COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

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1 LIMIT THEORY FOR MODERATE DEVIATIONS FROM A UNIT ROOT By Peter C.B. Phillips ad Tassos Magdalios July 4 COWLES FOUNDATION DISCUSSION PAPER NO. 47 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 88 New Have, Coecticut

2 Limit Theory for Moderate Deviatios from a Uit Root Peter C. B. Phillips Cowles Foudatio for Research i Ecoomics Yale Uiversity ad Uiversity of Aucklad & Uiversity of York, UK Tassos Magdalios Uiversity of York, UK Jue, 4 A first draft of the paper was writte i April, 3. Phillips thaks the NSF for research port uder Grat No.SES Magdalios thaks the EPSRC ad the Oassis Foudatio for scholarship port.

3 Abstract A asymptotic theory is give for autoregressive time series with a root of the form ρ +c/ α, which represets moderate deviatios from uity whe α (, ). The limit theory is obtaied usig a combiatio of a fuctioal law to a diffusio o D[, ) ad a cetral limit law to a scalar ormal variate. For c<, the results provide a (+α)/ rate of covergece ad asymptotic ormality for the first order serial correlatio, partially bridgig the ad covergece rates for the statioary (α )ad covetioal (α ) local to uity cases. For c>, the serial correlatio coefficiet is show to have a α ρ covergece rate ad a Cauchy limit distributio without assumig Gaussia errors, so a ivariace priciple applies whe ρ >. This result liks moderate deviatio asymptotics to earlier results o the explosive autoregressio proved uder Gaussia errors for α, where the covergece rate of the serial correlatio coefficiet is ( + c) ad o ivariace priciple applies. Keywords: Cetral limit theory; Diffusio; Explosive autoregressio, Local to uity; Moderate deviatios, Uit root distributio. AMS 99 subject classificatio: 6M; JEL classificatio: C

4 . Itroductio Regressio asymptotics with roots at or ear uity have played a importat role i time series ecoometrics over the last two decades. The limit theory makes extesive use of fuctioal laws of partial sums to Browia motio, fuctioal laws of weighted partial sums to liear diffusios ad weak covergece of discrete martigales to stochastic itegrals. Almost all this theory ivolves time series with autoregressive roots that are at uity (or o the uit circle) or roots that are local to uity i the sese that they have the form ρ +c/, where is the sample size. I the latter case, the situatio of primary importace occurs whe c<, so that ρ< ad the local asymptotics therefore seek to characterize alteratives to a uit root that lie i the statioary regio. The asymptotic theory turs out to be similar whether c or c<, ad the same rate of covergece i terms of the sample size applies i both cases. These results have bee useful i power evaluatios ad i cofidece iterval costructio. To characterize greater deviatios from uity we ca allow the parameter c to be large ad egative or eve cosider limits as c (Phillips, 987; Cha ad Wei, 988). While such aalysis has proved isightful, it does ot resolve all difficulties of the discotiuities of uit root asymptotics. I particular, it does ot effectively bridge the very differet covergece rates of the statioary ad uit root cases. The preset paper takes aother approach ad provides a asymptotic theory for time series with a autoregressive root of the form ρ +c/ α, where the expoet α lies o (, ). Such roots represet moderate deviatios from uity i the sese that they belog to larger eighborhoods of oe tha covetioal local to uity roots. The boudary value as α icludes the covetioal local to uity case, whereas the boudary value as α icludes the statioary or explosive AR() process, depedig o the sig of c. The limit theory for such time series is developed here usig a combiatio of a fuctioal law to a diffusio ad a cetral limit law. The paper provides limit results for a stadardized versio of such time series, for various sample momets i both the ear-statioary (c <) ad the ear-explosive (c >) cases, ad for the serial correlatio coefficiet. Whe there are ear-statioary moderate deviatios from uity, the cetred first order serial correlatio coefficiet bρ ρ is show to have a (+α)/ rate of covergece ad a limit ormal distributio, bridgig the ad asymptotics of the statioary (α )ad covetioal local to uity (α ) cases. For ear-explosive moderate deviatios from uity, the rate of covergece of bρ ρ is α ρ, which icreases with α from O() whe α to O(( + c) ) whe α, thereby bridgig the asymptotics of local to uity ad explosive autoregressios. A iterestig feature of the moderate deviatio explosive case (c >) is that the limit distributio theory is Cauchy eve for o-gaussia errors. This result differs from covetioal theory for the explosive case where the limit distributio is depedet o the distributio of the errors ad o ivariace priciple applies (Aderso, 959).

5 After these results were obtaied, we leart of some idepedet, related work by Park (3) o weak uit root asymptotics. Park cosiders autoregressive processes with a root that ca be writte i the form ρ m/ where m,. This (weak uit root) setup is aalogous to our formulatio (see () below) of moderate deviatios from uity of the form ρ + c for α (, ). However, the weak α uit root specificatio cosiders oly the statioary side of uity. Usig differet methods ad amog some other results, Park shows a rate of covergece of / m ad asymptotic ormality for the serial correlatio coefficiet i autoregressios with idepedet idetically distributed errors whe + m. Theorem 3.(d) of the m preset paper also establishes asymptotic ormality of the serial correlatio coefficiet with a rate of covergece + α, which correspods to / m, o the statioary side of uity (c <). As discussed above, this paper also provides a limit theory for the explosive side of uity (c >).. The moderate deviatios from uity model Cosider the time series y t ρ y t + u t, t,..., ; ρ + c,α (, ) () α iitialized at some y o p α/ ad where u t is a sequece of idepedet ad idetically distributed (,σ ) radom variables with fiite ν th absolute momet E u t ν < for some ν α. () These coditios suffice (cf. Phillips ad Solo, 99) to esure that partial sums S t P t i u i of u t satisfy the fuctioal law B k ( ) : S bk c k P bk c i u i k B( ) (3) for ay sequece (k ) N icreasig to ifiity, where b c sigifies iteger part ad B( ) is Browia motio with variace σ. A strog approximatio (e.g. Csörgõ ad Horváth, 993) to S t is also possible, accordig to which we ca costruct a expaded probability space with a Browia motio B ( ) for which S i B(i) o a.s. ( ν ) as. (4) i A straightforward calculatio (give i the Appedix) the shows that for each α (, )

6 µ B α (t) B (t) o a.s. as. (5) t [, α ] α ν I what follows, we will assume that the probability space has bee expaded as ecessary i order for (5) to apply. Note that the momet coditio ν i () α esures that o a.s. ³/ α ν o a.s. () i (5). Our approach to developig a limit theory for statistics arisig from model () is to segmet the series ito blocks. Specifically, we write the chroological sequece {t,..., } i blocks of size b α c as follows. Set t b α jc + k for k,..., b α c ad j,..., b α c, sothat y b α jc+k b α jc+k i ³ + c bαjc+k i ³ ui + + c bαjc+k y. α α This arragemet effectively partitios the sample size ito b α c blocks each cotaiig b α c sample poits. Sice the last elemet of each block is asymptotically equivalet to the first elemet of the ext block, it is possible to study the asymptotic behavior of the time series {y t : t,..., } via the asymptotic properties of the time series {y b α jc+k : j,..., b α c, k,...,b α c}. Lettig k b α pc, forsomep [, ], weobtai α/ y b α jc+b α pc α/ + b α jc+b α pc i ³ + c α bαjc+bαpc i ui ³ + c α bαjc+bαpc y α/. The radom elemet y b α jc+b α pc is cetral i the blockig method adopted i this paper. Most statistics of iterest such as the sample variace ad the sample covariace ca be expressed as fuctioals of y b α jc+b α pc, ad it will be coveiet to characterize its asymptotic behavior. We start with the ear statioary case c<. Notig that j + p [, b α c], α/ y b α jc+b α pc ca be writte i terms of the Stieltjes itegral V α (t) : e c(t r) db α (r) b α tc e c α/ α ( αt i) u i i the followig way: for each α (, ) ad c<, y α/ b α tc V α (t) o p (), (6) t [, α ] i 3

7 asshowitheappedix.(5)ad(6)togetherimplythatv α (t) coverges weakly to the liear diffusio J c (t) : e c(t s) db (s) o the Skorohod space D [,M] for every M> ad hece (e.g. Pollard, 984, Theorem V.3) o D [, ). I particular, we have the followig strog approximatio of V α (t) i terms of J c (t).. Lemma. For each α (, ) ad c< µ V α (t) J c (t) o a.s. t [, α ] α ν as (7) o the same probability space that (5) holds. A immediate cosequece of Lemma. ad (6) is that y α/ b α tc J c (t) o p (). (8) t [, α ] Therefore, for the origial radom variables y bα c (rather tha their distributioally equivalet copies for which (5) ad (8) hold) we obtai α/ y b α jc+b α pc Z j+p e c(j+p r) db (r) as (9) for all j,..., b α c ad p [, ]. Result (8) eables us to proceed with a limit theory for the ear statioary case where c<. 3. Limit theory for the ear statioary case This sectio develops the asymptotic properties of the serial correlatio coefficiet ˆρ ρ P t y t u t P t y t () whe ρ + c ad c<. Our approach is to use a segmetatio of the series ito α blocks i which we may utilize the embeddig (8) ad apply law of large umbers ad cetral limit argumets to the deomiator ad umerator of (). We start by cosiderig the sample variace P t y t. Usig Propositio A3 ad the idetity Z yb α jc+b pcdp b α c y α α b jc+k, α k 4

8 thesamplevariacecabewritteas +α yt t b α c b α c µ y +α b α jc+k + O p α j b α c α j b α c α j Z b α c α k b α c α Z k y b α jc+k µ y α/ b α jc+b α pc dp µ α/ y b α rc dr + o p (). By (8) ad Propositio A we obtai Z b α c µ Z b α c α y α/ b α rc dr J c (r) dr Z b α c µ α y α/ b α rc J c (r) dr Z b α c α y α/ b α rc J c (r) y α/ b α rc + J c (r) dr Ã! b α c α r [,b α c] y α/ b α rc J c (r) r [,b α c] y α/ b α rc + J c (r) r µ µ µ J c (r) o p + o r α p o ν α p. ν α ν Hece,thesamplevariacebecomes y +α t Z b α c µ J α c (r) dr + o p. () t α ν At this poit, it is coveiet to approximate the Orstei-Uhlebeck process J c (t) by its statioary versio J c (t) :e ct J c () + e c(t s) db (s) e ct J c () + J c (t), ³ where Jc () is a radom variable idepedet of B (.) that follows a N, σ distributio. ItiswellkowthatJc (t) is a strictly statioary process with autocovariace c fuctio give by γ J c (h) σ c ec h h Z. 5

9 Moreover, the followig approximatio of J c by Jc is established i the Appedix Z b α c J α c (r) dr Z b α c J α c (r) dr + O p ( α). () Combiig () ad () the sample variace i the ear-statioary case becomes +α yt t α α Z b α c b α c j σ c + o p µ Jc (r) dr + o p α ν Z j+ j µ α ν J c (r) dr + o p µ α ν by the weak law of large umbers for statioary processes, sice γ J c () σ / c. The limitig distributio of the sample covariace ca be obtaied by usig the fact that, as i the case of statioary asymptotics, the stadardized sample variace has a costat (o radom) probability limit. Defiig ξ t +α y t u t, (ξ t ) t N is amartigaledifferece sequece with respect to the filtratio F t σ (y,u,...,u t ). The coditioal variace of the martigale P t ξ t is give by t E Ft ξ t +α t σ +α E Ft y t u t t y t σ4 c + o p +α µ yt E Ft u t t α ν by (3), sice y t is F t measurable. By virtue of the Lideberg coditio t (3) E Ft ξ t { ξ t >η} o p (), η > (4) established i the Appedix, the martigale cetral limit theorem (e.g. (984), Theorem VIII.) yields +α y t u t N t µ, Pollard σ 4. (5) c Fially, the asymptotic distributio of the cetred least squares estimator bρ ρ P t y t u t / P t y t ca be derived by combiig (3) ad (5): + α (bρ ρ ) N (, c) as. We collect these results together as follows. 6

10 3. Theorem. For model () with ρ +c/ a, c< ad α (, ), the followig limits apply as : (a) α/ y b α jc+b α pc R j+p e c(j+p r) db (r), (b) α P t y t p σ c, (c) α P t y t u t N ³, σ4, c (d) + α (bρ ρ ) N (, c), where B is Browia motio with variace σ. 3. Remarks (i) Whe there are moderate deviatios from uity, the proofs above reveal that both a fuctioal law to a diffusio (part (a)) ad cetral limit theory (parts (b), (c) ad (d)) play a role i the derivatio of the results. The fuctioal law provides i each case a limitig subsidiary process whose elemets form the compoets that upo further summatio satisfy a law of large umbers ad a cetral limit law. While there is oly oe limitig process ivolved as, it is coveiet to thik of the fuctioal law operatig withi blocks of legth b α c ad the law of large umbers ad cetral limit laws operatig across the b α c blocks. The momet coditio i () esures the validity of the embeddig argumet that makes this segmetatio rigorous as. (ii) Results (b), (c) ad (d) match the stadard statioary limit theory for fixed ρ <. I particular, P t y σ t p, ρ P ³ t y σ t u t N, 4 (ˆρ ρ) N (, ρ ). ρ A heuristic argumet for the correspodece is that upo replacig ρ by +c/ a i each of the above results, a simple rescalig of the first order approximatio delivers (b)-(d) of Theorem 3.. Thus, for the serial correlatio coefficiet ˆρ, substitutig ρ c [ + o ()] ito the limit distributio of α (ˆρ ρ) gives the asymptotic approximatio µ (ˆρ ρ) d N, c α just as i part (d) of the theorem. 7 or + α (ˆρ ρ) d N (, c),,

11 4. Limit theory for the ear explosive case This sectio cosiders the limit behavior of the serial correlatio coefficiet ˆρ ρ whe ρ +c/ a ad c>. I this case the weak covergece of V α ³ (t) to J c (t) still holds o D [, ). However, the radom elemet J c (t) N, σ c (ect ) is o loger bouded i probability as t. For t [, α ], a further ormalizatio of O (exp { c α }) is eeded as to achieve a weak limit for V α (t). Itturs out that a similar ormalizatio is eeded for α/ y b α tc, amelyρ. For otatioal coveiece i what follows we defie κ α b α c ad q α b α c [, ). Two useful approximatio results for the ear explosive case follow. 4. Lemma. For each α (, ) ad c> µ ρ α s db α (s) e cs db (s) o p t [, α ] α ν as o the same probability space that (5) holds. 4. Lemma. For each α (, ) ad c> µ ρ (bα tc b α sc) db α (s) J c (t) o p t [, α ] α ν as o the same probability space that (5) holds. For the sample variace, ote first that, ulike the ear-statioary case, the limit theory is ot determied exclusively from the blocks {y b α jc+k : j,..., b α c, k,..., b α c}. Usig (3) i the Appedix, we ca write the sample variace as ρ κ α t yt ρ κ α b α c j b α c k yb α jc+k + ρ κ α tbκ c µ yt + O p. (6) α We deote by U ad U the first ad secod term o the right side of (6) respectively. Sice U is almost surely positive with limitig expectatio σ (e cq ) > 4c whe q>, we coclude that it cotributes to the limit theory wheever α is ot a iteger. We will aalyze each of the two terms o the right of (6) separately. The term 8

12 cotaiig the block compoets ca be writte as U ρ κ ρ κ ρ κ ρ κ b α c j b α c j b α c α Z k y b α jc+k + o p () µ y α/ b α jc+b α pc dp Z b α c µ y α/ b α rc dr + o p () Z b α c µz r ρ bα rc α s db α (s) dr + o p (). Takig the ier itegral alog [,r][, b α c] \ [r, b α c] we have, up to o p (), à Z! b α c Z b α c U ρ α s db α (s) ρ κ ρ bα rc dr + R, (7) where the remaider term R isshowitheappedixtobeo p (). The secod itegral o the right side of (7) ca be evaluated directly to obtai Z b α c ρ bα rc dr ρκ ( + o ()) as, (8) c as show i the Appedix. Usig Lemma 4., (7) becomes à Z! U b α c µ e cs db (s) + o p c α ν µz µ e cs db (s) + o p (9) c α ν o the same probability space that (5) holds. For the secod term o the right of (6), otig that b κ c b α qc, q [, ), we obtai U ρ κ α ρ κ α ρ κ α Z q bκ c i b α qc i Z q y i+bκ c µ yi+bκ c + O p α ybκ c+b pcdp ρ κ α α µ ρ κ µ y α/ bκ c+b α pc dp + O p 9 µ q bα qc α α y bκ c+b α qc. ()

13 Now for each p [,q], q [, ), we ca show (the details are icluded i the Appedix) that ρ κ Z µ y α/ bκ c+b α pc e cp e cs dw (s)+o p () α ν o the same probability space that (5) holds. Thus, applyig the domiated covergece theorem to () yields µz Z q µ U e cs dw (s) e cp dp + o p α ν µz e cs dw (s) e cq µ + o p c Lettig : R ³ e cs db (s) N the asymptotic equivalece ρ κ ρ α t, σ c e cq ρ α ν. (), we coclude from (6), (9), () ad [ + o ()] that y t c + o p µ α ν o the same probability space that(5)holds. This implies that the limitig distributio of the sample variace is give by, ρ α yt c (3) t o the origial space. As i the case of the sample variace, the asymptotic behavior of the sample covariace is partly determied by elemets of the time series y t u t that do ot belog to the block compoets y b α jc+k u b α jc+k : j,..., b α c,k,..., b α c ª. Obtaiig limits for the block compoets ad the remaiig time series separately i a method similar to that used for the sample variace will work. It is, however, more efficiet to derive the limitig distributio of the sample covariace by usig a P direct argumet o ρ α t y t u t.

14 Usig the iitial coditio y o p α/ ad (9) i the Appedix we ca write ρ α t y t u t ρ α ρ α ρ ρ ρ t µ ρ y t u t+ + o p α/ b α ( α t Z α α Z α α Z α α α )c y t u t+ µ y α/ b α rcdb α r + α y α/ b α (r )cdb α α (r) Z r a ρ bα rc α s db a (s) db α (r)+o p (). Takig the ier itegral alog,r [, α ] \ r, α we obtai, up a a to o p (), ρ α t Z α y t u t ρ where the remaider term I : ρ Z α ρ α s db a (s) ρ ( bαrc) db a (r) I, (4) α Z α Z α α r a is show i the Appedix to be o p (). Now,sice R α Lemma 4. implies that Z α α ρ bαrc αs db a (s) db a (r) ρ ( bα rc) ρ ( bαrc) db a (r) J c α + o p µ α ν ³ db a (r) O ρ p, α/ Sice J c (t) is a L -bouded martigale o [, ), the martigale covergece theorem esures the existece of a almost surely fiite radom variable Y such that Sice J c ( α ) N ³, σ c R e cs db (s) as i (3), (4) yields ρ α t J c α a.s. Y as. ³ c α e,itisclearthaty N y t u t Y + o p µ α ν as. ³, σ c.thus,if

15 o the same probability space that (5) holds. The latter strog approximatio implies that the asymptotic distributio of the sample covariace is give i the origial space by ρ α y t u t Y t,y N µ, σ. (5) c The asymptotic behavior of the serial correlatio coefficiet i the ear explosive case is a easy cosequece of (3), (5) ad the fact that the limitig radom variables ad Y are idepedet. 4.3 Theorem. For model () with ρ +c/ a, c> ad α (, ) α ρ c (ˆρ ρ ) C as (6) where C is a stadard Cauchy variate. 4.4 Remarks (i) Theorem 4.3 relates to earlier work (White, 958; Aderso, 959; Basawa ad Brockwell, 984) o the explosive Gaussia AR() process. For a Gaussia first order autoregressive process with fixed ρ > ad y, White showed that ρ ρ (ˆρ ρ) C as. (7) Replacig ρ by ρ + c/ a, we obtai ρ c [ + o ()]. Hece, α the ormalizatios i Theorem 4.3 ad (7) are asymptotically equivalet as ρ. Aderso (959) showed that (ˆρ ρ ρ) has a limit distributio that depeds o the distributio of the errors u t whe ρ > ad that o cetral limit theory or ivariace priciple is applicable. (ii) The limit theory derived i this sectio for the moderate deviatios case is ot restricted to Gaussia processes. I particular, the Cauchy limit result (6) applies for ρ +c/ a ad iovatios u t satisfyig () with α>, which icludes a much wider class of processes. At the boudary where α, Theorem 4.3 reduces to (7) with ρ +c, ad the the errors u t have ifiitely may momets as uder Gaussiaity. (iii) The limit theory for ear explosive moderate deviatios from uity is ivariat to the iitial coditio y beig ay fixed costat value or radom variable of smaller asymptotic order tha α/. This property is ot shared by explosive autoregressios where y does ifluece the limit theory, as show by Aderso (959).

16 5. Discussio The covergece rates of Theorem 3. bridge those for uit root or local to uity processes ad those that apply uder statioarity. Thus, i part (d) the covergece rate + α rages over /, for α (, ). However, the bridgig asymptotics are ot cotiuous at the boudaries of α. For example, whe α, part(d)becomes (ˆρ ρ ) N (, c), whereas the correct statioary result whe ρ +c is (bρ ρ) N (, c c ). Thus, part (d) as it stads overestimates the variace of bρ itheboudarycasewhereα. Cotiuityatthisboudarycabe achieved (for parts (b)-(d)) through replacemet of c by c + c / α, without affectig the asymptotic results for α>. For the limit as α, we have α, ad so b α c for α,iwhichcasej ecessarily ad part (a) becomes / y bpc J c (p), the usual local to uity limit result (cf. Phillips, 987). I that case, part (d) is replaced by the o-ormal limit (bρ ρ ) Z J c (q) db (q) / Z J c (q) dq. (8) Similarly, whe c >, the covergece rate of Theorem 4.3 takes values o (, ( + c) ) as α rages from to. Sice +c is the autoregressive root of a explosive AR() process whe α, there is a discotiuity due to the discrepacy betwee ρ c + O ( α ) whe α (, ) ad ρ c + c whe α. α As i the ear statioary case, cotiuity ca be achieved through replacemet of c by c + c / α without affectig Theorem 4.3. However, whe α, the blockig scheme is such that j ad agai the local to uity limit theory (8) applies. Thus, cotiuity is achieved at the outside boudaries with the statioary ad explosive case asymptotics, but ot at the iside boudaries with the covetioal local to uity asymptotics. 6. Notatio b c iteger part of : defiitioal equality B (t) Browia motio with variace σ t J c (t) Orstei-Uhlebeck process [] t quadratic variatio process of t κ : α b α c q : α b α c E F ( ) coditioal expectatio E ( F) P F ( ) coditioal probability P ( F) a.s. almost sure covergece p covergece i probability L p covergece i L p orm weak covergece distributioal equivalece d asymptotically distributed as o p () teds to zero i probability o a.s. () teds to zero almost surely { } idicator fuctio 3

17 7. Techical appedix ad proofs Propositio A. For each x [,M], M>, possibly depedig o, adreal valued, measurable fuctio f o [, ) bx α c f α/ i µ i u α i Z x f (r) db α (r). Proof. It is coveiet to reduce the iterval from [,M] to [, ]. If x [,M], the y : x [, ] ad m M : M α,sowecawrite sice bx α c f α/ i µ i u α i bym α c f α/ i M (M α ) / M Z x m / Z y bym c i µ i u α i bym α c f i µ f M i M α µ M i u i m M f (Ms) db m (s) Z x h ³ r i f (r) d MBm M f (r) db α (r) u i MBm ³ r M r M bm c M u m / i M ( α M) / α b α rc i i b α M r M c u i B α (r). i u i The followig itegral represetatio o [,M] is a immediate corollary of Propositio A: bx α c f α/ i µ i u α i+m Z x f (r) db α ³r + m α, m N. (9) Propositio A. For c<, t> J c (t) < a.s. 4

18 Proof. Sice t> J c (t) > a.s., it is eough to show that µ Z µ E J c (t) P J c (t) x dx <. t> Defie τ σ.sice[j c c] τa.s., µ µ P J c (t) x P J c (t) x, [J c ] τ t> t> µ P J c (t) x, [J c ] τ e x τ by Berstei s iequality (cf. Revuz ad Yor, 999, p.53 Exercise 3.6). Thus, µ Z r E J c (t) e x π τ dx τ< t> which completes the proof. Propositio A3. For each α (, ) ad c< x +α t b α c b α c µ x +α b α jc+k + O p α t j t> t> k as, o the same probability space that (5) holds. Proof. Deotig κ α b α c,otethat {yb α jc+k : j,..., α, k,..., b α c} {yt : t,..., }, where the maximal subscript b(b α c ) α c + b α c of the block compoets o the left takes values bκ c α α + b α c bκ c. (3) Also, (8) ad Propositio A give y k k yb α rc yb α rc α/ r [, α ] α/ r [, α ] J α/ c (r) + J c (r) O p () r> o the same probability space that (5) holds. Thus, the remaider term E of the propositio is give by E y +α t y +α t tb(b α c ) α c+b α c+ tbκ c µ µ y t bκ c O t α/ p, α which shows the propositio, sice E a.s.. 5

19 Proofof(5). For each α (, ) we have B α (t) B (t) t [, α ] α S b α tc B ( α t) t [, α ] α S b α tc B (b α tc) + α t [, α ] B ( α t) B (b α tc) t [, α ] α S i B (i) + α B (t) B (btc). (3) i t [,] For ay β, the Hölder cotiuity property of Browia motio sample paths (e.g., Revuz ad Yor, Theorem.) gives B (t) B (btc) t [,] t [,] t btc (, ) B (t) B (btc) t [,] t btc (, ) B (t) B (btc) t btc β < a.s., (3) so the secod term o the right side of (3) is of order O a.s. α. (3), (3) ad (4) give the stated result. Proof of (6). Write ρ +c/ α ad ote that y α/ t> ρ bα tc o p () sice t> ρ bα tc <. The t [, α ] y α/ b α tc V α (t) bt α c ρ btα c i t [, α ] α/ u i V α (t) i + o p () bt α c ρ bt α c i t [, α ] α/ e ª c α (bt α c i) u + o p () (33) i i Now bt α c ρ bt α c i t [, α ] α/ e ª c α (bt α c i) u i i m ρ m i e ª c α (m i) u o i p (), (34) m α/ i 6

20 sice Kolmogorov s iequality shows that for arbitrary η> à m P ρ m i α/ e ª! c α (m i) u >η i σ ρ i η α m i i e c α ( i) ª σ h c η α 6 + O α i, as, the last lie followig from direct calculatio of the sums. Combiig (33) ad (34) delivers the required result. Proof of Lemma.. we obtai Usig the itegratio by parts formula for Stieltjes itegrals t [, α ] t [, α ] t [, α ] V α (t) J c (t) e c(t r) db α (s) e c(t r) db (s) B α (t)+c e c(t s) B α (s) ds B (t) c B α (t) B (t) + c t [, α ] t [, α ] B α (t) B (t) t [, α ] + c à t [, α ] s [,t]! B α (s) B (s) t [, α ] e c(t s) B (s) ds e c(t s) B α (s) B (s) ds e c(t s) ds B α (t) B (t) + c B α (t) B (t) t [, α ] t [, α ] c e ct t B α (t) B (t) t [, α ] µ o a.s. α ν by the strog ivariace priciple (5). 7

21 Proofof(). Squarig Jc (r) J c (r)+e cr Jc () we obtai Z b α c J c (r) Jc (r) Z b α c dr e cr Jc () +Jc () e cr J c (r) dp by Propositio A. Thus, Z b α c Jc (r) J α c (r) ª dr t Z b Jc () α c Z b α c e cr dp + Jc () e cr J c (r) dp µ Z b Jc () α c + Jc () J c (r) e cr dp r < a.s. Z b α c J α c (r) Jc (r) dr O a.s. ( α). P Proofof(4). First, ote that sice +α i y i p σ / c ad E Ft ξ t { ξ t >η} o y t E +α Ft ³u t y t u t >η +α t ³ max E F t u t t y t u t >η +α o +α i y i, (4) holds if ³ o max E F t u t y t u t >η +α p. (35) t for each η>. To show (35), recall from the momet coditio () that the i.i.d. sequece (u t ) t N satisfies E u t ν < for some ν> >. Usig the Chebyshev α iequality ad the Hölder iequality with r ν >, r ν >, sothatr ν + r,weobtai o E Ft ³u t y t u t >η +α E Ft u t r r ³E Ft y t u t >η +α E Ft u t ν o ν ³P Ft y t u t >η +α ν ν (E u ν ) ν (E u ν ) ν (E u ν ) ν 8 ³P Ft " EFt µ σ η y t u t >η +α y t u t η +α ν ν # ν ν µ ν y ν t. +α o ν ν o r

22 Therefore, lettig C : (E u ν ) ν ³ ν σ ν η < we coclude that ³ o max E F t u t y t u t >η +α t C µ y C max t t +α " µ max t ν ν yt α/ # ν ν o p (), sice max yt t α/ s [, α ] s [, α ] y α/ b α sc y α/ b α sc J c (s) + J c (s) O p () s> by (8) ad Propositio A. ProofofLemma4.. t [, α ] t [, α ] + t [, α ] t [, α ] ρ α s db α (s) ρ α s db α (s) e cs db α (s) ρ α s db α (s) e cs db (s) e cs db α (s) e cs db (s) µ e cs db α (s) + o a.s. α ν by a argumet idetical to that used i the proof of Lemma.. So it is eough to show that ρ α s db α (s) e cs db α (s) o p (). (36) t [, α ] Sice bt ρ α s db α (s) e cs db α (s) α c ρ i t [, α ] t [, α ] α e ª c α i u i i m ρ i e ª c α i u, i m α i 9

23 Kolmogorov s iequality gives for ay ε> Ã m P ρ i e ª! c α i u >ε i m α i ε α σ ε α i i σ h c ε α ρ i ρ i e c α i ª E u i e c α i ª 6 + O α i as, agai by direct calculatio of the sums. It follows that m ρ i m α/ i e c α i ª u i o p (), (37) ad (36) ad the Lemma follow directly. ProofofLemma4.. The argumet is similar to that used for Lemma 4.. ρ (bα tc b α sc) db α (s) J c (t) t [, α ] ρ (bα tc b α sc) db α (s) e c(t s) db α (s) t [, α ] + e c(t s) db α (s) J c (t) t [, α ] µ ρ (bα tc b α sc) db α (s) e c(t s) db α (s) + o a.s., t [, α ] by Lemma. with c <. Sice ρ (b α tc b α sc) e c(t s)ª db α (s) α/ it is eough to show that bt α c ρ (b α tc i) t [, α ] α/ i bt α c ρ (b α tc i) t [, α ] α/ i m ρ (m i) m α/ i bt α c i ρ (b α tc i) e c α ( α t i) ª u i α ν e c α ( α t i) ª u i, e c α (b α tc i) ª u i + o p () e c α (m i) ª u i o p (), whichisprovedithesamewayas(37).

24 Proofof(8). Let s r ad sice b α c ρbκ c [+o ()] ρ κ Z b α c ρ bα rc dr α Z α ( Z κ bκ c α i ρ (i ) ρ bκ sc ds Z + κ κ Ã O! ρ bκ c α Z bκc κ bκc κ ρκ c we obtai Z ) + ρ bκ sc ds bκc κ [ + o ()]. Proofof(). Propositio A ad Lemma 4. give for each p [,q] ρ κ y α/ bκ c+b α pc ρ κ α/ bκ c+b α pc i ρ bκc+bα pc i u i + y pc α/ ρbα bκ + α pc ρ bα pc i α/ u i + o p () i b α (b α c+p)c ρ bα pc i α/ u i ρ bα pc e cp Z i Z b α c+p ρ α s db a (s) e cs db (s)+o p µ α ν o the probability space that (5) holds. Order of ρ κ. ³ log ρ κ κ log κ µ c α + O + c +log µ α +log α c α ( + o ()) + log c α +o () log c α c α ( + o ()), sice log o δ,forallδ>. Thus, ρ κ exp{ c α ( + o ())} o () ad ρ κ o as. (38)

25 Proof of asymptotic egligibility of R. Write R R R,where Z à b α c Z b α c R ρ κ ρ α (s r) db α (s)! dr where R ρ κ à Z b α c Z à b α c Z b α c à Z b α c Z b α c R : ρ κ From Propositio A R b α c r E bκ c b α rc ρ i α/ i u b α rc+i r r ρ α (s r) db α (s) ρ α (s r) db α (s) Z b α c r! ρ α (s r) db α (s)! ρ α (s r) db α (s) R,! dr ρ α (s r) db α (s) dr. σ bκ c b α rc α ρ σ + c P bκc b α rc α/ i ρ i i α ρ ρ i (bκ c b α rc) c + c α u b α rc+i, ad σ c ( + o ()) ρ (bκ c b α rc) O (), uiformly i r [, b α c] because ³ ρ (bκc bα rc) {+c/ α } ( α b α c b α rc) O e c (b c r) α O (). P Thus, bκ c b α rc α/ i ρ i u b α rc+i O p () uiformly i r b α c ad so Z b α c ρ α (s r) db α (s) O p (), uiformly i r α. r The, usig (38), we fid Z b α c R ρ κ O p () O O p ρ κ à à Z b α c r ρ κ Z! b α c dr µ α o p, α ρ α (s r) db α (s)! dr

26 ad Z b α c Z b α c R ρ κ r Z b O p Ãρ α c κ dr! ρ α (s r) µ o p, α db α (s) dr so that à Z b α c R ρ α (s r) db α (s)! µ R o α p, givig the required results. Proof of asymptotic egligibility of I. From Propositio A we have Z α r a ad so I ca be writte as Z α I ρ a ρ ρ ρ barc as db a (s) α/ Z α α Z α a j Usig the Cauchy-Schwarz iequality, E I ρ ρ α α ρ σ α r a α/ ib α rc ij j j j ib α rc ρ ba rc i u i, ρ barc as db a (s) db a (r) ρ j i u i u j. ij ij ij O ρ o (), ρ ba rc i u i db a (r) ρ j i E u i u j ρ j i Eu / i Eu / j ad so I o p () as required. ρ j i 3

27 Proof of Theorem 4.3. Havig established (3) ad (5), the oly thig that remais to be proved is that the zero mea Gaussia radom variables ad Y are idepedet, or equivaletly, that E (Y ). First, otethat Z α E e cs db (s) J à Z! α c α E e cs db (s) E J c α ³ c α σ e <. c R α Sice lim e cs db (s) a.s., Y lim J c ( α ) a.s. the domiated covergece theorem yields à Z α E (Y ) lim E e cs db (s) J! c α à Z α Z! α lim e c α E e cs db (s) e cr db (r) σ lim e c α Z α dr σ lim e c α α. Hece, ad Y are idepedet. 8. Refereces Aderso, T.W. (959), O Asymptotic Distributios of Estimates of Parameters of Stochastic Differece Equatios, Aals of Mathematical Statistics, 3, Basawa, I.V. ad P.J. Brockwell (984). Asymptotic coditioal iferece for regular oergodic models with a applicatio to autoregressive processes, Aals of Statistics, 6 7. Cha, N.H. ad C.Z. Wei (987). Asymptotic iferece for early ostatioary AR() processes, Aals of Statistics 5, Csörgő, M. ad L. Horváth (993). Weighted Approximatios i Probability ad Statistics. New York: Wiley. Park, J. Y. (3). Weak uit roots. Mimeographed, Departmet of Ecoomics, Rice Uiversity. Phillips, P. C. B. (987). Towards a uified asymptotic theory for autoregressio, Biometrika 74,

28 Phillips, P. C. B. ad V. Solo (99), Asymptotics for Liear Processes, Aals of Statistics, 97. Pollard, D. (984). Covergece of Stochastic Processes. Spriger-Verlag. Revuz, D. ad M. Yor (999). Spriger. Cotiuous Martigales ad Browia Motio. White, J. S. (958). The limitig distributio of the serial correlatio coefficiet i the explosive case, Aals of Mathematical Statistics 9,