically pivotal. More importatly, their secod-order terms are also exactly the same as the origial statistics except that the ukow parameters icluded i the expasios of the origial statistics are ow replaced by their sample aalogues, which strogly coverge to the correspodig populatio parameters. Cosequetly, usig the critical values obtaied by the bootstrap is expected to reduce the order of discrepacy betwee the actual fiite

3 sample) ad omial asymptotic) rejectio probabilities of the tests. The bootstrap thus provides a asymptotic refiemet for the tests. Though our asymptotic expasios for the uit root models are quite differet from the Edgeworth-type expasios for statioary models, the reaso that the bootstrap offers a refiemet of asymptotics is precisely the same. Through simulatios, we ivestigate how effective the bootstrap correctio is i small samples. We cosider both Gaussia ad o-gaussia uit root models. For the o- Gaussia models, we ivestigate models drive by iovatios that are distributed symmetrically ad asymmetrically. Our fidigs are geerally supportive of the theory developed i the paper. Moreover, they are cosistet with the simulatio results obtaied earlier by Nakervis ad Savi 996). Overall, the bootstrap does provide some obvious improvemets over the asymptotics. The tests based o the bootstrap critical values i geeral have rejectio probabilities that are substatially closer to their omial values. The actual magitudes of improvemets, however, somewhat vary depedig upo the distributioal characteristics of iovatios, the size of samples ad the presece of determiistic treds i the model. It appears i particular that the beefits from the bootstrap are more oticeable for the models with treds ad for the samples of small sizes. The rest of the paper is orgaized as follows. Sectio 2 itroduces the model, tests ad bootstrap method. The test statistics are itroduced together with the autoregressive uit root model ad the momet coditio, ad how to obtai bootstrap samples from such a model is explaied here. The asymptotic expasios are derived i Sectio 3. The sectio starts with the probabilistic embeddigs that are essetial for the developmet of our subsequet theory, ad preset the asymptotic expasios for the origial ad bootstrap tests. Some of their implicatios are also discussed. The asymptotic powers of the bootstrap tests agaist the local-to-uity model are cosidered i Sectio 4. Sectio 5 exteds the theory to the models with determiistic treds. The asymptotic expasios for the tests i models with costat ad liear time tred are preseted ad compared with the earlier results. The simulatio results are reported i Sectio 6, ad Sectio 7 cocludes the paper. Mathematical proofs are give i Sectio The Model, Tests ad Bootstrap Method 2. The Model ad Test Statistics We cosider the test of the uit root hypothesis i the ARp) uit root model y t = αy t + H : α = ) p α i y t i + ε t 2) 2

4 3 where is the usual differece operator. We defie αz) = p α i z i so that uder the ull hypothesis of the uit root ) we may write αl) y t = ε t usig the lag operator L. Assume Assumptio 2. Let ε t ) be a iid sequece with Eε t = ad E ε t r < for some r >. Also, we assume that αz) for all z. Uder Assumptio 2. with r 2) ad the uit root hypothesis ), the time series y t ) becomes a secod-order) statioary ARp) process. The uit root hypothesis is customarily tested usig the t-statistic o α i regressio 2). Deote by ˆα the OLS estimator for α i regressio 2). If we let ad defie x t = y t,..., y t p ) ) py t = y t y t x t x t x t ) x t, the we may explicitly write the t-statistic for the ull hypothesis ) as F = ˆα ) /2 3) σ py 2 t where σ 2 is the usual variace estimator for the regressio errors. The test is first proposed ad ivestigated by Dickey ad Fuller 979, 98), ad it is commoly referred to as the Dickey-Fuller test if applied to the regressios with o lagged differece term) or the augmeted Dickey-Fuller ADF) test if based o the regressios augmeted with lagged differece terms). We may also use the statistic G = ˆα ) α ) 4) to test the uit root hypothesis, where α ) = with the least squares estimators α i of α i for i =,..., p. The statistic G reduces to the ormalized coefficiet ˆα ) i the simple model with o lagged differece term. p α i

5 The asymptotic distributios of the statistics F ad G uder the presece of a uit root are well kow [see, e.g., Stock 994)], ad give by W t)dw t) F d F = ) /2, G d G = W t) 2 dt W t)dw t) W t) 2 dt where W is the stadard Browia motio. Sice F ad G do ot ivolve ay uisace parameter, the statistics F ad G are asymptotically pivotal. The distributios represeted by F ad G are however o-stadard, ad they are tabulated i Fuller 996). See Evas ad Savi 98, 984) for a detailed discussio o some of their distributioal characteristics. The iitializatio of y t ) is importat for some of our subsequet theories. I what follows, we let y,..., y p ) be fixed ad make all our argumets coditioal o them. If we let α = ad defie u t = y t, the we may equivaletly assume that y, u,..., u p+ )) are give. This covetio o the iitializatio of y t ) is crucial for the theory developed i Sectio 3 for the model with o costat term. It will however be uimportat for the model with costat or liear time tred cosidered i Sectio 4. Uder the uit root hypothesis, our statistics become ivariat with respect to the iitial values of y t ) i the regressio with itercept. 2.2 The Bootstrap Method Implemetatio of the bootstrap method i our uit root model is quite straightforward, oce we fit the regressio p y t = α i y t i + ε t 5) ad obtai the coefficiet estimates α i ) ad the fitted residuals ˆε t ). Sice our purpose is to bootstrap the distributios of the statistics uder the ull hypothesis of the uit root, it seems atural to resample from the restricted regressio 5) istead of the urestricted oe i 2). It is ideed well kow that the bootstrap must be based o regressio 5), ot o regressio 2), for cosistecy [see Basawas, et al. 99a)]. 2 The first step is to draw bootstrap samples for the iovatios ε t ) after mea correctio. As usual, we deote by ε t ) their bootstrap samples, i.e., ε t ) are the samples from ) ˆε t ˆε i which ca be viewed as iid samples from the empirical distributio give by ˆε t ˆε i/). Note that the mea adjustmet is ecessary, sice otherwise the mea of the bootstrap samples is ozero. 2 We may estimate α i) ad ε t) from regressio 2), as log as we set the value of α to uity istead of its estimated value) ad use regressio 6) to geerate bootstrap samples. The resultig differeces are of order o log ) a.s., ad therefore, will ot chage ay of our subsequet theory. 4

6 Oce the bootstrap samples ε t ) are obtaied, we may costruct the values for u t ) recursively from ε t ) as p u t = α i u t i + ε t 6) startig from u,..., u p+ ). Fially, the bootstrap samples y t ) for y t ) ca be obtaied just by takig partial sums of u t ), i.e., y t = y + give y. For the model with o itercept term, the iitializatios of u t ) ad yt ) are importat ad should be doe as specified here to make our theory applicable. However, they become uimportat for the models with determiistic treds icludig costat, as i the case of the iitializatios of u t ) ad y t ). The bootstrap versios of the statistics F ad G, which we deote by F ad G respectively, are defied from yt ) exactly i the same way that F ad G i 3) ad 4) are costructed from y t ). Of course, the distributios of the bootstrap statistics F ad G ca ow be foud by repeatedly geeratig bootstrap samples ad computig their values i each bootstrap repetitio. These distributios are regarded as approximatios of the ull distributios of F ad G. The bootstrap uit root tests use the critical values calculated from the distributios of the bootstrap statistics F ad G. 3. Asymptotic Expasios of Test Statistics 3. Probabilistic Embeddigs Our subsequet theoretical developmet relies heavily o the probabilistic embeddig of the partial sum process costructed from the iovatio sequece ε i ) ito a Browia motio i a expaded probability space. This will be give below. Throughout the paper, we deote by Eε 2 i = σ2, Eε 3 i = µ3 ad Eε 4 i = κ4, wheever they exist. Lemma 3. Let Assumptio 2. hold with r 2. The there exist a stadard Browia motio W t)) t ad a time chage T i ) i such that T ad for all, t W T i /) = d σ u i 5 i ε k 7) i =,...,, ad if we let i = T i T i, the i s are iid with E i = ad E i r/2 KE ε t r for all r 2, where K is a absolute costat depedig oly upo r. The reader is referred to Hall ad Heyde 98) for the explicit costructio of the time chage T i ) i. The result i Lemma 3. is origially due to Skorohod 965). If Assumptio k=

7 2. holds with r > 2, we have as show i Park ad Phillips 999) max T i i i s a.s. 8) for ay s > /2. I what follows, we will assume that ε i ) ad W, T i )) are defied o the commo probability space Ω, F, P). This causes o loss i geerality sice we are cocered oly with the distributioal results of the test statistics defied i 3) ad 4), yet it will greatly simplify ad clarify our subsequet expositio. The covetio will be made throughout the paper. From ow o, we would thus iterpret the distributioal equality i 7) as the usual equality. If we defie a stochastic process W o [, ] by W t) = /2 [t] ε i/σ, the it follows from the Hölder cotiuity of the Browia sample path ad the result i 8) that sup W t) W t) sup T [t] / t /2 ɛ = o /4+ɛ ) a.s. 9) t t for ay ɛ >. Therefore, we have i particular W a.s. W uiformly o [, ]. Throughout the paper, we let T i = T i /, i =,...,, for otatioal brevity. For the developmet of our asymptotic expasios, it is ecessary to defie additioal sequeces defied from the Browia motio W ad the time chage T i ) itroduced i Lemma 3.. We let δ i = i for i =,...,. Moreover, we defie η i = Ti T,i [W t) W T,i )]dw t) for i =,...,. Note that δ i ) ad η i ) are iid sequeces of radom variables. We also eed to cosider the sequece ξ i ) give by ξ i = x i ε i. Clearly, ξ i ) is a martigale differece sequece. Uder the ull hypothesis of the uit root, it has coditioal covariace matrix whose expectatio is give by σ 4 Γ, where Γ = Ex i x i /σ2. Fially, we let E δ 2 i = τ 4 /σ 4, which is fiite uder Assumptio 2.. Note that δ i, whe ad oly whe ε i ) are ormal. The parameter τ ca therefore be regarded as the o-ormality parameter. Subsequetly, we set τ = if ad oly if ε i ) are ormal. The parameters Γ ad τ 4 defied here, i additio to σ 2, µ 3 ad κ 4 itroduced earlier, will appear frequetly i the developmet of our asymptotic expasios. Now we defie v i = ε i /σ, δ i, η i, ξ i/σ 2 ) ad let B t) = [t] v i. ) The ivariace priciple holds, ad B d B for a properly defied vector Browia motio B. We preset this formally as a lemma. 6

8 Lemma 3.2 Let Assumptio 2. hold with r > 4. The B d B, where B is a vector Browia motio with covariace matrix Σ give by µ 3 /3σ 3 µ 3 /3σ 3 Σ = τ 4 /σ 4 κ 4 3σ 4 3τ 4 )/2σ 4 κ 4 /6σ 4 Γ where the parameters are defied earlier i this sectio. Followig our earlier covetio, we subsequetly assume that both B ad B are defied o the probability space Ω, F, P), ad that B a.s. B. It is well kow that ay weakly coverget radom sequece ca be represeted, up to the distributioal equivalece, by a radom sequece which coverges a.s. [see, e.g., Pollard 984)]. 7 Remark We make a partitio of the limit Browia motio B as B = W, V, U, Z ) coformably with v i ). Let W, W ) be a bivariate stadard Browia motio idepedet of W. Clearly, we may the write where U = ωw + ωw, V = ωw + ω W + ω W, ω = µ3 3σ 3, ω ) κ 4 /2 = 6σ 4 µ6 9σ 6, ) κ 4 /2 ω = 6σ 4 µ6 κ 4 3σ 4 9σ 6 ω = [ τ 4 σ 4 µ6 9σ 6 κ 4 2σ 4 τ 4 4σ 4 µ6 9σ 6 ), ) 6σ 4 µ6 κ 4 3σ 4 9σ 6 2σ 4 τ 4 ) 2 ] /2 4σ 4 µ6 9σ 6. The represetatios ca be greatly simplified for the Gaussia models, for which we have µ 3 = ad κ 4 = 3σ 4 as well as τ =. Cosequetly, we have ω = / 2 ad ω = ω = ω =, ad therefore, V ad U becomes idepedet of W. I additio to the represetatios of U ad V give above, we may write Z) = Γ /2 S usig a multivariate ormal radom vector S with the idetity covariace matrix. Sice Z is idepedet of W, V, U), so is S. Fially, our subsequet expasios also ivolve stochastic processes M ad N. We let M be a exteded stadard Browia motio o R idepedet of B ad

9 therefore all of the Browia motios ad ormal radom variates defied above), ad let N be aother exteded Browia motio o R defied by Nt) = W + t) W ). 3 The otatios defied here will be used throughout the paper without ay further referece. 3.2 Asymptotic Expasios We are ow ready to obtai asymptotic expasios for the distributios of the statistics F ad G itroduced i 3) ad 4). We defie P = ) ) ) y t ε t y t x t x t x t x t ε t, ) Q = ) ) ) 2 yt 2 2 y t x t x t x t x t y t. 2) 8 Also, we write the error variace estimate σ 2 as σ 2 = ) ) ) ε 2 t ε t x t x t x t x t ε t 3) ad write ) ) α ) = α) ι x t x t x t ε t. 4) Here ad elsewhere i the paper, ι deotes the p-vector of oes. The statistics F ad G ca ow be writte respectively as F = P σ Q ad G = P α )Q. 5) Here we assume that σ 2 ad α) are estimated uder the uit root restrictio. This assumptio is made purely for the expositioal purpose. All of our subsequet results also hold for the urestricted estimators of σ 2 ad α). To derive the asymptotic expasios for the statistics F ad G, we eed to cosider various sample product momets i ) 4). The asymptotics for some of them are preseted i Lemma 3.3, which ca be directly obtaied from the probabilistic embeddigs developed i the previous sectio. Propositio 3.4 is a direct cosequece of Lemma 3.3. To simplify the subsequet expositio, we use X to deote X), as well as the process itself, for Browia motio X. This should cause o cofusio. 3 The defiitio of N, of course, requires that W t) be defied for t < as well as for t. I the subsequet developmet of our theory, we assume that the ecessary extesio is made ad W is a exteded Browia motio defied o R.

10 9 Lemma 3.3 Let Assumptio 2. hold with r > 4. The we have a) σ 2 ε 2 t = + /2 V + 2U) + o p /2 ), b) /2 σ 2 x t ε t = Z + o p ), c) σ 2 x t x t = Γ + O p /2 ), for large. Propositio 3.4 Let Assumptio 2. hold with r > 4. The we have [ ] a) σ 2 = σ 2 + /2 V + 2U) + o p /2 ), b) α ) = α) /2 ι Γ Z + o p /2 ), for large. We ow obtai the asymptotic expasios for the sample product momets y t ε t, y 2 t ad x t y t. To effectively aalyze these product momets, we defie w t = t ε i for t ad w ad first cosider the asymptotic expasios for the sample product momets of w t ) ad ε t ). We let u t = y t as before, so that αl)u t = ε t uder the ull hypothesis of the uit root. Uder the uit root hypothesis, u t ) is just a liearly filtered process of ε t ), ad y t ) becomes a itegrated process geerated by such a process. Our subsequet asymptotic expasios ivolve various fuctioals of Browia motios. To ease the expositio, we let for Browia motios X ad Y, IX) = Xt)dt ad JX, Y ) = Xt)dY t), i the subsequet developmet of our theory. This shorthad otatio, together with X = X) itroduced above, will be used repeatedly for the rest of the paper. Lemma 3.5 Let Assumptio 2. hold with r 8. The we have a) /2 ε t = W + /4 MV ) + /2 NV ) + o p /2 ), σ b) 3/2 w t = IW ) + /2 [W V JW, V ) ω] + o p /2 ), σ c) 2 σ 2 wt 2 = IW 2 ) + /2 [ W 2 V JW 2, V ) 2ωIW ) ] + o p /2 ), d) σ 2 w t ε t = JW, W ) + /4 W MV ),

11 + /2 [ /2)MV ) 2 +W NV ) /2)V+2U) ] + o p /2 ), for large. The asymptotic expasios for y t ε t, y 2 t ad x t y t ca ow be obtaied usig the relatioships betwee y t ) ad w t ), ad betwee u t ) ad ε t ). To write dow more explicitly their relatioships, we eed to defie some ew otatio. We let π = /α) ad π i = p α j /α) j=i for i =,..., p, ad let We also defie ν = /πσ) ϖ = π,..., π p ). y + ) p π i u i. Note that we assume y, u,..., u p+ )) to be give. Therefore, we may ad will regard ν as a parameter i our subsequet aalysis. With the otatio itroduced above, we may write after some algebra ad subsequetly get u t = πε t + ϖ x t x t ) It is ow straightforward to deduce from Lemma 3.5 that y t = πσν + πw t ϖ x t. 6) Propositio 3.6 Let Assumptio 2. hold with r 8. The we have a) 3/2 y t = IW ) + /2 [W V JW, V ) + ν ω)] + o p /2 ), πσ b) π 2 σ 2 x t y t = ι[ + JW, W )] Γϖ/π 2 + o p ), c) 2 π 2 σ 2 yt 2 = IW 2 ), d) πσ 2 + /2 [ W 2 V JW 2, V )+2ν ω)iw ) ] + o p /2 ), y t ε t = JW, W ) + /4 W MV ), + /2 [/2)MV ) 2 +W NV )+νw /2)V+2U) ϖ Z/π] + o p /2 ), for large. The asymptotic expasios for the statistics F ad G ca ow be easily obtaied from 5), usig the results i Lemma 3.3 ad Propositios 3.4 ad 3.6.

12 Theorem 3.7 Let Assumptio 2. hold with r 8. The we have for large F = F + F / 4 + F 2 / + o p /2 ), G = G + G / 4 + G 2 / + o p /2 ), where F = W MV )/IW 2 ) /2, G = W MV )/IW 2 ) ad F 2 = /2)MV )2 + W NV ) + νw [ + JW, W )][V + 2U)/2 + πι Γ Z] IW 2 ) /2 JW, W )[W 2 V JW 2, V ) + 2ν ω)iw )] 2IW 2 ) 3/2, G 2 = /2)MV )2 + W NV ) + νw V + 2U)/2 πι Γ Z IW 2 ) JW, W )[W 2 V JW 2, V ) + 2ν ω)iw )] IW 2 ) 2 i otatio itroduced earlier i Lemma 3.5 ad Propositio 3.6. Naturally, the asymptotic expasios for the statistics F ad G have the leadig terms F ad G represetig their asymptotic distributios. For both F ad G, the secod terms F / 4 ad G / 4 i our expasios are of stochastic order O p /4 ). Their effects are, however, distributioally of order O /2 ). More precisely, we have P { F + F / 4 x } = P {F x} + O /2 ), P { G + G / 4 x } = P {G x} + O /2 ), uiformly i x. This is because the process M icluded i F ad G is a Gaussia process idepedet of W, V, U). Note that for ay fuctioals aw ) ad bw ) of W, we have aw ) + / 4 )bw )MV ) = d MN aw ), / )bw ) 2 V ) where MN stads for mixed ormal distributio. 4 Therefore, we call F = F / 4 + F 2 /, G = G / 4 + G 2 / the secod-order terms i our asymptotic expasios of F ad G. The remaider terms i the expasios are give to be of order o p /2 ). The results i Theorem 3.7 suggest that our secod-order asymptotic expasios of the statistics F ad G provide refiemets of their asymptotic distributios up to order o /2 ). This ca be show rigorously, if we assume higher momets exist. More precisely, if we let 2F = F + F, 2G = G + G, 7) the we have 4 The characteristic fuctios of F +F / 4 ad G+G / 4 ca therefore be expaded i itegral powers of /2 with the leadig terms beig the characteristic fuctios of F ad G, respectively. This shows that the secod terms F / 4 ad G / 4 have distributioal effects of order O /2 ).

13 2 Corollary 3.8 Let Assumptio 2. hold with r > 2. The we have P{F x} = P { 2 F x} + o /2 ), P{G x} = P { 2 G x} + o /2 ), uiformly i x R. It is thus expected i geeral that the actual fiite sample rejectio probabilities of the tests F ad G disagree with their omial values oly by order o /2 ), if the secodorder corrected critical values are used, i.e., a λ ad b λ such that P{ 2 F a λ } = λ ad P{ 2 G b λ } = λ for tests with omial rejectio probability λ. For both statistics, the secod-order terms F ad G ivolve various fuctioals of Browia motios. The fuctioals are depedet upo various model parameters, ot oly those icluded explicitly, but also those give implicitly by the variaces ad covariaces of W, V, U, Z) i Lemma 3.2. More precisely, if we represet V, U ad Z as suggested i Remark followig Lemma 3.2, the F ad G ca be writte explicitly as fuctioals of three idepedet Browia motios W, W, W ad aother idepedet ormal radom vector S. The fuctioals ivolve the parameter θ defied by θ = ν, π, σ 2, µ 3, κ 4, τ 4, Γ). 8) We deote by F θ) ad G θ) the resultig fuctioals respectively for F ad G. Symbolically, we write F θ) = F θ, W, W, W, S)), G θ) = G θ, W, W, W, S)) 9) to sigify such fuctioals. Our asymptotic expasios of the statistics F ad G provide some importat iformatios o their fiite sample distributios. For istace, our expasios make it clear that the iitial values have effects, which are distributioally of order O /2 ), o their fiite sample distributios. Note that they are parametrized as ν. Moreover, we may lear from the expasios that the presece of shortru dyamics, if it is correctly modelled, has distributioal effects also of order O /2 ). As is well kow, either the iitial values or the shortru dyamics affect the limitig distributios of F ad G. Though we will ot discuss the details i the paper, it is rather straightforward to obtai the secod-order asymptotic expasios for may other uit root tests usig our results here. For the tests cosidered i Stock 994, pp ), it is ideed ot difficult to see that the tests classified as ˆρ-class, ˆτ-class, SB-class, Jp, q), LMPI o-determiistic case) ad P T all have the asymptotic expasios that are obtaiable from the results i Lemmas 3.3 ad 3.5 ad Propositios 3.4 ad 3.6. This, of course, is true oly whe the uisace parameter is estimated from the ARp) model as for F ad G cosidered i the paper. The oparametric estimatio of the uisace parameter would fudametally chage the ature of asymptotic expasios, ad our results do ot apply to the uit root tests with uisace parameters estimated oparametrically. Our approach developed here ca also be used to aalyze the models with the local-to-uity formulatio of the uit root

14 hypothesis. The asymptotics for such models are quite similar to those for the uit root models, except that they ivolve Orstei-Uhlebeck diffusio process i place of Browia motio. Their asymptotic expasios ca be obtaied exactly i the same maer usig the probabilistic embeddig of Orstei-Uhlebeck process. 3.3 Bootstrap Asymptotic Expasios To develop the asymptotic expasios for the bootstrap statistics F ad G correspodig to those for F ad G preseted i the previous sectio, we first eed a probabilistic embeddig of the stadardized partial sum of the bootstrap samples ε i ) ito a Browia motio defied o a exteded probability space. Oce this embeddig is doe i a appropriately exteded probability space, the rest of the procedure to obtai the asymptotic expasios for F ad G is essetially idetical to that for F ad G. Followig the usual covetio i the bootstrap literature, we use superscript for the quatities ad relatioships that are depedet upo the realizatios of ε i ). Let W be a stadard Browia motio idepedet of ε i ), 5 ad assume that they are defied o the commo probability space Ω, F, P). Of course, there exists a probability space rich eough to support W together with ε i ), sice we assume that they are idepedet. We the let Ti ) i be a time chage defied o Ω, F, P) such that 3 W T i /) = d σ i k= ε k a.s. 2) where = d deotes the equivalece of distributio coditioal o a realizatio of ε i ). Note that, for each ad for ay possible realizatio of ε i ), we may fid a time chage Ti ) for which 2) holds with the same Browia motio W. The Browia motio W therefore is ot depedet upo the realizatios of ε i ). Just as the covetio made i Sectio 3., we idetify ε i ) oly up to their distributioal equivaleces so that we may assume ε i ) are also defied o the same probability space Ω, F, P), ad iterpret the equality = d i coditioal distributios as the usual equality i 2). Uder the covetio, we costruct the sequeces δi ) ad η i ) from W, T i )) for each realizatio of ε i ), aalogously as δ i ) ad η i ). We also let ξi ) be give similarly as ξ i ) for each realizatio of ε i ). Clearly, we may alteratively defie δi, η i ) to be the iid samples from the empirical distributio of δ i, η i ), which are draw together with ε i ) from ε i ). We may thus regard ε i, δ i, η i ) as the iid samples from the empirical distributio of ε i, δ i, η i ). To simplify the subsequet expositio, however, we will assume that δi, η i ) are defied from the embeddig 2) of ε i ) give a realizatio of ε i). Now we defie vi = ε i /σ, δi, ηi, ξi /σ) 2 5 The Browia motio W here is, of course, distict from the oe itroduced i Sectios 3. ad 3.2. We just use the same otatio here to make our results for bootstrap tests more directly comparable to those for asymptotic tests.

15 4 ad let as i ). It may be readily deduced that Bt) = [t] Lemma 3.9 Let Assumptio 2. hold with r > 4. The B d B a.s., where B is a vector Browia motio with covariace matrix Σ give by the sample aalogue estimator of Σ defied i Lemma 3.2. Aalogously as for B, we let v i B = W, V, U, Z ) ad further represet V, U ad Z i terms of idepedet stadard Browia motios W, W ad W, as i Remark below Lemma 3.2, with the coefficiets give by the sample aalogue estimators ω, ω, ω ad ω, say, of ω, ω, ω ad ω, i.e., U = ω W + ω W, V = ω W + ω W + ω W. Moreover, we may write Z ) = Γ /2 S, where Γ is the sample aalogue estimator of Γ. Note that we may use the same W, W ad S for all realizatios of ε i ) to represet V, U ad Z as above. Therefore, we may assume that W, W, S) are defied o the same probability space Ω, F, P) as ε i ) ad W, Ti )), ad idepedet of ε i) as well as W. We also let M, N) be defied as earlier, which we may also regard as beig idepedet of ε i ). Fially, correspodig to θ i 8), we defie θ = ν, π, σ 2, µ 3, κ 4, τ 4, Γ ) 2) where π = /α ), ad σ 2, µ 3, κ 4, τ 4 ad Γ are the sample aalogue estimators of σ 2, µ 3, κ 4, τ 4 ad Γ, respectively. As usual, P ad E refer respectively to the probability ad expectatio operators give a realizatio of ε i ). They ca be more formally defied as the coditioal probability ad expectatio operators P ε i )) ad E ε i )) o the probability space Ω, F, P) itroduced above. For the fuctioals of W, W, W, S) ad M, N), however, P ad E agree with P ad E respectively, sice they are idepedet of ε i ) by costructio. For the subsequet developmet of our theory, it is coveiet to itroduce the bootstrap stochastic order symbols. For a sequece of radom sequeces X ) o the probability space Ω, F, P), we let X = o p) if P { X > ɛ} a.s. for ay ɛ >. Likewise, we deote by Y = O p) for Y ) o Ω, F, P) if, for a.s. all realizatios of ε i ) ad for ay ɛ >, there exists a costat K such that P { Y > K} ɛ. The costat K may vary depedig upo the realizatios of ε i ). The symbols o p) ad O p) are the bootstrap versios of the stochastic order symbols o p ) ad O p ). For the radom sequeces whose distributios are idepedet of the realizatios of ε i ), the two otios become idetical. It is easy to

16 5 see that X = o p) if E X s a.s. for some s >. Moreover, o p) ad O p) satisfy the usual additio ad product rules that apply to o p ) ad O p ), as oe may easily check. Needless to say, the defiitios of o p) ad O p) aturally exted to o pa ) ad O pb ) for some umerical sequeces a ) ad b ). Theorem 3. Let Assumptio 2. hold with r 8. The we have for large F = F + F θ ) + o p /2 ), G = G + G θ ) + o p /2 ), where F ad G are itroduced i 9) ad θ is defied i 2). Corollary 3. Let Assumptio 2. hold with r > 2. The we have for large P {F x} = P{ 2 F x} + o /2 ) a.s., P {G x} = P{ 2 G x} + o /2 ) a.s. uiformly i x R, where 2 F ad 2 G are defied i 7). The asymptotics for the bootstrap statistics F ad G are completely aalogous to those for the correspodig statistics F ad G. Theorem 3. ad Corollary 3. are respectively the bootstrap versios of Theorem 3.7 ad Corollary 3.8. I Theorem 3., the parameters appeared i the asymptotic expasios of the origial statistics are replaced by their estimates, as i the bootstrap Edgeworth expasios for the stadard statioary models. Due to the law of iterated logarithm for iid sequeces, we have for ay ɛ > θ = θ + o p /2+ɛ ) uder the give momet coditio. We may therefore rewrite the results i Theorem 3. as F = F + F θ) + o p /2 ), G = G + G θ) + o p /2 ). Corollary 3. shows that these secod-order expasios of F ad G actually provide the refiemets of their asymptotic distributios a.s. Corollaries 3.8 ad 3. yield uder the required momet coditio P {F x} = P{F x} + o /2 ) a.s., P {G x} = P{G x} + o /2 ) a.s., uiformly i x R. Now we defie a λ ad b λ as P {F a λ } = P {G b λ } = λ

17 6 for tests with omial rejectio probability λ. The values a λ ad b λ are the bootstrap critical values for the λ-level tests based o the statistics F ad G. The it follows that P {F a λ }, P {G b λ } = λ + o /2 ) for large. The tests usig the bootstrap critical values a λ ad b λ probabilities with errors of order o /2 ). 6 thus have rejectio 4. Asymptotics uder Local Alteratives We ow cosider local alteratives H : α = c 22) where c > is a fixed costat, ad let y t ) be geerated as y t = αy t + p α i c y t i + ε t 23) where c = c/)l is the quasi-differecig operator. The model give by 22) ad 23) is commoly referred to as the local-to-uity model, ad itroduced here to ivestigate the asymptotic powers of the bootstrap tests. For the local-to-uity model, it is well kow [see, e.g., Stock 994)] that /2 F d F c) = c W c t) dt) 2 + G d Gc) = c + W c t)dw t) ) /2, 24) W c t) 2 dt W c t)dw t), 25) W c t) 2 dt where W c t) = W t) c t e ct s) W s)ds is Orstei-Uhlebeck process, which may be defied as the solutio to the stochastic differetial equatio dw c t) = cw c t)dt + dw t). As is well kow, P{F c) x} > P{F x}, P{Gc) x} > P{G x}, 26) for all x R, ad we may thus expect that the uit root tests relyig o F ad G have some discrimiatory powers agaist the local-to-uity model. 6 Note that the results here hold oly uder the assumptio that the uderlyig model is ARp) with kow p ad iid errors. For the model drive by more geeral, possibly coditioally heterogeeous, martigale differeces, oly the first-order asymptotics are valid. If p is ukow or give as ifiity, we may icrease p with the sample size ad apply the results for the sieve bootstrap established i Park 22).

18 The limitig distributios of the bootstrap statistics F ad G are, however, uaffected, i.e., their limitig distributios uder local alteratives are precisely the same as their limitig ull distributios. This is show below i Theorem 4.. We may ideed expect that the bootstrap samples asymptotically behave as the uit root processes uder may other alteratives as well, sice they are geerated uder the uit root restrictio regardless of the true data geeratig mechaism. It is therefore ot surprisig that the bootstrap statistics F ad G have the same limitig distributios uder both the exact-uit root ad local-to-uit root specificatios. 7 Theorem 4. model as. Let Assumptio 2. hold with r > 2. The we have uder the local-to-uity F d F a.s., G d G a.s. Uder the alterative of the local-to-uity model, we have i particular as, ad therefore, P{F a λ }, P{G b λ } λ lim P{F a λ } = lim P{F c) a λ } > λ, lim P{G b λ } = lim P{Gc) b λ } > λ due to 24), 25) ad 26). The bootstrap uit root tests would thus have o-trivial powers agaist the local-to-uity model. 5. Tests i Models with Determiistic Treds I this sectio, we ivestigate the uit root tests i the model y t = D t + αy t + p α i y t i + ε t 27) where D t is determiistic tred. I what follows, we oly explicitly cosider D t specified as D t = β or β + β t 28) sice they are most frequetly used i practical applicatios. Our theories ad methodologies here, however, apply to more geeral models with higher order polyomials possibly with structural chages, i.e., D t = q i= β it i or q i= β it i + q i= βi t i {t s i }, where s i, i =,... q, are kow break poits. We oly eed some obvious modificatios for such models. We eed to cosider model 27), istead of 2), whe it is believed that the observed time series y t ) icludes determiistic tred D t ad is geerated as y t = D t + y t 29)

19 where the stochastic compoet y t ) is assumed to follow 2). As a alterative to testig for the uit root i regressio 27), we may detred y t ) directly from the regressio give by 29) with 28) to obtai the fitted residuals ŷ t ), ad base the uit root tests o regressio 2) usig ŷ t ). It turs out that they are asymptotically equivalet ot oly i the first order, but also i the secod order. All our subsequet results are therefore applicable for both procedures. 7 To obtai the asymptotic expasios for the Dickey-Fuller tests i the presece of liear time treds, we eed the followig lemma ad the subsequet propositio. We deote by ı the idetity fuctio ıx) = x i what follows. Lemma 5. Let Assumptio 2. hold. The we have t a) /2 σ ε t = Jı, W ) + /4 MV ) /2 [W V JW, V ) NV ) ω] + o p /2 ), b) 3/2 σ for large. t w t = IıW ) + /2 [W V IW V ) JıW, V ) ω/2] + o p /2 ), Propositio 5.2 Let Assumptio 2. hold. The we have t 3/2 πσ y t = IıW ) for large + /2 [W V IW V ) JıW, V ) + ν ω/2)] + o p /2 ) We ow preset the asymptotic expasios of the Dickey-Fuller tests for the models with costat, D t = β, ad for the models with liear time tred, D t = β +β t. They are quite similar, ad we preset them together i a sigle framework. For both cases, we deote by F ad G the Dickey-Fuller statistics based o regressio 27), or equivaletly, the oes defied as i 3) ad 4) from the regressio 2) ru with the demeaed or detreded y t ). We deote by W the demeaed or detreded Browia motio, for the case of D t = β or D t = β + β t. Moreover, we let F ad G respectively be the fuctioals of Browia motios defied similarly as F ad G with W replaced by W. It is well kow that F ad G have the limitig distributios give by F ad G respectively. We also defie 2 F ad 2 G to be the secod-order expasios of F ad G, similarly as 2 F ad 2 G for F ad G. 7 We do ot cosider i the paper the GLS detredig proposed by Elliot, Rotheberg ad Stock 996) based o the local-to-uity formulatio of the uit root hypothesis. Such detredig i geeral yields asymptotics that are differet from those for the usual OLS detredig cosidered here. 8

20 9 Theorem 5.3 Let Assumptio 2. hold with r 8. The we have for large F = F + F / 4 + F 2 / + o p /2 ), G = G + G / 4 + G 2 / + o p /2 ), where F = W MV )/I W 2 ) /2, G = W MV )/I W 2 ) ad F 2 = /2)MV )2 + W NV ) [ + J W, W )][V + 2U)/2 + πι Γ Z] I W 2 ) /2 J W, W )[ W 2 V J W 2, V ) 2ωI W )] 2I W 2 ) 3/2, G 2 = /2)MV )2 + W NV ) V + 2U)/2 πι Γ Z I W 2 ) J W, W )[ W 2 V J W 2, V ) 2ωI W )] I W 2 ) 2. Moreover, if Assumptio 2. holds with r > 2, the for large uiformly i x R. P{ F x} = P{ 2 F x} + o /2 ), P{ G x} = P{ 2 G x} + o /2 ), The asymptotic expasios for F ad G i Theorem 5.3 are quite similar to those for F ad G i Theorem 3.7. We oly have two differeces. First, all of the terms i the expasios for F ad G represetig the depedecy o the iitial value ν disappear, ad are ot preset i the expasios of F ad G. This is aturally expected, sice the demeaig or detredig makes the statistics F ad G ivariat with respect to the iitial values. Secod, the Browia motio W is replaced by the demeaed or detreded Browia motio W i all of the expasio terms. The demeaig or detredig thus affects ot oly the first-order asymptotics, but also the secod-order asymptotics. Now we defie the secod-order expasio terms F = F / 4 + F 2 /, G = G / 4 + G 2 / for F ad G, ad let F θ) = F θ, W, W, W, S)), G θ) = G θ, W, W, W, S)), 3) aalogously as i 9). Moreover, we let 2 F = F + F, 2 G = G + G, 3) be the secod-order approximatios of F ad G correspodigly to 7).

21 2 Theorem 5.4 Let Assumptio 2. hold with r 8. The we have for large F = F + F θ ) + o p /2 ), G = G + G θ ) + o p /2 ), where F ad G are itroduced i 3) ad θ is the sample momet estimator of the parameter θ defied i 8). Moreover, if Assumptio 2. holds with r > 2, the for large P { F x} = P{ 2 F x} + o /2 ) a.s., P { G x} = P{ 2 G x} + o /2 ) a.s., uiformly i x R, where 2 F ad 2 G are defied i 3). The results i Theorem 5.4 make it clear that the mai coclusios o the asymptotic refiemets of the bootstraps i Sectio 3.3 cotiue to hold for the tests i models with costat ad liear treds. Usig bootstrap critical values would reduce the fiite sample distortio i rejectio probability to the order o /2 ) also i models with such determiistic treds. 6. Mote Carlo Simulatios We perform Mote Carlo simulatios to ivestigate the actual fiite sample performaces of the bootstrap tests. The model we use for the simulatios is specified as y t = αy t + β y t + ε t where α ad β are parameters ad ε t ) are iid iovatios. The parameter values are chose to be α = ad β =.4,.,.4. We set α = ad ivestigate oly the fiite sample sizes of the tests. 8 The iovatios are geerated as stadard ormal N, ), ormal-mixture N, ) ad N, 6) with mixig probabilities.8 ad.2, ad shifted chi-square χ 2 8) 8 distributios. 9 We thus cosider both ormal ad o-ormal iovatios, ad for the o-ormal iovatios we look at skewed oes as well as those that are ot skewed. The samples of sizes = 25, 5, are geerated. The rejectio probabilities for the tests with fitted mea ad time tred are give respectively i Tables ad 2. The omial rejectio probabilities of the test are 5%. The simulatio results reported i Tables ad 2 are geerally supportive of the theory developed i the paper. I particular, they make it clear that the bootstrap does provide asymptotic refiemets for the tests of a uit root i fiite samples. I all cases that we ivestigate here, the bootstrap tests, i.e., the tests based o the critical values computed by 8 We also looked at the fiite sample powers of the tests for various values of α. The bootstrap tests have essetially the same powers as the asymptotic tests. This cofirms the fidigs by Nakervis ad Savi 996). 9 To make our results more comparable to theirs, we look at the distributios cosidered by Nakervis ad Savi 996). However, we do ot follow them i stadardizig the distributios to have uit variace, sice the uit root tests cosidered here are ivariat with respect to scalig.

22 the bootstrap samplig, have rejectio probabilities that are closer to their omial values, if compared with the usual asymptotic tests. The actual magitudes of the refiemets, however, deped upo various factors such as the sample size, the model specificatio ad the distributio of iovatios. Overall, it appears that the bootstrap offers more sigificat refiemets for small samples ad for models with fitted time tred, respectively i terms of the sample size ad the model specificatio. The distributio of iovatios seems to have oly mior effects. Our simulatio results are largely comparable to those obtaied earlier by Nakervis ad Savi 996). Ideed, it ca be see clearly from Tables ad 2 that the bootstrap correctio i fiite samples is highly effective for the tests of a uit root. The rejectio probabilities of the bootstrap tests are quite close to their omial values regardless of the sample size, the model specificatio ad the distributio of iovatios. The discrepacies betwee the actual ad omial rejectio probabilities ever exceed more tha.5% i most cases. This is i cotrast with the asymptotic tests. For the asymptotic tests, the actual rejectio probabilities are larger tha % i several cases for the 5% tests. It seems clear that the use of asymptotic critical values ca seriously distort the test results i fiite samples, ad that the bootstrap provides a effective tool to prevet such a distortio. Our simulatios suggest that the bootstrap correctio is eeded more for the tests usig smaller samples ad based o models with maitaied time tred. The asymptotics provide poor approximatios especially whe the sample size is small ad the model icludes a maitaied time tred. 7. Coclusio I the paper, we develop asymptotic expasios for the uit root models ad show that the bootstrap provides asymptotic refiemets for the uit root tests. It is demostrated through simulatios that the bootstrap ideed offers asymptotic refiemets i fiite samples ad the bootstrap correctios are i geeral quite effective i elimiatig fiite sample biases of the test statistics. Though we cosider exclusively the Dickey-Fuller tests, it is made clear that our results are applicable for other uit root tests as well. Our methodology here ca also be exteded to aalyze the bootstrap for more geeral models, oliear as well as liear, with itegrated time series ad ear-itegrated time series. This will be reported i future work. 8. Mathematical Proofs We first preset some useful lemmas ad their proofs. They will be used i the proofs of the mai results i the text, which will follow subsequetly. Throughout the proof, deotes the Euclidia orm, ad K sigifies a geeric costat depedig possibly oly upo r, which may vary from place to place. 2

23 22 8. Useful Lemmas ad Their Proofs We write ε Ti i σ = dw t) = W T i ) W T,i ). T,i The it follows from Ito s formula that ) k+2 Ti εi σ = k + 2) [W t) W T,i )] k+ dw t) T,i + for k. Cosequetly, we have Lemma A a) ε 2 i /σ2 = δ i + 2η i. Moreover, it follows that b) E Ti k + )k + 2) 2 Ti Let Assumptio 2. hold with r 2. We have T,i [W t) W T,i )] k dt = for ay iteger k such that k r 2. T,i [W t) W T,i )] k dt 32) ) 2 k+2 k + )k + 2) σ E ε k+2 i Proof of Lemma A The result i part a) may easily be deduced from ) 2 Ti εi σ = 2 [W t) W T,i )]dw t) + T i T,i ), T,i which follows from Ito s formula 32) with k =. To derive part b), we rewrite the Ito s formula 32) as Ti T,i [W t) W T,i )] k dt = 2 k + )k + 2) 2 k + Ti ) k+2 εi σ T,i [W t) W T,i )] k+ dwt). The stated result follows immediately upo oticig that T,i [W s) W T,i )] k+ dw s) is a martigale, ad therefore E Ti T,i [W t) W T,i )] k+ dw t) =

24 23 due to the optioal stoppig theorem. Let u t = ϕl)ε t = ϕ i ε t i ad let υ 2 = i= ϕ2 i. Also, we defie Γ ij = Eu t i u t j, so that we have i particular Γ = σ 2 υ 2. Lemma A2 a) E u i r υ r E ε i r, i= If Assumptio 2. holds with r 2, the b) E δ i r/2 K + E ε i r ), E + δ i r/2 KE ε i r, c) E η i r/2 K + σ r )E ε i r. If Assumptio 2. holds with r 4, the r/2 d) E + δ k )u k i r/4 + υ r ) K k= r/2 e) E δ k u k i u k j r/4 υ r K k= f) E u k i u k j Γ ij ) k= for all i, j =,..., p. r/2 [ + E ε i r ) 2], [ + E ε i r ) 2], r/4 υ r K σ r + E ε i r ), Proof of Lemma A2 Part a) is well kow. Part b) is due to Lemma 3.. To prove part c), use part a) of Lemma A ad Mikowski s iequality to deduce E η i r/2 K E + δ i r/2 + σ r E ε i r) from which ad Lemma 3. the stated result readily follows. Give parts a) ad b), parts d) ad e) ca easily be deduced from the successive applicatios of Burkholder s iequality [see, e.g., Hall ad Heyde 98, Theorem 2.)] ad Mikowski s iequality. Ideed we have r/2 r/4 E + δ k )u k i KE + δ k ) 2 u 2 k i r/4 KE + δ i r/2 E u i r/2 k= k= ad part d) follows immediately, due to parts a) ad b). Note that υ r/2 + υ r ad E ε i r/2, E ε i r + E ε i r ) 2. The proof for part e) is etirely aalogous. For part f), we write u k i u k j Γ ij ) = ϕ p ϕ q ε k i p ε k j q σ 2 δ i+p,j+q ) k= k= p= q=

25 where δ ij is the usual Kroecker delta, i.e., δ ij = if i = j ad otherwise. The stated result ca the be easily obtaied as above by applyig Burkholder s ad Mikowski s iequalities successively. Our asymptotic expasios rely o a strog approximatio of B by B. It ivolves extedig the uderlyig probability space to redefie B, without chagig its distributio, o the same probability space as B, ad provides a explicit rate for the covergece of B to B. The relevat theory will be developed i the lemma give below. Followig our earlier covetio, we will ot distiguish B from its distributioally equivalet copy defied, together with B, i the ewly exteded probability space. We let B = A, Z ), B = A, Z ) 33) where A = W, V, U) ad A = W, V, U ) is defied coformably with A, i.e., W, V ad U are the partial sum processes that a.s. coverge respectively to W, V ad U. Lemma A3 If Assumptio 2. holds with r > 4, the we may choose A ad Z joitly such that { } P sup t A t) At) > c r/4 c r/2 + σ r )K + E ε i r ) for ay c /2+2/r, ad { } P sup Z t) Zt) > c r/4 c r + υ r )K [ + E ε i r ) 2] t for ay c /4. Proof of Lemma A3 The strog approximatio by Courbot 2) for geeral multidimesioal cotiuous time martigales is most directly applicable here, but his result oly provides the covergece rate that is far from optimal ad depeds also upo the dimesioality parameter. Therefore, we will develop a more direct embeddig for the martigale differece sequece ξ i ), ad subsequetly use the strog approximatio by Eimahl 987a, 987b, 989) for the iid radom vectors ε i, δ i, η i ). The first step embeddig for ξ i ) oly itroduces a limit process idepedet of W, ad therefore, does ot iterfere with the secod step embeddig for ε i, δ i, η i ), which are determied solely by W. O the other had, the distributios of ε i, δ i, η i ) fully specify those of ε i, δ i, η i, ξ i ), sice the values of ε i ) completely specify ξ i ). The secod step embeddig would therefore provide the desired strog approximatio for ε i, δ i, η i, ξ i ). Let C t) = x i {T,i t < T i } σ ad defie a cotiuous martigale Z t) = t C s)dw s) 24

26 for T. Notice that Z t) = Z T,i ) for i )/ t < i/. It follows that the quadratic variatio [Z ] of Z is give by where [Z ]t) = R a t) = σ 2 t C s) C s) ds = σ 2 x i x i {t T,i } + Rt) a δ i x i x i {t T,i } + t T,i ){T,i t < T i } for T. Moreover, the quadratic covariatio [W, Z ] of Z with W becomes where [W, Z ]t) = R b t) = σ t C s) ds = σ δ i x i {t T,i } + x i {t T,i } + Rt) b t T,i ){T,i t < T i } for T. We ow embed the cotiuous martigale Z, up to a egligible error, ito a vector Browia motio idepedet of W. Usig the represetatio of the cotiuous martigale W, Z ) as a stochastic itegral with respect to Browia motio [see, e.g., Theorem 3.9 i Revuz ad Yor 994, p75)], we may have Z t) = Γ /2 W t) + R t) 34) where W is a vector Browia motio idepedet of W, ad R is majorized i probability by ) /2 ) /2 sup [Z ]t) tγ + sup [W, Z ]t). t T t T Note that we may use a block lower triagular predictable process to represet W, Z ) as a Browia stochastic itegral. The represetatio 34) for Z is thus possible without chagig W. However, we have { } { P max i { P max i P max i δ i > c r/2 c r/2 K + E ε i r ), } i δ k x k x k > c r/4 c r/2 υ r K } i + δ k )x k > c r/4 c r/2 + υ r )K k= k= [ + E ε i r ) 2], [ + E ε i r ) 2], 25