We propose abootstrap resampling scheme for the least squares estimator of the parameter of an unstable first-order autoregressive model and we prove its asymptotic validity. This method is alternative to the invalid one studied by Basawa et al. (1991).Autoregressive processes; Bootstrapping least squares estimator; Unit root; Bootstrap invariance principle;

Ferretti, Nélida

Romo, Juan

English

Research Papers in Economics

Working Paper 92-22  Divisi6n de Economia 
May 1992  Universidad Carlos ill de Madrid 
Calle Madrid,  126 
28903 Getafe (Spain) 
Fax (341) 624-9849 
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["￿ 
BOOTSTRAPPING UNIT ROOT AR(1) MODELS 
Nt5lida  Ferretti and Juan Romo· 
Abstract  _ 
We propose a bootstrap resampling scheme for the least squares estimator ofthe 
parameter of an unstable first-order autoregressive model and we prove its asymptotic 
validity. This method is alternative to the invalid one studied by Basawa et al. (1991). 
Key  words:￿ 
Autoregressive processes, bootstrapping least squares estimator,  unit root,  bootstrap￿ 
invariance principle.￿ 
AMS  1980  subject  classsifications.  Primary  62M07,  62M09;  secondary  62MIO,￿ 
62E20.￿ 
·Ferretti, Universidad de la Plata, CONICET and  Universidad Carlos III de Madrid;￿ 
Romo,  Universidad Carlos III de  Madrid, Research partially supported by DGICYT￿ 
PB90-0266.￿ [~I
 
[  1  INTRODUCTION 
Let {Xd, t =  1,2, ... be a first-order autoregressive process defined by 
Xo =  0,  (1.1 ) 
where {Ut}  is  a sequence of independent and identically distributed random 
variables with E(ud = 0,  and V(ue) = (J'2  < 00.  Let 
In  = (t X~_l) -1 t XtXt- l 
t=l  t=l 
be  the least  squares  estimator of  {3,  based  on  a  sample of  n  observations 
(Xl,""  Xn ).  In  the stationary  case,  1 {3  1<  1,  Bose  (1988)  showed  the 
asymptotic validity of the bootstrap estimators corresponding to  ~n  and in 
the explosive case,  I  {3  I>  1,  this  has  been established by  Basawa,  Mallik, 
McCormick and Taylor (1989).  If 1{3  1=  1,  the unstable case,  ~n  has a non-
normal asymptotic distribution with a  complicated density (see,  e.g.,  Rao 
(1978)),  so  it  is  interesting  to  study the bootstrap  approximation  in  this 
situation.  Basawa,  Mallik,  McCormick,  Reeves  and  Taylor  (1991)  give  a 
bootstrap resampling scheme which  leads  to an asymptotic random distri-
bution showing,  in  this way,  that this bootstrap method is  asymptotically 
invalid even for  normally distributed errors.  In  this paper, we  introduce a 
different  bootstrap strategy and we  prove that it correctly approaches the 
asymptotic distribution of In. 
In  Section 2 we  describe the bootstrap resampling and we  establish its 
asymptotic validity.  Section 3 contains the proof of this result which needs 
a bootstrap invariance principle given in Proposition 3.1. 
2  MAIN  RESULT 
It is  known (see Anderson (1951)) that when {3 = 1, 
(2.2) 
L.  where 
2 
f__ and {W(t)}  is  a standard Wiener process. 
We  now  describe  our  bootstrap  resampling  scheme.  Let  Et  =  X t  -
~nXt-l'  t = 1, ... , n, and define ft = Et  - n-
1 '2:']=1  Ej,  the centered residuals. 
Denote by F n the empirical distribution function based on {ft:  t =  1, ... , n} 
and take a  random sample  {E~ t:  t  =  1, ... , n}  from  F n .  So,  the random , 
variables  {E~ t:  t =  1, ...  , n}  are i.i.d.  with distribution function F n ,  condi- , 
tionally on X  1, ••• ,Xn.  Then, the bootstrap sample  {X~,t:  t =  1, ... ,n} is 
recursively obtained from the model for  (3 = 1 
X~,t  =  X~,t_l +  E~,tl  t=l,  ... ,n  (2.3) 
with  X~,o =  0.  The bootstrap least squares estimate is  then given by 
[  n  )  -1  n￿ 
~~  =  ( L X~~t-l  L X~,tX~,t_l'
 
t= 1  t=1 
Let 
n  )  1/2 
Z~  =  (  LX~~-1  (~~ - 1)￿  (2.4) 
t=1 
be  the bootstrap  version  of  Zn  under  (3  =  1.  Our goal  is  to show  that 
Z~  ~w Z almost surely and so this bootstrap resampling approaches properly 
the correct limiting distribution.  Thus, our main result is  the following. 
[￿  Theorem 1 .  For  Z~  defined in (2.4), under the model (1.1) with (3 = 1,  we 
have that 
for almost all sample (X1 ,X2, .••)  where Z is  defined in (2.2). 
[ 
Remark.  Basawa et  al (1991) take Li.d.  {un with distribution N(O, 1)  and 
they obtain {Xn from 
3 [￿ 
[ 
x; = 0, 
where fin  is  the least squares estimate for  the AR(l) model; they show that 
for  ~~  = (L:~l  Xt~l)-l  L:~=l  X;X;_l' the sequence [ 
[ 
converges to a  random distribution not approaching the asymptotic correct 
one.  However, our resampling method works in the sense that the bootstrap 
distribution of  Z~  almost surely approximates the asymptotic distribution. 
3  PROOF  OF  THEOREM  1 
We  will  first  establish,  in  Proposition  3.1,  a  bootstrap  invariance  princi-
ple.  To  that end consider  the sequence of partial sums  S~,o  = 0,  S~,k  = 
L:j=l  €~,j'  k = 1, ... , n,  n  E  N.  A  sequence of continuous-time  process 
{Y n  * (t) : t E [0, 1]} ~=l  can be obtained from the sequence 
{  S~  k  : k =  1, ... , n} 00  ,  n=l 
[  by linear interpolation, i.e., 
Yn*(t)  =  O'~S~,[ntl + (nt - [nt]) O'~€~,[ntl+l'  t E [0,:1.],  nE N  (3.5) 
where [s]  denotes the greatest integer less than or equal to s. 
To prove Proposition 3.1,  we  need some lemmas.  In Lemma 3.1  we  will 
obtain the convergence of the finite-dimensional distributions; tightness will 
be proved in Lemma 3.2 and Lemma 3.3. 
Hereafter, P*, E", V* will  denote the respectively bootstrap probability, 
expectation and variance conditionally on the sample €l, ••. , €n' 
Lemma 3.1.  Conditionally on (€t, ... , En)  and for  almost all sample paths 
(€1l ... , €n), 
4 [￿ 
(Yt(tl), ... ,Y;(td))  --+w (W(td,···, W(td))  (3.6) 
for all (tl,.'"  td)  E [0, It 
[￿  Proof. It is enough to show that, for all s, t E [0,1], 
(Y;(s), Y;(t)) --+w (W(s), W(t))  a.s. 
Now, conditionally on (El, ... ,En),  since 
1  [nt]  1 
W;(t) - (j  C  LE~)::;  cl<,[nt]+1I, 
yn j =l  (jyn 
we obtain by  the Cebisev inequality that 
I  [nt]  }  V*(E*)
P*  1Y*(t) - -'"'E*.'  > 8 <  n,i.  (3.7) {  n  (j  !1i LJ  n,J  - 82(j2 n y  10  J=l 
But 
"n  "n €2  2
V*(E*  .) = L...Jj=1  j = L...Jj=l  Ej  +0(1) 
n,1  n n 
L:￿  =.!. tX]  - (.!.  tXJ-1)fi~  +0(1) 
n  j=1  n  j=1 
1 ~  2  '2  1  2  = (- LJ Xj _  )(1  - 13n ) +-Xn +0(1),  1 ::;  i  ::; n, 1 
n  j=1￿  n 
[￿  has a non degenerate limiting distribution when 13  =  1 and so the right hand 
side in (3.7)  converges to zero.  Therefore, 
1  [n8]  [nt] 
II(Y;(s), Y;(t)) - (jy'n(j; <,j, j; E~,j)11  ---+p. 0  a.s. 
and it suffices to prove that 
1  [n8]  [nt] 
-----r= (L  E~,j, L E~,j)  ---+w  (W(s), W(t ))  a. s. 
(jyn  j=l  j=1 
['￿  5 L 
This is  equivalent to show that 
1  [ns]  [nt] 
U  JTi(l:  E~,j,  l:  E~J  ----+w (W(s), W(t) - W(s))  a.s.;  (3.8) 
y  n  j==1  j=[ns]+1 
but, since the components on the left hand side are independent, (3.8) follows 
from the bootstrap central limit theorem (see, e.g., Singh (1981)).  0 
Lemma 3.2. For any 7]  > 0, 
limlimsup ~P" {  max  IS~jl  > 7]uvn} =  0 610  n-oo  v  1~j~[n6]+1  ' 
conditionally on (El, ... ,En)  and for  almost all sample paths (El"  .. , En). 
Proof.  By  the bootstrap  central  limit  theorem  (see,  e.g.,  Singh  (1981)), 
we  have  that  (1/uJ[n8] +  1)S~.[n6]+l  converges  weakly  almost  surely  to  a 
standard normal random variable V.  Fix A > 0 and let {'Pdk::l be a sequence 
of bounded, continuous functions on ~  with 'Pk  ll(-oo,.\]U[.\,oo)'  We have for 
each k, 
limsup P"  {IS~,[n,6]+ll  ~  AU~}  n-oo 
~  Ji~  E"  ('Pk  (u~S~,[n5]+l))  = E"('Pk(V)) 
Then, if k -? 00 we obtain 
We now define  T~  =min{j ~  1 :  IS~.JI  > 7]uvn}.  If 0 < 8 < ~,  we  have 
P" {  max  ISn"J·1  > 7]u..;Ti}  ~  P"  {IS~[n61+ll  ~  u..;Ti(7]  - J28)}
i~j~[n5]+l  . . 
['- 6 [ 
[n8] 
+ L: P"  {IS~,[n8]+11  < (jvn(T/ - VU)IT~ = j}  P"{T~  =n.  (3.10) 
j=1 
But for  T~  = j, 
[  IS~,[n8]+l1  < (jvn(T/ - VU) 
implies  IS~,j  - S~,[n8]+l1  > (jy'2n8,  and by Cebisev inequality it follows that 
P*  {IS~,[n8]+11  < (jvn(T/ - V28) 1< = j} 
~  2 ;  2 V*  ([n~1  t:~'i)'  1 ~  j  ~  [n8].  (3.11)
nU(j  i=j+l 
Moreover, the right hand side in (3.11)  is  bounded above by  2u 12 n  2:k =1 tZ. 
Therefore, going back to (3.10) 
[ 
-~ 
--~ 
P* {  max  IS~ j I > T/(jvn}
0:5i:5[n81+ 1  ' 
~  P* {I  S~,[n8]+11  ~  (jvn(T/  - V28)}
++ (t  t%)  P* {< ~  [n8]}
2(j  n  k=1 
~  P* {I  S~,[n8]+l1  ~ (jvn(T/  - V28)}
++ (t  tZ)  P* {  ~ax  IS~il  > T/(jvn}. 2(j  n  k=1  0:5J:5[n8]+l' 
[  It follows  that 
P* {  max  IS~ i I> T/(jvn}
0:5j:5[n8]+l  ' 
~  (1 --2 \  t t%) -1 P* {IS~,[n8]+11  ~ (jvn(T/ - V28)}.
(j  n  k=1 
Putting A =  (T/  - VU)/V8 in (3.9), we have 
[  7 L 
1 {  }  E(1V1 
3 )V6 limsup ,P"  max  IS~jl  > 1](jvn  < '  VU  R, 
n-oo  u  O~j ~[n8l+ I'  - (1] - 26)3 
where R is  the non degenerate limit of (1 - 2012 n  L:k=1  i%) -I (see the proof of 
Lemma 3.1).  Now  letting 6 t 0 the lemma follows.  D 
Lemma 3.3. For any 1]  > 0 and T > 0, 
limlimsupp,,{  max  IS~,k+j - S~,kl  > 1](jvn} =  0 
8!0  n-oo  1 5:.  j  5:.  [n6] +1 
o5:.  k 5:.  [nT] +1 
[ 
conditionally on (El,""  En)  for  almost all sample paths (El, ... , En). 
Proof.  Once we  have Lemma 3.2,  the proof follows  as  in Lemlna 4.19  of 
Karatzas and Shreve (1988), page 69, replacing Sk  by S~ k'  k = 1, ... ,n, n  E 
N.D·  , 
Now, we establish the bootstrap invariance principle. 
Proposition 3.1.  Let  {En}~l  be a sequence of independent and identically 
distributed random variables with mean zero and finite variance (j2  > 0 de-
fined on the probability space (f!, A, 'P).  Let F n  be the empirical distribution 
.  d t  L:;.l f}  - 1  d 1  "  . - 1  b . d A. -.  • assocIate  0  El  - El  - n  ,t - , ... , n an  et En,i'  t  - , ... , n  e In  e-
pendent random variables with distribution F n •  Define {Y;(t) : t E [0,  1]}~1 
by (3.5).  Then Y; --+w W  a.s.  in C[O,I],  where W  is  the standard one-
dimensional Brownian motion on [0, 1]. 
Proof.  For all the sample paths in f!1  n f!2  n f!a,  the proof in  page 71  of 
Karatzas  and  Shreve  (1988)  gives  the tightness of  {Y;}~=l;  this  and  the 
finite dimensional convergence in (3.6)  imply,  by theorem 4.15  in Karatzas 
and Shreve (1988), the weak convergence in C[O, 1].  D 
8 []￿ 
[ 
Finally, to prove Theorem 1 we  will  need the following  lemma. 
Lemma 3.4. Let 
[ 
Then, conditionally on  (~l' ...  ,~n) and for almost all sample paths (~1' ... , ~n) 
R~  ~p 0, 
as  n  -+ 00. 
Proof.  It is  straightforward from Proposition 3.1.  0 
Proof of Theorem 1.  Observe that 
[  Z~	  =  (L~1  X~~t_lr/2  (~~  - 1) 
=  (L~=l  X~~t_l) 1/2  (L~1  X~,t_l  ~~,t) 
Now, by  squaring (2.3)  and by summing we obtain 
~ X*  ~*  =  ~ X*2  _  ~  ~ ~*2  •  (3.12) L.J  n,t-l  n,t  2  n,n  2 L.J  n,t 
t=1  t=1 
Then, expressing the quantities  X~,t  in terms of Y;(t), defined in  (3.5), 
we  have 
(3.13) 
and 
(3.14) 
It follows  from (3.12),  (3.13) and (3.14)  that 
n)  (1 n1  (k))-1/2 Z*  =  ~  y*2(1) __1_" ~*2  _  ~ y*2￿  _ 
n  2 ( n  (72n  L.J  n,t  n  L.J  n n 
t=1  k=l 
9 
[.~ [1 
[ 
By the bootstrap weak  law  of large numbers  and  Proposition 3.1,  the nu-
merator converges  to  ~(W2(1) - 1).  Moreover,  from  Lemma 3.4,  Propo-
sition 3.1  and the continuous  mapping  theorem, the denominator tends to 
(Id W2(t)dtr/2.  Since can  be  easily proved  that  the bootstrap  version  of 
Slutsky's theorem holds, the theorem follows.O 
Remark. It is  straightforward to check that in the case f3  =  -1, the result 
also holds if the innovations distribution is  symmetric around zero. 
Acknowledgments. The authors wish to thank Ana Justel for some previ-
ous simulation work. 
10 [J 
REFERENCES 
[J 
Anderson, T.W. (1959).  On asymptotic distributions of estimates of pa-
rameters of stochastic difference equations.  Ann.  Math.  Statist.  30 
676-687. 
Basawa,  LV.,  Mallik,  A.K.,  McCormick,  W.P.  and Taylor,  R.L. 
(1989).  Bootstrapping explosive autoregressive processes.  Ann. Statist. 
17 1479-1486. 
Basawa, LV., Mallik,  A.K., McCormick, W.P., Reeves, J.H. and 
Taylor, R.L.  (1991).  Bootstrapping unstable first-order autoregres-
sive processes.  Ann.  Statist.  19 1098-1101 
Bose,  A.  (1988).  Edgeworth correction  by  bootstrap  in  autoregressions. 
Ann. Statist.  16 1709-1722. 
Karatzas, Land Shreve,  S.  (1988).  Brownian  Motion  and  Stochastic 
Calculus.  Springer-Verlag,  New  York. 
Rao, M.M. (1978).  Asymptotic distribution of an estimator of the boundary 
parameter of an unstable process.  Ann.  Statist.  6  185-190. 
Singh, K. (1981).  On the asymptotic accuracy of Efron's bootstrap.  Ann. 
Statist.  9  1187-1195. 
Departamento de Estadfstica y  Econometria 
U niversidad Carlos ITI  de Madrid 
28903 Getafe (Madrid)- Spain 
[ 
11 

Asymptotic distribution of an estimator of the boundary parameter of an unstable process.

Bootstrapping explosive autoregressive processes.

Bootstrapping unstable first-order autoregressive processes.

Brownian Motion and Stochastic Calculus.

On asymptotic distributions of estimates of parameters of stochastic difference equations.

On the asymptotic accuracy of Efron's bootstrap.

Bootstrapping unit root AR(1) models.

http://e-archivo.uc3m.es/bitstream/10016/2832/1/we9222.pdf

Bootstrapping unit root AR(1) models.

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