Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol96.Issue1.May2000:
Journal of Econometrics 96 (2000) 155}182
The spurious regression of fractionally
integrated processes
Wen-Jen Tsay!,*, Ching-Fan Chung"
!The Institute of Economics, Academia Sinica, Nankang, Taipei, Taiwan
"National Taiwan Univeristy, Taiwan
Received 1 September 1995; received in revised form 1 July 1999
Abstract
This paper extends the theoretical analysis of the spurious regression and spurious
detrending from the usual I(1) processes to the long memory fractionally integrated
processes. It is found that when we regress a long memory fractionally integrated process
on another unrelated long memory fractionally integrated process, no matter whether
these processes are stationary or not, as long as their orders of integration sum up to
a value greater than 0.5, the t ratios become divergent and spurious e!ects occur. Our
"nding suggests that it is the long memory, instead of nonstationarity or lack of
ergodicity, that causes such spurious e!ects. As a result, spurious e!ects might happen
more often than we previously believed as they can arise even between stationary series
while the usual "rst-di!erencing procedure may not completely eliminate spurious e!ects
when data possess strong long memory. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C22
Keywords: Fractionally integrated processes; Long memory; Spurious regression;
Spurious detrending
* Corresponding author. Tel.: 886-227822791 ext. 296, FAX: 886-27853946.
E-mail: wtsay@ieas.econ.sinica.edu.tw (W.-J. Tsay)
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 5 6 - 1
156
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
1. Introduction
The spurious regression was "rst studied by Granger and Newbold (1974)
using simulation. They show that when unrelated data series are close to the
integrated processes of order 1 or the I(1) processes, then running a regression
with this type of data will yield spurious e!ects. That is, the null hypothesis of no
relationship among the unrelated I(1) processes will be rejected much too often.
Furthermore, the spurious regression tends to yield a high coe$cient of determination (R2) as well as highly autocorrelated residuals, indicated by a very low
value of Durbin}Watson (D=) statistic. Granger and Newbold's simulation
results are later supported by Phillips' (1986) theoretical analysis. Phillips proves
that the usual t test statistic in a spurious regression does not have a limiting
distribution but diverges as the sample size approaches in"nity. He also shows
that R2 has a non-degenerate limiting distribution while the D= statistic
converges in probability to zero. Phillips' results has been generalized by
Marmol (1995) to cases with integrated processes of higher orders.
The history of the research on spurious detrending follows a similar thread.
Nelson and Kang (1981, 1984) "rst employ simulation to demonstrate that the
regression of a driftless I(1) process on a time trend produces an incorrect result
of a signi"cant trend. Extending the Phillips' (1986) approach, Durlauf and
Phillips (1988) derive the asymptotic distributions for the least squares estimators in such a regression. In particular, the latter authors show that the t test
statistics diverge and there are no correct critical values for the conventional
signi"cance tests.
Most studies of the spurious regression concentrate on the nonstationary I(1)
processes. It re#ects the widely held belief that many data series in economics are
I(1) processes, or near I(1) processes, as argued by Nelson and Plosser (1982).
Against this backdrop, we also witness in recent years fast growing studies on
fractionally integrated processes, or the I(d) processes with the di!erencing
parameter d being a fractional number. The I(d) processes are natural generalization of the I(1) processes that exhibit a broader long-run characteristics. More
speci"cally, the I(d) processes can be either stationary or nonstationary, depending on the value of the fractional di!erencing parameter. The major characteristic of a stationary I(d) process is its long memory which is re#ected by the
hyperbolic decay in its autocorrelations. A number of economic and "nancial
series have been shown to possess long memory. See Baillie (1996) for an
updated survey on the applications of the I(d) processes in economics and
"nance.
The objective of this paper is to extend the theoretical analysis of the spurious
regression from I(1) processes to the class of long-memory I(d) processes. We
establish and analyze conditions on the I(d) processes that in#ict the spurious
e!ect in a simple linear regression model. The nonstandard asymptotic distributions of various coe$cient estimators and test statistics are then derived.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
157
The main "nding from our study is that the spurious regression can arise
among a wide range of long-memory I(d) processes, even in cases where both
dependent variable and regressor are stationary. A few conclusions may then be
drawn. First, di!erent from what Phillips (1986) and Durlauf and Phillips (1988)
have suggested, the cause for spurious e!ects seems to be neither nonstationarity
nor lack of ergodicity but the strong long memory in the data series. As a result,
spurious e!ects might occur more often than we previously believed as they can
arise even among stationary series. Furthermore, the usual "rst-di!erencing
procedure may not be able to completely eliminate spurious e!ects if the data
series are not only nonstationary but possess strong long memory (such as in the
case where they are I(d) processes with d'1).
2. A general theory of spurious e4ects
Our analysis of the spurious e!ects are based on several simple linear
regression models in which the dependent variable and the single nonconstant
regressor are independent I(d) processes with d lying in di!erent ranges. Before
presenting these models, let us "rst brie#y review some basic properties of the
I(d) processes.
A process > is said to be a fractionally integrated process of order d, denoted
t
as I(d), if it is de"ned by (1!¸)d> "e , where ¸ is the usual lag operator, d is
t
t
the di!erencing parameter which can be a fractional number, and the innovation
sequence e is white noise with a zero mean and "nite variance. The fractional
t
di!erencing operator (1!¸)d is de"ned as follows: (1!¸)d"+= t ¸j, where
j
t "C( j!d)/[C( j#1)C(!d)] and C( ) ) is the gamma function.j/0
This process is
j
"rst introduced by Granger (1980, 1981), Granger and Joyeux (1980), and
Hosking (1981). They show that > is stationary when d(0.5 and is invertible
t
when d'!0.5. The main feature of the I(d) process is that its autocovariance
function declines at a slower hyperbolic rate (instead of the geometric rate found
in the conventional ARMA models):
c( j)"O( j2d~1),
where c( j) is the autocovariance function at lag j. When d'0, the I(d) process is
said to have long memory since it exhibits long-range dependence in the sense
that + =
c( j)"R. When d(0, then + =
Dc ( j)D(R and the process is
j/~=
j/~= process. See Chung (1994) for
sometimes referred to as an intermediate memory
other long-memory properties of the I(d) process.
Our analysis focuses on the class of long memory I(d) processes with d'0.
We are particularly interested in the distinction between the nonstationary
subclass of I(d) processes with d*0.5 and the stationary subclass with d(0.5.
To examine potentially di!erent types of spurious e!ects, we propose six
regression models for di!erent classes of I(d) processes, mainly based on
158
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
whether the fractional di!erencing parameter d is greater than 0.5 or not. The
exact speci"cations of these models can be conveniently expressed with four I(d)
processes. Let us "rst de"ne two stationary ones with di!erent di!erencing
parameters d and d whose values lie between !0.5 and 0.5:
1
2
(1!¸)d1 v "a and (1!¸)d2 w "b ,
t
t
t
t
where a and b are two white noises with zero mean and "nite variances p2 and
t
t
a
p2, respectively; that is, v and w are I(d ) and I(d ) processes, respectively, and
b
t
t
1
2
both of them are stationary and invertible. When these two processes are
employed in our later analysis, the values of their di!erencing parameters are
mostly assumed to be in (0, 0.5); i.e., the stationary processes v and w are often
t
t
assumed to have long memory. We can also de"ne two nonstationary I(1#d )
1
and I(1#d ) processes by integrating v and w :
2
t
t
y "y #v and x "x #w .
t
t~1
t
t
t~1
t
Obviously, the orders of integration of these two nonstationary fractionally
integrated processes lie between 0.5 and 1.5. Given these four fractionally
integrated processes, we consider the following six simple linear regression
models:
Model
Model
Model
Model
Model
Model
1:
2:
3:
4:
5:
6:
y is regressed on an intercept and x ,
t
t
v is regressed on an intercept and w , where d #d '0.5,
t
t
1
2
y is regressed on an intercept and w , where d '0,
t
t
2
v is regressed on an intercept and x , where d '0,
t
t
1
y is regressed on an intercept and the trend t,
t
v is regressed on an intercept and the trend t, where d '0.
t
1
In Model 1 the orders of integration of both the dependent variable and the
regressor lie between 0.5 and 1.5, and can be equal to 1. So Model 1 may be
considered a generalization of Phillips' (1986) spurious regression to the case of
fractionally integrated processes. Model 2 presents the most interesting case in
our analysis. In it both the dependent variable and the regressor are assumed to
be stationary, ergodic, and strongly persistent in the sense that their fractional
di!erencing parameters sum up to a value greater than 0.5. Following Phillips'
arguments, we tend to think no spurious e!ect should occur in such a model
where variables are ergodic. But our analysis of Model 2 presents a result to the
contrary. The analysis of Model 2 seems to go beyond the previous study of
spurious e!ects and allows us to gain new insight into the problem.
Models 3 and 4 di!er from Model 1 in that the order of integration in one
of the dependent variable and the regressor is reduced to the stationary
range between 0 and 0.5. We can conveniently view Models 3 and 4 as two
intermediate cases between Model 1 of nonstationary fractionally integrated
processes and Model 2 of stationary fractionally integrated processes. We thus
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
159
expect the analysis of these two new models to be a mixture of those of Models
1 and 2.
In Models 5 and 6 we consider the e!ect of detrending the nonstationary and
stationary fractionally integrated processes, respectively. Through these two
models, we generalize the results of Durlauf and Phillips (1988). Also, Models
5 and 6 can be regarded as variants of Models 1 and 4, respectively, with the
nonstationary regressor x replaced by the time trend. This similarity in the
t
model speci"cations will also be re#ected in their analytic results.
The following assumption on the two white noise processes a and b are made
t
t
throughout this paper.
Assumption 1. Each of the two processes a and b is independently and identit
t
cally distributed with a zero mean; and their moments satisfy the following
conditions: EDa Dp(R, with p*maxM4, !8d /(1#2d )N; and EDb Dq(R
t
1
1
t
with q*maxM4, !8d /(1#2d )N. Moreover, a and b are independent of each
2
2
t
t
other.
We also assume, without loss of generality, that the initial values of the
fractionally integrated processes v , w , y , and x are all zero. Hence, y and
0 0 0
0
t
x can be considered as the partial sums of v and w , respectively; i.e.,
t
t
t
y "+T v and x "+T w . The independent and identical distribution
T
t/1 tour analysis and could be relaxed, say, to the
t/1 tis made Tto simplify
assumption
case where a and b are short-memory processes. See Chung (1995).
t
t
Before presenting Lemma 1, which is the cornerstone of our analysis, let us
summarize two important asymptotic results on the partial sums y and x . First,
t
t
given the variances p2"Var(y ) and p2 "Var(x ), Sowell (1990, Theorem 1)
y
T
x
T
proves that
C(1!2d )
1
p2&p2
¹1`2d1
y
a (1#2d )C(1#d )C(1!d )
1
1
1
and
C(1!2d )
2
p2&p2
¹1`2d2 ,
b (1#2d )C(1#d )C(1!d )
x
2
2
2
where z &w means z /w P1 as ¹PR. Furthermore, Davydov (1970)
T
T
T T
shows that as ¹PR,
1
1
y NB 1 (r) and
x NB 2 (r),
*Tr+
d
d
p
p *Tr+
y
x
for r3[0, 1], where [¹r] denotes the integer part of ¹r, the notation N denotes
weak convergence, and B (t) is the normalized fractional Brownian motion
d
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W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
which is de"ned by the following stochastic integral:
S
(1#2d)C(1!d)
B (t),
d
C(1#d)C(1!2d)
GP
t
P
(t!s)d dB (s)#
0
H
0
[(t!s)d!(!t)d] dB ,
0
0
~=
for d3(!0.5, 0.5), where B (t) is the standard Brownian motion. See Mandel0
brot and Van Ness (1968). Our notation for the standard and the fractional
versions of Brownian motions suggests that the former is a special case of the
latter with d"0.
The independence between v and w and between y and x implies joint weak
t
t
t
t
convergence of y
and x . More speci"cally, let z "[p~1y , p~1x ]@, then
y t x t
*Tr+
*Tr+
t
z NB (r),
*Tr+
d
where B (r)"[B 1 , B 2 ]@ is a two-dimensional normalized fractional Brownian
d
d
d
motion, of which the two elements B 1 and B 2 are independent. This result
d
d
implies the following lemma:
Lemma 1. Let Assumption 1 hold. Then, as ¹PR, we have the following results:
1.
y
x
1
1
t N:1 B (s) ds and
t N:1 B (s) ds.
+T
+T
1
d
¹ t/1 p
¹ t/1 p
0
0 d2
y
x
1
y2
x2
1
t N:1 [B (s)]2 ds.
+T t N:1 [B 1 (s)]2 ds and
+T
d
¹ t/1 p2
¹ t/1 p2
0
0 d2
y
x
2
1
(y !y6 )2
t
3. +T
N:1 [B 1 (s)]2 ds! :1 B 1 (s) ds
and
¹ t/1 p2
0 d
0 d
y
2.
C
D
1
(x !x6 )2
t
+T
N:1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2.
¹ t/1 p2
0 d
0 d
x
v
4. +T t NB 1 (1)
d
t/1 py
5.
w
t NB (1).
and +T
d2
t/1 px
1
1 c (0)"C(1!2d1 ) p2 and
+T v2 P
v
¹ t/1 t
C(1!d )2 a
1
1
1 c (0)"C(1!2d2 ) p2.
+T w2 P
w
t
¹ t/1
C(1!d )2 b
2
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
6.
161
1
1 c (0) and 1 +T (w !w6 )2 P
1 c (0).
+T (v !v6 )2 P
v
w
t
¹ t/1
¹ t/1 t
G
"O (1),
if d #d '0.5,
1
1
2
v w
7. ¹ +T t t "O [(ln ¹)0.5],
if d #d "0.5,
1
1
2
t/1 py px
"O (¹0.5~d1 ~d2 ), otherwise.
1
1
y x
8. +T t t N:1 B 1 (s) ) B 2 (s) ds.
d
¹ t/1 p p
0 d
y x
y w
v x
9. +T t t "O (1), if d '0; and +T t t "O (1), if d '0.
1
2
1
1
t/1 py px
t/1 py px
tw
t v
t NB (1)!:1 B (s) ds and +T
t NB (1)!:1 B (s) ds.
10. +T
d1
d2
0 d1
t/1 ¹ px
0 d2
t/1 ¹ py
1
ty
tx
1
t N:1 sB (s) ds and
t N:1 sB (s) ds.
+T
11. +T
d1
¹ t/1 ¹ p
¹ t/1 ¹ p
0
0 d2
y
x
Moreover, joint weak convergence of 1}4, 8, 10, and 11 also applies. Here, B 1 (t)
d
and B 2 (t) are two independent normalized fractional Brownian motions, c ( j) and
d
v
c ( j) are the autocovariance functions of v and w , respectively, at lag j, and p2 and
a
w
t
t
p2 are the variances of the underlying white noises a and b , respectively. The
b
t
t
1 means convergence in probability.
notation P
All the theorem proofs are in the appendix. Note that, while the weak
convergence of z is in the space of functions, the weak convergence established
t
in the above lemma is in the real line, which is equivalent to convergence in
distribution. Following the convention in the literature, we use the same notation N for both types of weak convergence.
In the rest of this section the results of Lemma 1 will be used to develop the
theory of spurious e!ects, presented in a series of theorems and corollaries, for
the proposed six models. The "rst two models will be discussed separately
in Sections 2.1 and 2.2. These two models provide us with a framework
which facilitates the explanations of the other four models in Sections 2.3 and
2.5. One subsection } Section 2.4 } will be devoted to the analysis of an
important issue about how the orders of fractional integration are directly
related to the spurious e!ects. The results in Lemma 1 have also been used in
deriving the limiting distributions of the &modi"ed Durbin}Watson statistics' by
Tsay (1998).
We will adopt the following notation for the various statistics from the
Ordinary Least Squares (OLS) estimation. Let a( and bK denote the usual OLS
estimators of the intercept and the slope. Their respective variances are estimated by s2 and s2, from which we have the t ratios t "bK /s and t "a( /s .
a
b
b
a
a
b
Also, let s2 denote the estimated variance of the OLS residuals, R2 the coe$cient
162
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
of determination, and D= the Durbin}Watson statistic. Finally, in addition to
the autocovariance functions c ( j) and c ( j) of v and w , let o ( j) and o ( j) be
v
w
t
t
v
w
their respective autocorrelations at lag j.
2.1. Model 1 of nonstationary fractionally integrated processes
In Model 1 a nonstationary I(1#d ) process y is regressed on another
1
t
independent and nonstationary I(1#d ) process x . Since the permissible range
2
t
for the values of the fractional di!erencing parameters d and d is (!0.5, 0.5),
1
2
Model 1 generalizes Phillips' (1986) model of integrated processes in which
d "d "0. Not surprisingly, all the results we derive for Model 1 are straight1
2
forward generalization of Phillips' theory of the spurious e!ects. The results for
Model 1 are presented in the following theorem:
Theorem 1. Let Assumption 1 hold. Then, as ¹PR, we have the following results:
1.
p
:1 B (s)B 2 (s) ds![:1 B 1 (s) ds][:1 B 2 (s) ds]
x bK N 0 d1
d
,b .
0 d
0 d
H
p
:1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2
y
d
d
0
0
Note that p /p "O(¹d1 ~d2 ).
y x
2.
1
a( N:1 B 1 (s) ds!b :1 B 2 (s) ds,a , where b is dexned in 1. Note that
H 0 d
H
H
p
0 d
y
p "O(¹0.5`d1 ).
y
3.
1
s2N:1 [B 1 (s)]2 ds![:1 B 1 (s) ds]2!b2 M:1 [B 2 (s)]2 ds![:1B 2 (s) ds]2N
H 0 d
p2
0 d
0 d
0 d
y
,p2 , where b is dexned in 1. Note that p2"O(¹1`2d1 ).
y
H
H
4.
p2
¹p2
H
x s2 N
,p2 , where p2 is dexned in 3.
H
Hb
p2 b :1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2
y
0 d
0 d
Note that p2/¹p2 "O(¹2d1 ~2d2 ~1).
x
y
5.
G
H
[:1 B 2 (s) ds]2
¹
0 d
s2Np2 1#
,p2 , where p2 is dexned
H
H
Ha
:1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2
p2 a
y
0 d
0 d
in 3. Note that p2/¹"O(¹2d1 ).
y
6.
b
t N H , where b is dexned in 1 and p2 is dexned in 4.
Hb
b
H
p
J¹
Hb
1
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
7.
8.
9.
163
a
t N H , where a is dexned in 2 and p2 is dexned in 5.
Ha
a
H
p
J¹
Ha
b2 M:1 [B (s)]2 ds![:1 B 2 (s) ds]2N
0 d
R2N H 0 d2
, where b is dexned in 1.
H
:1 [B 1 (s)]2 ds![:1 B 1 (s) ds]2
0 d
0 d
1
1 0.
D= P
Here, B 1 (t) and B 2 (t) are two independent normalized fractional Brownian
d
d
motions.
The most important result in Theorem 1 is that, as the sample size ¹ increases,
the two t ratios t and t diverge at the same rate of J¹, which is independent of
b
a
the magnitudes of the fractional di!erencing parameters d and d . This result is
1
2
exactly the same as what Phillips (1986) has obtained for the case where
d "d "0. So even when the orders of integration in the dependent variable
1
2
and the regressor di!er from 1 by as much as 0.5, the usual problem in using the
t tests remains: the probability of rejecting the null hypothesis of b"0 or
a"0 based on t tests increases monotonically as the sample size increases. Also
note that Marmol (1995) generalizes Phillips' theory to cases where both y and
t
x are integrated processes of the same integer orders that are higher than one.
t
The limiting distributions in Theorem 1 for the special case where d "d are
1
2
also very similar to Marmol's results.
The limiting distributions of the t ratios, after normalized by J¹, are direct
generalization of those derived by Phillips (1986). The same conclusion also
holds for R2 and the D= statistics. In other words, when we compare our results
with Phillips', we observe a common feature in these four statistics; namely, the
nonzero values of d and d do not a!ect their convergence rates while the
1
2
e!ects on their limiting distributions are quite straightforward: all the standard
Brownian motions in Phillips' theory are replaced by fractional Brownian
motions. That the fractional di!erencing parameters d and d play a relative
1
2
minor role here is mainly because the four statistics are all ratios so that the e!ects
of d and d are cancelled out. In contrast, the results on the OLS estimators
1
2
bK and a( are a di!erent story. In Phillips' theory both bK and a( /J¹ converge to
some non-normal non-degenerate limiting distributions. But for the present
model of the fractionally integrated processes, the orders of bK and a( are ¹d1 ~d2 and
¹d1 `0.5, respectively. So while a( always diverges (though the rate can be slow if
d is close to !0.5), bK can be either divergent or convergent, depending on the
1
relative magnitudes of d and d . For example, if the order of integration in the
1
2
dependent variable y is smaller than that of the regressor x ; i.e., d (d , then
t
t
1
2
bK converges to zero, just like the conventional case of no spurious e!ects.
Moreover, if d !d "!0.5, then, similar to the case of no spurious e!ects,
1
2
J¹ ) bK has a limiting distribution, though its limiting distribution is not normal.
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W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
2.2. Model 2 of stationary fractionally integrated processes
In this section we consider Model 2 in which a stationary fractionally integrated process v is regressed on an independent and stationary fractiont
ally integrated process w . We show that, although both v and w are stationary,
t
t
t
the spurious e!ect in terms of the t tests could still exist under an additional
condition on the fractional di!erencing parameters: d #d '0.5. Loosely
1
2
speaking, this condition implies that the two processes v and w are both
t
t
strongly persistent.
Our analysis begins with a special case where we assume a set of more
stringent conditions which helps deriving the limiting distribution of the OLS
estimator. This theory is based on an important result of Fox and Taqqu (1987)
who show that the product of two highly persistent but stationary Gaussian
processes, if adequately normalized, can converge. After examining this special
case, we then show how the spurious e!ects may still exist in a more general
framework.
Let us "rst reproduce Fox and Taqqu's (1987) Theorem 6.1 here as Lemma 2.
Lemma 2. Let (X , > ) be a stationary jointly Gaussian sequence with
t t
E(X )"E(> )"0, E(X2)"E(>2)"1, and E(X > )"r. Suppose that p and
t
t t
1
t
t
t
p are two arbitrary real numbers and that there exist 0(d , d (0.5, such that
2
1 2
as jPR
E(X X )&p2 j~d1 ,
1
t t`j
op p b
E(X > )& 1 2 1 j~(d1 `d2 )@2,
t t`j
Ja a
1 2
E(> > )&p2 j~d2 ,
2
t t`j
op p b
E(> X )& 1 2 2 j~(d1 `d2 )@2,
t t`j
Ja a
1 2
where o is a constant between 1 and !1, while a "A(d , d ), a "A(d , d ),
1
1 1 2
2 2
b "A(d , d ), and b "A(d , d ) are four constants with A(d ,d ) being dexned
1
1 2
2
2 1
1 2
by :=x~(d1 `1)@2(x#1)~(d2 `1)@2 dx, then
0
1
*Ts+
+ (X > !r)NZ(s),
t t
¹1~(d1 `d2 )@2
t/1
where
p p
Z(s)" 1 2
Ja a
1 2
P PC
s
R2
D
2
< (u!x )~(di `1)@2IM i N du dM (x ) dM (x ).
x :u
i
1 1
2 2
0 i/1
Here, M and M are two Gaussian random measures with respect to Lebesgue
1
2
measure, having unit variances and covariance o.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
165
Note that the two processes X and > are not only strongly persistent, as
t
t
indicated by the hyperbolic convergence rates d and d in their autocorrela1
2
tions, but also highly correlated with each other, as indicated by the hyperbolic
convergence rates in their covariances. However, in our application we are only
interested in the case where X and > are independent so that r and o in the
t
t
above lemma are both zero. The above lemma o!ers us the convergence rate of
+T X > and its limiting process Z(t) given the Gaussian assumption and
t/1 t t
a narrower range for the parameters d and d . In order to apply this lemma, we
1
2
make the following assumption in addition to Assumption 1 made earlier.
Assumption 2. The two fractionally integrated processes v and w are both
t
t
Gaussian and the corresponding fractional di!erencing parameters d and
1
d are both in the range of (0.25, 0.5).
2
Given the facts that
C(1!d )
C(1!d )
1 j2d1 ~1 and o ( j)&
2 j2d2 ~1,
o ( j)&
v
w
C(d )
C(d )
1
2
it is straightforward to prove the following corollary in which X and > in
t
t
Lemma 2 are replaced by v /Jc (0) and w /Jc (0), respectively.
t
w
t
v
Corollary 1. Let Assumptions 1 and 2 hold. Then, as ¹PR,
1
v
T
w
t
t NZ(1),
+
¹d1 `d2
t/1 Jcv (0) Jcw (0)
where the limiting random variable Z(1) is dexned in Lemma 2 with d "1!2d ,
1
1
d "1!2d , p2 "C(1!d )/C(d ), and p2 "C(1!d )/C(d ). Consequently,
2
2 1
1
1
2
2
2
we have
T v w
¹ + t t NCJc (0) ) c (0) Z(1),
v
w
p p
t/1 y x
where
C2"C(1!2d )C(1!2d )/(1#2d )C(1#d )C(1!d )(1#2d )
1
2
1
1
1
2
C(1#d )C(1!d ).
2
2
The result of Corollary 1 supplements that of item 7 in Lemma 1. From these
results, we can then establish the following theorem about the spurious e!ect in
Model 2.
Theorem 2. Let Assumptions 1 and 2 hold. Then, as ¹PR, we have the following
results:
1.
¹2
bK "O (1). Note that p p /¹2"O(¹d1 `d2 ~1).
1
y x
p p
y x
166
2.
3.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
¹
a( NB 1 (1). Note that p /¹"O(¹d1 ~0.5).
d
y
p
y
1 c (0).
s2 P
v
4.
1 cv (0) .
¹s2 P
b
c (0)
w
5.
1 c (0).
¹s2 P
a
v
6.
¹3@2
t "O (1). Note that p p /¹3@2"O(¹d1 `d2 ~0.5).
1
y x
p p b
y x
7.
J¹
1
t N
B (1). Note that p /J¹"O(¹d1 ).
a
y
p
Jc (0) d1
y
v
¹4
R2"O (1). Note that p2p2/¹4"O(¹2d1 `2d2 ~2).
y x
1
p2p2
y x
1 2!2o (1)"2(1!2d1 ) .
9.
D= P
v
1!d
1
Here B 1 (t) is a normalized fractional Brownian motions.
d
8.
The most important result from this theorem is the divergence rates of the two
t ratios t and t , which are ¹d1 `d2 ~0.5 and ¹d1 , respectively. Recall that
b
a
d #d !0.5 is necessarily greater than 0 (and smaller than 0.5) under Assump1
2
tion 2. This result re#ects the spurious e!ect in the t tests. Since both
the dependent variable and the regressor are stationary and ergodic, the spurious e!ect is not really expected (see Phillips 1986, p. 318). The surprising
results we get here suggest that the cause for the spurious e!ect has more to do
with the strong persistence than stationarity and ergodicity of the variables
involved.
It is interesting to compare the divergence rates of the t ratios here with the
J¹ rate we observe in Model 1. We note that the divergence rates in the present
model depend on the magnitudes of the two fractional di!erencing parameters
d and d while those in Model 1 do not. Furthermore, the t ratios diverge more
1
2
slowly in the present model than in Model 1. In particular, the divergence rate of
t can become very slow when both d and d approach to their lower boundary
b
1
2
0.25.
Let us turn to the OLS estimators bK and a( . We note that both of them
converge in probability to zero as in the conventional case of no spurious e!ects.
However, their convergence rates are much slower than the usual ¹~1@2 rate. In
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
167
contrast to these irregular convergence rates of the OLS estimators, the
estimated variances s2 and s2 nevertheless converge at the standard ¹~1 rate. It
a
b
is such disparity in the convergence rates between the OLS estimators, which
converge at rates slower than ¹~1@2, and their standard errors, which converge
at the standard ¹~1@2 rates, that causes the resulting t ratios to diverge and
hence the spurious e!ect.
R2 in the present model converges to 0 as in the case of no spurious e!ects. It
is di!erent from what we observe in Model 1 where R2 converges to a random
variable. Consequently, as the sample size increases, the declining R2 in the
present model will correctly re#ect the fact that the regressor does not help
explain the variations in the dependent variable.
The D= statistic does not converge in probability to zero and this result is
also di!erent from that of Model 1. Its limit 2!2o (1) is similar to the one we
v
"nd in the conventional AR(1) case. This limit depends on the fractional
di!erencing parameter d of the dependent variable v and can only take value
1
t
in the range of (0,4/3), which is to the left of the value 2.
There is one technical detail that calls for some explanations: Unlike in
Theorem 1, we do not give an explicit expression for the limiting distribution of
bK in Theorem 2 because such an expression requires the joint weak convergence
of the sample averages of v , w , and v w , while the proof of the joint weak
t t
t t
convergence is beyond the scope of the present paper. However, lacking in an
explicit expression for the limiting distribution of bK does not hinder us from
evaluating the convergence rate of bK , which is all we need to show the spurious
e!ects in the t ratios. This same argument applies to some of the later analyses,
including the following one where we consider a less restricted speci"cation of
Model 2 that is de"ned by the following assumption.
Assumption 3. The sum of the two fractional di!erencing parameters d and
1
d is greater than 0.5.
2
Since the Gaussian distribution is not assumed while one of the fractional
di!erencing parameters d and d can be smaller than 0.25, Assumption 3 is thus
1
2
less stringent than Assumption 2.
Corollary 2. If Assumption 2 is replaced by Assumption 3 in Theorem 2, then all the
conclusions there remain true.
The analysis of Model 2 can thus be summarized as follows. The OLS
estimators bK and a( (as well as R2) do converge in probability to zero, correctly
re#ecting the lack of a relationship between the dependent variable and the
regressor. But the convergence rates of bK and a( are too slow in comparison with
those of their standard errors. Consequently, the t ratios diverge and the t tests
fail. The upshot is that the usual t tests can become invalid even when the
168
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
dependent variable and the regressor are both stationary and ergodic (so long as
they are su$ciently persistent).
A profound implication from Model 2 is as follows: If we begin with Model
1 where both the dependent variable and the regressor are nonstationary
fractionally integrated processes with the orders of integration 1#d and
1
1#d , respectively, where d #d '0.5, then "rst-di!erencing both variables
2
1
2
cannot completely eliminate the spurious e!ects. While R2 may be reduced and
the D= statistic may be increased, the t ratios may still be so large that we
cannot avoid making a spurious inference. This is a fairly serious problem with
the regression for the fractionally integrated processes. It implies that even the
popular "rst-di!erencing procedure might not prevent us from "nding a spurious relationship among highly persistent data series. One lesson we learn from
this discussion is that it is very important to check individual data series for
possible long memory before regression can be applied.
2.3. Two intermediate cases: Models 3 and 4
Models 3 and 4 can be considered as two intermediate models between
Models 1 and 2 in that one of the dependent variable and the regressor is
stationary while the other is not. We expect the asymptotic results for these two
new models to be hybrid of those of Models 1 and 2.
In Model 3 a nonstationary I(1#d ) process y is regressed on an indepen1
t
dent and stationary I(d ) process w . Note that the fractional di!erencing
2
t
parameter d for the regressor w here is assumed to be positive; i.e., w has long
2
t
t
memory. The asymptotic properties of the OLS estimators for Model 3 are given
in the following theorem:
Theorem 3. Let Assumption 1 hold. Then, as ¹PR, we have the following results.
1.
2.
3.
¹
bK "O (1). Note that p p /¹"O(¹d1 `d2 ).
1
y x
p p
y x
1
a( N:1 B 1 (s) ds,a . Note that p "O(¹0.5`d1 ).
H
y
p
0 d
y
1
s2N:1 [B 1 (s)]2 ds![:1 B 1 (s) ds]2,p2 . Note that p2"O(¹1`2d1 ).
H
y
p2
0 d
0 d
y
4.
¹
p2
s2N H , where p2 is dexned in 3. Note that p2/¹"O(¹2d1 ).
H
y
p2 b c (0)
y
w
5.
¹
s2Np2 , where p2 is dexned in 3. Note that p2/¹"O(¹2d1 ).
y
H
H
p2 a
y
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
6.
7.
8.
9.
169
J¹
t "O (1). Note that p /J¹"O(¹d2 ).
1
x
p b
x
a
t N H , where a is dexned in 2 and p2 is dexned in 3.
a
H
H
p
J¹
H
¹2
R2"O (1). Note that p2/¹2"O(¹2d2 ~1).
1
x
p2
x
1
1 0.
D= P
Here B 1 (t) is a normalized fractional Brownian motions.
d
Since both t ratios diverge, Model 3 also su!ers from the spurious e!ect in
terms of the t tests. Moreover, we "nd the results that the OLS estimator
a( diverge and that D= converges in probability to 0 are close to what we get in
Model 1, while the result of converging R2 is the same as that of Model 2. So
Model 3 is indeed a mixture of Models 1 and 2.
It should be pointed out that in Theorem 3 the range of the fractional
di!erencing parameter d of the regressor w is restricted to the positive half of
2
t
the original range (!0.5, 0.5). For the case of a negative d , it is quite
2
straightforward to show that the t ratios are convergent and there is no spurious
e!ect.
In Model 4 a stationary I(d ) process v is regressed on an independent and
1
t
nonstationary I(1#d ) process x . Similar to the restriction imposed on
2
t
Model 3, the fractional di!erencing parameter d of the dependent variable v is
1
t
assumed to be positive so that v has long memory. The asymptotic theory for
t
Model 4 is presented in the following theorem.
Theorem 4. Let Assumption 1 hold. Then, as ¹PR, we have the following results.
1.
¹p
x bK "O (1). Note that p /¹p "O(¹d1 ~d2 ~1).
1
y
x
p
y
2.
¹
a( "O (1). Note that p /¹"O(¹d1 ~0.5).
1
y
p
y
3.
1 c (0).
s2 P
v
4.
c (0)
v
¹p2 ) s2N
,p2 .
x b :1 [B (s)]2 ds![:1 B (s) ds]2
Hb
2
2
d
d
0
0
Note that 1/¹p2"O(¹~2~2d2 ).
x
170
5.
6.
7.
8.
9.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
[:1 B 2 (s) ds]2
0 d
¹ ) s2Nc (0)M1#
N,p2 .
a
v
Ha
:1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2
0 d
0 d
J¹
t "O (1). Note that p /J¹"O(¹d1 ).
1
y
p b
y
J¹
t "O (1). Note that p /J¹"O(¹d1 ).
1
y
p a
y
¹2
R2"O (1). Note that p2/¹2"O(¹2d1 ~1).
1
y
p2
y
1 2!2o (1)"2(1!2d1 ) .
D= P
v
(1!d )
1
Here B 2 (t) is a normalized fractional Brownian motions.
d
Since both t ratios diverge (at the same rate), the spurious e!ect in terms of
failing t tests again exists in Model 4. But contrary to the results in Model 3, the
OLS estimators bK and a( , together with R2, all converge in probability to zero,
while the D= statistic converges to 2!2o (1). These "ndings obviously bring
v
Model 4 closer to Model 2 than to Model 1.
2.4. The relationship between the orders of integration and the divergence rates
The divergent t ratios in the above four models and the resulting failure of
the t tests are referred to as the spurious e!ects. In this section we compare the
divergence rates of t ratios across the four models and investigate how they are
related to the respective model speci"cations.
First note that the divergence rates of the t ratio t are ¹0.5, ¹d1 `d2 ~0.5, ¹d2 ,
b
and ¹d1 , respectively, in Models 1}4. Let us also compare the speci"cations of
the four models using Model 1 as the benchmark: Model 3 di!ers from Model
1 in that the order of integration in the regressor is reduced from above 0.5 to
below 0.5 (but above 0); Model 4 di!ers from Model 1 in that the order of
integration in the dependent variable is reduced from above 0.5 to below 0.5 (but
above 0); and, "nally, Model 2 di!ers from Model 1 in that the orders of
integration in both the dependent variable and the regressor are reduced from
above 0.5 to below 0.5 (but their sum is assumed to be greater than 0.5). By
associating these changes in the orders of integration with the changes in the
divergence rates of t , we can conclude that reducing the order of integration in
b
the dependent variable causes the divergence rate of t to decrease by the order
b
of ¹d1 ~0.5 and reducing the order of integration in the regressor causes the
divergence rate of t to decline by the order of ¹d2 ~0.5, while these two e!ects
b
are cumulative as in Model 2.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
171
Recall that in Models 2}4 restrictions have been imposed on the usual ranges
(!0.5, 0.5) of the fractional di!erencing parameters d and d . In Model 3 the
1
2
range of d is restricted to be (0, 0.5), which is also the range of d in Model 4,
2
1
while the sum of d and d must be greater than 0.5 in Model 2. From the
1
2
analysis in the previous paragraph, particularly the fact that the divergence rates
are directly related to the magnitudes of d and d , we come to realize that
1
2
the restricted ranges of d and d in Models 2}4 ensure the reduction in the
1
2
divergence rates of t from the ¹0.5 level is not too great so that t remains
b
b
divergent (in which case the spurious e!ects occur). Although we did not
explicitly consider the asymptotic theory for cases where the fractional di!erencing parameters lie outside their prescribed ranges, it is readily seen that the
conditions we impose on the ranges are not only su$cient but also necessary for
the existence of the spurious e!ect in terms of divergent t .
b
From a similar analysis for the divergence rates of the t ratio t we also "nd
a
that reducing the order of integration in the dependent variable causes the
divergence rate of t to decrease by the order of ¹d1 ~0.5, while reducing
a
the order of integration in the regressor does not cause the divergence rate of
t to change, as we probably should have expected.
a
It is also interesting to see how the changes in the orders of integration of the
dependent variable and the regressor a!ect the large-sample property of R2.
Recall that in Model 1 R2 converges to a random variable and such asymptotic
behavior of R2 is considered part of the spurious e!ect by Phillips (1986). But
when we examine Models 2}4, we note that reducing the order of integration in
the dependent variable helps to increase its convergence rate by the order of
¹1~2d1 while reducing the order of integration in the regressor helps to increase
the convergence rate by the order of ¹1~2d2 . As a result, in Models 2}4, R2 all
converge to 0, correctly re#ecting the fact that there is no relationship between
the regressor and the dependent variable. This "nding implies that the spurious
e!ects in Models 2}4 are con"ned to the two t ratios while the asymptotic
tendency of R2 toward zero is not a!ected by the spurious e!ects (though the
convergence rates are).
The sharp di!erence in the asymptotic behavior between the t ratios and R2 in
Models 2}4 actually o!ers us an opportunity to diagnose the spurious e!ect in
these models. That is, when we "nd two highly signi"cant t ratios coexisting with
a completely contradictory near-zero R2, we are e!ectively reminded of the
possibilities that one of the Models 2}4 may be at work and that the dependent
variable and the regressor may possess strong long memory, while one of them
may even be nonstationary. With the possibility of such an informal diagnosis, it
seems that the spurious e!ects in Models 2}4 are less damaging than those in
Model 1 in the sense that in Model 1 there is no internal inconsistency among
the OLS estimates to indicate the spurious e!ects.
Finally, let us brie#y state a few more results about the asymptotic tendency
of the OLS estimators bK and a( and the D= statistic. First, we note that bK will
172
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
converge unless the dependent variable is nonstationary and its order of integration is su$ciently large. Secondly, whether a( diverges or not and whether the
D= statistic converges in probability to zero or not depend entirely on whether
the dependent variable is nonstationary or not. Note that, as mentioned earlier,
even though the OLS estimators bK and a( can converge in probability to zero in
the four proposed models, the corresponding t ratios always diverge and it is
these divergent t ratios that are referred to as the spurious e!ects.
2.5. Models 5 and 6: Detrending fractionally integrated processes
As has been pointed out by Nelson and Kang (1981, 1984) and Durlauf and
Phillips (1988), detrending integrated processes results in the spurious e!ect of
"nding a signi"cant trend. In this section we extend their analysis by considering
the potential problems in detrending fractionally integrated processes. It turns
out that the spurious e!ect of divergent t ratios exists as long as the fractional
di!erencing parameter is larger than zero. The implication is that whenever
there is long memory in the process, the routine procedure of detrending can
produce misleading results. It appears that the spurious e!ect in detrending
occurs more often than we previously thought.
In our analysis of detrending fractionally integrated processes, we separate
the nonstationary case from the stationary case. In Model 5 we examine the
regression of a nonstationary I(1#d ) process y on a time trend t. The
1
t
asymptotic theory for the OLS estimation is given in the following theory.
Theorem 5. Let Assumption 1 hold. Then, as ¹PR, we have the following results.
1.
¹
bK N12:1 sB 1 (s) ds!6:1 B 1 (s) ds,b . Note that p /¹"O(¹d1 ~0.5).
H
y
p
0 d
0 d
y
2.
1
a( N4:1 B 1 (s) ds!6:1 sB 1 (s) ds,a .
H
p
0 d
0 d
y
3.
1
1
s2N:1 [B 1 (s)]2 ds![:1 B 1 (s) ds]2!12[:1 sB 1 (s) ds! :1 B 1 (s) ds]2
d
d
d
p2
2 0 d
0
0
0
y
,p2 .
H
4.
¹3
s2N12p2 , where p2 is dexned in 3. Note that p2/¹3"O(¹2d1 ~2).
y
H
H
p2 b
y
5.
¹
s2N4p2 , where p2 is dexned in 3. Note that p2/¹"O(¹2d1 ).
y
H
H
p2 a
y
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
6.
7.
8.
173
1
t Nb /J12p , where b is dexned in 1 and p2 is dexned in 3.
H
H
H
H
J¹ b
1
t Na /2p , where a is dexned in 2 and p2 is dexned in 3.
H
H H
H
J¹ a
b2
H
R2N
,
12:1 [B 1 (s)]2 ds!12[:1 B 1 (s) ds]2
d
d
0
0
where b is dexned in 1.
H
1 0 and p2D=Nc (0)/p2 .
D= P
H
y
v
Here, B 1 (t) is a normalized fractional Brownian motion.
d
9.
The results on detrending a stationary long memory I(d ) process v , which is
1
t
our Model 6, are presented in the following theorem.
Theorem 6. Let Assumption 1 hold. Then, as ¹PR, we have the following results.
1.
¹2
bK N6B 1 (1)!12:1 B 1 (s) ds,b . Note that p /¹2"O(¹d1 ~1.5).
d
H
y
p
0 d
y
2.
¹
a( N6:1 B 1 (s) ds!2 B 1 (1),a . Note that p /¹"O(¹d1 ~0.5).
d
H
y
p
0 d
y
3.
1 c (0).
s2 P
v
4.
1 12c (0).
¹3s2 P
b
v
5.
1 4c (0).
¹s2 P
a
v
6.
J¹
t Nb /J12c (0), where b is dexned in 1. Note that p /J¹"O(¹d1 ).
H
v
H
y
p b
y
7.
J¹
t Na /2Jc (0), where a is dexned in 2. Note that p /J¹"O(¹d1 ).
H
v
H
y
p a
y
8.
¹2
R2Nb2 /12c (0), where b is dexned in 1. Note that p2/¹2"O(¹2d1 ~1).
y
H
v
H
p2
y
174
9.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
1 2!2o (1)"2(1!2d )/(1!d ).
D= P
v
1
1
Here, B 1 (t) is a normalized fractional Brownian motion.
d
In terms of the convergence (or divergence) rates of the various OLS estimators, Models 5 and 6 can be conveniently viewed as &special cases' of Models
1 and 4, respectively. More speci"cally, if we replace the term p by ¹ in those
x
normalizing factors in Theorems 1 and 4, then we immediately get all the
normalizing factors in Theorems 5 and 6. For example, while the normalizing
factor for bK in Theorem 1 is p /p , the one in Theorem 5 is ¹/p . Similarly, while
x y
y
the normalizing factor for bK in Theorem 4 is ¹p /p , the one in Theorem 6 is
x y
¹2/p . (Also note that the fractional Brownian motion B 2 (t) does not appear in
y
d
Theorems 5 and 6 since the regressors in Models 5 and 6 are the time trend and
are unrelated to the I(d ) process w .) Given these observations, we then
2
t
conclude that all the analyses about Models 1 and 4 can be readily extended to
Models 5 and 6. In particular, the divergence rates of the t ratios, which
respectively are in the orders of ¹0.5 and ¹d1 in Models 1 and 4, are also the
rates in Models 5 and 6. (Note that in both Models 4 and 6 the same condition
d '0 is imposed on the stationary dependent variable v so that the resulting
1
t
t ratios are divergent.) As a result, the type of spurious e!ects we observe in
Models 1 and 4 occur again in Models 5 and 6. That is, detrending a fractionally
integrated process with a positive fractional di!erencing parameter, certainly
including the usual case of the I(1) process, will result in the spurious "nding of
a signi"cant trend. One important inference we draw from Models 5 and 6 is
that the cause for the spurious e!ect in detrending a process is neither nonstationarity nor lack of ergodicity but long memory in the process.
From Models 5 and 6 we also note the following result: If the data series are
nonstationary with the order of integration greater than 1, then the spurious
e!ect can happen to the detrending procedure even after the series are "rst
di!erenced. What "rst di!erencing does to the detrending procedure in such
a case is simply reducing R2, increasing the value of the D= statistic, and
slowing down the divergence of the two t ratios from the ¹0.5 rate to the
¹d1 rate. Based on this observation, it seems that the spurious e!ects in
detrending may occur more often than we previously thought.
3. Conclusion
In our analysis of spurious regressions for the long memory fractionally
integrated processes, we "nd that no matter whether the dependent variable and
the regressor are stationary or not, as long as their orders of integration sum up
to a value greater than 0.5, the t ratios become divergent. So it is the long
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
175
memory, instead of nonstationarity or lack of ergodicity, that causes the spurious e!ects in terms of failing t tests. Nonstationarity in one or both of the
dependent variable and the regressor only helps to accelerate the divergence
rates of the t ratios. We thus learn that spurious e!ects might occur more often
than we previously believed as they can arise even among stationary series and
the usual "rst-di!erencing procedure may not be able to completely eliminate
spurious e!ects when data possess strong long memory. It is interesting to note
Phillips (1995) recently has o!ered some contrast thoughts about spurious
regressions (and he argues that spurious regressions may not be as serious as
many researchers have been led to believe).
In Section 2.4 we have carefully examined the exact relationships between the
orders of integration in the fractionally integrated processes and the divergence
rates in the t ratios. From this analysis we gain many insights into the problem
of spurious e!ects which are not available in Phillips' (1986) classical study of
I(1) processes. In short, it is found that the extents of spurious e!ects are directly
related to the degrees of long memory in the data. Our results on detrending
fractionally integrated processes also greatly broaden Durlauf and Phillips'
(1988) theory of spurious detrending in which the relationship between the
orders of integration and the divergence rates of the t ratios again plays a useful
role in the analysis.
A fairly extensive Monte Carlo study has also been conducted to verify the
theoretical results, especially those of convergence rates, we have established in
the paper. We do not report the simulation results here other than pointing out
the fact that almost all our theoretical results are well supported by simulation.
A few generalizations of our study are worthy of further consideration.
A natural extension is to consider the multiple regression where there are more
than one nonconstant regressor. Another one is to allow the fractionally integrated processes to have nonzero means. Based on Phillips' (1986) work, we
expect most, if not all, of the asymptotic results we obtain from the simple
regression case to hold in the multiple regression of fractionally integrated
processes with drifts. These issues have been examined by Chung (1995).
One aspect of our study that is slightly more restricted than Phillips, (1986)
and Durlauf and Phillips' (1988) analysis is that the fractionally integrated
processes we consider are built on white noises a and b that are required to
t
t
satisfy the relatively stringent conditions as speci"ed in Assumption 1. These
conditions e!ectively rule out the possibility of allowing short-run dynamics
such as the ARMA components in the fractionally integrated processes we have
studied. Chung (1995) has studied the case where Assumption 1 is relaxed to
incorporate the short-run dynamics and found no substantial changes in the
analysis of spurious e!ects.
Finally, our study of spurious regression can serve as the basis for the analyses
of &fractional cointegration' where the dependent variable and regressors are
related I(d) processes. This line of the work appears to be quite important and
176
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
has attracted a lot of attention in the literature recently. One of the pioneer
works in this area is by Cheung and Lai (1993).
Acknowledgements
We are very grateful for two referees and an associate editor for their valuable
suggestions.
Appendix A. Proof of Lemma 1
The proofs of items 1}3 are straightforward applications of the continuous
mapping theorem to the Davydov's results. They are omitted here. Item 4 follows directly from Davydov's result, while items 5 and 6 are due to ergodicity of
the two stationary processes v and w .
t
t
To prove item 7, we note, since v and w are assumed to b
The spurious regression of fractionally
integrated processes
Wen-Jen Tsay!,*, Ching-Fan Chung"
!The Institute of Economics, Academia Sinica, Nankang, Taipei, Taiwan
"National Taiwan Univeristy, Taiwan
Received 1 September 1995; received in revised form 1 July 1999
Abstract
This paper extends the theoretical analysis of the spurious regression and spurious
detrending from the usual I(1) processes to the long memory fractionally integrated
processes. It is found that when we regress a long memory fractionally integrated process
on another unrelated long memory fractionally integrated process, no matter whether
these processes are stationary or not, as long as their orders of integration sum up to
a value greater than 0.5, the t ratios become divergent and spurious e!ects occur. Our
"nding suggests that it is the long memory, instead of nonstationarity or lack of
ergodicity, that causes such spurious e!ects. As a result, spurious e!ects might happen
more often than we previously believed as they can arise even between stationary series
while the usual "rst-di!erencing procedure may not completely eliminate spurious e!ects
when data possess strong long memory. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C22
Keywords: Fractionally integrated processes; Long memory; Spurious regression;
Spurious detrending
* Corresponding author. Tel.: 886-227822791 ext. 296, FAX: 886-27853946.
E-mail: wtsay@ieas.econ.sinica.edu.tw (W.-J. Tsay)
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 5 6 - 1
156
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
1. Introduction
The spurious regression was "rst studied by Granger and Newbold (1974)
using simulation. They show that when unrelated data series are close to the
integrated processes of order 1 or the I(1) processes, then running a regression
with this type of data will yield spurious e!ects. That is, the null hypothesis of no
relationship among the unrelated I(1) processes will be rejected much too often.
Furthermore, the spurious regression tends to yield a high coe$cient of determination (R2) as well as highly autocorrelated residuals, indicated by a very low
value of Durbin}Watson (D=) statistic. Granger and Newbold's simulation
results are later supported by Phillips' (1986) theoretical analysis. Phillips proves
that the usual t test statistic in a spurious regression does not have a limiting
distribution but diverges as the sample size approaches in"nity. He also shows
that R2 has a non-degenerate limiting distribution while the D= statistic
converges in probability to zero. Phillips' results has been generalized by
Marmol (1995) to cases with integrated processes of higher orders.
The history of the research on spurious detrending follows a similar thread.
Nelson and Kang (1981, 1984) "rst employ simulation to demonstrate that the
regression of a driftless I(1) process on a time trend produces an incorrect result
of a signi"cant trend. Extending the Phillips' (1986) approach, Durlauf and
Phillips (1988) derive the asymptotic distributions for the least squares estimators in such a regression. In particular, the latter authors show that the t test
statistics diverge and there are no correct critical values for the conventional
signi"cance tests.
Most studies of the spurious regression concentrate on the nonstationary I(1)
processes. It re#ects the widely held belief that many data series in economics are
I(1) processes, or near I(1) processes, as argued by Nelson and Plosser (1982).
Against this backdrop, we also witness in recent years fast growing studies on
fractionally integrated processes, or the I(d) processes with the di!erencing
parameter d being a fractional number. The I(d) processes are natural generalization of the I(1) processes that exhibit a broader long-run characteristics. More
speci"cally, the I(d) processes can be either stationary or nonstationary, depending on the value of the fractional di!erencing parameter. The major characteristic of a stationary I(d) process is its long memory which is re#ected by the
hyperbolic decay in its autocorrelations. A number of economic and "nancial
series have been shown to possess long memory. See Baillie (1996) for an
updated survey on the applications of the I(d) processes in economics and
"nance.
The objective of this paper is to extend the theoretical analysis of the spurious
regression from I(1) processes to the class of long-memory I(d) processes. We
establish and analyze conditions on the I(d) processes that in#ict the spurious
e!ect in a simple linear regression model. The nonstandard asymptotic distributions of various coe$cient estimators and test statistics are then derived.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
157
The main "nding from our study is that the spurious regression can arise
among a wide range of long-memory I(d) processes, even in cases where both
dependent variable and regressor are stationary. A few conclusions may then be
drawn. First, di!erent from what Phillips (1986) and Durlauf and Phillips (1988)
have suggested, the cause for spurious e!ects seems to be neither nonstationarity
nor lack of ergodicity but the strong long memory in the data series. As a result,
spurious e!ects might occur more often than we previously believed as they can
arise even among stationary series. Furthermore, the usual "rst-di!erencing
procedure may not be able to completely eliminate spurious e!ects if the data
series are not only nonstationary but possess strong long memory (such as in the
case where they are I(d) processes with d'1).
2. A general theory of spurious e4ects
Our analysis of the spurious e!ects are based on several simple linear
regression models in which the dependent variable and the single nonconstant
regressor are independent I(d) processes with d lying in di!erent ranges. Before
presenting these models, let us "rst brie#y review some basic properties of the
I(d) processes.
A process > is said to be a fractionally integrated process of order d, denoted
t
as I(d), if it is de"ned by (1!¸)d> "e , where ¸ is the usual lag operator, d is
t
t
the di!erencing parameter which can be a fractional number, and the innovation
sequence e is white noise with a zero mean and "nite variance. The fractional
t
di!erencing operator (1!¸)d is de"ned as follows: (1!¸)d"+= t ¸j, where
j
t "C( j!d)/[C( j#1)C(!d)] and C( ) ) is the gamma function.j/0
This process is
j
"rst introduced by Granger (1980, 1981), Granger and Joyeux (1980), and
Hosking (1981). They show that > is stationary when d(0.5 and is invertible
t
when d'!0.5. The main feature of the I(d) process is that its autocovariance
function declines at a slower hyperbolic rate (instead of the geometric rate found
in the conventional ARMA models):
c( j)"O( j2d~1),
where c( j) is the autocovariance function at lag j. When d'0, the I(d) process is
said to have long memory since it exhibits long-range dependence in the sense
that + =
c( j)"R. When d(0, then + =
Dc ( j)D(R and the process is
j/~=
j/~= process. See Chung (1994) for
sometimes referred to as an intermediate memory
other long-memory properties of the I(d) process.
Our analysis focuses on the class of long memory I(d) processes with d'0.
We are particularly interested in the distinction between the nonstationary
subclass of I(d) processes with d*0.5 and the stationary subclass with d(0.5.
To examine potentially di!erent types of spurious e!ects, we propose six
regression models for di!erent classes of I(d) processes, mainly based on
158
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
whether the fractional di!erencing parameter d is greater than 0.5 or not. The
exact speci"cations of these models can be conveniently expressed with four I(d)
processes. Let us "rst de"ne two stationary ones with di!erent di!erencing
parameters d and d whose values lie between !0.5 and 0.5:
1
2
(1!¸)d1 v "a and (1!¸)d2 w "b ,
t
t
t
t
where a and b are two white noises with zero mean and "nite variances p2 and
t
t
a
p2, respectively; that is, v and w are I(d ) and I(d ) processes, respectively, and
b
t
t
1
2
both of them are stationary and invertible. When these two processes are
employed in our later analysis, the values of their di!erencing parameters are
mostly assumed to be in (0, 0.5); i.e., the stationary processes v and w are often
t
t
assumed to have long memory. We can also de"ne two nonstationary I(1#d )
1
and I(1#d ) processes by integrating v and w :
2
t
t
y "y #v and x "x #w .
t
t~1
t
t
t~1
t
Obviously, the orders of integration of these two nonstationary fractionally
integrated processes lie between 0.5 and 1.5. Given these four fractionally
integrated processes, we consider the following six simple linear regression
models:
Model
Model
Model
Model
Model
Model
1:
2:
3:
4:
5:
6:
y is regressed on an intercept and x ,
t
t
v is regressed on an intercept and w , where d #d '0.5,
t
t
1
2
y is regressed on an intercept and w , where d '0,
t
t
2
v is regressed on an intercept and x , where d '0,
t
t
1
y is regressed on an intercept and the trend t,
t
v is regressed on an intercept and the trend t, where d '0.
t
1
In Model 1 the orders of integration of both the dependent variable and the
regressor lie between 0.5 and 1.5, and can be equal to 1. So Model 1 may be
considered a generalization of Phillips' (1986) spurious regression to the case of
fractionally integrated processes. Model 2 presents the most interesting case in
our analysis. In it both the dependent variable and the regressor are assumed to
be stationary, ergodic, and strongly persistent in the sense that their fractional
di!erencing parameters sum up to a value greater than 0.5. Following Phillips'
arguments, we tend to think no spurious e!ect should occur in such a model
where variables are ergodic. But our analysis of Model 2 presents a result to the
contrary. The analysis of Model 2 seems to go beyond the previous study of
spurious e!ects and allows us to gain new insight into the problem.
Models 3 and 4 di!er from Model 1 in that the order of integration in one
of the dependent variable and the regressor is reduced to the stationary
range between 0 and 0.5. We can conveniently view Models 3 and 4 as two
intermediate cases between Model 1 of nonstationary fractionally integrated
processes and Model 2 of stationary fractionally integrated processes. We thus
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
159
expect the analysis of these two new models to be a mixture of those of Models
1 and 2.
In Models 5 and 6 we consider the e!ect of detrending the nonstationary and
stationary fractionally integrated processes, respectively. Through these two
models, we generalize the results of Durlauf and Phillips (1988). Also, Models
5 and 6 can be regarded as variants of Models 1 and 4, respectively, with the
nonstationary regressor x replaced by the time trend. This similarity in the
t
model speci"cations will also be re#ected in their analytic results.
The following assumption on the two white noise processes a and b are made
t
t
throughout this paper.
Assumption 1. Each of the two processes a and b is independently and identit
t
cally distributed with a zero mean; and their moments satisfy the following
conditions: EDa Dp(R, with p*maxM4, !8d /(1#2d )N; and EDb Dq(R
t
1
1
t
with q*maxM4, !8d /(1#2d )N. Moreover, a and b are independent of each
2
2
t
t
other.
We also assume, without loss of generality, that the initial values of the
fractionally integrated processes v , w , y , and x are all zero. Hence, y and
0 0 0
0
t
x can be considered as the partial sums of v and w , respectively; i.e.,
t
t
t
y "+T v and x "+T w . The independent and identical distribution
T
t/1 tour analysis and could be relaxed, say, to the
t/1 tis made Tto simplify
assumption
case where a and b are short-memory processes. See Chung (1995).
t
t
Before presenting Lemma 1, which is the cornerstone of our analysis, let us
summarize two important asymptotic results on the partial sums y and x . First,
t
t
given the variances p2"Var(y ) and p2 "Var(x ), Sowell (1990, Theorem 1)
y
T
x
T
proves that
C(1!2d )
1
p2&p2
¹1`2d1
y
a (1#2d )C(1#d )C(1!d )
1
1
1
and
C(1!2d )
2
p2&p2
¹1`2d2 ,
b (1#2d )C(1#d )C(1!d )
x
2
2
2
where z &w means z /w P1 as ¹PR. Furthermore, Davydov (1970)
T
T
T T
shows that as ¹PR,
1
1
y NB 1 (r) and
x NB 2 (r),
*Tr+
d
d
p
p *Tr+
y
x
for r3[0, 1], where [¹r] denotes the integer part of ¹r, the notation N denotes
weak convergence, and B (t) is the normalized fractional Brownian motion
d
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W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
which is de"ned by the following stochastic integral:
S
(1#2d)C(1!d)
B (t),
d
C(1#d)C(1!2d)
GP
t
P
(t!s)d dB (s)#
0
H
0
[(t!s)d!(!t)d] dB ,
0
0
~=
for d3(!0.5, 0.5), where B (t) is the standard Brownian motion. See Mandel0
brot and Van Ness (1968). Our notation for the standard and the fractional
versions of Brownian motions suggests that the former is a special case of the
latter with d"0.
The independence between v and w and between y and x implies joint weak
t
t
t
t
convergence of y
and x . More speci"cally, let z "[p~1y , p~1x ]@, then
y t x t
*Tr+
*Tr+
t
z NB (r),
*Tr+
d
where B (r)"[B 1 , B 2 ]@ is a two-dimensional normalized fractional Brownian
d
d
d
motion, of which the two elements B 1 and B 2 are independent. This result
d
d
implies the following lemma:
Lemma 1. Let Assumption 1 hold. Then, as ¹PR, we have the following results:
1.
y
x
1
1
t N:1 B (s) ds and
t N:1 B (s) ds.
+T
+T
1
d
¹ t/1 p
¹ t/1 p
0
0 d2
y
x
1
y2
x2
1
t N:1 [B (s)]2 ds.
+T t N:1 [B 1 (s)]2 ds and
+T
d
¹ t/1 p2
¹ t/1 p2
0
0 d2
y
x
2
1
(y !y6 )2
t
3. +T
N:1 [B 1 (s)]2 ds! :1 B 1 (s) ds
and
¹ t/1 p2
0 d
0 d
y
2.
C
D
1
(x !x6 )2
t
+T
N:1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2.
¹ t/1 p2
0 d
0 d
x
v
4. +T t NB 1 (1)
d
t/1 py
5.
w
t NB (1).
and +T
d2
t/1 px
1
1 c (0)"C(1!2d1 ) p2 and
+T v2 P
v
¹ t/1 t
C(1!d )2 a
1
1
1 c (0)"C(1!2d2 ) p2.
+T w2 P
w
t
¹ t/1
C(1!d )2 b
2
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
6.
161
1
1 c (0) and 1 +T (w !w6 )2 P
1 c (0).
+T (v !v6 )2 P
v
w
t
¹ t/1
¹ t/1 t
G
"O (1),
if d #d '0.5,
1
1
2
v w
7. ¹ +T t t "O [(ln ¹)0.5],
if d #d "0.5,
1
1
2
t/1 py px
"O (¹0.5~d1 ~d2 ), otherwise.
1
1
y x
8. +T t t N:1 B 1 (s) ) B 2 (s) ds.
d
¹ t/1 p p
0 d
y x
y w
v x
9. +T t t "O (1), if d '0; and +T t t "O (1), if d '0.
1
2
1
1
t/1 py px
t/1 py px
tw
t v
t NB (1)!:1 B (s) ds and +T
t NB (1)!:1 B (s) ds.
10. +T
d1
d2
0 d1
t/1 ¹ px
0 d2
t/1 ¹ py
1
ty
tx
1
t N:1 sB (s) ds and
t N:1 sB (s) ds.
+T
11. +T
d1
¹ t/1 ¹ p
¹ t/1 ¹ p
0
0 d2
y
x
Moreover, joint weak convergence of 1}4, 8, 10, and 11 also applies. Here, B 1 (t)
d
and B 2 (t) are two independent normalized fractional Brownian motions, c ( j) and
d
v
c ( j) are the autocovariance functions of v and w , respectively, at lag j, and p2 and
a
w
t
t
p2 are the variances of the underlying white noises a and b , respectively. The
b
t
t
1 means convergence in probability.
notation P
All the theorem proofs are in the appendix. Note that, while the weak
convergence of z is in the space of functions, the weak convergence established
t
in the above lemma is in the real line, which is equivalent to convergence in
distribution. Following the convention in the literature, we use the same notation N for both types of weak convergence.
In the rest of this section the results of Lemma 1 will be used to develop the
theory of spurious e!ects, presented in a series of theorems and corollaries, for
the proposed six models. The "rst two models will be discussed separately
in Sections 2.1 and 2.2. These two models provide us with a framework
which facilitates the explanations of the other four models in Sections 2.3 and
2.5. One subsection } Section 2.4 } will be devoted to the analysis of an
important issue about how the orders of fractional integration are directly
related to the spurious e!ects. The results in Lemma 1 have also been used in
deriving the limiting distributions of the &modi"ed Durbin}Watson statistics' by
Tsay (1998).
We will adopt the following notation for the various statistics from the
Ordinary Least Squares (OLS) estimation. Let a( and bK denote the usual OLS
estimators of the intercept and the slope. Their respective variances are estimated by s2 and s2, from which we have the t ratios t "bK /s and t "a( /s .
a
b
b
a
a
b
Also, let s2 denote the estimated variance of the OLS residuals, R2 the coe$cient
162
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
of determination, and D= the Durbin}Watson statistic. Finally, in addition to
the autocovariance functions c ( j) and c ( j) of v and w , let o ( j) and o ( j) be
v
w
t
t
v
w
their respective autocorrelations at lag j.
2.1. Model 1 of nonstationary fractionally integrated processes
In Model 1 a nonstationary I(1#d ) process y is regressed on another
1
t
independent and nonstationary I(1#d ) process x . Since the permissible range
2
t
for the values of the fractional di!erencing parameters d and d is (!0.5, 0.5),
1
2
Model 1 generalizes Phillips' (1986) model of integrated processes in which
d "d "0. Not surprisingly, all the results we derive for Model 1 are straight1
2
forward generalization of Phillips' theory of the spurious e!ects. The results for
Model 1 are presented in the following theorem:
Theorem 1. Let Assumption 1 hold. Then, as ¹PR, we have the following results:
1.
p
:1 B (s)B 2 (s) ds![:1 B 1 (s) ds][:1 B 2 (s) ds]
x bK N 0 d1
d
,b .
0 d
0 d
H
p
:1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2
y
d
d
0
0
Note that p /p "O(¹d1 ~d2 ).
y x
2.
1
a( N:1 B 1 (s) ds!b :1 B 2 (s) ds,a , where b is dexned in 1. Note that
H 0 d
H
H
p
0 d
y
p "O(¹0.5`d1 ).
y
3.
1
s2N:1 [B 1 (s)]2 ds![:1 B 1 (s) ds]2!b2 M:1 [B 2 (s)]2 ds![:1B 2 (s) ds]2N
H 0 d
p2
0 d
0 d
0 d
y
,p2 , where b is dexned in 1. Note that p2"O(¹1`2d1 ).
y
H
H
4.
p2
¹p2
H
x s2 N
,p2 , where p2 is dexned in 3.
H
Hb
p2 b :1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2
y
0 d
0 d
Note that p2/¹p2 "O(¹2d1 ~2d2 ~1).
x
y
5.
G
H
[:1 B 2 (s) ds]2
¹
0 d
s2Np2 1#
,p2 , where p2 is dexned
H
H
Ha
:1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2
p2 a
y
0 d
0 d
in 3. Note that p2/¹"O(¹2d1 ).
y
6.
b
t N H , where b is dexned in 1 and p2 is dexned in 4.
Hb
b
H
p
J¹
Hb
1
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
7.
8.
9.
163
a
t N H , where a is dexned in 2 and p2 is dexned in 5.
Ha
a
H
p
J¹
Ha
b2 M:1 [B (s)]2 ds![:1 B 2 (s) ds]2N
0 d
R2N H 0 d2
, where b is dexned in 1.
H
:1 [B 1 (s)]2 ds![:1 B 1 (s) ds]2
0 d
0 d
1
1 0.
D= P
Here, B 1 (t) and B 2 (t) are two independent normalized fractional Brownian
d
d
motions.
The most important result in Theorem 1 is that, as the sample size ¹ increases,
the two t ratios t and t diverge at the same rate of J¹, which is independent of
b
a
the magnitudes of the fractional di!erencing parameters d and d . This result is
1
2
exactly the same as what Phillips (1986) has obtained for the case where
d "d "0. So even when the orders of integration in the dependent variable
1
2
and the regressor di!er from 1 by as much as 0.5, the usual problem in using the
t tests remains: the probability of rejecting the null hypothesis of b"0 or
a"0 based on t tests increases monotonically as the sample size increases. Also
note that Marmol (1995) generalizes Phillips' theory to cases where both y and
t
x are integrated processes of the same integer orders that are higher than one.
t
The limiting distributions in Theorem 1 for the special case where d "d are
1
2
also very similar to Marmol's results.
The limiting distributions of the t ratios, after normalized by J¹, are direct
generalization of those derived by Phillips (1986). The same conclusion also
holds for R2 and the D= statistics. In other words, when we compare our results
with Phillips', we observe a common feature in these four statistics; namely, the
nonzero values of d and d do not a!ect their convergence rates while the
1
2
e!ects on their limiting distributions are quite straightforward: all the standard
Brownian motions in Phillips' theory are replaced by fractional Brownian
motions. That the fractional di!erencing parameters d and d play a relative
1
2
minor role here is mainly because the four statistics are all ratios so that the e!ects
of d and d are cancelled out. In contrast, the results on the OLS estimators
1
2
bK and a( are a di!erent story. In Phillips' theory both bK and a( /J¹ converge to
some non-normal non-degenerate limiting distributions. But for the present
model of the fractionally integrated processes, the orders of bK and a( are ¹d1 ~d2 and
¹d1 `0.5, respectively. So while a( always diverges (though the rate can be slow if
d is close to !0.5), bK can be either divergent or convergent, depending on the
1
relative magnitudes of d and d . For example, if the order of integration in the
1
2
dependent variable y is smaller than that of the regressor x ; i.e., d (d , then
t
t
1
2
bK converges to zero, just like the conventional case of no spurious e!ects.
Moreover, if d !d "!0.5, then, similar to the case of no spurious e!ects,
1
2
J¹ ) bK has a limiting distribution, though its limiting distribution is not normal.
164
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
2.2. Model 2 of stationary fractionally integrated processes
In this section we consider Model 2 in which a stationary fractionally integrated process v is regressed on an independent and stationary fractiont
ally integrated process w . We show that, although both v and w are stationary,
t
t
t
the spurious e!ect in terms of the t tests could still exist under an additional
condition on the fractional di!erencing parameters: d #d '0.5. Loosely
1
2
speaking, this condition implies that the two processes v and w are both
t
t
strongly persistent.
Our analysis begins with a special case where we assume a set of more
stringent conditions which helps deriving the limiting distribution of the OLS
estimator. This theory is based on an important result of Fox and Taqqu (1987)
who show that the product of two highly persistent but stationary Gaussian
processes, if adequately normalized, can converge. After examining this special
case, we then show how the spurious e!ects may still exist in a more general
framework.
Let us "rst reproduce Fox and Taqqu's (1987) Theorem 6.1 here as Lemma 2.
Lemma 2. Let (X , > ) be a stationary jointly Gaussian sequence with
t t
E(X )"E(> )"0, E(X2)"E(>2)"1, and E(X > )"r. Suppose that p and
t
t t
1
t
t
t
p are two arbitrary real numbers and that there exist 0(d , d (0.5, such that
2
1 2
as jPR
E(X X )&p2 j~d1 ,
1
t t`j
op p b
E(X > )& 1 2 1 j~(d1 `d2 )@2,
t t`j
Ja a
1 2
E(> > )&p2 j~d2 ,
2
t t`j
op p b
E(> X )& 1 2 2 j~(d1 `d2 )@2,
t t`j
Ja a
1 2
where o is a constant between 1 and !1, while a "A(d , d ), a "A(d , d ),
1
1 1 2
2 2
b "A(d , d ), and b "A(d , d ) are four constants with A(d ,d ) being dexned
1
1 2
2
2 1
1 2
by :=x~(d1 `1)@2(x#1)~(d2 `1)@2 dx, then
0
1
*Ts+
+ (X > !r)NZ(s),
t t
¹1~(d1 `d2 )@2
t/1
where
p p
Z(s)" 1 2
Ja a
1 2
P PC
s
R2
D
2
< (u!x )~(di `1)@2IM i N du dM (x ) dM (x ).
x :u
i
1 1
2 2
0 i/1
Here, M and M are two Gaussian random measures with respect to Lebesgue
1
2
measure, having unit variances and covariance o.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
165
Note that the two processes X and > are not only strongly persistent, as
t
t
indicated by the hyperbolic convergence rates d and d in their autocorrela1
2
tions, but also highly correlated with each other, as indicated by the hyperbolic
convergence rates in their covariances. However, in our application we are only
interested in the case where X and > are independent so that r and o in the
t
t
above lemma are both zero. The above lemma o!ers us the convergence rate of
+T X > and its limiting process Z(t) given the Gaussian assumption and
t/1 t t
a narrower range for the parameters d and d . In order to apply this lemma, we
1
2
make the following assumption in addition to Assumption 1 made earlier.
Assumption 2. The two fractionally integrated processes v and w are both
t
t
Gaussian and the corresponding fractional di!erencing parameters d and
1
d are both in the range of (0.25, 0.5).
2
Given the facts that
C(1!d )
C(1!d )
1 j2d1 ~1 and o ( j)&
2 j2d2 ~1,
o ( j)&
v
w
C(d )
C(d )
1
2
it is straightforward to prove the following corollary in which X and > in
t
t
Lemma 2 are replaced by v /Jc (0) and w /Jc (0), respectively.
t
w
t
v
Corollary 1. Let Assumptions 1 and 2 hold. Then, as ¹PR,
1
v
T
w
t
t NZ(1),
+
¹d1 `d2
t/1 Jcv (0) Jcw (0)
where the limiting random variable Z(1) is dexned in Lemma 2 with d "1!2d ,
1
1
d "1!2d , p2 "C(1!d )/C(d ), and p2 "C(1!d )/C(d ). Consequently,
2
2 1
1
1
2
2
2
we have
T v w
¹ + t t NCJc (0) ) c (0) Z(1),
v
w
p p
t/1 y x
where
C2"C(1!2d )C(1!2d )/(1#2d )C(1#d )C(1!d )(1#2d )
1
2
1
1
1
2
C(1#d )C(1!d ).
2
2
The result of Corollary 1 supplements that of item 7 in Lemma 1. From these
results, we can then establish the following theorem about the spurious e!ect in
Model 2.
Theorem 2. Let Assumptions 1 and 2 hold. Then, as ¹PR, we have the following
results:
1.
¹2
bK "O (1). Note that p p /¹2"O(¹d1 `d2 ~1).
1
y x
p p
y x
166
2.
3.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
¹
a( NB 1 (1). Note that p /¹"O(¹d1 ~0.5).
d
y
p
y
1 c (0).
s2 P
v
4.
1 cv (0) .
¹s2 P
b
c (0)
w
5.
1 c (0).
¹s2 P
a
v
6.
¹3@2
t "O (1). Note that p p /¹3@2"O(¹d1 `d2 ~0.5).
1
y x
p p b
y x
7.
J¹
1
t N
B (1). Note that p /J¹"O(¹d1 ).
a
y
p
Jc (0) d1
y
v
¹4
R2"O (1). Note that p2p2/¹4"O(¹2d1 `2d2 ~2).
y x
1
p2p2
y x
1 2!2o (1)"2(1!2d1 ) .
9.
D= P
v
1!d
1
Here B 1 (t) is a normalized fractional Brownian motions.
d
8.
The most important result from this theorem is the divergence rates of the two
t ratios t and t , which are ¹d1 `d2 ~0.5 and ¹d1 , respectively. Recall that
b
a
d #d !0.5 is necessarily greater than 0 (and smaller than 0.5) under Assump1
2
tion 2. This result re#ects the spurious e!ect in the t tests. Since both
the dependent variable and the regressor are stationary and ergodic, the spurious e!ect is not really expected (see Phillips 1986, p. 318). The surprising
results we get here suggest that the cause for the spurious e!ect has more to do
with the strong persistence than stationarity and ergodicity of the variables
involved.
It is interesting to compare the divergence rates of the t ratios here with the
J¹ rate we observe in Model 1. We note that the divergence rates in the present
model depend on the magnitudes of the two fractional di!erencing parameters
d and d while those in Model 1 do not. Furthermore, the t ratios diverge more
1
2
slowly in the present model than in Model 1. In particular, the divergence rate of
t can become very slow when both d and d approach to their lower boundary
b
1
2
0.25.
Let us turn to the OLS estimators bK and a( . We note that both of them
converge in probability to zero as in the conventional case of no spurious e!ects.
However, their convergence rates are much slower than the usual ¹~1@2 rate. In
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
167
contrast to these irregular convergence rates of the OLS estimators, the
estimated variances s2 and s2 nevertheless converge at the standard ¹~1 rate. It
a
b
is such disparity in the convergence rates between the OLS estimators, which
converge at rates slower than ¹~1@2, and their standard errors, which converge
at the standard ¹~1@2 rates, that causes the resulting t ratios to diverge and
hence the spurious e!ect.
R2 in the present model converges to 0 as in the case of no spurious e!ects. It
is di!erent from what we observe in Model 1 where R2 converges to a random
variable. Consequently, as the sample size increases, the declining R2 in the
present model will correctly re#ect the fact that the regressor does not help
explain the variations in the dependent variable.
The D= statistic does not converge in probability to zero and this result is
also di!erent from that of Model 1. Its limit 2!2o (1) is similar to the one we
v
"nd in the conventional AR(1) case. This limit depends on the fractional
di!erencing parameter d of the dependent variable v and can only take value
1
t
in the range of (0,4/3), which is to the left of the value 2.
There is one technical detail that calls for some explanations: Unlike in
Theorem 1, we do not give an explicit expression for the limiting distribution of
bK in Theorem 2 because such an expression requires the joint weak convergence
of the sample averages of v , w , and v w , while the proof of the joint weak
t t
t t
convergence is beyond the scope of the present paper. However, lacking in an
explicit expression for the limiting distribution of bK does not hinder us from
evaluating the convergence rate of bK , which is all we need to show the spurious
e!ects in the t ratios. This same argument applies to some of the later analyses,
including the following one where we consider a less restricted speci"cation of
Model 2 that is de"ned by the following assumption.
Assumption 3. The sum of the two fractional di!erencing parameters d and
1
d is greater than 0.5.
2
Since the Gaussian distribution is not assumed while one of the fractional
di!erencing parameters d and d can be smaller than 0.25, Assumption 3 is thus
1
2
less stringent than Assumption 2.
Corollary 2. If Assumption 2 is replaced by Assumption 3 in Theorem 2, then all the
conclusions there remain true.
The analysis of Model 2 can thus be summarized as follows. The OLS
estimators bK and a( (as well as R2) do converge in probability to zero, correctly
re#ecting the lack of a relationship between the dependent variable and the
regressor. But the convergence rates of bK and a( are too slow in comparison with
those of their standard errors. Consequently, the t ratios diverge and the t tests
fail. The upshot is that the usual t tests can become invalid even when the
168
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
dependent variable and the regressor are both stationary and ergodic (so long as
they are su$ciently persistent).
A profound implication from Model 2 is as follows: If we begin with Model
1 where both the dependent variable and the regressor are nonstationary
fractionally integrated processes with the orders of integration 1#d and
1
1#d , respectively, where d #d '0.5, then "rst-di!erencing both variables
2
1
2
cannot completely eliminate the spurious e!ects. While R2 may be reduced and
the D= statistic may be increased, the t ratios may still be so large that we
cannot avoid making a spurious inference. This is a fairly serious problem with
the regression for the fractionally integrated processes. It implies that even the
popular "rst-di!erencing procedure might not prevent us from "nding a spurious relationship among highly persistent data series. One lesson we learn from
this discussion is that it is very important to check individual data series for
possible long memory before regression can be applied.
2.3. Two intermediate cases: Models 3 and 4
Models 3 and 4 can be considered as two intermediate models between
Models 1 and 2 in that one of the dependent variable and the regressor is
stationary while the other is not. We expect the asymptotic results for these two
new models to be hybrid of those of Models 1 and 2.
In Model 3 a nonstationary I(1#d ) process y is regressed on an indepen1
t
dent and stationary I(d ) process w . Note that the fractional di!erencing
2
t
parameter d for the regressor w here is assumed to be positive; i.e., w has long
2
t
t
memory. The asymptotic properties of the OLS estimators for Model 3 are given
in the following theorem:
Theorem 3. Let Assumption 1 hold. Then, as ¹PR, we have the following results.
1.
2.
3.
¹
bK "O (1). Note that p p /¹"O(¹d1 `d2 ).
1
y x
p p
y x
1
a( N:1 B 1 (s) ds,a . Note that p "O(¹0.5`d1 ).
H
y
p
0 d
y
1
s2N:1 [B 1 (s)]2 ds![:1 B 1 (s) ds]2,p2 . Note that p2"O(¹1`2d1 ).
H
y
p2
0 d
0 d
y
4.
¹
p2
s2N H , where p2 is dexned in 3. Note that p2/¹"O(¹2d1 ).
H
y
p2 b c (0)
y
w
5.
¹
s2Np2 , where p2 is dexned in 3. Note that p2/¹"O(¹2d1 ).
y
H
H
p2 a
y
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
6.
7.
8.
9.
169
J¹
t "O (1). Note that p /J¹"O(¹d2 ).
1
x
p b
x
a
t N H , where a is dexned in 2 and p2 is dexned in 3.
a
H
H
p
J¹
H
¹2
R2"O (1). Note that p2/¹2"O(¹2d2 ~1).
1
x
p2
x
1
1 0.
D= P
Here B 1 (t) is a normalized fractional Brownian motions.
d
Since both t ratios diverge, Model 3 also su!ers from the spurious e!ect in
terms of the t tests. Moreover, we "nd the results that the OLS estimator
a( diverge and that D= converges in probability to 0 are close to what we get in
Model 1, while the result of converging R2 is the same as that of Model 2. So
Model 3 is indeed a mixture of Models 1 and 2.
It should be pointed out that in Theorem 3 the range of the fractional
di!erencing parameter d of the regressor w is restricted to the positive half of
2
t
the original range (!0.5, 0.5). For the case of a negative d , it is quite
2
straightforward to show that the t ratios are convergent and there is no spurious
e!ect.
In Model 4 a stationary I(d ) process v is regressed on an independent and
1
t
nonstationary I(1#d ) process x . Similar to the restriction imposed on
2
t
Model 3, the fractional di!erencing parameter d of the dependent variable v is
1
t
assumed to be positive so that v has long memory. The asymptotic theory for
t
Model 4 is presented in the following theorem.
Theorem 4. Let Assumption 1 hold. Then, as ¹PR, we have the following results.
1.
¹p
x bK "O (1). Note that p /¹p "O(¹d1 ~d2 ~1).
1
y
x
p
y
2.
¹
a( "O (1). Note that p /¹"O(¹d1 ~0.5).
1
y
p
y
3.
1 c (0).
s2 P
v
4.
c (0)
v
¹p2 ) s2N
,p2 .
x b :1 [B (s)]2 ds![:1 B (s) ds]2
Hb
2
2
d
d
0
0
Note that 1/¹p2"O(¹~2~2d2 ).
x
170
5.
6.
7.
8.
9.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
[:1 B 2 (s) ds]2
0 d
¹ ) s2Nc (0)M1#
N,p2 .
a
v
Ha
:1 [B 2 (s)]2 ds![:1 B 2 (s) ds]2
0 d
0 d
J¹
t "O (1). Note that p /J¹"O(¹d1 ).
1
y
p b
y
J¹
t "O (1). Note that p /J¹"O(¹d1 ).
1
y
p a
y
¹2
R2"O (1). Note that p2/¹2"O(¹2d1 ~1).
1
y
p2
y
1 2!2o (1)"2(1!2d1 ) .
D= P
v
(1!d )
1
Here B 2 (t) is a normalized fractional Brownian motions.
d
Since both t ratios diverge (at the same rate), the spurious e!ect in terms of
failing t tests again exists in Model 4. But contrary to the results in Model 3, the
OLS estimators bK and a( , together with R2, all converge in probability to zero,
while the D= statistic converges to 2!2o (1). These "ndings obviously bring
v
Model 4 closer to Model 2 than to Model 1.
2.4. The relationship between the orders of integration and the divergence rates
The divergent t ratios in the above four models and the resulting failure of
the t tests are referred to as the spurious e!ects. In this section we compare the
divergence rates of t ratios across the four models and investigate how they are
related to the respective model speci"cations.
First note that the divergence rates of the t ratio t are ¹0.5, ¹d1 `d2 ~0.5, ¹d2 ,
b
and ¹d1 , respectively, in Models 1}4. Let us also compare the speci"cations of
the four models using Model 1 as the benchmark: Model 3 di!ers from Model
1 in that the order of integration in the regressor is reduced from above 0.5 to
below 0.5 (but above 0); Model 4 di!ers from Model 1 in that the order of
integration in the dependent variable is reduced from above 0.5 to below 0.5 (but
above 0); and, "nally, Model 2 di!ers from Model 1 in that the orders of
integration in both the dependent variable and the regressor are reduced from
above 0.5 to below 0.5 (but their sum is assumed to be greater than 0.5). By
associating these changes in the orders of integration with the changes in the
divergence rates of t , we can conclude that reducing the order of integration in
b
the dependent variable causes the divergence rate of t to decrease by the order
b
of ¹d1 ~0.5 and reducing the order of integration in the regressor causes the
divergence rate of t to decline by the order of ¹d2 ~0.5, while these two e!ects
b
are cumulative as in Model 2.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
171
Recall that in Models 2}4 restrictions have been imposed on the usual ranges
(!0.5, 0.5) of the fractional di!erencing parameters d and d . In Model 3 the
1
2
range of d is restricted to be (0, 0.5), which is also the range of d in Model 4,
2
1
while the sum of d and d must be greater than 0.5 in Model 2. From the
1
2
analysis in the previous paragraph, particularly the fact that the divergence rates
are directly related to the magnitudes of d and d , we come to realize that
1
2
the restricted ranges of d and d in Models 2}4 ensure the reduction in the
1
2
divergence rates of t from the ¹0.5 level is not too great so that t remains
b
b
divergent (in which case the spurious e!ects occur). Although we did not
explicitly consider the asymptotic theory for cases where the fractional di!erencing parameters lie outside their prescribed ranges, it is readily seen that the
conditions we impose on the ranges are not only su$cient but also necessary for
the existence of the spurious e!ect in terms of divergent t .
b
From a similar analysis for the divergence rates of the t ratio t we also "nd
a
that reducing the order of integration in the dependent variable causes the
divergence rate of t to decrease by the order of ¹d1 ~0.5, while reducing
a
the order of integration in the regressor does not cause the divergence rate of
t to change, as we probably should have expected.
a
It is also interesting to see how the changes in the orders of integration of the
dependent variable and the regressor a!ect the large-sample property of R2.
Recall that in Model 1 R2 converges to a random variable and such asymptotic
behavior of R2 is considered part of the spurious e!ect by Phillips (1986). But
when we examine Models 2}4, we note that reducing the order of integration in
the dependent variable helps to increase its convergence rate by the order of
¹1~2d1 while reducing the order of integration in the regressor helps to increase
the convergence rate by the order of ¹1~2d2 . As a result, in Models 2}4, R2 all
converge to 0, correctly re#ecting the fact that there is no relationship between
the regressor and the dependent variable. This "nding implies that the spurious
e!ects in Models 2}4 are con"ned to the two t ratios while the asymptotic
tendency of R2 toward zero is not a!ected by the spurious e!ects (though the
convergence rates are).
The sharp di!erence in the asymptotic behavior between the t ratios and R2 in
Models 2}4 actually o!ers us an opportunity to diagnose the spurious e!ect in
these models. That is, when we "nd two highly signi"cant t ratios coexisting with
a completely contradictory near-zero R2, we are e!ectively reminded of the
possibilities that one of the Models 2}4 may be at work and that the dependent
variable and the regressor may possess strong long memory, while one of them
may even be nonstationary. With the possibility of such an informal diagnosis, it
seems that the spurious e!ects in Models 2}4 are less damaging than those in
Model 1 in the sense that in Model 1 there is no internal inconsistency among
the OLS estimates to indicate the spurious e!ects.
Finally, let us brie#y state a few more results about the asymptotic tendency
of the OLS estimators bK and a( and the D= statistic. First, we note that bK will
172
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
converge unless the dependent variable is nonstationary and its order of integration is su$ciently large. Secondly, whether a( diverges or not and whether the
D= statistic converges in probability to zero or not depend entirely on whether
the dependent variable is nonstationary or not. Note that, as mentioned earlier,
even though the OLS estimators bK and a( can converge in probability to zero in
the four proposed models, the corresponding t ratios always diverge and it is
these divergent t ratios that are referred to as the spurious e!ects.
2.5. Models 5 and 6: Detrending fractionally integrated processes
As has been pointed out by Nelson and Kang (1981, 1984) and Durlauf and
Phillips (1988), detrending integrated processes results in the spurious e!ect of
"nding a signi"cant trend. In this section we extend their analysis by considering
the potential problems in detrending fractionally integrated processes. It turns
out that the spurious e!ect of divergent t ratios exists as long as the fractional
di!erencing parameter is larger than zero. The implication is that whenever
there is long memory in the process, the routine procedure of detrending can
produce misleading results. It appears that the spurious e!ect in detrending
occurs more often than we previously thought.
In our analysis of detrending fractionally integrated processes, we separate
the nonstationary case from the stationary case. In Model 5 we examine the
regression of a nonstationary I(1#d ) process y on a time trend t. The
1
t
asymptotic theory for the OLS estimation is given in the following theory.
Theorem 5. Let Assumption 1 hold. Then, as ¹PR, we have the following results.
1.
¹
bK N12:1 sB 1 (s) ds!6:1 B 1 (s) ds,b . Note that p /¹"O(¹d1 ~0.5).
H
y
p
0 d
0 d
y
2.
1
a( N4:1 B 1 (s) ds!6:1 sB 1 (s) ds,a .
H
p
0 d
0 d
y
3.
1
1
s2N:1 [B 1 (s)]2 ds![:1 B 1 (s) ds]2!12[:1 sB 1 (s) ds! :1 B 1 (s) ds]2
d
d
d
p2
2 0 d
0
0
0
y
,p2 .
H
4.
¹3
s2N12p2 , where p2 is dexned in 3. Note that p2/¹3"O(¹2d1 ~2).
y
H
H
p2 b
y
5.
¹
s2N4p2 , where p2 is dexned in 3. Note that p2/¹"O(¹2d1 ).
y
H
H
p2 a
y
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
6.
7.
8.
173
1
t Nb /J12p , where b is dexned in 1 and p2 is dexned in 3.
H
H
H
H
J¹ b
1
t Na /2p , where a is dexned in 2 and p2 is dexned in 3.
H
H H
H
J¹ a
b2
H
R2N
,
12:1 [B 1 (s)]2 ds!12[:1 B 1 (s) ds]2
d
d
0
0
where b is dexned in 1.
H
1 0 and p2D=Nc (0)/p2 .
D= P
H
y
v
Here, B 1 (t) is a normalized fractional Brownian motion.
d
9.
The results on detrending a stationary long memory I(d ) process v , which is
1
t
our Model 6, are presented in the following theorem.
Theorem 6. Let Assumption 1 hold. Then, as ¹PR, we have the following results.
1.
¹2
bK N6B 1 (1)!12:1 B 1 (s) ds,b . Note that p /¹2"O(¹d1 ~1.5).
d
H
y
p
0 d
y
2.
¹
a( N6:1 B 1 (s) ds!2 B 1 (1),a . Note that p /¹"O(¹d1 ~0.5).
d
H
y
p
0 d
y
3.
1 c (0).
s2 P
v
4.
1 12c (0).
¹3s2 P
b
v
5.
1 4c (0).
¹s2 P
a
v
6.
J¹
t Nb /J12c (0), where b is dexned in 1. Note that p /J¹"O(¹d1 ).
H
v
H
y
p b
y
7.
J¹
t Na /2Jc (0), where a is dexned in 2. Note that p /J¹"O(¹d1 ).
H
v
H
y
p a
y
8.
¹2
R2Nb2 /12c (0), where b is dexned in 1. Note that p2/¹2"O(¹2d1 ~1).
y
H
v
H
p2
y
174
9.
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
1 2!2o (1)"2(1!2d )/(1!d ).
D= P
v
1
1
Here, B 1 (t) is a normalized fractional Brownian motion.
d
In terms of the convergence (or divergence) rates of the various OLS estimators, Models 5 and 6 can be conveniently viewed as &special cases' of Models
1 and 4, respectively. More speci"cally, if we replace the term p by ¹ in those
x
normalizing factors in Theorems 1 and 4, then we immediately get all the
normalizing factors in Theorems 5 and 6. For example, while the normalizing
factor for bK in Theorem 1 is p /p , the one in Theorem 5 is ¹/p . Similarly, while
x y
y
the normalizing factor for bK in Theorem 4 is ¹p /p , the one in Theorem 6 is
x y
¹2/p . (Also note that the fractional Brownian motion B 2 (t) does not appear in
y
d
Theorems 5 and 6 since the regressors in Models 5 and 6 are the time trend and
are unrelated to the I(d ) process w .) Given these observations, we then
2
t
conclude that all the analyses about Models 1 and 4 can be readily extended to
Models 5 and 6. In particular, the divergence rates of the t ratios, which
respectively are in the orders of ¹0.5 and ¹d1 in Models 1 and 4, are also the
rates in Models 5 and 6. (Note that in both Models 4 and 6 the same condition
d '0 is imposed on the stationary dependent variable v so that the resulting
1
t
t ratios are divergent.) As a result, the type of spurious e!ects we observe in
Models 1 and 4 occur again in Models 5 and 6. That is, detrending a fractionally
integrated process with a positive fractional di!erencing parameter, certainly
including the usual case of the I(1) process, will result in the spurious "nding of
a signi"cant trend. One important inference we draw from Models 5 and 6 is
that the cause for the spurious e!ect in detrending a process is neither nonstationarity nor lack of ergodicity but long memory in the process.
From Models 5 and 6 we also note the following result: If the data series are
nonstationary with the order of integration greater than 1, then the spurious
e!ect can happen to the detrending procedure even after the series are "rst
di!erenced. What "rst di!erencing does to the detrending procedure in such
a case is simply reducing R2, increasing the value of the D= statistic, and
slowing down the divergence of the two t ratios from the ¹0.5 rate to the
¹d1 rate. Based on this observation, it seems that the spurious e!ects in
detrending may occur more often than we previously thought.
3. Conclusion
In our analysis of spurious regressions for the long memory fractionally
integrated processes, we "nd that no matter whether the dependent variable and
the regressor are stationary or not, as long as their orders of integration sum up
to a value greater than 0.5, the t ratios become divergent. So it is the long
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
175
memory, instead of nonstationarity or lack of ergodicity, that causes the spurious e!ects in terms of failing t tests. Nonstationarity in one or both of the
dependent variable and the regressor only helps to accelerate the divergence
rates of the t ratios. We thus learn that spurious e!ects might occur more often
than we previously believed as they can arise even among stationary series and
the usual "rst-di!erencing procedure may not be able to completely eliminate
spurious e!ects when data possess strong long memory. It is interesting to note
Phillips (1995) recently has o!ered some contrast thoughts about spurious
regressions (and he argues that spurious regressions may not be as serious as
many researchers have been led to believe).
In Section 2.4 we have carefully examined the exact relationships between the
orders of integration in the fractionally integrated processes and the divergence
rates in the t ratios. From this analysis we gain many insights into the problem
of spurious e!ects which are not available in Phillips' (1986) classical study of
I(1) processes. In short, it is found that the extents of spurious e!ects are directly
related to the degrees of long memory in the data. Our results on detrending
fractionally integrated processes also greatly broaden Durlauf and Phillips'
(1988) theory of spurious detrending in which the relationship between the
orders of integration and the divergence rates of the t ratios again plays a useful
role in the analysis.
A fairly extensive Monte Carlo study has also been conducted to verify the
theoretical results, especially those of convergence rates, we have established in
the paper. We do not report the simulation results here other than pointing out
the fact that almost all our theoretical results are well supported by simulation.
A few generalizations of our study are worthy of further consideration.
A natural extension is to consider the multiple regression where there are more
than one nonconstant regressor. Another one is to allow the fractionally integrated processes to have nonzero means. Based on Phillips' (1986) work, we
expect most, if not all, of the asymptotic results we obtain from the simple
regression case to hold in the multiple regression of fractionally integrated
processes with drifts. These issues have been examined by Chung (1995).
One aspect of our study that is slightly more restricted than Phillips, (1986)
and Durlauf and Phillips' (1988) analysis is that the fractionally integrated
processes we consider are built on white noises a and b that are required to
t
t
satisfy the relatively stringent conditions as speci"ed in Assumption 1. These
conditions e!ectively rule out the possibility of allowing short-run dynamics
such as the ARMA components in the fractionally integrated processes we have
studied. Chung (1995) has studied the case where Assumption 1 is relaxed to
incorporate the short-run dynamics and found no substantial changes in the
analysis of spurious e!ects.
Finally, our study of spurious regression can serve as the basis for the analyses
of &fractional cointegration' where the dependent variable and regressors are
related I(d) processes. This line of the work appears to be quite important and
176
W.-J. Tsay, C.F. Chung / Journal of Econometrics 96 (2000) 155}182
has attracted a lot of attention in the literature recently. One of the pioneer
works in this area is by Cheung and Lai (1993).
Acknowledgements
We are very grateful for two referees and an associate editor for their valuable
suggestions.
Appendix A. Proof of Lemma 1
The proofs of items 1}3 are straightforward applications of the continuous
mapping theorem to the Davydov's results. They are omitted here. Item 4 follows directly from Davydov's result, while items 5 and 6 are due to ergodicity of
the two stationary processes v and w .
t
t
To prove item 7, we note, since v and w are assumed to b