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THESIS

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dissertation entitled

SPURIOUS REGRESSION WITH
FRACTIONALLY INTEGRATED PROCESSES

presented by

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of the requirements for

Ph . D . . Economics
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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SPURIOUS REGRESSION WITH
FRACTIONALLY INTEGRATED PROCESSES

By

Wen-J en Tsay

A DISSERTATION

Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1995

ABSTRACT

SPURIOUS REGRESSION WITH
FRACTIONALLY INTEGRATED PROCESSES

By

VVen-J en Tsay

This dissertation considers the spurious effect in a simple linear
regression model of I (d) processes. In Chapter 2 we find that when we
regress a fractionally integrated process on a constant and another in-
dependent fractionally integrated process, spurious effects could arise.
The most interesting finding is that the spurious effect could occur
when both the dependent variable and regressor are stationary frac-
tionally integrated processes. This implies the usual procedure to
avoid the spurious effect by differencing may be questionable and em-
pirical results based on such time series regressions may be misleading.
In Chapter 3 we consider the asymptotic distributions of the regres-
sion coefficient estimators and the corresponding test statistics when
the regressor and disturbance term are independent fractionally inte-
grated processes. The main finding is that a long memory disturbance
term could cause the null hypothesis to be overly rejected. The main
conclusion of this dissertation is that careful study of the properties of
the regressor and residuals are necessary before a regression is used.
Otherwise, we could incorrectly find two independent time series to
be correlated or two correlated time series to be independent, due to

the persistence in data series.

ACKNOWLEDGMENTS

First of all, I would like to thank the members of my disser-
tation committee for their precious advice to help me prepare and
complete this dissertation. Without their assistance the completion of
this dissertation will not be possible. Especially I would like to thank
Professor Richard T. Baillie, the committee chair, who inspires me to
work on the subjects of this dissertation and provides me invaluable
comments. Also I would like to thank Professor Peter Schmidt who
reads my manuscript patiently and suggests numerous corrections and
improvement. Moreover, my greatest debt of gratitude goes to Pro-
fessor Ching-Fan Chung, who reads much of the near-final draft and
offers substantial mathematical and editorial comments on it.

There are so many people who have been important for me dur-
ing the whole program. I gratefully acknowledge Professor Christine
E. Amsler who is especially generous to me when I was her Teaching
Assistant. Also I am glad that I met Hailong Qian and Keshin Tswei
who entertain my life in Michigan State University. Especially I am
indebted to Hailong Qian for his patience and wisdom to help me solve
many problems. I also want to thank Dan Hansen, Bill Horrace, Jess
Reaser and Jim Zolnierek who have been my officemate and classmate.
They are kind and teach me a lot of American culture. Without them,

studying in my office room would not have been fun anymore.

iii

Special appreciation also goes to the administrative staff in the
department. Especially I acknowledge Mrs. Ann Feldman who is
kind and offers me helpful advice to help me get through my program
smoothly.

Finally, I would like to thank my wife, Ching-Yi Hsieh, for her
sacrifice, understanding and encouragement. Without her help, it is
impossible for me to finish the whole program. I also want to thank
my parents for their love and support. Unfortunately, my father died
two years ago. I could never repay the debt of gratitude I owe him.
Also I wish to express my thanks to the family of my parents in law,
they offer me a lot of help and I am indebted to them forever. My
son, Kevin Tsay, who gives me the most wonderful joy, I would like

to thank him for his presence in my life.

TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION ....................... 1

CHAPTER 2 THE SPURIOUS REGRESSION OF
FRACTIONALLY INTEGRATED PROCESSES .. 6

2.1. Introduction ..................................... 6
2.2. A General Theory of Spurious Effects ......... 8
2.3. Monte Carlo Experiments ..................... 41
2.4. Conclusion ...................................... 43
2.5. Mathematical Proof ............................ 46

CHAPTER 3 THE SPURIOUS EFFECT WHEN
REGRESSOR AND DISTURBANCE ARE
FRACTIONALLY INTEGRATED PROCESSES 72

3.1. Introduction .................................... 72
3.2. The Four Classes of Models ................... 73
3.3. The Class A Models ........................... 75
3.4. The Class B Models ........................... 81
3.5. The Class C Models ........................... 87
3.6. The Class D Models ........................... 89
3.7. Monte Carlo Experiments ..................... 92
3.8. Conclusion ...................................... 94
3.9. Mathematical Proof ............................ 94
CHAPTER 4 CONCLUSION ........................ ' 145

LIST OF REFERENCES ............................... 146

LIST OF TABLES

TABLE 2—1 REJECTION PERCENTAGES AND MEAN
|tg| UNDER MODEL 1 ............................ 66

TABLE 2—2 THE DIV ERGENCE RATE OF MEAN |t5|
UNDER MODEL 1 ................................. 67

TABLE 2—3 REJECTION PERCENTAGES AND MEAN
|t3| UNDER MODEL 2 ............................ 68

TABLE 2—4 THE DIVERGENCE RATE OF MEAN |tfi|
UNDER MODEL 2 ................................. 69

TABLE 2—5 REJECTION PERCENTAGES AND MEAN
|t3| UNDER MODELS 3 AND 4 .................. 70

TABLE 2—6 THE DIVERGENCE RATE OF MEAN |t/3|
UNDER MODELS 3 AND 4 ....................... 71

TABLE 3—1 REGRESSION OF C. = 1 + 231+ 5. ON 1 AND
cot, WHERE x. = 1(1) AND a. -.= I(d) .............. 141

TABLE 3—2 THE DIVERGENCE RATE OF MEAN |tg|
UNDER MODELS A—l AND B—l ............... ‘ 142

vi

TABLE 3—3 REGRESSION OF C. = 1 + 33,0 + 5. ON 1 AND
x3, WHERE x: = t+ 1(1) AND a. = I(d) .......... 143

TABLE 3—4 THE DIVERGENCE RATE OF MEAN |tg|
UNDER MODELS C—l AND D—l ............... 144

vii

CHAPTER 1

INTRODUCTION

It is a 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). In recent years we also witness fast growing studies on
fractionally integrated processes, or the I (d) processes with the differ-
encing parameter d being a fractional number. The I (d) processes are
natural generalizations of the I (1) process that exhibit broader long-
run characteristics. More specifically, the I (d) processes can be either
stationary or nonstationary, depending on the value of the fractional
differencing parameter. The major characteristic of a stationary I (d)
process is its long memory which is reflected by the hyperbolic decay
in its autocorrelations. A number of economic and financial series
have been shown to posses long memory.

The I (d) process is not the only model that displays the hy—
perbolic decay in its autocorrelations. The long range dependence in
time series data can be traced back to Hurst (1951) who found the
long term persistence in hydrology data which were referred to as a
Hurst effect. The work of Hurst has attracted a lot of attention while
the phenomenon of the hyperbolic decay in autocorrelations can be
observed in many other fields. Many models have thus been proposed
to characterize the long range persistence in time series. The two most
famous models are the fractional Gaussian noise model proposed by
Mandelbrot and Van Ness (1968) and the fractionally integrated pro-

cess which is the subject of this dissertation.

l

2
A process Yt is said to be a zero mean fractionally integrated

autoregressive-moving average process of order p,d,q or ARFIMA

(p, d, q) if it is defined by
“MU—LWKFAXMQ an

where L is the lag operator, <I>(.) is a 11‘" order polynomial, d is the
differencing parameter, O(.) is a qth order polynomial, the roots of
<I>(.) and @(.) are outside the unit circle, <I’(.) and @(.) have no com-
mon roots, 6, is a white noise process with finite variance of and the
fractional differencing operator (1 —— L)d has the following binomial

series expansion:

(1 — L)" = Z «xv-Li (1.2)
j=0
where

, PU—d) I—l—d .
u.= , = .____g zotga , 13
J in¢+nrpa) Ali f 3 ( )

 

and F(.) is the gamma function.

Given the above definition, we note a random walk is ARFIMA
(0,1,0) and a zero mean fractionally integrated process of order d, I(d)
is ARFIMA(0,d,0) and has the following form:

u—Lflxzm. (1%

This process is first introduced by Granger (1980, 1981), Granger and
J oyeux (1980), and Hosking (1981). They show that I ( d) is stationary
and invertible when d E (—0.5,0.5).

The main feature of the I (d) process is that its autocovarance
function declines at a slower hyperbolic rate (instead of the geometric

rate found in the conventional ARMA models). When (I > 0, the series

3
are positively correlated and we call it a long memory process since

it exhibits long range dependence 1n the sense of Z J_ 00 (j) = 00,
where r(j) is the autocovariance function of I (d) at lag j. When
—0.5 < d < 0, the series are negatively correlated and such that
2:40 | 7’( j) |< 00, and the process is sometime referred to as an

intermediate memory process.
If (I E (—0.5, 0.5), then an I(d) process has the following AR(oo)
and MA(oo) representations:

 

 

Y, = :¢,1';_,~+ e. j =1,2,3,... (1.5)
.21
where
_ PU - d) _ / .
Q” “ P<j+1>P<—d> ‘ ’f’” (1'6)
and 00
”,2 2 Am, = (1 — L)‘%. (1.7)
j=0
where
I‘(j + d) E — 1+ (1 . ,
.2 , = —, )=0,1,2,.... 1.8
J I‘(] + 1)I‘(d) 011g)“ f ( ‘)

Let us denote the spectral density, autocorrelation function and
partial autocorrelation of a stationary I ((1) process by f ( .), p(.) and
cr(.), respectively, then

02 02
f(/\)=|1-€"A r)";— =I 2....(1/2) I 2" 2— —7r 3 ,\ s 7r, (1.9)

W’

 

._I‘(j+d)I‘(1—d)_ €—1+d ._
P(J)—F(j_d+1)r(d)— H fl )_1,2,..., (1.10)

0<£gj

 

and (1
Applying Stirling’s formula to (6), (8) and (11) gives that, as j —> oo,
1 -—d—l
1 :d—1 .
and F( d)
- 1 — -2d—1
(2(1) I(d) J , (1.14)

where a,- ~ bj means lim 11.00 aj/bj = 1. Moreover, we note sin/\ ~ A
as )1 —> 0. Therefore,

f(/\) ~ r“. (1.15)

as A ——> 0. This result suggests that the spectral density of a long
memory process has a singularity at 0 frequency.

This dissertation considers the spurious effect in a simple lin-
ear regression of I (d) processes. We first study the spurious effect
when we regress a fractionally integrated process on a constant and
another independent fractionally integrated process and then investi-
gate the spurious effect when the disturbance term and regressor are
fractionally integrated processes.

This dissertation is organized as follows: In Chapter 2 we extend
Granger and Newbold’s (1974) work to study the spurious effect of re—
gressing a fractionally integrated process on a constant and another
independent fractionally integrated process. We also examine the spu-
rious effect of detrending an I ((1) process. We specify the conditions
under which the spurious effect can occur. An important implication
from our results is that the usual procedure to avoid spurious effect

by differencing may be questionable.

5
In Chapter 3 we show the spurious effect when the dependent

variable and the regressor are correlated while both regressor and dis—
turbance term are I (d) processes. We consider the issue of the sta-
tistical inference regarding the regression coefficient estimators. The
conclusion is that many empirical testing results should be interpreted
more carefully because the usual t tests may not be valid when long
memory is present in the disturbance term.

The Conclusion and some extensions are presented in Chapter

CHAPTER 2

THE SPURIOUS REGRESSION OF
FRACTIONALLY INTEGRATED PROCESSES

2.1. Introduction

The spurious regression was first studied by Granger and N ew-
bold (1974) using simulation. They show that when unrelated data
series are integrated processes of order 1 or the I (1) processes, then
running a regression with this type of data will yield spurious effects.
That is, the null hypothesis of no relationship among the unrelated
I (1) processes will be rejected much too often. Furthermore, the spuri-
ous regression tends to yield a high coefficient of determination (R2) as
well as highly autocorrelated residuals, indicated by a very low value
of Durbin—Watson (DW) statistic. Granger and N ewbold’s simulation
results are later supported by Phillips’s (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
(T) approaches infinity. He also shows that R2 has a non-degenerate
limiting distribution while the DW statistic converges in probability
to zero.

The history of the research on spurious detrending follows a
similar thread. Nelson and Kang (1981, 1984) first employ simulation
to demonstrate that the regression of a driftless I (1) process. on a
time trend produces an incorrect result of a significant trend. Ex-

tending the Phillips’s (1986) approach, Durlauf and Phillips (1988)

derive the asymptotic distributions for the least squares estimators in

6

7
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 significance tests.

All these studies of the spurious regression concentrate on the
nonstationary I (1) processes. It reflects the widely held belief that
many data series in economics are I ( 1) processes, or near I (1) pro-
cesses, as argued by Nelson and Plosser (1982). Against this backdrop,
we also witness in recent years fast growing studies on fractionally inte-
grate processes, or the I (d) processes with the differencing parameter
d being a fractional number. The I ( d) processes are natural general-
ization of the I (1) processes that exhibit a broader long-run charac-
teristics. More specifically, the I (d) processes can be either stationary
or nonstationary, depending on the value of the fractional differencing
parameter. The major characteristic of a stationary I ((1) process is its
long memory which is reflected by the hyperbolic decay in its autocor-
relations. A number of economic and financial series have been shown
to possess long memory. See Baillie (1995) for an updated survey on
the applications of the I (d) processes in economics and finance.

The objective of this chapter is to extend the theoretical analy-
sis 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 inflict the spurious effect in a simple linear re-
gression model. The nonstandard asymptotic distributions of various
coefficient estimators and test statistics are then derived.

The main finding 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, different from what

8
Phillips (1986) and Durlauf and Phillips (1988) have suggested, the

cause for spurious effects seems to be neither nonstationarity nor lack
of ergodicity but the strong long memory in the data series. As a
result, spurious effects might occur more often than we previously
believed as they can arise even among stationary series. Furthermore,
the usual first-differencing procedure may not be able to completely
eliminate spurious effects 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).
2.2. A General Theory of Spurious Effects

Our analysis of the spurious effects are based on several simple
linear regression models in which the dependent variable and the single
non-constant regressor are independent I (d) processes with d lying
in different ranges. Before presenting these models, let’s first briefly
review some basic properties of the I (d) processes.

A process Yt is said to be a fractionally integrated process of

order (I, denoted as I (d), if it is defined by
(1 — L)d)/t : Eta

where L is the usual lag operator, (1 is the differencing parameter which
can be a fractional number, and the innovation sequence 6) is white
noise with a zero mean and finite variance. The fractional differencing
operator (1 — L)d is defined as follows: '

PU — d)
(1+ 1)F(—d)’

 

(1 _. L)d = E 1/1ij, where 1/Jj = I‘
i=0

9
and F() is the gamma function. This process is first introduced

by Granger (1980, 1981), Granger and Joyeux (1980), and Hosking
(1981). They show that Y} is stationary when d < 0.5 and is invertible
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):

70') = 0(1'2‘1‘1),

where 7(3) is the autocovariance function at lag j. When (1 > 0, the
I (d) process is said to have long memory since it exhibits long range
dependence in the sense that 22:4» y( j) = 00. When (I < 0, then
2;:_OO |7(j)| < 00 and the process is sometimes referred to as an
intermediate memory 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 2 0.5 and the
stationary subclass with d < 0.5. To examine potentially different
types of spurious effects, we propose six regression models for differ-
ent classes of I (d) processes, mainly based on whether the fractional
differencing parameter (1 is greater than 0.5 or not. The exact specifi-
cations of these models can be conveniently expressed with four I (d)

processes. Let’s first define two stationary ones with different differ-

encing parameters (11 and d2 whose values lie between —0.5 and 0.5:
(1 — L)d‘v) 2 at and (1 — L)d2wt = b),

where at and bt are two white noises with zero mean and finite vari-

ances 0:“: and 0?, respectively; that is, v) and wt are I ((11) and I (d2)

10
processes, respectively, and both of them are stationary and invert-

ible. When these two processes are employed in our later analysis,
the values of their differencing parameters are mostly assumed to be
in (0, 0.5); i.e., the stationary processes vi and wt are often assumed
to have long memory. We can also define two nonstationary I ( 1 + d1)

and I (1 + (12) processes by integrating v) and w):
311 = 311—1 +1}: and It = 131—1 + wt-

Obviously, the orders of integration of these two nonstationary frac-
tionally 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 1: .111 = 0' +131?) + 11),
Model 2: v) = (r + [311), + u), where d1 + (L; > 0.5,
Model 3: y), = a + flwt + u), where d2 > 0,
Model 4: v) = a + (it) + at, where d1 > 0,
Model 5: 3/1 = a + fit + 11),
Model 6: 1!) = a; + {it + at, where (I) > 0.

In Model 1 the orders of integration of both the dependent vari-
able 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

11
the dependent variable and the regressor are assumed to be station-

ary, ergodic, and strongly persistent in the sense that their fractional
differencing parameters sum up to a value greater than 0.5. Following
Phillips’ arguments, we tend to think no spurious effect 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 effects and allow us to gain
new insight into the problem.

Models 3 and 4 differ from Model 1 in that the order of inte-
gration 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 non-
stationary fractionally integrated processes and Model 2 of stationary
fractionally integrated processes. We thus 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 effect of detrending the
nonstationary and stationary fractionally integrated processes, respec—
tively. 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 1:)
replaced by the time trend. This similarity in the model specifications
will also be reflected in their analytic results.

The following assumption on the two white noise processes a.)

and b) are made throughout this paper to simplify our analysis.

Assumption 1. The two white noises a) and b, are independently and
identically distributed with zero means, and their moments satisfy the

following conditions: E|at|p < 00, with p Z max{4, —8(l1/(1 + 2d1)};

12
and Elbtlq < 00 with q 2 max{4, —8d2/(1 + 2d2)}. Moreover, at and

b.) are independent of each other.

We also assume, without loss of generality, that the initial values
of the fractionally integrated processes v0, wo, yo, and 3:0 are all zero.
Hence, yt and cm can be considered as the partial sums of v) and wt,
respectively; i.e., 3],. = 23:11)) and 117,. = 2321M-

Before presenting Lemma 1, which is the cornerstone of our anal-
ysis, let’s summarize two important asymptotic results on the partial

sums yt and :13). First, given the variances of yT and car:

a: — —Var( (y)T =Var (Z Vt) and a: —— Var(x =Var (2: wt)

Sowell (1990, Theorem 1) proves that

02 = O(T1+2d‘) and 02 = O(T1+2d'~’).

y .1:

Furthermore, Davydov (1970) shows that as T —> 00,
1 . 1 ,
— yr", => B0,5+d,('r) and 0— ;rml :> B0,5+d2(r),
y 1'
for r E [0, 1], where [T 7'] denotes the integer part of Tr, the notation
=> denotes weak convergence, and B0,5+d(t) is the fractional Brownian

motion which is defined by the following stochastic integral

1 t
Bo.5+d(t) _=_ ) f (t — s)d dB0,5(s), for d E (—0.5, 0.5),
0

m
where 30,5(1?) is the standard Brownian motion. See Mandelbrot 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. These two well-known results help

establishing the following lemma.

13
Lemma 1. Given that Assumption 1 holds, then, as T —> 00, we

have the following results:

T .1
1 y; 1
1. T- E 3— 2> /0 B0.5+dl(5) (IS and

t=l y
1 T - '1
l
— — :> B05 (12(8) (IS.
T Z a /0 +

1 T 312 1 2
2. E —-’- => [[Bo,5+d,(s)] ds and
.0

1 T If 1 ,, 2
TED"? fi .0 [B()_5+d2(.5)] (18.

£21
I” “It
4 -— => B0 5+d1(1) and — => B05+d(1)
t:1 04" 1:1 (71.

 

"BIH

—_— 0,,(Td1+d2), if d1+ d2 2 0.5,

T
E ”01101

:1 .
t _<_ 0,9(TO‘5), otherWISe,

or, equivalently,

T 0p(Tf—l), If (11+ d2 2 0.5,
’Ut w)

0p(T‘_0'5’d1’d'~’), otherwise,

for any 6 > 0.

T
1 M It

1
T E—- 0— => /0Bo.5+d1(3)'Bo.5+d2(3)d19-
t=l y 1:

3| E

1
=> / B0.5+d1(8)dB0.-5+(12(S)7 for d2 > 0,
0

Tyt
23;...

15

T j 1
231.2 2. / BO,,,,,(.9)dBO,+.,,(s), for d1>0.
i=1 . 0

T 1
t
10. E TE => B0,5+d1(1)—/ B0_5+d,(s) ds and
0

T 1
t
255— :> Bo.5+d2(1)—/ B0,5+d,(s)ds.
. 0

T 1
1 t
11. TE T-g—t- => /0.9-BO,5+(1,(3) ds and

1 2T t :r 1
It ..
5"— T.-0_—x. : A QS'BO5+d2(QS) dog.

Here, BO,5+d,(t) and BO,5+d,(t) are two independent fractional Brown-
ian motions, 7,, (j) and 7,; (j ) are the autocovariance functions of vt and
wt, respectively, at lag j, and 02 and of are the variances of the un—
derlying white noises at and b), respectively. The notation -—p——> means

convergence in probability.

All the theorem proofs are in the Mathematical Proof. In the
rest of this section the results of Lemma 1 will be used to develop
the theory of spurious effects, presented in a series of theorems and
corollaries, for the proposed six models. The first two models will be
discussed separately in subsections 2.1 and 2.2. These two models

provide us with a framework which facilitates the explanations of the

16
other four models in subsections 2.3 and 2.5. One subsection — subsec-

tion 2.4 — will be devoted to the analysis of an important issue about
how the orders of integration of the fractionally integrated processes
are directly related to the spurious effects.

We will adopt the following notation for the various statistics
from the Ordinary Least Squares (OLS) estimation. Let 62 and ,6
denote the usual OLS estimators of the intercept and the slope. Their
respective variances are estimated by 3% and 33,, from which we have
the t ratios t5 = 3/83 and to, = (Ar/sq. Also, let 32 denote the estimated
variance of the OLS residuals, R2 the coefficient of determination,
and DW the Durbin-Watson statistic. Finally, in addition to the

autocovariance functions 7,,(j) and yw(j) of U) and 11),, let p,,(j) and

pw( j ) be their respective autocorrelations at lag j.

2.2.1. Model 1 of Nonstationary Fractionally Integrated

Processes

In Model 1 a nonstationary I (1 + d1) process yt is regressed on

another independent and nonstationary I ( 1 + d2) process 1:):
yt = CY +/3It + Ut-

Since the permissible range for the values of the fractional differenc-
ing parameters d1 and d2 is (—0.5, 0.5), Model 1 generalizes Phillips’
(1986) model of integrated processes in which d1 2 d2 = 0. Unsur-
prisingly, all the results we derive for Model 1 are straightforward
generalization of Phillips’ theory of the spurious effects. The results

for Model 1 are presented in the following theorem:

17
Theorem 1. Given that Assumption 1 holds, then, as T —> 00, we

have the following results:

0' A
1. 1,3 =>
031

1 1 l
/ Bo.5+d1(3)'Bo.5+d-.>(3)(13— [] Bo.5+d.(8)d8] U Bottoms]
0 0 7 0

l 7 1 2
/ [B0,5+(12(.9)]2 ds — [/ B()_5+d,,(s) (15]
0 0

E (’3...

Note that cry/or = O(T"1_d'~’).

1 A 1 1 ,
2. — CY 2} B0.5+d1(3) d3 — [3*] B()_5+d._,(8) (18 E (1*,
0

0y 0

where [3... is defined in 1. Note that 0,, = ()(T0'5+"').

1 ‘ 1 ‘ 1 ‘2
3. —,— 82 => / [B0_5+d1(8)]2 (f8 - [/ B0_5+d1(8) d8]
0 O

05
1 2 1 2
_ [33 / [BO.S+d3(3)] d3 - [/ B0.5+d2(3) d8] } E 03,
0 0

where (3,. is defined in 1. Note that of, = ()(T’Hdl ).

To2 . 02
4. ——xs,); : * 2

1 1 2 E 0,1,3,
y / [BO‘S+"2(S)l2 d3 _ [/ B0.5+dg(5) ds]
0 0

 

7.

8.

9.

18
where of is defined in 3. Note that 05/110: = 0(T2d1—2d2‘1).

1
[/ BO_5+d2(S) d9]
0
1 2 1
/ [30.5+(12(8)] (18 - [/ B0.5+d2(3) d8]
0 0

‘2

 

T .
—s‘2 => of 1+ 2

 

where fl. is defined in 1 and 033 is defined in 4.

1 a...
ta => ,

:77: i 0*0

 

where oz... is defined in 2 and 0:0 is defined in 5.

1 1 2
16:12: {/ [B0.5+d2(3)]2 d3 — [/ BO.5+d2($) d8] }
. 0 0
R2 => 1 1 2 ,
/ [30.5+d1(3)l2 d3 " [/ BO.5+d1(3) (13]
0 0

where 1’3... is defined in 1.

 

DW—p—>O.

19
Here, BOI5+d,(t) and B0.5+d2(t) are two independent fractional Brown-

ian motions.

The most important result in Theorem 1 is that, as the sam-
ple size T increases, the two t ratios tfl and to diverge at the same
rate of x/T, which is independent of the magnitudes of the fractional
differencing parameters d1 and d2. This result is exactly the same as
what Phillips (1986) has obtained for the case where (11 = (12 = 0. So
even when the orders of integration in the dependent variable and the
regressor differ 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
,{3 = O or a = 0 based on t tests increases monotonically as the sample
size increases.

The limiting distributions of the t ratios, after normalized by
x/T, are direct generalization of those derived by Phillips. The same
conclusion also holds for R2 and the DW 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 (11 and
d2 do not affect their convergence rates while the effects on their limit-
ing distributions are quite straightforward: all the standard Brownian
motions in Phillips’ theory are replaced by fractional Brownian mo-
tions. That the fractional differencing parameters d1 and (12 play a
relative minor role here is mainly because the four statistics are all
ratios so that the effects of (11 and d2 are canceled out. In contrast,
the results on the OLS estimators [If and a are a different story. In
Phillips’ theory both 3 and 31/ x/T converge to some non-normal non-
degenerate limiting distributions. But for the present model of the

fractionally integrated processes, the orders of fl and (.7 are T"“d2 and

20
Td1+0'5, respectively. So while 6 always diverges (though the rate can

be slow if all is close to —0.5), B can be either divergent or convergent,
depending on the relative magnitudes of (11 and d2. For example, if the
order of integration in the dependent variable y; is smaller than that
of the regressor rt; i.e., (11 < (12, then 3 converges to zero, just like the
conventional case of no spurious effects. Moreover, if d1 — d2 = —0.5,
then, similar to the case of no spurious effects, x/T-B has a limiting

distribution, though its limiting distribution is not normal.

2.2.2. Model 2 of Stationary Fractionally Integrated

Processes

In this section we consider Model 2 in which a stationary frac-
tionally integrated process vt is regressed on an independent and sta-

tionary fractionally integrated process 111,:
1’1 = (Id-X3101 + Ut-

We show that, although both vi and wt are stationary, the spurious
effect in terms of the t tests could still exist under an additional condi-
tion on the fractional differencing parameters: d1 + d2 > 0.5. Loosely
speaking, this condition implies that the two processes vi and wt are
both strongly persistent.

Our analysis begins with a special case where we assume a set
of more stringent conditions which helps deriving the exact limiting
distributions for the various OLS estimators. 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

21
case, we then show how the spurious effects may still exist in a more

general framework even though the exact limiting distributions cannot
be readily defined in such a case.
Let’s first reproduce Fox and Taqqu’s (1987) Theorem 6.1 here

as Lemma 2.

Lemma 2. Let (X1,Yt) be a stationary jointly Gaussian sequence
with E(Xt) = E(Yt) = 0, E(X%) = E(Yt2) = 1, and E(Xth) = r.
Suppose that 01 and 02 are two arbitrary real numbers and that there
exist 0 < 61, (52 < 0.5, such that asj —> 00

- -_1 {”10sz ._ , .,
E(XtXt+j) N Ui'] 6 1 E(Xth+j) N W] (6 “SJ/2,

‘ ._,, 0102b2 ,_ ~1 ., .
E(Y1Yt+1) N 05.] 6-, E(YtXt+j) N ”751?;3 (a +6-)/2,

where p is a constant between 1 and —1, while (11 = A(61,61), a2 =
A(62,62), bl = A(61,62), and b2 = A(62,61) are four constants with
A(61,62) being defined by foo $M1+1’/2( +1) “um/2111:, then

0

[T3]
1 . ,
Tl—(51+62)/2 :(At}t — 7’) i Z(S)’
i=1

 

where Z(s

0 U
¢;_1:_2/3:S/OS [HUNT —i(6 +1)/QI{J <10] (111 (“1410131) dfl’[2(l'2)

i=1

 

Here, M1 and M2 are two Gaussian random measures with respect to

Lebesgue measure, having unit variances and covariance p.

Note that the two processes Xt and Yt are not only strongly

persistent, as indicated by the hyperbolic convergence rates 61 and 62

22
in their autocorrelations, 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 t and Yt are independent so that r and p in the above lemma are both
zero. The above lemma offers us the convergence rate of 2le Xth
and its limiting process Z (t) given the Gaussian assumption and a
narrower range for the parameters 61 and 62. In order to apply this
lemma, we make the following assumption in addition to Assumption

1 we have made earlier.

Assumption 2. The two fractionally integrated processes vt and wt
are both Gaussian and the corresponding fractional differencing pa-

rameters d1 and d2 are both in the range of (0.25, 0.5).

Given the facts that

. Pl—d ._ . P1—d ._
PleN—(‘P—(Cm‘lllml ’ and pw(J)~—(1:(712)—2)J"” 1.

it is straightforward to prove the following corollary in which X t and Y,

in Lemma 2 are replaced by vt/ \/7,,(0) and wt/1/7w(0), respectively.

Corollary 1. Given that Assumptions 1 and 2 hold, then, as T —-> oo,

 

1 T W wt
W: Mo? W Z} 2“”
Where the limiting random variable Z (1) is defined in Lemma 2 with
51 = 1 — 2d1, 62 = 1 — 2(12, of = I’(1 — d1)/P(d1), and 0% = I’(1 —
d2)/P(d2).

23
With Lemma 1 and Corollary 1, we can then establish the fol—

lowing theorem about the spurious effect in Model 2.

Theorem 2. Given that Assumptions 1 and 2 hold, then, as T —> 00,

we have the following results:

 

 

 

 

T2 ’2‘ 79(0) 1 .
1. => ZI)— B_5d,l-B.5.,1.
03/01: 710(0) [ ( \/7v(0).7w(0) 0 + ( ) 0 +d,( )
Note that oyol./T2 = O(Td1+d2").
T A
2. — CY z) B0.5+d1(1).
0y

Note that oy/T = 0(Td1‘0'5).

3. 82 -p—> 7,,(0).

 

V’ 7 71.10).
5. T-si —p—> 7,,(0).

3/2 1
T tg 2} Z(1)

0,... _ x/7v(0)'7w(0)

 

 

B0.5+1il(1)'Bo,5+d2(1).

 

Note that oyox/T3/2 = O(Td1+d2‘0'5).

24

 

Bo.5+d1 (1 ).

Note that cry/VT = 0(Td1).

T4

2
0y

1

2 _
R 1” Z(1) x/vv(0)-7w(0)

 

 

 

02 B0_5+d1(1)'B0.5+d2(1)

Note that ogog/T" = O(T2d1+2"2—2).

20—2a)

P
0 2—2v1:
9 DW——* p() 1_d1

 

Here B05”, (t) and B0_5+d,(t) are two independent fractional Brownian

motions, and Z (1) is a random variable defined in Corollary 1.

The most important result from this theorem is the divergence
rates of the two t ratios 15;; and ta, which are le‘Ldz‘O'5 and T“, re-
spectively. Recall that d1 + (12 — 0.5 is necessarily greater than 0 (and
smaller than 0.5) under Assumption 2. This result reflects the spuri-
ous effect in the t tests. Since both the dependent variable and the
regressor are stationary and ergodic, the spurious effect is not really
expected (see Phillips 1986, p.318). The surprising results we get here
suggest that the cause for the spurious effect 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 \/T rate we observe in Model 1. We note that the

25
divergence rates in the present model depend on the magnitudes of

the two fractional differencing parameters d1 and (12 while those in
Model 1 do not. Furthermore, the t ratios diverge more slowly in the
present model than in Model 1. In particular, the divergence rate of
t3 can become very slow when both d1 and (12 approach to their lower
boundary 0.25.

Let’s turn to the OLS estimators 1’1 and a. We note that both
of them converge in probability to zero as in the conventional case of
no spurious effects. However, their convergence rates are much slower
than the usual T‘l/2 rate and, if they are normalized appropriately,
their limiting distributions are not standard normal either. In contrast
to these irregular convergence rates of the OLS estimators, the esti-
mated variances 3% and sf, nevertheless converge at the standard T ‘1
rate. It is such disparity in the convergence rates between the OLS

1/2, and their stan-

estimators, which converge at rates slower than T—
dard errors, which converge at the standard T’l/2 rates, that causes
the resulting t ratios to diverge and hence the spurious effect.

R2 in the present model converges to 0 as in the case of no
spurious effects. It is different 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 reflect
the fact that the regressor does not help explain the variations in the
dependent variable.

The DW statistic does not converge in probability to zero and
this result is also different from that of Model 1. Its limit 2 — 2pv( 1) is

similar to the one we find in the conventional AR(1) case. This limit

depends on the fractional differencing parameter d1 of the dependent

‘26
variable vt and can only take value in the range of (0, 4/3), which is

to the left of the value 2.
We now consider a less restricted specification of Model 2 which

is defined by the following assumption.

Assumption 2A. The sum of the two fractional differencing param-

eters d1 and d2 is greater than 0.5.

Since the Gaussian distribution is not assumed while one of the
fractional differencing parameters d1 and (12 can be smaller than 0.25,
Assumption 2A is thus considerably less stringent than Assumption 2.
The price we pay for such generality is that we are not able to express
the limiting distribution of some statistics in closed form as indicated

in the following corollary.

Corollary 2. If Assumption 2 is replaced by Assumption 2A in T he-
orem 2, then all the conclusions there remain true except that, while
T2fi/oyoz, T 3/ Qtfl / oyox, and T 4R2 / 0303, still converge weakly, the ex-
act specifications of their limits are unknown. In particular, the pro-
cess Z (1) in Theorem 2 will be replaced by a process of an unknown

form .

The main finding in this corollary is that, even though the lim-
iting distributions of some statistics are not readily expressible, all the
discussions of the spurious effects following Theorem 2 still apply to
the more general specification of Model 2. The analysis of Model 2
can thus be summarized as follows. The OLS estimators 3 and d (as

well as R2) do converge in probability to zero, correctly reflecting the

27
lack of a relationship between the dependent variable and the regres-

sor. But the convergence rates of 3 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 dependent variable and the regressor are both
stationary and ergodic (so long as they are sufficiently 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+d1 and 1+d2, respectively, where d1+d2 > 0.5, then first-
differencing both variables cannot completely eliminate the spurious
effects. While B2 may be reduced and the DW statistic may be in-
creased, the t ratios may still be so large that we cannot avoid making
a spurious inference. This is a fairly serious problem with the regres-
sion for the fractionally integrated processes. It implies that even the
popular first-differencing procedure might not prevent us from finding
a spurious relationship among highly persistent data series. One les-
son 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.
2.2.3. Two Intermediate Cases: Model 3 and Model 4

Model 3 and Model 4 can be considered as two intermediate
models between Model 1 and Model 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 Model 1 and Model 2.

28
In Model 3 a nonstationary I ( 1 + d1) process yt is regressed on

an independent and stationary I ( ([2) process wt:
yt = (1" + [3101 + 11,, where (12 > 0.

Note that the fractional differencing parameter d2 for the regressor
wt here is assumed to be positive; i.e., wt has long memory. The

asymptotic properties of the OLS estimators for Model 3 are given in

the following theorem:

Theorem 3. Given that Assumption 1 holds, then, as T —> 00, we

have the following results.

A

 

 

1. /3 =>
ago],
1 1
{/ BO.5+d1(5) dBo.5+d2(8) - [/BO.5+d1(3)d3:|'B0.5+d2(1)}
’1’w(0) 0
E 13,...
Note that oon/T = O(T"‘+d'~’).
1 1
2. — a => B0,5+d1(8) (is E CY...
0y 0

Note that a, = O(T°-5+d1).

1 1 1 2
3. -—2 82 11> / [B0,5+d1(3)]2 d8 - [/ B0.5+d1(8) d3] E 03.
0y 0 0

7.

29
Note that a; = O(T’+2"1).

T 2 2; o
-—1- S " 3
05 ‘3 7.1,(0)

*N

 

where of is defined in 3. Note that og/T = 0(T2d1).

 

 

where [3* is defined in 1 and of is defined in 3. Note that O'x/\/'_ =
0(Td2).

1 a...
to Z?) —a

7? a.

where CY... is defined in 2 and of is defined in .3.

_R2 2) ill/1140);}:

o? of

where [3,. is defined in 1 and of is defined in 3. Note that JE/T2 =

30
0(T2d2‘1).

9. DW 1. 0.

Here B05”, (t) and B0.5+d,(t) are two independent fractional Brownian

motions.

Since both t ratios diverge, Model 3 also suffers from the spuri-
ous effect in terms of the t tests. Moreover, we find the results that
the OLS estimator a diverge and that DW 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
Model 1 and 2.

It should be pointed out that in Theorem 3 the range of the
fractional differencing parameter d2 of the regressor wt is restricted to
the positive half of the original range (-0.5, 0.5). For the case of a
negative d2, it is quite straightforward to show that the t ratios are
convergent and there is no spurious effect.

In Model 4 a stationary I ((11) process is regressed on an inde-

pendent and nonstationary I (1 + ([2) process
1), = a + [317, + 11‘, where (11 > 0.

Similar to the restriction imposed on Model 3, the fractional differenc-
ing parameter d1 of the dependent variable fut is assumed to be positive
50 that v; has long memory. The asymptotic theory for Model 4 is pre-

Sented in the following theorem.

31
Theorem 4. Given that Assumption 1 holds, then, as T —> 00, we

have the following results.

T01- 7‘

 

 

 

 

 

1. )3 =>
01/
1 ~ 1 -
/ B0.5+d3(3) dB0.5+d1(3) - BO.5+d1(1)' / B0.5+d2(3) d8
0 1 1 h 0 2 i Z ’3’"
/ [B0.5+d2(3)]2 d8 — [/ B0.5+d2(8) d9]
0 0
Note that oy/Tox = 0(Td1‘d2—1).
T 1
2. — 8 => B0,5+d1(1) — 13*] B0.5+d2(3) (13 E CY,”
0y 0
where [3... is defined in 1. Note that oy/T = O(Td1_0'5).
3 82 —£-> 711(0)
4. Tog-3,2, 7””(0) = 02.3.

2) 1 1 2 — *1
/ [B0.5+d2(8)l2 €13 — [/ B0.5+d2(3) d8]
0 0

Note that 1/Taf, = O(T“2—?d2).

32
‘2

1/ d1 .

/01 [B0.5+d2(8)l2 d3 — [/01305+d2(3) d3] 2

x/T 1:3.

-—ts => .
0y 0*);

III
q

711(0) 1 +

 

where [3... is defined in 1 and 03,, is defined in 4. Note that oy/ \/— :-

0(Td1).

 

0(Td1).

T2 2
8 —.—,R2 z.» 2*,

0y */3

where fl... is defined in 1 and 03,, is defined in 4. Note that ”ii/T? =
O(T2d1_’).

9. DW ii. 2—2p,(1)=2(i_:"".
_ 1

 

33
Here B0.5+d1 (t) and B0,5+d,(t) are two independent fractional Brownian

motions.

Since both t ratios diverge (at the same rate), the spurious effect
in terms of failing t tests again exists in Model 4. But contrary to the
results in Model 3, the OLS estimators g and 3, together with R2, all
converge in probability to zero, while the DW statistic converges to
2 — 2,0,,(1). These findings obviously bring Model 4 closer to Model 2
than to Model 1.

2.2.4. 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 effects. 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
specifications.

First note that the divergence rates of the t ratio t3 are T05,
Td1+d2—0'5, T ”’2, and T“, respectively, for Models 1 to 4. Let’s also
compare the specifications of the four models using Model 1 as the
benchmark: Model 3 differs from Model 1 in that the order of inte-
gration in the regressor is reduced from above 0.5 to below 0.5 (but
above 0); Model 4 differs from Model 1 in that the order of integra-
tion in the dependent variable is reduced from above 0.5 to below 0.5
(but above 0); and, finally, Model 2 differs 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

34
integration with the changes in the divergence rates of t1}, we can con-

clude that reducing the order of integration in the dependent variable
causes the divergence rate of t5 to decrease by the order of T""“"5 and
reducing the order of integration in the regressor causes the divergence
rate of t5 to decline by the order of Td2’0'5, while these two effects are
cumulative as in Model 2.

Recall that in Models 2, 3 and 4 restrictions have been imposed
on the usual ranges (—0.5, 0.5) of the fractional differencing parame-
ters d1 and d2. In Model 3 the range of d2 is restricted to be (0, 0.5),
which is also the range of (11 in Model 4, while the sum of d1 and d2
must be greater than 0.5 in Model 2. From the analysis in the pre—
vious paragraph, particularly the fact that the divergence rates are
directly related to the magnitudes of d1 and d2, we come to realize
that the restricted ranges of all and d2 in Models 2, 3, and 4 ensure
the reduction in the divergence rates of tfi from the T 0'5 level is not too
great so that t3 remains divergent (in which case the spurious effects
occur). Although we did not explicitly consider the asymptotic theory
for cases where the fractional differencing parameters lie outside their
prescribed ranges, it is readily seen that the conditions we impose on
the ranges are not only sufficient but also necessary for the existence
of the spurious effect in terms of divergent t5.

From a similar analysis for the divergence rates of the t ratio to
we also find that reducing the order of integration in the dependent
variable causes the divergence rate of ta to decrease by the order of
Td1—0'5, while reducing the order of integration in the regressor does
not cause the divergence rate of ta. to change, as we probably should

have expected.

35
It is also interesting to see how the changes in the orders of

integration of the dependent variable and the regressor affect 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 effect by Phillips (1986). But when we examine
Models 2, 3, and 4, we note that reducing the order of integration in
the dependent variable helps to increase its convergence rate by the
order of T1’2"1 while reducing the order of integration in the regressor
helps to increase the convergence rate by the order of T1‘2d2. As a
result, in Models 2, 3, and 4, R2 all converge to 0, correctly reflecting
the fact that there is no relationship between the regressor and the
dependent variable. This finding implies that the spurious effects in
Models 2, 3, and 4 are confined to the two t ratios while the asymptotic
tendency of R2 toward zero is not affected by the spurious effects
(though the convergence rates are).

The sharp difference in the asymptotic behavior between the t
ratios and R2 in Models 2, 3, and 4 actually offers us an opportunity to
diagnose the spurious effect in these models. That is, when we find two
highly significant t ratios coexisting with a completely contradictory
near-zero R2, we are effectively reminded of the possibilities that one
of the Models 2, 3, and 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 effects in Models 2, 3,
and 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 effects.

36
Finally, let’s briefly state a few more results about the asymp-

totic tendency of the OLS estimators 3 and a and the DW statistic.
First, we note that Z3 will converge in probability to zero unless the
dependent variable is nonstationary and its order of integration is suf-
ficiently large. Secondly, whether a diverges or not and whether the
DW statistic converges in probability to zero or not depends entirely
on whether the dependent variable is nonstationary or not. Note that,
as mentioned earlier, even though the OLS estimators 3 and 61 can
converge in probability to zero in the four proposed models, the corre-
sponding t ratios always diverge and it is these divergent t ratios that

are referred to as the spurious effects.

2.2.5. Model 5 and Model 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 effect of finding a significant trend. In this section we
extend their analysis by considering the potential problems in detrend-
ing fractionally integrated processes. It turns out that the spurious
effect of divergent t ratios exists as long as the fractional differencing
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 effect 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

37
5 we examine the regression of a nonstationary I (1 + d1) process y, on
a time trend t:
y, = a +/3t +11,.
The asymptotic theory for the OLS estimation is given in the following
theory.

Theorem 5. Given that Assumption 1 holds, then, as T —> 00, we

have the following results.

T 6 l l
1. —/3 Z} 12/ 82805441109) d3 ‘- 6/ B0.5+d1(3) (13 E 13*.
0 0

0y

Note that oy/T = 0(Td1—0'5).

1 A 1 1
2. — 01 => 4/ B0_5+d,(s) (ls — 6 / s-BO_5+d,(s) ds E ca...
01/ 0 . 0
1 1 2 1 2
3 ——2- 52 2) [B0 5+d,(8)] (18 — [/ B0.5+d1(8) (19]
0y 0 0
l 1 l 2
_12 [/ 8B05+d1(9) 613—5] B05+d1(8) (13] E03
0 0
3
4 [-2— sf, : 1203,
all

where of is defined in 3. Note that (IE/T3 = 0(T2d1’2).

2 *’
0y

38
where of is defined in 3. Note that og/T = 0(T2d1).

6 It: ’3"
'fi” «1‘20:

 

where 13’... is defined in 1 and of is defined in 3.

1 (1,.
7. —— ta ,
20,.

\/T

 

where a... is defined in 2 and of is defined in 3.

2
13..

8. R2 => 1 1
12/ [B0,5+d1(8)]2 d8 — 12 [/ B0.5+d1(3) d8]
0 0

 

29

where [3,. is defined in 1.

9. DW —p—> 0 and oi-DI/V =>

 

Here, B0,5+d,(t) and BO,5+d,(t) are two independent fractional Brown-

ian motions.

The results on detrending a stationary long memory I ( (11) pro-

cess vi

111 = O, + [it + at, where all > 0,

which is our Model 6, are presented in the following theorem.

39
Theorem 6. Given that Assumption 1 holds, then, as T —> 00, we

have the following results.

T2 A 1
1' _fl : 6B0.5+d1(1) — 12/ BO.5+d1(S) d8 E 18*-
~ 0

0y

Note that Oy/T2 = 0(Td1‘1‘5).

T A l
2. —- 06 => 6/ B0.5+d1(8) d8 — 2 B0.5+d1(1) E C13...
0

03/

Note that oy/T = O(Td1_0'5).
3. 32 —”—> 70(0).
4. T3-sf, L) 12%(0).

5. T-s: —p—+ 4%(0).

x/T fl...

6. —t/3

=> 3
0y \/127v(0)

where fl... is defined in 1. Note that oy/x/T = 0(Td1).

 

21’ _9
y 2 712(0)

40
where 01* is defined in 2. Note that oy/x/T = 0(Td1).

 

T2 [32
.7122 : * . ,
8 Of, 12%(0)

where ,6... is defined in 1. Note that (Ii/T2 = 0(T2dl—l).

zu—zm)
1—d1 '

 

9. DW —"_> 2—2p,,(1)=

Here, B0.5+d,(t) and BO.5+d,(t) are two independent fractional Brown-

ian motions.

In terms of the convergence (or divergence) rates of the various
OLS estimators, Models 5 and 6 can be conveniently viewed as “spe-
cial cases” of Models 1 and 4, respectively. More specifically, if we
replace the term 0:8 by T in those normalizing factors in Theorems
1 and 4, then we immediately get all the normalizing factors in The—
orems 5 and 6. For example, while the normalizing factor for 1; in
Theorem 1 is ox/oy, the one in Theorem 5 is T / 0y. Similarly, while
the normalizing factor for 3 in Theorem 4 is Tax/0y, the one in The-
orem 6 is T 2 / 0y. Given this observation, we then 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 re-
spectively are in the orders of TO'5 and T d1 in Models 1 and 4, are also
the rates in Models 5 and 6. (Note that in both Model 4 and Model
6 the same condition d1 > 0 is imposed on the stationary dependent

variable 2), so that the resulting t ratios are divergent.) As a result, the

41
type of spurious effects 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 differencing parameter, certainly including
the usual case of the I (1) process, will result in the spurious finding of
a significant trend. One important inference we draw from Models 5
and 6 is that the cause for the spurious effect 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 effect can happen to the detrending proce-
dure even after the series are first-differenced. What first-differencing
does to the detrending procedure in such a case is simply reducing
R2, increasing the value of the DW statistic, and slowing down the
divergence of the two t ratios from the T 0‘5 rate to the le rate. Based
on this observation, it seems that the spurious effects in detrending

may occur more often than we previously thought.

2.3. Monte Carlo Experiments

After the theoretical analysis of the spurious effect, we now con-
duct an extensive Monte Carlo experiment to investigate the relevance
of the theory in small sample applications. The design of the Monte
Carlo study is standard. The Monte Carlo experiment for each model
is based on 10,000 replications with different sample sizes (T). To
construct T values of the stationary I ((1) process, we first generate T

independent values from the standard normal distribution and form

42
a T x 1 column vector e. We then calculate the T analytic auto-

covariances of the I ((1) process, from which we construct the T x T
variance-covariance matrix 2 and compute its Cholesky decomposi-
tion C (i.e., E = C C ’ ) Finally, the vector p of the T realized values
of the I ((1) process is defined by p 2 Ce. This algorithm was suggested
by McLeod and Hipel (1978) and Hosking (1984).

To verify item 6 of Theorem 1 that the t ratio t3 for the slope
coefficient fl diverges at the \/T rate, we test the null hypothesis

Hoz/3=0

at different level of significance (N) using the traditional two—tailed t
test. Tables 2—1 contains the information about the rejection percent-
ages and the averages of the absolute value of t;; under Model 1, where
both dependent variable and regressor are nonstationary.

If we treat the result of hypothesis testing in every replication
as a binomial trial and define the null hypothesis being rejected as a
success, then each hypothesis testing is a binomial trial with probabil-
ity of success N. Therefore, the 95% confidence interval for N = 0.05

is

 

0.05 x 0.95
10000

As shown in Table 2—1, the rejection percentages at every value of N

 

0.05 i 1.96\/ 2 (4.57%, 5.43%).

are outside their corresponding 95% confidence intervals. Moreover,
we find the average absolute value of t); increases with the sample size
increases. These results support Theorem 1 that t); diverges with T
increases.

To further verify the theory that the t ratio t5 for the slope
coefficient 6 diverges at the x/T rate, we calculate the ratios of the

averages of lip! for two sample sizes T1 and T 2. These ratios can be

43
compared with y/Tl/TQ. As shown in Table 2—2, these ratios are all

very close to the corresponding \/T1_/T2 .

Table 2—3 contains the simulation results for Model 2, where
both dependent variable and regressor are stationary. Once again,
we find the rejection percentages at every value of N increases as T
increases and the average of the absolute value of t5 diverges with
T increases. We also calculate the ratios the averages of |th for two

)d1+d2—0.5 in

sample sizes T1 and T2 and compare them with (T 1 /T 2
Table 2—4. We find these ratios are very close to 1.

The simulation results in the lowest block of Table 2—3 and Table
2—4 are based on a chi-square distribution with degree of freedom 1
x? instead of the standard normal distribution. More specifically, the
innovations of the fractionally integrated processes are generated as
independent x? —- 1 random variables. The conclusions we draw from
these simulations are the same, which implies that the spurious effects
will occur to Model 2 irrespective of the distributional assumption.

Table 2—5 and Table 2—6 contain simulation results for Models
3 and 4. Again, the rejection percentages at every N value and the

averages of |tfl| diverge as sample sizes increase, which convincingly

support Theorems 3 and 4.
2.4. Conclusion

In our analysis of spurious regressions for the long memory frac-
tionally integrated processes, we find that no matter whether the de-
pendent 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

44
ratios become divergent. So it is the long memory, instead of non-

stationarity or lack of ergodicity, that causes the spurious effects in
terms of failing t tests. N onstationarity in one or both of the depen-
dent variable and the regressor only helps to accelerate the divergence
rates of the t ratios. We thus learn that spurious effects might occur
more often than we previously believed as they can arise even among
stationary series and the usual first-differencing procedure may not be
able to completely eliminate spurious effects when data possess strong
long memory.

In subsection 2.4 we have carefully examined the exact relation-
ships 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 effects which are
not available in Phillips’ (1986) classical study of I (1) processes. In
short, it is found that the extents of spurious effects are directly related
to the degrees of long memory in the data. Our results on detrend-
ing fractionally integrated processes also greatly broaden Durlauf and
Phillips’ (1988) theory of spurious detrending in which the relation-
ship 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 this chapter. Our theoretical results are well
supported by simulation.

A few generalizations of our study are worthy of further con-
sideration. A natural extension is to consider the multiple regres-
sion where there are more than one non-constant regressor. Another

one is to allow the fractionally integrated processes to have non-zero

45
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.

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
at and b, that are required to satisfy the relatively stringent condi-
tions as specified in Assumption 1. These conditions effectively rule
out the possibility of allowing short-run dynamics such as the ARMA
components in the fractionally integrated processes we have studied.
Although relaxing Assumption 1 to incorporate the short—run dynam-
ics does not seem to pose too great technical difficulty and we do not
expect substantial changes in the analysis of spurious effects, some
modification in the theorem proofs is nevertheless necessary and is
beyond the scope of this paper.

Finally, our study of spurious regression can serve as the basis
for the analyses of “fractional cointegration” where the dependent
variable and regressors are M I (d) processes. This line of the
work appears to be quite important and has attracted a lot attention
in the literature recently. One of the pioneer works in this area is
by Cheung and Lai (1993). The research on this topic has also been
conducted in Chapter 3 of this dissertation.

46
2.5. Mathematical Proof

A.1. Proof of Lemma 1

The proofs of items 1, 2, and 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 1),
and 11),.

To prove item 7, we note, since v, and wt are assumed to be
independent and have zero means, the autocovariance of the product
vtwt at lag j is the product of their respective autocovariance at lag
j: 7,,(j)’yw(j). Also, it is well- known that 7,,(3 )= 0(j2"1 1) and
7w(j) = 0(j2d2 1) if d1 7Q 0 and d2 7% 0. Consequently, we have

ar (‘2: mm): Te; 1) (— ’)’J’11)(J)’Yw(j)

T

j=—(T—1>
= O(T2d1+2d'~’), if all + d2 2 0.5,
: O(T)C(2 — 2611 — 2d2), 1f (11+ d2 < 0.5, and d1,d2 > 0,

S 0(T), otherwise,
where C () is Riemann’s zeta function. Given this result and the fact
that E(vtwt) = 0, then Chebyshev’s inequality implies that, for any
6 > 0,

T
1
P ( Tc+d1+d2 Z vtw‘
i=1

 

T
1
> 6) < éVar (WZVtwt) : 0(1),
t:

 

47
when d1 + (12 _>_ 0.5, and

 

 

 

T T
1 1
P ( T5+0'5 :1 mm, > (5) < 32' Var (W thwt) = 0(1
t: t=l
when (11 + (12 < 0.5. Consequently,
op(Tf+d1+d2), if d1 + d2 2 0.5,

T
E ”()1 w, =

(:1 r' .
op(T‘+0"’), otherW1se.

To prove item 8, we note

T y a: T y a: T
%Z 1:: t—l t—l Zl(
— _— _ ' (111$(_1+ wtyt— 1+ Utwt)
_ 0,, .130 :T 0y 0,, +Tolyo _

 

 

l
=>/ BO.5+d1(S).B0.5+d-2(S)d3-
0

To Show the second term at the end of the first line is 0,,(1), we note

that ‘
T T 1/2 T 1/2
Emu—1 s (ng) (2x?)
t=l t=l

t=l

: Op(TO'5)'Op(T1+d2) : OP(T1'5+d2),

T T 1/2 T 1/2
2w. 3 (2y?) (wa)
i=1 i=1

: Op(Tl+d1)'0p(T0'5) : 01)(T1.5+d1).

48
The orders of the four sums of squares are based on the results of

items 2 and 5. We also note that T ago, 2 0(T 2+11%“). These results,

together with item 7, imply

 

 

 

11’ 1"
l . 1 .

, _ < O T—0.5—d1 a _— < O T—0.D—d2
T03101- glad} I _ p( ) Tag/Um 1:1 yt lwt — P( )’
and

7"
1 (— c— — —
thwt : max{o,,(T 2la 0p(T 15 d1 (12)},

T oyoz i=1
for any 6 > 0. So the above three terms all approach to 0 and the
proof of item 8 is completed. To prove item 9, we first note that we

are considering the case (12 > 0 only. Now, for a sufficiently large T,

 

 

 

we have
alt lit — $t_l dirt_1 diffs] t— 1 t
__ : —— z = , for g 3 < —.
0,, at 0,, or T T

This observation, together with Davydov’s result, the continuous map-

ping theorem, and item 7, imply

T y- w ow
t
210 — —+§T13—

/T ,, .
H:/t y[7’s] dJ/[Ts] + 0 (1)
l)
(

t 1)/T 01: 0y

T

i=1

 

1 l
y’l’s d1: 3
/ [1, [Tl-+0.41) :> / Bo,5+d,(s>dBo.5+d.(s).
0 0

 

at 0,,

Note that the condition (12 > 0 is needed to ensure that

H) t

—) : 0p(1l-

0,, 0,,

49
The result for 2:1(vt/oy)(xt/ox) given d1 > 0 can be proved in a

similar fashion. To prove item 10, we first note that

i=1
can be derived using a similar argument. To prove item 11, we note
T T
£21 ytzlzt_1.y‘—1+_i_ yt—l+_1_Ti.fi
2
T 0,, T 1:1 T 0,, T t:1 0,, T i=1 T 0,,

 

_Z/LT [is] gm ds+op(1)

1)/TT

1
T3
:/ Lfl'yns] d3+0P(1)
0

1
=>/ 3'BO.5+(11(8) d5-
0

The orders of the last two terms at the end of the first line are based on
the results of items 1 and 10. The result for (1/T)Z,T___1(t/T)(:rt/oz)

can be proved by a similar argument.

 

 

h

 

50
A.2. Proof of Theorem 1

Let’s first summarize the formulas for all relevant statistics in

the simple linear regression model yt 2 oz + [in + at:

 

 

 

 

 

 

 

 

T 1 T T
244—42424 1T 1.
fl: t=1 T i=1 i=1 , 62T2yt—If TELL],
_ i=1 i=1
2% - iv)
i=1
12t - yt — a — fix,
1 T 1 T A 1 T
32:? fif‘fZWt—filz-flszWt—«Fla
i=1 1:1 tzl
2 — 2
2 _ 3 2 _ 2 l (1‘)
8,6 — T 7 80! _ 8 a; + T ’
Em - 5) Ba - 2I?)
i=1 L i=1 _
A T T
.82 Z(x. — 35)? Z(a. — 31—02
R2 : t=1 DW/ _ i=2

 

T ’ — T
(yt — m2 2 1222

i=1 i=1

To prove item 1, we note

 

 

 

 

51
where the weak convergence is due to items 1, 3, and 8 of Lemma 1.

To prove item 2, we note

T

szt : 0*,

t=1 T’tzl

 

where the weak convergence is based on item 1 above and item 1 of

Lemma 1. To prove item 3, we have

1 2 1 T —2 __2:"2 T 2
253 ZfiZWt—y) — T102ZI(It—a:)2 0*,
y

y (:1

 

where the weak convergence is based on item 1 above and item 3 of

Lemma 1. To prove item 4, we note

 

 

'2 _ 2
8/3 -— i 02“}?

T
:W — 5):?

t=l

 

2
ToJr
where the weak convergence is based on item 3 above and item 3 of

Lemma 1. To prove item 5, we see

 

 

 

 

 

.— 2 _
T
1
33t
£S2__1_82 1+ T03t=1 => 2
02 ' a — 02 1 T 0"“
a Z:
_ 1” i=1 1

where the weak convergence is based on item 3 above and items 1 and

3 of Lemma 1. Items 6 and 7 are direct results from items 1, 2, 4, and

5. To prove item 8, we note

 

 

. 1 ‘ 1 ‘2
1’33 {0 [30.5+d2(8)12 (18 — U Bo.5+d2(8)d8] }
O
2 a
/O[BO.(5+<118)12 d9- [/0 Bo.5+d1(3) ([5]

where the weak convergence is based on item 1 above and item 3 of

 

Lemma 1. To prove item 9, we note

(at — €21-12): [<y1— a -131,)_(yt_1— a Wit—1)]?

: ('Ut — BU)(,)2—_ — fill? _ ngtuu + /32wt7

and, based on itemsS and 7 of Lemma 1, we have

 

 

1 T
2
—— v = 0 (1) 1:102 =0
2 t P 1 2 t P(
Toy t=2 T01. (2 2
and
1 T
Z = Mom—2), <1}
Tayaz t2?

for any 6 > 0. Consequently,

 

 

 

53
Here we note that the denominator of DVV is 32/03 which converges

weakly to 03 by the result of item 3.

A.3. Proof of Theorem 2

To prove item 1, we note

T

 

 

 

 

 

2) \/7v(0)7w(0)'z(1) - Bo.5+d1(1)'Bo.5+d2(1)

710(0) 3
where the weak convergence is based on Corollary 1, and items 4 and
6 of Lemma 1. To prove item 2, we note that from item 1 above and

item 4 of Lemma 1, we have

 

T . T
T A 1 03 T2 7 l.
‘0‘ = — ’“t ‘ T7“ 13—2 :wt 2» Bo.5+d1(1)a
0y 0y t=1 UyUx 01. t:1

where the second term converges in probability to zero since (Ii/T? =
0(T2d2"1) which converges to zero for 0.25 < d2 < 0.5. To prove item

3, we note that from item 1 above and item 6 of Lemma 1, we have

2 1 T 2 03/01; 2 T2 A 2 1 T 2 p
. rim-’2 — < T2 > (My) Tam—u» _, .10),

 

 

where the second term converges in probability to zero since agar /T2 =
O(Td1+d2‘1) which converges to zero for 0.25 < (11, d2 < 0.5. To prove

item 4, we have

32 P ) 711(0)

T 7
. 7111(0)
t=1

 

,2 _
T'bfi —

54
where the weak convergence is based on item 3 above and item 6 of

Lemma 1. To prove item 5, we have

 

 

 

where the second term in the bracket converges in probability to zero
since its numerator does. Items 6 and 7 are straightforward conse-
quence of items 1, 2, 4, and 5. To prove item 8, we have

.1 T
T4 (Us/:2: Bi?— 2( wt — u

i=1
R2 =
0202

y I 1 _.
aim—w
i=1

 

 

 

 

 

MM Z(1)—Bow5+d1(1)30 3+d( ) 2111(1)
710(0) 711(0),

where the weak convergence is based on item 1 above and item 6 of

Lemma 1. To prove item 9, we first note

(at — {it—1V = [(Ut — a — 311%) — (Ut—l — a — 3101—1)]?

A

=[(vt — vt_1)— [3(fw, — 2014)]?

A

= (”t — fut—1)? — 2/3(Ut — 'Ut_1)('wt _ wt—l)

+ 1,3201% — wt-l)2-

55
Also, from item 7 of Lemma 1 and the assumption that d1 + d2 > 0.5,
we have

f Y

T 1
M — 111— 1) (wt — U’t— 1) " Utwt + '- ’Ut—lwt—l
T 2‘ ’1': T 21-2

i=2

T

1
01— lwt“_E:2 Utwt— 1
_th2

= 0p(Tc—1+d1+d2)

7

for any 6 > 0. Moreover, we have 13 = 0,,( 1) from item 1 above and

1T 1T

T
. 1 1 2
T (M - 01—02 = ff ”12+ TE :th__1— T ”tut—1
i=2 i=2

—”+ 71(0) + 71(0) — 271(1).
All these results then imply DW statistic is

T T T
1 A ‘ 1
r Z(vt — 21-112 — 213? Z(1)! - 21-1mm — ”wt-1) + ‘32?sz
t=2 t:2 t:2
1 T
7252
i=1

1" 2 _ 2pv(1)7

where the second and the third terms in the numerator converge in
probability to zero while the denominator converges in probability to

15(0)-

A.4. Proof of Corollary 2
It suffices to show that T 213 / ayax = 0,,(1). But from the proof
for item 7 of Lemma 1 we know that if (11 + (12 > 0.5, then

T
T
:o

t:l

 

56
If we denote its weak limit by Z * (of which the exact specification is

unknown), then, by the same argument as in the proof of item 1 of

Theorem 2, we have

 

 

 

 

2* — B0.5+d1(1)°B0.5+d2(1)
7111(0)

That is, T2B/ayax is indeed 0,,(1). Given this result, then all other

2}

conclusions in Theorem 2 can be established by the same analysis as
in the proof of Theorem 2. The only change required in the proof is
that all Z (1) in Theorem 2 be replaced by the Z * process.

A.5. Proof of Theorem 3

To prove item 1, we have

 

 

 

where the weak convergence is based on items 1, 4, 6, and 9 of Lemma

1. To prove item 2, we see

 

1 1 a2 T A 1
_A=__ _i. 3._§ u,“
aya T0311 yt ‘ / , wt Z) (1'

57
where the weak convergence is based on item 1 above and items 1

and 4 of Lemma 1, as well as the fact that ai/T2 = 0(T2d2_1) which

converges to zero. To prove item 3, we see

1 2_ 1 T _.2 0.3. T ”‘21 T _2 2
1; W211.-.) -— My Tim—u» =>

t:1

 

where the weak convergence is based on item 1 above and items 3
and 6 of Lemma 1, as well as the fact that (IE/T2 = 0(T 2‘12—1) which

converges to zero. To prove item 4, we see

 

 

where the weak convergence is based on item 3 above and item 6 of

Lemma 1. To prove item 5, we have

a2 1 T 2—
, —(—:w)
2

T
73/232751 1+ => (T2
0y 0y 1

 

 

 

where the weak convergence is based on item 3 above and items 4
and 6 of Lemma 1, as well as the fact that ag/T:2 = O(T2d2_1) which
converges to zero so that the second term in the bracket does too.
Items 6 and 7 are straightforward results of items 1, 2, 4, and 5. To

prove item 8, we have

 

 

58
where the weak convergence is based on items 1 and 3 above and items

3 and 6 of Lemma 1. To prove item 9, we first note

fit —- E4)? = [(y1 — E? — 3W) — (yt—l — 3’ — Qua—1)]?

= [111 "" fflwt '— lUt_1)]2

A

= "U? — 232141121 — U}t_1) + /32(wt — ’U)t_1)2.

Also, from item 7 of Lemma 1, we have

1 T 1 T
fi;v(wt—wg_1) ZTU—évtwt_§1_;vtwt—l

: Inax{0p(T€—l.5+d2), 0P(T€—l—d1)}7

for any 6 > 0. Moreover, we know that from item 1 above 13/03] =
0,,(Td2—0'5) converges in probability to zero. From item 5 of Lemma
1, we also know that Z: 2 v, 2/T -p—> 7,,(0) and that

T

T
1 . 1 . 1 . 2
T (wt—'wt—l)‘2 Zfzwtz'l'fiwf—1—Tzwiwt—l
t2? i=2 t:2

i=2

_P_) 710(0) + 710(0) — 2 710(1)-
All these results imply Dl/V statistic: is

 

—->

where the three terms in the numerator all converge in probability to
zero while the denominator converges weakly to 03 from the result of

item 3 above.

59
A.6. Proof of Theorem 4

To prove item 1, we have

1 T 1 1 T
T... M: 7.23” m2“

T
”" 12
(JR—.202
T02._

where the weak convergence is based on items 1, 3, 4, and 9 of Lemma

 

 

 

1. To prove item 2, we see

T T
TA 1 TOIA 1
—oz=;— ’Ut— ,6 211%: 0*,

0y y tzl 0y T03 i=1

 

 

where the weak convergence is based on item 1 above and items 1 and

4 of Lemma 1. To prove item 3, we see

To. 1 T _2 ,,
,92—_-—Z(vt—v)2— %TT(0y T); T02: Z(xt—m) -—> 7,,(0),

i=1

 

 

where the weak convergence is based on item 1 above and items 3
and 6 of Lemma 1, as well as the fact that ag/T2 = 0(T2‘T1‘1) which

converges to zero. To prove item 4, we see
32 2
i 0*33

 

2 2_
Taxsfi— T

(mt — Z5)2

1

 

0'2 i=1
Where the weak convergence is based on item 3 above and item 3 of

Lemma 1. To prove item 5, we have

1 T T
t: 2

=> am,
(291— (13)
T02 Z}

ztzl

 

T-32 = .92 1+

0

 

 

 

 

60
where the weak convergence is based on item 3 above and items 1 and

3 of Lemma 1. Items 6 and 7 are straightforward results of items 1,

2, 4, and 5. To prove item 8, we have

T
To,B
(as/fl 121,10) 22:;(T:-T)2 ’2

—‘R2: : 2*,
0'

T

31 *fi
—E (U, — '6)
i=1

Where the weak convergence is based on item 1 above and items 3 and

 

 

 

6 of Lemma 1. To prove item 9, we first note

(a. — am)2 = [(v. — a — flan) — (vz—l - a _Bx._1)]2
: (”Ut — ’Ut_1 — EUQ)2

: (Ut — Ut_1)2 — 23(vt — vt—l)wt +1232th-

Also, from item 7 of Lemma 1, we have

T
TE(vt—TTT1TUT—T E vtw —T-ET vt_1wt
=2 T=t2

: m&X{0p(TT_l+d1T—d2), 0p(T6—0.5)},

for any 6 > 0. Moreover, we know from item 1 above that 3 =
O (T d1"d2"1) converges in probability to zero. From item 5 of Lemma
1, we also know that Z,_ _2 w, 2/T -—> ~w(0) and that

T

1 2 1 T 2 1 T 2 2 T
T2011 - vt—I) = TZTT‘ + TZUt—l — TZW’H

Finally,

1 T 1 T 1 T
_ _ 2_ ’5._ _ "2._ 2
17;“): Ut—l) 215 Tgwt ’Ut._1)’wt+fl Tzwt

DW t=2
1 T
T Z a?
i=1

i" 2pv(0) _ 210v“)?
where the second and the third terms in the numerator converge in

probability to zero and the denominator converges in probability to

yv(0) from the result of item 3 above.

A.7. Proof of Theorem 5

To prove item 1, we note

2; Tioy‘it'yt-(%iyt)(%2t)

0'

 

Q
H| I-|
05
E a
A
H-
I
'fll r=-'
E ~3
94.
V”

where the weak convergence is due to items 1 and 11 of Lemma 1 and

the facts that

and

62
where the weak convergence is based on item 1 above and item 1 of

Lemma 1. To prove item 3, we have

TA 21 T 1 T 2
_ 2_T10.3t§:(: _ _ _2: __§: 3 2
0y s - yt y)2 —(0y ,6) T3 tzl (t T t) 0*,

where the weak convergence is based on item 1 above and item 3 of

Lemma 1. To prove item 4, we note

 

where the weak convergence is based on item 3 above. To prove item

5, we see

=> 402

*3

 

 

 

 

where the weak convergence is based on item 3 above. Items 6 and 7

are direct results from items 1, 2, 4, and 5. To prove item 8, we note

_F-Bfiit-i)

 

T

T3
‘10—yZ(3/t—

:1

(N-

2
18*

=> 2,

12/01 [B0.5+d1(3)]2 d8 - 12 [/01305+d1(8) d8]

 

63
where the weak convergence is based on item 1 above and item 3 of

Lemma 1. To prove item 9, we note

(at - 8—12)=[(yt— a — B-t) — (311—1 — a — Bu— 1))12

= (vi — B)? = ”0,2 — 23m + 32.

Then, by the results of item 1 above and items 4 and 5 of Lemma 1,

 

 

we have
T 2 T 2 2
_ _ 2 _ _y_ __ __ _y
T2321 03/, 20,:vt+(,,fl) 2 (0)
Ug'DW—J t:2 T t:2 fi 71):: ’
1 A2 0...
T0221“
y tzl

where a: / T 2 = 0(T2d1‘1) which converges to zero so that the second
and the third terms in the numerator also converge in probability to
zero. The denominator is 32/03 which converges weakly to of by the

result of item 3.

A.8. Proof of Theorem 6

To prove item 1, we have
T T T
1 1 1
2 Ta— t'“t‘(;‘zvt)(fizt)
T a—yfl_ yt=1 yt=1 i=1
1 T 1 T 2
— t—— t

where the weak convergence is based on items 4 and 10 of Lemma 1.

 

=> 5*,

To prove item 2, we see

T
TA 1 T2
0710/20ny ’Ut—Zfi T—2tEI t 2} (1*,

t: 1

64
where the weak convergence is based on item 1 above and item 4 of
Lemma 1. To prove item 3, we see

02 T2 A 2 1 T
aim—z- wk fl) fiat—4;) ——> M
where the weak convergence is based on item 1 above and item 6 of

Lemma 1, as well as the fact that a; / T 2 = 0(T2d1‘1) which converges

to zero. To prove item 4, we see
2
s

T3-sf, = => 127.,(0),

my

where the weak convergence is based on item 3 above. To prove item
5, we have

 

, T 2
(5-2 Z t)

T3: = s2 1 + T => 47,,(0),

% z. (t- - Z .)

where the weak convergence is based on item 3 above. Items 6 and 7

are straightforward results of items 1, 2, 4, and 5. To prove item 8,
we have

Th2 1 T 1 T 2
T2 (ET) EEG—T2?) [3‘2

_ R2 : i=1 2} —————*
02 1 T 127v(0)’

y T20” __ a)?

where the weak convergence is based on item 1 above and item 6 of
Lemma 1. To prove item 9, we first note

(a. — a._1)2=[(v,_ a — 31> — (UH — a — Eu — 1»?

 

 

 

= (v. — 22H — W

A

—-—(’Ut—Ut 1)‘2 —2%5(Ut—’—Ut 1)+/32-

65
Now, from item 4 of Lemma 1, we have

Moreover, we know from item 1 above that B = Op(Td‘_1'5) converges

in probability to zero. Also, from item 5 of Lemma 1, we have

._1 — — ’Ut’Ut—l

Finally,

 

'L’ 2Pv(0) _ 2m“):
where the second and the third terms in the numerator converge in

probability to zero while the denominator converges in probability to

711(0) from the result of item 3 above.

66

TABLE 2—1

REJECTION PERCENTAGES AND MEAN |t3|
UNDER MODEL 1

 

 

 

 

 

y. x. T 1% 5% 10% 20% 30% |t5|
125 0.487 0.594 0.652 0.721 0.774 3.0171
I(0.7) I(0.7) 250 0.615 0.703 0.747 0.798 0.838 4.2187
500 0.710 0.776 0.811 0.853 0.882 5.7634
125 0.678 0.749 0.788 0.831 0.864 5.0990
I(0.7) I(1.3) 250 0.765 0.819 0.849 0.882 0.907 7.0356
500 0.824 0.863 0.885 0.909 0.927 9.7107
125 0.664 0.746 0.787 0.837 0.868 5.0258
I(1.3) I(0.7) 250 0.762 0.822 0.850 0.882 0.905 7.0125
500 0.825 0.866 0.887 0.912 0.930 9.7648
125 0.860 0.892 0.910 0.930 0.946 13.5955
I(1.3) I(1.3) 250 0.902 0.925 0.937 0.952 0.963 19.5147
500 0.935 0.951 0.957 0.966 0.972 27.6573

 

Note: the critical values of the two-tailed t tests are 21:2.576 for
N = 0.01, :1: 1.96 for N = 0.05, :1: 1.645 for N = 0.10, :1: 1.282 for N =

0.20, :t 1.0326 for N = 0.30. Ith is the average absolute value of t3

of the simulation.

67

TABLE 2—2
THE DIVERGENCE RATE OF MEAN ltgl

UNDER MODEL 1

For 3;; = I(0.7) and 11:, = I(0.7),
4.2187 _ 5.7634

 

 

 

 

 

 

3.0171 " MT“, 4.2187 = W' 206'
For 3}, = I(0.7) and act 2 I(1.3),
gag—33:9;9—7—5—7'20'5 %=MQQ-ZO5
For yt = I(1.3) and as, = I(0.7),
__ _—
For yt = I(1.3) and 27¢ = I(1.3),
12:23:; = —-—- 33:2i13= ——

Note: the above numbers are taken from the last column of

Table 2—1.

68

TABLE 2—3

REJECTION PERCENTAGES AND MEAN Itfil

UNDER MODEL 2

 

111

wt

T

1%

5%

10%

20%

30%

”91

 

[(0.4)

[(0.4)

125
250
500
1000

0.117
0.164
0.234
0.312

0.229
0.289
0.361
0.436

0.312
0.378
0.445
0.511

0.430
0.492
0.550
0.604

0.526
0.577
0.629
0.678

1.3110
1.4887
1.7423
2.0569

 

[(0.3)

[(0.3)

125
250
500
1000

0.047
0.061
0.082
0.101

0.129
0.150
0.177
0.214

0.203
0.230
0.260
0.301

0.321
0.346
0.385
0.422

0.429
0.447
0.484
0.517

1.0363
1.0901
1.1766
1.2670

 

[(0.4)

[(0.2)

125
250
500
1000

0.040
0.051
0.068
0.087

0.115
0.137
0.161
0.194

0.183
0.213
0.238
0.269

0.304
0.324
0.356
0.393

0.413
0.423
0.456
0.490

1.0005
1.0440
1.1177
1.2040

 

[(0.2)

[(0.4)

125
250
500
1000

0.043
0.052
0.069
0.089

0.121
0.140
0.166
0.196

0.195
0.218
0.247
0.277

0.312
0.330
0.368
0.400

0.417
0.441
0.467
0.498

1.0134
1.0616
1.1309
1.2141

 

[(0.3)

[(0.3)

125
250
500
1000

0.048
0.060
0.078
0.105

0.123
0.151
0.180
0.217

0.192
0.229
0.253
0.296

0.311
0.348
0.374
0.409

0.416
0.450
0.473
0.504

1.0192
1.0912
1.1612
1.2627

 

Note: the critical values of the two-tailed t tests are :1:2.576 for
N = 0.01, :1: 1.96 for N = 0.05, i: 1.645 for N = 0.10, :1: 1.282 for N =

0.20, :1: 1.0326 for N = 0.30. For the first four rows of data, at and bt
are independent N (0, 1). For the last row of data, at and b, are inde-

pendent X? -—— 1. |tg | is the average absolute value of t3 of the simu~

lation.

69

TABLE 2—4

THE DIVERGEN CE RATE OF MEAN lit/3|
UNDER MODEL 2

For vi = [(0.4), wt 2 [(0.4), at and I), are independent N(0,1),

1.4887 03 1.7423 0, 2.0569 03
—= .22 .2- —=0.9506-2" ———=0.9 .2-.
1.3110 09 3 1.4887 1.7423 589

For vt = [(0.3), wt = [(0.3), at and bt are independent N(0,1),

1.0901 _ 1.1766 —1.0071.2051 1.2670 _

_ _ _1. 2‘“.
1.0363 1.0901 —— 1.1766 0047

0.9815 . 20-1

 

 

 

 

For vt = [(0.4), wt 2 [(0.2), at and (2,; are independent N(0,1),

1.0440
=0.973 20-1 —_. . - __ _ .
1.0005 —-—§ 1.0440 -— 1.1177 _—

 

1.0616
:09 74.20-1 —— . - - . .
1.0134 “—7—- 1.0616 _— 1.1309 _—

 

 

For 211 = [(0.3), wt 2 [(0.3), at and bt are independent x? — 1,

1.0912 0, 1.1612
2 .989-2- —=0.9929-2~ _—
1.0192 0 9 1.0912 1.1612

 

 

Note: the above numbers are taken from the last column of

Table 2—3.

70

TABLE 2—5

REJECTION PERCENTAGES AND MEAN |tfi|
UNDER MODELS 3 AND 4

 

T

1%

5%

10%

20%

30%

113/31

 

[(0.7) [(0.3)

125
250
500

0.172
0.251
0.320

0.294
0.378
0.450

0.375
0.458
0.525

0.487
0.560
0.621

0.577
0.638
0.688

1.4985
1.7864
2.0863

 

[(1.3) [(0.3)

125
250
500

0.274
0.357
0.449

0.401
0.483
0.561

0.477
0.556
0.626

0.580
0.645
0.703

0.653
0.710
0.760

1.8791
2.2475
2.7299

 

[(0.3) [(0.7)

125
250
500

0.167
0.245
0.329

0.293
0.372
0.459

0.374
0.454
0.535

0.486
0.556
0.633

0.574
0.633
0.703

1.4936
1.7801
2.1336

 

[(0.3) [(1.3)

125
250
500

0.273
0.369
0.453

0.406
0.491
0.565

0.486
0.561
0.628

0.590
0.649
0.708

0.662
0.710
0.768

1.9003
2.2892
2.7747

 

Note: the critical values of the two-tailed t tests are 21:2.576 for
N = 0.01, :1: 1.96 for N = 0.05, :1: 1.645 for N = 0.10, :t 1.282 for N =

0.20, :1: 1.0326 for N = 0.30. |t5 | is the average absolute value of t);

of the simulation.

TABLE 2—6

THE DIVERGENCE RATE OF MEAN |t5|
UNDER MODELS 3 AND 4

For yt = [(0.7) and wt 2 [(0.3),

 

 

 

 

 

 

 

 

1:132: = M81203 gig: = 0.9486 - 20-3.
For yt = [(1.3) and wt 2 [(0.3),
For 71, = [(0.3) and x, = [(0.7),

3533-.....03
For I), = [(0.3) and :1:) = [(1.3),

3333: = 0.9785 - 20-3 3:3; = 0 9845 20 3

 

 

Note: the above numbers are taken from the last column of

Table 2—5.

CHAPTER 3

THE SPURIOUS EFFECT WHEN
REGRESSOR AND DISTURBANCE ARE
FRACTIONALLY INTEGRATED PROCESSES

3.1. Introduction

This chapter derives the asymptotic distributions for the OLS
estimators and corresponding test statistics in the following simple

linear regression model:
Yt=a+/3Xt+€ta t:1,2,...’

where the regressor Xi and the disturbance term 51 are both fraction-
ally integrated long memory processes and independent of each other.
We further assume that X, is always nonstationary while 51 can be
either stationary or nonstationary, and in the latter case the order of
integration of at is smaller than that of Xi. In other words, the order
of integration of the disturbance term at must be smaller than that
of X). We also assume 6 aé 0 to get rid of the possibility of spurious
regression which we have discussed in Chapter 2.

Similar models have been analyzed by Robinson and Hidalgo
(1995) where they assume Xt and at are both stationary long mem-
ory processes and then prove central limit theorems for a number of
estimators of the slope coefficient 6. Kramer ( 1986) and Phillips [and
Park (1988) have studied the asymptotic properties of a model with

nonstationary I (1) regressors and [ (0) disturbance. Moreover, Park

72

73
and Phillips (1988, 1989) study the multiple regression case where Xt

is a m — dimensional process and may be cointegrated.

The main finding of this chapter is that the t ratio for the slope
coefficient 6 diverges as the sample size increases, as long as the order
of integration of the disturbance term at is positive; i.e., St has long
memory. Consequently, if the traditional critical values are adopted,
the null hypothesis for testing any finite value of 6 tends to be overly
rejected and this is what we call the spurious effect. Moreover, it
is found that the inclusion of an intercept or a time trend in the
regression model does not change the convergence rates of the t ratios
even though the asymptotic distributions of the t ratios are different.
The cases where the regressor Xt contains a drift are also considered
and we find that the convergence rates of the OLS estimators largely
the same as those from the cases with a driftless X t.

The asymptotic behavior of R2 and DW is as follows: R2 —-p—> 1
for all the cases while DW ——p—> 2 — 2,0,,(1) when at is stationary, and
DW L 0 when 6; is nonstationary.

Monte Carlo study is also included to evaluate the small sample
properties of the t ratios, R2, and DW. The Monte Carlo results

support the theory quite well.
3.2. The Four Classes of Models

Given the stationary [ ((11) and I(dg) processes vi and 10,, and

their respective partial sums yt 2 23:1 v,- and 23¢ 2 23:1 111,-, defined

74
in Chapter 2, we considered the following two classes of simple linear

regression models:

Model A-0: Ct = fisct + vvt, where d1 > 0,

Model A-1: Ct = a + 61:, + 1)), where d1 > 0,
Model A-2: C; 2 oz + 7t + [3.1% + m, Where all > 0,
Model B—0: Ct = flxt + yt, where 611 < (12,

Model B-1: C; = oz + 6231+ yt, where (11 < d2,

Model B-2: C, = a + 7t + [311+ yt, where d1 < (12.

The disturbance terms vt of the class A models are stationary long
memory [ (d1) processes, while in the class B models the disturbance
terms yt are nonstationary I (1 + d1) processes and their orders of
integration are assumed to be smaller than those of the regressor act
which is also nonstationary [ (1 + d2) processes. The two models with
the label “1” do not include the intercept term, while the two models
with the label “3” contain a time trend.

To study the effect of a regressor with a drift, we consider
1'? = 7’ + $111+ wt,

and, without loss of generality, 7’ can be set at 1. Based on such an

56;), we have two additional classes of models that correspond to the

75
previous two classes of models:

Model C-0: C1 = 6372’ + ’Ut, where d1 > 0,

Model C-1: C1 = a + 7311:? + v), where all > 0,
Model C-2: Ct = a + 7t + 5.7:? + v), where d1 > 0,
Model D-0: Ct = 731'? + 3}), where all < d;

Model D-1: Ct 2 oz + 733:? + y, Where d1 < d2

Model D—2: Ct = a + 7t + [32,0 + 3],, where ([1 < d2.

3.3. The Class A Models

In this section we derive the asymptotic distributions of the OLS
estimators and the corresponding test statistics for the class A models
where the regressor 23¢ is a driftless nonstationary I (1 + 012) processes
and the disturbance term is a stationary long memory I (d1) process

With d1 > 02

Model A-0: C, 2 fix, + 2;),
Model A-1: Ct = oz + [333, + 1),,

Model A-2: Ct = a + 7t + 732') +12).

76
Let their OLS estimation be denoted as follows:

Model A-OI C1 : B(0)$t + ’12),
Model A-li C; = (37(1) +,B(1).’17t + E,

Model A-Z: C; = 8(2) + $(Q)t + 3(313 + fit.

We also adopt the following notation for the various statistics from the
OLS estimation. Let 8%(0), 3%,“) and 3%”), respectively, the variances of
the corresponding OLS estimators of 6, from which we have the t ratios
tflw) : (3(0) _fll/Sfiww tfio) = (3(1) —fi)/Sfl(1)’ and tfim) = (am—fll/Sflmr
Furthermore, the notation introduced in Chapter 2 will be used repeat-
edly throughout this chapter. In particular B0,5+d,(s) and BO,5+d,(s)
denote, unless otherwise stated, two independent fractional Brown-
ian motions. It will substantially simplify our subsequent formulae to
write these as 81(8) and B2(s), respectively. Thus, we will frequently

use 81(5) and 82(3) in place of B0,5+d,(s) and B0,5+d,(s).

Theorem 1. Given that Assumption 1 of Chapter 2 holds, then as

T —> 00, we have the following results:

A B2(.S') dB1(S')

 

 

T0,: A .
1. (73(0) — (3) => , a 00,...
(79 2 1
/ [32(8)] ds
. 0
Note that (Ty/T0,, = 0(Td1—d2_1).
T A
2. — (04(1) — CY) fi

03/

”q

 

77

131(1)/0 [B2(s)] d8 — 1132“") 8H], B2<S>dBl< )l E a ,
[[B2<> (WM {/132‘3 ”3f *
To

mam—x3) =>

 

 

—B1(1) {/013209 NH] A182“ 3)d281(8) E /31*.

/18lB2( (8)] d8 -[/0 B29 95”]

Note that Uy/T = 0(Td1—0'5)‘

 

h
imam—a) => Elm
T2 (A ) i E—
3; 7(2) 7 13—72

1
[B2(3)d3
h3=B1(1) — 0

 

 

 

 

 

+ [31(1)—/0131(s)d3] f0 Bi<8>d8_/01332(s)ds

 

 

 

Note that ay/T2 = 0(Td1‘1°5).

Theorem 1 indicates that the OLS estimators of the regression
coefficients are consistent in all three setups of the class A models. The
convergence rates of the OLS estimators of a and fl are independent
of whether the time trend is included in the model or not while their
asymptotic distributions are different among the three different setups.
It is also interesting to note that the asymptotic distributions of (8(1) —
a) and (3(1) — 5) in Model A—l are the same as those of a and 3 in
Model 4 of Chapter 2, since the latter are nothing but the special cases
of the former with 04 = ,8 = 0.

The asymptotic distributions of the t ratios for the OLS pesti-
mators of 5, R2, and the Durbin-Watson statistic DW are presented
in the next theorem. We refer to these nonstandard results as the

spurious effects in class A models.

80
Theorem 2. Given that Assumption 1 of Chapter 2 holds, then as
T —> 00, we have the following results:

gtflm, => fi0*{;;}@/01[Bg(3)]2d3}1/2,

where 130... is defined in item 1 of Theorem 1.
R2 Ji.) 1.

DW i. 2 — 2M1)-

2- gtfim =15 51*{fi{/01l32(3)l2d5 — {/01 32(3) d5]2}}1/2,

where 51* is defined in item 2 of Theorem 1.
R2 L 1.
DW L 2 — 2,0,,(1).

fl 12A
. _‘t =0: —3
3 a, ‘3": 2} ‘32 71(0)

where A and fig* are defined in item 3 of Theorem 1.
R2 —”—> 1.
DW _L 2 — 2,1,,(1).

The t ratios for the OLS estimators of B in all three setups
diverge at the same le rate. This finding implies spurious effects
in t tests in that the null hypothesis for testing any finite value of fl
tends to be overly rejected. Theorem 2 also shows that R2 L 1 and
DW —p—+ 2 — 2,0,,(1) for all three setups.

81
3.4. The Class B Models

In this section we derive the asymptotic distributions of the OLS
estimators and the corresponding test statistics for the class B models
where the regressor mt is a driftless nonstationary I (1 + d2) processes
and the disturbance term is a nonstationary long memory I (1 + ([1)

process with d1 < (12:

Model B-OI Ct = ,8th + yta
Model B-l: Gt 2 at + [3131+ yta

Model B—2: Gt 2 oz + 7t + flan + yt.

The notation for the OLS estimators and test statistics will be the

same as that used for the class A models in the previous section.

Theorem 3. Given that Assumption 1 of Chapter 2 holds, then, as

T -—+ 00, we have the following results:

0:1:

1
1. —-(§(0)—/3) => /0B1(8)B2(8)d8
U

1 [30*-
1’ / [132(3)]2 d3
0

III

 

Note that cry/ax = 0(Td1’d2).

1 A
2- a(a(1)"0‘) =>

[ [013...] ft... 1 . _[/1.. mil] [ 41......)4
1‘3” 10’: MB?“ ”8]

CY”.

:(a— 3) =>

/01 31(3) B.(.) d. — U01 31(3) d3] U01 32(3) d3]
f01[32(s)12 d3 — U01 132(3) d.9]2

(5'9

2 [31,...

. Ely-(3(2) — a) => g- E (12
T A C2 _
0—y(7(2) —’Y) => K =72

where A is defined in item 3 of Theorem 1.

(I E 1131(3) d3 {/0 [32:9]? d3 — [[01332(3)d3]2}

 

84

1
f0 B.(.)B,(.)d.
12 '

 

+

In the class B models the OLS estimators of 13 and 7 converge
while those of the intercept a diverge. That is, the OLS estimators
of fl and ”y are consistent but those of oz are not. These results are
different from what we have derived for the class A models, where
all the OLS estimators are consistent. Also note that the asymptotic
distributions of (6(1) — a) and (3(1) — B) in Model B-l are identical to
that of a and 3 in Model 1 of Chapter 2 since Model 1 of Chapter 2
is simply a special case of Model B-1, just like Model 4 of Chapter 2
is a special case of Model A-l.

The following theorem gives the asymptotic distributions of t

ratios, R2 and DW for the class B models.

Theorem 4. Given that Assumption 1 of Chapter 2 holds, then as
T —> 00, we have the following results:

3*
1. -—1—t /"

fl; 13(0)

 

2 1/2’

=.
/0 1[131(3)]2ds _ [01B1(3)B2(s)ds
/()1l132(5)l2 d3 /01lB2(3)l2 d3

 

 

where 130... is defined in item 1 of Theorem 3.

321.1.

85

 

DW 1. 0.
1 [31*
2. 177.]:th => 1/2,
0?.

 

/01[132(s)12 d3 — U01 32(3) dsr

where [31... is defined in item 2 of Theorem 3.

2

oi. :—: [11311.112... — U01 B11.) ds]
- 5f. {/01 [132(8)]2 (18 — [[0132(3)d8]2}-

122—191.

DW—p—>0.
3 1 t => 13 12A
' «r 2* 03.3

where A is defined in item 3 of Theorem 1.

 

03*E/Ol[Bl(s)]2ds— {/0131(3)d32] +732*+/3: A1321.) )1‘ d9

86

1 2 1 1
—13§.[/ 3.1.1.1.] —233./ sBl(s)ds+32./ Bl<s3ds
0 0 0

— 213.2,. {AIBI(S)B.(S).13 — U01 131(3) .13] U01 34.1.1.1}
+ 27.2.13... [[313219133 — é]; 1321.91.13] ,

where ”1’2... and 132... are defined in item 3 of Theorem 3.
R2 —"—. 1.
DW i3 0.

The t ratios for fl diverge at the same Tl/2 rate, which indicates
that reducing the order of integration in the disturbance term from
above 0.5 as in the class A models to below 0.5 as in the class B
models causes the divergence rate of t ratio to decrease by the order
of le’l/g. It is the same as the conclusion we arrived in Chapter 2.
Theorem 4 also shows that R2 —p> 1 and DW —p—> O for all three
setups of the class B models. That is, while R2 still grows to become
1 as T increases, DW reduces to 0, instead of 2 — 210.0(1) as in the class
A models.

From the results of Theorems 2 and 4 for both class A and class
B models, we conclude that the t tests for the slope coefficients are
affected by the spurious effects due to the long memory in the regressor

and in the disturbance term.

 

87
3.5. The Class C Models

In this section we derive the asymptotic distributions of the OLS
estimators and the corresponding test statistics for the class C models
where the regressor .731 is a nonstationary I (1+d2) processes with a drift
and the disturbance term is a stationary long memory I (d1) process

with d1 > 0:

Model C-O: C1 = 13:13? + 131,
Model C-1: C1 = a +/3:1;to +131,

Model C-2: C1 = a + 7t + [3:10; + 131.

Theorem 5 presents the asymptotic distributions of OLS esti-
mators of oz, 7, and fl, while Theorem 6 shows the asymptotic distri-

butions of t ratios, R2, and DW.

Theorem 5. Given that Assumption 1 of Chapter 2 holds, then as

T -> 00, we have the following results:

T2 A 1

1. —- (5(0) — 13) => 381(1) — 3/ 31(9) (18 E 130,...
0y 0
T 1

2. -— (3(1) — 0!) => 6/ B1(8) d8 — 2131(1) E 031*.
0y 0
T2

(311) —fl) => 6B1(1) — 12/01B1(s) ds E [31,...

0y

88

T
3.—
11

(3121-0) => “2*3

T2 A
—(’Y(2)—’Y) => ”12*,
0y

 

Tax (3(2) — #3) => X32“

01/

where (12..., 72... and 132... are defined in item 3 of Theorem 1.

Theorem 6. Given that Assumption 1 of Chapter 2 holds, then as

T -—+ 00, we have the following results:

1 fit z} 360*
. — 130 —_—3
03/ ( ) V 3711(0)

where 130... is defined in item 1 of Theorem 5.
R2 L1.
DW —p_) 2 — 2pv(1).

fi 161*

2. ——t => .
0., fl” 127,10)

 

where 31* is defined in item 2 of Theorem 5.
R2 .1. 1.

DW —”. 2 — 2p.(1).

89

\/T 12A

3. —t => .. —,
0y ‘3‘” 132 7.3(0)

where 132... and A is defined in item 3 of Theorem 1.
R2 11.3 1.
DW L3 2 — 2p.(1).

By comparing the results of Theorems 5 and 6 for the class C
models with those of Theorems 1 and 2 for the class A models, we
learn how the drift in the regressor affects the asymptotic behavior of
the OLS estimators and t test statistics. Two interesting findings are
worth mentioning. First, the asymptotic distributions for the statistic
1333(2) are identical in both Models A—2 and C-2 where the time trend
is included as a regressor. Secondly, the asymptotic distributions of
62(1) —a, 3(1) —fl and 1313(1) in Model C-l are the same as those of Model 6
in Chapter 2 in which the regressor is the time trend. The intuition is
that an nonstationary fractionally integrated process 1:1 with a nonzero
drift in Model C-l behaves asymptotically like a deterministic trend
as in Model 6 of Chapter 2.

3.6. The Class D Models

In this section we derive the asymptotic distributions of the OLS
estimators and the corresponding test statistics for the class D models

where the regressor x, is a nonstationary I(1 + ([2) processes with a

90
drift and the disturbance term is a nonstationary I(1 + d1) process

with all < d2:

Model D-O: C1 = 8.1:? + 131,
Model D-l: C1 = 01 + 131:? + 311,

Model D-2: C1 = a + ”11+ 132:? + yt.

Theorem 7. Given that Assumption 1 of Chapter 2 holds, then

T —> 00, we have the following results:

T T? - 1 _ ,
1. 3; (13(0) -- 13) => 3/() 3 131(8) dS : 130*.
l 1
2. 3—(&(1)—oz) => 4/ B1(s)ds—6/ sBl(s)ds E (11,...
0 0

A 1 1
1103(1) “13) => 12/ 8B1(S) ds—6/ B1(s)ds E 131*-
0' 0 0

3. —(oz(2)—Oz) => 02*,

where (12..., 72* and 132* are defined in item 3 of Theorem 3.

91
Theorem 8. Given that Assumption 1 of Chapter 2 holds, then as

T —> 00, we have the following results:

380*

{3/[Bl(s)]2ds — 9 [/3B1(s)ds]2}

where 130... is defined in item 1 of Theorem 7.

 

1
1. —-t13(10) =>

x/T

1/2’

RQ—Ll.

DW—p-30.

1 381*

—tfi(1) => ,
‘ff 3 111203”,

where fll... is defined in item 2 of Theorem 7.

of, 2 [0113119112 d3 — [foleMSY

p

1
1 [B1(.S‘)d8
—12 /3B1(s)ds— 0
0 2

 

«2

 

 

 

Elia.

DW—LO.

92

3 1 t => '3 12A
. fl 13(2) X21: 03* a
where 132... is defined in item 3 of Theorem 3, A is defined in item
3 of Theorem 1, and 03* is defined in item 3 of Theorem 4.
R2 —”_. 1.
DW _”_3 0.

By comparing the results of Theorems 7 and 8 for the class D
models with those of Theorems 3 and 4 for the class B models, we
learn how the drift in the regressor affects the asymptotic behavior of
the OLS estimators and t test statistics. Three interesting findings are
worth mentioning. First, the asymptotic distributions for the statistic
tflm) are identical in both Models B-2 and D-2 where the time trend is
included. Secondly, the asymptotic distributions of 31(1) — oz, 3(1) - 13
and t3“) in Model D-l are the same as those of Model 5 in Chapter
2 in which the regressor is the time trend. The intuition is that an
nonstationary fractionally integrated process art with a nonzero drift
in Model D-l behaves asymptotically like a deterministic trend as in
Model 5 of Chapter 2. Thirdly, if d2 < d1, then the OLS estimator 3(2)
in Model D—2 diverges so that 3(2) can be an inconsistent estimator of

fl, in which case B2 will not converge in probability to 1.
3.7 . Monte Carlo Experiments

Monte Carlo experiments are conducted to investigate the rel-

evance of the theory in small sample applications. The Monte Carlo

93
experiment for each model is based on 10,000 replications with three

different sample sizes (T). The algorithm for simulating the stationary
fractionally integrated processes v1 and wt is the same as in Chapter
2, while the nonstationary series yt and wt are constructed as their
partial sums. Given the four series 111, wt, 3],, and 111;, the series for the
dependent variable C1 can be easily computed based on various model
specifications. In our Monte Carlo study we focus on the specification
thata=13=1 and7=0.

For two model specifications: 1:1 = 1(1), at = I(0.3); and
.131 = I(1), 51 = I (0.7), Table 3—1 contains the results on the rejec-
tion percentages of the two-tailed t test for the null hypothesis

H.513 = 1

at various levels of significance (N). Table 3—1 also contains the average
R2 and the average DW and the average of the absolute value of t3.
The Monte Carlo results on another pair of specifications: at? = :1:, +t
51 = I(0.3); and :1:? = wt + t, 81 = I(0.7), are in Table 3—3.

Theorems 2 and 6 indicate that t ratios diverge at the T d1 rate
irrespective of whether there is a drift in the regressor $1 or not. Fur-
thermore, Theorems 4 and 8 show that t ratios diverge at the TO'5 rate
irrespective of whether there is a drift in the regressor 1:1 or not. So
the probability of rejecting null hypothesis of fl = 1 should increase
as T increases. All of the rejection percentages at every value of N in
Table 3—1 and Table 3—3 support these theoretical results. Moreover,
based on the simulated results on ltfil we estimate their divergence
rates in Table 3—2 and Table 3-4. It is found that the divergence rates
are quite close to the theoretical rates T 0'3 and T05, respectively.

Let us now consider the asymptotic behavior of R2 and Durbin-

Watson statistic DW. Our theory suggests that R2 L 1 for all twelve

94
models. As shown in Tables 3—1 and 3—3, the average R2 increases with

T increases and they are all very close to 1. Our theory also says that
when the disturbance term is stationary, then DW —p—> 2 — 2,0,,(1),
which is 1.1429 when 131 = I (0.3). The results in Tables 3—1 and 3*—
3 indicate that DW approaches the theoretical value as T increases.
Finally, when the disturbance term is nonstationary, Theorems 4 and
8 suggest that DW L 0. Our simulation results show that DW does

decrease as T increases.
3.8. Conclusion

In this chapter we derive the asymptotic distributions of the
OLS estimators and the corresponding test statistics for four classes
of simple linear regression models where the regressor and the dis-
turbance term are both fractionally integrated processes. The main
finding is that the t ratios for the slope coefficients 3 are divergent and
therefore the null hypothesis for testing any finite value of 13 tends to
be overly rejected. This latter result is referred to as the spurious

effect.
3.9. Mathematical Proof
Most of the proofs of our theorems are based on the functional

central limit theorem (CLT) and the continuous mapping theorem

(CMT).

A.1. Proof of Theorem 1

To prove item 1, we see

 

 

 

 

where the weak convergence is based on items 2 and 9 of Lemma 1 of
Chapter 2 and the CMT.

To prove item 2, we see

3(1) — (1

3(1) __ fl T T T
L _ ELL} :1)? thvt

 

 

 

 

 

 

 

 

Therefore,

96

 

 

 

T A _
3; (0(1) — a)
T0,, ’7‘ , —
L ay ([30) — ’6) _

 

 

01*

0:11;.) 1:112:11

T
T02 2 (1“ _ if
L 1:

 

 

 

 

 

 

where the weak convergence is based on items 1, 2, 3, 4 and 9 of

Lemma 1 of Chapter 2 and the CMT.

To prove item 3, we see

 

 

 

 

 

r T T l
, l T Zt 251:1 :vt — ,
3(2) _ a 1:1 1:1 1:1 ’91
T T ‘ T T 1
3(2)—1 = Zt :12 Zn. Ztv. =5 k2 .
1:1 1:1 1:1 1:1
ETD—f3 T T T T k3
‘ ‘ 2:171 Zia :23? 2:131:11, ‘ ‘
1:1

 

 

I
911.
ll
11.—11
9..
ll
p—n

 

where

(3) (mm-(3)313)

. (Z 1) {(212 (1 (2)) - (2212}

3)1-(3)(3)+(>;:)(3))

()()() (3)

(3)1—(3) ()() (3))

(3) (3) ()() (3)

.12) {($12) 1le

98

+-+13)13)+21>2)13

(ZHZHZH

 

 

 

 

 

 

 

 

Therefore,

F 11%;, _a) - —_k_1__

0y (2) ’ T4030y
T211 ) — 1 ’“2 2»
0,, 7(2) 7 1 D T3030},

T502
Tax (,8 fl) .1: k3
0y (2) (T401701!

 

[>|'-*

 

)1

 

3)

t=l

)131

(12*
=> 72* a

182*

 

 

where the weak convergence is based on items 1, 2, 4, 9, 10 and 11 of

Lemma 1 of Chapter 2 and the CMT.

A.2. Proof of Theorem 2

To prove item 1.1, we see

 

 

 

T am (A
)3 w)
«T a) _ a, 10>
‘ 10) — 1/2 '
0 2 2
y (To,t 8%))
For the denominator, we note
2
2 2 90
T0,, 85(0) — ,

 

where
1 T . 1 T A 2
312) Z T: “22 : ff: (Ct — 13(0):“) 9

and A A
C: — ,{3(0).’I't = fll't + ’Ut — 13(0).”17;

= ’Ut — (3(0) — ’13) flit.
And from item 1 of Theorem 1, we have [3(0) — /3 = 0,,(Td1’d2’1).
Moreover, we know from items 2, 5 and 9 of Lemma 1 of Chapter 2
that

T T
Z . 1 Z Z
tzl i=1

Therefore, we have

‘fi

because 2d1 — 1 < 0. Moreover, we have

T 1
1 2 2
2 E :‘Tt : / [B0+5+d2(5)] (18,
T01. tzl 0

where the weak convergence is based on item 2 of Lemma 1 of Chapter

 

2. So we prove
2 711(0)

(310) :> 1 2 '
/[B0.5+d2(3)] dS
0

T03. s

 

100
For the numerator, we note

 

T0; ’5 . ,
(13, — /3) => 50.
0y
by using item 1 of Theorem 1. Combining the asymptotic distributions
of the numerator and denominator, item 1.1 is proved.

To prove item 1.2, we see

 

 

From items 1, 3, 4, 6 and 9 of Lemma 1 of Chapter 2, we have

T T

Z(wt 4)? = 0P<T2+2da Z111 — v) =0 (T)

i=1 £21

and

(xt — .7:) (11¢ — 71)— — E .1:,121— —
T

m”
9
H
A
b*3
33
V
A
Ma
S
V
||
0
1
_+_
S
+
E“

 

 

t: l t: i=1
Consequently,
7‘2 '1- 2— 2
2 2 N(O) : [1'3 + OP(Td d 1)] p ; 1
,82 + 0p(T—1+d1—dg) + Op(T—l—2d2) 182 + 0p(1) ,

where the convergence in probability is based on 3(0) 2 13 + 0,,(1)

and the second and the third terms in the denominator coverge in

101
probability to zero because —1 + d1 — (12 < 0 and —1 — 2112 < 0. To

prove item 1.3, we first note

(at ‘“ at—l)2 = (Ct " 3(0).”171 - 01—1 + 3(0)33t—1)2
= [1), — vt_1 — (23(0) — /3) wt] 2
:(Ut — Tit—112‘ 2 (13(0) "- 13) (12,— v14)th

’7 2 2
+ (13(0) — H) w..
And from item 7 of Lemma 1 of Chapter 2, we have

T T

1 1
— (’Ut — 111— 1) wt: —Z ’01“).— ’Ut—lwt
T i=2 T T t: 2

: max {0P(T(_1+d1+d2), 0})(Jfi€—O.5)}7

for any 6 > 0. Moreover, we know from item 1 of Theorem 1 that
3(0) — [3 = 0,,(Td1‘d2‘1) converges in probability to zero. From item
5 of Lemma 1 of Chapter 2, we also know that 2322 wig/T L 7,,(0)

and that
T T

1 T 1 T 1 2
TZ(I)t—Ut_l)2:TZUtZ+TZUt2_ _l—T vt'Ut—l

—p-+ 112(0) + 11(0) — 211(1)-
Finally,
DW .1; 2 — 2p.,(1),

because the second and third terms in the numerator of DW all con-
verge in probability to zero and the denominator Z,_ 111, 2/T=

converges in probability to 7,,(0) from the above results.

102
To prove item 2.1, we see

 

 

a 1/2
11 2 .2
For the denominator, we note
.3

 

 

where

and

C1 — a(1) — 1’3(1)117t = 0’ +/3~T1 + ”U. — 5(1) — flaw
= (111 — l7)— (/3(1)—/3) (331— 4E)-

And from item 2 of Theorem 1, we have [3(1) — 13 = 0,,(le'd2‘l).

Moreover, we know from items 1, 3, 4, 6 and 9 of Lemma 1 of Chapter

2 that

103
Therefore, we have

(111—012—— T(13(,,—13) Z (sup-.1:)2

i=1

2_
1_

8

Ms

1
T

t

ll
p—n

(m — €02 — 0(T “10,,(T‘2d1-‘M2—2 )-O,,(T2+2d2 )

||
'fllH
1111

W11.

T
(’01 — U) — OP(T2d1_l)= —%;( Ut — 102+ 0p(1)
"113'7v(0)1

Where the convergence in probability to ”1,,(0) is based on item 6 of

Lemma 1 of Chapter 2. Moreover, we have

T 1 1 2
1 _ 2 / 2
T02 2( t ) 0 i + l 0 +

1: i=1

 

where the weak convergence is based on item 3 of Lemma 1 of Chapter

2. So we prove

7v ( 0)

1 1 ‘2'
/ [30.5+d2(8)]248- [/ Bo.5+d2(8) (1.9]
0 0

For the numerator, we note

To <~ ,_
1? (IL/3(1) — ’13) fi [131*

0y

 

T0329

I310)

 

by using item 2 of Theorem 1. Combining the asymptotic distributions

of the numerator and denominator, item 2.1 is proved.

104
To prove item 2.2, we see

 

 

2 _
R — 1 — T
Z (a — C12
i=1
= 1 _ T312
T T T ’
[32 Z (331 — EL")2 + 213 Z (111 — 7) (.1: — .73) + Z (111 — '71)2
1:1 t=1 i=1 .1
given there is a constant included in the regression. From items 1, 3, :3
4, 6 and 9 of Lemma 1 of Chapter 2, we have -.
T T ‘_
Z(Tt " 9‘5)‘2 = OP(T2+2dB)v Z(t)‘ _ m2 = 0,,(T), i

 

i=1 i=1 ' 3‘-

T

20:1 — 5:101 — o = 01(T‘+d1+“=').

t=l

and

We also note .9? —> 7,,(0), consequently,
0,,(T)

R2 = 1 — . . ..
01(T 2+‘2‘“) + 0p(Tl+d‘+d"’) + 0p(T)

 

0 (1) p
: 1 — ._ P . = 1 + 0 1 ‘—> 1,
OP(T1+2d2) + 0p(Td1+d2) + 012(1) 1)( )
where the first term in the denominator of the second term diverge
because 1 + 2112 > 0.

To prove item 2.3, we first note

 

A r. 2
(at — 7121—1? = (Ct — a(1) — /3(1)171 — Ct-1+ (1(1) + 13(1)1'1—1)

A ‘2
1‘ [Ut — ’Ut_1 — (13(1) — 13) 1111]

=(111— 111_1)2 — 2 (1311, — 13) (111 — 111411111

’5 2 2
+ (13(1) — 113) “’1-

105
And from item 7 of Lemma 1 of Chapter 2, we have

T

1 T 1 T 1
— 2 (vi — Ut_1)U-’t = _thwt — — ’Ut—l’wt
T i=2 T i=2 T

i=2
___ max {0p(Tc—l+d1+d2)’ 0p(Tc—O.5)},

for any 6 > 0. Moreover, we know from item 2 of Theorem 1 that
73(1) — [3 = 0,,(T d1“124) converges in probability to zero. From item
5 of Lemma 1 of Chapter 2, we also know that 2322 "wt2 / T J; 71,)(0)
and that

Finally,

DW —”—> 2 — 2,0,,(1),
because the second and third terms in the numerator of DW all con-
verge in probability to zero and the denominator 2le fig/T = .9?
converges in probability to 7,, (0) from the above results.

To prove item 3.1, we see

fit _ T: (1% - /3)
g: 73(2) _ (T0232 )1/2 '
For the denominator, we note
T T 2
T272 — (:1:)
T02 82 93 t:1 t:1 1

13(3— _—

 

 

 

 

 

106
where D is defined in the proof of item 3 of Theorem 1 and

. A 1 2
8% = — T2112 ut —— —T —2 (Ct— (Y(2 7(2)t — flay“) ,
Tt=l

and

A

Ct — (3(2) — 75(2)?" -- ,8(2);I)t = (Y + "yt + [3.171 + vt — 6(2) — 312% — Barri
= W — '17) —‘ (73(2) — 7) (t — 1‘)

And from item 3 of Theorem 3, we have
23(2) _ ,3 = 0,)(Td1‘d2‘1) and 7(2) — 7 = 0,,(Td1‘1'5).

From items 3 and 6 of Lemma 1 of Chapter 2, we have

T T

( o ') 1 _
E (lit — T) _ —OP(T2+2d‘) and T E (’Ut —- U)? —p—> ’71,(0).
i=1 i=1

From items 4 and 10 of Lemma 1 of Chapter 2, we have

T

_ _ T 1 T T
Z(1), — v) (t — t) = Z; vtt — T (21),) (Z; t)

i=1 i=1

OP(T1.5+d1) _ 019(1-F1)'Op(Ti0'5+dl )O(T2)

: OP(T1.5+d1).

From items 1, 4, 9 and 11 of Lemma 1 of Chapter 2 and the same

arguments above, we have

"i

(Ut _ ’U) (mt _ 51-?)_ Op(Tl+d1+d2 )(,Zt— (xi _ (1.?) : Op(Td2+2.5).

t=l i=1

Me

107
Therefore, we have

+ 0(T—1)'OP(T2d1_2(12—2)‘OP(T2+2d2)
__ 0(T_1)'Op(le_l'5)'0p(Tl'5+dl)
__ 0(T—l)'Op(le_d2—l)'OP(T1+d1+d2)

+ 0(T—1).Op(Td1—l.5).0p(Td1——dg—l).OP(T2.5+d2)

vt_ 17) 2+ OP(T‘2d1—l) + OP(T‘2d1—l)

Ms

t(=l
_ OP(T2d1—l) _ OP(T2d1—l) + OP(T'2d1—l) i) 71(0),

where convergence in probability is based on item 6 of Lemma 1 of

Chapter 2. Therefore, we prove

2 2 7v(0)
T01, 5’3“!) => 12A,

 

 

108
where the weak convergence is based on item 3 of Theorem 1 and the

above results. For the numerator, we note

 

T0” (3(2) — 5) => 132.

y

by using item 3 of Theorem 3. Combining the asymptotic distributions
of the numerator and denominator, item 3.1 is proved.
To prove item 3.2, we see

T 53 T53

T =1— T
Z(Ct_C)2 21/303:—57)+7(t—?)+(v,—o)]2
tzl

i=1

122:1—

 

 

‘J

given there is a constant included in the regression. The order of the
denominator of the second term is 0(T3) since the term 23:1 (t — i)2

has higher order than the other terms and 231:1 (t —— D2 = 0(T3). We

also note .93 —p—-> 7,,(0), consequently,

0(T) _ p
32:1—Ji—=1—0 1‘2 ——>1.

To prove item 3.3, we first note

.. A, A 2
(at - at—1)2 = [(1% — 1’1—1) — (13(2) — ,3) wt — (7(2) — 7)]

c A , ‘ A 02
W ‘_ vt—I)2 + 03(2) — 15) “152+ (7(2) — 7)
— 2 (Ba) — l3) ('Ut. — Ut—1)wz

— 2 (2(2) — 7') (Pt — ”Ur—1) + 2 (13(2) — 5) W — 7) wt-

And from item 7 of Lemma 1 of Chapter 2, we have

T
Z (21, — vt_1) w, = max {01,(Td‘+d2), OP(TO"’)}.
i=2

109
Moreover, we know from item 3 of Theorem 3 that

23(2) — fl = Owl‘s-0’24) and 3(2) - 7 = owl‘s-1'5).

From item 5 of Lemma 1 of Chapter 2, we know that

:11? =()p( and £111,220 ,(

From item 4 of2 Lemma 1 of Chapter 2, we 2know that

T

Z: 1): = OP(T0.5+d1) and 2: wt 2 O,,('T0 5+dr, ,)

i=2

All these results imply

TZ< U1“Ut 1

1

(1),—11,4)+0(T—l)OP(T2d‘_2d2_2)OP(T)

Ms

Tt=2

+ 0(T-1)0,,(T'2d1-3) -O(T)
— 0(T‘1>-0p<le‘d2'1> max {0..1Tds‘“), 0.11105}
_ 0(T_1)'OP(le—l'5)-OI,(T0'5+d1)

+ 0(T—l)0p(Td1—d2-l)_OP(T(11—l.).5 OP(TO 5+d2)

('Ut _ 722—1)? + OP(T‘2dl—‘2dg—2) + OP(T2d1—4)

SI“
Ms

i=2

_ max {01,(T2d1—2), OP(Td1—dg—l.5)} _ OP(T2d1—‘2)

+OP(T2d1—3)
1 T 1 T T
iv +-zv2.1—2zw—1+op<1>
i=2 Tt=2 t:2

_L 27,,(0) —— 27,,(1).

1 10
Therefore,

DW i) 2 — 2,0,,(1),
because 231:1 ”fig/T = 8322 —p—> 7,,(0).
A.3. Proof of Theorem 3

To prove item 1, we see

1

 

171%

To a
0x ’,‘ y 17 =
—‘ (5(0) — ,3) = tTl => 30*,
t:

1
T03 Z 513,2

1

 

 

where the weak convergence is based on items 2 and 8 of Lemma 1 of
Chapter 2 and the CMT.

To prove item 2, we see

I

_ T
F - T a: E
A t yt
(1(1) _ CY £21

A

13(1) — 13 in

 

 

 

 

 

 

 

 

 

1 11
Therefore,

 

 

F i(511)— a’) -

0y

a A _
_1: 1'3 — 3)

0y ( (1) 1

 

tzl 01*

W111) (1:1—51:11

 

 

 

 

 

 

where the weak convergence is based on items 1, 2 and 8 of Lemma 1

of Chapter 2 and the CMT.

To prove item 3, we see

 

 

 

 

 

 

 

' T T ' _1 F T '
F q T 2t :1:) 23/, .. .
64(2) — a £21 £21 i=1 kl]
T T T T 1
3(2) — 7 = Zt ZR 215331 Ztyt = E k'2 7
1:1 1:1 1:1 1:1
13(2) — /3 :r T T T k5
‘ ‘ 2 art 2 tat) Z :1:? Z arty) ‘ ‘
_1:1 1:1 1:1 j _1:1

 

(

112311) (:1) (DJ-1121”) (:11)

@wfll‘étfi-(étYi

Th

113

 

 

 

 

 

 

 

 

 

 

' 1 (A 1‘ h k" 1 ' ‘ ‘ 1

— a — a -—

0y (2) T5030?! (1 02*

T (A ) __ 1 k'2 : 1 C

0y 7(2) 7 -" 1 D T4030.” A 2 => 72* a

T503 ,

93. (3(2) _ ,3) k3 C3 fi2=t

L 0y _ T515031 - - L -

where the weak convergence is based on items 1, 2, 8 and 11 of Lemma

1 of Chapter 2 and the CMT.

AA. Proof of Theorem 4

To prove item 1.1, we see

 

a ’5
1 If (13(0) _ fl)
— 1,3 : 3’ .
/— (0) 1/2
T To: 2
— s
0: law)
For the denominator, we note
To: 2 3% 1

 

— ,3 ,. :
2 1310) 2
0y 0y

 

where

and

= yt — (3(0) — 13) 5311'

1 14
Therefore, we have

 

2

{/01BO‘5+d1(5)BO.5+dg(s)d3]
/01[BO.5+112(3)]2 d3

 

1
=> / [Bmusws —-

2
Z} 00*:

where the weak convergence is based on item 1 of Theorem 3 and

items 2 and 8 of Lemma 1 of Chapter 2. Moreover, we have

*3

 

1 1
T0221? => /0 130.5+12(s)12ds,

:1: i=1
where the weak convergence is based on item 2 of Lemma 1 of Chapter

2. So we prove

 

 

 

1 1 2
To: 2 /O[BO-5+d1(3)]2 d5 /0 BO.5+d1(3)BO.5+d2(3)d8
02 83(0) 2} l 2 — 1 2
y /[B0-5+d2(5)i d3 / [30.5+d2(8)] d8
0 0

For the numerator, we note

0’ A

i (13(0) — ,5) => 130*

0y

by using item 1 of Theorem 3. Combining the asymptotic distributions

of the numerator and denominator, item 1.1 is proved.

 

T
3120) Z (331 _ 5L")?

T T T '
52:111—212HfiZ1x1—i)(yt—y>+Z<yt—y)2
1:1 1:1

1:1
From items 1, 3 and 8 of Lemma 1 of Chapter 2, we have

T T
2111 — :1)? = Op(T2+2"‘*), Z111 — :7)? = Op<T2+2dl),
i=1

t=l

and
T

Z122 — soot — p) = 0,.1T2+dl+d'~’>.

t:l
Consequently,
A . _ 2
2 #30) [1‘3 + 011(le d2)l P .

= f .1 __ , 1,
[32 +Op(Td1—d2) +OP(T2(11—2dg) ,62 +0p(1)

 

 

where the convergence in probability is based on 3(0) 2 {3 + 010(1)
and the second and the third terms in the denominator coverge in
probability to zero because d1 — d2 < 0.

To prove item 1.3, we first note

. A 45 2
(521 — 21-1)2 = (Ct — 13(0)th — C1—1 + 13(01171—1)

A A 2 ..
= 17? — 2 (18(0) — ,6) "Ut’lUt + (flm) — ’13) "(1722.

116
And from item 7 of Lemma 1 of Chapter 2, we have

T
1
—T—0_—y- ; vtwt : max {0P(Tc—1.5+d2), 0P(Tc-l—d1)} ,

for any 6 > 0. Moreover, we know from item 1 of Theorem 3 that
(3(0) — fl) /0y = Op(T"d'~’”‘0'5) converges in probability to zero. From
item 5 of Lemma 1 of Chapter 2, we also know that 23:2 113/T —p—>
7.,(0) and that 2322 111,2 /T —p—> 7“,,(0). All these results imply DW

statistic is

 

where the three terms in the numerator all converge in probability
to zero while the denominator 2;, 52,2 / T a; = 38/03 => 03* from the
above results.

To prove item 2.1, we see

01. ’5 ,
1 — (13(1) — H)
_t, 03/
\/T '13“) i.

 

 

2 2
Tax 2 = 31 1
:1: —:r)
2Z< ’
T0,,

Where

117
and A 4
C1 — 0(1)—/3(1)131= CY +13171+ 311— 3(1) — 13ml}

= (311— g) " (13(1) — 5) (371— 375)-
Therefore, we have

s? 1 T .2 01. 6 2 1 T 2
7 2 T7: (yt — 37) _ [3— 03(1) ’13)] T03 Z (371 — 5?)

t:l

 

$ 01*,

where the weak convergence is based on item 2 of Theorem 3 and
items 1, 3 and 8 of Lemma 1 of Chapter 2. Moreover, we have

T 2

l l
T102 Z (Slit — 53)? fi /0 [B0.5+d2(8)]2 d8 — {/0 BO.5+d2(3) d8] ,

:1: i=1

 

Where the weak convergence is based on item 3 of Lemma 1 of Chapter
2- So we prove
T0": 2 02*
2 8311)

0y :7} 1 2 l
/ [B0.5+d2(8)] d3 — I:/ B0.5+d2(8) d3]
0 . 0

For the numerator, we note

 

2.

023 7‘ ,, ,

_ (#0) — 1‘3) :> 131*

0 y

by using item 2 of Theorem 3. Combining the asymptotic distributions

of the numerator and denominator, item 2.1 is proved.

118
To prove item 2.2, we see

 

Tsf

T T
1221.1. — 212+2521y1 — 1111.1 — .1) +2131 — 1)?

i=1

 

given there is a constant included in the regression. From items 1, 3
and 8 of Lemma 1 of Chapter 2, we have
T

T
:0“ _ 1,) OP(T2+2(12), Z(yt _ 9):; ___ Op(T2+2dl)a

1:1 t:1

and

(ft _ x) y) : OP(}T‘2+d1+d2).

:Ms

i=1

We also note 81/03 2) 01*, consequently,

011T”)

R2 : — .- . .
OP(T2+2(I2) + OP(T2+d1+d2) +OP(T2+2d1)

 

_ _ 0111)
ogre-Ml) + 0,,(Td2-dl) + 0,11)

 

:1+o,,(1)—5’—>1,

Where the first and the second terms in the denominator of the second
term diverge because d2 — d1 > 0.

To prove item 2.3, we first note

A A 2
(121— 7121—1)? = (Ct — 0(1) — 13(111’1 — C1—1 + 65(1) + 13(1)3?1—1)

= 11,2 — 2 (13(1) — 13) 1111111 + (13(1) — 1021112

119
And from item 7 of Lemma 1 of Chapter 2, we have

T
1
m ; Ugwt : max {()p(Tc—1..‘3+d2)7 0p(T(—l—d1)} ’

for any 6 > 0. Moreover, we know from item 2 of Theorem 3 that
(3(1) — 3) /0y 2 0,,(T’d2‘0'5) converges in probability to zero. From
item 5 of Lemma 1 of Chapter 2, we also know that 2:2 113/T —p—>
7.,(0) and that 2:2 11,1,2/T L yu,(0). All these results imply DW

statistic is

 

where the three terms in the numerator all converge in probability
to zero while the denominator 2:1 12,2 / T0; = 33/03 => 0%,, from the
above results.

To prove item 3.1, we see

 

0.1: ’F /
1 3’ (15(2) _ ‘3)
—— t/3( ) = y 1
2 /2
fl: T0: 2
0: 51")

For the denominator, we note

Ti12—(it>2 1

 

 

2 2
T02: S2 _ f2 1: 1:1
0,2 ' 13(2) 0.2 T4 1 ’
y y 5 21)
T 0,,

Where D is defined in the proof of item 3 of Theorem 1 and

1 T 1 T 2
2 _ A2 _ A A 6 j
82 — T Z; ’11, — T 2 (Ct — (1(2) — ”7(2)t — 13(2):“) ,

120

and
Ct — (32(2) — fiat - 131211} = CY + ”115+ 13131 + yt — a(2) — 212% — /3(2)1131

= (311 — 17) - (3(2) — 7) (t - Z) - (13(2) - 13) (It - .1).

Therefore, we have

32 1 T T 2 1 T

2 —2 A '2

172227032 :(1/1-11) +[;y'(7(21—7)] 72:05—15)
:9 i=1 i=1

 

03 ¢~ . 2 1 T _ 2
+ [33" (13(2) — 5)] 7103:0171“ 41?)

 

 

 

2
z) 02*?

Where the weak convergence is based on item 3 of Theorem 3 above

and items 1, 3, 4, 8 and 11 of Lemma 1 of Chapter 2. Therefore, we

 

prove
T05. 2 022*
__T_'S. 2?
2 13(2) 1213’

01/

Where the weak convergence is based on item 3 of Theorem 1 and the
ab ove results. For the numerator, we note

$(3121-13) => 132.

y

121
by using item 3 of Theorem 3. Combining the asymptotic distributions

of the numerator and denominator, item 3.1 is proved.

To prove item 3.2, we see

2
Ts2

:(a 4‘2)?

i=1 i=1

,2
T52

 

 

122:1— =1—

[3(xt—i)+v(t—t)+(yt—y)l2

Me

given there is a constant included in the regression. The order of the
denominator of the second term is 0(T3) since the term 2le (t — f)?

has higher order than the other terms and 2le (t — D2 = 0(T3). We

also note 53/03 :> 03*, consequently,

Op(T2+2d1)

=1—0 T2d1‘1 L1
0(T3) P( ) 9

R221—

 

because 2(11 — 1 < 0.

To prove item 3.3, we first note
A A 2 ’,‘ . A 2
(714 — ut_1) = [Wt — (ll/3(2) — /3) wt — (7(2) — A0]
2 ’-‘ 2 2 A 2 Ar
= w + 03(2) — 3) wt + (7(2) — 7) — 2 (13(2) — 3) WU:

— 2 (11(2) — 7)1’t+ 2 (3(2) — fl) (”7 — 7) wt.

And from item 7 of Lemma 1 of Chapter 2, we have

T
Z W, = max {OP(T"1+“'~’), OP(T0'5)} .

1:2

Moreover, We know from item 3 of Theorem 3 that

3(2) -fi=0p(T"""2) and ,(2,_,:OP(T.1._0.5,

122
From item 5 of Lemma 1 of Chapter 2, we know that

T
v, = 0,,(T) and Z w? = O T

From item 4 of Lemma 1 of Chapter 2, we know that
T

Z vt : OP(TO‘5+d‘) and 2; wt: 0,,(T0 5M2).

i=2

All these results imply

T

A A 2
E (“t — ut—l)
i=2

_ OP(T) + OP(T2dl—2d2) 0P(T) + OP(T2dl—1)OO(T)

_ Op(Td1_d .,) max {Op( (Td1+(12) OP(TO.5)} _ Op(le—0'5)°OP(TO'5+dl)

+ Op(Td1—d2) . Op(Td1—O.5) . OP(TO'5+d2)

= 0m + 0p(T2"“2d'-’+‘) + 0.41”“)

— max {op(:r‘2d1).0,,(Td1-d2+0-5)} — 0,,(T2d1 ) + 0,,(T2d‘ ).

T A
We also know 21:1 u? = T33 = OP(T2+2‘11). Therefore, we note

T
E( (Ht—UFO
i=2
T
2% 2

t: l

— marl—“1) + OAT—1‘2"") + OAT—2)

— max {OAT—2), Op(T_1'5—d1_d2)}

— OAT—‘2) + OAT-'2).

123
Consequently, DW —p—> 0 because —1 — 2111, < O, —1 — 2d2 < 0 and

—1.5—d1 —d2 <0.

To prove Theorem 5, let us first present the following lemma

which will shorten our presentation considerably.

Lemma A.1. Given that Assumption 1 of Chapter 2 holds, then, as

T ——> 00, we have the following results:

 

T 1
1 0
1. mtg-1 .Tt'Ut : B0.5+d1(1) — A B0.5+d1(3) d8.
1 T 1
2. . => B d
Tzayt21:xtyt 0 9 05+d1(5) 9

124
A.5. Proof of Lemma A.1

To prove item 1, we see

1 T 1 T 1 T 1 T
——§‘0x =—§:(t — ——§:t ——2
Toy _ ‘T‘ To _1 +3”) ”‘ Tayt_1 ”+110, 1—1 T‘T‘
T
:-—E t 1
Tag t=1 TUT+0P()

1
=> BO.5+d1(1)—/ Bo.5+d1(8)d8,
0

where the weak convergence is based on item 10 of Lemma 1 of Chapter
2 and the fact that Toy = 0(T1'5+d1) and 23:, rim 2 Op(T1+d1+d2)
by using item 9 of Lemma 1 of Chapter 2. The remaining items in

Lemma A.1 can be proved by the same arguments.

A.6. Proof of Theorem 5

To prove item 1, we see

 

T2 y =
—(/3(0)-/3) = ‘ 1 => fie...
0'

y 1 T 02
fizxt

(:1

Where the weak convergence is based on items 1 and 3 of Lemma A.1

and the CMT.

To prove item 2, we see

125

am — a

,2 _ 1 T T
13(1) l3 2: :1:? Z 55;)?
i=1

 

 

 

 

Therefore,

 

 

 

 

 

 

 

:1 7
1 - _
7: (1'? — 1'0)?
(11*

T210, [" (t; “3?) (:11) +T (2: $3.10] 131:

where the weak convergence is based on items 1 and 3 of Lemma
A.1 and item 4 of Lemma 1 of Chapter 2 and the CMT.

To prove item 3, we see that the introduction of a nonzero drift

 

 

 

 

 

in the trending variable when a time trend is included in the regression
causes the covariance matrix to be singular. Therefore, we transform
Ct = a+7t+flmf+vt to be Ct = a+(7+fi)t+flxt+vt. The
remaining proof can follow the same arguments in the proof of item 3

of Theorem 1.

A.7. Proof of Theorem 6

To prove item 1.1, we see

 

0y

fl.— T—: (1%» -13)
(T

T1310): 1/2 °
STUD)

For the denominator, we note

88

T 7
12,02
T3 ‘

i=1

 

3 2 _
T 81310) —

 

where

and

And from item 1 of Theorem5 w,e have (310)—13 = Op(T‘T1_1'5). More-
over, we know from items 1 and 3 of Lemma A.1 and item 5 of Lemma

1 of Chapter 2 that

T T

0‘ 1 ‘
2:th :O(T3), 5;va L7,,(0), 223,111: O,,‘(T15+T‘).
i=1 (:1

Therefore, we have
T
1 . 1
2 _ _ 2 _ _
8° _ T 1; T‘ T

”012 - 0(T"1)°OP(T2T‘_3)-0(T3)

||
HIH
1M3

T T
_ 1 1
=TZvE—01TMI 1): T —va + 010(1) 31» 1.10).
1:1 1— 1
because 2d1 — 1 < 0. Moreover, we have 2L 1:1:2/T3 —p—> 1/3 by
using item 3 of Lemma A. 1. So we prove T3330 ) 193740). For the

numerator, we note

T2
—(13(0)— 13) => 130*
U

y

by using item 1 of Theorem 5. Combining the asymptotic distributions

of the numerator and denominator, item 1.1 is proved.

 

128
To prove item 1.2, we see

T
1‘ o -0 2
13301811- T )
t=1
T

Z(Ct—C‘)

[:1

R2:

 

T
’1'“ 0 -0 2
13(20) 2 (x, — 1‘ )
i=1

T T T '
12 Z (1:? — W) + 213 21121 — 5101121 — 11+ 2111— 1—11‘2
t:1 i=1 i=1

Horn items 4 and 6 of Lemma 1 of Chapter 2 and items 1 and 3 of

Lemma A.1, we have
T T

211? — :2”) = 0<T311 Z111 — 1112 = 011T).

t=l i=1

 

 

 

and
T
23111—10111—1 = 0.11511
t:l
Consequently,
A _ 2
2: _ (3120) f =15+0p(Td‘ 1'51] P.
fl2+0p(Td1-1.5)+Op(T—2) 182+0p(1) ’

where the convergence in probability is based on 3(0) = 13 + 0,,(1)
and the second and the third terms in the denominator coverge in
probability to zero because d1 — 1.5 < 0.

To prove item 1.3, we first note

.A A 6 2
(U1 — 111—1)? = [”1 — 111—1 — (13(0) — {3) (1 + Ml]

:: ('Ut — vt—l)2 __ 2 (3(0) - 113) ('Ut — ’Ut_1)(1 + 11,11)

+(1A3(0)— 13)? (1 + 1002‘

 

129
And from items 4 and 7 of Lemma 1 of Chapter 2, we have

T
%Z( vt — 11t_1)(1 +113)
t=2

= max {<TT1 011211-151} ,

for any 6 > 0. Moreover, we know from item 1 of Theorem 5 that
13(0) — fl = OP(T d1‘1'5) converges in probability to zero. From items 4
and 5 of Lemma 1 of Chapter 2, we also know that

 

—”—+ 1101+ 11(0) — 211(1).
Finally, DW statistic is

DW —> 2 — 2pv(1)

because the second and third terms in the numerator all converge in

probability to zero and the denominator 21-1 at / T — 83 converges in
probability to ”10(0) from the above results.

To prove item 2.1, we see
2

1T 5T (73“ — 13)

——t13 — .
1111 1 2
0y /
813m

For the denominator, we note

 

.11

1 T
EEC”? —
i=1

 

3 2 _
T 3‘3“) —

130

1 T 1 T A 2
=TZQE=TZ(Ct—a(1)—fl(1)xf) ,
t=l t=l

where

and A A
Ct — a'(1) — 13(1)??? = a + 13513? + 01 — 8t(1)'-/3(1)~"31

: (”Ut — 17)— (3(1) — ,8) (513;) — To) .
And from item 2 of Theorem 5, we have 3(1) — 13 = Op(Td1—1'5). More-

over, we know from items 1 and 3 of Lemma A.1 and items 4 and 6

of Lemma 1 of Chapter 2 that

z:(33t“io)2 :

gms

(11t — 17) —p+ 7,,(0),

and

T
Z(CL‘O (11, — 11)— _ OP(T1'5+d1).

t=1

Therefore, we have

(11— a -%— (B11) — #211: — W

”Mi—L]

(vt - 1'02 - 0(T_1)°0p(T2d‘_3)'0p(T3)

||
HIP“
1.11%

T T
1
:T;( 1),—1))2 — OP(T2d‘ I):TZI( (1),—~11).2 +0p( (1)

'3‘) 7v(0)v

where the convergence in probability to 7,,(0) is based on item 6 of

Lemma 1 of Chapter 2. Moreover, we have 2:1 (.17? — 47:0)2 /T3 _p,

131
1/12 by using item 3 of Lemma A.1. So we prove T332?” —p—+ 1270(0).

For the numerator, we note

T2 A

— (K311) — 13) => 51*

0y

by using item 2 of Theorem 5. Combining the asymptotic distributions
of the numerator and denominator, item 2.1 is proved.

To prove item 2.2, we see

T3?

2:101—

i=1

112:1—

 

:1_ T3?

T
1322:1111 ——0) +212] 111—17) c-c")+2::1v1—v)
i=1

given there is a constant included in the regression. From items 1 and

 

3 of Lemma A.1 and items 4 and 6 of Lemma 1 of Chapter 2, we have

T T

2:11;) — :20)? = 01T3), Z111 — 17>? = 011T),

t2] i=1
and
T .
Z122: — 101111 — a) = Opal-51"“).
i=1
We also note 3% ——p—> 7,,(0), consequently,
0p(T)
01T3> + 0p(T1'5+"1) + 011T)

 

122:1—

0,,(1)

z 1 _ 011"?) + 011Td1+°-5)+ 0111)

 

=1+o,,(1)—L1,

 

132
where the first two terms in the denominator of the second term di-

verge because all + 0.5 > 0.

To prove item 2.3, we first note

. A 2
(’th — [fit—02 : [Ut — ’Ut_1 - (/3(1)—1’3) (1+ 100]
= (’Ut — ’Ut_1)2 '— 2 (22(1) ‘- 13) (’Ut — 01.4) (1 + 1110+

(3(1) — fl)? (1 + “’02-

And from items 4 and 7 of Lemma 1 of Chapter 2, we have

Ms

(1), — vt_ 1)( (1 + 111,)
i=2

: max {0P(T6—0.5+d17TE-l+d1+d2)’ 0p(T6—0.5)} ,

for any 6 > 0. Moreover, we know from item 2 of Theorem 5 that
3(1) — fl = Op(Td1"1'5) converges in probability to zero. From items 4
and 5 of Lemma 1 of Chapter 2, we also know that

T
Elf—2;;(Hwt) ZTT +TZwt+— 1221113—411+1w(0)
and that
1 T 2 p
T’ Z (,0, _ 111-1) _1 7,111) + 7.10) — 211(1)-
Finally, Dl/V statistic is
Dw J; 2 — 2,0,,(1)

because the second and third terms in the numerator all converge in

probability to zero and the denominator 2:1 12,2 / T = 3? converges in

probability to 70(0) from the above results.

133
The proof of item 3 of this theorem is identical to that of item

3 of Theorem 2, we won’t repeat here.

A.8. Proof of Theorem 7

To prove item 1, we see

 

 

T A z
0— 03(0) — [3) = 1 T => (30*,
y 02
:73: 2 “It
i=1
where the weak convergence is based on items 2 and 3 of Lemma A.1

and the CMT.

To prove item 2, we see

-a(1)_a‘ T :33? EM

 

 

 

 

 

 

 

 

 

134

 

 

 

Therefore,
_ 1 A -
3(0(1)—a)
T é _
W>
' 1 ‘ T T T
- t: t: = t: 7

 

 

 

 

where the weak convergence is based on items 2 and 3 of Lemma
A.1 and item 1 of Lemma 1 of Chapter 2 and the CMT.

To prove item 3, we have to transform C; = a + 7t + [333,0 + yt
to be Ct = 01 + (7 + fl) t + [35m + vt by the same arguments in item 3

of Theorem 5. The remaining proof can follow the same steps in the

proof of item 3 of Theorem 3.

A.9. Proof of Theorem 8

To prove item 1.1, we see
T A
_ . _ .3)
1 0,, (5(0) ’1

«TM 2 T3 1/2'
2
(7 83(0))

 

 

 

01*

[81*

 

135
For the denominator, we note

 

where

and

2

where the weak convergence is based on item 2 of Lemma 1 of Chapter
2 and items 2 and 3 of Lemma A.1 and item 1 of Theorem 7. Moreover,
we have 231:1 acid/T3 —p—> 1/3 by using item 3 of Lemma A.1. So we

prove

T3 ‘ l ‘ 1 2
—2— 3223(0) :> 3/ [B0.5+d1(5)]2 (IS _ 9 [/ 3B0.5+d1(9) C13] .
0y ‘ 0 0

For the numerator, we note

I- (10) — 13) => 130*

0y

136
by using item 1 of Theorem 7. Combining the asymptotic distributions

of the numerator and denominator, item 1.1 is proved.
To prove item 1.2, we see
T
’72 0 —,0 '2
(3(0) 2 (“3t — l )
2 (Ct — 6)

i=1

 

T
T 0 —o '2
(330)2(371 “ 37 )
t=l

 

: T T T '
52 2 (5'3? — i’of + 25: (xi) — 50) (yt — 3?) + Z (yt - .72)?
i=1 i=1 t=1

From items 2 and 3 of Lemma A.1 and items 1 and 3 of Lemma 1 of

Chapter 2, we have

"3

and

Consequently,

2 (3(20) [3 + OP(Td1_O'5)l2 p

_ [(32 +OP(Td1—0.5) +OP(T2d1—l) — [82 + 0p(1)

 

 

,

where the convergence in probability is based 011 29(0) = [3 + 0,,(1)
and the second and the third terms in the denominator coverge in

probability to zero because (11 — 0.5 < 0.

137
To prove item 1.3, we first note

A

2
(at _ [fit—1)? = [vt _ (3(0) — 5) (1+ 1111)]
A A 2
= ”12- 2 (13(0) - fl) ’Ut (1 + wt) + 03(0) — fl) (1 + wt)2.
And from items 4 and 7 of Lemma 1 of Chapter 2, we have

1 T
—_ ’Ut (1 + wt)
Toy Z;

: max {01)(TC—l)’ 0p(Tc—l.5+d2), 0P(T€-1—d1)} ,
for any 6 > 0. Moreover, we know from item 1 of Theorem 7 that
(3(0) — fl) /0y 2 Op(T’1) converges in probability to zero. From

items 4 and 5 of Lemma 1 of Chapter 2, we also know that

T U2
t p
_ 1v 0
t: T 7 ( )
and that
1 T T +1 T
i=2 i=2 i=2
All these results imply
DW —”—> 0,

becuse the three terms in the numerator all converge in probability
to zero while the denominator 2;] ’12? / T03 = 33/03, => 03* from the
above results.

To prove item 2.1, we see

1 I;(B(‘)_’3)

Tit/3(1) “ T3 1/2'
2

y

 

138
For the denominator, we note

 

T3 2 s? 1
—.—' 8 = -—
0.2 13(1) 0: 3

y 1 T
0 —o 2
51—32631 _ 5’7 )
i=1
where

1 T 1 T A 2
3i:_ZaiZ—Z(Ct—3(I)—fi(l)$?) 1
Tt:l Tt=1
and

A

Ct — 51(1) — /3(1)33? = 0’ + 31‘? + yt — 5(1) — flay”?
--= (y. — 37> — (x30) — fl) (an? — sis").
Therefore, we have

1 _-)2-[2:@ waitofioy
yt y 0y (1) T3 i=1 t

2

=> l[130,5+dl(s)]2 d3 — U01 BO_5+d,(s) (13]

1 ' 2
/ B0.5+d1(3) d8
0
2

 

1
— 12 / 380,5”,(8) d8 —
. 0

 

 

:3 0f...
where the weak convergence is based on item 2 of Theorem 7 and
items 1 and 3 of Lemma 1 of Chapter 2 and items 2 and 3 of Lemma
A.1. Moreover, we have 2:1 (:1:? — 57:")2 /T3 —3—> 1/12 by using item

3 of Lemma A.1. So we prove

T3 2 2
ngfiU) => 1201*.
y

139
For the numerator, we note

a 91(/7(1)—fl) :5 51*

y
by using item 2 of Theorem 7. Combining the asymptotic distributions
of the numerator and denominator, item 2.1 is proved.

To prove item 2.2, we see

 

Ts?

T T
322(221— 27:0)? + 2321m— :1) (x0 — 52°) + [(1. —

i=1 i=1 t:1

 

given there is a constant included in the regression. From items 1 and
3 of Lemma 1 of Chapter 2 and item 3 of Lemma A.1 we have

T

T
Z (x: — 5:0)? = 0<T3>, Z(yt — y)? = 0p<T2+2d11
t-_-1

i=1
and
T
2e: ><y.— y>=0p<T25+d11
i=1

We also note 31/03 => 01*, consequently,

 

R2 : 1 — ‘ 0P(T2+2dl)
0(TJ) + OP(T2.5+d1) + OP(T2+2d1)

 

owl-“1) + 0p<T°-5-d1> + 0.0)

=1+op(1)i>1,

140
where the first and the second terms in the denominator of the second

term diverge because 0.5 — (11 > 0.

To prove item 2.3, we first note

(a. — @412 = [W - (3w “ *3) (1 + ma]?

32—23343) mam—1(2.1.—311+w32.

And from items 4 and 7 of Lemma 1 of Chapter 2, we have

T
1 Z — ‘ 6— 5 9 6— —
T3; 1:2 ”1 (1 + wt) = max {014716 1)10P(T 1' +d“)’0P(T 1 dl)}’

for any 6 > 0. Moreover, we know from item 2 of Theorem 7 that
(13(1) — 5) /0y = 0,,(T'1) converges in probability to zero. From
items 4 and 5 of Lemma 1 of Chapter 2, we also know that

T 2

”11?
___,v0
:21. M)

and that

T T
1
TZ(1+wz)2=—T+2—ZQIU1+ T2“); i’1'1"‘r’w(0)°

All these results imply

DIV—Lo,

because the three terms in the numerator all converge in probability
to zero while the denominator Z,_ 111, 2,/T02 — _32/03 2) 01* from the
above results.

The proof of item 3 of this theorem is identical to that of item

3 of Theorem 4, we won’t repeat here.

141

TABLE 3—1

REGRESSION OF C, = 1 + x, + 5. ON 1 AND 1,,
WHERE x. = I(1) AND a, = I(d)

 

d T 1% 5% 10% 20% 30% R2 DW WI
125 0.237 0.364 0.446 0.557 0.635 0.9158 1.375 1.7553
0.3 250 0.327 0.454 0.529 0.626 0.692 0.9542 1.312 2.1111

500 0.416 0.532 0.602 0.684 0.744 0.9753 1.266 2.5547
125 0.604 0.696 0.742 0.794 0.830 0.8062 0.458 4.0987

0.7 250 0.708 0.773 0.808 0.851 0.881 0.8551 0.333 5.7237
500 0.782 0.839 0.864 0.894 0.915 0.8986 0.243 8.1469

 

 

 

Note: the critical values of the two-tailed t tests are :1:2.576 for
N = 0.01, :1: 1.96 for N = 0.05, :1: 1.645 for N = 0.10, :t 1.282 for N =

0.20, :1: 1.0326 for N = 0.30. R2 denotes the average R2 of the simula-
tion. DW denotes the average DW of the simulation. | t3 | is the aver-

age absolute value of t5 of the simulation.

142

TABLE 3—2
THE DIVERGENCE RATE OF MEAN |tg|

UNDER MODELS A-l AND B—l

FOI‘ 5} = [(0.3),
2.1111

 

 

 

 

__ , 0.3 = . 03
1.7553—0.9769 2 2.1111 09829 2
For (it = I(0.7),
5.7237 8.1469
=0.9875~20'5 —=1. . 0-5.
4.0987 — 5.7237 0065 2

 

Note: the above numbers are taken from the last column of

Table 3—1.

143

TABLE 3—3

REGRESSION OF Ct = 1 + :1:? + at ON 1 AND If,

WHERE x3 = t + I(1) AND a. = I(d)

 

d T 1%

5%

10%

20%

30%

R2

DW

ltfil

 

125 0.317
0.3 250 0.407
500 0.483

0.448
0.526
0.594

0.519
0.589
0.653

0.614
0.672
0.724

0.682
0.735
0.776

0.9991
0.9998
0.9999

1.392
1.322
1.275

2.0598
2.4628
2.9763

 

125 0.735
0.7 250 0.802
500 0.863

0.798
0.850
0.896

0.829
0.874
0.914

0.867
0.901
0.932

0.889
0.920
0.945

0.9981
0.9993
0.9998

0.503
0.366
0.269

6.1141
8.4465

11.8042

 

Note: the critical values of the two-tailed t tests are i2.576 for
N = 0.01, :1: 1.96 for N = 0.05, d: 1.645 for N = 0.10, :1: 1.282 for N =

0.20, :1: 1.0326 for N = 0.30. R2 denotes the average R2 of the simula-
tion. DW denotes the average DW of the simulation. | t5 | is the aver-

age absolute value of t3 of the simulation.

144

TABLE 3—4

THE DIVERGENCE RATE OF MEAN |t5|

UNDER MODELS C—l AND D—l

 

 

 

 

 

 

For 81 = [(0.3),
3:323: = 0.9712 203 3:2: = 0 9816 203

For at = I(0.7),
31111:: = m . 20-5 ““8042 = 0.9882 - 20-5.

 

8.4465

Note: the above numbers are taken from the last column of

Table 3-3.

CHAPTER 4

CONCLUSION

In this dissertation we consider spurious effects in a simple linear
regression model of I (d) processes. In Chapter 2 we find that spurious
effects could occur when we regress a fractionally integrated process
on a constant and another independent fractionally integrated process.
The most interesting finding is that spurious effects could occur even
when both the dependent variable and regressor are stationary. This
implies the usual differencing procedure may not be sufficient for a
complete avoidance of the spurious effect. The recent findings of the
existence of long memory in many macroeconomic and financial time
series remind us of the possibility that spurious effects may present in
some previous empirical work.

In Chapter 3 we consider the asymptotic theory for the OLS
estimators and the conventional test statistics when the regressor and
disturbance term are independent fractionally integrated processes.
The main finding is that nonstationarity in the regressor and long
memory in the disturbance term may result in over-rejection of the
null hypothesis.

From the analysis in Chapter 2 and Chapter 3, we conclude
that the possible presence of fractionally integrated processes in the
regression model may render the usual asymptotic theory for the OLS
estimation useless. Before estimating the regression model, any sus-

picion of long memory in the variables should be investigated.

145

_41

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146

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