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'l is to certify that the

dissertation entitled

Analysis of Economic Time Series with Long Memory
presented by
Hyung Seung Lee
has been accepted towards fulfillment

of the requirements for

Ph . D degree in Economics

 

 

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Date July 27, 1995

 

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W

”3-93

J 4. ..__, .-......___ __ __fi ,

ANALYSIS OF ECONOMIC TIME SERIES WITH LONG MEMORY
By

Hyung Seung Lee

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

ANALYSIS OF ECONOMIC TIME SERIES WITH LONG MEMORY

By

Hyung Seung Lee

This dissertation focuses on economic time series that follow a general
ARFIMA(p,d,q) process with 0< d <1, which is intermediate between short memory
(d =0) and unit root ((1 =1). Chapter 2 considers the unit root test proposed by
Kwiatkowski, Phillips, Schmidt and Shin (KPSS, 1992) against I(d) alternatives. We
show that the KPSS unit root test is consistent against stationary long memory
processes (d <l/2) but is not consistent against nonstationary long memory processes
(d >1/2). Therefore, the KPSS test only can distinguish short memory processes
(d=0), stationary long memory processes and nonstationary processes. Simulation
results are provided to support our asymptotic findings.

Chapter 3 considers the non-parametric estimation of the differencing
parameter in the ARFIMA(p,d,q) process using the Adjusted Minimum Distance
Estimator (AMDE) of Chung and Schmidt (1995). We compute the asymptotic bias
of the AMDE and the MDES that occur if we ignore short-run dynamics and estimate

the (0,d,0) model. Our computational results for the ARFIMA(1,d,O) and

ARFIMA(0,d,l) models show that the asymptotic bias is larger when the short-run
dynamics are stronger and when the number of ignored low-order autocorrelations is
smaller.

Chapter 4 considers the estimation of the cointegrating coeficient in the case
of fractional cointegration. We derive the asymptotic distribution of the OLS
estimator under fairly strong assumptions and find that its order in probability is T‘“,
for -1/2< d <3/2 except (1 = 1/2. Also we derive the asymptotic distribution of OLS in
difl‘erences and find that it is not consistent unless the error and the regressor are

uncorrelated. We provide simulation results that support our asymptotic findings.

ACKNOWLEDGMENTS

This dissertation can be finished by guidance, support and encouragement from
many people around me. Especially, the wonderful guidance fiom Professor Peter
Schmidt makes it possible for me to complete the dissertation smoothly and quickly.
His careful comments and advice, which are essential to improve this dissertation from
the beginning to the end, are highly appreciated. I also wish to thank my other
committee members, Professor Richard Baillie, Professor Ching-Fan Chung and
Professor Jefi'ey Wooldridge for their useful comments.

The financial support from Korean government allowed me to start this
program and I thank everyone who involved it. ll am grateful to many people who are
important to me during the whole process and would like to name some of them who
are special. The aid from Dr. Jaeyoun Hwang, Dr. Dongin Lee and Dr. Kyungso Im is
valuable in studying econometrics. The fiiendship with Inkyo Cheong, Youngmin
Kwon and Hailong Qian has made the sailing enjoyable.

Lastly, I express thanks to encouragement and support from my grandma,
parents, parents-in-law, aunts, brothers and sister, brother and sister-in-law, and
Maeng’s family. The love from my wife, Jeongeun, and two daughters, Jimin and

Jiwon is always with me and I am indebted to them forever.

iv

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................ viii
CHAPTER 1. INTRODUCTION ...................................................................... 1

CHAPTER 2. CONSISTENCY OF THE KPSS UNIT ROOT TEST AGAINST
FRACTIONALLY INTEGRATED ALTERNATIVES

1. INTRODUCTION ................................................................................. 14
2. THEORETICAL RESULTS
A. The KPSS Test Under Short Memory and Unit Root .......................... 19
B. Asymptotics and Consistency of the KPSS Unit Root Test Under I(d). 22
3. SIMULATION RESULTS ..................................................................... 27
4. CONCLUSION ..................................................................................... 31
APPENDIX ........................................................................................... 43
CHAPTER 3. ASYMPTOTIC BIAS OF THE MDE WHEN SHORT-RUN
DYNAMICS ARE IGNORED
1. INTRODUCTION ................................................................................. 52
2. MOMENT CONDITIONS FOR MDE .................................................... 57
3. CALCULATION OF ASYMPTOTIC BIAS ........................................... 63
4. RESULTS ............................................................................................. 65
5. CONCLUDING REMARKS .................................................................. 68

CHAPTER 4. REGRESSION IN FRACTIONAL COINTEGRATION

1. INTRODUCTION ................................................................................. 83
2. COINTEGRATION ............................................................................... 85
3. ASYMPTOTICS FOR OLS ESTIMATES OF COINTEGRATING
VECTORS
A. Short Memory Case (d = O) ............................................................... 88
B. Spurious Regression Case ((1 = 1) ...................................................... 91
C. Stationary Long Memory Case (O< d <1/2) ........................................ 92
D. Nonstationary Long Memory Case (1/2< (I <1) .................................. 95
E. Remarks ........................................................................................... 97
4. SIMULATION RESULTS ..................................................................... 99
5. CONCLUDING REMARKS .................................................................. 100

TABLE 2.1

TABLE 2.2-]

TABLE 2.2-2

TABLE 2.3-]

TABLE 2.3-2

TABLE 2.4

TABLE 3.1-1
TABLE 3.1-2
TABLE 3.1-3
TABLE 3.2-1
TABLE 3.2-2

TABLE 3.3-1

TABLE 3.3-2

TABLE 3.3-3

TABLE 3.4

LIST OF TABLES

Power of KPSS Short Memory Test against I(d), de[0.0, 1.5) ...... 33
Power of KPSS Lower Tail Unit Root Test against

I(d), de[0.0, 1/2) ........................................................................ 36
Power of KPSS Lower Tail Unit Root Test against

I(d), de[l/2, 3/2) ....................................................................... 37
Power of KPSS Two Tail Unit Root Test against

I(d), de[0.0, 1/2) ........................................................................ 39
Power of KPSS Two Tail Unit Root Test against

I(d), de[1/2, 3/2) ....................................................................... 40
Power Comparison with Dickey-Fuller Type Tests ....................... 42

Asymptotic Bias ofMDE in ARFIMA(1,do,0), [do = 0.2, d) = 0.4] . 69

Asymptotic Bias of MDE in ARFIMA(1,d0,0), [do = 0.2, d) = 0.8] . 70

Asymptotic Bias of MDE in ARFIMA(1,do,0), [do = 0.4, d) = 0.4] . 71
Asymptotic Bias of MDE in ARFIMA(0,do,1), [do = 0.4, 0 = -0.4] 72
Asymptotic Bias of MDE in ARFIMA(0,do,1), [do = 0.4, 0 = -0.8] 73
Asymptotic Bias of MDE in ARFIMA(1,do,0) in GLS (F'),

d) = 0.4 ....................................................................................... 74
Asymptotic Bias of MDE in ARFIMA(1,do,0) in Ratios (F2),

4) = 0.4 ....................................................................................... 76
Asymptotic Bias of MDE in ARFlMA(l,do,0) in Common
Denominator Ratios (F3), 4) = 0.4 ................................................ 78
Asymptotic Bias of MDE in ARFIMA(l,do,O), do = 0.2 ................. 80

TABLE 4.1 Mean and Standard Deviation of OLS .......................................... 102

TABLE 4.2 Mean and Standard Deviation of OLS in Difi‘erences .................... 103

viii

CHAPTER 1

INTRODUCTION

2

Classical methods of time series analysis assume stationarity, so that the series
fluctuates around its mean level (or a trend) without changes in its autocovariance
structure over time. Stationary series are often assumed to follow an ARMA process,
which implies that the series has short memory in the sense that its autocorrelations
and impulse response weights decay at a geometric rate. Models which can deal with
time series that are more persistent than a short memory process have focused
primarily on the existence of a unit root process.

However, the classification of time series into either unit root or stationary
ARMA processes is too extreme and restrictive. Between these two types of
processes, a long memory process can be considered to cover intermediate cases
which are not well fit by either short memory or unit root models. In the data, when
the sample autocorrelations do not decay quickly for long lags and yet the low order
autocorrelations are not close to unity, we can suspect a long memory process. That
is, a long memory process displays autocorrelations that are too small at low orders
for a unit root, but too persistent at long lags for a stationary ARMA process.

There are many cases of long memory models in the physical sciences. Data
with hyperbolically decaying autocorrelations and impulse response weights were
firstly observed by Hurst (1951, 1956) and Mandelbrot and Wallis (1968) in
hydrology and climatology. In economics, many financial data, such as forward
premiums, interest rate differentials and inflation rates, have recently been found to
display long memory characteristics. Baillie (1995) provides a survey of the

application of long memory models in economics.

3

There are several possible definitions of the concept of long memory. McLeod
and Hipel (1978) defined a stochastic process to be long memory if the
autocorrelation function is not summable; that is, with p,- = j-th autocorrelation,

T
(1) limp”, lejli finite.
j=-T
A more specific definition of long memory is that the process has autocorrelations that

decline hyperbolically at large lags:
(2) 7,. ~ MC“, (2» >0, 0< 0t <1), as k—)oo.
(Here at ~ bk means at /bk —> 1 as k —) 00.) In (1) above, It can be a constant, or more

generally it can be a function of k that is slowly varying at infinity (i.e., A(Ck)/A(k)

—>1, as k —> 00, for any c >0). Long memory can be defined equivalently in terms of
the behavior of the spectral density as one approaches the zero frequency. A long
memory process has infinite spectral density at zero frequency, as does a unit root
process; however, unlike a unit root process, the spectral density at zero of the first
difference of a long memory process vanishes.

Rosenblatt (1956) has defined long memory based on the dependence between
two points of a process. Mandelbrot and Van Ness (1968) and Mandelbrot (1970)
formalized Hurst’s empirical findings and defined fractional Gaussian noise which is
designed to account for the long run behavior of a long memory time series. Granger
and Joyeux (1980) and Hosking (1981) proposed an alternative long memory process,

the fractionally integrated process. Geweke and Porter-Hudak (1983) proved that the

4
definition of fractional Gaussian noise by Mandelbrot and Van Ness and the

fractionally integrated process are equivalent.
We will consider in detail the fi'actionally integrated process of Granger
(1980), Granger and Joyeux (1980) and Hosking (1981). A time series {y} is a

fractionally integrated process of order (1, I(d), if
(3) (1- WE = 2.,
where L is the lag operator, d is the differencing parameter and {8,} is a white noise

process with zero mean and finite variance 0%. For any d > -1, y. is invertible (Odaki

(1993)) and (l-L)d can be expressed via the binomial expansion:

(4) (1- L)‘1 = it-ntfi) = £1:ij = F(-d,1;1;L),
i=0 1 i=0

where forj = 0, l, 2, ...,

 

d) = d(d—1)---(d—j+1)

(5A) (. .
J J!

 

 

_ F(j—d) _ k—l—d
(5B) TCj - F(j+l)F(—d) — 01:19 k 9

l

w
Itx—le-tdt, x > 0,
0

(5C) F(x) = gamma function =<

oo, x=0,

F(l+x)

l x<0,
x

 

(5D) F(a,b;c;z) = hypergeometric fimction

5

: 1+3—b— +a(a+l)b(b+1)z2+
c-l c(c+l)-l-2

 

F__(_c) __Z F(a + j)F(b + j) zj .
I‘(a)1“(b) j—o NO + Dr (J + 1)

 

Therefore, the infinite AR representation is the following:

(6) yt = Z¢jyt_j +8t, where ¢j = —1Cj.
j=l

y. is a stationary process for d <1/2. Its infinite MA representation can be

expressed as follows:
(7) Yt = ZejSt—j ’
j=0

where

F(j+d)
I“(J'tl)1“((1)’

 

(8) Oj = = 0,1, 2,... .

Its variance-covariance structure is as follows:

2
(9) 0'3, = 08F(1—2d)

 

 

F2(l—d) ’
._I‘(j+d)l‘(1-d)_j k—l+d j=1
(10) pJ-l“(d)l‘(j—d+l)= LII—:— , ,2, 3

Thus for -1< d <l/2, y is a stationary and invertible fractionally integrated process.
Since Oj represents the impact of SH on y, the cumulative impulse response (=0(1))

which is the total effect of a unit innovation can be obtained by summation of Oj:

Nd 0, d<0
ll 01 0 =1 ,0 OJ- =1° w———=
( ) ( )= Z] 1mN_, £1 1mN—> I‘(l+d) {(13, d > O.

For more details, see Sowell (1990).
The AR weights, MA weights and autocorrelations all decay hyperbolically,

though at different hyperbolic rates:

(12A)TC “Aral—(1W —d_l asj —)<X),

(12B) Oj ~——jd lasj—)oo,
Nd)

(12C) pj ~Iq—(l-—d)j2d‘l asj —>oo.

I‘(d)

This is in contrast to the case of a stationary ARMA process, for which the
autocorrelations decrease rapidly (at an exponential rate rather than at the hyperbolic
rate for an I(d) process). Since 0,- is close to zero for large j as long as d <1, an I(d)
process with 1/2_<. (I <1 is still mean-reverting, even though it is not stationary. Baillie
(1995) showed this result using the cumulative impulse response functions of the first
difference of an I(d) process. For further details see Chung (1994b).

The spectral density at zero frequency is another measure of persistence in a

time series. The spectral density of an I(d) process is: for «S to Sat,

2

. —2d 2 _
(13) f(m)=38—1-e“°’| =E£|2sin(to/2)| 2“
21c 21c

 

The spectral density at (0 =0 is infinite for (1 >0, finite for d =0, and zero for <1 <0.
More specifically, because sine) ~03 as (1) ->0, f(0) ~ ((52 /21t)(t) "2“ as a) —) 0. Thus

0, d<0,

(l4) f(0)={oo d>0

0, dS-l/2,
(15) f'(0)= -oo, -l/2<d<0,
w, d>Q

Therefore, the differencing parameter d is not identified by the level or derivative of
its spectral density at zero frequency (Sowell, 1992b).

An I(d) process can be extended to cover more general economic time series
models when 8. in (3) is allowed to follow a general stationary ARMA process. A
time series {y} is an autoregressive fractionally integrated moving average process of

order p, (1, q, or ARFIMA(p,d,q), if it satisfies:

(16) (1— my. = a. = <D(L)"<~>(L)u. [so ¢(L)e. = mm];

(17) <D(L)= l+¢1L+¢2L2+~+¢pr and O(L)=1+01L+02L2+~+0qu;

where all the roots of <D(L) and @(L) lie outside the unit circle, (I) and (9 have no
common roots, and {ut} is white noise. For -1< d <l/2 the ARFIMA(p,d,q) process is
stationary and invertible. For the general ARFIMA(p,d,q) process, Sowell (1992a)
and Chung (1994a) show how to compute the autocovariance and autocorrelation
functions. In a stationary and invertible ARFIMA(p,d,q) model the autocorrelations
decay at the same hyperbolic rate as in the corresponding I(d) process; the rate of
decay is independent of the ARMA parameters. Specifically, if pk“, k = 0, 1, ...,

represent the autocorrelations of the ARFIMA(p,d,q) process,
(1 8) p; ~ Ckz‘H, as k—)oo.

Here C¢0 is a constant that depends on the ARMA parameters.

8
Since the ARFIMA process is an I(d) process with ARMA error or an ARMA

process with an I(d) error, its spectral density function is:

|®(e’i‘” )l2

f
|<D(e‘i‘” ){2 (0))

0' _
_§___ 1-

_ 21t|¢(e-im)|2

(19) f*(0))=

       

=E:|®(e e—im )lz
2n |¢(e—i (1)2)!

At zero frequency the spectral density is similar to the expression for the I(d) case:

2[2|l— COS((D)|]2d , -1t <(l)< 1t.

()(1)

Turn” awn—>0.
(9(1)

(20) f *(0)~ T;2——[

The asymptotic distribution for many statistics based on data generated by an
I(d) process will be established using a fimctional central limit theorem involving the

fi-actional Brownian motion of Mandelbrot and Van Ness (1968):

 

(21) Wd(t)= I(r —s) dW(s),re[0,1],

l
F(d+ 1)0
where W(s) is the standard Brownian motion. To state this functional central limit

theorem, suppose that v. is an I(d) process and V. is its cumulation:

t
(22) V.=Zvj.

i=1

9

Thus (1— L)d vt = ut where u. is short memory. We follow Lee and Schmidt (1995)
in assuming the following (Assumption A):

(A1) v, is I(d) with |d| < 1/2.

(A2) 11, is iid N(0, 02“).

This assumption is somewhat stronger than others have made, and probably stronger

than necessary; see Sowell (1990), Lo (1991) and Hosking (1984).

Define the variance of the partial sum process as in Sowell (1990):

T
(23) 0% = Var(VT) = Vamzvj).
j=l

Then when v. follows Assumption A, Sowell ( 1990) shows that:

 

(24) 0.2 = 0’2 F(1-2d) [r(l+d +T) _ f‘(1+d)]
T “ r(1+2d)r(1+d)r(t—d) r(T-d) F(-d)

and, as T—>oo,

0% 2 r(1-2d)

(25) T1+2d " C“ (1+ 2d)r(1+ d)r(1— d)

5.03.

 

 

Furthermore, under Assumption A and using results of Davydov( 1970), Sowell (1990,

p.498) shows the following invariance principle, for re [0, 1]:

V
(26) [rT]

 

=>wd(f),
0T

or, equivalently,

‘1er}

=> (ode(r).

10

The discussion above has focused on the case of a stationary long-memory
process, and specifically on the I(d) process with |d|<1/2. However, we will be
interested primarily in positive values of d, because an I(d) process with d <0 is anti-
persistent, which is not of empirical relevance. In some cases it may be useful,
theoretically and empirically, to consider nonstationary long memory processes. A
specific type of nonstationary long memory process is the I(d) process with 1/2< (1 <1.
An I(d) process with 1/2< (1 <1 is nonstationary but still mean-reverting. This
contrasts with a stationary long memory process, which is stationary and mean-
reverting; also with a unit root process, which is nonstationary and not mean-
reverting. The discussion above applies to such series after differencing, since ‘3', is
I(d) with 1/2< (1 <1” is equivalent to “Ayt is I(d) with -1/2< (1 <0.”

In this dissertation we will investigate how we can distinguish among three
difl‘erent kinds of processes, namely short memory, long memory and unit root
processes. This is empirically relevant because for some data one can reject both the
null hypothesis of a unit root and the null hypothesis of short memory. It is possible
that such series may follow a long memory process, and we need to test this
possibility. Also we will review estimation of the differencing parameter which
determines the main stochastic properties of a long memory process. Lastly the case
that the error in a cointegrating relationship of unit root series is I(d) will be
considered.

The plan of this dissertation is as follows. In Chapter 2 we will check whether

the KPSS test, introduced by Kwiatkowski, Phillips, Schmidt and Shin (1992), is

1 l
useful in distinguishing short memory, long memory and unit root processes. We will

show that the KPSS test can not consistently distinguish a unit root from a
nonstationary long memory process, since the order in probability of the KPSS
statistic is equivalent under these two processes. However, the KPSS statistic can
distinguish consistently between short memory, stationary long memory, and either
unit root or nonstationary long memory. We provide simulation results which support
our asymptotic results and we compare our results with other results of Diebold and
Rudebusch ( 1991) and Hassler and Walters (1994) for Dickey-Fuller type tests.

In Chapter 3 we will consider the problem of Minimum Distance Estimation
(MDE) of the differencing parameter of a fractionally integrated long memory
process. The simple MDE proposed by Tieslau, Schmidt and Baillie (1995), which
minimizes the difference between sample and population autocorrelations, is useful
because it does not require a distributional assumption and it is easy to compute,
which is also true in the Adjusted MDE of Chung and Schmidt (1995). Furthermore
in the general ARFIMA model it provides a way to estimate the differencing
parameter separately from the ARMA parameters which determine short-run
dynamics. However, such a non-parametric treatment of short-run dynamics will
cause asymptotic bias, and we investigate ways of decreasing the bias due to ignored
short-run dynamics. We investigate how the size of the bias is afl‘ected by the value
of d and of the ARMA parameters, the number of moment conditions used, and the
order of autocorrelations considered. Our computations show that, for certain

methods of expressing the moment conditions suggested by Chung and Schmidt

12

(1995), the asymptotic bias becomes small when only high-order autocorrelations are
used or the short-run dynamics are not strong.

In Chapter 4 we consider the estimation of the cointegrating vector in the case
that the error in a cointegrating relationship is I(d) with 0< (1 <1, rather than in the
usual case with 1(0) errors. We find that OLS in this case is still consistent and its
order in probability depends on the value of (1. Specifically, for 0< (1 <1 OLS is
Op(T1'd). For comparison we also consider OLS in difi‘erences. We find that it is not
consistent if the errors and regressors are correlated, and it converges at the usual rate
T”2 for all values of de[0, 1]. We provide some simulation results that support these
asymptotic results.

Finally in Chapter 5 we summarize our results and make some suggestions for

firture research.

CHAPTER 2

CONSISTENCY OF THE KPSS UNIT ROOT TEST AGAINST

FRACTIONALLY INTEGRATED ALTERNATIVES

l3

l4
1. INTRODUCTION

Since Nelson and Plosser (1982), there has been an enormous body of
theoretical and empirical work seeking to distinguish whether economic time series
are trend stationary or have a unit root. This distinction is important for both
economic and statistical reasons. For a survey, see Diebold and Nerlove (1992).

There are two main approaches to this problem. The most traditional
approach is to test the null hypothesis of a unit root against the alternative hypothesis
of trend stationarity. For this problem, the Dickey-Fuller tests were introduced by
Dickey (1976), Fuller (1976), and Dickey and Fuller (1979). The standard Dickey-
Fuller tests are extended to allow general ARMA error processes by Said and Dickey
(1984), Phillips (1987) and Phillips and Perron (1988). Dejong, Nankervis, Savin and
Whiteman (1992) found that the standard Dickey-Fuller tests and the extensions of
Said-Dickey, Phillips-Perron, and Choi-Phillips (1991) have trouble distinguishing unit
root processes with substantial short-run dynamics fiom trend stationary alternatives.

Conversely, a more recent approach is to test the null hypothesis of stationarity
against the alternative of a unit root. Tests of the null of stationarity have been
suggested by Park and Choi (1988), Kwiatkowski, Phillips, Schmidt, and Shin (1992)
(hereafter, KPSS), Saikkonen and Luukkonen (1993) and Leybourne and McCabe
(1994). In this chapter we will consider the KPSS test, which is a test of the null
hypothesis of stationarity around a deterministic trend, and which controls for short-
run dynamics using a non-parametric correction similar to those used by Phillips and

Perron (1988) or Schmidt and Phillips (1992). Since many simulation results show

15
that the traditional Dickey-Fuller tests are not reliable in the presence of MA errors

whose coefficient is not close to zero [for details see Agiakloglou and Newbold
(1992), Schwert (1989), Pantula (1991)], Saikkonen and Luukkonen (1993) and
Leyboume and McCabe (1994) suggest tests of the stationary null hypothesis that are
similar to KPSS, but which differ from KPSS in the way they deal with
autocorrelation under the null hypothesis.

The asymptotic analysis of the Dickey-Fuller type unit root tests, including
those extended versions which allow error autocorrelation, shows that those tests are
consistent against stationary alternatives. Also, the KPSS stationarity test is
consistent against unit root alternatives. Although the KPSS test was originally
intended as a test of the null of stationarity against the unit root alternative, it can also
be used as a test of the unit root null against the alternative of stationarity. This has
been suggested by Shin and Schmidt (1992) and Stock (1990). Shin and Schmidt
(1992) show that the KPSS unit root test is consistent against the alternative
hypothesis of stationarity.

A common empirical puzzle is what to conclude when one rejects both the null
of a unit root (e.g., using the Dickey-Fuller tests) and the null of stationarity (e.g.,
using the KPSS test). To understand this outcome, suppose that z. (t = 1, 2, ...) is the

series in question and that Z. is its cumulation (partial sum), i.e.,

t
Z; : ZZj .
j=l

16
Then we follow Lee and Schmidt (1995) in saying that z. is a short memog process if

it satisfies the following two requirements (Assmtion B):

(B1) 02 =1imT_,,,o T'1E(Z%~) exists and is non-zero.
(132) W 6 [0,1], T‘1’22[,T]:> oW(r).

Here [rT] denotes the integer part of rT, => denotes weak convergence, and W(r) is
the standard Wiener process (Brownian motion). The concept of short memory is
important because the asymptotic analysis of the KPSS test actually assumes that
under the null the series is short memory, and the asymptotic analysis of unit root
tests actually assumes that under the null the first difference of the series is short
memory. Thus we can rationalize rejections of both null hypotheses by postulating
series that are not short memory either in levels or in first differences.

These arguments lead to the consideration of long memory processes that are
more persistent than a short memory process, but less persistent than a unit root
process. Accordingly, they are not short memory either in levels or in first
differences. The consideration of such long memory time series has mostly taken
place in the physical sciences. They have been applied extensively in hydrology
(Hurst, 1951, 1956) and have also been used to model data on temperatures and
growth of tree rings (Seater, 1993). The Beveridge wheat price index from 1500
through 1869 (Beveridge, 1921) and US. monthly consumer price index inflation
rates are examples of economic data that exhibit typical long memory features. There

are also studies of long memory in a spatial context; e. g., Whittle (1956) and Beran

17

(1992). A good survey of long memory from the point of view of economics and
econometrics is given by Baillie (1995).

We will use the fiactionally integrated process defined by Granger (1980),
Granger and Joyeux (1980) and Hosking (1981), and considered by Lee and Schmidt

(1995), which is introduced in Chapter 1:

(1) (l—L)d}’t =3t
where L is the lag operator, (1 is the differencing parameter and {8.} is a short memory
process with zero mean and finite variance of.

There has been some recent research on tests related to the fractionally
integrated long memory process. Lo (1991) finds that his “rescaled range” test, for
which the null hypothesis is short memory, is consistent against I(d) processes with
de(-1/2, 1/2). Cheung (1993) investigated the finite sample performance of the GPH
test, the modified rescaled range test and two LM type tests of the null of short
memory against the alternative of fi'actional integration. Lee and Schmidt (1995)
show that the KPSS “stationarity” test is actually a test of the null hypothesis of short
memory, and that it is consistent against stationary long memory alternatives (I(d) for
-l/2< d <1/2 and d¢0). They also provide simulation results on the power of the
KPSS test. They found the power of the KPSS short memory test in finite samples to
be comparable to that of Lo’s rescaled range test. Their results suggest that the
KPSS test can be used to distinguish a short memory and stationary long memory

processes but a rather large sample size is required to do so reliably.

18

There are several studies on the power of unit root tests against fractionally
integrated alternatives. Diebold and Rudebusch (1991) give Monte Carlo evidence of
the low power of the Dickey-Fuller test against fractionally integrated alternatives
with d >1/2; that is, nonstationary long memory alternatives. Sowell (1990) derives
the asymptotic distribution of the Dickey-Fuller tests under the hypothesis of an I(d)
process with 1/2< d <3/2 and shows the consistency of these tests against
nonstationary long memory alternatives. Hassler and Wolters (1994) show that the
Dickey-Fuller type tests, including the Said-Dickey and Phillips-Perron extensions,
have low power in finite samples against I(d) alternatives with 0< d <1, and especially
that the augmented Dickey-Fuller test works poorly.

The purpose of this chapter is to investigate whether the KPSS test is useful in
distinguishing short memory, long memory and unit root processes. Specifically, we
want to ask whether the KPSS test can distinguish the following four types of
processes: (i) short memory ((1 =0); (ii) stationary long memory (|d| <1/2, d¢0); (iii)
nonstationary long memory (l/2< (1 <1); and (iv) unit root ((1 =1). Asymptotics for
the KPSS statistic are previously known for cases (i), (ii) and (iv), but not for (iii).
Therefore we need to derive the asymptotic distribution of the KPSS statistic when
1/2< (1 <1.

In the following it will be shown that the asymptotic distribution of the KPSS
statistic in the case of a nonstationary long memory process (1/2< (1 <1) is difierent
from the other cases, but its order in probability is the same as in the case of a unit

root. Therefore, the KPSS unit root test is inconsistent against nonstationary long

19
memory alternatives. More generally, the KPSS test can not consistently distinguish a

unit root fi'om a nonstationary long memory process. Using the KPSS statistic we can
only distinguish consistently between the following three cases: (i) short memory; (ii)
stationary long memory; and (iii) either nonstationary long memory or unit root.

Some Monte Carlo evidence on finite sample power is also provided. It is

generally in agreement with the asymptotic results.

2. THEORETICAL RESULTS
A. The KPSS Test Under Short Memory and Unit Root
We consider the data generating process:
(2) yt=d>+§t+et,t=l, 2, ..,,T
where {y.} is the observed series and {8.} is the deviation from deterministic linear
trend. Let e. be the residuals from a regression of y. on intercept and trend (t), and let

S. be the partial sum of the e.:
t
St: 261' .
j=1

Let 0'2 be the long-run variance of the at, as in (B1) above and let 52(8) be the

Newey-West estimator of oz:

(3) s2(e) = liez +32!st 3) i6 e
T t ’ t t-S'

t=1 T s=l t=s+l

20

Here w(s, K) = 1-€—:—1, and E is chosen so that Z —->oo but €/T—>0 as T—-)oo. We will
later also consider the case that 2 =0, in which case the second term on the right hand

T
side of (3) is set to zero and s2 (O) = %Ze.2 .
t=1

The KPSS statistic is then defined as:

T
T4283
4 " g =__ti_
( ) n.( ) 93(3)

The KPSS statistic 1‘1”“) is defined similarly except that we set i=0 in (2), which

implies use of the residuals e. = yt — y in defining S. and 52(6).

Under the hypothesis that a. is a short-memory process, KPSS show that

T 1
T'ZZsf => jv2(t)2dt,
t=1 0

where V2(r) is a second-level Brownian bridge, as defined by KPSS, equation (16).

Also s2(€) is a consistent estimator of 02. Therefore,

Mt): j;V2(f)2dI-
Similar statements hold for 1111(3), with V2(r) replaced by the standard Brownian
bridge, V1(r)=W(r)-rW(l). For the purpose of the present chapter, the important
result is that 131(6) and 1].,(6) are 0,,(1) when a. is short memory.

Next consider the case that e. is a unit root process, in the sense that As. is

short-memory. In this case KPSS show that

21

T l a 2
(5) T428.2 :>oZI[JW*(s)ds] da,
t=1 o o

where W*(s) is a demeaned and detrended Wiener process, as defined in Park and

Phillips (1988, p.474), and 02 is the long run variance of Ae.. Furthermore,

 

82(3) 1 1t
(6) —€—T—:>oz_([W (s)2ds.
This implies that
l a 2
I(IWYSXIS] da
(7) WWW) => ° °, .

IW‘(S)2ds
0

Therefore fi1(€) is 01) (T/ 8) when a. is a unit root process. Ifwe set §=0 in (2), then
fin“) is also Op (T/ E): in fact, we have the same result as in (7) except that W*(s) is

replaced by the demeaned Brownian motion, \_lV(s):
1
Ms) = W(s) [wean
o

The KPSS unit root test suggested by Shin and Schmidt (1992) sets I? = 0,
since the distribution in (7) is independent of the nuisance parameter 02 for all values

of E , including 6 = 0, under the unit root hypothesis. Then T' l{11(0) has the same
distribution as on the right hand side of (7) above, and 131(0) is OP(T) under the

hypothesis that a. is a unit root process.

22
These results are easy to summarize. (1) When 6 = 0, fi,(0) and 1],,(0) are

0.,(1) if a. is short memory and OP(T) if a. has a unit root. (2) If I! -—> 00 but Z/T -—> 0
as T—)oo, {11(3) and 1],,(2) are 09(1) ifs. is short memory and Op(T/€) ifs. has a

unit root. Thus, in either case, the KPSS statistic distinguishes consistently (correctly

with probability one as T—>oo) between short memory and unit root processes.

B. Asymptotics and Consistency of the KPSS Unit Root Test Under I(d)

First we will show the consistency of the lower tail KPSS unit root test against
the stationary long memory alternative hypothesis (-l/2< d <1/2). Thus we suppose
that (1— L)dat =ut, with -1/2< d <1/2, and with Assumption B satisfied. Under
these assumptions, Lee and Schmidt (1995) derive the asymptotic distribution of the
KPSS statistics. In the level stationary case (e. = y. -y), from their Lemma 1,

Theorem 1 and Theorem 3:

T 1
(8A) T‘(Zd+‘>2s3 2 to: ] Bd(r)2dr, where 13,.(r) = Wd(r) - rWd(l);
t=1 0
r(1— 2d)
(8B) 52(0)—p——)o2 =Var(e )=oZ———— (8:0);
. ‘ “ {1"(1— <1)}2
82(3) 2
(8C) “ET—L)(Dd (€—)oo but €/T—>0 as T900).

Therefore, the asymptotic distributions of the KPSS statistics in the level stationary

case, when a. is I(d) with -1/2< d <l/2, are as follows:

23

21
(9A) fitttom: ng Bdtrfdr «=0)
680

(9B) (— $2“ (2):]Bd(r)2dr (€—)oobut€/T—)OaST—)oo)
0

Furthermore, f], has the same orders in probability as f1“, and its asymptotic

distribution is exactly the same as in (9) except that Bd(r) should be replaced by

Vcl (r), defined as:
(10) v,(r) = w,(r) + (21‘ - 3t2 )w,(1) + (—6r + 6r2 )jw,(s)ds.

Thus the KPSS statistics 1A1,1 and fit are OP(T2d) for 8 = 0 and 0.,( (T/€)2d) for

8 -—) 00 under the stationary long memory alternative, while they are respectively

Op(T) and Op( T/€ ) under the null hypothesis of a unit root.

This implies the following result.
THEOREM 1:
Suppose that a. is I(d) with de(-1/2, 1/2) and Assumption B is satisfied. Then

1 A ..
gum—ho, %n.(0)-P—>0 (2:0)

6 . (Z ..
¥np(€)—P—)O, ¥n1(€)——>O (£—>oobut €/T—>0 as T—->oo)

Proof: The KPSS test statistics (l,/T)fi,l and (€/T)f1,1 (also, (l/T)1':|t and (€/T)fi,)

are each Op(T2d'1) and O..( (T/K) 2d"), and 2d is less than 1 because |d|<1/2. I

24

Theorem 1 implies that the lower tail KPSS unit root test is consistent against
the stationary long memory alternative hypothesis. However, as d approaches 1/2, the
order in probability of the KPSS statistic under the I(d) alternative approaches the
same order in probability as under the unit root null hypothesis. This suggests that
when dis close to 1/2 the power of KPSS unit root test would be small. There is also
an issue of the continuity of the power of the KPSS test against I(d) alternatives as d
——)1/2.

We now turn to the main theoretical contribution of this chapter, which is the
derivation of the asymptotic distribution of the KPSS statistics when a. is a
nonstationary long memory process. Thus we wish to consider the case that a. is I(d)
with l/2< d <3/2.

Define d* =d-l, so that As. is I(d*) with Id“ I <1/2; that is, As. is a stationary
long memory process. We assume that Assumption A in Chapter 1 holds with v. =
As.. Then a. is the cumulation of the stationary I(d“) variables As., and we have the
invariance principle:

8 T
(11) fiaodmpm.

Note that this is really the same invariance principle as equation (27) in Chapter 1,
with d* replacing (1 because 1/2< d <3/2 and |d*| <l/2.

We will first consider the fiLl test. Thus we consider e. =yt —y=s, -E and

t
St = Zej . Then we can derive the following results.
j=1

25

LEMMA 1
Suppose (I — L)d a. = ut for l/2< d <3/2, d*=d-l, and A8. satisfies Assumption

A in Chapter 1. Then

[ 'Tl

. 1
(l) Td*+3/2t_ get :> mdeIWda(a)da,
0

(ii) T “13/2 81“] :wdsjwda(a)da, where Wd___.t_(a)= Wde(a)— JWda(b)db;
o

l r

2
T
(iii) $28.2 amiaJ‘thuMa] dr
t=1

00

Proof: See Appendix. I

THEOREM 2:

Under the same assumptions as in LEMMA 1,
. 1
(I) When 8= 0, then Tl+ +2d‘ ——S 2(0) => (Ode {dee(a)2 d8}

(ii) When 6 —>oo and K /T—)0 as T—)oo, then

6—:—_21£:‘)P =9 (Ode {IWde(a)2 (18}.

Proof: See Appendix. I

Then we can prove the following theorem.

26
THEOREM 3:

Under the same assumptions as in LEMMA 1,

(no).

1

 

 

 

 

e o o _
¥ p(0):> l (for 6—0)
l‘lnmzda
o
l r 2
, l llamda a.
-T-fi..(€)=> ° ‘1 (when €—>oo and €/T—>0 as T—)oo)
{Insets}
o
(2d*+4) T 2 -(2d*+4) T 2
T_ S T S
Proof: Since if] (0): E t and if] (E)= E t the
' T ” T-(2d*+l)82(0) T 1* €T’<2d*+l)s2(€)’

asymptotic distribution of the numerator is given by part (iii) of LEMMA 1 and that of

each denominator is given by part (i) and (ii) of THEOREM 2. I

The analysis of f], is very similar. We just need the generalization of LEMMA

1 for the case of the residuals from OLS of y. on constant and t, t = l, 2, ..., T.
LEMMA 3: Let e. be the residuals from an OLS regression of y. on (1, t),

t = 1, 2, ..., T. Then, under the same assumptions as in LEMMA 1,

l'
T—(d*+3/2)S[fl~] :> mdsIWJa(a)da ,
0

27

l l
where W;a(a) = de(a) + (63 — 4)dee(b)db + (—123 + 6)! des (b)db .
0 0

Proof: See Appendix. I

Given LEMMA 3, it is easy to establish the same asymptotic results for the

KPSS fit statistic under the nonstationary long memory process as are given for fin

in THEOREM 2 and 3. All that is necessary is to replace the demeaned fi'actional

Brownian motion, &(a) , with the demeaned and detrended fi'actional Brownian

motion, W5. (a), in THEOREM 2 and 3.

Those theorems have several interesting implications. First, even though the
KPSS unit root test is consistent against stationary long memory alternatives, I(d) for
-1/2< d <1/2, the KPSS unit root test is not consistent against nonstationary long
memory alternatives, I(d) for 1/2< d <3/2, because the KPSS statistics have the same
orders in probability under both the null and alternative hypothesis. This is the main
theoretical result of this chapter. Second, Lee and Schmidt (1995) show that the
KPSS short memory test is consistent against a stationary long memory process (-1/2
< d <l/2), and here we can now see that it is also consistent against a nonstationary
long memory process (1/2< d <3/2). Below we will show higher power in finite
samples against nonstationary long memory alternatives than against stationary long
memory alternatives, as we should expect. Third, under the hypothesis of a non-
stationary long memory process, the orders in probability of the KPSS statistics are

independent of the value of (1, even although the form of their asymptotic distributions

28

are affected by the value of d. This is in contrast to the case of a stationary long
memory process, where both the order in probability and the form of the asymptotic
distribution depends on (1. Also by way of contrast, the order in probability of the
Dickey-Fuller statistics depends on d for (1 <1 but not for 1< d <3/2; see Sowell

(1990)

3. SIMULATION RESULTS

In this section we provide simulation evidence on the power of the KPSS
stationarity (short memory) and unit root tests. The computations are done in
FORTRAN using the normal random number generator GASDEV/RAN3 of Press,
Flannery, Teukolsky and Vetterling (1989), as in Lee and Schmidt (1995). The data
on an I(d) process for d <l/2 are generated using the Levinson algorithm [Levinson
(1947), Durbin (1960), Whittle (1963), Brockwell and Davis (1991)]. For a
nonstationary long memory process (1/2< d <3/2) the data are generated by
cumulating I(d*) random variates, where d* =d-l. Given the I(d) process 8., data on
the observable series y. are generated according to equation (2) with d) = g = 0. The
value of d) and i do not matter for the power of any of the tests that we consider,

except that the fin test requires a = 0.

We have considered only positive values of (1 because we are primarily
interested in testing unit roots against nonstationary long memory processes. The lag
truncation parameters are chosen as (’0: 0, €4= integer [4(T/100)“4], and £12:

integer [ma/100)“) as in Schwert (1989), KPSS (1992) and Lee and Schmidt

29
(1995). We consider sample sizes 50, 150, 250, 500 and 1000, and the number of

iterations is 10000. All of our tests are based on the 5% significance level.

Table 2.1 gives the powers of the 5% upper tail KPSS short memory tests against
the alternatives d =0.0, .1, .2, ..., .9, 1.0, and also (1 =.45 and .499. These are similar
to the values considered by Lee and Schmidt (1995), except that we add some cases
with (1 >1. Where they overlap, our results are very similar to those of Lee and
Schmidt. There are no surprise in these results, so we will not discuss them in detail.
Power increases with T for fixed d or with d for fixed T.

Table 2.2-1 gives the power of the lower-tail KPSS unit root test against I(d) for
05 d <1/2, and Table 2.3-1 does the same for the two-tailed KPSS unit root test.
Basically these results are as we would expect from our asymptotics and from the
previous limited simulations of Shin and Schmidt (1992). (i) The lower tail tests are
more powerful than the corresponding two-tailed tests. (ii) For a given (1, power
increases with T. This reflects the consistency of the tests against stationary long
memory alternatives. (iii) Power is largest when 6= 0 and smallest when 6: 6 12.
(There are a few exceptions, for small values of T, due to large size distortions.)
Again, this is consistent with the relevant asymptotics, which indicate that power
depends on 6/T, even asymptotically, for d <1/2. (iv) Power is larger when d is

farther fiom unity. (v) The power of flu and fit are similar.

Table 2.2-2 gives the power of the lower tail KPSS unit root test against I(d)
processes with l/ZS d <3/2, while Table 2.3-2 does the same for the two tailed test.

The most important result is that, with d fixed, power does not approach one as T

30

increases. This is a reflection of our theoretical result that the KPSS unit root test is
not consistent against nonstationary long memory processes. For example, for (1 =7
and 6 = 0, and for the one-tailed test, power grows from .169 with T = 50 to only
.256 with T = 1000, and would not be expected to approach one even for arbitrarily
large values of T.

Some other results in Tables 2.2-2 and 2.3-2 are as follows. (i) For the lower tail
test, power is always lower when 6 is larger, and is very small for 612 for T S 250.
This is due to the large size distortion of the test (too few rejections) for T S 250.

For example, the size of the lower tail test based on fiu(612) is zero for T = 50, and

still only .026 for T = 250. However, for the two tail test, power first decreases as
the number of lags grows (64 ), and then increases with more lags (61.2). Again, this

is due to large size distortions in the two tail test. (ii) The lower tail fin and f]. tests

have similar powers against I(d) for any d in the range (1/2, 3/2). However, the two
tail tests have similar powers only against I(d) with (1 <1. If d is greater than 1 there

is a distinct difference in the powers of two statistics. The flu test is generally more

powerful. (iii) For a given sample size and number of lags, power increases

monotonically with ll — d] for both the lower and two tail tests for (1 <1, and the lower

tail test has little power against (1 >1. The power of two tail test is asymmetric around
d =1, as in Diebold and Rudebusch (1991) and Lee (1994).

Lastly, our results can be compared with the previous results of other authors
for Dickey-Fuller type unit root tests against I(d) alternatives. These comparisons are

given in Table 2.4. Diebold and Rudebusch (1991) show that Dickey-Fuller test is not

3 1
very powerful against I(d) alternatives for d >0.6. This is despite the fact that Sowell

has proved that the test is consistent against such alternatives. Lee (1994) gives the
power of Dickey-Fuller tests against I(d), and finds it to be small for d >1/2;
essentially, his results are the same as those of Diebold and Rudebusch. He also
found a discontinuity of the power fimction of the simple (no constants and no trends)
Dickey-Fuller tests at d =1/2. Hassler and Wolters (1994) show that the Phillips-
Perron and Dickey-Fuller tests have similar power against I(d) processes but the
augmented Dickey-Fuller test is not powerful; they argue that it is inconsistent. Our
results are not directly comparable with the others because our data is demeaned
while the others’ are not. However, two statements seem correct. First, the KPSS
unit root test has lower power than Dickey-Fuller tests (except the augmented
Dickey-Fuller tests) against nonstationary long memory processes. Second, the
unsurprising implication of all of these results is that it is very diflicult to distinguish
between a unit root and nonstationary long memory. This is unfortunate, because
these two types of series differ in fundamental ways, notably their degree of mean

reversion.

4. CONCLUSION

In this chapter we have asked whether the KPSS unit root test can be used to
distinguish long memory processes from unit root processes. We have shown that the
KPSS unit root test is consistent against stationary long memory alternatives, namely

I(d) processes for de(-l/2, 1/2); but it is not consistent against nonstationary long

32
memory alternatives, I(d) for de(1/2, 3/2). This implies that the KPSS statistic can

not distinguish nonstationary long memory processes from unit root processes, even
though it can consistently distinguish between short memory processes, stationag
long memory processes, and nonstationary processes. Also, we have provided the
simulation results on power in finite samples. These support the relevance of our
asymptotic results.

Dickey-Fuller tests can consistently distinguish a unit root fi'om an I(d) process
with l/2< <1 <1, but not from an I(d) process with 1< d <3/2; see Sowell (1990). Thus
distinguishing a unit root from nonstationary long memory is a difficult and not

completely solved problem that is worthy of further attention.

33

TABLE 2.1

Power of KPSS Short Memory Test against I(d), de [0.0, 1.5)

 

fip Test

fit Test

 

50 100 150 250

500 1000

50

100

150 250

500 1000

 

 

d=0.0
60
64
612
=.l
60
64
612

60
64
212

60
64
612

60
64
612
d=.45
60
64
612
d=.49
60
64
612
d=.499
60
64
612
=.5
60
64
612

.044
.036
.012

.124
.072
.020

.240
.123
.032

.387
.193
.047

.543
.278
.072

.613
.320
.088

.664
.357
.101

.659
.352
.095

.667
.356
.101

 

.046
.044
.041

.053
.047
.033

.050
.046
.039

.165
.100
.055

.195
.119
.075

.216
.131
.085

.356
.182
.093

.402
.221
.121

.485
.265
.158

.542
.281
.139

.631
.351
.193

.730
.413
.251

.707
.384
.198

.801
.468
.255

.884
.549
.329

.773
.43 l
.224

.858
.531
.297

.930
.621
.381

.833
.466
.245

.893
.572
.324

.951
.669
.419

.826
.479
.250

.902
.589
.336

.959
.663
.416

.837
.476
.254

.901
.588
.333

.960
.680
.433

.052
.051
.049

.262
.169
.117

.592
.348
.211

.845
.555
.343

.958
.714
.455

.981
.775
.515

.989
.821
.556

.990
.819
.559

.991
.826
.570

.053
.051
.048

.323
.195
.141

.693
.418
.282

.928
.636
.425

.990
.797
.576

.997
.850
.645

.999
.887
.680

.999
.850
.697

.999
.898
.696

 

.051
.041
.044

.140
.071
.049

.267
.121
.060

.418
.168
.064

.574
.235
.072

.639
.259
.077

.687
.293
.078

.700
.289
.083

.704
.306
.088

.052
.045
.034

.185
.098
.055

.380
.166
.086

.608
.267
.115

.775
.362
.154

.837
.417
.168

.880
.467
.192

.885
.470
.192

.888
.465
.187

.052
.048
.040

.215
.122
.071

.461
.224
.113

.714
.361
.159

.870
.490
.219

.921
.559
.257

.944
.599
.267

.949
.606
.275

.951
.616
.278

.053
.053
.044

.265
.145
.092

.577
.280
.149

.829
.444
.226

.941
.600
.316

.972
.661
.352

.986
.717
.389

.988
.721
.388

.984
.721
.392

.052
.050
.048

.331
.196
.121

.722
.407
.218

.932
.625
.345

.991
.794
.476

.051
.050
.050

.409
.226
.152

.839
.488
.295

.983
.735
.473

.999
.877
.626

.997 1.000
.848 .919
.536 .685

.999 1.000
.886 .945
.578 .737

.999 1.000
.893 .947
.597 .742

.999 1.000
.891
.593 .748

.951

 

 

34

TABLE 2.1, CONTINUED

 

fin Test

fi, Test

 

50

100 150 250 5001000

50

100 150 250 5001000

 

 

d=.51
60
64
612

60
64
612

60
64
£12

60

64

612

=9

60

64

612
d=.95

60

64

612
d=.99

60

64

612
d=l.0

60

64

612
d=l.l

60

64

612

 

.688
.370
.103

.778
.437
.134

.848
.507
.171

.905
.586
.229

.935
.652
.286

.946
.676
.310

.960
.701
.341

.959
.712
.346

.975
.759
.421

.845
.493
.268

.908
.560
.315

.952
.643
.384

.978
.714
.455

.990
.776
.523

.992
.800
.548

.994
.823
.574

.994
.821
.583

.910
.602
.343

.962
.690
.439

.992
.833
.579

.999
.905
.713

.961
.689
.414

.987
.771
.509

.999 1.000
.900 .951
.661 .791

.981
.766
.491

.997 1.000 1.000
.835 .946 .975
.583 .737 .855

.992
.823
.553

.999 1.000 1.000
.891 .967 .991
.644 .806 .907

.996 1.000 1.000 1.000
.865 .928 .984 .995
.609 .713 .861 .938

.998 1.000 1.000 1.000
.892 .942 .990 .997
.636 .739 .884 .951

.999 1.000 1.000 1.000
.906 .948 .991 .998
.666 .751 .892 .962

.999 1.000 1.000 1.000
.913 .954 .992 .998
.669 .766 .898 .960

.997 1.000 1.000 1.000 1.000
.871
.648 .732 .808 .933 .974

.934 .967 .995 .999

 

.711
.315
.086

.807
.379
.791

.872
.450
.121

.924
.511
.131

.953
.574
.150

.967
.606
. 169

.971
.623
.173

.973
.628
.177

.987
.685
.205

.901
.483
.196

.951
.556
.101

.976
.645
.283

.992
.714
.335

.954
.617
.282

.990
.723
.410

.999 1.000
.898 .953
.604 .752

.983
.709
.234

.998 1.000 1.000
.812 .949 .984
.340 .484 .681

.995 1.000 1.000 1.000
.792 .882 .978 .994
.392 .564 .776 .897

.999 1.000 1.000 1.000
.849 .928 .990 .998
.461 .636 .825 .936

.997 1.000 1.000 1.000 1.000
.771
.365 .514 .692 .876 .963

.892 .950 .9951.000

.998 1.000 1.000 1.000 1.000
.810 .908 .961
.400 .545 .717 .896 .972

.998 1.000

.999 1.000 1.000 1.000 1.000
.814 .922 .969 .998 1.000
.414 .561

.732 .905 .975

.999 1.000 1.000 1.000 1.000
.824 .925 .970 .998 1.000
.421

.570 .746 .911 .977

.999 1.000 1.000 1.000 1.000
.859 .946 .982 .999 1.000
.451

.616 .783 .933 .985

 

 

35

TABLE 2.1, CONTINUED

 

fin Test

r], Test

 

50 100 150 250 5001000

50 100 150 250 5001000

 

d=l.2
60
64
612
d=l.3
6O
64
612
d=l.4
60
64
612
d=l.45
60
64
612
d=l.499
60
64
612

.810

.851

.985 .999 1.000 1.000 1.000 1.000
.903 .960 .982 .9991.000
.504 .706 .784 .854 .952 .985
.991 1.000 1.000 1.000 1.000 1.000
.932 .970 .988 .9991.000
.598 .771 .828 .891 .967 .991
.995 1.000 1.000 1.000 1.000 1.000
.902 .958 .983 .994 1.000 1.000

.719 .846 .887 .931 .982 .995

.997 1.000 1.000 1.000 1.000 1.000
.933 .972 .991 .995 1.000 1.000
.805 .887 .925 .954 .989 .997

1.000 1.000 1.000 1.000 1.000 1.000

.991
.975

.997 .999 1.000 1.000 1.000
.988 .991 .993 9981.000

 

.991 1.000 1.000 1.000 1.000 1.000
.685
.205 .451

.859 .946 .982 .9991.000
.616 .783 .933 .985

.996 1.000 1.000 1.000 1.000 1.000
.768 .917 .971
.265 .561

.992 1.000 1.000
.707 .856 .968 .995

.996 1.000 1.000 1.000 1.000 1.000
.809 .939 .983 .995 1.000 1.000
.304 .598 .744 .881

.979 .997

.998 1.000 1.000 1.000 1.000 1.000
.809 .939 .983 .995 1.000 1.000
.304 .598 .744 .881

.979 .997

.999 1.000 1.000 1.000 1.000 1.000
.834 .948 .986 .996 1.000 1.000
.353 .629 .774 .903 .984 .999

 

 

 

 

36

TABLE 2.2-1

Power of KPSS Lower Tail Unit Root Test against I(d), de [0.0, 1/2)

 

fin Test

it, Test

 

50

100

150 250 500

1000

50

100 150 250 500

1000

 

 

d=01)
60
64
612
=a1
60
64
612

60
64
612

60
64
612

60
64
612
(Fa45
60
64
612
(Pa49
60
64
612
d=.499
60
64
612

 

.964
.623
.000

.890
.514
.000

.780
.425
.000

.640
.338
.000

.486
.260
.000

.412
.226
.000

.361
.201
.000

.365
.205
.000

.976
.695
.114

.896
.553
.087

.751
.449
.071

.579
.351
.051

.502
.304
.045

.435
.269
.041

.424
.266
.037

.996 L
.799
.156

000 L000 L000

.915 .964 .998
.395 .617 .851

.993 .999 L000
.816 .886 .975
.305 .504 .735

.945 .977 .997
.694 .761 .897
.234 .390 .602

.804 .867 .934
.558 .615 .748
.176 .294 .459

.637 .694 .757
.435 .478 .591
.127 .223 .346

.537 .587 .647
.372 .404 .509
.111 .182 .298

.474 .498 .554
.334 .358 .448
.102 .162 .260

.451 .503 .548
.319 .362 .443
.095 .161 .256

L000
L000
.954

L000
.989
.865

L000
.929
.725

.963
.798
.585

.806
.622
.432

.690
.525
.362

.608
.477
.328

.570
.452
.312

 

.969
.286
.000

.903
.222
.000

.790
.170
.000

.656
.124
.000

.499
.091
.000

.435
.083
.000

.380
.073
.000

.374
.069
.000

.831 .890 .944 .982
.344 .537 .630 .824
.000 .006 .119 .386

.666 .730 .798 .877
.261 .406 .477 .658
.000 .005 .085 .263

.572 .617 .693 .777
.223 .343 .414 .569
.000 .004 .070 .225

.488 .551 .608 .680
.191 .307 .355 .502
.000 .004 .057 .191

.478 .533 .589 .661
.182 .297 .351 .481
.000 .004 .052 .186

.999 L000 L000 L000 L000
.708 .910 .969 L000 L000
.000 .013 .333 .815

.964

.990 .999 L000 L000 L000
.585 .808 .898 .990
.000 .010 .248 .667

.997
.878

.942 .977 .996 L000 L000
.467 .682 .784 .942
.000 .010 .173 .514

.975
.744

.996
.877
.574

.922
.714
.419

.832
.618
.355

.738
.537
.299

.719
.532
.297

 

 

37

TABLE 2.2-2

Power of KPSS Lower Tail Unit Root Test against I(d), de [112, 3/2)

 

fin Test

i], Test

 

50

100

150

250

500

1000

50

100

150

250

500

1000

 

 

612
d=.51

612

612

612

612
d=.95

60

64

612
d=.99

60

64

612
d=l.0

60

64

612

 

.358
.197
.000

.338
.189
.000

.243
. 147
.000

.169
.112
.000

.108
.080
.000

.072
.059
.000

.063
.053
.000

.048
.043
.000

.048
.042
.000

.426
.259
.037

.397
.248
.035

.295
.195
.027

.190
.139
.021

.119
.099
.012

.074
.070
.009

.059
.059
.008

.050
.051
.006

.049
.049
.006

.450
.327
.097

.433
.310
.090

.298
.230
.063

.193
.166
.048

.126
.124
.037

.082
.088
.025

.061
.069
.021

.049
.059
.017

.045
.056
.016

.485
.346
.155

.465
.336
.155

.322
.252
.113

.207
.183
.084

.125
.125
.054

.075
.083
.037

.059
.068
.030

.050
.060
.028

.044
.055
.026

.535
.430
.252

.515
.420
.243

.347
.309
.180

.213
.212
.121

.129
.145
.082

.078
.093
.055

.058
.072
.042

.049
.066
.038

.045
.061
.036

.576
.454
.310

.549
.434
.293

.364
.315
.214

.223
.213
.148

.127
.134
.096

.078
.091
.064

.059
.071
.051

.045
.055
.040

.048
.059
.042

 

.360
.073
.000

.350
.068
.000

.250
.046
.000

.169
.035
.000

.107
.026
.000

.068
.018
.000

.049
.013
.000

.043
.013
.000

.042
.011
.000

.499
. 186
.000

.456
.173
.000

.328
.130
.000

.209
.094
.000

. 122
.062
.000

.073
.041
.000

.056
.034
.000

.044
.028
.000

.040
.026
.000

.528
.291
.004

.510
.286
.004

.351
.209
.003

.212
.138
.002

.130
.100
.001

.076
.066
.001

.058
.055
.000

.044
.045
.001

.042
.043
.000

.590
.347
.057

.566
.339
.054

.379
.242
.038

.226
.160
.021

.130
.104
.015

.075
.072
.010

.060
.060
.007

.046
.049
.006

.044
.047
.007

.647
.481
.182

.634
.469
. 180

.430
.344
.125

.240
.220
.076

.144
.151
.052

.078
.095
.035

.055
.073
.025

.048
.067
.025

.041
.057
.019

.714
.523
.290

.687
.509
.283

.456
.359
.195

.256
.228
.125

.150
.151
.081

.077
.092
.047

.057
.072
.037

.049
.062
.033

.044
.056
.030

 

 

38

TABLE 2.2-2, CONTINUED

 

f1” Test

fit Tea:

 

50

100

150 250

500

1000

50

100

150 250

500 1000

 

 

 

d=l.1
60
64
612
d=lL2
60
64
612
d=lL3
60
64
612
d=131
60
64
612
1&=L45
60
64
612
d=l.499
60
64
612

.030
.029
.000

.019
.021
.000

.011
.013
.000

.006
.007
.005

.004
.006
.000

.000
.001
.000

.028
.032
.005

.017
.022
.003

.008
.012
.001

.005
.007
.001

.004
.005
.001

.000
.000
.000

.029 .030
.038 .037
.010 .017

.017 .015
.025 .021
.006 .010

.012 .010
.019 .015
.006 .007

.006 .004
.009 .008
.002 .003

.003 .003
.005 .006
.001 .002

.001 .000
.001 .001
.000 .000

.028
.039
.023

.015
.024
.014

.010
.016
.009

.005
.009
.004

.003
.005
.003

.000
.000
.000

.029
.040
.028

.017
.023
.017

.010
.013
.010

.005
.008
.005

.003
.005
.003

.000
.001
.000

.022
.006
.000

.014
.005
.000

.008
.004
.000

.006
.003
.000

.004
.002
.000

.003
.002
.000

.026
.020
.000

.015
.013
.000

.009
.009
.000

.004
.005
.000

.003
.004
.000

.003
.004
.000

.026 .024
.030 .029
.000 .005

.013 .013
.018 .019
.000 .003

.010 .009
.015 .012
.000 .002

.005 .005
.008 .009
.000 .001

.004 .004
.006 .007
.000 .001

.004 .004
.007 .006
.000 .001

.025
.040
.013

.015
.025
.009

.008
.015
.005

.005
.009
.003

.005
.009
.003

.003
.007
.002

.025
.036
.019

.013
.021
.011

.008
.013
.007

.005
.008
.004

.003
.006
.003

.003
.004
.002

 

 

 

 

39

TABLE 2.3-1

Power of KPSS Two Tail Unit Root Test against I(d), de [0.0, 1/2)

 

fin Test

fi, Tea:

 

50

100

150 250 500

1000

50

100 150 250 500 1000

 

 

d=01)
60
64
612

60
64
612

60
64
612

60
64
612

60
64
612
(#245
60
64
612
(Fa49
60
64
612
d=.499
60
64
612

 

.919
.456
.011

.808
.358
.017

.672
.276
.027

.518
.211
.042

.367
.150
.063

.301
.130
.019

.254
.108
.089

.257
.112
.084

.989
.679
.034

.940
.551
.024

.817
.421
.020

.649
.323
.017

.468
.239
.013

.391
.199
.014

.326
.169
.014

.318
.170
.014

.999 L000 L000
.843 .916 .991
.228 .469 .753

.975 .997 L000
.709 .800 .938
.164 .353 .608

.887 .946 .985
.571 .647 .819
.116 .256 .467

.712 .788 .877
.427 .493 .639
.082 .176 .332

.520 .586 .656
.312 .359 .471
.059 .125 .231

.426 .463 .537
.260 .291 .396
.049 .100 .194

.360 .390 .446
.231 .255 .335
.042 .089 .161

.335 .391 .434
.216 .251 .327
.040 .084 .163

L000
.998
.903

L000
.971
.771

.998
.862
.610

.932
.706
.452

.719
.508
.321

.582
.413
.255

.497
.362
.223

.465
.344
.211

 

.932
.119
.472

.828
.090
.464

.689
.064
.484

.536
.045
.481

.379
.029
.501

.320
.027
.504

.271
.025
.510

.262
.022
.506

.894 .951 .987
.317 .549 .676
.019 .001 .082

.738 .820 .901
.220 .410 .505
.028 .002 .052

.553 .619 .707
.158 .287 .352
.045 .007 .034

.455 .507 .590
.127 .237 .301
.053 .010 .027

.381 .436 .497
.107 .202 .243
.065 .014 .020

.367 .423 .480
.106 .197 .236
.069 .013 .020

.530

.891
.379

.958
.730
.260

.802
.542

L67

.680
.454
.136

.579
.391
.114

.547
.366
.108

.998 L000 L000 L000 L000
.556 .833 .933 .998 L000
.006 .000 .169 .702 .922

.974 .994 L000 L000.L000
.423 .700 .825 .972 .991
.011 .004 .119 .790
.998 L000
.943
.632

.988
.800
.442

.870
.606
.302

.752
.507
.244

.639
.421
.203

.619
.416
.199

 

 

40

TABLE 2.3-2

Power of KPSS Two Tail Unit Root Test against I(d), de [1I2, 3/2)

 

fin Test

fit Test

 

50

100

150 250 5001000

50

100

150

250 500 1000

 

 

612
d=.51
60
64
612
=.6
60
64
612

60
64
612

60
64
612

60

64

612
d=.95

60

64

612
d=.99

60

64

612
d=l.0

60

64

612

.245
.110
.092

.237
.106
.093

.162
.079
.121

.105
.059
.155

.063
.038
.212

.046
.025
.267

.046
.023
.291

.050
.016
.324

.049
.017
.325

 

.311
.163
.013

.291
.154
.015

.199
.115
.019

.121
.077
.029

.073
.052
.055

.047
.035
.088

.042
.027
.111

.050
.023
.135

.053
.024
.144

.344
.223
.040

.322
.211
.038

.206
.143
.025

.122
.098
.021

.077
.072
.028

.051
.047
.043

.047
.037
.053

.051
.030
.072

.051
.026
.076

.372
.239
.081

.358
.232
.082

.228
.163
.054

.135
.110
.038

.073
.070
.027

.047
.043
.030

.045
.034
.035

.049
.030
.043

.049
.027
.042

.422
.322
.160

.403
.311
.154

.251
.214
.106

.139
.135
.067

.076
.084
.045

.046
.052
.030

.041
.037
.024

.046
.032
.023

.049
.032
.025

.466
.343
.212

.439
.320
.197

.267
.217
.138

.146
.139
.091

.077
.082
.054

.049
.049
.033

.044
.037
.030

.044
.027
.025

.047
.029
.027

 

.257
.026
.517

.251
.021
.521

.164
.012
.538

.105
.009
.573

.064
.007
.581

.046
.004
.604

.044
.003
.617

.047
.002
.622

.047
.002
.625

.364
.097
.066

.347
.098
.072

.233
.065
.091

.138
.046
.122

.076
.028
.156

.050
.017
.191

.047
.013
.219

.047
.013
.233

.047
.009
.239

.415
.193
.011

.403
.188
.015

.258
.133
.025

.144
.080
.039

.079
.052
.068

.051
.033
.094

.048
.026
.110

.048
.021
.127

.045
.018
.133

.480
.243
.022

.459
.236
.022

.283
.157
.016

.153
.095
.020

.078
.057
.027

.053
.038
.048

.048
.030
.060

.047
.022
.074

.051
.022
.076

.548
.369
.107

.526
.357
.104

.325
.245
.067

.166
.144
.037

.088
.091
.028

.051
.055
.030

.047
.038
.036

.050
.035
.040

.047
.029
.043

.613
.414
.191

.582
.397
.191

.354
.265
.122

.183
.155
.072

.097
.093
.046

.051
.048
.035

.046
.038
.034

.049
.033
.038

.053
.030
.043

 

 

41

TABLE 2.3-2, CONTINUED

 

f1" Test

fit Tea:

 

50

100

150 250

500

1000

50

100

150 250

500 1000

 

 

d=l.1
60
64
612
d=lJ2
60
64
612
d=lL3
60
64
612
d=111
60
64
612
d=l.45
60
64
612
d=l.499
60
64
612

 

.082
.012
.403

.146
.006
.487

.249
.003
.582

.409
.003
.708

.556
.002
.796

.936
.000
.973

.086
.014
.213

.154
.009
.304

.248
.003
.415

.417
.002
.574

.551
.002
.698

.931
.000
.958

.084 .086
.017 .017
.129 .083

.150 .150
.011 .010
.213 .148

.252 .248
.008 .005
.321 .251

.417 .413
.003 .002
.500 .417

.554 .560
.001 .002
.628 .573

.928 .932
.000 .000
.942 .934

.085
.019
.047

.146
.010
.086

.250
.006
.175

.414
.003
.332

.553
.002
.477

.931
.000
.917

.081
.018
.042

.147
.011
.081

.253
.005
.166

.414
.003
.316

.561
.002
.462

.934
.000
.913

 

.066
.001
.661

.100
.001
.684

.141
.001
.693

.190
.001
.722

.213
.001
.739

.242
.000
.747

.071
.008
.275

.104
.005
.332

.147
.003
.381

.197
.002
.422

.212
.002
.434

.239
.001
.465

.070 .072
.013 .014
.175 .113

.105 .098
.007 .007
.217 .143

.145 .141
.006 .006
.261 .185

.193 .195
.003 .006
.311 .239

.215 .218
.002 .006
.327 .259

.238 .237
.003 .000
.348 .283

.071
.018
.066

.103
.011
.096

.143
.006
.132

.190
.004
.179

.218
.004
.204

.234
.003
.222

.070
.020
.059

.102
.014
.086

.149
.019
.125

.191
.030
.162

.219
.036
.193

.241
.044
.210

 

 

42
TABLE 2.4

Power Comparison with Dickey-Fuller Type Tests

 

(1) (2) (3) (4)Lowfi1L (5)Two fin

 

 

d T ‘r P 1: P to ADF PP 60 612 60 612
d=.45 100 .88 .86 .88 .87 .999 .111 .999 .502 .045 .391 .014
250 .99 .99 .99 .99 1.00 .336 1.00 .587 .182 .463 .100
=.6 100 .71 .71 .71 .71 .927 .069 .926 .295 .027 .199 .019
250 .90 .90 .90 .90 .999 .186 .996 .322 .113 .228 .054
=.9 100 .10 .10 .09 .09 .138 .038 .136 .074 .009 .047 .088
250 .14 .14 .13 .14 .199 .060 .173 .075 .037 .047 .030
d=l.0 100 .05 .05 .04 .04 .051 .045 .053 .049 .006 .053 .144
250 .06 .05 .05 .05 .046 .045 .050 .044 .026 .049 .042
d=l.3 100 .54 .22 .54 .21 -- -- -- .008 .010 .248 .415
250 .62 .25 .62 .25 -- -- -- .001 .007 .248 .251

 

 

 

 

 

 

 

(1) Lee’s dissertation (1994): two tail D-F test (5%)
(2) Diebold and Rudebusch (1991): two tail D-F test (5%)
(3) Hassler and Wolters (1994): one-sided (5%)
To = t-type simple D-F test
ADF = t-type Augmented D-F test with lags (612)
PP = t-type Phillips-Perron test with lags (612)
(4) KPSS unit root test: lower tail (5%)
(5) KPSS unit root test: two tail (5%)

 

43
APPENDIX

Proof of LEMMA 1: Using equation (1 l),
1 [rT] 1 [rT] 8t r
d*+3/2 Zst = $21 d*+l/2) D md‘lwd'(a)da’
T H = T o

1 l

r
_. z . e—a>=1‘:zjt——)-I—lit 1
Td*+3/2 [rleTtle*+1/2 t T Td*+1/2 T =1 Td*+l/2

r l r l
D (Ode [fwde (a)da -' rIst(a)da] = CDdeI|:wda (a) -’ IWd *(b)db:|da
0 0 0

0

which proves parts (i) and (ii). For part (iii),

t 2 ..
2:54 —_—'SZZ " ——;[W:l :> (0‘11?! IW£(a)da dl' , by (11) 811d the
T t=1 =1 T o 0
continuous mapping theorem. I

Proof of THEOREM 2: For the proof we use the following Lemmas. In each, we

make the same assumptions as in LEMMA 1 of the main text, and results are as T—>oo.

LEMMA 2.1:
1 T
W ZSt-lAst —p—>0 .

t=1

1‘
Proof: Since Za._1As.= —(eT- 80- 2(A8t)2)
t=1

2 2
1_1_ 81 1 1 so
2+—Zst—12d*1Aet= d*+1/2 ““ d*+l/2 l+2d"‘ %Z(A3 alt)
T 2T T 2 T T 2 T

_p_)0

been

where

Proc

sine

bec

usin

(Soy

44

2 T
because [FE—:13] :>m§.Wd.(l)2, [-;:Z(Aet)2:l—£—>yo and so is 0,,(1),
t=1

where y,- is the j-th autocovariance of A81.

LEMMA 2.2:
1 T
71357, E aHAeHs —p—)0 for any nonnegative integer s.

t=1
Proof: LEMMA 2.1 is the special case of 5 =0. For any given positive integer s,
T T s—l T

Zst—IASHS = Zst+s—1A8t+s ’ Z ZA8t+jA8t+s ,
t=1 t=1 j=o t=1

smce 8H = SHH _(A8t+s—1 +Ast+$_2 + ...... + A8,). Then,

1 1 1 s— 1 T
—_Z(8t—2+2d*lA8t+S)= —tz:;(8t+S—1A8t+5)-T2+2d*—Z(TI_~ZA8t+jA3t+s)l+2d*
T T j=0 t=1

_P_)(),

T
because Yflszst+s—lAat+s—p—’o by LEMMA 2.1 and
t=1

 

8-1 8—1
1
P x
Tl+2d‘j_o —Z(% t2::A8t+jA8t+s) 7 T1+2dt [ZYS-j] _) 0:
j=0

 

 

. s-1 8 r l-2d* s F (1* -

i=0 i=0 i=1

 

 

_[ I‘(l—2d*) 02] 1[I‘(1+d*+s)_I‘(l+d*)]
_ I‘(d*)I‘(l—d*) " 2d* F(1—d*+s) F(1-d*) ’

(Sowell, 1990).

45

LEMMA 2.3:
1 5—1 T
772372 ZSHASH ——p—>O for any nonnegaitve integer s.
T '=0t=s+l
. 1 5—1 T l T
Proof. W2 ZeHAeH = 7523; Z[8t—sAst +8t—sA8t—l+ ...... +st_sAst_s+1].
T j=0t=s+l T t=s+l
T
Then, $27127, Z:[t:t_,,Ast +3t—sA3t—1+ ...... +et_sAst_s+2]—L—>O by LEMMA 2.2 and
t=s+l
l T
7:237 Z[e._,Ae,_,,,]—P—>o by LEMMA 2.1.
T t=s+l
LEMMA 2.4:
1 T 2 l 2
W Zetet-s => mdadeJa) d3 .
T t=s+l o
PIOO . W ZetSt-s — :I—‘EIZd—‘l 28t_s +WZ ZSt—s St-j ,
t=s+l t=s+1 j=0t=s+l

because at = 8H — (AeH + Aat_s+1+ ...... +Aet). Also,

2 l
1 T 2 1 T 3t— 2 2
W Est—s =¥ Z (W) :mdajwdda) (18 and
t=s+l t=s+l o
1 8-1 T
W2 ZSHAeH—i—w by LEMMA 2.3.
T j=0t=s+l

LEMMA 2.5:

46

For any nonnegative integer E ,

Z[1+2Z(1-——)]=€ 63—1

2

Proof:

1
€(€+1)

€+__1

 

 

l)[(e+1)+2(e+(e—1)+ ...... +2+1)]= [(e+1)+2—21—e(e+1)]=

«21+

We now will use Lemmas 2.1-2.5 to prove Theorem 2. First consider part (i), with

6 =0. Then,

1 T 1 T 2
s2(0)=—Z(st—§)T= %Zet—[—Ze J and
Tt=l Tt=1

1 1 T s 2 1 T e 2
2 __ __t_ _ _ __t_
T2d‘+l S (O) — TZ(Td‘+l/2) (TZTd‘+1/2]

t=1

1 1 2
3 (Din: de*(a)2d8 —[det(b)db]
0 0

1
= (o (21. {Ififiaf da}
0

2
l

1 1 2 1 1
since I&(a)2da= [[wd.(a)- jwd.(b)db] da= fwd.(a)2da-[[wd.(b)db] .
0 O 0 0

0

For proving part (ii), from (3) and et = 8t — E,

T
s2(€)=%2(et—E)2+.i —Zw(s,e)z(st— eXets— s)
t=1

Ts=l t: s+l

47

1T 1T 2 t 1 T _ _ _2
= —ZStz-(TIT 281) +22W(5,€){¥ 2(8t81_s—818-881—s—8 )}

s=l t=s+]

Thus,

 

T e 8 _s a E 58 _s 52
{321 [(13:sz ' (12:sz — (12:sz +(T2+2d‘]]}
l
= (A)+2§w(s,€) x (B).

Part (A) is the same as s2 (0) above. For fixed 6 , (B) equals

T as l T e ' 1 T - - 2
z W—2 ‘(f )-—z ( f )+(———f )-

t=s+ t=s+ t=s+

 

 

 

1 T 1
From LEMMA 2.4, W 23.5.4, => w§.jwd.(a)2da,
t=s+l o

 

1 T at E _ 1 T at l T at
f 2 Td*+l/2 Td*+l/2 ’ f X T1114112 f 2 d*+l/2

t=s+l t=s+] t=s+l T

 

 

2
l
=m§.[jwd.(b)db] ,
0
ili X 5 H5 llii—s‘ l
d* 1/2 (1* 1/2 (1* 1/2 d"' 112
T t=s+lT + T + Tt=s+-lT + Tt=s+lT +

l 2
=> (031*[det(b)db] ,
0

48

E 2 1 T 8 2 l 2
__ t
0

t=s+l T

1 1 2
ThCl'CfOl'C, (B) :> (0‘2”! Wd.(a)2 — [de*(b)db] d8 .
0 0

Since 6 —>oo and (K /T)—>O, as T—)oo,

l

2
Z—:T—_21£2* => (oat! Wan-(8)2 - [IW‘p (b)db:| da , by the above argument and
o

LEMMA 2.5. I

Proof of LEMMA 3: Let a) and E be the coeflicients of intercept and trend in the OLS

regression of y on (1, t). Then

H =[ T 2‘ Niel
§ —§ 2t th Ztst
and by the same algebra as in Lee and Schmidt (1994) we can show the following:

(lb 4’): (7:: +—'-)28 8t- T2 -§2-Zt8t+op(l), then

A

M = (4 +3) li(__8t_) _Ei(i_§_t__) +_°£(_l)_
Td“+1/2 T T Td‘+1/2 T t=1 T Td"+l/2 Td‘+1/2

t=1

l 1
=> 0) d“ {4! W6. (a)da — 6! aWd. (a)da}

0 0

49

l l a
= wd.{-2jwd.(a)da + 6 j [ j wd.(b)db]da},
0

0 0

l l l a
since Iawd.(a)da = JWd.(a)da—I[de.(b)db]da.
o o o o

, —6 T 12 T
Also, (§—§)=—228t+—;Zt2t+op(l),
T t=1 T t-l

 

 

( T \
,. Zet
£3,451 in_ +12 ii_‘3t_ +551
Td*—l/2 - T Td*+l/2 T t=1TTd'“+l/2 Td*—l/2
\ J

l 1
=> (0 d“ {-6de: (a)da +12IaWd.(a)da}
0 0

l l a
= md.{6IWd.(a)da - 12]. (de*(b)d ]} .
0

0 0

Th _ [YT] . _ [YT] . 1 *
en SITT] — t2::(8t -8t) - get — [rT](¢ — (b) — §[rTK[rT]+1)(E, —§),

1

 

 

 

Ser] -133] 8: _[rT]($-¢]_1[rTl(erl+l)[é-a]

Td*+3/2 — T t=1 Td*+l/2 T Td*+l/2 2 T T Td*—l/2
r l 1

=> (0 d“ IWd.(a)da — 1' 4J'Wd*(a)da — 6". aWd*(a)da
0 0 0

l l
— g 1'2 [-6I Wd. (a)da + 12! aw” (a)da]}
0 0

50

1'
= md.IW;.(a)da ,
0

which is similar to the result of Shin and Schmidt (1992, p.388).

CHAPTER 3

ASYMPTOTIC BIAS OF THE MDE

WHEN SHORT-RUN DYNAMICS ARE IGNORED

51

52
1. INTRODUCTION

Suppose that an observed series {yr} follows an ARFIMA(p,d,q) process:
(1) (1—L)‘y. =2“ wast =9(L)u.,
where ¢(L)=1—¢1L—¢2L2— ...... —¢pr, 9(L) = 1+91L+92L2+ ...... +eqL‘l, all ofthe

roots of ¢(L) and 9(L) lie outside the unit circle, (ML) and 9(L) have no common
roots, and {ut} is white noise. This is the same model and the same notation as in
Chapter 1. When at itself is white noise, y; is a fractionally integrated white noise, or
ARFIMA(O,d,O), process, also called an I(d) process. In the ARFIMA model, the
differencing parameter d determines the long run properties of the series, such as its
persistence and the persistence of its autocorrelations, while 9 and d) influence short-
run dynamics. In this chapter we will consider the estimation of d, with particular
attention to whether we can estimate d separately fiom the ARMA parameters that
determine short-run dynamics.

The first systematic treatment of estimation of d was by Geweke and Porter-
Hudak (1983), hereafter GPH, who suggested a simple semi-parametric two step
procedure for estimating d. Their estimator is based on a spectral regression and

linear filter theory. If {y.} follows the ARFIMA process (1), its spectral density is:

0'2

"2“ . 2 (D -d
fe((°)=§;{4sm (3)} fem),

 

2 .
(2) rye») = %|1— e‘m’

where f8 ((0) is the spectral density of at, which is finite, bounded away from zero and

continuous on the interval [-1t, 11:]. Taking logarithms,

53

2
(3) 1°g{fy((°)}=1°g{%t—fe(0)}- <110g{4sin2 (— 02))}+l og{f —((——0))t:}

Then GPH suggested an OLS regression based on [(0) j), which denotes the
periodogram at the harmonic ordinate, a) j=27tj/T for j = l, ..., m, where T is the

sample size. The number of ordinates used is m = g(T), where g(T) —) 00 but g(T)/T

—-) 0 as T—) 00. The regression model is:
4 1 I — d1 4 ' 2 a)"
() og{(co,-)}—a— og sm (—2—) +vj,
where a = constant and
[(0%)
vJ-=10g{f (00)}.
y J

This implies that E(Vj) = 0, var(vJ-) = 1:2/6, j = l, 2, ..., m, and that cov(v;, Vj) = 0, i¢j.

 

The GPH estimator, say d, is then defined as the OLS estimator of d in (4). GPH

show that d is consistent, for d <0, and Robinson (1990) shows consistency for 05 d

<1/2. Under the further condition limp”, {(log(T)2)/ g(T)} = 0, d is asymptotically

normal. However it is not JT -consistent; asymptotically its variance is of order m",
not 1". The important features of the GPH estimator is that we can disregard the last
term in (3), which involves the unknown short-run dynamics parameters (<b’s and 9’s),
because it is asymptotically nearly constant for sufliciently low fi'equencies.

However, the GPH estimator can be badly biased even for moderately large

sample sizes, especially when there is substantial autocorrelation in the a process.

54

Agiakloglou, Newbold and Wohar (1993) show that the GPH estimator of (1 under
AR(1) or MA(1) errors with quite large short-run dynamics, is seriously biased when
m = T”. They conclude that tests based on the GPH estimator are significantly
misleading and we may need joint estimation of d and short-run dynamics. A further
difficulty is that the sampling distribution of the estimated ARMA parameters after (1
is replaced with the GPH estimator is currently unknown.

Maximum Likelihood Estimation (MLE) under the assumption of normality has
been suggested by a number of authors, apparently starting with Hosking (1984).
Sowell (1992) suggested the exact MLE of the general ARFIMA(p,d,q) process with

normal disturbances. This estimator maximizes the log likelihood function:
T 1 1 , -1
(5) logL = ——log(21t)-— —log|Q| ——Y (2 Y,
2 2 2
where Qij = yli—J'I ( with y,- = j-th order autocovariance of {y,}), and Y is the Txl

vector of observations. In the MLE any ARMA parameters in 0(L) and ¢(L) must be
estimated jointly with d. The exact MLE is computationally dificult, and probably
not feasible for sample sizes larger than 1000 or so, because of the need to invert the
TxT covariance matrix Q. Several approximate MLEs which do not require the
inversion of the covariance matrix 0 have been suggested. A conditional sum-of-
squares estimator (CSS) was proposed by Li and McLeod (1986). It truncates the
infinite sum in (l—L)d to a finite sum for estimation. Chung and Baillie (1993)

investigated the small sample performance of the CSS estimator and showed that for

the I(d) process the CSS estimator is very close to Sowell’s exact MLE, even for

55
T<100. They show that the estimation of the mean can make a considerable
difference to the small sample bias. Fox and Taqqu (1986) and Dalhaus (1989) used
an approximation formula for the spectral density given by Whittle (1951, 1953),
where the autocovariance matrix is diagonalized by transforming the {y} process into
the frequency domain and the asymptotic properties of Toeplitz matrices and an
equicontinuity property of quadratic forms are used to show consistency and

asymptotic normality. The approximate log likelihood is as follows:

 

T—l T—l
(6) logL = Zlog{21t -f(mJ-)}+ :1“
j=l 1:21“ 1')

where I( (01-) is the periodogram and {(0) j) is a spectral density of y as above.

The main focus of this chapter will be on minimum distance estimation (MDE).

Let

I

(7) p: [p1, p2, ..., pn]
be the vector of the first 11 population autocorrelations, and let {5 be the

corresponding vector of estimated (sample) autocorrelations. Tieslau, Schmidt, and

Baillie (1994), hereafter TSB, suggest minimizing the distance between p and I).

More precisely, let
(8) K =[d, 411, ..., ‘11,), 91, ...,Oq]'

so that p depends on 7». TSB suggest minimization of a criterion function

{fa—MM} W {fi-p(}t)}, where W is a positive definite matrix. This estimator is

56

consistent, and it is JT-consistent for -l/2< d <l/4. A similar GMM estimator is
suggested by Dueker and Startz (1992).

Chung and Schmidt (1995) provide a modification of the TSB estimator,
“adjusted MDE” (AMDE), to achieve JT -consistency for -l/2< d <l/2, not just for -

l/2< d <1/4. Define the vector

I

(9) 5 = [5,, 82, 6“]

where 81:1—p1, 82 = p, —p2, ..., 8n = Pn-1“Pn- The information in 5 is the same
as in p and the MDE based on 8 is asymptotically the same as the MDE based on p.
However, Chung and Schmidt suggest an MDE based on (n-l) functions of mics of

elements of 8. More specifically, they consider ratios of the form

= a38/b38, j = 1, 2, ..., n-l,
where a,- and b, are vectors of known constants; they also allow general differentiable
functions of these ratios. Using results from Hosking (1995), they show that such an
MDE, which they call an AMDE, is JT -consistent for -1/2< d <l/2. Its asymptotic
variance does not depend on the choice of a,, bi, nor on which function of the ratios
(10) are taken. For d <1/4, the AMDE is less emcient than TSB’s MDE because it
does not use information on the levels of the 8’s (or p’s).

Now consider the problem of estimating the differencing parameter (1
separately from the ARMA parameters that determine short-run dynamics. Every

element of 8 (or p) depends on a non-trivial way on the ARMA parameters.

However, ratios like 8/qu (or pj/pH) depend only on d, in the limit as j —) 00. This

57
suggests that we could ignore short-run dynamics by considering ratios of sufficiently
high-order autocorrelations, and this is easy to do in the AMDE of Chung and
Schmidt. Let n be the number of 8’s considered, and let I? be the number of elements

of 5 that are not considered, so that we consider only

(11) 511,3 E[5e+1a 8H2, ..., 86+n] -
We will consider the AMDE based on 6a,) , ignoring short-run dynamics; that is, the

AMDE assuming the I(d) or ARFMA(O,d,O) model. When the data are generated by
the ARFIMA(p,d,q) model, the AMDE based on the (0,d,0) model will yield
asymptotically biased (inconsistent) estimates. However, we expect this asymptotic
bias to be small when 6 is large. In this chapter we will calculate this asymptotic bias,
as a function of n, e and the parameters. The idea is to see whether approximately
unbiased estimates of d can be obtained, without the need to model short-run
dynamics. The idea of doing so using high-order autocorrelations obviously is similar
to the GPH idea of using the periodogram at low frequencies. Lobato and Robinson

(1993) used the same idea.

2. MOMENT CONDITIONS FOR MDE
In this section, we give the explicit form of the fimctions of 6 that are used in
the AMDE estimators that we will consider. To do so, we first need to give a little

more detail on the TSB and Chung-Schmidt procedures. Consider the case of an I(d)

58

A

process; that is, a fi'actionally integrated white noise. The MDE of d, say d, is

defined as the value which minimizes the following criterion function:

I

(12) S(d)={F[13]— F[p(d)]} w {ml-How},

where p(d) = [p1(d), p2(d), ..., pn(d)] is a vector of population autocorrelations that
depends on the value of d, p is the corresponding vector of sample autocorrelations,

F is a m-dimensional vector of transformation fimctions of the autocorrelations (m.<_n),
and W is a symmetric and positive-definite weighting matrix. The asymptotically

optimal weighting matrix is as follows:

(13) w = AV(F(p))'l = [P C P1".

Here P = 6F/6p (mxn). C is the asymptotic variance-covariance matrix of the

sample autocorrelations (nxn). For -1/2< d <1/4 it can be defined by Bartlett’s

formula,

00
(14) Ci,j = 203m +Ps_i ‘ZPiPsXPs+j +Ds—j ‘29jps), I: 1, 2, ..., 11,

s=l

from Hosking (1995, 1984) and Brockwell and Davis (1991). Define D = 6p/ 6d
(nxl). Then the asymptotic variance of the MDE d defined in (12) above is:

(15) AV(d) = {[D'P’WPDrlD’P’W(PCP')WPD[D'P’WPD]—l}.

When W = [PCP']_1, this simplifies to

(16) AV(d) = [D’P'(PCP')'1PD]-l.

59

We can note that P cancels if it is nonsingular, which will be the case if n = m and the
elements of F are functionally independent. In this case the transformation F is
asymptotically irrelevant. However, F clearly matters when m < n.

This MDE is similar to the GMM estimator introduced by Hansen (1982). The
GM estimator of a parameter, say [3 (hxl), is based on conditions of the form:

E[gi(y,, [3)] = O, i = 1, 2, ..., k (h S k).
The GM estimator minimizes a criterion function:

min 8(3) = {§(3)'W i(B)}
’1‘
where 81(B)=::tzgi()’u B). Therefore, the above MDE of d is not a GM
=1

estimator because F(p) is not an average, but it still has the basic properties of a

GM estimator.
TSB (1995) introduced the simplest case of the MDE using the trivial

transformation F{p(d)} = p(d) , so that m = n. Then the criterion in equation (12) can

be simplified as:

(17) S(d)={b-p(d)}'w{a—p(d)}.

The optimal weighting matrix is W = C'1 since P = I. The asymptotic distribution of
w/T(d—d), for -1/2< d <1/4, is N(O, [D'CD]'1). For (1 =l/4, d converges to a
normal distribution, but at a rate of (T/lnT)”2; for 1/4< d <1/2, the MDE converges to

a non-normal asymptotic distribution at a rate of T‘l'z‘”. Thus for (1 21/4 convergence

is slower than the usual T”2 rate.

60

These asymptotic results basically depend on the asymptotic distribution of p

(for details, see Hosking (1995)). However, even for de[1/4, 1/2), Hosking (1995)

has shown that the following normalized differences of the sample autocorrelations

are T -consistent and asymptotically normal:

 

(18) fi[5k-Pk _13£"Pe].
I‘m 1‘9:

Therefore, TSB (1995) suggested an MDE based on such normalized differences of

the sample autocorrelations, which is JT -consistent for the whole stationary and

invertible range (-1< d <1/2) of the long-memory process.

Also, Chung and Schmidt (1995) introduced an AMDE that is also JT -
consistent for the entire range -l< d <1/2. The MDE based on the quantity in (18) is

an AMDE. More generally, Chung and Schmidt rely on Hosking’s result that
differences of sample autocovariances are JT -consistent: s/T[(~?i — j? j) — (7i — 71)] is

asymptotically normal with variance given by Hosking (1995, equation (16)). Ratios
of difl‘erences of autocorrelations are fimctions of differences of autocovariances; for
example,(p3 — p2)/(l—p1) = (73 —72)/(yo ~71). With 8 as defined in (9) above,

the ratios r-

J =a35/b38 given in (10) above are ratios of difl‘erences of

autocorrelations and provide the basis for a JT -consistent AMDE.
Chung and Schmidt (1995) provide several different but asymptotically

equivalent forms of the AMDE. They correspond to different choices of the constants

aj, bj, and different functions of the ratios rj. Formally, they also correspond to

61
choices of the function F in (12) above. In all cases, if p (or 8) has n elements, F is of
dimension
(n-l). The formulas given earlier in this section depend on the matrix C, which is
defined only for d <1/4, but Hosking (1995) and Chung and Schmidt (1995) give the
necessary modifications that are well-defined for d in the entire range -1< d <l/2.

We will discuss three different specializations of F. Because we will be
interested in the general ARFIMA(p,d,q) process, we will take the parameter vector
as:

(19) k =[d, 411, ..., op, 91, ..., Oq]' =[d, @']'

as in (8) above, with ®=[¢1,...,¢p,01,...,Gq]’; hence we distinguish the

differencing parameter (1 from the ARMA parameters G).
The GLS estimator introduced by Chung and Schmidt (1995) is based on the

function F 1 defined by:
(20) F}[p(d, ®)]=1-job,-(d, 9), j= (6+1), (6+2), (em).

where

5j(d, (9)—5,41“, ('9)
5N1, ®)+5j+1(d, 9) ’

 

(21) b,-(d, ®)= with 6,-(4, ®)= pj-1(d, (9)—pad, 9).

Thus b M, 6)) =1p,--1(d, @1—2p,(d, ®)+p,-.1(d, @11/1p,-_1(d, ®)-p,-+l(d, on. As
in the previous section, 3 is the lowest-order autocorrelation used in the calculation;

(n+3 +1) is the highest-order autocorrelation used; and estimation is based on n

moment conditions, derived from n ratios of linear combinations of 8M1! as defined

62
in (1 1) above. Chung and Schmidt (1995) consider only the case 3 = 0 (which is well

defined, with the convention p0 = 1). For this pure I(d) process without short-run
dynamics (i.e., 9:0), Fjllp(d, 0)]: 1— j~b ,.(d, O)=d for all j, and the MDE is

expressible as a GLS regression of F1 on a vector of ones. See Chung and Schmidt
(1995) for more detail. They refer to this as the GLS version of the AMDE.

A second possibility is to use the following transformation F2:

(22) F-2[p(d, (wk—fl, j=(e+1),(e+2),...,(e+n).
J-’l(d (’9)

For the pure I(d) process without short-run dynamics we have:

(23) Fj2[p(d)] = 9%:9.

A third possibility is similar to the second, but uses the same denominator for

each ratio. That is, the function F3 is defined by:

3 p(d 9)
(24) Fj,[p(d @)]=p—’—— Md (9) j=<€+1),(e+2),...,(e+n).
pt

For the pure I(d) process without short-run dynamics we have:

1 d
Fflpmfl: k§1((k— 3))'

There is presumably an emciency loss from ignoring low-order

autocorrelations, so that the asymptotic variances of the AMDE or the MDE grow

with K . However, larger values of I? are useful to avoid the asymptotic bias that

63
results fi'om ignoring or misspecifying short-run dynamics. We now turn to the

calculation of this asymptotic bias.

3. CALCULATION OF ASYMPTOTIC BIAS
Suppose that {y,} follows an ARFIMA(p,d,q) process as in (1) above. Let

2t=[d, 4)], ..., ‘11,), 91, ..., Oq]'=[d, G’]' as above. We can apply the MDE or

AMDE estimators described in the previous section to obtain a consistent estimator of
2». However, we now consider the case that we assume (incorrectly) that {y.} follows

a pure I(d), or ARFIMA(O,d,O), process and we calculate the MDE or AMDE
estimator of d. This estimate will in general be inconsistent. Let d represent the

probability limit of the estimate d, so that the asymptotic bias is (d - do), where do is
the true (population) value of d. We wish to evaluate this asymptotic bias.
The MDE based on the assumed (O,d,0) model minimizes the criterion

fimction:

I

(26) min S(d)={Plfil-F[p(d, on} w {PIfi]-Flp(d, on}-

We can calculate d = plimd by invoking the general principle that 3 should minimize

A

in the population the same criterion that d minimizes in the sample. Since

plimp = p(do, 6)), d minimizes the criterion function:

I

(27) min 8(3) ={F[p(do, eon—PW, 0)]} w {Home 90)]— 11ml 01]}.

64
Thus we can calculate d by a numerical minimization of the criterion (27), and from it
calculate the asymptotic bias (d - do). This will depend on the fimction F, the
weighting matrix W, the differencing parameter do, and the ARMA parameters 6).
The transformation F[p] = p defined the (ordinary) MDE of TSB. Suppose

that the MDE is based on (p g, ..., p 5+“) , where the standard case of TSB is K = 1.

For this MDE there is no reason to expect the asymptotic bias to decrease when E
increases, since every autocorrelation pj(d, 6)) depends on (1‘), no matter how large j is.
However, ratios like pj(d, ®)/pj-1(d, G) depend only on d, not on ('9, for suficiently
large j. Thus, for the AMDE based on the function F1 or the MDE based on F2 and F3
of the previous section, each of which involves ratios, we do expect the asymptotic
bias to decrease as 6 increases.

For the MDE based on F2 or F3, defined in equations (22) and (24) above, d
must be calculated by a numerical minimization. For the GLS version of the AMDE,
based on F 1 defined in equation (20), there is a closed form (GLS) solution for d:
(28) E1 = i;,w Fl[p(do, @)]/[i;,w1,,],
where in is a vector of ones, of dimension 11.

Finally, we need to discuss the relevant form of the weighting matrix, W. The
optimal weighting matrix, (PCP')_1 above, depends on population quantities and
would typically be estimated using the results of some initial estimation. Since the

initial estimates will also generally be biased when short-run dynamics are

misspecified, there is some ambiguity in how we should handle the weighting matrix

65
W in our bias calculations. Therefore we will consider three different cases
corresponding to different treatment of the weighting matrix. In Case 1, we use the
identity matrix as the weighting matrix, so W = I... In Case 2, we use the weighting
matrix in which P and C are evaluated using the true autocorrelations, p(do, @0).
This is unambiguous, but does not correspond to any feasible method of estimation

that misspecifies ('9 as zero. In Case 3, we assume that the weighting matrix is

evaluated in the sample based on the initial estimate 3 =5,/(1+(31), which would be
consistent if the (0,d,0) model were true. When the (p,d,q) model is true,

'5 —> d *Ep1(do, ®)/[1+p1(d0, (9)] and we evaluate P and C using d = (1*.

4. RESULTS

In this section, we provide the results of our calculations of the asymptotic
bias of the MDE of d, when we ignore short-run dynamics in the true
ARFIMA(p,do,q) process. We consider three forms of transformation fimctions F 1,
F2, and F3, and three cases that differ in the evaluation of the weighting matrix, as
described at the end of last section. The minimization problem (27) that defined d
was solved (for the MDE based on F2 and F3) using the optimization procedure in
GAUSS 2.0. For simplicity we consider only ARFIMA(1,do,O) and ARFIMA(O,do,1)
processes, where just one short-run parameter exists (4) or 9).

Tables 3.1-1, 3.1-2 and 3.1-3 give the asymptotic bias of the AMDE and the

MDE for the case of an ARFIMA(1,d0,0) process, for three parameter values: (do =

66

.2, ¢ = .4), (do: .2, d) = .8) and (do = .4, d) = .4). Tables 3.2-l and 3.2-2 do the same
for the case of an ARFIMA(O,do, 1) process, for two parameter values (do = .2, 9 =-.4)
and (do = .2, 9 = -.8). We consider numbers of moment conditions (r1) equal to 1, 3, 5
and 10. We consider lags (I! , equal to the lowest order autocorrelation used) equal to
0, 2, 3, 4, 5, 6, 7, 8, 10, 20 and 30. We consider the AMDE based on F1 and the
MDE based on F2, and F3, and three cases corresponding to the treatment of the
weighting matrix, as we discussed previously.

Table 3.1 and 3.2 give us some clear and interesting results. (1) For fixed 11,
the asymptotic bias of the AMDE or the MDE that ignore short-run dynamics
decreases and becomes close to zero as we use higher order of autocorrelations
(larger 6 ). This is as expected given the characteristics of autocorrelations of our
long memory process. This is our main result. It essentially implies that semi-
parametric estimation of d through the MDE principle is possible, if we choose an
appropriate form of transformation function of the autocorrelations and use high order
autocorrelations. (2) The three different transformations (F1, F2, F3) show similar
results for larger values of e. This especially true for F2 and F3, for which the results
are quite similar even when K is not very large. The absolute bias for the estimator
based on F1 is generally larger than for F2 or F3. (3) The choice of method of
evaluating the weighting matrix (Case 1, 2 or 3) does not usually make much
difference. It matters more for F1 that for F2 or F3. (4) The asymptotic bias depends

more on the order of autocorrelations used (6) than on the number of moment

67
conditions (11). Thus increasing n with fixed 6 does not decrease asymptotic bias very
much. Especially for large I? it has almost a negligible effect.

Tables 33-], 3.3-2 and 33-3 give the asymptotic bias for many values of do (=
-.49, -.4, -.3, -.2, -.1, .1, .2, .24, .25, .3, .4, .49) and two difl‘erent values ofn (= 1,
10), with d) = .4. For n =1 the estimation problem is “exactly identified” in the sense
that the number of moment conditions equals the number of parameters estimated.
Therefore the choice of weighting matrix does not matter, and the results are the same
for Cases 1, 2 and 3. For n =10, however, the choice of weighting matrix matters.

These results show that the asymptotic bias decreases as e increases, as
expected. The pattern of absolute bias as a function of d, holding constant n and é , is
complicated when 6 is small. For larger values of e (and both values of n), absolute
bias decreases as d increases.

Table 3.4 provides the opposite comparison as in Table 3. It gives the
asymptotic bias for many values of d) (= -.6, -.4, -.2, .2, .4, .6, .8, .9) for two difi‘erent
values of n (= 5, 10), with do = .2. Results are given only for two relatively large
values of e, (=20 and €=30. The asymptotic bias is generally larger in absolute
value when (I) is larger in absolute value, as we would expect. For large |1|)| (i.e.,
strong short-run dynamics), the asymptotic bias is discouragingly large even for
€=30. For example, for «1) =9 the asymptotic bias is about -0.1 or -O.2 with [=30
(and do = .2). This reflects the fact that any non-parametric treatment of short-run
dynamics will have problems if they are strong enough; it is intrinsically diflicult to

distinguish long-run properties of the model from very strong short-run dynamics.

68

5. CONCLUDING REMARKS

In this chapter we have considered the MDE including the adjusted MDE
(AMDE) estimator of Chung and Schmidt (1995) for the differencing parameter in the
general ARFIMA model. In applying the MDE, one can estimate the ARFIMA
model, which amounts to modeling short-run dynamics with an ARMA model; or one
can estimate the pure I(d) model, but not using low-order autocorrelations, which is a
non-parametric treatment of short-rim dynamics. This non-parametric treatment is
similar in spirit to the frequency-domain approach of Geweke and Porter-Hudak,
based on the periodogram at low frequencies only. We expect a non-parametric
treatment of short-run dynamics to have some cost in terms of efficiency, and Chung
and Schmidt’s results show that this is so. We also expect a non-parametric treatment
to lead to finite sample bias, especially when the nuisance parameters (ARMA
parameters) take on extreme values; i.e., when short-run dynamics are very strong. In
this chapter, we do not evaluate finite samples biases, but we evaluate the asymptotic
bias that results from ignoring a fixed number (6) of low-order autocorrelations. The
asymptotic bias is larger when short-run dynamics are stronger and when I? is smaller.
This is as expected. It supports the conjecture of Tieslau, Schmidt and Baillie (1995)
and Chung and Schmidt (1995) that a consistent nonparametric estimate of the
differencing parameter results from letting I! grow with the sample size. A rigorous
proof of this conjecture, and a derivation of the asymptotic properties of the estimate

when I grow with T, are important topics for future research.

69
TABLE 3.1-1

Asymptotic Bias of MDE in ARFIMA(I,d.,0)
[do = 0.2, 9 = 0.4]

GLS (F1)

 

n=d.

n=3

rr=5

n=10

 

 

oo-q O\Ln-h caro<= “9

10
20

 

30

Cbsel Chse21Case3

.5348 .5348 .5348
20466 20466 20466
22123 22123 22123
22851 22851 22851
22838 22838 22838
22410 22410 22410
21863 21863 21863
21371 21371 21371
20721 20721 20721
20115 20115 20115
20050 20050 20050

 

(kweJ_Chse21Cama3

.2323 .4798 .4788
21813 20997 21307
22604 22376 22491
22699 22790 22770
22370 22561 22491
21881 22057 21990
21408 21536 21487
21027 21111 21079
20556 20588 20576
20104 20106 20105
20047 20047 20047

 

Camal<kme2£Cama3

.0399 .4573 .4418
22137 21203 21668
22416 22401 22509
22266 22638 22541
21894 22314 22163
21471 21802 21672
21096 21321 21229
20806 20948 20888
20452 20506 20483
20096 20098 20097
20044 20045 20045

camel ChmaZACama3

20748 .4287 .3920
21467 21395 21825
21611 22292 22229
21432 22295 22032
21173 21889 21620
20912 21408 21205
20690 21006 20871
20520 20715 20630
20311 20387 20354
20079 20084 20082

 

20039 20040 20039

 

Ratios (F2)

 

n=1.

rr=3

rr=5

n=10

 

oo~4<m Uisstato c>9§

10
20
30

Case. 1 Case.2 Case.3

.1861 .1861 .1861
21126 21126 21126
21355 21355 21355
21166 21166 21166
20864 20864 20864
20593 20593 20593
20403 20403 20403
20275 20275 20275
20142 20142 20142
20013 20013 20013
20000 20000 20000

Casel Case2.Chse3

.1372 .2209 .3642
21204 21072 21199
21184 21252 21207
20936 21005 20952
20665 20690 20674
20453 20434 20457
20308 20277 20329
20212 20172 20214
20115 20068 20115
20015 20000 20011
20002 .0000 .0000

(kweJ Chse2.Cama3

.1200 .2211 .3701
21113 21058 21159
21034 21181 21101
20790 20904 20830
20553 20593 20572
20375 20378 20383
20254 20234 20259
20171 20144 20180
20099 20057 20093
20016 20000 20009
20000 .0000 20001

CkweJ.Chse2mCasa3

.1101 .2213 .3627
20954 21053 21216
20847 21092 21072
-0628 20780 20755
20431 20494 20495
20290 20312 20318
20198 20182 20210
20139 20108 20144
20079 20040 20075
20018 20000 20008
20005 .0000 20001

 

Common Denominator Ratios (F3)

 

n=1.

rr=3

n=5

n=10

 

 

 

casel CaseZ,Case3
.1861 .1861 .1861
21126 21126 21126
21355 21355 21355
21166 21166 21166
20864 20864 20864
20593 20593 20593
20403 20403 20403
20276 20276 20276
20142 20142 20142
20019 20019 20019
20007 20007 20007

 

Casel Chse2,Chse3
.1449 .2072 .2378
21197 21128 21092
21236 21219 21419
20989 21008 21115
20706 20694 20757
20481 20470 20487
20326 20314 20319
20225 20168 20216
20120 20074 20115
20025 20000 20023
20011 .0000 .0000

 

Camal(kwe21Cama3
.1244 .2055 .2394
21159 21106 21242
21118 21190 21600
20863 20923 21243
20605 20638 20792
20410 20398 20482
20279 20268 20194
20194 20185 20101
20106 20033 20099
20023 20000 20019
20010 .0000 20009

 

(kmeJ.Chse2.CamL3
.1023 .2128 .2397
21031 21081 21384
20922 21112 21680
20683 20808 21371
20469 20520 20902
20316 20339 20531
20216 20224 20314
20152 20154 20195
20086 20056 20091
20021 20000 20010
20010 .0000 20001

 

 

 

70
TABLE 3.1-2

Asymptotic Bias of MDE in ARFIMA(I,d.,0)
[(1. = 0.2, 4) = 0.8]

GLS (F‘)

 

n=1.

1r=3

1r=5

n=10

 

 

N

...-i
OOOOQO‘UIthO

N

30

 

Camal Chm22.Chsa3
.8472 .8472 .8472
.6419 .6419 .6419
.5471 .5471 .5471
.4563 .4563 .4563
.3695 .3695 .3695
.2869 .2869 .2869
.2085 .2085 .2085
.1349 .1349 .1349
.0026 .0026 .0026
23129 23129 23129
22228 22228 22228

 

(kweJ,C3562:Cama3
.7435 .8204 .8246
.5484 .5986 .5857
.4576 .5014 .4863
.3709 .4096 .3937
.3883 .3229 .3069
.2101 .2411 .2254
.1365 .1641 .1491
.0679 .0920 .0785
20534 20349 20462
23138 23136 23139

 

22085 22104 22092

Chsal Casa2,Chsa3
.6468 .8078 .8079
.4603 .5749 .5409
.3737 .4752 .4363
.2962 .3820 .3415
.2132 .2946 .2544
.1398 .2125 .1739
.0714 .1360 .0992
.0082 .0648 .0313
21016 20588 20861
23100 23112 23113
21947 21996 21964

 

Case.1 Case.2 Case.3
.4300 .7908 .7780
.2659 .5428 .4557
.1909 .4390 .3413
.1209 .3430 .2422
.0558 .2540 .1544
20040 .1713 .0761
20583 .0952 .0060
21070 .0253 20554
21872 20934 21567
22174 22995 .2952
21641 21774 .1687

 

Ratios (F2)

 

n=1.

1r=3

IF=5

n=10

 

<\

t-nl
OOMQQUIAWNO

N

30

CkweJ.Cama24Cama3
.2783 .2783 .2783
.1623 .1623 .1623
.1062 .1062 .1062
.0549 .0549 .0549
.0094 .0094 .0094
20299 20299 20299
20628 20628 20628
20896 20896 20896
21261 21261 21261
21040 21040 21040
20371 20371 20371

Casal Cam221C3923
.2677 .2955 20841
.1269 .2256 20011
.0702 .1790 .0419
.0210 .1292 .0095
20210 .0783 20264
20561 .0288 20587
20843 20171 20860
21065 20575 21074
21344 21121 21348
-l.003 20856 21008
20394 20074 20404

(kweJ Cama2‘Cama3
.2616 .2955 21472
.1030 .2243 24822
.0455 .1778 20807
20022 .1284 20415
20415 .0782 20586
20731 .0292 20813
20979 20165 21020
21167 20570 21185
21379 21149 21385
20937 20816 20941
20363 20217 20373

Camal<kwe21€ama3
.2535 .2954 22293
.0697 .2235 28270
.0117 .1768 25908
20330 .1278 21879
20673 .0778 21184
20931 .0291 21142
21117 20166 21209
21245 20572 21283
21360 21143 21361
20799 20729 20804
20307 20184 20316

 

Common Denominator Ratios (19)

 

n=1.

1r=3

1r=5

n=10

 

 

 

Casal Chs62.Chsa3
.2783 .2783 .2783
.1623 .1623 .1623
.1062 .1062 .1062
.0549 .0549 .0549
.0094 .0094 .0094
20299 20299 20299
20628 20628 20628
20896 20896 20896
21261 21261 21261
21040 21040 21040
20371 20371 20371

 

Cam&1 CamaZ1Cama3
.2617 .2968 .2890
.1318 .2209 .2513
.0764 .1122 .1928
.0275 .1030 .1235
20149 .0476 .0585
20508 20016 .0020
20801 20435 20445
21033 20778 20804
21331 21227 21260
21025 20953 21023
20412 20243 20407

 

Case. 1 Case.2 Case.3
.2480 .2968 .2892
.1090 .2207 .2672
.0544 .1617 .2238
.0074 .1025 .1586
20324 .0472 .0906
20654 20021 .0293
20917 20442 20232
21120 20784 20653
21365 21221 21219
20972 20872 21000
20388 20352 20382

 

(kweJ.Cbsa2iCama3
.2224 .2987 .2892
.0709 2192 .2743
.0187 .1598 .2343
20242 .1006 .1587
20589 .0455 .0842
20863 20036 .0210
21071 20452 20317
21221 20786 20744
21375 21202 21344
20860 20896 21084
20339 20271 20356

 

 

 

71
TABLE 3.1-3

Asymptotic Bias of MDE in ARFIMA(l,d.,0)
[do = 0.4, ¢ = 0.4]

GLS (F1)

 

n=1.

1r=3

n=5

n=10

 

<\

y—s
OOOOQONUI-bWNO

“N
O

Casel Chse2.Chme3
.4902 .4902 .4902
.0483 .0483 .0483
20568 20568 20568
21010 21010 21010
21067 21067 21067
20941 20941 20941
20761 20761 20761
20593 20593 20593
20355 20355 20355
20072 20072 20072
20032 20032 20032

Case] Chsa2.Case3
.2560 .4415 .4378
20365 .0105 20062
20882 20741 20805
21001 21014 21019
20923 20976 20960
20765 20819 20800
20603 20646 20630
20468 20498 20487
20287 20300 20295
20065 20066 20066
20030 20030 20030

Casel Cam12,Cas&3
.1221 4208 .4061
20621 20040 20297
20869 .0774 .0854
20874 20964 20954
20764 20890 20851
20621 20729 20690
20489 20568 20537
20382 20436 20454
20241 20265 20255
20060 20061 20061
20028 20028 20028

Casel Case2.Came3
.0228 .3937 .3658
20555 20184 20458
20626 20756 20796
20588 20848 20790
20502 20742 20664
20409 20548 20522
20326 20451 20401
20260 20344 20310
20172 20211 20195
20050 20053 20052
20024 20025 20025

 

Ratios (F1)

 

n=fl.

n=3

1r=5

n=10

 

oo~q axnn.n babJ<D "s

10
20
30

Casel Chm224Casa3
.0671 .0671 .0671
20143 20143 20143
20196 20196 20196
20173 20173 20173
20132 20132 20132
20096 20096 20096
20070 20070 20070
20050 20050 20050
20030 20030 20030
20005 20005 20005
20000 20000 20000

Camel Caae2.Cama3
.0574 .0937 22947
20164 .0047 20145
20176 20324 20194
20143 20245 20160
20106 20148 20114
20077 20099 20081
20055 20067 20057
20041 20048 20041
20024 20026 20024
20006 20000 20006
20000 .0000 20000

Camel Cam22lCmm23
.0547 .0932 22981
20156 20007 20180
20157 20388 20226
20124 20287 20173
20091 20179 20118
20066 20108 20080
20048 20072 20054
20036 20049 20039
20022 20025 20023
20004 20000 20005
20000 .0000 20002

Casal Chse2.Cama3
.0529 .0926 22092
20139 20080 20320
20133 20489 20330
20102 20371 20237
20073 20234 20151
20053 20149 20094
20038 20088 20060
20029 20058 20041
20018 20028 20022
20004 20000 20004
20000 .0000 20000

 

Common Denominator Ratios (F3)

 

n=1.

1r=3

1F=5

n=10

 

 

c>oo~q O\Ln-h unbaca "a

Nv—o
O

30

 

Canal Chat2,Cbse3
.0671 .0671 .0671
20143 20143 20143
20196 20196 20196
20173 20173 20173
20132 20132 20132
20096 20096 20096
20070 20070 20070
20050 20050 20050
20030 20030 20030
20005 20005 20005

 

20000 20000 20000

Casel case2.Came3
.0526 .0996 .0834
20164 .0039 20023
20180 20268 20335
20148 20186 20296
20111 20124 20189
20080 20084 20117
20058 20062 20075
20044 20046 20050
20026 20027 20027
20006 20003 20005

 

20001 20000 20000

Camel Chme2<Cama3
.0451 .0996 .0838
20160 .0010 20165
20162 20338 20440
20129 20204 20408
20096 20126 20277
20069 20086 20166
20051 20056 20099
20038 20042 20062
20023 20025 20030
20006 20000 20006

 

20003 .0000 20000

ChseJ_Chse2.Case3
.0366 .0997 .0839
20140 20060 20284
20132 20470 20466
20102 20303 20430
20075 20158 20326
20054 20098 20221
20040 20065 20142
20030 20043 20091
20019 20019 20041
20005 20003 20006
20002 20000 20000

 

 

72
TABLE 3.2-1

Asymptotic Bias of MDE in ARFIMA(O,d.,1)
[d. = 0.2, 0 = -0.4]

GLS (F11

 

n=1.

1r=3

n=5

n=10

 

 

N

OOOOQQm-bWNO

30

 

Case] ChasZwCase3
.6103 .6103 .6103
21285 21285 21285
20624 20624 20624
20376 20376 20376
20253 20253 20253
20182 20182 20182
20138 20138 20138
20108 20108 20108
20072 20072 20072
20019 20019 20019
20007 20007 20007

 

CuelCanCwe3
20153 .5947 .4591
20761 21061 20957
20471 20513 20480
20270 20311 20297
20191 20212 20204
20143 20154 20150
20111 20118 20116
20089 20093 20092
20061 20063 20062
20018 20018 20018
20008 20008 20008

 

CwelCmeZCue3
20291 .5640 .4070
20544 20929 20797
20314 20445 20402
20211 20270 20251
20154 20185 20175
20117 20136 20130
20093 20104 20101
20075 20083 20081
20053 20057 20055
20016 20017 20017
20008 20008 20008

 

CuelCueZCae3
20222 .5178 .3548
20318 20753 20610
20195 20353 20305
20137 20213 20191
20103 20146 20134
20081 20107 20100
20066 20083 20078
20055 20067 20063
20040 20046 20044
20014 20014 20014
20007 20007 20007

 

Ratios (F2)

 

n=1

1r=3

1r=5

n=10

 

 

 

Casel CbseZ,Chse3
.1575 .1575 .1575
20411 20398 20409
20182 20176 20178
20104 20059 20102
20068 20029 20059
20046 20020 20036
20035 20014 20011
20027 20008 20009
20011 20008 20007
20000 20084 20001
20000 20200 20000

 

CuelCue2Cue3
.0754 .2162 .3050
20283 20328 20298
20133 20145 20135
20079 20066 20080
20053 20008 20051
20039 20004 20038
20029 20002 20029
20023 20001 20015
20015 20009 20006
20000 .0000 20000
20000 .0102 20000

 

Casel Chm321Came3
20678 .2192 .2981
20240 20297 20265
20113 20129 20117
20068 20059 20067
20046 20022 20045
20034 20004 20033
20026 20002 20026
20020 20001 20020
20014 20000 20006
20000 20000 .0000
20002 .0074 .0000

 

CuelCanCae3
.0628 .2193 .2853
20202 20224 20250
20093 20099 20103
20056 20049 20057
20038 20010 20038
20027 20004 20027
20021 20002 20021
20017 20001 20017
20011 .0000 20006
20000 .0000 20000
20002 .0000 .0000

 

 

Common Denominator Ratios (F’)

 

n=1.

1r=3

1r=5

n=10

 

 

 

CuelCueZCae3
.1575 .1575 .1575
20411 20405 20409
20182 20176 20178
20104 20083 20102
20068 20009 20059
20046 20005 20036
20035 20003 20025
20024 20001 20009
20011 20008 20000
20000 .0114 .0001
20000 .0200 .0000

 

CaelCueZCae3
.0957 .2015 .2026
20310 20344 20376
20142 20144 20149
20084 20083 20082
20056 20023 20054
20041 20012 20038
20031 20005 20029
20024 20002 20017
20016 .0003 20011
20004 .0006 20002
20002 .0168 .0005

 

(kwe1.Chse2LCama3
.0801 .2021 .2028
20263 20310 20399
20122 20123 20145
20073 20069 20075
20049 20046 20048
20036 20033 20033
20027 .0006 20025
20022 .0003 20015
20015 .0000 20013
20004 .0012 20002
20002 .0259 .0003

 

(kwel.Came21Cam33
.0656 .1985 .2011
20209 20274 20422
20097 20107 20154
20059 20052 20076
20040 20026 20044
20029 20011 20029
20023 20004 20019
20018 20009 20014
20012 20003 20009
20004 .0024 20002
20002 .0018 20003

 

 

 

73
TABLE 3.2-2

Asymptotic Bias of MDE in ARFIMA(O,d.,1)
[d. = 0.2, 0 = -0.8]

GLS (F’)

 

n=1.

1r=3

n=5

n=10

 

 

N

...-a
OOOOQQUI-hUJNO

N

30

 

CmelCueZCae3
.8469 .8469 .8469
21513 21513 21513
20745 20745 20745
20451 20451 20451
20304 20304 20304
20220 20220 20220
20166 20166 20166
20130 20130 20130
20086 20086 20086
20023 20023 20023
20011 20011 20011

 

(kwel CameZlCaae3
.0403 .9504 .6385
20903 21318 21127
20500 20639 20573
20325 20386 20356
20230 20261 20246
20172 20190 20181
20134 20145 20139
20107 20114 20111
20073 20077 20075
20021 20021 20022
20010 20010 20010

 

CmelCmeZCme3
.0002 .9867 .5670
20647 21151 20939
20377 20569 20480
20254 20342 20301
20185 20232 20210
20142 20169 20156
20112 20129 20121
20091 20102 20097
20064 20069 20067
20020 20020 20020
20009 20010 20009

 

CwelCueZCme3
200911 0164 .4956
20379 21004 22768
20234 20465 20366
20165 20276 20230
20124 20186 20161
20098 20136 20121
20080 20104 20095
20066 20083 20077
20048 20057 20054
20017 20017 20017
20008 20008 20008

 

Ratios (F2)

 

n=d,

1r=3

1r=5

n=10

 

 

c>oo~q O\\h-h uaro<3 N:

30

CuelCanCae3
.1970 .1970 .1970
20494 20479 20493
20219 20213 20218
20125 20082 20125
20082 20023 20078
20056 20014 20056
20043 20010 20042
20033 20007 20033
20015 20012 20011
20000 .0083 20000

 

20000 .0058 20000

casel Chm32<Came3
.1110 .2656 .4270
20340 20428 20359
20161 20186 20164
20096 20086 20097
20065 20037 20063
20047 20013 20046
20036 20009 20035
20028 20006 20028
20019 20004 20019
20001 .0080 20001
20000 .0056 20000

Case 1 Case 2 Case 3
.1012 .2714 .4125
20289 20410 20322
20137 20172 20144
20082 20079 20083
20056 20034 20056
20041 20012 20041
20031 20008 20031
20025 20006 20025
20017 20003 20017
20001 .0000 20001

 

20000 .0000 .0000

case] case2,Chse3
.0945 .2740 .3973
20243 20398 20312
20113 20157 20130
20067 20069 20072
20046 20030 20047
20033 20011 20034
20025 20007 20026
20020 20005 20020
20014 20003 20014
20001 20000 20001
20000 .0000 20000

 

 

Common Denominator Ratios (F’)

 

 

n=1.

1r=3

1r=5

n=10

 

 

oo~q Oxtn $>t» 59¢: “5

10
20
3()

 

CuelCueZCwe3
.1970 .1970 .1970
20494 20494 20493
20219 20213 20218
20125 20103 20125
20082 20065 20081
20056 20015 20056
20043 20010 20042
20033 20007 20033
20015 20012 20015
20000 .0113 20000
20000 .0078 20000

 

CuelCueZCue3
.1191 .2712 .2405
20372 20444 20557
20172 20186 20201
20102 20105 20107
20068 20052 20068
20049 20022 20048
20037 20013 20036
20029 20007 20024
20019 20001 20016
20005 .0005 20003
20002 .0080 20004

 

CaelCme2Cue3
.0995 .2894 .2414
20316 20424 20602
20148 20173 20217
20088 20092 20108
20060 20060 20065
20043 20042 20044
20033 .0001 20031
20026 .0005 20025
20018 .0008 20016
20005 .0010 20003
20002 .0009 20002

 

CuelCueZCue3
.0814 .2963 .2410
20252 20399 20585
20118 20148 20234
20071 20074 20119
20048 20043 20070
20035 20022 20045
20027 20017 20033
20022 20010 20023
20015 20007 20014
20004 .0022 20003
20002 .0018 20001

 

 

 

74
TABLE 3.3-1

Asymptotic Bias of MDE in ARFIMA(1,d.,0) in GLS (F')

¢=o.4

[n=1]

 

d. -0.49 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.24 0.25

0.3 0.4 0.49

 

6 = 0
I .6359 .6262 .6144 .6014 .5872 .5541 .5348 .5266 .5245
II .6359 .6262 .6144 .6014 .5872 .5541 .5348 .5266 .5245
III .6359 .6262 .6144 .6014 .5872 .5541 .5348 .5266 .5245

.5137 .4902 .4668
.5137 .4902 .4668
.5137 .4902 .4668

 

6 = 2
I 1.908 4.009 -5.974 -l.282 25697 21362 20466 20207 20149
I] 1.908 4.009 -5.974 -l.282 25697 21362 20466 20207 20149
III 1.908 4.009 -5.974 -l.282 25697 21362 20466 20207 20149

.0107 .0483 .0707
.0107 .0483 .0707
.0107 .0483 .0707

 

€= 5
I .1688 .2654 .4174 .7338 2.489 25301 22838 22301 22188
H .1688 .2654 .4174 .7338 2.489 25301 22838 22301 22188
111 .1688 .2654 .4174 .7338 2.489 25301 22838 22301 22188

21713 21067 20687
21713 21067 20687
21713 21067 20687

 

(=10
I 21633 21264 20907 20542 .0051 21210 20721 20617 20595
11 21633 21264 20907 20542 .0051 21210 20721 20617 20595
II] 21633 21264 20907 20542 .0051 21210 20721 20617 20595

20499 20355 20259
20499 20355 20259
20499 20355 20259

 

(=20
I 20355 20314 20272 20235 20200 20141 20115 20106 20103
II 20355 20314 20272 20235 20200 20141 20115 20106 20103
HI 20355 20314 20272 20235 20200 20141 20115 20106 20103

20092 20072 20056
20092 20072 20056
20092 20072 20056

 

(=30
I 20147 20131 20115 20100 20086 20061 20050 20046 20045
I] 20147 20131 20115 20100 20086 20061 20050 20046 20045
HI 20147 20131 20115 20100 20086 20061 20050 20046 20045

 

20040 20032 20025
20040 20032 20025
20040 20032 20025

 

Note: I (=Case 1), II (=Case 2) and HI (=Case 3).

 

75

TABLE 3.3-1, CONTINUED

[n = 10]

 

do

-0.49 -0.4 -0.3 -0.2 -0.1

0.1 0.2 0.24 0.25 0.3 0.4 0.49}

 

Env—

.1721 .5791 22207 2.714 -1.448
.9486 -L457 .5816 .5671 .5260
.6779 .6236 .4388 .8594 .2579

(=0
.2075 20748 20453 20391 20128 .0228 .0430
.4506 .4287 .4211 .4192 .3837 .3937 .3793
.3985 .3920 .3878 .3866 .3657 .3658 .3504

 

.2630 .5569 22777 2.646 -l.517
.6601 .6371 1.464 27537 22933
.8593 1.562 29605 .6082 -3.122

€= 2
23017 21467 21304 21237 20953 20555 20314
22613 21395 21058 20984 20658 20184 20109
23478 21825 21427 21342 20975 20458 20149

 

20903 20441 .0119 .0945 .3993

21095 .0156 .0932 .2102 .4810 23674 21889 21534 21461 21154 20742 20497

21027 20142 .0615 .1812 .6322

Z: 5
22136 21173 20973 20931 20752 20502 20348

23039 21620 21330 21269 21014 20664 20452

 

20892 20740 20594 20455 20277
21095 20895 20706 20526 20289
21027 20847 20676 20514 20306

(=10
20452 20311 20275 20267 20231 20172 20130
20579 20387 20341 20331 20285 20211 20159
20518 20354 20313 20304 20262 20195 20146

 

20237 20211 20184 20159 20136
20253 20224 20196 20169 20146
20247 20220 20192 20166 20142

[=20
20096 20079 20073 20071 20063 20050 20038
20102 20084 20077 20075 20067 20053 20041
20100 20082 20075 20074 20066 20051 20040

 

 

20112 20101 20089 20077 20066
20116 20104 20091 20079 20068
20115 20103 20090 20078 20067

(=30
20047 20039 20035 20035 20031 20024 20019
20048 20040 20037 20036 20032 20025 20020
20048 20039 20036 20035 20032 20025 20019

 

 

76
TABLE 3.3-2

Asymptotic Bias of MDE in ARFIMA(1,d.,0) in Ratios (F2)

 

 

 

 

 

 

 

 

o = 0.4
[n = 1]

d. on -o.4 .03 .02 .0.1 0.1 0.2 0.24 0.25 0.3 0.4 0.49!
e: o

I .4613 .4353 .4033 .3671 .3289 .2382 .1861 .1639 .1582 .1291 .0671 .0069

II .4613 .4353 .4033 .3671 .3289 .2382 .1861 .1639 .1582 .1291 .0671 .0069

III .4613 .4353 .4033 .3671 .3289 .2382 .1861 .1639 .1582 .1291 .0671 .0069
e: 2

I .7322 .8796 1.258 20700 -2.120 22401 21126 20813 20747 20474 20143 20008

II .7322 .8796 1.258 20700 -2.120 22401 21126 20813 20747 20474 20143 20008

III .7322 .8796 1.258 20700 -2.120 22401 21126 20813 20747 20474 20143 20008
e: 5

I 20106 .0401 .0986 .1803 .3999 22030 20864 20623 20574 20378 20132 20010

II 20106 .0401 .0986 .1803 .3999 22030 20864 20623 20574 20378 20132 -.0010

[II 20106 .0401 .0986 .1803 .3999 22030 20864 20623 20574 20378 20132 20010
2:10

I 20900 20694 20509 20347 20171 .0062 20142 20107 20101 20074 20030 20001

II 20900 20694 20509 20347 20171 .0062 20142 20107 20101 20074 20030 20001

III 20900 20694 20509 20347 20171 .0062 20142 20107 20101 20074 20030 20001
2:20

I 20168 20148 20102 20077 20038 .0001 20013 20020 .0020 20014 20005 20001

H 20168 20148 20102 20077 20038 .0001 20013 20020 .0020 20014 20005 20001

III 20168 20148 20102 20077 20038 .0001 20013 20020 .0020 20014 20005 20001
e=3o

I 20060 20037 20018 .0008 20006 .0010 20000 20009 20009 20006 20000 -.0001

II 20060 20037 20018 .0008 20006 .0010 20000 20009 20009 20006 20000 2000]

III 20060 20037 20018 .0008 20006 .0010 20000 20009 20009 20006 20000 20001

 

 

77

TABLE 3.3-2, CONTINUED

 

 

 

 

 

 

 

 

[n=10]

d. -0.49 -0.4 -0.3 -0.2 -O.1 0.1 0.2 0.24 0.25 0.3 0.4 0.49
(=0

I .9485 28312 .2217 -L$12 .5731 .0842 .1101 .1063 .1046 .0921 .0529 .0057

II .4357 .4807 .5512 27021 .1923 .2620 .2213 .2021 .1970 .1678 .0926 .0099

III .3767 .3708 .3692 9999 .2847 .3865 .3627 .3395 .3331 .2992 22092 .0094
(=2

I .2955 .4097 .6994 2TZ40 .L640 22135 20954 20696 20642 20419 20139 20009

I] .3775 .4229 .3904 .3860 .4067 22247 21053 20768 20724 20472 20080 .0064

III .3573 .4301 .6850 9999 1 819 22320 21216 20940 20878 20664 20320 20009
€=5

I 20741 20399 20044 .0380 .1316 20986 20431 20319 20296 20200 20073 20006

11 - 0682 20257 .0011 .0524 .1460 20951 20494 20394 20374 20292 20234 .0038

III .0317 .0281 20076 .0262 .1033 20938 20495 20384 20359 20275 20151 20008
[=10

I 20526 20422 20326 20241 20155 20133 20079 20064 20059 20043 20018 20001

11 -20154 .0055 20033 20090 .0060 .0107 20040 20066 20063 20049 20028 20003

III .0600 20066 20024 20089 20082 20116 20075 20066 20060 20046 20022 20002
(=20

I 20125 20106 20086 20068 20054 20025 20018 20016 20015 20010 20004 20001

I] .0600 20300 .0300 .0400 20094 .0103 20000 20038 20015 20011 20000 20000

III .0600 20300 20068 20008 20021 20024 20008 20016 20015 20010 20004 20000
€=30

I 20058 20049 20038 20026 20014 20012 20005 20008 20007 20005 20000 20000

I] .0600 20300 .0300 20700 20700 .0079 20000 20007 20007 20005 20000 20000

.0600 20300 .0300 .0008 .0002 20012 20001 20008 20007 20005 20000 20000

 

 

78
TABLE 3.3-3

Asymptotic Bias of MDE in ARFIMA(1,d.,0)
in Common Denominator Ratios (1‘4)

 

 

 

 

 

 

 

 

¢==041
ln= 1]
d. 4149 .02: -03 412 -01 (11 02 (124 025 (13 011 049
e==o
I .4613 .4353 .4033 .3671 .3289 .2382 .1861 .1639 .1582 .1291 .0671 .0069
II .4613 .4353 .4033 .3671 .3289 .2382 .1861 .1639 .1582 .1291 .0671 .0069
III .4613 .4353 .4033 .3671 .3289 .2382 .1861 .1639 .1582 .1291 .0671 .0069
e==2
I .7322 .8796 1.258 20700 -2.120 22401 21126 20813 20747 20474 20143 -.0008
II .7322 .8796 1.258 20700 -2.120 22401 21126 20813 20747 20474 20143 20008
III .7322 .8796 1.258 20700 -2.120 22401 21126 20813 20747 20474 20143 20008
e==5
I 20106 .0401 .0986 .1803 .3999 22030 20864 20623 20574 20378 20132 20010
II 20106 .0401 .0986 .1803 .3999 22030 20864 20623 20574 20378 20132 20010
III 20106 .0401 .0986 .1803 .3999 22030 20864 20623 20574 20378 20132 20010
3:10
I 20900 20694 20509 20347 20171 .0062 20142 20107 20101 20074 20030 20001
II 20900 20694 20509 20347 20171 .0062 20142 20107 20101 20074 20030 20001
HI 20900 20694 20509 20347 20171 .0062 20142 20107 20101 20074 20030 20001
e=2o
I 20168 20148 20102 20077 20038 .0001 20019 20020 .0020 20014 20005 20001
I] 20168 20148 20102 20077 20038 .0001 20019 20020 .0020 20014 20005 20001
1]] 20168 20148 20102 20077 20038 .0001 20019 20020 .0020 20014 20005 20001
3:30
I 20060 20037 20018 .0008 20006 .0010 20007 20009 20009 20006 20000 20001
II 20060 20037 20018 .0008 20006 .0010 20007 20009 20009 20006 20000 20001
III 20060 20037 20018 .0008 20006 .0010 20007 20009 20009 20006 20000 20001

 

 

79
TABLE 3.3-3, CONTINUED

[n= 10]

 

d. -0.49 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.24 0.25 0.3 0.4 0.49

 

E = 0
I .4018 .3596 .3109 .2624 .2159 .1360 .1023 .0894 .0862 .0700 .0366 .0038
II .3124 .2997 .2849 .2692 .2528 .2181 .2128 .1964 .1944 .1795 .0997 .0100
III .3892 .3756 .3594 .3412 .3200 .2816 .2397 .2110 .2036 .1652 .0839 .0084

 

€= 2
I .4114 .4653 .5960 -24.45 -2.189 22458 21031 20736 20676 20432 20140 20000
II .3197 .3594 .4121 .2500 -l.468 22426 21081 20791 20731 21179 20060 .0076
III .4321 .4180 .5319 -14.88 -1.496 22718 21384 21055 20982 20686 20284 -9999

 

€= 5
I 20673 20282 .0109 .0562 .1525 21164 20469 20341 20315 20209 20075 .0006
II 20733 20334 .0016 .0490 .0859 21057 20520 20395 20368 20264 20158 .0044
III .0600 .0107 .0018 .0351 .1090 21662 20902 20711 20670 20541 20326 20021

 

(=10
I 20627 20492 20371 20269 20170 20150 20086 20069 20065 20047 20019 20000
II 20347 20034 20277 .0216 .0219 20122 20056 20068 20054 20043 20019 20000
H] .0600 .0700 20271 20088 20024 20128 20091 20077 20075 20063 20041 20000

 

[=20
I 20137 20116 20094 20075 20059 20031 20021 20017 20016 20012 20005 -.0000
II .0600 .0700 20300 20700 20700 .0251 20000 20016 20011 20003 20003 20000
III .0600 .0700 20300 20023 .0181 20004 20000 20015 20011 20012 20006 20000

 

 

€=30
I 20062 20052 20043 20034 20027 20015 20010 20008 20007 20006 20002 20000
H .0600 .0700 20300 20700 20700 .0699 .0000 .0000 .0002 .0002 20000 20000
III .0600 .0700 20300 .0398 .0141 20004 .0001 20001 20004 .0001 20000 20000

 

 

80

TABLE 3.4

Asymptotic Bias of MDE in ARFIMA(I,d.,0)

 

 

 

 

 

d. = 0.2
[n = 5]
Q -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0.9
GLS (F‘)
K = 20
I 22849 .0016 .0011 .0000 20025 20096 20461 23100 .0113
H 20652 .0016 .0011 .0000 20026 20098 20483 23112 .0283
III -.0813 .0016 .0011 .0000 20026 20097 20471 23113 .0163
6 = 30
I 20042 .0008 .0005 20000 20012 20044 20165 21947 22147
11 .0002 .0008 .0005 20000 20012 20045 20168 21996 22065
111 20006 .0008 .0005 20000 20012 20045 20166 21964 22128
Ratios (F2)
2 = 20
I 20057 .0000 .0000 20000 20000 20016 20092 20937 21261
11 20000 .0000 .0000 .0000 .0000 20000 .0017 20816 20876
III -.0000 .0000 .0000 .0000 .0000 20009 20092 20941 21267
8 = 30
I .0000 .0000 .0000 .0000 20001 20000 20028 20363 -.1411
II .0000 .0000 .0000 .0000 20000 20000 .0000 20217 -.1182
III .0000 .0000 .0000 .0000 20000 20001 20033 20373 -. 1433
Common Denominator Ratios (F’)
l? = 20
I 20071 .0004 .0003 .0000 20006 20023 20097 20972 21242
11 .0000 .0000 .0000 .0000 .0000 20000 20030 20872 2 1062
III .0000 .0000 .0000 .0000 .0000 20019 20092 21000 21199
E = 30
I .0001 .0001 .0000 .0000 20003 20011 20038 20388 21436
11 .0000 .0000 .0000 .0000 .0000 20000 .0000 203 52 2 1290
III .0000 .0000 .0000 .0000 .0000 20009 20030 203 82 21440

 

 

81

 

 

 

 

 

TABLE 3.4, CONTINUED
[n = 10]
4. -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0.9
GLS (F’)
6 = 20
I -. 1208 .0013 .0009 .0000 20021 20079 20357 22174 20535
11 20168 .0014 .0010 .0000 20022 20084 20398 22995 20035
III -.0095 .0014 .0010 .0000 20022 20082 20373 22952 20389
2 = 30
I 20015 .0007 .0005 20000 20011 20039 20142 -. 1641 22461
II .0006 .0007 .0005 20000 20011 20040 20148 21774 22247
111 .0006 .0007 .0005 20000 20011 20039 20145 21687 22419
Ratios (F2)
8 = 20
I 20030 20000 .0000 20000 20004 20018 20077 20799 21310
H -.0000 .0000 .0000 .0000 20000 20000 20017 20729 -.0862
HI .0000 .0000 .0000 .0000 20000 20008 20077 20804 -. 1137
2 = 30
I .0000 .0000 .0000 20000 20000 20005 20025 20307 21356
H -.0000 .0000 .0000 .0000 .0000 20000 .0000 20184 21164
[H -.0000 .0000 .0000 .0000 .0000 20001 20029 20316 21377
Common Denominator Ratios (F3)
6 = 20
I 20038 .0004 .0002 .0000 20006 20021 20084 20860 -. 1310
II 20000 20000 20000 .0000 .0000 20000 .0007 20896 21032
111 .0000 .0000 .0000 .0000 20000 20010 20084 21084 21136
I? = 30
I .0002 .0002 .0001 20000 20003 20010 20035 20339 21401
11 .0000 .0000 20000 .0000 20000 20000 .0000 20271 -. 1342
III .0000 .0000 .0000 .0000 20000 20001 20029 20356 21444

 

 

CHAPTER 4

REGRESSION IN FRACTIONAL COINTEGRATION

82

83
1. INTRODUCTION

It is now well established that many economic time series contain a unit root.
Such series are I(l) in the sense that they are nonstationary but their first difference is
stationary. Such series can move in random directions over arbitrarily long time
periods. However, in some cases economic theory indicates that certain pairs of
series are related and should not diverge too much from each other. This may be
reasonable because, when certain economic variables begin to diverge, market forces
or government intervention may reestablish their long run relationship.

Suppose that {in} and {y} are nonstationary unit root processes where y. is a
scalar but 1:. may be a vector. Consider a linear combination of those processes:

zt = Yt ‘ ng ,
where A is a nonrandom vector. Generally such a linear combination 2, will also be a
unit root process. As long as z. is a unit root process, whether A is zero or not does
not make much difl‘erence (as we will see later) and we need first-differencing to‘deal
with such cases.

However, when a linear combination of the unit root processes y. and x; is an
I(d) process with d <1, it is said that {xi} and {yt} are ‘cointegrate ’ and A is a
cointegrating vector Lot coefficient); see Engle and Granger (1987). An alternative
definition of cointegration is that a linear combination of the unit root processes y. and
x. is stationary. This would rule out the case 1/2 S d < l. CointegLation implies that

although there are permanent changes in the individual series x and y over time, there

84
is some long-run equilibrium relationship tying them together, which is represented by

the linear combination 2,.

There are several standard examples of cointegration relationships. Davidson,
Hendry, Srba and Yeo (1978) show that even though both consumption and income
are unit root processes, in the long run the difference between the log of consumption
and the log of income appears to be a stationary process. Kremers (1989) proposes
that the difi‘erence between the log of government debt and the log of GNP is a
stationary process even though each is not stationary. Also, although many empirical
studies show significant deviations from the Purchasing Power Parity (PPP)
hypothesis in the short run, it is argued that the PPP hypothesis works in the long run,
in the sense that a cointegration relationship exists among the foreign price index, the
domestic price index and the nominal exchange rate; alternatively between a relative
price index and the nominal exchange rate. For further details, see Cheung and Lai
(1993) and Baillie and Selover (1987).

The standard statistical treatments of cointegration deal with the case of a
short memory error; i.e., a linear combination of nonstationary I(l) processes
becomes a stationary 1(0) process. Under short memory error, the properties of
cointegrating coemcient estimates are well known. OLS is consistent and converges
in probability at the rate of T rather than the usual rate of Tm. However, in general
OLS is asymptotically biased, and it does not lead to asymptotically valid inference.
There are many other emcient estimates of cointegrating coefficients. For example,

see Johansen (1988, 1991), Stock and Watson (1988, 1993), Phillips and Hansen

85
(1990), Saikkonen (1991) and Park (1992). These methods also lead to

asymptotically valid inference.
However, it is possible that the linear combination of nonstationary unit root
processes may be an I(d) process with 0< (I <1. This case is referred to as fi'actional

cointegration in Baillie and Bollerslev (1994) and Cheung and Lai ( 1993). Ifthe error

 

in the cointegrating relationship is I(d) with 0< d <1/2, it is still stationary but it has
more persistent autocorrelations than in the usual short memory case. If 1/2< (1 <1,
the error in the cointegrating relationship is not stationary but it is mean-reverting, so
that a shock in a given time period will finally disappear in the long run.

Therefore, in this chapter our interest is in the case that the error in a
cointegrating relationship is I(d) with 0< d <1, rather than in the usual model of
cointegration with errors that are 1(0). For this case, we derive the asymptotic
distribution of the least squares estimator. Least squares is consistent, and has a rate
of convergence to its asymptotic distribution that depends on d. This can be
compared to least squares in differences, which is not consistent if the errors and
regressors are correlated, and which converges at the usual T”2 rate for all values of d
in the range 0< <1 <1. We also provide some simulations that support the relevance of

these asymptotics in samples of moderate size.

2. COINTEGRATION
First we define cointegration in a way similar to that in Engle and Granger

(1987)

86

Definition of Cointegration: The components of an le vector 24 are said to be
cointegrated of order a, b, i.e. 2. ~ CI(a,b), if
(i) z. is 1(a) for a >1/2 and

(ii) there exists a non-zero vector (1 such that (it’zt ~ I(a-b), a 2 b > 0.

If 2. has more than 2 components (N 2 2), then we may have more than one
cointegrating vector. When there exist h linearly independent cointegrating vectors
with h S N—l, we can combine them to make a cointegrating matrix C (Nxh). The
rank of C is h and is called as the cointegrating rank.

There are at least two reasons why cointegration is important. First, in a
regression with nonstationary variables, cointegration is a useful way of distinguishing
a meaningful regression from a ‘nonsense’ (Yule (1926)) or ‘spurious’ (Granger and
Newbold (1978)) regression. In a spurious regression the error is [(1) and least
squares does not have useful properties. Second, the Error Correction Representation
(ECR) exists only when the nonstationary variables are cointegrated. This is
important since the ECR provides a sensible way of combining the information
contained both in levels and differences. The ECR models the dynamics of both
short-run changes and the long-run adjustment process simultaneously.

In the short memory case (i.e., a = b =1), the problem of estimation of the
cointegrating vector has been studied by many economists. OLS is consistent and
converges at rate T, which is faster than the usual T“2 rate; see Stock (1987) and
Phillips and Durlauf (1986). Phillips and Park (1988) showed that when the error in

cointegrating relationship follows a stationary AR process, OLS and Generalized

87
Least Square (GLS) are asymptotically equivalent. There are many asymptotically

emcient estimates, which also lead to asymptotically valid inference. Johansen (1988)
derived a maximum likelihood estimator of the dimension of the space of the
cointegrating vectors and tests of linear hypotheses on those vectors. Phillips and
Hansen (1990) suggested a ‘fully modified’ least squares estimator. The method of
adding leads and lags was suggested by Saikkonen (1991), Phillips and Loretan
(1991) and Stock and Watson (1993). Park (1992) proposed an OLS procedure after
transforming both regressors and dependent variables.

There are many usefiil ways of representing cointegrated variables, including
the ECR The Vector Autoregressive Representation (VAR) is a basic tool for
analyzing nonstationary variables by making them stationary through first-
difl‘erencing. For details, see Engle and Yoo (1987) and Ogaki and Park (1992).
Johansen (1988) provided the Interim Multiplier Representation (IMR) by modifying
the error correction representation. The Triangular Representation (TR) by Phillips
(1991) divides the cointegrated system into exactly cointegrated variables and other
non-cointegrated variables. The Common Trend Representation (CTR) in Stock and
Watson (1988) decomposes the cointegrated nonstationary system into a stationary
component plus linear combinations of common deterministic trends and common
random walk variables. The Granger Representation Theorem in Engle and Granger
(1987) and Johansen (1991) gives several interesting results on the representation of

the cointegrated system.

88

3. ASYMPTOTICS F OR OLS ESTIMATES OF COINTEGRATING
COEFFICIENTS

We consider the following data generating process:

(1) yt =XtB+ut, t=1,2,ooo,T,
(2) xt = xt-I +vta
(3) (1— L)“u. = 8..

where {x.} and {y.} are the observed unit root series and {Vt} and {at} are assumed to
be short memory processes. For simplicity we consider the case that x. is a scalar;
thus there is at most one cointegrating relationship between y. and x.. In general, the
error process {u.}, which is a linear combination of unit root processes, may be
another unit root process. However, when B is not zero and {m} is an I(d) process
with 0S. (1 <1, {x.} and {y.} are cointegrated as in Engle and Granger (1987). This
model does not include intercept or deterministic time trend. To do so is a feasible

but non-trivial extension of this analysis.

A. Short Memory Case ((1 = 0)

We first consider the case that d =0 in (3) above. Therefore u. = a. in (3)
above, and we have the standard case of cointegration considered in the literature.
We will give a brief summary of the results for this case, for purposes of comparison

with our results for the case of fractional cointegration.

89
For simplicity, and to ensure comparability with our treatment of the fractional

case, we consider the special case in which the errors {v1} and {at} are only

2
8‘} iid(0, 2) with 2=[°8 9.2.].

contemporaneously correlated; i.e., [
Vt ve 0v

We begin with the following lemma which can be obtained from Phillips (1988) and

Phillips and Durlauf (1986):

LEMMA 1:

T l
(i) $12.12.: [1310193201
t 0

.. 1 T 1
(n) 72x? => IBi(r)dr,
T t o

1T
(111) 'T‘thst _p)0ve’
t

Br“)

where B(r) E [B (r)
2

]= Brownian motion with covariance matrix 2.

We note that, in a more general setting, the Brownian motion B(r) would have

as its covariance matrix the “long-run covariance matrix” of v. and 8., defined as:

1

T
1

SI

Q = 1iHIT—>00 E

 

 

 

r

2‘“
i

2‘4
t

fil

T

b —- —

90

However, because we have assumed that v. and a. are iid, the long-run covariance
matrix Q and the contemporaneous covariance matrix 2 are equal.
The OLS estimate of the cointegrating coefiicient B is:
T T‘
Z xtyt 12,1 xtut

(4) fi=%—=B+ "

T
2th 2x3
t=1 t=1

T
thSt
= B+——‘=I .
2x3
t=1

Thus

 

T‘1 Z Xt_18t + T—1 thst

T4222 T4222 ’

since th8. =th_lat+2vte,. Then, using LEMMA 1, one obtains the

(5) T(B-B)=

following asymptotic result:

 

l
. [Bitrnazunow
(6) T(B-l3)=> ° ,
IBimdr
0

Therefor 8-3 is Op(l/T) when the error 11. is short memory. OLS is

consistent whether or not v. and e. are correlated (i.e., whether or not (so, = 0), but
there is a bias in the asymptotic distribution when 0., ¢ 0. These are well-known,

standard results.

B. Spurious Regression Case (d =1)

91

We next consider the case of a spurious regression, as defined in Granger and
Newbold (1978), for which a rigorous asymptotic analysis was given by Phillips
(1986). This is the case in which the error process u. in (1) above is I(d) with d = 1;
i.e., a unit root process. Thus we write:

“1 = ut—1 + Tlt ,

where 11. is 1(0). We will consider that {Vt} and {m} are only contemporaneously

2
0' o
correlated, so that [m]~ iid (0, £1), 21 = 11 v2" .
Vt Onv 0v

Then the following lemma can be obtained from Philips (1986):

Br“)

LEMMA 2: Define
B3(r)

]= Brownian motion with covariance matrix 21.

Then

1 T 1
Fthut 3 IB1(I)B3(I)dI .
t 0

Again, in a more general setting the Brownian motion in LEA/[114A 2 would have as its

covariance matrix of v. and 1]., say (21 defined as

I

_ T _

sl—

T
Tth

9]: 1iInT—MO E 1T;
lez

t

 

 

 

3|

- d-

However, in the iid case 01 and 21 are the same.

Using LEMMA 1 and LEMMA 2, we obtain Phillips’ result:

92

1 T 1
. .1322... IB1(r)B3(r)dr
(7) (B-B)=--l—‘T—=>°l
sztz IBIUW
t o

 

This result implies that B is not consistent, since (B —- B) is 0,,(1) and therefore does

not diminish as T—)oo.
The case discussed difi‘ers slightly from the spurious regression in Granger and

Newbold (1978), in that B is not necessarily zero. However, this is not important,

since the asymptotic distribution of (B - B) does not depend on B.

C. Stationary Long Memory Error Case (0< d <1l2)

We now turn to the case that the error 11. is I(d) 0< d <1/2, so that it is a
stationary, long memory process. This is the leading case of fractional cointegration.
An asymptotic analysis of: OLS or other methods of estimation of the cointegrating
vector B has not previously been done.

As above, we consider the case that the innovations (v., at)' are iid (0, X),

with 2 as in section 3.A above. We have the following (standard) result for the joint
convergence of partial sums of v. and st:

(til ‘
Vt
[trI‘l] =B(r)=[Bl(r)].

(3)
B20)
8

al~

 

 

93

where B(r) is a Brownian motion with covariance matrix 2. The basic result that we
need, however, is an expression for the joint limit of the partial sums of v. and u.. The
problem is the need for a joint limit. The marginal limiting distributions follow from

existing results in the literature. For v., we have

[11']

I
(9) 7—,]? évt 3 B10'),

where Bl(r) is a Brownian motion with variance 0%,; this is the marginal statement

corresponding to (8) above. For u., we have convergence to a fractional Brownian
motion, as given by equation (27) of Chapter 1:

[le
(10) W Zut => mdwd“),
t=1

where of, = aim —- 2d) / [(1+ 2d)F(1+ d)F(1— d)] and mm is the solution to

l

(11) Wd(f)=l.(d+l)

 

[(r — s)d dW(s),
0

with W(s) a standard Wiener process.

With these marginal results in hand, the only question is how to express the
joint result so that it properly reflects the covariance between the two limiting
processes. This covariance is also reflected in the covariance between B1(r) and B2(r)

in (8) above. Thus the standard Wiener process W(d) is in (1 1) above should in fact

be the specific process ong2(r), to capture this covariance. More specifically,

define

94

(12) F.(r)=——MI(r-s>dd32(s1

I‘(1+d)0

where 1(a) = of, /o§ = m- 2d)/[(1+2d)F(1— d)F(1+ <1)]. Then we have the joint
convergence result as given in the following lemma.
LEMMA 3: Suppose that (vt, at)' are iid (0, 2), that [B1(r), B2(r)]’ is a
Brownian motion with covariance matrix 2, that Fd(r) is the fractional

Brownian motion defined in (12), and that the model (1)-(3) above holds with

O< d <l/2. Then

 

— V
T :> .
1 [r ] Fd(1’)
d+1l2 Zut
-T t=1 .

 

Perhaps surprisingly, the joint convergence result of LEMMA 3 for a vector of
ordinary and fractional Brownian motions does not seem to exist in the statistical
literature. Our argument leading to LEMMA 3 was somewhat heuristic, but we
believe that it captures the essential ideas that would be part of a more rigorous proof.
In any case, with LEMMA 3 in hand we can proceed to the analysis of least squares
for the fractionally integrated model with 0< d <1/2.

LEMMA 4: Let the same conditions hold as in LEMMA 3. Then
1 T 1
Fthut => jBr(T)dFd(f)-
t 0

Proof: See Appendix. I

95
Therefore,under the conditions of LEMMA 3, we have (using LEMMA 1, part

(ii), and LEMMA 4) the following result for the asymptotic distribution of the OLS
estimate B in (1):

1 T 1
...,; 2...... I Bi(r)dFs(r)
(14) T“*"(B—B)= T ; => 0

1 1
$32th jBimdr
t o

 

 

Thus, for the case that the errors in the cointegrating relationship are I(d) with 0< d

<l/2, B- B is CPU/TN) = Op(T""). In particular, the value of d affects the order in

probability of the OLS estimate.

D. Nonstationary Long Memory Error Case (1/2< (I <1)

We now suppose that u., the error in the cointegrating relationship (1), is I(d)

with 1/2< d <1. Define d* = d-l, so -1/2< d* <0. Then Aut 2 pt is I(d*). Let the
innovations be represented by a. as in (3) above, so that

(15) (1— L)‘u. =(1—L)“p. = 2..

As in section C, we assume that the innovations (vt, at)' are iid (0, 2), so that the

partial sums of v. and e. converge jointly to [Bl(r), Bz(r)]' as in (8) above. We

define:

m

(16) Fd*(r)=I‘—_(l(+ (““0

I(f J5) (132(5)

corresponding to (12) above. Then

96

 

 

 

 

 

" X 1 — 1 [IT]
[1’1“] _‘th
(17) fi = fi,=, 3 Bio)
u[rT] 1 [EP Fat-(f)
”our/w LTow/2 H t—

which is similar to the result of LEMMA 4.

LEMMA 5: Under the assumptions listed above in this section,

1 T 1 T 1
d*+2 thut = d+1 thut 2) I B1(r)Fd‘(r)dr'
T t T t 0

Proof: See Appendix. I

From LEMMA 1 and LEMMA 5, we therefore have the following result for the

asymptotic distribution of the OLS estimate B:

 

 

 

1 T I
~ 1 .. sztut jBr(f)Fd*(f)df
08) T““"(B—B)=Td.(B—B)= 1 15 =0 1
p222 [Bfmdr
‘ 0

Thus, for the case that the errors in the cointegrating relationship are I(d) with 1/2< (I

<1, B— s is o,(1/T"") = 0,0“).

E. Remarks
We have considered the regression of y. on x., where y. and x. are 1(1), and
where the error is I(d). We have considered the cases: (1 = 0, d = 1, 0< d <1/2, and

1/2< d < 1. These correspond to all values of d in [0, 1] except d = 1/2, for which the

97

necessary convergence result for partial sums of an I(d) process is apparently not
available. While the form of the asymptotic distribution varies across cases, we have
the interesting result that B— B is CPU/T”) for all d in [0, 1] (except perhaps d=1/2).
Our findings can be compared with the result of Cheung and Lai (1993). They
showed that T(1_d_5)(B— B) converges in probability to 0 for all 8 >0. Our findings

confirm their results and provide the exact asymptotics of the OLS estimates in
fractional cointegration relationships.

It is interesting to compare these results to those for another simple estimator;
namely, least squares in first differences. Thus suppose that (1) is difi‘erenced to yield
(19) Ayt = AxtB + Aut

and a least-squares estimator

T T
Z AxtAyt ZAxtAut
" _ L .. i:2__.__.___
(20) B - T — B + T °
2 Ax,2 Z Ax,2
t=2 =2

If u. is I(d) with 0< (1 <1, then Aut is I(d*) with -1< d* <0. From Odaki (1993), this

is a stationary and invertible process. Since Axt is also a stationary and invertible
process, standard results indicate the following. First, TTIZAth converges in
probability to yxx a B(Axtz). Second, if qu a E(AxtAut), then TTIZAxtAut
converges in probability to y x“ , and TTUZZMXtAut — Tim) is asymptotically normal

with zero mean. This implies that B converges in probability to B»: E B +qu / Yxxa

93
and JT(B— B.) is asymptotically normal with zero mean. Thus B-B. is 0,,(T'm).

Thus, unlike the OLS estimator in levels (B), B is inconsistent when x. and u. are
correlated. Also unlike B, the rate of convergence of B does not depend on d. B

converges faster than B when d >l/2 (since 1/2> l-d) but slower than B when d

<l/2. Thus difi‘erencing is a poor idea when d <1/2, but it may be a good idea when d

>1/2.

4. SIMULATION RESULTS

In this section we provide some simulation results that support the relevance of
our asymptotic results of the previous section. The data are generated according the
equations (1)-(3) above. We choose B = 1 but this choice is not substantive. We also

choose (5‘,8 = 0 so that the v. and a. processes are not correlated, even

contemporaneously. The sample sizes considered are T = 50, 100, 250, 500, 1000
and 1500. The number of iterations in the simulation was 10000. The computations
were done in FORTRAN, using the normal random number generator
GASDEV/RAN 3 as in Chapter 2.

Table 4.1 gives results for the OLS estimator of B. It presents the mean, the
standard deviation, and the standard deviation multiplied by T”. Since
asymptotically the least squares estimator has estimation error that is Op (Td'l), we
expect the normalized standard deviation to approach a limit as T increases. This

appears to be true in Table 4.1, and in fact the normalized standard deviation does not

99
change much over the range from T = 100 to T = 1500. This supports the relevance

of our asymptotic theory, even for only moderate sample sizes. All of the estimates
are essentially unbiased, as would be expected given the strict exogeneity of the
regressors. As d increases with T fixed, the standard deviation of the estimate
increases and the normalized standard deviation of the estimate decreases.

Table 4.2 gives similar results, for a smaller set of values of d, for the estimate
of B obtained by least squares in differences. Once again the estimates are essentially
unbiased. Now the normalized standard deviation is the standard deviation multiplied
by Tm, since asymptotically least squares in differences has an estimation error that is
OP(T""2). The relevance of the asymptotic theory is supported again, since the
normalized standard deviation is more or less constant over difi‘erent values of T for
any given d. For given T, the standard deviation of the estimate does not depend
strongly on d.

Comparing results in Tables 4.1 and 4.2, we see that, in terms of the standard
deviation of the estimates, least squares in levels dominates least squares in
difl‘erences for d < .5, while the opposite is true for d > .5. The estimators have
similar variability when d is close to .5, but the difference between them increases as (1
moves away from .5 in either direction. This result is also as expected fiom the

asymptotics, based on the differing rates of convergence of the two estimators.

5. CONCLUDING REMARKS

100

This section has considered the case of fractional cointegration, defined as the
case in which a set of variables is I( l) but the regression error is I(d), d <1. This case
is empirically relevant, and very little is previously known about the properties of
estimates of the regression. We have derived the asymptotic distribution of the
ordinary least squares estimate, under a fairly strong set of assumptions, and
performed simulations that support the relevance of the asymptotic theory.

We assume that similar results would hold under weaker assumptions. In
particular, it would be worthwhile to extend these results to a more general model in
which there are multiple regressors, possibly including intercept and trend, and in
which the innovations are a general short memory process rather than white noise.

The results that we have derived are similar to the results for the usual
cointegration model, in that least squares is consistent, but the asymptotic distribution
is not necessarily centered at zero, and there is no reason to think that the estimator is
efficient or that it leads to asymptotically valid inference. In the cointegration
literature, these findings for the least squares estimator were followed by a large
volume of research that established asymptotically efiicient estimators and
asymptotically valid methods of inference. The same considerations should apply to
the case of fractional cointegration, and this would appear to be a valuable firture line

of research.

101
TABLE 4.1

Mean and Standard Deviation of OLS

Mean

 

d=.0 d=.1 d=.3 d=.49 d=.51 d=.7 d=.9 d=l.0

 

 

T=50 1 .0000 l .0019 .9998 .9992 1.0008 .9934 .9992 .9998
T=100 1.0000 1.0005 .9999 1.0000 .9994 1.0036 .9997 1.0037
T=250 .9999 1.0001 .9997 1.0002 1.0009 1.0022 .9970 1.0010

=500 1.0000 1.0001 1.0001 .9998 .9993 1.0013 1.0005 .9873
T=1000 1.0001 1.0001 1.0001 .9999 .9994 .9993 .9995 .9942
T=1500 1.0000 1.0000 1.0001 .9996 1.0002 .9994 .9957 .9982

 

 

 

 

 

 

 

Standard Deviation
d=.0 d=.1 d=.3 d=.49 d=.51 d=.7 d=.9 d=l.0
T=50 .0668 .0757 .1092 .1612 .1703 .2703 .4694 .6517

T=100 .0326 .0411 .0675 .1125 .1197 .2181 .4339 .6295
T=250 .0130 .0177 .0350 .0705 .0766 .1658 .3989 .6230
T=500 .0066 .0094 .0216 .0494 .0546 .1335 .3689 .6255
T=1000 .0033 .0052 .0133 .0345 .0389 .1081 .3356 .6355
T=1500 .0022 .0036 .0098 .0282 .0319 .0972 .3298 .6341

 

 

 

 

T“- Standard Deviation

 

d=.0 d=.l d=.3 d=.49 d=.51 d=.7 d=.9 d=1.0

 

 

T=50 3.34 2.56 1.69 1.19 1.16 .874 .694 .652
T=100 3.26 2.59 1.70 1.18 1.14 .868 .688 .630
T=250 3.25 2.55 1.67 1.18 1.15 .869 .693 .623
T=500 3.30 2.52 1.67 1.18 1.15 .861 .687 .626
T=1000 3.30 2.61 1.67 1.17 1.15 .859 .670 .636
T=1500 3.30 2.60 1.64 1.18 1.15 .872 .685 .634

 

 

 

 

 

 

 

102

TABLE 4.2

Mean and Standard Deviation of OLS in Differences

 

 

 

 

 

 

 

 

 

 

 

 

 

Mean

d=.l d=.3 d=.49 d=.51 d=.7 d=.9 d=1.0

T=50 1.0029 1.0072 1.0000 1.0007 1.0028 1.0004 1.0003
T=100 1.0016 1.0012 1.0012 .9987 1.0016 1.0007 1.0008
T=250 .9997 1.0003 1.0005 .9999 .9991 1.0014 .9999
T=500 .9989 .9997 .9998 1.0001 1.0006 1.0002 .9995
T=1000 .9996 .9997 1.0009 1 .0002 .9996 1 .0002 .9995
T=1500 .9995 .9999 .9997 1.0003 1.0006 1.0001 .9999

Standard Deviation

d=.l d=.3 d=.49 d=.51 d=.7 d=.9 d=1.0

=50 .1992 .1828 .1694 .1690 .1560 .1505 .1470
T=100 .1394 .1260 .1156 .1160 .1091 .1033 .1017
T=250 .0856 .0773 .0717 .0715 .0666 .0644 .0638
T=500 .0614 .0545 .0508 .0503 .0468 .0453 .0447
T=1000 .0432 .0390 .0360 .0356 .0340 .0318 .0316
T=1500 .0350 .0320 .0296 .0292 .0274 .0258 .0259

Tm- Standard Deviation
d=.l d=.3 d=.49 d=.51 d=.7 d=.9 d=1.0

T=50 1.41 1.29 1.20 1.20 1.10 1.06 1.04
T=100 1.39 1.26 1.16 1.16 1.09 1.03 1.02
T=250 1.35 1.22 1.13 1.13 1.05 1.02 1.01
T=500 1.37 1.22 1.14 1.12 1.05 1.01 1.00
T=1000 1.37 1.23 1.14 1.13 1.08 1.01 1.00
T=1500 1.36 1.24 1.15 1.13 1.06 1.00 1.00

 

 

 

 

 

103
APPENDIX

Proof of LEMMA 4: Note that x. is the partial sum of the innovations v.. Define Zt to

t
be the partial sum of the u.: Zt = Zuj . For re[0, 1], define the sample version of
i=1

the processes B.(r) and Fd(r) as follows:

 

 

[ 0, r<l
x (r)—__l_x —4__1_x _t_<r<.t:].
T JT [rT] fl ta T— T
I
—XT, 1' I
NT
1
0, r<l
I I 1 1+1
ZT(T)=WZM] =1 Flt, $ST<—T-
1
—Z , r=l.
NT T
Then
[Xr(f)] [BAD]
3
Zr“) Fa“)
1 1
I xs(r)dzs(r) 2 IBi(r)dF.(r>.
o 0
Thus,

1 T 1 1 . .
{XTUWZHU = §(—JT MAX-{3:13 ut) (e.g., Phillips (1986’ 9327))

104

1 T 1
= Tm szu, :s I B1(r)dFd(r).
t 0

But ill—+12%“: = #ZxHut #thut. So we need to show that
t , t t

l .
Wthut —) 0. To do so, we write
t

2
l l 2 l 2
— vu S — v — x
(T; ‘ ‘I (T; ‘IIT; ‘I
as in Cheung and Lai (1993, p106). Then %th2 —> 03,, %Zuf -> 0,2,, where the
t t

first result is standard and the second result follows from Hosking (1995). So

%thut is bounded in probability and T—rlfifzvtut —-> 0 for d >0. I
t 1

Proof of LEMMA 5: Define xT(r) as above, and

1
UT“) = mum]-

Then

1

1
I xT(r)UT(r)dr :> I B1(r)Fds(r)dr
0 0

because of the joint convergence result (17). But

1 '1' t/T

ij(T)Ur(f)df = Z ij(T)UT(T)dI

0 t=l(t—1)/T

105

Hi __1__,,
MT xt Tour/2 t

1
=> I Bl(r)Fds(r)dr

CHAPTER 5

CONCLUSIONS

106

107
This dissertation considered the ARFIMA(p,d,q) process, which can apply to

many economic time series. The long-run characteristics of an ARFIMA(p,d,q)
process, such as stationarity, mean-reversion and persistence of autocorrelations, are
determined by the differencing parameter value (1. We consider the case that the value
ofd is in the range -l< d _<_1. Ifd = 1 such a series is a unit root process and if-l< d
<1 and d at 0 it is a fractionally integrated process or long memory process. When
d=0, it is a usual stationary short memory process.

In this dissertation we showed how the KPSS unit root test works in
distinguishing long memory processes from unit root processes. Our asymptotic
findings indicate that the KPSS unit root test is consistent against stationary long
memory alternatives with -1/2< d <1/2, but it is not consistent against nonstationary
long memory alternatives with 1/2< d <3/2. This implies that the KPSS statistic can
consistently distinguish between short memory processes, stationary long memory
processes and nonstationary processes. Dickey-Fuller type tests can consistently
distinguish a unit root from an I(d) process with -1/2< (1 <1 but not from an I(d)
process with l< d< 3/2. Further work is needed on ways to distinguish unit root
processes fi'om nonstationary (but mean-reverting) long memory processes.

The estimation of the difl‘erencing parameter d is an interesting problem, and
there are many difl‘erent estimators, such as the GPH estimator, the maximum
likelihood estimator, the CSS estimator and the MDE. The MDE does not require
distributional assumptions and is relatively simple in computation. We considered

MDES including the AMDE of Chung and Schmidt (1995) for the general ARFIMA

108
model. In applying the MDES, we can estimate the model by letting short-run

dynamics follow an ARMA model, or we can estimate the pure I(d) model but omit
the first 8 low-order autocorrelations, which is a nonparametric approach. In this
nonparametric method we can expect some bias, especially when the ARMA
parameters have extreme values, due to misspecifying the short-run dynamics. We
compute the asymptotic bias that results from ignoring a fixed number (3 ) of low-
order autocorrelations in the case of simple ARFIMA(1,d,0) and ARFIMA(O,d,l)
processes. The asymptotic bias of the AMDE or the MDE is small when Z is large.
A derivation of the asymptotic properties of the MDE when E grows with T is an
important future task.

Fractional cointegration, defined as the case in which a set of variables is [(1)
but the regression error is I(d) with (1 <1, is empirically important but little is known.
We found that OLS is consistent and derived its asymptotic distribution. Its order in
probability and asymptotic distribution are afi‘ected by the value of d. The asymptotic
distribution of the OLS estimate in the case of fractional cointegration is not
necessarily centered at zero and there is no reason to think that OLS is eflicient or
that it leads to asymptotically valid inference. Finding an emcient estimate that leads

to asymptotically valid inference is another important topic for further research.

LIST OF REFERENCES

109

1 10
REFERENCES

Agiakloglou, C. and P. Newbold (1992), Empirical Evidence on Dickey-Fuller-Type
Test, Journal of Time Series Analysis, 13, 471-483.

Agiakloglou, C., P. Newbold and M. Wohar (1993), Bias in an Estimator of the
Fractional Difference Parameter, Journal of Time Series Analysis, 14, 23 5-246.

 

Andrews, D.W.K. (1991), Heteroskedasticity and Autocorrelation Consistent
Covariance Matrix Estimation, Econometrica, 59, 817-858.

 

Baillie, RT. (1995), Long Memory Processes and Fractional Integration in
Econometrics, Journal of Econometrics, forthcoming.

Baillie, RT. and T. Bollerslev (1994), Cointegration, Fractional Cointegation and
Exchange Rate Dynamics, Jouml of liin_ance. 49, 737-745.

Baillie, RT. and DD. Selover (1987), Cointegration and Models of Exchange Rate
Determination, International Journal of Forecasting, 3, 43-51.

Beran, J .A. (1992), Statistical Methods for Data with Long-Range Dependence,
fitistical Science. 7, 404-427.

Beveridge, W.H. (1921), Weather and Harvest Cycles, Economic Journal, 31, 429-
452.

Boes, D.C., RA. Davis and SN. Gupta (1989), Parameter Estimation in Low Order
Fractionally Difi‘erenced ARMA Models, Stothastic HydrologLand Hydrolics.
3, 97-1 10.

Brockwell, P.J. and RA. Davis (1991), Time Series: Theta and Methodg, 2nd
edition, Springer-Verlag, New York.

Campbell, J .Y. and N.G. Mankiw (1987 ), Are Output Fluctuations Transitory?,
Qaarterly Joumalfiof Economics, 102, 857-880.

Cheung, Y.-W. (1993), Tests for Fractional Integration: A Monte Carlo Investigation,
Journal of Time Series Analysis, 14, 331-345.

Cheung, Y.-W. and F.X. Diebold (1994), On Maximum Likelihood Estimation of the
Difl‘erencing Parameter of Fractionally Interated Noise with Unknown Mean,
Journal of Econometrics, 62, 301-316.

111

Cheung, Y.-W. and KS. Lai (1993), A Fractional Cointegration Analysis of
Purchasing Power Parity, Journal of Buainess and Economic Statistics, 11,
103-1 12.

Choi, I and PCB. Phillips (1993), Testing for a Unit Root by Frequency Domain
Regression, Journal of Eccarometrica, 59, 263-286.

Chung, C.-F. (1994a), A Note on Calculating the Autocovariances of Fractionally
Integrated ARMA Models, Ecopomic Letters, 45, 293-297.

Chung, C.-F. (1994b), Calculating and Analyzing Impulse Responses and Their
Asymptotic Distributions for the ARFIMA and VARMA Models, unpublished
manuscript, Michigan State University.

Chung, C.-F. and RT. Baillie (1993), Small Sample Bias in Conditional Sum of
Squares Estimators of Fractionally Integrated ARMA models, Empirical
Economics, 18, 791-806.

Chung, C.-F. and P. Schmidt (1995), The Minimum Distance Estimator for
Fractionally Integrated ARMA Models, unpublished manuscript, Michigan
State University.

Cox, DR (1984), Long-Range Dependence: A Review, Statistics: An Appraisal,
Proceedings 50th Anniversary Conference (Iowa State Statistical Laboratory).

Dalhaus, R (1989), Efficient Parameter Estimation for Self Similar Processes, Annals
of Statistics. 17, 1749-1766.

 

Davidson, J.E.H., D.F. Hendry, F. Srba and S. Yeo (1978), Econometric Modelling of
the Aggregate Time-series Relationship between Consumer’s Expenditure and
Income in the United Kingdom, Economic Jouma_l, 88, 661-692.

Davydov, Y.U. (1970), The Invariance Principle for Stationary Processes, Theory of
Probability and Its Applications, 15, 487-498.

Dejong, D.N., J.C. Nankervis, N.E. Savin and CH. Whiteman (1992), The Power
Problems of Unit Root Tests in Time Series with Autoregressive Errors,
Journal of Econometrig, 53, 323-343.

Dickey, DA. (1976), Estimation and Hypothesis Test'mg in Nonstationary Time
Series, Ph.D. dissertation ( Iowa State University, Ames, IA).

Dickey, DA. and WA. Fuller (1979), Distribution of the Estimators for
Autoregressive Time Series with a Unit Root, Journal of the America
Statistical Association, 74, 427-431.

112

Diebold, F.X. and M. Nerlove (1990), Unit Roots in Economic Time Series: A
Selective Survey, Advances in Econometrics. 8, 3-69.

 

Diebold, F.X. and GD. Rudebusch (1991), On the Power of Dickey-Fuller Type
Tests against Fractional Alternatives, Economics Lettera, 35, 155- 160.

Durbin, J. (1960), The Fitting of Time Series Models, Review of the Iptemational
flitisticg Institute. 28, 233-243.

Engle, RF. and C.W.J. Granger (1987), Co-integration and Error Correction:
Representation, Estimation, and Testing, Econometrica, 55, 251-276.

Engle, RF. and BS. Yoo (1987), Forecasting and Testing in Co-integrated Systems,
Journal of Econometrics. 35, 143-159.

Fox, R and MS Taqqu (1986), Large Sample Properties of Parameter Estimates for
Strongly Dependent Stationary Gaussian Time Series, Annals of Statistics, 14,
5 17-532.

Fuller WA. (1976), Introduction to Saatisticza Tim; Series, Wiley, New York.

Geweke, J. and S. Porter-Hudak (1983),The Estimation and Application of Long
Memory Time Series Models, Journal of Time Seriea Analysis, 4, 221-23 8.

Granger, C.W.J. (1980), Long Memory Relationships and the Aggregation of
Dynamic Models, Journal of Econometrica, 14, 227-238.

Granger, C.W.J. and R Joyeux (1980), An Introduction to Long-Memory Time Series
Models and Fractional Difl‘erencing, Journal of Time Series Analysis, 1, 115-
129.

Granger, C.W.J. and P. Newbold (1974), Spurious Regressions in Econometrics,
Journal of Econometrica, 2, 111- 120.

Hansen, LP. (1982), Large Sample Properties of Generalized Method of Moments
Estimators, Econometrica, 50, 1029-1054.

Hassler, U. and J. Wolters (1994), On the Power of Unit Root Tests against
Fractional Alternatives, Economics Letters. 45, 1-5.

Hauser, MA. (1992), Long Range Dependence in International Output Series: A
Reexamination, unpublished manuscript, University of Economics and Business
Administartion, Vienna, Austria.

1 l3
Hosking, J.RM. (1981), Fractional Differencing, Biometrika, 68, 165-176.

Hosking, J .RM. (1984), Modeling Persistence in Hydrological Time Series Using
Fractional Difi‘erencing, Water Resources Research. 20, 1898- 1908.

Hosking, J.RM. (1995), Asymptotic Distributions of the Sample Mean,
Autocovariance and Autocorrelations of Long-memory Time Series, Journal of
Econometrica, forthcoming.

Hurst, HE. (1951), Long-term Storage Capacity of Reservoirs, Transactions of the
American Society of Civil Engineers, 116, 770-779.

 

Hurst, HE. (1956), Methods of Using Long Term Storage in Reservoirs, Proceedings
of the Institute of Civil Eng1_n' eers, 1, 519-543.

Johansen, S. (1988), Statistical Analysis of Cointegrating Vectors, Journal of
Ecopomic Dynamics and Control, 12, 231-254.

Johansen, S. (1991), Estimation and Hypothesis Testing of Cointegration Vectors in
Gaussian Vector Autoregressive Models, Econometrica, 59, 1551-1580.

Kremers, J.J.M. (1989), US. Federal Indebtness and the Conduct of Fiscal Policy,
Journal of Monetary Economics. 23, 219-238.

Kwiatkowski, D., P.C.B. Phillips, P. Schmidt and Y. Shin (1992), Testing the Null
Hypothesis of Stationarity against the Alternative of a Unit Root: How Sure
Are We that Economic Time Series Have a Unit Root 7, Journal of
Econometrics, 54, 159-178.

Lee, D. (1994), Asymptotic Theory for Long-Memory Time Serfi, Ph.D. dissertation
(Michigan State University, East Lansing, MI).

Lee, D. and P. Schmidt (1995), On the Power of the KPSS Test of Stationarity
against Fractionally-Integrated Alternatives, Jouraal of Econometrics,
forthcoming.

Levinson, N. (1947), The Wiener RMS Criterion in Filter Design and Prediction,
Jougal of Mathematics a_nd Physifi, 25, 261-278.

Leyboume, BJ. and B.P.M. McCabe (1994), A Consistent Test for a Unit Root,
Journal of Business and Economic Statistics, 12, 157-166.

Li, W.K. and AI. McLeod (1986), Fractional Time Series Modeling, Biometrika, 73,
2 17-22 1.

114

L0, AW. (1991), Long-Term Memory in Stock Market Prices, Econometrica, 59,
1279-13 13.

Lobato, I. and RM. Robinson (1993), Averaged Periodogram Estimation of Long
Memory, unpublished manuscript, London School of Economics.

Mandelbrot, BB. (1971), A Fast Fractional Gaussian Noise Generator,Water
Resources Research. 7, 543-553.

Mandelbrot, BB. and J.W. Van Ness (1968), Fractional Brownian Motions,
Fractional noises and Applications, SIAM Review. 10, 422-437.

Mandelbrot, BB. and J.Wallis (1968), Noah, Joseph and Operational Hydrology,
\Jfllter Resources Research. 4, 909-918.

McLeod, AI. and KW. Hipel (1978), Preservation of the Rescaled Adjusted Range
1: A Reassesment of the Hurst Phenomenon, later Resources Research. 14,
491-508.

Nelson, CR and C.I. Plosser (1982), Trends Versus Random Walks in
Macroeconomic Time Series: Some Evidence and Implications, Journal of
Monetary Economics, 10, 139-162.

Newey, W.K and KD. West (1987), A Simple Positive Semi-Definite
Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,
Econometrg, 55, 703-708.

Odaki, M. (1993), On the Invertibility of Fractionally Difi‘erenced ARIMA Processes,
Biometrika, 80, 703-709.

Ogaki, M. and J.Y. Park (1992), A Cointegration Approach to Estimating Preference
Parameters, unpublished manuscipt, University of Rochester.

Pantula, S.G. (1991), Asymptotic Distributions of Unit-Root Tests When the Process
is Nearly Stationary, Journal of Business m Economic Statistics, 9, 63-71.

Park, J.Y. (1992), Canonical Cointegrating Regressions, Econometrica, 60, 119-143.

Park, J.Y. and B. Choi (1988), A New Approach to Testing for a Unit Root, Working
paper No. 88-23 (Center for Analytical Economics, Cornell University, Ithaca,
NY).

Park, J.Y. and P.C.B. Phillips (1989), Statistical Inference in Regressions with
Integrated Processes:Part 2, Econometric Theory. 5, 95-131.

115

Phillips, P.C.B. (1986), Understanding Spurious Regressions in Econometrics, Journal
of Econometrics, 33, 311-340.

Phillips, P.C.B. (1987), Time Series Regression with a Unit Root, Econometrica, 55,
277-301.

Phillips, P.C.B. (1988a), Weak Convergence of Sample Covariance Matrix to
Stochastic Integrals via Martingale Approximations, Econometric Theory. 4,
528-533.

Phillips, P.C.B. (1988b), Weak Convergence to the Matrix Stochastic Integral
1
IOBdB', Jounatl of Multivaaite Analysis, 24, 252-264.

Phillips, P.C.B. (1991), Optimal Inference in Cointegrated Systems, Econometrica 59,
283-306

Phillips, P.C.B. and SN. Durlauf (1986), Multiple Time Series Regression with
Integrated Porcesses, Review of Economic Studies, 53, 473-495.

Phillips, P.C.B. and BE. Hansen (1990), Statistical Inference in Instrumental
Variables Regression with 1(1) Processes, Review of Economic Studies. 57,
99- 125.

Phillips, P.C.B. and J.Y. Park (1988), Asymptotic Equivalence of Ordinary Least
Squares and Generalized Least Squares in Regressions with Integrated
Regressors, Journal of the American Statistical Association. 83, 111-115.

Phillips, P.C.B. and M. Loretan (1991), Estimating Long-Run Economic Equilibria,
Review of Economic Studies, 58, 407-436.

Phillips, P.C.B. and P. Perron (1988), Testing for a Unit Root in Time Series
Regression, Biometrika, 75, 33 5-346.

Press, W.H., B.P. Flannery, S.A. Teukolsky and WT. Vetterling ( 1989), Numerical
Recipes: the Art of Scientific Computing, Cambridge University Press.

Robinson, RM. (1994), Time Series with Strong Dependence, Advmes in
Econometrics. 6th World Congress. vol. 1, Cambridge University Press.

Rosenblatt, M (1956), A Central Limit Theorem and a Strong Mixing Condtion,
Proceedings of the National Aczgemy of Sciences, 42, 43-47.

Said, SE. and DA. Dickey (1984), Testing for Unit Roots in Autoregressive-Moving
Average Models of Unknown Order, Biometrika, 71, 599-608.

116

Saikkonen, P (1991), Asymptotically Efficient Estimation of Cointegrating
Regression, Econometric Theory. 7, 1-21.

Saikkonen, P. and R. Luukkonen (1993), Testing for a Moving Average Unit Root in
Autoregressive Integrated Moving Average Models, Journal of the American
Sgtistical Association, 88, 596-601.

 

Schmidt, P. and P.C.B. Phillips (1992), LM Tests for a Unit Root in the Presence of
Deterministic Trend, Oxford Bulletin of Economics_ and Sfitistics. 54, 257-
287.

Schwert, G.W. (1989), Tests for Unit Roots: A Monte Carlo Investigation, Journal of
Business and Economic Studies, 7, 147-159.

Seater, J .J . (1993), World Temperature-Trend Uncertainties and Their Implications
for Economic Policy, Journal of Busiawss and Economic Studies, 11, 265-277.

Shin, Y. and P. Schmidt (1992), The KPSS Stationary Test as a Unit Root Test,
Economics Letters. 38, 387-392.

 

Sowell, F. (1990), The Fractional Unit Root Distribution, Econometrica, 58, 495-505.

Sowell, F. (1992a), Maximum Likelihood Estimation of Stationary Univariate

Fractionally Integrated Time Series Models, Journal of Economatrics, 53, 165-
188.

Sowell, F. (1992b), Modelling Long-run Behavior with the Fractional ARIMA model,
Journal of Monetary Economics, 29, 2770302.

Stock, J .H. (1987), Asymptotic Properties of Least Squares Estimators of
Cointegrating Vectors, Economatrica, 55, 1035-1056.

Stock, J .H. (1990), A Class of Tests for Integration and Cointegration, unpublished
manuscript, Harvard University.

Stock, J .H. and M.W. Watson (1988), Testing for Common Trends, Journal of the
American Statistical Association, 83, 1097-1107.

Stock, J .H. and M.W. Watson (1993), A Simple Estimator of Cointegrating Vectors
in Higher Order Integrated Systems, Econometrica, 61, 783-820.

Tieslau, M.A., P. Schmidt and RT. Baillie (1995), A Minimum-Distance Estimator
for Long Memory Processes, Journal of Econometrics, forthcoming.

117

Whittle, P. (1951), Hypothesis Testing in Time Series Analysis, Almquist and
Wiksells, Uppsala, Sweden.

Whittle, P. (1956), Variation of Yield Variance with Plot Size, Biometrika, 43, 337-
343.

Whittle, P. (1963), On the Fitting of Multivariate Autoregressions, and the
Approximate Canonical Factorization of a Spectral Density Matrix,
Biometrika, 50, 129-134.

Yule, G.U. (1926), Why Do We Sometimes Get Nonsense Correlations Between Time
Series? A Study in Sampling and the Nature of Time Series, Journal of the
Royal Statistical Society. 89, 1-64.

    

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