with a unit root is discussed. We propose a modication of the Block Bootstrap which

Transcription

1 The Continuous-Path Bock-Bootstrap E. PAPARODITIS and D. N. POLITIS University of Cyprus and University of Caifornia, San Diego Abstract - The situation where the avaiabe data arise from a genera inear process with a unit root is discussed. We propose a modication of the Bock Bootstrap which generates repicates of the origina data and which correcty imitates the unit root behavior and the weak dependence structure of the observed series. Vaidity of the proposed method for estimating the unit root distribution is shown. Research supported by NSF Grant DMS and by a University of Cyprus Research Grant.. INTRODUCTION Consider time series data f();();:::;(n)g arising from the process (t) = (t, ) + U(t); () for t =;;:::, where () = 0, is a constant in[,;], and Assumption A fu(t);t g is a stochastic process satisfying U(t) = j"(t, j) () where 0 =, P jj j j <, P j 6= 0 and f"(t);t Zg is a sequence of independent identicay distributed random variabes with mean zero and 0 <E(" ()) <. We wi be especiay concerned with the nonstationary (integrated) case where =. Note that if j = 0 for j>we are in the case of a random wak, i.e., () aows for a wide range of weak dependence of the dierenced process (t), (t, ). A number of papers in the econometrics iterature has deat with mode (); see e.g. Hamiton (994) or Fuer (996) and the references therein. The traditiona approach so far has been based on the Dickey and Fuer (979) pioneering work and consists of conducting a test of the nu hypothesis that

2 E. Paparoditis and D. N. Poitis there is a unit root; in this connection see aso Phiips and Perron (988), and Ferretti and Romo (996). Recenty however, there has been some interest in the attempt to go beyond the simpe unit root test. Stock (99) managed to deveop condence intervas for in the equation (t) =(t,)+u(t) based on `oca-to-unity' asymptotics. Hansen (997) proposed the `grid-bootstrap' to adress this situation, and reports improved performance. Finay, Romano and Wof (998) appied the genera subsamping methodoogy of Poitis and Romano (994) to the AR() mode with good resuts; see Poitis et a. (999) for more detais. In the paper at hand, we present a dierent approach towards inference under the presence of a unit root; our approach is based on a modication of the Bock-Bootstrap (BB) of Kunsch (989), and for reasons to be apparent shorty is termed \Continuous-Path Bock-Bootstrap (CBB)". To motivate the CBB, et us give an iustration demonstrating the faiure of the BB under the presence of a unit root. Figure (a) shows a pot of (the natura ogarithm of) the S&P 500 stock series index recorded annuay from year 87 to year 988, whie Figure (b) shows a reaization of a BB pseudo repication of the S&P 500 series using bock size 0. It is obvious visuay that the bootstrap series is quite dissimiar to the origina series, the most striking dierence being the presence of strong discontinuities (of the `jump' type) in the bootstrap series that {not surprisingy- occur every 0 time units, i.e., where the independent bootstrap bocks join. Figure (c) suggests a way to x this probem by forcing the bootstrap sampe path to be continuous. A simpe way to do this is to shift each ofthe bootstrap bocks up or down with the goa of ensuring (i) the bootstrap series starts o at the same point as the origina series, and that (ii) the bootstrap sampe path is continuous. Notaby, the bootstrap bocks used in Figure (c) are the exact same bocks featuring in Figure (b). At east as far as visua inspection of the pot can discern, the series in Figure (c) coud just as we have been generated by the same probabiity mechanism that generated the origina S&P 500 series. In other words, it is pausibe that a bootstrap agorithm generating series such as the one in Figure (c) woud be successfu in mimicking important features of the origina process; thus, the \Continuous-Path Bock-Bootstrap" of Figure (c) is expected to `work' in this case. Of course, the actua yeary S&P 500 data are in discrete time, and taking about continuity is -stricty speaking- inappropriate. Nevertheess, an underying continuous-time mode may aways be thought to exist, and the idea of continuity of sampe paths is powerfu and intuitive; hence the name \Continuous- Path Bock-Bootstrap" (CBB for short) for our discrete-time methodoogy as we. The CBB is described in detai in the next Section, and some of its key properties are proven.

3 The CBB for Nonstationary Processes 3 a b c Figure : Pot of the natura ogarithm of the S&P500 stock index series (a), of a BB reaization (b) and of a CBB reaization (c) with bocksize 0.. THE CONTINUOUS-PATH BLOCK-BOOTSTRAP (CBB) Before introducing the Continuous-Path Bock-Bootstrap (CBB) Method we review Kunsch's (989) Bock-Bootstrap (BB). The BB agorithm is carried out conditionay on the origina data f();();:::;(n)g, and thus impicity denes a bootstrap probabiity mechanism denoted by P? that is capabe of generating bootstrap pseudo-series of the type f? (t);t=;;:::g. Bock-Bootstrap (BB) agorithm:. First chose a positive integer b(< n), and et i 0 ;i ;:::;i be drawn i.i.d. with distribution uniform on the set f; ;:::;n,b+g; here we take k =[n=b], where [] denotes the integer part, athough dierent choices for k are aso possibe. The BB constructs a bootstrap pseudoseries? ();? ();:::;? (), where = k as foows.. For m =0;;:::;, et? (mb + j):=(i m +j,) for j =;;:::;b.

4 4 E. Paparoditis and D. N. Poitis The Continuous-Path Bock-Bootstrap (CBB) agorithm is now dened in the foowing three steps beow. As before, the agorithm is carried out conditionay on the origina data f();();:::;(n)g, and impicity denes a bootstrap probabiity mechanism denoted by P that is capabe of generating bootstrap pseudo-series of the type f (t);t =;;:::g. In the foowing we denote quantities with respect to P with an asterisk. Continuous-Path bock-bootstrap (CBB) agorithm:. First cacuate the centered residuas bu(t) =(t),(t,), n, n ((t), (t, )) for t =;3;:::;n. Attention now focuses on the new variabes (t) e dened as foows: e(t) = 8 < : () for t = () + P t j= b U(j) for t =;3;:::;n.. Chose a positive integer b(< n), and et i 0 ;i ;:::;i be drawn i.i.d. with distribution uniform on the set f; ;:::; n, bg; here, we take k = [n=b] as before. The CBB constructs a bootstrap pseudo-series ();:::; (), where = k as foows. 3. Construction of the rst bootstrap bock. Let for j =;:::;b:to eaborate: (j):=() + [ e (i0 + j, ), e (i0 )] () := () () := () + [ e (i0 +), e (i0 )] (3) := () + [ e (i0 +), e (i0 )]. (b) := ()+[e (i0 +), e (i0 )]: 4. Construction of the (m + )-th bootstrap bock from the m-th bock for m =;:::;. Let (mb + j) := (mb)+[e (im +j), e (im )]

5 The CBB for Nonstationary Processes 5 for j =;:::;b:to eaborate: (mb +):= (mb)+[e (im +), (im e )] (mb +):= (mb)+[e (im +), (im e )]. (mb + b) := (mb)+[e (im +b), (im e )]: An intuitive way to understand the CBB construction is based on the discussion regarding Figure (c) in the Introduction and goes as foows: (i) construct a BB pseudo-series f (t);t = ;;:::g based on bocks of size equa to b + from the series e (t); (ii) shift the rst bock (of size b +)by an amount seected such that the bootstrap series starts o at the same point as the origina series; (iii) shift the second BB bock (of size b +)by another amount seected such that the rst observation of this new bootstrap bock matches exacty the ast observation of the previous bootstrap bock; (iv) join the two bocks but deete the ast observation of the previous bootstrap bock from the bootstrap series; (v) repeat parts (iii) and (iv) unti a the generated BB bocks are used up. Note that the CBB is appied to f e (t)g and not to f(t)g. The reason is that athough the (t) series is produced via the zero mean innovations U(t), the observed nite-sampe reaization of the innovations wi ikey have nonzero (sampe) mean; this discrepancy has an important eect on the bootstrap distribution eectivey eading to a random wak with drift in the bootstrap word. Fortunatey, there is an easy x-up by recentering the innovations; a simiar necessity for residua centering has been recommended eary on even in reguar inear regression see Freedman (98). Note that a CBB series using bock size b is associated to a BB construction with bock size b +. This phenomenon is ony due to the fact that we are deaing with discrete-time processes; it woud not occur in a continuous-time setting. The reason for this is our step (iv) above: athough we are eecting the matching of the rst observation of a new bootstrap bock to the ast observation of the previous bootstrap bock, it does not seem advisabe to eave both occurrences of this common (matched) vaue to exist side-by-side; one of the two must be deeted as step (iv) suggests. 3. ESTIMATION OF THE UNIT ROOT DISTRIBUTION In this section the properties of the CBB in estimating the distribution of the rst order autoregressive coecient in the presence of a unit root are considered. Reca that based on the observations f();();:::;(n)g a

6 6 E. Paparoditis and D. N. Poitis common estimator of the rst order autoregressive coecient in () is given by ^ = The CBB version of is given by ^ = T (t)(t, ) n (t, ) (t) (t, ) (t, ) : (3) and the distribution of the statistic (^,) is used to estimate the distribution of n(^, ) Consider rst the basic random wak case. For this case the foowing resut can be estabished. Theorem Let (t) =(t,) + "(t), t =;;:::where (0) = 0, "(t) IID(0; ) and E(" 4 ()) <. Ifb!as n!such that b= p n! 0 then in probabiity. sup xr P (^, ) x, P n(^, ) x! 0 (4) The asymptotic vaidity of the CBB for the genera case where the stationary process fu(t)g satises Assumption A is estabished in the foowing theorem which is our main resut. Theorem Let (t) =(t,) + U(t), t =;;::: where fu(t);t Zg satises Assumption A and E(" 4 ()) <. If b!as n!such that b= p n! 0 then in probabiity. sup xr P (^, ) x, P n(^, ) x! 0 It is noteworthy that, under the assumptions of Theorem, the asymptotic distribution of ^ has a compicated form, depending on many unknown

7 The CBB for Nonstationary Processes 7 parameters such as the innite sum P j ; see Hamiton (994) or Fuer (996). The CBB eortessy achieves the required distribution estimation, and provides an attractive aternative as compared to the asymptotic distribution with estimated parameters. Notaby, estimation of the sum P j is tantamount to estimating the spectra density of the dierenced series (evauated at the origin) which is a highy nontrivia probem. 4. PROOFS Proof of Theorem : Reca that b U(im + s) = e (im +s), e (im +s,). It is easiy seen that for t =;3;:::; (t)=() + [(t,)=b] B bu(i m + s) (5) where B = minfb, 0;m ;t,mb, 0;m g, i;j is Kronecker's deta, i.e., i;j = if i = j and zero ese. Aternativey, we can write (t) = 8 < : () for t = (t,) + b U(im + s) for t =;3;:::; where m =[(t,)=b] and s = t, mb, 0;m. Now, assume without oss of generaity that =. Furthermore, set bu(i m + s) e(i m + s) and note that in the random wak case considered here we haveby the denition of U(i m + s) that e(i m + s) = "(i m + s), n, "(t): (7) n, By the centering of the b U(t)'s we have E (e(i m +s)) = Substituting expression (6) we get (^, ) = n, b n,b t= = O P (b = n, ): "(t + s)+ n, n (t), (t, ) (t, ) (t, ) "(t) (6)

8 8 E. Paparoditis and D. N. Poitis =, + (t, ), b m= h Appying (5) we get after some simpe agebra that Thus h e(i 0 + s) (s)+ b m= = "()h e(i 0 + s)+ +, e(i 0 + s)+ e (i 0 + s)+ = h "() e(i 0 + s)+, (^, ) = + p, +, e(i 0 + s) (s) e(i m + s) (mb + s, ) i : m= e(i m + s) (mb + s, ) i b b m= m= e (i 0 + s)+,, e(i 0 + s)+ e(i m + s) i e(i m + s) b e (i m + s) b m= m= e(i m + s) i b m= e (i m + s), b,nh p (t, ), h (t, ) (t, ),h (8) e(i m + s), : (9) 0;m 0;m 0;m e(i m + s) i, o e (i m + s), i e(i m + s)"() i : (0) Because of (0) and in order to estabish the desired resut we have to show that the foowing three assertions are true: T ;n 0;m e(i m + s)"() = o P (); ()

9 The CBB for Nonstationary Processes 9 and in probabiity, where T ;n (t, ); G = i= 0;m p e (i m + s), = o P () () 0;m i U i ; G = e(i m + s) d! i= p i U i ; G ; G (3) i =(,) i+ =[(i,)] and fu i g i=;;::: is a sequence of independent standard Gaussian variabes. The assertion of the Theorem foows then because under vaidity of () to (3) and by Sutsky's theorem we get o d K L n(^, )j ; ;:::; n ;Ln (G ), (G, )o! 0 in probabiity, which is the asymptotic distribution of n(^, ); cf. Fuer (996). Here d K denotes Komogorov's distance d K (P; Q) = sup xr jp( x),q( x)jbetween probabiity measures P and Q. We proceed to show that () to (3) are true. To see () note that T ;n = = "() 0;m 0;m e(i m + s)"() = e T;n + O P (n,= ): "(i m + s)+o P (n,= ) Note that e T;n is a bock bootstrap estimator of the mean E(" t ) based on the i.i.d. sampe "();"(3);:::;"(n). Thus assertion () foows because and E ( e T;n )! 0 and Var (e T;n )=O P (, ): To estabish () verify rst using E, b m= E, m= "(i m + s) b = O P ((n, b),= ); "(i m + s) = OP ((n, b), )

10 0 E. Paparoditis and D. N. Poitis that 0;m e (i m + s) = 0;m " (i m + s)+o P (n,= (n,b),= ): The desired resut foows then by recognizing that the rst term on the right hand side of the above equation is a bock bootstrap estimator of E(" ()) = based on bocks from the i.i.d. sequence "();"();:::;"(n). Consider (3). Let e i0 =(e(i 0 +);e(i 0 +);:::;e(i 0 + )), e im = (e(i m +);e(i m +);:::;e(i m +b)) for m =;;:::;k, and e i =(e(i + );e(i +);:::;e(i +)). Denote by e be the -dimensiona random vector e =("(); e i0 ; e i ; :::; e i )0 (4) and by A the (, ) (, ) matrix given by A = P, I s where I s is the (, ) (, ) matrix with (i; j)th eement equa to one if i; j s and zero ese. Note that A = Q Q 0 where the (i; j)th eement of the orthogona matrix Q is given by q i;j = (, ),= cos[(4, ), (j, )(i, )] and the i-th eement of the diagona matrix = diag( ; ; ; ;:::;,; )isgiven by i; =0:5sec [(, i)(, )]; cf. Fuer (996). Using this decomposition of the matrix A wehave (t, ) = e0 A e =, i= i; U i where the random variabe U i is given by U i = q i "() + V i;m ; and V i;m = = 0;m,;m q i;mb+s+0;m e(i m + s) 0;m,;m q i;mb+s+0;m "(i m + s)+o P (b = k,= n,= ): Therefore, E (V i;m )=O P(b = k,= n,= ) for i =;;:::;, and m = 0; ; ;:::; and E (U i )=O p(k,= ).

11 The CBB for Nonstationary Processes Let J be the (, )-dimensiona vector J =(;;:::;) 0. Using the fact that (U ;U ;:::;U, )0 =Q e wehave,= 0;m e(i m + s) =,= J 0 e =, i= k i; U i where k i; =,= J 0 Q, i and i is the (, ) vector with one in the ith position and zero esewhere. Thus we have for the term on the eft hand side of (3) that (,, (t, );,= 0;m, e(i m + s))=( i=, i; U, i ; k i; U i ): To estabish the desired asymptotic distribution consider rst the asymptotic behavior of the bootstrap variabe U i. Since i= E b q i;mb+s e(i m + s) = = n, b n, b n,b b t= s ;s = n,b b t= s ;s = +O P (bk, (n, b),= n,= ) q i;mb+s q i;mb+s e(t + s )e(t + s ) q i;mb+s q i;mb+s "(t + s )"(t + s ) we get using E (V i;m )=O P(b = k,= n,= ) that n,b Var (U i ) = n,b t= 0;m,;m s ;s = q i;mb+s + 0;m q i;mb+s + 0;m "(t + s )"(t + s )+O p (b(n,b),= n,= ) = Var(" ), r= = Var(" )+o p (): q i;r + o p() The ast equaity above foows since b=n! 0, P, r= q i;r = and q i;r = O(,= ) uniformy in r. Furthermore, and because E(U i )=O P(k,= )we have Cov (U i ;U j )=E (U i U j )+O P(k, ) and by the independence of the

12 E. Paparoditis and D. N. Poitis V i;m Using and E (U i U j ) = = E (V i;m V j;m )+o P() n,b n, b t= 0;m,;m s ;s = "(t + s )"(t + s )+O P (b 3= n,3= ): n,b (n, b), " (t + s)! Var("()) t= q i;mb+s + 0;m q j;mb+s + 0;m n,b (n, b), "(t + s )"(t + s )=O P ((n, b),= ) t= for s 6= s uniformy in s and s,we get by the property P, r= q i;rq j;r =0 for i 6= j that E (U i U j ) = Var(" ) r= q i;r q j;r + O P (b(n, b),= ) = O(, )+O P (b(n,b),= ): Thus, Cov (U i ;U j )!0 in probabiity for i 6= j. Consider next the asymptotic distribution of the U i 's and reca that U i = P V i;m + o P () where the V i;m are independent (but not identicay distributed) zero mean random variabes. Appying a CLT for trianguar arrays of independent random variabes (see Coroary of Sering (98, p. 3)) we can show that d K (L(U i ); L(Z))! 0 in probabiity asn!, where Z denotes a standard Gaussian distributed random variabes. To eaborate and because P j Var (V i;m ), j = o P () it suces to show that P E jv i;m j = o P () for some >. This, however, foows since E jv i;mj = n, b n,b t= b q i;mb+s+0;m "(t + s) + o() = O P (b =,= ) and therefore, P E jv i;m j= = O P (k,= )! 0. Therefore, and because Cov (U i ;U j )!0 in probabiity we get that for 0 <N<xed U 0 ;U ;:::;U d 0 N! U ;U ;:::;U N (5)

13 The CBB for Nonstationary Processes 3 in probabiity asn!, where (U ;U ;:::;U N ) 0 is a random vector having a N-dimensiona Gaussian N(0;I N ) distribution and I N is the N N unity matrix. The rest of the proof proceeds aong the ines ofpthe proof of Theorem 0.. of Fuer (996, p. 550). Briey, since im,! i= j, i;, i j =0 we get that, ( i= i; U i ;, i= k i; U i ) = ( N i= +( i U i ;, i=n + N i= i U i ; p i U i ), i=n + = M ; + M ; + o P () p i U i )+o P () with an obvious notation for M ; and M ;.Now, by the summabiity of the sequence f i g we have that M ;L = o P () as N!uniformy in. From this, equation (5) and Lemma 6.3. of Fuer (996) we then get, i= i; U i ;, i= k i; U i d! G ;G in probabiity which concudes the proof of (3) and of the theorem. Proof of Theorem : that and, (t, ) = (), We ony give asketch of the proof. We rst show (t, )( (t), (t, )) =, [(t,)=b] M () e(i m + s) + op () (6) h p 0;m e(i m + s) i j = () + o P (); (7) where e(i m + s) is dened in (7). To see this et C = P j and note that by assumption A and using a Beveridge-Neson decomposition (cf. Hamiton (994), Proposition 7.), P n U(t) can be written in the form n n U(t) =C "(t)+(n),()

14 4 E. Paparoditis and D. N. Poitis where (t)= P j "(t, j) and j =, P i=j+ i. Since P j j j < we have n n (n, ), U(t) =C (n,), "(t)+o P (n, ): (8) Furthermore, a same type of decomposition can be appied for the observations within each bootstrap bock, i.e., b U(i m + s) =C b "(i m + s)+(i m +b),(i m ): (9) Now, using (5), (8) and (9) we get [(t,)=b] B (t) =()+ C e(i m +s)+((i m +B),(i m ))+O P (n, ) (0) for t =;3;:::; where B = minfb, 0;m ;t,mb, 0;m g. Substituting the above expression for (t)in,p (t), assertion (6) foows after some straightforward cacuations. To see (7) note rst that using arguments identica to those in (8) we have (t, )( (t), (t, )) = U() =, + h p h p 0;m 0;m 0;m (7) foows then using (0) and 0;m bu (i m + s), bu(i m + s), bu(i m + s), 0;m j j i j bu(i m + s) = ()+o p (): i bu(i m + s) + o P (): Under vaidity of (6) and (7) and aong the same ines as in the proof of Theorem it foows that, (t, );, (t, )( (t), (t, )) d! ()G ;, ()[G, j = ()]

15 The CBB for Nonstationary Processes 5 in probabiity. Now, since n(^, )! G, G, j = () in distribution as n!(cf. Fuer (996), Hamiton (994)), the proof of the theorem is concuded by appying Sutsky's theorem. REFERENCES. D. A. Dickey and W. A. Fuer (979), Distribution of the estimators for autoregressive time series with a unit root. Journa of the American Statistica Association 74, N. Ferretti and J. Romo (996), Unit root bootstrap tests for AR() modes, Biometrika 83, D. A. Freedman (98), Bootstrapping regression modes, Annas of Statistics 9, W. Fuer (996), Introduction to Statistica Time Series, (nd Ed.), John Wiey, New York. 5. J. D. Hamiton (994), Time Series Anaysis, Princeton University Press, Princeton, New Jersey. 6. H. R. Kunsch (989), The jackknife and the bootstrap for genera stationary observations, Annas of Statistics 7, P. C. B. Phiips and P. Perron (988), Testing for a unit root in time series regression, Biometrika, 75, D. N. Poitis and J. P. Romano (994), Large sampe condence regions based on subsampes under minima assumptions, Annas of Statistics, D. N. Poitis, J. P. Romano and M. Wof (999), Subsamping, Springer, New York. 0. J. P. Romano and M. Wof (998), Subsamping condence intervas for the autoregressive root, Technica Report, Department of Statistics, Stanford University.. R. Sering (980), Approximation Theorems of Mathematica Statistics. John Wiey, New York.. J. H. Stock (99), Condence intervas for the argest autoregressive root in U. S. macroeconomic time series, Journa of Monetary Economics 8,