First order autoregression is shown to satisfy a limit theory which is uniform over stationary values of the autoregressive coefficient rho = rho_{n} in [0,1) provided (1 - rho_{n})n approaches infinity. This extends existing Gaussian limit theory by allowing for values of stationary rho that include neighbourhoods of unity provided they are wider than O(n^{1}), even by a slowly varying factor. Rates of convergence depend on rho and are at least squareroot of squareroot of n but less than n. Only second moments are assumed, as in the case of stationary autoregression with fixed rho.Autoregression, Gaussian limit theory, local to unity, uniform limit

Liudas Giraitis

Peter C.B. Phillips

English

Research Papers in Economics

 
 
 
UNIFORM LIMIT THEORY FOR STATIONARY AUTOREGRESSION 
 
By 
Liudas Giraitis and Peter C.B. Phillips 
 
July 2004 
 
 
 
 
 
 
 
 
 
COWLES FOUNDATION DISCUSSION PAPER NO. 1475 
 
 
 
 
 
COWLES FOUNDATION FOR RESEARCH IN ECONOMICS 
YALE UNIVERSITY 
Box 208281 
New Haven, Connecticut 06520-8281 
 
http://cowles.econ.yale.edu/ Uniform Limit Theory for Stationary Autoregression
Liudas Giraitis and Peter C. B. Phillips
University of York, UK and Cowles Foundation, Yale University, CT, USA,
University of Auckland & University of York
July 12, 2004
Abstract
First order autoregression is shown to satisfy a limit theory which is uniform over
stationary values of the autoregressive coeﬃcient ρ = ρn ∈ [0,1) provided (1 − ρn)n →
∞. This extends existing Gaussian limit theory by allowing for values of stationary ρ
that include neighbourhoods of unity provided they are wider than O
¡
n−1¢
, even by a
slowly varying factor. Rates of convergence depend on ρ and are at least
√
n but less
than n. Only second moments are assumed, as in the case of stationary autoregression
with ﬁxed ρ.
AMS Subject Classiﬁcations: Primary 62M10; Secondary 60F15
JEL Classiﬁcations: C22
Keywords and Phrases: Autoregression, Gaussian limit theory, local to unity, uni-
form limit.
Corresponding author: P e t e rC .B .P h i l l i p s
Cowles Foundation, Yale University, Box 20828, New Haven CT 06520-8281, USA; email:
peter.phillips@yale.edu
11I n t r o d u c t i o n
In pioneering work on limit theory for autoregressions, Mann and Wald (1943) showed
consistency and asymptotic normality of least squares regression in stationary models. An-
derson (1959) conﬁrmed that these results hold in scalar models under weaker conditions
requiring only second moments. Lai and Wei (1982) extended the results further to stochas-
tic regression models with martingale diﬀerence errrors having homoscedastic variance and
moments of order greater than two. In contrast, it is well known that in unit root autore-
gressions (White, 1958) and in models whose roots are local to unity (Phillips, 1987; Chan
and Wei, 1987) the limit distribution is non-Gaussian and involves functionals of stochastic
processes.
The present note shows that in stationary regions that are further removed from unity
than O
¡
n−1¢
for samples of size n the Gaussian limit theory still applies. In particular,
in ﬁrst order autoregression a Gaussian limit theory holds uniformly over stationary values
of the autoregressive coeﬃcient ρ = ρn ∈ [0,1) which includes local vicinities of unity that
satisfy (1 − ρn)n →∞ . Thus, even for ρn =1− Ln/n where Ln →∞is slowly varying at
inﬁnity the usual Gaussian limit theory applies. Rates of convergence depend on ρn and are
at least
√
n but less than n. Only second moments are assumed, as in the case of stationary
autoregression with ﬁxed ρ.
The results given here provide a supplement to those of Lai and Wei (1982). Theorem 3
of Lai and Wei gives the asymptotic normality of a suitably standardized and centred least
squares estimator in regression models with stochastic regressors under known conditions
that enable the use of a standard martingale CLT. These conditions, which need to be
checked in individual cases, involve a stability condition on the sample variance of the
regressors, a uniform negligibility condition on the standardized regressors and uniform
error moments of order greater than two. The present work provides a direct proof of
asymptotic normality under primitive conditions on ρn and the errors in an autoregression,
allowing for roots in the vicinity of unity of the form ρn. These conditions appear to be
near minimal for Gaussianity in an autoregression. Our proof of asymptotic normality uses
an asymptotic truncation argument and a martingale CLT that applies when only second
moments are ﬁnite.
2M a i n r e s u l t s
We consider the model
yt = ρnyt−1 + ut,t =1 ,...n (2.1)
where ut,t∈ Z+ is a stationary and ergodic martingale diﬀerence sequence with respect to
the natural ﬁltration Ft−1 = σ (ut−1,u t−1,...) with ﬁnite conditional variance E(u2
t|Ft−1)=
σ2 a.s. and initialization y0.
The coeﬃcient ρ is ﬁtted least squares, giving the estimator ˆ ρ =
Pn
j=1 ytyt−1/
Pn
j=1 y2
t−1,
and
ˆ ρ − ρn =
Pn
j=1 utyt−1
Pn
j=1 y2
t−1
. (2.2)
The following conditions are imposed on ρn and ut throughout the remainder of the paper.
2A.1. ρn ∈ [0,1) m a yd e p e n do nn a n di ss u c ht h a tvn =1− ρn has property
vnn →∞ . (2.3)
A.2. y0 is independent of {ut : t =1 ,2,...} and
Ey0
2 = o(n1/2). (2.4)
Note that distribution of ut does not depend on n.
Theorem 2.1 Under A.1-A.2,
n1/2
(1 − ρ2
n)1/2(ˆ ρ − ρn) ⇒ N(0,1). (2.5)
and
n1/2
|1 − ˆ ρ2|1/2(ˆ ρ − ρn) ⇒ N(0,1). (2.6)
In both cases the limit distribution is Gaussian uniformly in ρn satisfying A1, although
the convergence rate depends directly on how close ρn is to unity. The asymptotic distri-
bution of the sample mean of yt is similarly Gaussian, again with convergence rate that
depends on ρn.
In what follows, ⇒ denotes convergence in distribution and →p convergence in proba-
bility.
Theorem 2.2 Under A.1-A.2,
(1 − ρn)
n1/2
n X
t=1
yt ⇒ N(0,σ2). (2.7)
and
[(1 − ρn)/(1 + ρn)]1/2
Pn
t=1 yt
¡Pn
t=1 y2
t
¢1/2 ⇒ N(0,1). (2.8)
The proofs use the following lemmas.
Lemma 2.1
(1 − ρ2
n)1/2
n1/2
n X
t=1
utyt−1 ⇒ N(0,σ4). (2.9)
Lemma 2.2
1 − ρ2
n
n
n X
t=1
y2
t−1 →p σ2. (2.10)
Result (2.10) proves the ‘stability’ condition which Lai and Wei (1982, theorem 3, con-
dition (4.2)) use for their limit theory corresponding to (2.5).
33P r o o f s
P r o o fo fT h e o r e m2 . 1 . The convergence (2.5) follows from Lemmas 2.2 and 2.1. To
prove (2.6), note from (2.5) that
ˆ ρ − ρn = OP(v1/2
n n−1/2)=oP(vn),
since n−1 = o(vn) by (2.3). Thus
|1 − ˆ ρ|−1/2 = |1 − ρn + oP(vn)|−1/2 = |1 − ρn|−1/2|1+oP(vn)|−1/2.
This and (2.5) imply (2.6).
P r o o fo fL e m m a2 . 1 . Denote by ζt = v
1/2
n n−1/2utyt−1 t h es e q u e n c eo fm a r t i n g a l e
diﬀerences with respect to the sigma algebra Ft−1 generated by u1,...,u t−1. Recall that
vn =1− ρn.
a) We ﬁrst prove convergence (2.9) in the case where fourth moments are ﬁnite: Eu4
t <
∞ and Ey4
0 < ∞.I ts u ﬃces to show that, as n →∞ ,
(1 + ρn)
n X
t=1
E[ζ2
t|Ft−1] →p σ4, (3.1)
and
n X
t=1
E[|ζt|21{|ζt|≥δ}|Ft−1] →p 0, (3.2)
for all δ>0, which in turn implies the convergence
(1 + ρn)1/2
n X
t=1
ζt ⇒ N(0,σ4)
and yields (2.9). We note that (3.2) corresponds to the uniform negligibility condition used
by Lai and Wei (1982, condition (4.3)). To check (3.1), we show that
(1 + ρn)
n X
t=1
Eζ2
t → σ4, (3.3)
and
E
³ n X
t=1
E[(ζ2
t − Eζ2
t)|Ft−1]
´2
→ 0. (3.4)
Since
E[ζ2
t|Ft−1]=( vn/n)E[u2
ty2
t−1|Ft−1]=( vn/n)σ2y2
t−1,
where vn =1− ρn,w eh a v e
(1 + ρn)E
n X
t=1
E[ζ2
t|Ft−1]=( 1− ρ2
n)n−1
n X
t=1
σ2E[y2
t−1] → σ4
by Lemma 3.1 below.
4To show (3.4) note that E[(ζ2
t −Eζ2
t)|Ft−1]=σ2(vn/n)(y2
t−1−Ey2
t−1), t =1 ,2,.... Then,
using C to denote a generic constant in what follows, we have by Lemma 3.2 below
Ey4
t−1 ≤ Cv−2
n , (3.5)
and therefore
E
³ n X
t=1
E[(ζ2
t − Eζ2
t)|Ft−1]
´2
= v2
nn−2
n X
t,s=1
E[(u2
t − σ2)(u2
s − σ2)]E[y2
t−1y2
s−1]
= v2
nn−2
n X
t=1
E[(u2
t − σ2)2]E[y4
t−1]
≤ Cv2
nn−2
n X
t=1
(1 − ρn)−2 ≤ Cn−1 → 0,
proving (3.4).
Finally, to show (3.2), note that by (3.5),
qn := E
n X
t=1
E[|ζt|21{|ζt|≥δ}|Ft−1] ≤ δ−2
n X
t=1
E|ζt|4 ≤ Cδ−2v2
nn−2
n X
t=1
E|yt−1|4 ≤ Cn−1 → 0.
(3.6)
b) Suppose now that either or both of Eu4
1 = ∞ and Ey4
0 = ∞ apply. Let K>0 be a
ﬁxed constant. Set
u
(1)
t = ut1(|ut| ≤ K)−E[ut1(|ut| ≤ K)|Ft−1],u
(2)
t = ut1(|ut| >K )−E[ut1(|ut| >K )|Ft−1],
y
(1)
0 = y01(|y0| ≤ K)−E[y01(|y0| ≤ K)|Ft−1],y
(2)
0 = y01(|y0| >K )−E[y01(|y0| >K |Ft−1],
and then by recursion deﬁne
y
(1)
t = ρny
(1)
t−1 + u
(1)
t ,y
(2)
t = ρny
(2)
t−1 + u
(2)
t ,t =1 ,2,...
Note that
E[ut1(|ut| ≤ K)|Ft−1]+E[ut1(|ut| >K )|Ft−1]=E[ut|Ft−1]=0
and therefore y
(1)
t + y
(2)
t = yt,t=0 ,1,2,... . Moreover, the random variables u
(j)
t and y
(j)
0
satisfy A.2 when j =1 ,2.T h e nw ec a nw r i t e
n X
t=1
utyt−1 =
n X
t=1
u
(1)
t y
(1)
t−1 +
n X
t=1
u
(2)
t y
(1)
t−1 +
n X
t=1
uty
(2)
t−1
=: Sn,1 + Sn,2 + Sn,3.
Since y
(1)
t is a sequence of martingale diﬀerences, by part a) we have,
(1 − ρ2
n)1/2n−1/2Sn,1 ⇒ N(0,σ4
K), (3.7)
5where
σ2
K = E(u
(1)
t )2 = Eu2
t1(|ut| ≤ K) → σ2 = Eu2
t,
as K →∞ . We show that for l =2 ,3, uniformly in n ≥ 1,
(1 − ρ2
n)n−1ES2
n,l ≤ δK, (3.8)
where δK → 0 as K →∞which together with (3.7) proves convergence (2.9) in case b).
Set δK = E(u
(2)
t )2.N o t et h a t
δK → 0, as K →∞ . (3.9)
By Lemma 3.1,
n X
t=1
E(y
(1)
t−1)2 ≤ Cnv−1
n ,
n X
t=1
E(y
(2)
t−1)2 ≤ CδKnv−1
n
where C does not depend on n,K.T h u s , s i n c e u
(2)
t y
(1)
t−1 and uty
(2)
t−1 are uncorrelated se-
quences,
ES2
n,2 + ES2
n,3 ≤
n X
t=1
[E(u
(2)
t )2E(y
(1)
t−1)2 + Eu2
tE(y
(2)
t−1)2]
≤ CδK2nv−1
n ,
which, together with (3.9), implies (3.8).
Proof of Lemma 2.2. Since yt = ρnyt−1 + ut we can write
n X
t=1
y2
t = ρ2
n
n X
t=1
y2
t−1 +2 ρn
n X
t=1
yt−1ut +
n X
t=1
u2
t.
Thus
n X
t=1
y2
t−1 =( 1− ρ2
n)−1
³
−y2
n + y2
0 +2 ρn
n X
t=1
yt−1ut +
n X
t=1
u2
t
´
. (3.10)
By Lemma 2.1,
Zn := (1 − ρ2
n)1/2n−1/2
n X
t=1
yt−1ut ⇒ N(0,σ4).
Moreover, because ut is an ergodic sequence with a ﬁnite second moment σ2,w eh a v e
n X
t=1
u2
t = nσ2 +
n X
t=1
(u2
t − σ2)=nσ2 + oP(n).
Therefore
n X
t=1
y2
t−1 =( 1− ρ2
n)−1
³
−y2
n + y2
0 +2 ρn(1 − ρ2
n)−1/2n1/2Zn + nσ2 + oP(n)
´
=( 1− ρ2
n)−1n
³
−n−1(y2
n − y2
0)+2 ρn((1 − ρ2
n)n)−1/2Zn + σ2 + oP(1))
6=( 1− ρ2
n)−1n(σ2 + oP(1)),
because (1−ρ2
n)n →∞by assumption (2.3), and therefore ((1−ρ2
n)n)−1/2Zn →p 0,w h e r e a s
n−1y2
n = OP((nvn)−1) →p 0,
by (3.12), and n−1y2
0 →p 0 by Assumption A.2, proving (2.10).
Proof of Theorem 2.2. We can write
Pn
t=1 yt = ρn
Pn
t=1 yt−1 +
Pn
t=1 ut. Thus
n X
t=1
yt−1 =( 1− ρn)−1
³
−yn + y0 +
n X
t=1
ut
´
(3.11)
=( 1− ρn)−1n1/2
³
(y0 − yn)n−1/2 + n−1/2
n X
t=1
ut
´
.
Then
(1 − ρn)n−1/2
n X
t=1
yt =( y0 − yn)n−1/2 + n−1/2
n X
t=1
ut.
Since ut is a martingale diﬀerence sequence with Eu2
t < ∞,
n−1/2
n X
t=1
ut ⇒ N(0,1).
By A.2, n−1/2E|y0| = o(1). Writing
yt = ρnyt−1 + ut = ρ2
nyt−2 + ρnut−1 + ut = ρt
ny0 +
t−1 X
j=0
ρj
nut−j,
we have
n−1/2E|yt| = n−1/2E
¯ ¯ ¯ρt
ny0 +
t−1 X
j=0
ρj
nut−j
¯ ¯ ¯
≤ n−1/2E|y0| + n−1/2
³
E
³t−1 X
j=0
ρj
nut−j
´2´1/2
≤ o(1) + n−1/2
³t−1 X
j=0
ρ2j
n Eu2
t−j
´1/2
≤ o(1) + Cn−1/2(1 − ρ2
n)−1/2 ≤ o(1) + C(nvn)−1/2 = o(1),
by (2.3). Thus n−1/2(y0 − yn) →p 0 and
(1 − ρn)n−1/2
n X
t=1
yt = oP(1) + n−1/2
n X
t=1
ut ⇒ N(0,σ2),
proving (2.7). Finally, (2.8) follows from (2.7) and Lemma 2.2.
7Lemma 3.1 Suppose A.1-A.2 hold. Then
Ey2
n ≤ C(1 − ρn)−1, (3.12)
where C does not depend on ρn.F u r t h e r ,
(1 − ρ2
n)n−1
n−1 X
j=0
Ey2
j = σ2 + o(1). (3.13)
Proof of Lemma 3.1. Since Eysut =0for t>s , by (2.1), for t ≥ 1
Ey2
t = ρ2
nEy2
t−1 + Eu2
t = ρ2
nEy2
t−1 + σ2 = ρ4
nEy2
t−2 +( ρ2
n + ρ0
n)σ2
= ρ2t
n Ey2
0 +( ρ2(t−1)
n + ... + ρ0
n)σ2 = ρ2t
n Ey2
0 +
1 − ρ2t
n
1 − ρ2
n
σ2, (3.14)
by virtue of the sum
ρ2(t−1)
n + ... + ρ0
n =
1 − ρ2t
n
1 − ρ2
n
.
Note that inequality log(1 − v) ≤− v (0 ≤ v<1) implies that
ρn
n =e x p ( nlogρn)=e x p ( nlog(1 − vn)) ≤ exp(−nvn) ≤ (vnn)−1 → 0, (3.15)
since nvn →∞by (2.10). Since 0 ≤ ρn < 1 and, under A.1 and A.2, ρ2n
n Ey2
0 ≤
C(vn)−1Ey2
0 = o(v−1
n ), it follows that (3.14) implies (3.12).
To show (3.13), note that
(1 − ρ2
n)n−1
n−1 X
j=0
Ey2
j =( 1− ρ2
n)n−1
³1 − ρ2n
n
1 − ρ2
n
Ey2
0 + σ2 1
1 − ρ2
n
(n −
1 − ρ2n
n
1 − ρ2
n
)
´
= o(1) + σ2,
by virtue of the assumption Ey2
0 = o(n1/2), to prove (3.13).
Lemma 3.2 Suppose Eu4
t < ∞, Ey4
0 < ∞.T h e n
Ey2
t ≤ C(1 − ρn)−1, (3.16)
Ey4
t ≤ C(1 − ρn)−2,t =1 ,2,... (3.17)
uniformly in 0 ≤ ρn < 1 and t.
Proof of Lemma 3.2. Since Ey2
0 < ∞, (3.16) follows from (3.14).
By (2.1),
Ey4
t = E(ρnyt−1 + ut)4 = E(ρ2
ny2
t−1 +2 ρnyt−1ut + u2
t)2
= E(ρ2
ny2
t−1 + σ2 +[ 2 ρnyt−1ut + u2
t − σ2])2 = E(ρ2
ny2
t−1 + σ2)2 + E(2ρnyt−1ut + u2
t − σ2)2
= ρ4
nEy4
t−1 +2 ρ2
nEy2
t−1σ2 + σ4 +4 ρ2
nEy2
t−1Eu2
t +4 ρnEyt−1Eu3
t + E(u2
t − σ2)2
8≤ ρ4
nEy4
t−1 + C(Ey2
t−1 + |Eyt−1| +1 ) ,
for some C>0 since ρn < 1. Since by (3.16), Ey2
t ≤ C/(1 − ρn), then
Ey2
t−1 + |Eyt−1| +1≤ C/(1 − ρn)
uniformly in ρn,a n d
Ey4
t ≤ C(ρ4t
n +
t X
j=1
ρ4(j−1)
n (1 − ρ2
n)−1) ≤ C(ρ4t
n +( 1− ρ4
n)−1(1 − ρ2
n)−1)
≤ C(1 + (1 − ρn)−2) ≤ C(1 − ρn)−2
proving (3.17).
Acknowledgements
Giraitis thanks the ESRC for research support under Grant R000239538. Phillips thanks
the NSF for research support under Grant SES 04-142254.
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9

Asymptotic inference for nearly nonstationary AR(1) processes.

Least squares estimates in stochastic regression models with applications to identiﬁcation and control of dynamic systems.

On Asymptotic Distributions of Estimates of

On the statistical treatment of linear stochastic diﬀerence equations.

P h i l l i p s

The limiting distribution of the serial correlation coeﬃcient in the explosive case.

Uniform Limit Theory for Stationary Autoregression

http://cowles.econ.yale.edu/P/cd/d14b/d1475.pdf

Uniform Limit Theory for Stationary Autoregression

Abstract

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