hum!

. burglar.“

It 1},

if
a [flaw
31;; .
. {VIII A“.
.wu........i fl 1

:3 fig.

a '

mafia.
.33.,
3:!“

 

5.1.15
.33.: .
11.13-

7.5.3.“ ”mi.

9:
.v

c.232r

{'30. ..«

. I}: :5:
il’

 

 

'MES’Q

lUlNiallliflfllllllllllll'llllllllllllIll "

01770 4341

This is to certify that the

dissertation entitled

Essays on Economic Time Series:

Theory and Application

presented by
Jen-Je Su

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Economic s

WSQgQX/

Major professor

Date EebruargLL6_,_LQ_Q9

MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771

 

LIBRARY

Michigan State
University

 

 

 

PLACE lN RETURN BOX to
TO AVOID FINE return
MAY BE RECALLED with earlier d

remove this checkout from your record.
on or before date due.
ue date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

@131ng

972809

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1M COMM“

ESSAYS ON ECONOMIC TIME SERIES: THEORY AND APPLICATION
by

Jen-je Su

A DISSERTATION
Submitted to
Michigan State University
In partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1 999

ABSTRACT
ESSAYS ON ECONOMIC TIME SERIES: THEORY AND APPLICATION
by

Jen-je Su

This thesis is made up of three essays, each of which is related to the KPSS test
(Kwiatowski, Phillips, Schmidt and Shin (1992)). The first essay gives two extensions of
Schmidt (1993) which shows that the KPSS test becomes inconsistent against unit root
alternatives if it is based on data detrended in differences (instead of detrended in level).
First, we find that the same result holds for the Leyboume-McCabe (1994) modification
of the KPSS test. We also find the same result for the KPSS test when long memory
alternatives are considered. The second essay provides an extension of the KPSS test to a
multivariate setting. The resulting statistic is a recognizable algebraic generalization of
the KPSS statistic. We find that the test based on this statistic is consistent against long
memory and unit root alternatives, and simulations show that there is a non-trivial power
gain from using the multivariate test instead of applying the KPSS test separately to each
series. The third essay applies the multivariate KPSS test to the so-called convergence
question. By applying the multivariate stationarity test, we reject the hypothesis of joint
convergence for the entire set of 15 OECD countries. We then use a clustering algorithm
to construct "convergence clubs." There appear to be four or five clubs of moderate size.
We also consider the question of convergence in growth rates. We find that the entire

sample can be split into two convergence clubs.

Dedicated to Shu-shyan

ACKNOWLEDGMENTS

First of all, I would like to thank my dissertation advisor Professor Peter Schmidt,
for his guidance, support (both academically and financially) and patience. I am so
privileged to have an advisor like him. I also wish to thank my other committee
members, Professor Jeff Wooldridge and Professor Robert DeJong for their helpful
comments.

There are many people important to me during this whole process. Among them, I
would never forget the spiritual nourishment from Jim and Karen, the encouragement
from Hailong and the fellowship with Ching-cheng, Hung-liang and Enoch. Also, I value
the friendship of Toshihiro, Chung-Jung and Wei-Kuo.

My special appreciation goes to my wife, Shu-shyan. Her love meltdowns the snow
in Michigan and warms my heart. I also would like to express my thanks to my parents
for their unconditional support.

Finally, I give my foremost thanks to Jesus Christ, my Savior and Lord.

Jesus said, "Come to me, all of who are weary and burdened, and I will give you
rest. Take my yoke upon you and learn from me, for I am gentle and humble

in heart, and you will find rest for your souls. For my yoke is easy and my
burden is light." (Matthew 11:28-30)

TABLE OF CONTENTS

CHAPTER 2 ON THE ASYMPTOTICS OF SOME TESTS USING DATA

DETRENDED IN DIFFERENCES .................................................. 4
2.1 Introduction .......................................................................... 4
2.2 Preliminaries ......................................................................... 5

2.3 On the Asymptotics of the Revised KPSS Test against I(d) Alternatives 10

2.4 On the Asymptotics of the Revised Leyboume/McCabe Test ............... 13
2.5 Conclusion .......................................................................... 17
Appendices .............................................................................. 19

CHAPTER 3 A SCORE-BASED TEST OF THE NULL OF MULTIVARIATE

SHORT MEMORY AGAINST UNIT ROOT AND LONG MEMORY

ALTERNATIVES ................................................................... 27
3.1 Introduction ....................................................................... 27
3.2 Derivation of the Statistic ........................................................ 29
3.3 Asymptotic Theory (1): Short Memory and Unit Roots ..................... 32
3.4 Asymptotic Theory (1): Long Memory ........................................ 37
3.5 Simulations: Unit Roots ......................................................... 43
3.6 Simulations: Long Memory ..................................................... 48
3.7 Conclusion ........................................................................ 51
Appendices . ........................................................................... 65
CHAPTER 4 CONVERGENCE CLUBS AMONG OECD COUNTRIES ...... 72
4.1 Introduction ........................................................................ 72

4.2 Convergence Hypotheses ......................................................... 74

4.3 Convergence Measures ............................................................ 77
4.4 Finding Convergence Clubs ...................................................... 81
4.5 Data Descriptions and Preliminary Results .................................... 83
4.6 Empirical Results on Finding Convergence Clubs ............................ 85
4.7 Conclusion .......................................................................... 88
Appendix ................................................................................ 96
LIST OF REFERENCES ................................................................... 97

vi

Table 3-1

Table 3-2

Table 3-3

Table 3-4

Table 3-5

Table 3-6

Table 3-7

Table 3-8

Table 3-9

Table 4-1

Table 4-2

Table 4-3

Table 4-4

Table 4-5

LIST OF TABLES

Critical Values for the MKPSS Test .......................................... 53
Size ................................................................................... 54
Power against [(1) Alternatives (I) ............................................. 55
Power against [(1) Alternatives (II) ............................................ 56
Power against Cointegrated 1(1) Alternatives ............................... 57
Power against Long Memory Alternatives (d1 = d2) ....................... 58
Power against Long Memory Alternatives (d, ¢ d2) ....................... 60
Size with AR(1 ) Errors ............................................................ 63
Power against Long Memory Alternatives .................................. 64
Test of Stationarity ............................................................... 90
Number of Convergence Pairs (out of 105 pairs) ........................... 91
Multivariate Convergence Clubs with MKPSS Measures ................. 91
Multivariate Convergence Clubs with Hobijn-Franses Measures ....... 93
Comparisons ........................................................................ 94

vii

CHAPTER 1

INTRODUCTION

The KPSS test of Kwiatowski, Phillips, Schmidt and Shin (1992) was proposed as a
test of the null hypothesis that an economic time series is stationary against the
alternative that it has a unit root. Lee and Schmidt (1996) noted that the asymptotic
distribution theory on which the test is based actually assumes that the series has short
memory, and they therefore proposed that the KPSS statistic could be used to test the null
hypothesis that an economic time series has short memory against the alternative that it
has long memory (e. g. is fractionally integrated) or has a unit root. In conjunction with
unit root tests, the KPSS statistic can then be used to distinguish short memory, long
memory and unit roots.

This thesis is made up of three essays, each of which is related to the KPSS test.
The first essay considers the effect of the method of removing deterministic trend on the
power of the test. Unit root tests typically remove trend by linear regression in levels, but
some unit root tests (e.g. Schmidt and Phillips (1992)) detrend in differences. The
resulting tests are consistent against stationary alternatives. Conversely, Schmidt (1993)
Showed that the KPSS test becomes inconsistent against unit root alternatives if it is
based on data detrended in differences. The first essay gives two extensions of this
result. First, it shows that the same inconsistency result holds for the Leyboume—McCabe
(1994) modification of the KPSS test. Second, it shows that the KPSS test is also
inconsistent against long memory alternatives when it is based on detrending in

differences.

The second essay provides an extension of the KPSS test to a multivariate setting.
In some cases we may have a set of variables for which we wish to distinguish between
stationarity and unit root. Univariate unit root tests can be applied to each series in turn,
or there are multivariate unit root tests that can be used to test the null hypothesis that
each series has a unit root against the alternative that one or more are stationary.
Similarly, the KPSS test can be applied to each series separately, or one might wish to
use a multivariate test to test the null hypothesis that each series is stationary (actually,
short memory) against the alternative that one or more series have unit roots or long
memory. This essay provides such a multivariate test. It is derived as the LM test of the
hypothesis that the variances of the random walk components of the series have zero
variance, under a restriction on the covariances. The resulting statistic is a recognizable
algebraic generalization of the KPSS statistic. The essay shows that the test based on this
statistic is consistent against long memory and unit root alternatives, and simulations
show that there is a non-trivial power gain from using the multivariate test instead of
applying the KPSS test separately to each series.

The third essay applies the multivariate KPSS test to the so-called convergence
question. Certain theories of economic growth imply that countries' output levels Should
converge over time, and one definition of this convergence is that differences of output
levels should be stationary. Bernard and Durlauf (1995) applied multivariate unit root tests
to data on 15 OECD countries, and failed to reject the hypothesis of non-convergence, but
this raises the issue of whether one can reject the hypothesis of convergence. Hobijn and
Franses (1997) tested the stationarity of output differences for pairs of countries and

generally failed to reject the convergence hypothesis. This essay, like Hobijn and Franses

(1998), applies the multivariate stationarity test to the entire set of 15 countries, and rejects
the hypothesis of joint convergence for the entire set of countries. It then uses a clustering
algorithm to construct "convergence clubs" within which the convergence hypothesis
cannot be rejected. There appear to be four or five clubs of moderate size. The essay also
considers the question of convergence in growth rates, which is the hypothesis that the

growth rates have the same mean for each country. This hypothesis is tested using a
modification of Hotelling's T 2 test, and is rejected, but the entire sample can be split into
two convergence clubs within which the hypothesis of convergence of grth rates is not

rejected.

CHAPTER 2
ON THE ASYMPTOTICS OF SOME TESTS USING DATA DETRENDED IN

DIFFERENCES

2.1 Introduction

Tests of either the stationarity hypothesis or of the unit root hypothesis are typically

based on residuals from the following data generating process (DGP):
y,=a+,61+g,,t=l,2, ,T, (1)
where {y, } L. is the observed series and {6, },:1 represents the unobserved deviations from

the deterministic (linear) trend. However, in practice, different types of residuals can be
used, based on different methods of estimating the parameters or and B in ( l ). Below are
two examples:

[1]. OLS residuals from the regression of y, on (1, t);

[2]. BSP (Bhargava-Schmidt-Phillips) residuals -- residuals that are based on

parameters (or, B) estimated in differences (Ay, = ,6 + As, ):
£3“ = mean<Ay,) = (YT — y. )/(T — 1)

§=yl—fi,

=(yp’yl)"(t—1)(yr ‘y|)/(T—1)

= (a, — £,)—(t-1)(8T — 5,)/(T — 1) , t=l,2, ,T.

KPSS (1992), Lee and Schmidt (1996), Leyboume and McCabe (1994), and Dickey-
Fuller tests are based on [1], while Bhargava (1986), Schmidt and Phillips (1992) and
Schmidt (1993) are based on [2].

Unit root tests that are consistent against trend stationary alternatives can be based on
either type of residuals. See Schmidt and Phillips (1992), for example. However, tests of
stationarity against unit root alternatives seem not to Share this kind of flexibility. Schmidt
(1993) shows that the revised KPSS (stationarity) test based on BSP residuals is
inconsistent against unit root alternatives.

In this chapter, we generalize Schmidt’s result in two ways. First, we consider long
memory alternatives. Lee (1995) and Lee and Schmidt (1996) have Shown that the KPSS
test, viewed as a test of the null hypothesis of short memory, is consistent against long-
memory alternatives. In section 2.3 we Show that the revised KPSS test is inconsistent
against long memory alternatives. Second, we consider the test of Leyboume and McCabe
(1994), which is similar to the KPSS test but handles short-memory autocorrelation in a
different way. In section 2.4 we Show that, if their test is revised to be based on detrending

in differences, it becomes inconsistent against unit root alternatives.

2.2 Preliminaries

KPSS describe their test as a test of the null of trend stationarity. More precisely,
they test that the deviations of a series from deterministic trend are short memory (as

defined in Assumption Ho below). The DGP considered by KPSS is equation (I ).

Let é, be the residuals from a regression on intercept and time (t), and let S, be the
partial sum process of the é,: S, = Z’Flé, , t =1,2, ,T. Let 77 be the long run variance of
the errors 5,, and consider the Newey-West (1987) estimator of 0'2 :

52(1)- — T 'Lléz + 2T '1; W(s 1):: ,_,_,é,é, _,,
where w(s,l) =1—s/(l+1), which guarantees that 52(1) 2 O. For consistency of .30) ,

throughout this chapter we will assume that the lag truncation parameter 1 satisfies: 1 —> 00
but l/T—>0 as T—)oo.

The KPSS statistic is defined as follows:
I): T42,"=l sf /s “2( (2)
Following Schmidt (1993), the revised-KPSS statistic based on BSP residuals E, (as in [2])

can be defined similarly:
5 = T42; S? Ma), (3)
where S, ,§2(l) are the same as S, ,S2 (1) except replacing é, by 'e', .

To analyze the properties of the KPSS and revised-KPSS tests, we make the

following (alternative) assumptions about a, in the DGP above.

Assumption Ho (Short memory) (i) The long run variance 0'2 , 0'2 = lim T"E[2,r=| a 2
exists. (ii) An invariance principle holds for the partial sums of the 5,. That is.
Til/22:8 I :> W (r) for re [0 1]. (Here [rT] denotes the integer part of rT, “2)” denotes

weak convergence, and W(r) is the standard Wiener process.)

Assumption Hl (Unit root) An invariance principle holds for the 8,. That is,

714/28,,“ 2) O'W(r) for re[0,l], with 0' > 0.

Following Schmidt (1993), for expositional simplicity, we have assumed just the
necessary central limit theorems. However, for the consistency of the Newey-West
estimate, we need to specify some regularity conditions like, for example, Phillips and
Perron (1988, p.336). Throughout this chapter, we implicitly assume that these conditions
hold as necessary. We summarize some of the asymptotic properties of the KPSS statistic

and the revised-KPSS statistic as the following four propositions.

Proposition 1 (KPSS (1992, p.167)) Under H0 (short memory), I] :> I; V2 (r)2dr , where

V,(r) = W(r) + (2r -— 3r2)W(l) + (—6r + 6r2) 1,; W(s)ds is the second level Brownian bridge.

Proposition 2 (KPSS (1992, pp.l68—69)) Under Hl (unit root),
(1 / T)7’j => [qugW‘(s)ds)2da]/[jgW*(s)2ds], where W'(s) is the demeaned and detrended

Wiener process: W’(s) = W(s) + (65 — 4) Wow + (—-12s + 6) 1,; rW(r)dr .

Proposition 3 (Schmidt (1993, p.4)) Under HO (short memory),

(I / T)fi :> (363, + 95008, + 8.912)/[20(a,,2O + 6008' + 8'2 )]. Here a“, is the weak limit of 6., as

T—-)oo.

Proposition 4 (Schmidt (1993, p.3)) Under H, (unit root),
(1 / T)t7 2 [[3 ([6 V(s)ds)2dr] /[[,1) V(s)2ds] , where V(s)=W(s)—sW(1) is a standard Brownian

bridge.

Comparing Propositions l and 2, we see that, since T /l —> 00 as T —9 co, the KPSS

test is consistent. However, comparing Propositions 3 and 4, since i] is 0,,(T/ I) under

both the null and alternative hypotheses, we conclude that the revised-KPSS test is
inconsistent. The results show us that data detrending procedures are important, even
asymptotically.

Lee and Schmidt (1996) and Lee (1995) apply the KPSS statistic to test the null of
short memory against the alternative that 8, follows a fractionally integrated, or I(d),
process in the sense of Granger (1980), Granger and Joyeux (1980) and Hosking (1981). A
process a, is said to be I(d) if (1— L)" a, = u, , where L is the usual lag operator, d is the
differencing parameter (which can be a fractional number) and u, is white noise. More
generally, one could allow u, to be a short memory process (as defined in Assumption HO ),
but we do not consider this generalization in this chapter.

An I(d) process is stationary and invertible for d in the range of (-1,1/2). Its
autocovariance function decays slowly, at a hyperbolic rate rather than at the usual
exponential rate found in conventional ARMA models. For -l/2<d<1/2, an I(d) process is
said to have “stationary long memory” since it is stationary, but it exhibits long range
dependence. And, for l/2<d<3/2, the process is called “nonstationary long memory”. Lee
and Schmidt assume that (1 belongs to a range (-l/2,l/2), while Lee (1995) assumes that d

belongs to the range (l/2,3/2).

For the purposes of asymptotics under the hypothesis that a, is I(d), we make the

following assumptions.

Assumption H: (Stationary long memory) a, is I(d) with de(-1/2,l/2) but d¢0.

(1— L)"£, = u, where u, is i.i.d. N(O,0'2).

Assumption H ," (Nonstationary long memory) a, is I(d) with de(l/2,3/2).

(1 — L)"e, = (1 — L)"'(1- L)e, = u, where d‘ = d -1 e(-l/2,l/2) and u, is i.i.d. N(0,o~2).

As noted above, we could consider a, to be a long-memory process so long as u, is a
short-memory process in the sense of Assumption H0 above. However, stronger

assumptions on u, are needed to justify the relevant limit theory. The assumption that u, is

i.i.d. N(O, 02) is the same as in Lee and Schmidt (1996), and is stronger than necessary.

See Chung (1996) for central limit theorems under somewhat weaker assumptions.

Proposition 5 (Lee and Schmidt (1996), pp.29l-92) Under H,’ (stationary long memory),
(I / T)2dfi 2 [A V d (r)2 dr where V,, (r) is a second order fractional Brownian bridge:
Vd(r) = Wd(r) + (2r — 3r2 )Wd(1) + (—6r + 6r2 )1; Wd(s)a's . where

Wd(r) -—_- [go — s)"dW(s)/r(1 + d).

Comparing Proposition 1 and Proposition 5, since i] is 0,,(1) under H0 and
0,,[(T/l)2"] under H: (so that 797—), 00 for 0<d<1/2 and $74,, 0 for -1/2<d<0), we

conclude that the KPSS test is consistent against the stationary long memory.

Proposition 6 (Lee (1995, pp26-27)) Under H 1” (nonstationary long memory),
(1/T)rj -_—> [13, (ngd’. (s)ds)2da]/[ t), Wd‘. (s)ds], where d‘ = d -1 [with d‘ e(—l / 2,1 / 2)]
and W;. (s) is the demeaned and detrended fractional Wiener process:

Wd‘. (s) = Wd. (3) + (65 — 4) j), Wd. (r)dr + (—l2s + 6) jngd. (r)dr .

Comparing Proposition 2 and Proposition 6, we see that the KPSS statistic is of the

same order, 0,,(T/l) , in the range of (1/2,3/2). Since T/I —> 00 as T —> co, the KPSS test

is consistent against nonstationary long memory alternatives.

2.3 On the asymptotics of the revised KPSS test against I(d) alternatives

In this section, we are concerned with the asymptotic properties of the revised-KPSS
test under I(d) alternatives. The asymptotic distribution of the revised-KPSS test under the

null of short memory was given in Proposition 3 above.

Lemma 1 Denote the weak limit of 8,. as T —> 00 by 5,0. Then, under Assumption H: , we

have

10

(i) 7—1151“: —r3, — r2(5,,, — 8,) / 2 ,
(ii) T42; S? :> (3.9,, + 98008, + 85,2) I60.
Proof: T"§,,,, = T-' 2:38, _ T"[rT]£, — T“(T—1)"(a,. — 5,)2'1’130- 1)
1. T" We —),, 0 since the partial sum of a, is of order 0,,(T‘m’2). (See Lee and

J=1 J

Schmidt (1996), Lemma 1.)

2. T"[rT]£, :> rs,.

3. T"'(T—1)*'(a,- — soz‘jjo—I)

= T"(T—1)"(s.,. ~— s,){([rT]-1)[rT]/2} :> %r2(£w - 3,).

By 1,2, and 3, the result (i) follows. And, (ii) can be easily proved in the same way as

in Schmidt (1993), Lemma 3. 0

Lemma 2 Under Assumption H: , [432(1) :> (53;, + 8,08, + 8,2 ) / 3.

Proof: See Appendix 1.6

Lemma 1 and 2 show that the statistic based on the BSP residuals will have an
asymptotic distribution that depends on the marginal distribution of .9, even under the
alternative hypothesis of stationary long memory. This is different from the result of
Schmidt (1993) where the revised-KPSS statistic has this kind of novel asymptotic property

only under the null hypothesis.

w 353, + 96005, + 8e,2
1452(1) 20(83, + 6003, + 8,2)

 

Theorem 1 Under Assumption H: , (l / T )Fj 2

Proof: The result directly follows from Lemma 1 and Lemma 2. 0

Comparing Proposition 3 and Theorem 1, we see that 17 = 0,,(T/I) under both the

null of short memory and the alternative of stationary long memory. We conclude that the

revised-KPSS test is inconsistent against long-memory alternatives.

Lemma 3 Under Assumption H ," ,

(i) T‘3/2‘d SW, :> wd. [53,. (s)ds,
.. —4—2d' T ~2 I r . 2
(11) T ES, :> wd.[0[j0 Bd.(.s)ds] dr.

Proof: See Appendix [1.0

Lemma 4 Under Assumption H," , I"T"'2d 372(1) :> (0:. [3 Bd. (s)2ds.

Proof: See Appendix 111.0

Theorem 2 Under Assumption H ,l l , (I / T )77 :9 [[3 (ngd. (.s')ds)2dr]/[j,l, Bd. (s)2ds]

Proof: By Lemma 3 and Lemma 4, we have

T—z-zd‘ (T—z 27‘ 3:12 )

t=l

(1/ T)” = .
n l-lT-l-Zd ’52(1)

2 113,053,.(sown/[$13,(srdsi ..

Comparing Proposition 3, Theorem 1 and Theorem 2, we see that the revised-KPSS

statistic is of the same order, 0,,(T / I), for d in the range of (-1/2,3/2). In other words, the

revised-KPSS test is inconsistent no matter whether we test “short memory against

stationary long memory” or “short memory against nonstationary long memory”.

2.4 On the asymptotics of the revised Leyboume/McCabe test

Leyboume and McCabe (1994) propose a test of the null of short memory that is
similar to the KPSS test. Like the KPSS test, the Leyboume/McCabe test can be derived as
a one-sided LM test and is based on detrending in levels. However, the tests differ in their
treatment of the stationary component existing in the detrended data. That is, they differ in
the way that they allow for short-memory autocorrelation in 8,.

The KPSS test makes only the weak assumption H0 about 8,, and allows for short-

memory autocorrelation through a nonparametric estimate of the long-run variance.
Leyboume and McCabe make a parametric assumption, as follows. They assume the

model
(DP(L)y,=a+,Bt+v, (4)
where (1),,( L) = l— (15, L — a), L2 —...—¢,, L” is a pth order polynomial with roots outside the

unit circle, and v, is i.i.d. (0, o2 ). Thus, defining y, = (1),,(L)y, , we could apply the KPSS

statistic with [=0 (i.e. with no autocorrelation correction) if y, were known, or equivalently

if (1),,(L) were known.

13

The assumption that v, is i.i.d. in equation (4) is equivalent to the assumption that, in
equation (1), 8, = v, / (1),,(L). This is just the assumption that 8, follows a stationary AR(p)

process.

AssumptionHé (Stationary AR(p)) 8, = v, / (1),,(L), where v, is i.i.d.(0,02) and
(1),,(L) = 1—— ¢, L — ¢2L2 —...-—¢,, L”, with the roots of (I),,( L) outside the unit circle. That is,

8, is a stationary AR(p) process.

AssumptionH,’ (Unit root) 8, = v, /(D,,(L) with Av, i.i.d.(0,o‘2).

The Leyboume/McCabe test statistic is based on the “locally best test” (LBT)
statistic, which is the same as the one sided LM statistic. It is proportional to the quadratic

form e'Ms, where 8=[ 8, , 82 ,..., 8., ]' is 3 Tx 1 vector and M is a Tx T matrix with the i,jth

element equal to the minimum of i and j. In practice, {8, },’;, is unknown and must be

replaced with a residual. The residuals are calculated as follows:

(1). Construct y,‘ = y, —Z:, ¢,'y,_, where ¢,' are the normal quasi maximum likelihood

estimates of 4’, from the fitted ARIMA (p,l,l) model: A y, = ,B+ 2:, ¢,Ay,_, + A 8,.

According to Potscher (1991), ¢,' is a consistent estimate of ¢,.
(2). Calculate the residuals, denoted by e, , from the least square regression of y: on an

intercept and time trend.

Note that procedure (I) is the elimination of stationary “AR(p) components” in y, 9 and
procedure (2) is the elimination of the “trend components” in the remaining part of y, (i.e.

y: ). So, a, are estimators of the i.i.d. process v,. The Leyboume/McCabe test statistic is

defined as:

.., . 2

'r'=——(e..M.‘f’)/T . (5)
(e'e)/T

If we define S, = 2,1,8, , then é'A/fe' = 2,1,5? . Also are“ / Tcorresponds to s2(l)

when l = 0. Thus the Leyboume/McCabe statistic is of the same form as the KPSS statistic

with 1 =0, except for the difference between a, (in Leyboume/McCabe) and (3, (in KPSS) 8,

corresponds to 8', with (I),,( L) =1.

Proposition 7 (Leyboume and McCabe (1994, p.159)) Under Hg (short memory),

t :13, V2(r)2dr.

This is the same asymptotic distribution as was given for i] in Proposition 1.

Proposition 8 (Leyboume and McCabe (1994, p. 1 59)) Under H,’ (unit root), (l/T)'r

converges to some distribution.

According to the above two propositions, it is easily to conclude that the test
proposed by Leyboume and McCabe is consistent against unit root alternatives. Also,

although Leyboume and McCabe do not explicitly give the distribution to which (l/T)'i

converges under H,’ it is the same as the distribution given for (l / T) i] in Proposition 2
above.

Data detrending may be carried out in differences instead. Thus, we can define a
revised Leyboume/McCabe test, based on the BSP residuals constructed from y,’ . These

residuals, denoted by E, , are obtained by the following procedure.

(1)'. The same as (1); that is, construct y,‘ = y, — 2:, ¢,‘y,_, .
(2)'. Calculate e, by y,' — 8 ~31 where ,3: mean(Ay,' ) and Er: y; —,3. That is,

E. = (y: —y.')—(t —1><y; -.vI )/(T—l). t=1,2, ,T-

(e'M‘é)/T2

The revised Leyboume/McCabe test is 'r = JV
(e e)/ T

, and its asymptotic

distribution will be derived as follows.

Lemma 5 Under H5, (i) (é'M‘é)/ T3 :>(3v: +9v,vt +8vf)/60;
(ii) (é'é)/T:>o2 +(v3, +v,2 +v,v,)/3.
Proof: See Appendix IVA
Theorem 3 Under H6, (1/ T)? :> (3": + 9vwv, + 8v,2)/[600’2 + 20(v: + vmv, + v,2 )].

é'ME/Tz _ é'Mé/T3

. ) , the result follows from Lemma 3. O
E e /T

1
Proof: Since 1/ T f: — .
( ) (TX éé/T

The result in Theorem 3 is essentially the same as the result in Schmidt (1993, p.4)

for the KPSS statistic with [=0 (and white noise errors).

Lemma 6 Under H,', (i) (12"Mé)/T4 :> 0'2 [,1,[j0'V(s)ds]2dr ;
(ii) (Ea/T2 => 82 ng(s)2ds.
Proof: The proof follows the same lines as for Lemma 5. In the expression (A12) for é,,

the term 17, dominates the other term asymptotically, and the proof then follows as in

Theorem 1 of Schmidt (1993), p.3. 0

Theorem 4 Under 11;, (HT)? :> {[,',[[,;V(s)ds]2dr} /[j,‘, V(s)2ds].

é'Mé/T2)_é'Me'/T‘

l s 1
Proof: Since —— T: —— . — .
(T) (TX éé/T e'é/TZ

, the result follows directly from Lemma

6.0

The result in Theorem 4 is essentially the same as the result in Schmidt (1993, p.3) for
the KPSS statistic with (=0.

Comparing Theorem 3 and Theorem 4, we see that T: 0,,(T) under both H5 and H,'.

Thus, we conclude that the revised Leyboume/McCabe test of stationarity based on BSP
residuals is inconsistent against unit root alternatives.

2.5 Conclusion

The KPSS test is based on data detrended in levels and is known to be consistent
against unit root alternatives and also against fractionally integrated alternatives. Schmidt

(1993) showed that a revised KPSS test based on data detrended in differences (BSP

l7

residuals) is inconsistent against unit root alternatives. In this chapter, we Show that the
revised KPSS test is also inconsistent against fractionally integrated alternatives.

Leyboume and McCabe (1994) have suggested a test that is similar to the KPSS test,
but that differs in the way it allows for short-run error autocorrelation. Their test is known
to be consistent against unit root alternatives. This chapter considers a revised version of
their test, based on BSP residuals, and shows that it is inconsistent against unit root
alternatives.

It is interesting that the consistency of unit root tests is not affected by the choice of
detrending procedure, while the consistency of tests of the null of stationarity (or short
memory) is affected. Further research is needed to understand the connections between
hypotheses, test statistics and types of detrending, so that we can move beyond case-by-

case results.

18

APPENDICES

Appendix I (Proof of Lemma 2)

I
52(1): rim) + 22 w(s.l)r_; where 7,: %
3:}

First term: 8'2 (O)

3' (0)=T§(8 -6,2) +T§[(-T-_—,)(£— '1 81)] —-T-§[(T

 

[ngt ‘31)(81'_ 81)]- (A1)

First term on the right hand side of (A1 ):

1T 17‘ 2 2 2 T 2 F(1-2d)
-7:§(8, ‘g|)2 27§51+£|—781§6t20:+8| Where 0i =W

7' — T
%Z 8,2 :> ————Il:g (2;? 02 , where F(')is the nga function, and %Z 5: —>,, 0
(=1 —

(=1

0'2 . For

(see Lee and Schmidt (1996), Lemmal).

Second term on the right hand side of (Al):

 

 

 

 

 

1T t—I 2 (8i—2£w£,+£,)’f.o 1
__ ,.-— 3 .
T§1(T_,)(€I 81)] 31(T.1_)3
Third term on the right hand side of (A1):
2 7’ t-I
7§[(T—1)(£' “51)(87‘ _£/)l
2 1 I
= ,. {—1 —2 .. 00—5 .
(£11)_T(T 1)§( )5,“ 51(51" 1)].(7.__11)th:> I)
Fro

. 7‘
2(t — I)8, —),, 0, since T“(‘“3"”Zt8, :> [:rdW,,(r) -- see Tsay
T(T1— 1),-/ (=1

19

and Chung (1995), Lemma 1; and

 

t—>1/2.
T(T-1)§

Therefore, collecting terms, we obtain:

83, + 8008, + 8,2

3 (A2)

32(0):>a§ +

 

I
Second term: 2 w(s,l)"r:

s: I

Let r; = a!” + 7,”! + rs!“ where

'1‘” =— Eta -£1)(£,_. ~61),

T‘FS'H

~42]- —S'-l 5' __£, 2
r‘ T,§.1(T— 1X T— 1XT ‘)

 

and

—l t

rs“ =-— z 1<———>(e.—s.)+(—T%',>(ar_.-a)l<£r*1)

T 1: 3+1 TS — 1
Below, we will discuss the asymptotic distributions of 22:, MM)?!” (i=l,2,3) each by

each.

I
2w(s,l)?;“' :
s=l

 

zw(s 1)r“' zw(s 1)( 25,5,_,)

TI: s+l

— ZIWU I)[-— 2(6 +8, .)e.1+ 2|,W(s lx— 26.2) (A3)

TI: s+l TI: s+l

For any I such that l/T—rO, the first and second terms on the right hand side of (A3)

1 T
converge in probability to zero since we have 2 w(s,l)(l Z £,£,_s) = 0,, (12") (see

s=l t=s+l

20

T
Lemma 2 in Chapter 3) and ‘71—” 2(8, + 8,_3) = 0,,(T‘H/2). And, obviously, ifthe third

t=s+l
term on the right hand side of (A3) is rescaled by l"l , it will converge in distribution to

6,2/2. Thus,

I"(22w(s 1);“) => .92 (A4)
3:] ’ s 1 °

1
2w(s,l)?;m :
s=l

 

 

1"(2élwts,l)?;”'>=l' {22w(s 111— z<")<‘:“,‘>1(er—a. )thées-s. )2,

 

 

                 

Tl: s+l 77"!
(A5)
t—s—l
because] l[22,w(s ’1)Tr§+t(T—ll)( T—l )]—)1/3.
inst)?!“ :
3:1
I ~[3]= fl _ 2(I—l)—S _
“W ’31)" T1: 3+1 _ ,151+(T_,)5r—s (”T—l )31]}(€T 51)-

Note that, for any 5 such that s/T—>O, we have— T 2 [—————- T—1—) S8 8,20,,(1)

1: 3+1 —1

 

 

2(t-l)— d— 1/2
be —>1.Al — + — 0 T
cause Tr)::+1[__ T— 1 1 so T1511“ T— Ilg’) (T—_ law A )
1 T T d+3/2
because1 — 2 [(t— )8 +(— :118) 8,_ ,]~ —2T 22t8, ,and 218, =0,,(T );see Tsay
Tr: —s+l T— 1 1:1 1:1

and Chung (1995, Lemma 1). This implies that the latter term is asymptotically

negligible. Therefore,

2]

S6) 61]}(8r-81)=> -€.(6‘- 6).

1"(22':w(s,l)z”‘)~1"{2iw(s,1)[———T— z (———-— 2’,‘ 1')
3:1 3:1

1: 3+] _1
(A6)

By (A2) and (A4)~(A6), we conclude that

l'§2(l):>— 3,3,(8 +£,£w +8,2). 9

Appendix 11 (Proof of Lemma 3)

Denote 8,. = (I — L)8, , then

”I . -1 'I‘ .
a[rT] =(5[r'/'_] 5;“) (“Tl— I81)( 67—51) :25: “([LTTJ_T)§5. -

 

. r’l‘] .
Since T‘W’d ‘28, :> wd. W4 (r) , we have

1:2
T‘W d Fm, 2 wd.[Wd. (r) —rWd.(1)] = wd.Bd. (r) .

By the continuous mapping theorem, the results follow. 0

Appendix 111 (Proof of Lemma 4)

I T
32(1) = 32(0) + 22 w(s,1)'r: where Z— — 22211,.

i
.\'=1 . . Tl:
First term: 3'2 (0)

2 T I
3' (0)=%;<a,— 8,) +T;[(i{i,)<e— , ,)1 ——T-§t<’_

 

)(8, - £1)(£T- 81)]

22

1XI.2__2_T.TL__1_1.
, X22891 T<r§e,)2[(,_,)(§a,>]. (A7)

(=1

1-

1" I.
_?§(§32/)

II
TE“T

First term on the right hand side of (A7):

T’Hd.2(:8 )2 :> wj. 5W; (S)ds.

I: I j=
Second term on the right hand side of (A7):

T‘Z 2" 2[(’T—:1—)(2g )] :> wj. f5 2W2(1)ds.

Third term on the right hand side of (A7):

t— I
—1)

_2-T—2 2d. (:8:

J

)2[(T )(25‘ )] :> —2wj.W,.(1)jj sWd.(s)ds

for T2” d.:[1(28 )]:> wd. 53W. (s)ds, see Tsay and Chung (1995), Lemma].
I: -I j= —2

Therefore, collecting terms, we obtain:

T-’-2""§2(0) => wj. £13,. (s)2ds. (A8)

I
Second term: 2 w(s,1)r:

x: I

First we note that 7‘; is defined to be the sum of 7;“! , 73m and 75m , as in Appendix

I. We can rewrite 73“] as:

”r;“‘=— 2e —a.)(s,,— 8):; 2: (:5; x2192)

les+l t=s+lj=2

=-[Z (28,2) - Z (ZEJX ZED]

Tt=’=s+l121 l='=s+l12 k=—!s+l

Note that, for any 5 such that s/T—20, we have

23

T‘2 2" 2 (242) 260‘onch (5)2615

(2 s+l j: 2

and

T222 2: [dam 2801—», 0

r=s+lj=2 k=—ts+l

becauseT22" 2[(2a;)( 2a;)~] T“"[Z(Zs)l(28,p+n)

(2 3+1 1: 2 k: l- -s+l t: s+l F —2

T2’32 2 (252:) 0,,‘“'(1)andT'/2‘2 p2s,’_pp+,—> 0.

I: s+l F 2

Thus,
—i—2d’~|11 2 l 2

T rs => (0d. [0 Wd. (3) ds.
Also, by similar arguments as above, we can show that

—1—2d'~[2] 2 1 2 2

T rs :> cod. [Os Wd. (1) ds
and
—l-2d‘~[3] _ 2 1 ,
T rs :> de.Wd. (l)jOsWd. (s)ds .

Then, we obtain

1 'T" 2" 22w(s,1)r :>a)2 103d. (s)2 ds (A9)

3: l

where Bd. (5) = Wd. (s) — sWd. (1).
By (A8) and (A9), we conclude that

1"T"‘2‘2 32(1)2>w:. (13d. (s)2ds.o

Appendix IV (Proof of Lemma 5)

24

. J
é ME=ZS2 and S] =Zé,,where

1:] (=1

é,=<y, —y,')— (;_ ’1)(y;— yi) t=1,2,...,T (A10)

. p O
with y, = yl — Z¢, y,_, . We note that
i=1

yr --il¢:yt-z -:(¢I‘ _¢I)yt—: : a+fl +v1_$(¢:_¢l)yt—I’
so that
y,‘ —y.' =(v. -V.)+fl(t-1)+:(¢, —¢,‘)(y,-, —y,-,). (A11)
By substituting (A10) into (Al 1), we obtain
s t—l P .
e: :(v: _V1)—(}—_T)(V1_V|)+Z(¢l —¢.){(yu —yl-:) ”(T—(1Xy'r-q yl-:)}'

But

 

y,= (a+flt+v,)=§+yt+£,

<1) (L)

P

with a, = v, / d) p( L), for some 8, 7 that depend on a, [3 and ¢p(-). Therefore

(y,,— y..)— <’—— :11wa y.-,)=(s,_,—el_,>—(-;—‘_—‘I>(a,--,—s,_,).

We can write

v

8:17—

EM‘a

(¢ -,¢ )€,_, (A12)

where V, = (v( — v,) — (%-—]1)(v.,. — v,) and similarly for 2",.
To establish part (i) of Lemma 5, we note that

25

Thus,
—/ _l[r'l'l~ p . _llr7'l~
T S[r'l'] : T 2v} _ Z(¢l _¢1)(T Zgj—l)'
.l=1 I=l j=l

w] WI
Here T" 29’] :> —rv, —r"(vw — v,) / 2 as in Schmidt (1993, p.4). Also T" 23,-:
j=l

j=l
converges to a well-defined limit. Since ( (15: —¢, —>p O), we have
T48“,l :> —rv, —r2(vco -— v,) / 2.
The proof of part (i) of Lemma 5 then follows exactly as in the proof of Schmidt

(1993), p.4. The proof of part (ii) of Lemma 5 is similar. In (A12), the term '17,

p 0 ~ 0 o c
dominates the term 2 (¢, — (15, )8, asymptotically, and the proof [8 essentially the same
(:1

-I

as the proof of Lemma 4 of Schmidt (1993, p.4). 0

26

CHAPTER 3
A SCORE-BASED TEST OF THE NULL OF MULTIVARIATE SHORT MEMORY

AGAINST UNIT ROOTS AND LONG MEMORY ALTERNATIVES

3.1 Introduction

Since the influential article of Nelson and Plosser (1982), testing for unit roots has
become a routine procedure in the research agenda of economists analyzing the economic
time series. Two types of tests have been used most frequently. First, tests of the null of
a unit root against the alternative of stationarity are considered; examples are the standard
Dickey-Fuller test (Dickey and Fuller (1979)) and its augmented (Said and Dickey
(1984)) or nonparametrically corrected (Phillips and Perron (1988)) versions. Second,
there are tests considering stationarity as the null and unit root as the alternative.
Examples are KPSS (1992), Saikkonen and Luukkonen (1993) and Leyboume and
McCabe (1994).

A possible empirical puzzle emerges if both the null of a unit root and the null of
stationarity are rejected. One possible and reasonable solution is long memory. Long
memory is often defined by the condition that the autocorrelations decay hyperbolically,
as opposed to geometrically (which, loosely speaking, is short memory.) The standard

long memory model is the fractional differencing model of Granger (1 980), Granger and
Joyeux (1980) and Hosking ( 1981): (1 — L)" y, = ,u, where ,u, is short memory. More

precisely, we may define stationary long memory as the case that the fractional

27

parameter, d, is in the range of (-1/2,1/2) with "0" not included, and define nonstationary
long memory as d in the range of (1/2,3/2). We note that a unit root is the special case of
long memory when d=1. The sense in which long memory is an explanation for the
empirical puzzle is as follows. The KPSS test is called a test of stationarity, but its
distribution under the null actually assumes short memory. Both "stationarity" tests and
unit root tests are generally consistent against long memory alternatives. Thus, rejections
of both null hypotheses may simply reflect the existence of long memory.

There has also been interest in multivariate tests of the unit root or stationarity
hypotheses. Multivariate tests may be preferred to the application of a univariate test to
each of a number of series, both because the size of the overall testing procedure can be
better controlled and because power may be higher. For example, Phillips and Durlauf
(1986) propose a Wald statistic as a test of the null hypothesis of "all the time series
contain unit roots" against "at least one of the time series does not has a unit root". This
test can be seen as an extension of the Dickey-Fuller test to multiple time series. See also
Park and Phillips (1988, 1989) and Sims, Stock, and Watson (1990) for examples of
multivariate unit root tests. On the other hand, Choi and Ahn (1999) proposed several
consistent tests of the multivariate stationarity (short memory) hypothesis, and these were
extended in Choi and Ahn (1995) to the problem of testing multiple equations for
cointegration. Basically these procedures amount to applying multivariate unit root tests
to the cumulated data.

In this chapter we derive an LM test of the null of stationarity for a multiple time
series. Our test is a multivariate generalization of the KPSS test. Interestingly, despite a

completely different derivation, our test is the same as one of the tests of Choi and Ahn

28

(1999). The same test has also been proposed in a recent paper by Hobijn and F ranses
(1998). We show that the test is consistent against both unit root alternatives and long-
memory alternatives, and we give simulation evidence to show the gain in power that is
obtained from a multivariate approach.

The outline of this chapter is as follows. In the next section, we derive a
multivariate extension of the KPSS test. In sections 3.3 and 3.4, along the lines of KPSS
(1992), Lee and Schmidt (1996) and Lee (1995), we establish the asymptotic distribution
of this new statistic under quite general assumptions, including short memory, stationary
and nonstationarey long memory, and unit root. In sections 3.5 and 3.6, we provide
Monte Carlo evidence comparing the new test with the univariate KPSS test, in terms of
size and in terms of power against long memory and unit root alternatives. Finally,

section 3.7 gives our conclusions.

3.2 Derivation of the Statistic

Let y), , i=1,2,...,l( and t=1,2,...,T, be the observed time series (T observations on
each of K series). We assume that each series, {y,., );l , can be decomposed into a
deterministic part {4,}le and a stochastic part {SH LC] :

ya =dit+sit (1)
In this chapter, we consider three cases that differ in their assumptions about the

deterministic part. CASE A (zero-mean): d” = 0; CASE B (level): d" = a, ; CASE C

(trend): d,, = a, + ,8} .

29

We need to make rather strong assumptions such as normality to derive our LM
statistic. Our derivations of the asymptotic properties of the statistic, given in the next
two sections, will proceed under much more general assumptions.

Proceeding with our setup of the model for which we will derive the LM statistic,

we assume a components representation of 5,, :
s,, = ,u,, +6,,. (2)
Here we assume that ,u,, is a random walk and 5,, is an iid process. Defining
p,, — ,u,,_I = v,, , we take ”(.0 = 0 which entails no loss of generality in CASE A (zero-
mean), CASE B (level) or CASE C (trend).
In order to allow correlation across series, define V, = (VI, ,..., V,“ )' and

E,=(£,,,...,£,,,)'. Then we assume that V, and E, are iid N(0, 2’) and N(0, 2 ),

respectively, and that V, and E, are mutually independent. Here 2’ and Z are arbitrary
positive definite matrices, for the moment.

We wish to test the multivariate null hypothesis that each of the 5,, is short
memory. In this model, this corresponds to ,u,, E O (in which case 3,, = 8,, is an iid
process), and therefore to 2‘ = 0. This raises some issues. The matrix 2' contains
K(K+1)/2 distinct elements, but it must be positive definite, and therefore the K-
dimensional statement "every diagonal element of 2' equals zero" is sufficient to imply
the K(K+1)/2-dimensional statement " 2' =0." Thus the most general hypothesis we

would consider is that every diagonal element of 2' is equal to zero. Since this is a
multivariate one-sided hypothesis we may apply, for example, the LMMP (locally most

mean powerful) statistic of King and Wu (1997). This is essentially a statistic based on

30

the sum of scores. The problem is that there are K(K-l)/2 nuisance parameters (the off-

diagonal terms in 2') which are identified only under the alternative hypothesis. We
might address this difficulty by applying results from King and Shively (1993) or

Andrews and Ploberger (1994).

In this chapter, we simplify matters by assuming (for purposes of deriving the LM
statistic) that 2' =12 where 720 is a scalar. This is essentially an assertion that the long
run and short run components of the series have the same cross correlation structure. We

then simply test the null hypothesis H 0 : 1:0 against the alternative H , : A>O. In a sense,

our test is not a true multivariate test, but a univariate test for multiple time series. It
looks for deviations from stationarity in one particular direction in K-dimensional space.
Appendix I gives the derivation of the LM statistic under the assumptions given

above. The statistic equals

27‘ =%[vec(1§")'(2‘l_l ® AT)vec(E')]

=izfz. 23—42“, 72:45:. 2,2924]. (3)
Here, E is the KxT matrix whose ith row contains {y,, )L, (CASE A), the OLS residuals
from the regression of {y,, },T:I on an intercept (CASE B) or an intercept and time trend
(CASE C); 2, is the Kx] vector defined as the partial sum of the columns of E ; so that

2, = 22:, E, where E, (the sth column of E ) is a le vector with typical element é,s;

E = T"EE' is an estimate of 2; and A, is a TxT matrix with if element equal to

max(T—i+1,T-—j+1).

31

For K=1 this statistic equals the LM statistic of KPSS (1992, p.163, equation (6)).
We will refer to it as the modified KPSS (MKPSS) statistic. It was also suggested as a
multivariate generalization of the KPSS test by Hobijn and Franses (1998); but they did
not derive it as the LM test (or from other general principle of testing). Interestingly, the
MKPSS statistic is also the same as the SBDH (SBDHm for CASE A and CASE B,
SBDH: for CASE C) statistic of Choi and Ahn (1999). The MKPSS and SBDH
statistics are derived from different approaches. The SBDH statistic is a multivariate
analog of the Sargan and Bhargava (1983) and Durbin-Hausman tests for a (cumulated)

AR unit root, while the MKPSS statistic is based on the LM (score) principle.

Nevertheless the statistics turn out to be the same.

3.3 Asymptotic Theory (1): Short Memory and Unit Roots

We first consider the asymptotic distribution of the MKPSS statistic under the null

of short memory. In the previous section, the MKPSS statistic was derived under strong
assumptions, notably that the errors E, were iid normal. Following KPSS (1992), we
acknowledge that these assumptions are too restrictive to be realistic in an empirical
setting, and we will proceed under the weaker assumption that the errors are short
memory, as defined below.

As a matter of notation, we denote weak convergence as "=>", convergence in

probability as " —>,, ", and the integer part of x as [x].

32

Assumption 1 (Short memory) (i): Existence of the long run covariance matrix.

p lim,._No T"E[Z.,.Z.,.'] = Q where Z, = 2,2, E, and Q is a finite positive definite KxK
matrix. (ii): Multivariate invariance principle. T’WZW, :> QUZBU‘) where B(r) is a

standard multivariate Brownian motion (i.e., the variance matrix of B(r) is rIK ).

In the literature, several sets of sufficient conditions have been provided for such a
multivariate invariance principle (or functional central theorem) to hold. Examples are,
the strong mixing conditions of Phillips and Durlauf (1986), the linear process conditions
of Phillips and Solo (1992) (see Choi and Ahn (1999) also), and the near epoch
dependent (NED) conditions of De Jong and Davidson (1997).

Under Assumption 1, ‘2 is not a consistent estimate of the long-run covariance Q ,
and so a different estimate is needed. Following Newey and West (1987), we define a

consistent estimate of Q as:

62(1) = 620 + éwwxé, + 52;)

J=|

where w(j,l)=l-j/(l+l) with 1—> 00 as T—> co and 1/ T—> o, and f2, = 23;,” (13:, E,_,')/T .
Thus, by replacing f: in (3) by 62(1), we may redefine the MKPSS statistic to be
1 ~ ~ — ~.
77 = F[vec(E' )' (9(1) ' ® AT)vec(E )1

=L

T, ”[01. Z‘,Z‘,')f2<l>“'1. <4)

23;.2;é(1)"23
From Assumption 1, it follows that T "“ 22,,“ :> QWB‘U) where B'(r) is a

Brownian motion in CASE A (zero-mean):

33

B‘(r) = B(r).
a Brownian bridge CASE B (level):
B.(r) = B(r) — rB(l),
or a second-level Brownian bridge in CASE C (trend):
B.(r) = B(r) — (2r — 3r2 )B(1) + (—6r + 6r2 )1; B(s)ds.
As above, B(r) is a standard multivariate Brownian motion. We may note that different
elements of B‘(r) , say B,‘(r) and B;(r) , are independent. Then, applying the continuos
mapping theorem, we obtain
T‘2 2,1, 2,2”; :> own},13‘(r)B‘(r)'arr]Q“2 .
And, since OH) —>,, 0 , we arrive at the asymptotic distribution of the MKPSS statistic

under the null, as the in following theorem. This distribution was also given by Choi and

Ahn (1999, Theorem 1) and Hobijn and Franses (1998, proposition 3).

Theorem 1 Under Assumption 1, we have

n :> was B‘(r)B‘(r)’dr) =25; US, Bf (rm.

The critical values of 2,2 [1,; B,‘(r)2 ] are given in Table 3-1, for K s 16. They are

calculated via a direct simulation, using a sample size of 1000, 50000 replications, and
the random number generator RNDN of GAUSS. These critical values agree fairly

closely with those given by KPSS (1992) when K=1, by Choi and Ahn (1999) for K S 6

and by Hobijn and Franses (1998) for KS 5 and K=10.

34

We next consider the asymptotic distribution of the MKPSS statistic under the

alternative hypothesis of a unit root. In the previous section, this would correspond to

7L>0 (2. ¢ 0 ), but we can proceed under the more general alternative defined in

Assumption 2.

Assumption 2 (Unit root) T"’ 2E,,,, :> Q'”B(r) where Q is the long-run covariance of

the difference of the E, , and B(r) is a standard multivariate Brownian motion.

We can now derive the asymptotic distribution of 17 under Assumption 2. This
distribution was also given by Choi and Ahn (1999, Theorem 2). A somewhat less
powerful result (establishing its order in probability) was given by Hobijn and Franses

(1998, Proposition 4).

Theorem 2 Under Assumption 2, we have
(I / Tm => I; [I;§(s)ds1'u:, E0)? or drr‘ua’fi (s)ds1da,
where: (CASE A), E(s) is a le column vector Brownian motion, (CASE B), B(s) is
the demeaned Brownian motion (KPSS (1992, p168)):
E (S) = 8(5) — I; B(r)dr ,
(CASE C), E(s) is the demeaned and trended Brownian motion (KPSS (1992, p169)):
B(s) = 8(3) — (6s — 4) 1;, B(r)a'r + (—12s + 6) 13 rB(r)dr .

Proof: See Appendix 11.6

35

We may also consider the situation that not all the series of interest contain a unit

root, but rather some of the series are [(1) and others 1(0).

Assunytion 3 (A mixture of 1(1) and 1(0)) T'” 213,,“ :> Q” 2B(r) , where

EU") EU")
= [r7] 0 _ ’ . o o o n n
FE,” [Zl'TllE(K_m)] With E, — [E‘K’m , Q Is the posrtive definite long run covanance
I: I I

matrix of the AF, and B(r) is a K-dimensional standard Brownian motion.

In Assumption 3, F, is a multivariate (K-dimensional) time series with its first m

components being 1(1) and the remaining K — m components being I(O).

Theorem 3 Under Assumption 3, we have
(I / T»; :> IAIISE‘“’(s>ds1'UlE‘“’(r)§""’(r)'dr1"{I;§‘”’(s)ds1da ,
where E‘””(s) is an mxl column vector with each element defined as in Theorem 2.

Proof: See Appendix III. 0

Basically, Theorem 3 says the following. If F, contains one or more unit root

components, only these unit root components affect the asymptotic distribution of I].
A comparison of Theorem 1 and Theorem 2 shows that the MKPSS test is

consistent as a test of short memory against the alternative that all series have unit root

components. Theorem 3 indicates that the test is also consistent against the less

restrictive alternative that one or more of the series have a unit root.

36

Furthermore, Theorem 3 implies that the MKPSS test will be consistent against the
alternative that each of the series have a unit root, but there may be one or more
cointegrating relationships among the series. This is so because the MKPSS test is
invariant to non-singular linear transformations of the vector of time series, and a
cointegrated vector can be transformed into a vector of the form given in Assumption 3

by an appropriate linear transformation.

3.4 Asymptotic Theory (II): Long Memory

In this section we consider the asymptotic distribution of the MKPSS statistic under

the assumption that the errors E, are long memory. Along the lines of Lee and Schmidt

(1996), who show that the KPSS test is consistent as a test of univariate short memory
against univariate stationary long memory alternatives, we show that the MKPSS test as a
test of multivariate short memory is consistent against multivariate stationary long
memory alternatives. Moreover, along the lines of Lee (1995), who shows that the KPSS
test is inconsistent as a test of univariate unit root against univariate nonstationary long
memory alternatives, we show that the MKPSS test is inconsistent as a test of
multivariate unit root against multivariate nonstationary long memory alternatives.

Based on the work of Chung (1996) and De Jong and Davidson (1997), we define

B,,(r) = (B d, (r),..., 8,, (r))' to be a multivariate fractional Brownian motion if each

fractional parameter, d, , is in the open interval (—1/2,1/2), and each element of 8,, (r)

is a univariate fractional Brownian motion:

37

 

1 m {I60 - s)"" dB.(s> + 13.,[0 — sf" — (—s)"" 1dB.-(s)}

B_ r =
d'( ) F(d, +1)v,,,

where B,(r) is a standard Brownian motion and

_ 1 1 d,_ 2
_ F(d, +1)2 {2d, +1 +g°[(1+r) 1] dr}.

 

V d,

We allow the possibility that B, (r) and B, (r) , for i¢j, are not independent, and we define
w, = E[B,(1)B , (1)] . Also, based on De long and Davidson (1997, Theorem 6.1) or

Chung (1996, p.7), we have E[B,,(1)B,,(1)'] = E with the elements of 5 defined by

.. W.) 1 d. d. d. d.
I: = + 1+ '_ I 1+ 1 _ 1 (1T ,
” F(d,+l)F(d, +1){d,+d,+1 LT“ I) I 1K 1) T ] }

 

for i,j=1,...,K.

We also define a diagonal scaling matrix D(T) as
D(T) =diag(T"' ,sz T"*‘ ).

Then, we may define a stationary long memory process as the following assumption.

Assumption 4 (Multivariate stationary long memory) D(T)‘l (T "H 221:] E,) :9 30 (r) 9

with d, e (—1/2,1/2), for all i=1,2,...,K, and there exists at least one d, at 0.

In De Jong and Davidson (1997), conditions involving the NED of the mixing
process conditions have been provided for Assumption 4 to hold. Other sufficient

conditions were considered in Chung (1996) but only in the case when d, 6 (0,1 / 2) , for

all i=1,2,. . .,K. We also note that the stationary and invertible VARFIMA (vector

38

fractional integrated autoregressive moving average) process is an example of

multivariate stationary long memory defined in Assumption 4.

Lemma 1 Under Assumption 4, we have
D(T>"(T'2 2,1. 232,300)“ 2 IABL(r)B;(r)'dr
where BI, (r) is a column vector with each element B; (r) defined as a fractional

Brownian motion in CASE A, a fractional Brownian bridge in CASE B (Lee and Schmidt

(1996, p.291)):
13;, (r) = 3.1, (r) - r34, (1) ,

or a second-level fractional Brownian bridge in CASE C (Lee and Schmidt (1996,

p.292)):
13;, (r) = 13,, (r) — (2r — 3r2 )ij (1) + (—6r + 6r2)j,‘,13,,, (s)ds .
Proof: Since D(T)'l (T"”2Z,,,.,) :> B,‘,(r) , by applying the continuous mapping theorem

the result follows. 9

In order to derive the result of Lemma 2 (below), a more restrictive assumption of
multivariate stationary long memory is given as Assumption 4'. We note that, since
Assumption 4' is a special case of Assumption 4, Lemma 1 still holds under Assumption
4'; see Chung (1996, Theorem 1) for example. This is an extension to the multivariate

case of Assumptions (1), (2), (3) and (4b) of Hosking (1996).

39

Assumption 4' (Multivariate stationary long memory) (i) Each fractional parameter
belongs to the same range given in Assumption 4. (ii) E, = 23:0 A ,.X,_ j . Here, A ,— is a
sequence of KxK matrices ( A, = [aab (j)],':,,:,) such that aab(j) ~ pabj‘md" “1”)” ,
p0,, > 0 if a=b, and aab (j) ~ 0 otherwise, as j—>oo. And, {X ,} is a sequence of K-
dimensional innovations such that X, ~ IID(0, 2) with Z = [00,, 15M , and
E(Xa‘st,,Xc,,,X,,‘v) = 77,,de ifs=t=u=v, E(Xa,st,,Xc,,,Xd‘v) = 00,0“, ifs=t¢u=v and
E(Xa,st‘,Xc‘,,Xd,v) = 0 otherwise (XM: the ath element of X, ).

~1+da +db

(iii) rab(m) ~ labm , lab > 0, as m—>oo. (Here, fab (m) = E(£,,,,8,,,,_,,,) With 5a,:

the ath element of E, .)

Lemma 2 Under Assumption 4', we have

[I

D(I)“[S‘2(I)ID(1)" 4,, "
where E is defined as above.

Proof: See Appendix IV. 0

Theorem 4 Denote dmax = max(d, ) and d1min = min(d,) . (i) Under Assumption 4' with

d, e[0,1/2) for all i and dmm, #0, we have n=0,,((T/l)2""“‘). Thus 77—)” 00. (ii)

Under Assumption 4' with d, e (—1/2,0] for all i and d at 0 , we have

min

77 = 0,,((T/l)2""""). Thus I] —>,, 0. (iii) Under Assumption 4' with d, e (—1/2,1/2) for

all i and dmax >0 but dmin < 0, we have 77=0,,((T/l)2""“‘). Thus 77—), oo.

40

Proof: Using Lemma 1 and Lemma 2, we have
tr{D(T/1)"([T‘zz,:,2,2,']f2(1)“')D(T/1)"}:>tr(j,',B;,(r)'_=."B;,(r)'dr).

Then the results easily follow. 0

A comparison of Theorem 1 and Theorem 4 implies that the upper tail test is
consistent against stationary long memory alternatives with the fractional parameters in
the range [0,1/2), while the lower tail test is consistent against stationary long memory
alternatives with the fractional parameters in the range (-1/2, 0], so long as one or more of
the series has d¢0. Consistency holds even if some of the series are short memory, so
long as one or more of the series are long memory. Theorem 4 also implies that the

upper tail test is consistent in the case that some of the d, are negative and some are

positive.

Assumption 5 (Multivariate nonstationary long memory) D(T)‘l (T'”2E,,.,.,) :> B,, (r)

with d, e (—1 / 2,1/ 2) for all i=1,2,...,K, and BD(r) as defined above.

We note that, if E, is nonstationary long memory as defined in Assumption 5, then
AE, is stationary long memory as defined in Assumption 4. Thus, we may define the
fractional parameters of the nonstationary long memory simply as d; (= d,- + 1) e (l/ 2,3/ 2) .

We also note that d: =1 for all i is the special case of unit root as defined in Assumption

2.

41

Lemma 3 Under Assumption 5, we have
00")“ (T4 2,1, Z,Z,')D(T) —_—> [Aug §D(a)da][j,; BD(a)da]'dr ,
where 130(3) is a le column vector with each element defined as the fractional

Brownian motion in CASE A (zero-mean), the demeaned fractional Brownian motion

(Lee (1995, p.25)):
E, (s) = B, (s) - [33,, (r)dr
in CASE B (level), or the demeaned and trended fractional Brownian motion (Lee (1995,
p.27)):
11,. (s) = 3,, (s) — (6s — 4) I; 3,, (r)dr + (—12s + 6) 1:, r3, (r)dr ,
in CASE C (trend).

Proof: (omitted). 0

Lemma 4 Under Assumption 5, we have
D(T)"[(IT)"£°2(I)]D(T)“ => 1; §s(r)§s(r)'dr.

Proof: See Appendix V. 0

Theorem 5 Under Assumption 5, we have 77 = 0,,(T / 1).
Proof: By Lemma 3 and Lemma 4, we have
”(042.1. 23230062" (1») 2 ”([1305 §D(a)da>(lg §D(a)da>'dr1(h', §D(r)§s(r)'dr)“} .

Thus, the results easily follow. 9

42

We note that Theorem 2 is a special case of Theorem 5 corresponding to d: =1 or
d, =0, for all i=1,2,...,K. Theorem 5 implies that the MKPSS test is a consistent test of

the null hypothesis of short memory against nonstationary long memory alternatives.
However, similarly to Lee (1995), we find that under nonstationary long memory, the rate
of divergence of the MKPSS statistic is independent of the fractional parameters. This is
different from the results of the stationary long memory case. The implication of this
finding is that if we use the MKPSS statistic to test the null of unit roots, it is inconsistent

against nonstationary long memory alternatives.

3.5 Simulations: Unit Roots

In this section we report the results of Monte Carlo simulations designed to
investigate the size and power of the MKPSS test in finite samples. More specifically,
we wish to compare the performance of the MKPSS test with that of the univariate KPSS
test applied to each series. In this section, the alternatives against which we will calculate
power involve unit roots.

For simplicity, we will consider only bivariate systems. Also, we will consider only
the tests that allow for "level" but not "trend."

Simulations were performed in GAUSS and use the random number generator
RNDN. The results were calculated using 5000 iterations at sample sizes T=75, 200, and

400. For long-run variance-covariance estimation, we considered lag lengths 10=0,
14: int[4(T /100)”‘] , and 112: int[12(T/100)'”].
Data are generated by the following bivariate I(l) process:

43

[ylt]=['u":|+[g"] for t=1,...,T, (5)
Y2: #2: 32:

,U ,u _ V . v '
[ It]=l: I! l]+[ 1’] With(.ui,oa/12,0)=(0’O) °
#2, fizz—I V2!

Here, (8,, , £2, )', (v,, , v2, )' are independent iid bivariate normal with zero mean and

where

variance matrices 2 and 2’ , respectively. This is a bivariate unit root process except

when 2‘ =0, in which case it is a bivariate short memory process.

We report results for three types of testing procedures. First, we apply the MKPSS
test at the 10% level. Second, we apply the KPSS test at the 10% level to each of the
series separately, calculate the power (rejection frequency) for each of the two separate
KPSS tests, and report the larger of the two powers, which we denote KPSSmax. For
many of our experimental designs, the two series have the same characteristics and so the
two univariate KPSS tests should have the same power. The "max" operation is over two
estimates of the same quantity (power), and will induce some upward bias, but this is
very small because we have a large number of replications. For other experimental
designs, one series violates the null more severely (e. g., a larger unit root component),
and KPSSmax is essentially the power of the univariate KPSS test applied to that series.
Third, we consider a Bonferroni procedure in which we apply the KPSS test at the 5%
level to each of the series separately, and reject the bivariate null if either of the two tests
rejects its univariate null. By the Bonferroni inequality, this procedure has size
(asymptotically) no larger than 10%. (Its maximal size is 0.0975, for independent series.)

This is a feasible but conservative testing procedure.

Our first experiment is done by considering 2' =12 with 2 defined by

2r (i
p 1

where we allow p=0, p=0.5 and p=0.9. We consider It=0, 0.001, 0.01, 0.1, 1. We can

note, following KPSS ( 1992, p163), that in this case the model can be rewritten as:

)’2: km 9'2: 9‘21—1
where g, = v,, + (8,, — £,,_,) and where (9 = —{(/1 + 2) - [21(1 + 4)]”2}/2. Thus,
corresponding to l=0, 0.001, 0.01, 0.1, 1, we have 0=-1, -0.969, -0.905, -0.730 and

-0.3 82. In particular, the null of short memory ()t=0) corresponds to a moving average

unit root in Ay,.

Table 3-2 gives the size of the tests, and corresponds to 1:0 so that the null
hypothesis is true. Consider first the case that l =10=0, which is sufficient because we
have white noise errors. The size of each of the tests is quite close to the nominal size
0.10. The exception is that the Bonferroni test (BKPSS) is conservative, as expected,
when the series are strongly correlated. When we use I =14 or 1:112, the sizes of the tests
are generally too small. This problem is serious when T=75 and [=112, especially for the
MKPSS and BKPSS tests. It is not surprising to find serious size distortions when T is
small and l is large. Except in this case, the sizes of the tests seem reasonably accurate.

Table 3-3 gives the power of the tests against the alternatives with 7t=0.001, 0.01,
0.1, 1.0. For all of the tests, power is larger (other things held constant) when T is larger;

and when It is larger; and when l is smaller. These results are as expected. In particular,

45

the dependence of the power on I, even for large T, reflects the fact that the asymptotic
distribution of each statistic depends on (T/l).

The power of the MKPSS test and of the univariate KPSS tests (KPSSmax) are more
or less independent of the correlation between the series (p). The Bonferroni test has
lower power when p is large than when p is small, as expected.

The most interesting comparison is between the MKPSS test and the other two
tests. The MKPSS test is more powerful than the others in every case except one, namely
T=75, [=112, k=0.001. The MKPSS test is usually slightly better than the Bonferroni test,
though its superiority is more noticeable when the series are highly correlated (so the
Bonferroni test is too conservative). There is a clear gain from using the bivariate test
(MKPSS) instead of the univariate test (KPSSmax). Naturally, this is most evident in
cases when power is neither close to one nor close to zero. For example, when T=75,
1:14, 7t=0.01, p=0, compare power of 0.657, 0.604 and 0.505 for MKPSS, BKPSS, and
KPSSmax, respectively.

The case considered in Table 3-3 can be considered favorable to the MKPSS test,
because the data generating process matches the one that was assumed in deriving the test
as an LM test. Our second experiment therefore allows for unit roots of different strength

in the two series. Specifically, we still have 2 = I2 (i.e. p=0) but now we allow

2' =diag( vl , v2 ). We allow different combinations of VI and v2 from the same set of

values: 0.001 , 0.01, 0.1, 1.0, 10. Table 3-4 gives our results for this case.
The results corresponding to T and l are the same as in our first experiment: power
is higher when T is larger or I is smaller. The comparison of power across tests is

somewhat more ambiguous than in the previous experiment. The MKPSS test is clearly

46

best when T=400, and almost always when T=200. However, when T=75 it does not
clearly dominate the KPSSmax test. The parameter values for which the KPSSmax test is
more powerfiJl are those for which v2 is very small (v2 =0.001), so that effectively the
unit root component in the first series is very large relative to the unit root component in
the second series. In such cases there is little gain to be expected from a bivariate test as
opposed to the univariate test applied to the first series.

In our third experiment, we allow the two I(l) series to be cointegrated. The setup

is similar to that of the first experiment with p=0, so (5,, , £2, ) have variance matrix

2 = 12 . However, now the long-run components of the two series are the same:

,u,, = ,u2, = ,u, , with ,u, = ,u,_l +v, , with vO =0 and var(v,) = ,1. Since y,, and y2, have
the common stochastic trend ,u, , they are cointegrated ( yl, — y,, is stationary). We

considered It=0.001, 0.01, 0.1, l, 10. The results are given in Table 3-5.

Once again, power is higher when T is larger and when I is smaller. Power is
generally higher when I). is larger. However, for T=75 and T=200, the power of all of the
tests is actually l_o_\1e_r for 7t=10 than i=1 , an unexpected result for which asymptotic
theory or intuition does not provide an apparent explanation.

Comparing tests, the MKPSS test is generally best for small values of It (say, It s
0.01), but the univariate KPSS test is generally best for larger values of I. (say, It. 2 0.1).
When a single, strong trend dominates the two series, there seems to be no need for a
bivariate test. This is not surprising. In this case, the MKPSS statistic and the KPSS

statistic have the same asymptotic distribution (see Theorem 3). And, since the critical

47

value of the MKPSS test is larger than that of the KPSS test (e. g., O.608>0.350, for the

10% level), the KPSS test should be more powerful asymptotically.
3.6 Simulations: Long Memory

In this section, as in the previous section, we wish to compare the finite sample
power of the MKPSS test with that of the univariate KPSS test. However, in this section
we will consider stationary long memory series. More specifically, we will consider I(d)
with d e (0, 1/2).

Thus we consider the bivariate process ( y, , , yz, ):

[h]: (1—L>"" 0 [at]
yz: o (,_L)-d, 52: '

We will generate I(d) observations using the Cholesky decomposition of the error

covariance matrix. Let (6,, , £2, ) be iid with covariance matrix 2(2x2). Then the

covariance of the y series is F(2Tx2T) defined as follows:

 

 

 

’ r,(0) r,(1) r,,(T—1))
r. —l r 0 r T-2
1“,, Ft» . ,( ) .,() ,_,( ) ..
F: ‘ wrth F, = for 1,]=1,2,
I‘21 22
\r,(—T+l) r,,(—T+2) r,,(0) ,
where
(—1)"l“(1-d,—d )
7,,(k)= ’ 0,,
F(1+k—d,)F(1—k—d,)
and

48

r,, (—k) = r,, (k) for k=0,1,.. .,T-l,
where 0,, is the (i,j)th term of 2 in (6). See Sowell (1989, p14 and appendix II). The

Cholesky decomposition of F yields Stzrsz), lower triangular, such that F=SS'. Then we
create observations having variance matrix F as (88,), where E, is a (2Tx1) vector of iid
N(0, 1) random deviates. We consider 2 as in equation (6) above, with p=0, 0.5, 0.9. We
consider d=0.1, 0.2, 0.3, 0.4 and 0.45. (Also, the "size" results in Table 3-2 correspond to
d=0.)

We will first consider the case with the same fractional parameter (d, = d2) in each
series. The power of the MKPSS, BKPSS, and KPSSmax tests is given in Table 3-6.

With other things held constant, we see that power is higher when d is larger, when T is
larger, and when l is smaller. These results are similar to those for the unit root case, and
are as expected from the asymptotic theory for the long memory case.

The MKPSS test and the univariate KPSS test have power that does not depend
perceptibly on the correlation (p) between the series. The Bonferroni test (BKPSS) has
lower power for large p than small p, as it should.

Comparing the various tests, the MKPSS test is almost always the most powerful.
The only exceptions are for small sample T and large I, where univariate KPSS is more
powerful; this is presumably a reflection of the smaller size distortion of the univariate
KPSS test for small T and large I, as found in Table 3—2. The gain in power from using
the bivariate test can be considerable. For example, for T=200, I=4, d=0.3, compare

power of 0.705 for MKPSS to 0.534 for the univariate KPSS test.

49

In Table 3-7 we provide results for cases with d, ¢ a'2 . The general results on the

effects of changing T, l or the differencing parameters are the same as in Table 3-6. Once
again, the MKPSS test is generally most powerful. There are non-trivial exceptions when
the values of d are sufficiently different for the two series. For example, for T=200, p=0,
1 =10, the univariate KPSS test applied to the second series (KPSSmax) is more powerful
than the MKPSS test when d, =0.0 and d2 =0.4 or 0.45. As in the case of unit root
alternatives, there is no gain to a bivariate procedure if the null hypothesis is violated in
one series much more strongly than in the other. However, we can note that the MKPSS
is still generally better, even if d2 is much larger than d, , if the series are very strongly
correlated (e.g. p=0.9). In that respect the results of this experiment are different from
those of the last experiment (with d, = d2 ). Now the power of the MKPSS test increases
with the correlation between the series.

In our final two experiments, we compare the power of the three tests against

multivariate I(d) alternatives to their size in the presence of short memory

autocorrelations. We model the short memory process as a stationary VAR( 1 ):

[Mt] 2 [PI 0 ][ylt-l ] +|:3u :|

)’2: 0 P2 yZI—l 52:

For meaningful comparison, we pick parameters that match the value of the one-period
autocorrelations: p, = ,02 = p = 0.2, 0.5, 0.8, 0.95 and correspondingly, d, =d2 = d =0.l67,
0.333, 0.433, 0.487. The size and power for these values are given in Tables 3-8 and 3-9,
respectively.

In Table 3-8 we see that short-run autocorrelation causes size distortions for all of

the tests. For large values of the AR( 1) coefficients these size distortions are quite

50

severe, even when T and l are moderately large. The MKPSS test exhibits larger size
distortions than the univariate KPSS test, for given values of T, l and the AR(l)
parameters. This means that, for a given T and a given level of short run autocorrelation,
we would have to pick a larger value of I for the MKPSS test than for the KPSS test, in
order to avoid serious size distortions. A larger value of I will reduce power. This
implies that the power advantage of the MKPSS test over the univariate KPSS test,
clearly seen in Table 3-6 for the case of no short-run dynamics, may be smaller or
nonexistent when short-run dynamics are properly controlled for. Further consideration
of this point is the subject of future research.

Comparing corresponding entries in Tables 3-8 and 3-9, we see that power against
long memory alternatives does exceed size in the presence of short-memory
autocorrelation (where corresponding entries match the one-period autocorrelation, as
discussed above). For example, for T=400 and the one period autocorrelations equal to
0.5, the choice [=14 yields the size for the MKPSS test (in the presence of AR(l) with

p, = p2 =0.5) of 0.188, and power (against I(d) with d, = a’2 =l/3) of 0.857. Thus the test

can successfully distinguish long memory from short memory autocorrelation. However,

it is obvious from Tables 3-8 and 3-9 that this will require a rather large sample size.

3.7 Conclusion

In this chapter, we develop a score-based test statistic for multiple time series. We
show that this new statistic is consistent as a test for the null of multivariate short

memory against the alternative of unit root and the alternative of multivariate long

51

memory. We also show that, in most cases, there is a non-trivial finite-sample power
improvement using the new statistic rather than applying its univariate counterpart to
each component of a multiple time series. This optimistic conclusion must be qualified
in two ways. First, the multivariate test is more susceptible than the univariate test to size
distortions in the presence of short run dynamics. Second, the multivariate test is not
more powerful than the univariate test in some cases where the null hypothesis is violated
essentially in only one of the series. Examples would be cases in which only one series
has a sizable unit root or long memory component, or cases of cointegration in which the

series share a single stochastic trend.

52

CASE A: Zero-Mean

Table 3-1

Critical Values for the MKPSS Test

 

 

 

K 20% 10% 5% 2.50% 1%

1 0.764 1.199 1.676 2.182 2.794

2 1.502 2.086 2.654 3.198 3.982
3 2.193 2.872 3.493 4.125 4.948
4 2.821 3.570 4.266 4.987 5.864
5 3.450 4.256 5.031 5.768 6.674
6 4.049 4.933 5.745 6.486 7.410
7 4.662 5.577 6.428 7.190 8.187
8 5.252 6.211 7.097 7.922 8.993
9 5.847 6.830 7.767 8.608 9.661

10 6.433 7.482 8.401 9.325 10.374
11 7.023 8.096 9.063 9.959 11.070
12 7.590 8.694 9.717 10.614 11.732
13 8.152 9.304 10.343 11.244 12.481
14 8.717 9.906 10.923 11.882 13.063
15 9.284 10.496 11.521 12.522 13.757
16 9.838 11.087 12.148 13.191 14.422

CASE B: Level

K 20% 10% 5% 2.50% 1%

1 0.242 0.350 0.461 0.581 0.745
2 0.468 0.608 0.750 0.891 1.089
3 0.679 0.843 1.005 1.059 1.357
4 0.879 1.062 1.235 1.404 1.622
5 1.082 1.284 1.469 1.653 1.813
6 1.275 1.491 1.694 1.884 2.120
7 1.471 1.695 1.909 2.115 2.355
8 1.660 1.904 2.124 2.325 2.576
9 1.848 2.100 2.332 2.543 2.806
10 2.037 2.298 2.537 2.752 3.038
11 2.223 2.490 2.740 2.958 3.259
12 2.406 2.690 2.947 3.175 3.462
13 2.590 2.887 3.141 3.890 3.677
14 2.773 3.076 3.348 3.607 3.908
15 3.957 3.267 3.543 3.809 4.120
16 3.139 3.460 3.748 4.012 4.336

 

53

Table 3-1, continued

 

 

 

 

 

 
  
 
 

   

 

     

 

 

 

 
 
 
    

    

CASE C: Trend
K 20% 10% 5% 2.50% 1%
1 0.092 0.120 0.147 0.175 0.214
2 0.174 0.210 0.245 0.280 0.325
3 0.252 0.295 0.336 0.376 0.426
4 0.328 0.378 0.423 0.465 0.519
5 0.404 0.457 0.505 0.552 0.609
6 0.479 0.536 0.588 0.637 0.699
7 0.552 0.614 0.667 0.722 0.784
8 0.626 0.690 0.748 0.803 0.872
9 0.700 0.765 0.826 0.885 0.956
10 0.772 0.841 0.905 0.966 1.038
11 0.884 0.917 0.985 1.045 1.120
12 0.915 0.923 1.060 1.120 1.198
13 0.986 1.066 1.134 1.199 1.278
14 1.059 1.140 1.210 1.274 1.359
15 1.130 1.214 1.287 1.355 1.442
16 1.200 1.287 1.362 1.433 1.521
Table 3-2
Size
m
lag I0 I12
p BKPSS KPSSmax
0 0.105 0.104 0.088 0.079 0.098 0.047 0.047 0.090
0.5 0.097 0.089 0.082 0.076 0.092 0.041 0.045 0.091
0.9 0.102 0.076 0.087 0.064 0.098 0.047 0.040 0.098
m
lag I0 I12
p KPSSmax BKPSS KPSSmax
0 0.107 0.102 0.101 0.096 0.092 0.097
0.5 0.096 0.092 0.096 0.092 0.083 0.092
0.9 0.103 0.074 0.102 0.095 0.069 0.101
114.09
lag I0 I12
p BKPSS KPSSmax
0 0.096 0.094 . 0.092 .
0.5 0.097 0.094 0.104 0.092 0.088 0.102 0.065 0.077 0.096
0.9 0.102 0.071 0.101 0.100 0.069 0.098 0.089 0.065 0.095

 

   

54

Table 3-3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Power against I(1) Alternatives ( 2' = ,1): , 2' = [2) fl)
T=75
lag I0 I4 112
I. p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 1.000 0.999 0.986 0.975 0.958 0.879 0.795 0.754 0.649
1.0 0.5 0.999 0.998 0.986 0.976 0.952 0.869 0.798 0.727 0.645
0.9 1.000 0.999 0.988 0.975 0.912 0.880 0.776 0.632 0.640
0 0.986 0.981 0.907 0.934 0.914 0.806 0.741 0.710 0.610
0.1 0.5 0.988 0.976 0.904 0.935 0.903 0.796 0.740 0.681 0.610
0.9 0.988 0.953 0.910 0.936 0.838 0.801 0.738 0.571 0.610
0 0.742 0.714 0.560 0.673 0.648 0.513 0.454 0.449 0.426
0.01 0.5 0.721 0.674 0.548 0.657 0.604 0.505 0.446 0.410 0.419
0.9 0.723 0.580 0.563 0.659 0.512 0.512 0.447 0.337 0.419
0 0.233 0.217 0.194 0.197 0.192 0.182 0.111 0.115 0.164
0.001 0.5 0.235 0.207 0.196 0.197 0.183 0.183 0.104 0.107 0.168
0.9 0.235 0.165 0.195 0.201 0.142 0.182 0.111 0.080 0.156
T=200
lag IO I4 I12
I. p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 1.000 1.000 1.000 0.999 0.997 0.973 0.950 0.923 0.811
1.0 0.5 1.000 1.000 1.000 1.000 0.997 0.971 0.948 0.912 0.819
0.9 1.000 1.000 1.000 0.998 0.986 0.970 0.945 0.835 0.799
0 1.000 1.000 0.996 0.998 0.995 0.965 0.946 0.921 0.816
0.1 0.5 1.000 1.000 0.996 0.998 0.994 0.961 0.939 0.904 0.801
0.9 1.000 1.000 0.995 0.998 0.978 0.956 0.936 0.828 0.794
0 0.984 0.978 0.890 0.966 0.954 0.841 0.890 0.867 0.719
0.01 0.5 0.985 0.973 0.897 0.963 0.944 0.849 0.891 0.847 0.724
0.9 0.984 0.937 0.897 0.966 0.891 0.850 0.885 0.752 0.723
0 0.675 0.649 0.505 0.641 0.612 0.487 0.549 0.536 0.437
0.001 0.5 0.666 0.606 0.493 0.639 0.584 0.475 0.556 0.507 0.443
0.9 0.663 0.496 0.489 0.632 0.471 0.467 0.552 0.408 0.428
=4OO
lag 10 14 I12
kl. p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 1.000 1.000 1.000 1.000 1.000 0.992 0.991 0.983 0.928
1 .0 0.5 1.000 1.000 1.000 1.000 1.000 0.993 0.991 0.974 0.923
0.9 1.000 1.000 1.000 1.000 0.998 0.994 0.988 0.952 0.921
0 1.000 1.000 1.000 1.000 1.000 0.991 0.993 0.983 0.920
0- 1 0.5 1.000 1.000 1.000 1.000 0.999 0.991 0.989 0.972 0.915
0.9 1.000 1.000 1.000 1.000 0.997 0.991 0.989 0.951 0.920
0 1.000 1.000 0.987 0.997 0.995 0.964 0.981 0.969 0.891
0-01 0.5 1.000 0.999 0.989 0.999 0.996 0.967 0.982 0.958 0.895
\ 0.9 1.000 0.996 0.985 0.997 0.997 0.962 0.981 0.930 0.890
0 0.925 0.907 0.779 0.905 0.886 0.749 0.858 0.838 0.693
0001 0.5 0.926 0.900 0.778 0.910 0.879 0.751 0.860 0.818 0.695
0.9 0.931 0.824 0.770 0.914 0.797 0.743 0.869 0.727 0.694

 

 

 

55

 

Power against I(1) Alternatives (2 = I , z' = diag(vl, v2) )

Table 3-4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_T=_75
oov I lag IO I4 I12
v1 v2 MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
10 1 1.000 0.999 0.995 0.974 0.956 0.886 0.782 0.753 0.646
10 0.1 1.000 0.998 0.994 0.962 0.944 0.890 0.766 0.731 0.638
10 0.01 0.993 0.991 0.994 0.902 0.881 0.877 0.652 0.633 0.646
10 0.001 0.985 0.987 0.994 0.829 0.823 0.877 0.528 0.538 0.643
1 0.1 0.998 0.996 0.986 0.962 0.944 0.876 0.766 0.730 0.649
1 0.01 0.988 0.985 0.986 0.900 0.882 0.877 0.655 0.631 0.654
1 0.001 0.978 0.976 0.987 0.820 0.813 0.873 0.538 0.536 0.649
0.1 0.01 0.937 0.922 0.905 0.855 0.833 0.796 0.615 0.606 0.604
0.1 0.001 0.884 0.879 0.911 0.761 0.748 0.798 0.498 0.506 0.622
0.01 0.001 0.539 0.523 0.548 0.473 0.466 0.500 0.282 0.289 0.415
EL.”
oov / lag I0 I4 I12
v1 v2 MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
10 1 1.000 1.000 1.000 0.998 0.996 0.970 0.950 0.926 0.804
10 0.1 1.000 1.000 1.000 0.999 0.998 0.974 0.950 0.929 0.808
10 0.01 1.000 1.000 1.000 0.992 0.989 0.972 0.928 0.905 0.816
10 0.001 1.000 1.000 1.000 0.971 0.967 0.976 0.839 0.815 0.811
1 0.1 1.000 1.000 1.000 0.998 0.996 0.971 0.945 0.924 0.805
1 0.01 1.000 1.000 1.000 0.994 0.989 0.970 0.924 0.902 0.807
1 0.001 1.000 1.000 1.000 0.972 0.968 0.973 0.841 0.813 0.810
0.1 0.01 0.999 0.998 0.993 0.990 0.983 0.952 0.917 0.898 0.797
0.1 0.001 0.996 0.995 0.996 0.963 0.954 0.960 0.835 0.817 0.803
0.01 0.001 0.924 0.912 0.898 0.883 0.867 0.850 0.771 0.754 0.727
E409
cov / lag I0 I4 I12
v1 v2 MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
10 1 1.000 1.000 1.000 1.000 1.000 0.994 0.991 0.982 0.923
10 0.1 1.000 1.000 1.000 1.000 1.000 0.994 0.990 0.980 0.927
10 0.01 1.000 1.000 1.000 0.999 0.999 0.992 0.984 0.974 0.922
10 0.001 1.000 1.000 1.000 0.997 0.994 0.991 0.956 0.941 0.915
1 0.1 1.000 1.000 1.000 1.000 1.000 0.993 0.989 0.981 0.922
1 0.01 1.000 1.000 1.000 1.000 1.000 0.994 0.988 0.979 0.931
1 0.001 1.000 1.000 1.000 0.997 0.994 0.995 0.962 0.950 0.926
0.1 0.01 1.000 1.000 1.000 0.999 0.999 0.993 0.988 0.975 0.919
0.1 0.001 1.000 1.000 1.000 0.997 0.996 0.995 0.960 0.944 0.922
0.01 0.001 0.996 0.991 0.986 0.985 0.979 0.964 0.943 0.927 0.894

 

 

 

56

 

Table 35

Power against Co-integrated 1(1) Alternatives

 

 

 

 

 

 

 

 

 

 

 

 

1:15
lag I0 I4 I12
1 MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
10 0.985 0.986 0.993 0.804 0.804 0.878 0.505 0.519 0.649
1 0.976 0.979 0.988 0.811 0.818 0.874 0.508 0.528 0.645
0.1 0.918 0.907 0.904 0.759 0.768 0.795 0.480 0.527 0.613
0.01 0.643 0.612 0.565 0.558 0.547 0.515 0.354 0.383 0.437
0.001 0.217 0.202 0.191 0.185 0.177 0.178 0.105 0.110 0.167
B99
lag 10 I4 I12
1. MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
10 1.000 1.000 1.000 0.943 0.946 0.959 0.737 0.726 0.810
1 0.999 1.000 1.000 0.945 0.950 0.971 0.749 0.747 0.816
0.1 0.995 0.994 0.996 0.935 0.942 0.957 0.728 0.743 0.798
0.01 0.928 0.909 0.894 0.859 0.850 0.843 0.683 0.695 0.721
0.001 0.570 0.543 0.490 0.534 0.514 0.469 0.449 0.442 0.427
Iifl
lag 10 I4 112
It MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
10 1.000 1.000 1.000 0.985 0.988 0.994 0.864 0.865 0.922
1 1.000 1.000 1.000 0.983 0.985 0.994 . 0.862 0.872 0.924
0.1 1.000 1.000 1.000 0.980 0.982 0.989 0.862 0.870 0.922
0.01 0.991 0.987 0.985 0.959 0.958 0.959 0.843 0.856 0.888
0.001 0.834 0.802 0.758 0.788 0.766 0.731 0.700 0.686 0.677

 

 

57

 

Table 3-6

Power against long memory alternatives (d, = d,)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T=75
lag I0 I4 I12
d,=d2 p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.296 0.276 0.229 0.187 0.176 0.168 0.074 0.080 0.128
0.1 0.5 0.304 0.252 0.236 0.189 0.165 0.174 0.073 0.073 0.134
0.9 0.309 0.201 0.230 0.195 0.133 0.172 0.081 0.062 0.134
0 0.564 0.52 0.422 0.348 0.326 0.277 0.149 0.146 0.182
0.2 0.5 0.565 0.487 0.419 0.342 0.295 0.275 0.137 0.130 0.187
0.9 0.583 0.395 0.419 0.353 0.231 0.278 0.136 0.098 0.188
0 0.794 0.742 0.602 0.502 0.462 0.374 0.202 0.020 0.238
0.3 0.5 0.786 0.699 0.593 0.495 0.417 0.375 0.204 0.182 0.235
0.9 0.783 0.592 0.597 0.500 0.34 0.372 0.208 0.156 0.238
0 0.909 0.869 0.740 0.635 0.595 0.474 0.284 0.272 0.299
0.4 0.5 0.911 0.850 0.749 0.634 0.555 0.468 0.282 0.258 0.286
0.9 0.904 0.757 0.733 0.643 0.460 0.476 0.289 0.200 0.295
0 0.948 0.921 0.805 0.712 0.661 0.537 0.338 0.321 0.332
0.45 0.5 0.946 0.903 0.800 0.702 0.626 0.538 0.337 0.300 0.326
0.9 0.950 0.832 0.805 0.706 0.527 0.532 0.346 0.233 0.329
I:2_0_9
lag I0 14 I12
dl=d2 p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.411 0.374 0.302 0.273 0.243 0.220 0.172 0.165 0.168
0.1 0.5 0.401 0.329 0.299 0.258 0.223 0.208 0.161 0.149 0.159
0.9 0.416 0.273 0.312 0.264 0.182 0.225 0.170 0.121 0.170
0 0.762 0.700 0.559 0.491 0.453 0.367 0.289 0.270 0.244
0.2 0.5 0.749 0.664 0.565 0.491 0.410 0.370 0.291 0.247 0.252
0.9 0.758 0.554 0.570 0.499 0.340 0.371 0.293 0.201 0.246
0 0.933 0.899 0.785 0.705 0.650 0.534 0.437 0.401 0.345
0.3 0.5 0.934 0.884 0.782 0.695 0.606 0.523 0.433 0.365 0.331
0.9 0.938 0.800 0.789 0.703 0.508 0.532 0.417 0.289 0.345
0 0.991 0.978 0.904 0.839 0.789 0.652 0.546 0.502 0.411
0.4 0.5 0.990 0.969 0.906 0.832 0.765 0.653 0.563 0.484 0.419
0.9 0.989 0.930 0.905 0.835 0.659 0.648 0.551 0.390 0.422
0 0.996 0.989 0.941 0.880 0.842 0.705 0.612 0.565 0.454
0.45 0.5 0.996 0.987 0.941 0.889 0.819 0.704 0.616 0.546 0.460
0.9 0.996 0.967 0.942 0.883 0.714 0.712 0.609 0.421 0.452

 

 

 

58

 

Table 3-6, continued

 

 

 

 

 

 

 

 

m0
lag Io I4 I12
d,=d, p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.499 0.446 0.356 0.321 0.295 0.246 0.212 0.202 0.188
0.1 0.5 0.490 0.410 0.354 0.327 0.268 0.247 0.227 0.196 0.190
0.9 0.500 0.326 0.355 0.334 0.211 0.250 0.222 0.149 0.190
0 0.859 0.806 0.679 0.592 0.538 0.440 0.390 0.360 0.302
0.2 0.5 0.864 0.778 0.674 0.607 0.518 0.444 0.411 0.353 0.309
0.9 0.859 0.678 0.670 0.596 0.420 0.440 0.396 0.271 0.308
0 0.982 0.966 0.875 0.813 0.789 0.622 0.582 0.538 0.429
0.3 0.5 0.983 0.953 0.881 0.793 0.716 0.616 0.572 0.490 0.427
0.9 0.987 0.904 0.887 0.808 0.610 0.627 0.577 0.397 0.426
0 0.999 0.995 0.967 0.923 0.893 0.765 0.727 0.671 0.554
0.4 0.5 0.998 0.994 0.966 0.929 0.870 0.765 0.715 0.644 0.544
0.9 0.999 0.992 0.971 0.923 0.786 0.763 0.723 0.532 0.545
0 1.000 0.998 0.985 0.951 0.923 0.811 0.769 0.717 0.588
0.45 0.5 1.000 0.998 0.984 0.959 0.915 0.812 0.776 0.699 0.599
0.9 1.000 0.992 0.985 0.950 0.831 0.811 0.773 0.576 0.587

 

 

 

59

 

Power against long memory alternatives (d, :4 d2 )

Table 3-7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

60

 

T=75
lag I0 14 112
d, d, p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.197 0.191 0.238 0.127 0.129 0.172 0.060 0.680 0.132
0.0 0.1 0.5 0.208 0.183 0.240 0.145 0.132 0.177 0.064 0.660 0.137
0.9 0.262 0.165 0.250 0.167 0.116 0.183 0.070 0.560 0.140
0 0.441 0.399 0.411 0.258 0.244 0.267 0.109 0.113 0.178
0.1 0.2 0.5 0.433 0.373 0.404 0.262 0.230 0.260 0.105 0.103 0.173
0.9 0.497 0.332 0.423 0.306 0.205 0.284 0.129 0.097 0.194
0 0.693 0.645 0.597 0.423 0.391 0.380 0.171 0.166 0.246
0.2 0.3 0.5 0.691 0.603 0.595 0.428 0.363 0.377 0.173 0.160 0.241
0.9 0.716 0.522 0.598 0.443 0.305 0.382 0.179 0.133 0.243
0 0.859 0.815 0.739 0.581 0.540 0.480 0.243 0.240 0.303
0.3 0.4 0.5 0.862 0.789 0.739 0.577 0.513 0.478 0.245 0.216 0.292
0.9 0.874 0.703 0.747 0.593 0.416 0.487 0.250 0.166 0.290
0 0.928 0.896 0.802 0.681 0.633 0.535 0.314 0.295 0.322
0.4 0.45 0.5 0.927 0.883 0.798 0.671 0.594 0.535 0.304 0.281 0.335
0.9 0.925 0.796 0.804 0.672 0.490 0.538 0.320 0.217 0.327
0 0.356 0.350 0.423 0.226 0.223 0.277 0.093 0.104 0.189
0.0 0.2 0.5 0.375 0.345 0.424 0.228 0.206 0.282 0.093 0.094 0.180
0.9 0.483 0.303 0.416 0.261 0.177 0.269 0.100 0.080 0.189
0 0.596 0.561 0.592 0.355 0.338 0.380 0.143 0.146 0.236
0.1 0.3 0.5 0.594 0.531 0.586 0.347 0.305 0.366 0.142 0.139 0.231
0.9 0.679 0.493 0.589 0.399 0.278 0.375 0.164 0.118 0.240
0 0.796 0.754 0.737 0.507 0.492 0.472 0.206 0.205 0.289
0.2 0.4 0.5 0.802 0.727 0.744 0.510 0.449 0.477 0.216 0.196 0.294
0.9 0.850 0.676 0.754 0.553 0.396 0.488 0.239 0.138 0.293
0 0.896 0.857 0.807 0.612 0.561 0.533 0.271 0.250 0.322
0.3 0.45 0.5 0.896 0.834 0.804 0.618 0.545 0.531 0.273 0.240 0.324
0.9 0.906 0.753 0.795 0.619 0.449 0.529 0.278 0.198 0.315
0 0.514 0.505 0.599 0.302 0.300 0.380 0.124 0.126 0.238
0.0 0.3 0.5 0.529 0.485 0.593 0.310 0.281 0.372 0.120 0.119 0.234
0.9 0.669 0.480 0.593 0.376 0.270 0.379 0.155 0.118 0.239
0 0.729 0.698 0.740 0.459 0.446 0.492 0.188 0.189 0.313
0.1 0.4 0.5 0.735 0.685 0.745 0.449 0.407 0.482 0.192 0.176 0.289
0.9 0.825 0.663 0.759 0.517 0.389 0.495 0.219 0.159 0.294
0 0.835 0.798 0.797 0.557 0.513 0.530 0.237 0.237 0.322
0.2 0.45 0.5 0.841 0.785 0.807 0.570 0.509 0.545 0.251 0.227 0.342
0.9 0.878 0.732 0.802 0.586 0.412 0.532 0.266 0.188 0.322
0 0.672 0.670 0.743 0.405 0.399 0.490 0.164 0.175 0.296
0.0 0.4 0.5 0.691 0.659 0.739 0.412 0.384 0.487 0.168 0.169 0.299
0.9 0.787 0.642 0.737 0.471 0.375 0.480 0.192 0.157 0.298
0 0.772 0.746 0.795 0.481 0.457 0.519 0.202 0.209 0.317
0.1 0.45 0.5 0.793 0.744 0.802 0.508 0.465 0.532 0.222 0.208 0.333
0.9 0.851 0.716 0.793 0.543 0.420 0.527 0.235 0.193 0.326
0 0.726 0.730 0.803 0.445 0.436 0.535 0.190 0.201 0.329
0.0 0.45 0.5 0.747 0.721 0.801 0.465 0.438 0.533 0.195 0.196 0.335
0.9 0.822 0.709 0.793 0.502 0.423 0.531 0.208 0.175 0.327

Table 37, continued

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m
leg 10 I4 112
d, d, p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.258 0.252 0.308 0.184 0.176 0.225 0.125 0.122 0.172
0.0 0.1 0.5 0.251 0.227 0.300 0.177 0.160 0.208 0.120 0.114 0.161
0.9 0.350 0.214 0.309 0.219 0.142 0.221 0.143 0.101 0.171
0 0.616 0.571 0.574 0.393 0.356 0.367 0.233 0.219 0.239
0.1 0.2 0.5 0.603 0.525 0.560 0.383 0.328 0.363 0.224 0.196 0.238
0.9 0.668 0.474 0.569 0.412 0.282 0.369 0.241 0.165 0.248
0 0.882 0.839 0.790 0.609 0.564 0.526 0.374 0.344 0.342
0.2 0.3 0.5 0.878 0.807 0.779 0.620 0.536 0.521 0.369 0.321 0.342
0.9 0.878 0.715 0.771 0.606 0.431 0.515 0.364 0.250 0.329
0 0.970 0.950 0.904 0.773 0.721 0.659 0.497 0.463 0.423
0.3 0.4 0.5 0.974 0.943 0.906 0.777 0.696 0.652 0.492 0.432 0.426
0.9 0.978 0.894 0.904 0.789 0.613 0.660 0.504 0.361 0.434
0 0.994 0.986 0.949 0.877 0.830 0.721 0.587 0.540 0.471
0.4 0.45 0.5 0.993 0.982 0.945 0.873 0.806 0.720 0.596 0.514 0.466
0.9 0.994 0.953 0.940 0.863 0.384 0.703 0.574 0.408 0.458
0 0.490 0.476 0.570 0.297 0.285 0.366 0.180 0.179 0.246
0.0 0.2 0.5 0.500 0.460 0.564 0.302 0.284 0.370 0.186 0.174 0.248
0.9 0.668 0.450 0.569 0.384 0.258 0.363 0.224 0.148 0.247
0 0.792 0.751 0.788 0.511 0.481 0.524 0.295 0.283 0.331
0.1 0.3 0.5 0.796 0.734 0.784 0.529 0.468 0.526 0.312 0.277 0.338
0.9 0.868 0.696 0.783 0.552 0.414 0.519 0.327 0.245 0.338
0 0.940 0.913 0.903 0.705 0.658 0.649 0.435 0.400 0.416
0.2 0.4 0.5 0.942 0.898 0.908 0.701 0.625 0.647 0.426 0.367 0.411
0.9 0.962 0.864 0.908 0.732 0.577 0.655 0.456 0.328 0.426
0 0.983 0.968 0.939 0.820 0.774 0.711 0.530 0.480 0.459
0.3 0.45 0.5 0.983 0.962 0.946 0.806 0.734 0.710 0.530 0.461 0.467
0.9 0.989 0.926 0.943 0.814 0.633 0.695 0.525 0.373 0.446
0 0.703 0.704 0.785 0.442 0.439 0.518 0.263 0.263 0.349
0.0 0.3 0.5 0.725 0.698 0.779 0.449 0.416 0.529 0.259 0.242 0.335
0.9 0.840 0.682 0.776 0.514 0.400 0.514 0.301 0.226 0.331
0 0.898 0.880 0.902 0.622 0.601 0.646 0.383 0.368 0.418
0.1 0.4 0.5 0.902 0.869 0.906 0.636 0.582 0.657 0.389 0.347 0.426
0.9 0.944 0.845 0.905 0.670 0.550 0.650 0.408 0.316 0.424
0 0.963 0.947 0.943 0.747 0.702 0.716 0.470 0.435 0.457
0.2 0.45 0.5 0.966 0.940 0.940 0.758 0.686 0.709 0.479 0.422 0.465
0.9 0.975 0.910 0.944 0.758 0.629 0.703 0.494 0.380 0.470
0 0.857 0.863 0.911 0.572 0.571 0.657 0.328 0.326 0.433
0.0 0.4 0.5 0.866 0.850 0.908 0.577 0.559 0.651 0.347 0.332 0.423
0.9 0.926 0.85 0.907 0.642 0.546 0.660 0.376 0.314 0.424
0 0.930 0.912 0.942 0.677 0.649 0.715 0.415 0.399 0.468
0.1 0.45 0.5 0.941 0.915 0.949 0.692 0.637 0.713 0.434 0.399 0.466
0.9 0.963 0.902 0.945 0.724 0.613 0.717 0.454 0.354 0.460
0 0.900 0.908 0.944 0.616 0.620 0.714 0.374 0.365 0.463
0.0 0.45 0.5 0.917 0.905 0.947 0.639 0.624 0.720 0.386 0.360 0.470
0.9 0.951 0.899 0.942 0.673 0.602 0.707 0.421 0.356 0.460

 

 

 

61

 

Table 3-7, continued

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I109
leg 10 I4 112
d, d, p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.293 0.285 0.356 0.203 0.200 0.244 0.154 0.145 0.187
0.0 0.1 0.5 0.306 0.277 0.366 0.206 0.196 0.244 0.151 0.141 0.190
0.9 0.422 0.261 0.361 0.264 0.173 0.254 0.182 0.123 0.202
0 0.728 0.676 0.679 0.473 0.483 0.453 0.310 0.285 0.316
0.1 0.2 0.5 0.733 0.647 0.672 0.487 0.417 0.449 0.322 0.277 0.319
0.9 0.780 0.581 0.674 0.501 0.347 0.450 0.334 0.227 0.306
0 0.952 0.919 0.886 0.722 0.675 0.627 0.491 0.451 0.432
0.2 0.3 0.5 0.942 0.896 0.879 0.718 0.642 0.628 0.490 0.417 0.429
0.9 0.961 0.838 0.882 0.735 0.551 0.616 0.499 0.345 0.423
0 0.996 0.989 0.972 0.879 0.837 0.761 0.654 0.613 0.550
0.3 0.4 0.5 0.994 0.981 0.965 0.881 0.816 0.767 0.649 0.575 0.550
0.9 0.996 0.963 0.969 0.876 0.719 0.761 0.649 0.467 0.537
0 0.999 0.997 0.985 0.945 0.912 0.812 0.750 0.701 0.596
0.4 0.45 0.5 0.999 0.997 0.987 0.945 0.894 0.811 0.749 0.665 0.592
0.9 0.998 0.991 0.987 0.945 0.813 0.822 0.742 0.560 0.598
0 0.575 0.568 0.654 0.360 0.356 0.426 0.247 0.242 0.299
0.0 0.2 0.5 0.615 0.575 0.677 0.384 0.352 0.447 0.254 0.233 0.306
0.9 0.778 0.564 0.675 0.470 0.325 0.441 0.288 0.196 0.303
0 0.879 0.846 0.875 0.617 0.576 0.617 0.408 0.380 0.424
0.1 0.3 0.5 0.887 0.843 0.879 0.623 0.558 0.623 0.416 0.372 0.425
0.9 0.928 0.805 0.872 0.654 0.506 0.615 0.444 0.326 0.423
0 0.986 0.974 0.970 0.809 0.768 0.757 0.581 0.541 0.538
0.2 0.4 0.5 0.985 0.960 0.968 0.815 0.758 0.764 0.581 0.502 0.539
0.9 0.989 0.946 0.968 0.828 0.679 0.761 0.598 0.440 0.530
0 0.998 0.994 0.986 0.905 0.866 0.818 0.686 0.638 0.598
0.3 0.45 0.5 0.997 0.991 0.985 0.898 0.839 0.812 0.684 0.612 0.589
0.9 0.999 0.983 0.987 0.908 0.768 0.813 0.686 0.513 0.583
0 0.814 0.813 0.877 0.520 0.519 0.604 0.339 0.332 0.420
0.0 0.3 0.5 0.843 0.819 0.887 0.554 0.525 0.629 0.359 0.341 0.424
0.9 0.928 0.809 0.880 0.621 0.509 0.621 0.402 0.316 0.427
0 0.966 0.955 0.973 0.741 0.716 0.759 0.512 0.490 0.542
0.1 0.4 0.5 0.969 0.954 0.972 0.757 0.706 0.764 0.524 0.482 0.549
0.9 0.984 0.939 0.964 0.782 0.674 0.760 0.548 0.435 0.540
0 0.992 0.987 0.985 0.848 0.812 0.810 0.629 0.589 0.591
0.2 0.45 0.5 0.995 0.986 0.989 0.857 0.796 0.817 0.615 0.547 0.590
0.9 0.995 0.971 0.986 0.860 0.737 0.810 0.622 0.497 0.584
0 0.937 0.941 0.966 0.682 0.684 0.766 0.454 0.440 0.541
0.0 0.4 0.5 0.949 0.942 0.969 0.690 0.679 0.764 0.462 0.437 0.539
0.9 0.976 0.938 0.969 0.746 0.666 0.774 0.504 0.428 0.544
0 0.982 0.977 0.985 0.793 0.769 0.818 0.568 0.536 0.596
0.1 0.45 0.5 0.984 0.974 0.987 0.805 0.770 0.829 0.574 0.531 0.604
0.9 0.992 0.965 0.985 0.819 0.734 0.812 0.583 0.478 0.587
0 0.967 0.969 0.986 0.735 0.746 0.821 0.494 0.496 0.597
0.0 0.45 0.5 0.974 0.972 0.986 0.752 0.737 0.817 0.508 0.489 0.594
0.9 0.988 0.965 0.986 0.775 0.721 0.807 0.545 0.483 0.592

 

 

 

62

 

Table 3-8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

63

 

Size with AR(1) errors
1:75.
leg 10 I4 112
p. - p2 p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.273 0.243 0.208 0.119 0.115 0.120 0.057 0.057 0.104
0.2 0.5 0.258 0.215 0.209 0.116 0.103 0.125 0.056 0.053 0.105
0.9 0.261 0.164 0.201 0.122 0.087 0.126 0.058 0.047 0.106
0 0.659 0.580 0.481 0.222 0.199 0.185 0.076 0.079 0.116
0.5 0.5 0.656 0.542 0.473 0.225 0.186 0.183 0.070 0.070 0.113
0.9 0.663 0.463 0.487 0.233 0.152 0.203 0.081 0.063 0.129
0 0.982 0.957 0.879 0.608 0.531 0.446 0.174 0.160 0.204
0.8 0.5 0.980 0.939 0.879 0.592 0.486 0.440 0.172 0.150 0.205
0.9 0.980 0.885 0.877 0.596 0.387 0.427 0.177 0.120 0.195
0 1.000 0.998 0.984 0.926 0.887 0.771 0.530 0.482 0.444
0.95 0.5 0.999 0.998 0.986 0.915 0.853 0.769 0.537 0.446 0.440
0.9 1.000 0.994 0.986 0.917 0.768 0.769 0.542 0.357 0.444
Ifl
lag 10 I4 112
pl -- p; p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.268 0.245 0.207 0.117 0.118 0.114 0.083 0.088 0.104
0.2 0.5 0.275 0.229 0.218 0.131 0.118 0126 0.092 0.089 0.106
0.9 0.276 0.182 0.212 0.123 0.092 0.127 0.086 0.656 0.108
0 0.689 0.602 0.508 0.202 0.188 0.169 0.106 0.100 0.115
0.5 0.5 0.687 0.585 0.496 0.210 0.173 0.175 0.111 0.098 0.115
0.9 0.685 0.458 0.495 0.200 0.136 0.167 0.111 0.078 0.117
0 0.992 0.978 0.919 0.550 0.478 0.394 0.205 0.193 0.179
0.8 0.5 0.992 0.964 0.915 0.548 0.442 0.401 0.199 0.165 0.188
0.9 0.993 0.921 0.918 0.548 0.350 0.391 0.198 0.132 0.180
0 1.000 1.000 0.999 0.959 0.927 0.820 0.620 0.555 0.459
0.95 0.5 1.000 1.000 0.999 0.962 0.914 0.830 0.623 0.524 0.462
0.9 1.000 1.000 0.998 0.966 0.841 0.828 0 626 0.417 0.453
1:499
lag I0 14 112
p, = p2 p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.261 0.236 0.204 0.112 0.113 0.111 0.093 0.092 0.098
0.2 0.5 0.278 0.224 0.218 0.124 0.106 0.113 0.101 0.092 0.103
0.9 0.281 0.180 0.216 0.128 0.092 0.122 0.104 0.077 0.111
0 0.703 0.612 0.516 0.188 0.177 0.169 0.118 0.111 0.121
0.5 0.5 0.672 0.560 0.491 0.187 0.158 0.153 0.115 0.100 0.116
0.9 0.679 0.460 0.489 0.188 0.117 0.152 0.110 0.074 0.110
0 0.993 0.972 0.924 0.482 0.420 0.357 0.200 0.187 0.171
0.8 0.5 0.995 0.976 0.930 0.493 0.395 0.362 0.194 0.168 0.174
0.9 0.994 0.932 0.924 0.491 0.316 0.366 0.193 0.128 0.171
0 1.000 1.000 1.000 0.965 0.928 0.836 0.638 0.558 0.467
0.95 0.5 1.000 1.000 1.000 0.969 0.907 0.825 0.639 0.532 0.463
0.9 1.000 1.000 1.000 0.960 0.827 0.838 0.632 0.427 0.464

Table 3-9

Power against long memory alternatives

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TED
leg 10 I4 I12
8I =8, ,0 MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.487 0.444 0.359 0.288 0.261 0.239 0.110 0.112 0.164
0.167 0.5 0.480 0.414 0.357 0.283 0.246 0.238 0.121 0.115 0.171
0.9 0.478 0.317 0.346 0.293 0.189 0.238 0.113 0.090 0.162
0 0.833 0.784 0.650 0.551 0.509 0.413 0.220 0.219 0.252
0.333 0.5 0.827 0.747 0.649 0.546 0.471 0.412 0.239 0.208 0.254
0.9 0.843 0.672 0.669 0.568 0.405 0.427 0.242 0.177 0.270
0 0.941 0.912 0.792 0.685 0.637 0.511 0.329 0.305 0.309
0.433 0.5 0.939 0.893 0.787 0.688 0.608 0.519 0.321 0.275 0.323
0.9 0.932 0.811 0.794 0.686 0.503 0.522 0.329 0.229 0.329
0 0.963 0.939 0.840 0.749 0.698 0.574 0.373 0.344 0.349
0.487 0.5 0.961 0.930 0.838 0.743 0.662 0.547 0.361 0.322 0.340
0.9 0.966 0.864 0.839 0.751 0.555 0.565 0.367 0.251 0.347
122m
leg 10 I4 112
d, =8, p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.662 0.601 0.481 0.420 0.385 0.316 0.250 0.235 0.212
0.167 0.5 0.657 0.569 0.486 0.422 0.368 0.334 0.260 0.229 0.229
0.9 0.667 0.469 0.485 0.428 0.288 0.327 0.254 0.171 0.228
0 0.967 0.939 0.836 0.764 0.701 0.577 0.490 0.443 0.369
0.333 0.5 0.962 0.921 0.837 0.753 0.666 0.574 0.475 0.405 0.377
0.9 0.967 0.853 0.831 0.755 0.557 0.565 0.473 0.319 0.355
0 0.995 0.986 0.934 0.870 0.825 0.699 0.603 0.559 0.464
0.433 0.5 0.993 0.982 0.934 0.866 0.798 0.691 0.600 0.516 0.451
0.9 0.995 0.958 0.933 0.873 0.695 0.687 0.605 0.425 0.460
0 0.998 0.995 0.962 0.916 0.874 0.752 0.658 0.608 0.491
0.487 0.5 0.998 0.993 0.959 0.910 0.854 0.741 0.652 0.575 0.479
0.9 0.998 0.978 0.962 0.910 0.769 0.756 0.650 0.473 0.499
11499
leg 10 I4 I12
8I =8, p MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax MKPSS BKPSS KPSSmax
0 0.762 0.701 0.573 0.507 0.460 0.382 0.340 0.311 0.270
0.167 0.5 0.764 0.664 0.573 0.504 0.435 0.374 0.339 0.296 0.269
0.9 0.769 0.567 0.583 0.501 0.342 0.379 0.340 0.228 0.270
0 0.989 0.980 0.914 0.857 0.807 0.667 0.628 0.574 0.464
0.333 0.5 0.991 0.974 0.917 0.852 0.772 0.668 0.634 0.549 0.460
0.9 0.990 0.937 0.921 0.850 0.676 0.668 0.621 0.439 0.475
0 1.000 0.999 0.982 0.945 0.920 0.791 0.756 0.703 0.572
0.433 0.5 0.999 0.997 0.980 0.946 0.905 0.813 0.772 0.691 0.600
0.9 1.000 0.989 0.978 0.942 0.812 0.784 0.745 0.558 0.563
0 1.000 1.000 0.992 0.971 0.950 0.855 0.817 0.765 0.635
0.487 0.5 1.000 0.999 0.991 0.971 0.943 0.854 0.813 0.742 0.644
0.9 1.000 0.997 0.993 0.972 0.876 0.850 0.807 0.625 0.632

 

 

 

64

 

APPENDICES

Appendix I (Derivation of the MKPSS Statistic)

We define Y =(yu),...,y(K))' with y“) = (y,,,...,y,T), X = (x,,...,xT)' with x, =0
(CASE A), x, =1 (CASE B) or x, =(l,t) (CASE C), and B = (b“’,...,b‘K’) with b“) = 0
(CASE A), b‘” = a, (CASE B) or b‘” = (8,., 6,) (CASE C). We also define
S = (S(l),...,S(K))' with s“) = (s,,,...,s,T) where 3,, = ELM, + 8,,. Then we may rewrite
(1) and (2) as

vec(Y') = (1K 81 X)vec(B) + vec(S')

OI'

Y.=X.B.+S.. (A1)
We have E(S.) = 0 and E(S.S.') = 2(2 (8 VT) + 2 ® I, 5 11(2) where V7 is a TxT
matrix with (i,j)th element equal to min(i,j) . The log-likelihood of (Al) is

L(2, 2, B.; Y.) = const.—%ln|1‘1(2)|—(Y. — X.B. )'I'I(/1)'I (Y. — X.B.). (A2)

The first derivative w.r.t. A of (A2) is

1 6111(4)
d2

l 6111(4)

—__1_ T _ 1 -_ —l _
77— 2tr(l"l(/l.) )+(Y. X.B.)H(/i) 6111 11(4) (Y. X43.)

where 4111(2) / d). = 2 <8 VT. Under the null hypothesis of A=0, we have 121(0) = 2 ® 1,.

and 80(2 =0)/82 =E®v,, and so
f7 = const.+%E.'(2‘l ® V,)E.,

65

where E. = Y. — X .B. , and B. and 2 are the restricted maximum likelihood estimates
(MLE). We note that, since (A1) is a seemingly unrelated (SUR) model with exactly the
same regressors in each equation, the MLE is just OLS. The score-based test statistic can

be considered as the non-constant part of 7} , and, for convenience in deriving its

asymptotic distribution, we scale it by l/T2 . Thus, we have

’7‘ : £2 EA¢'(i-l ® VT)EQI . (A3)

 

Noting that in CASE B (level) and CASE C (trend), since 2,2, E, = 0 , it is
equivalent to replace V7 by A, (A, is a TxT matrix with (i,j)th element equal to
max(T-i+l,T-j+l)) in (A3):

I]. = 732 E¢'(i—l ® AT)EA‘0 . (A4)

 

For CASE A (zero-mean), since 2,1, E, at 0 , (A4) no longer holds. That is, the

expression in (A3) and (A4) are not the same in CASE A. However, they do have the
same asymptotic distribution. Therefore, for reasons of simplicity and similarity to the

other two cases we will suggest (A4) as our test statistic for the case of zero-mean.

Appendix 11 (Proof of Theorem 2)

First, we have
74/212,”, => o'”§(r).
Then

74/223113", = 773/22,,“ 2 o"21,;§(s)8s.

66

Therefore,
74‘ 2,1, 2,2“; :> 521/21 131 138 (s)ds][ 1;,’i§(s)8s]'88):)“2 . (A5)
We also have
(lT)“‘ {2(1) :5 o"2[ 11', B(a)B(a)’ 8819'”, (A6)

the multivariate version of the result of KPSS (1992, p.168, equation (23)). (A5) and

(A6) imply that
(I I T»; 2» ma”2 (131138(s)dsiu:§(s)ds1'4489'”19'”(1.; F<a>§(a)'dax2”21")

=13[13§(s)dsl'[l$§(r)§(r)'8r]"[138(s)8s]8a.4

Appendix III (Proof of Theorem 3)

First, we have

B‘"”(r) ,

A(K—m) (711)
['TI

- ,. - £11171)
T WENT] =T l/2[ [rT] 30112

/2

where Q: is the upper-left mxm matrix of 0'”. Then, by the same argument as in

II!)
Appendix II, we have
1422.2,12‘= 911.311.; 11:8 ”494111;? (“1041489231 . (47)
We also have
(IT)" (2(1) :5 Q: (3,11,; 73’ (“’(a)§ (””(a)' 8819} g}, . (A8)

By (A7) and (A8), the result directly follows. 0

Appendix IV (Proof of Lemma 2)

67

For the proof of Lemma 2, we need the following lemma.

Lemma (A) Define fab(m) = T"l 2,1,”, 80‘,g,,,,_,,, , the mth period sample autocovariance
of the ath and b"1 series of {E,}. Let Kabcd(m,n) = cov(r"ab (m), 08 (11)) and

d ' = d,, + db + d, + dd. Then, under Assumption 4', we have

x,,,.,,(m, 71) ~ C,T‘2+2d. if 8‘ e (1 I 2,1),
~ C,T"1og(T) if 8‘ = 112,
~ C,T" if 8‘ e (=1,1 I 2) .
Here, A~B means that A/B—->l when T——)oo, and C, (i=1,2,3) are finite constants

independent of m and 11.

Proof: By Harman (1980, p.209), we have

Kabcd(m, n) = l 2 (l — lgflra, (i)r,,,,(i + n — m) + rad (i + n)rb,. (i — m) + 6),,th (i, m, n)] ,

T |i|<T
where 6),,de (i, m, n) is a term involving the fourth cumulant of 5,. After some algebra,
it can be shown that (9,,de (i, m, n) is dominated by the other terms asymptotically. So,

we may write

Kabcd (m, 71) ~ 213 (1 ‘ gm... (0’68 (i + n - m) + rad(i 1" ’0"ch ‘ m)].

By Assumption 4', which gives the asymptotic properties of fab(m) , and by similar logic

as is applied in L0 (1991, p.1310), the results directly follow. 9

68

(Proof of Lemma 2)

Let
I
(2(1) = 00 + le(j,l)(Q, + Q,’)
P
with Q, (the jth autocovariance of E,) defined as Q, = E(E,E,_,'). We note that

(I+l)Q(l)= (I+l)QO + Z(l+l—j)Qj ='E(Z,,,Z,+, ),

j: 1
where Z,+, =2’:'E,. And, asl—->oo,
D(I + 1)" Q(1)D(1 + 1)" = D(I + 1)"[(I + 1)" E(Z,,,Z,,, ’)]D(I + 1)" =5 E .

Let (2(1) as the Newey-West estimate for CASE A (zero-mean). Then, if we can

show that
D(I)"i<‘2(I)—9(I)10(I)" —>, 0.

we complete the proof (for CASE A).

Let r 0,,(1) and r 0,,(1) be the (a,b)th elements of (2(1) and (2(1), respectively. We

note that

1
Ida +db

. l
Er...(I) —r... (0| s——— 1,0,,

 

 

Erab(1)- E(rab(1))l+l —TbE|E(r.,-s(l)) ras(1)| (A9)

For the first term on the right hand side of (A9):

1
Ida +611,

 

Erab(l)— E(r ab(l))|

=,———, .,, Err..(01—E(r‘..(0»1+2250111150)-£02.00]
,-

69

l
-lda+db

   

l?éw(0)“£(fiz

                    

f85(1') - E (988(1' ))Il

Sold 1+8), —[Vvar(rab(0) ) +2ZIW(J l)\/V3r(rab(J) )-I

Then, by Lemma (A) and similar logic as applied in L0 (1991, p.1310), we obtain

1
l1d0+db

5986(0- E(585(1))I —>, 0.

 

For the second term on the right hand side of (A9):
Ida l+db —IE(r ra—rb(l)) ra:b(1)| T—Iat'db jZlW(J.91)[%rab(./O)I—)0-
The result for CASE A directly follows.

Let (2(1) be the Newey-West estimate for CASE B (level) or CASE C (trend) and

70,, (1) be the (a,b)“1 element of 5(1). We have

1 1
1d,, +81, Elr ra-ba) rab(l)l——— [,8 +8 El? I‘d—5(1) "011(1)|Jr 1,,8 1+8),

 

Efab(1)-rab(l)|°

For the completion of the proof for CASE B and CASE C, we need to show

l
[‘8— .8, E’la‘bU) MOI—50.

To do so, first we modify condition V3(iii) in Hansen (1992, p.969) as
06/714010“ — 6.15" = 0,,(1)
and keep conditions V3(i) and V3(ii) unchanged. Then, we note that the sample mean for
the case of stationary long memory has probability order given as " 5,, = OAT—”2“!" ) "

(see Hosking (1996, Theorem 1)), so the above conditions are satisfied for CASE B.

Also, since the OLS estimates of level and trend in CASE C have probability orders

70

given as " 62,, = 0P(T_1/2+d")" and " ,6", = 0,,(T'3/2+d" ) " (see Lee and Schmidt (1996,

p292)), the above conditions are satisfied. Now, we can copy the proof of Theorem 3 in

Hansen (1992) to complete the proof. 0

Appendix V (Proof of Lemma 4)

For CASE A (zero-mean), we note that {2(1) = O, + 23:, w(j,1)(§2, + f), ') where
O, = ZL,,,(E,E,_,')/T. Since E, = Z§=,E; (15;: multivariate stationary long

memory), we have

éj= —2(2E)(2E)——2(2E)-—2(2E)( 2 E;-)

Tt=j+lp=lp t=j+lp=l I=j+lp=l q=—lj+l

Note that for any j such that j/T-—>0 as T—)oo,

D(T)—l[; 2 (2 E ,2) lD(T) Dlo30(61)1-’3o(a)'dai

l=j+lp=l

and

D(T) [— 2 (2E, )( 2 E;)]D(T)l -*p 0.

(T=j=+lpl q=-—t}+l

So,
D(T)"(T"6,)D(T)" :> 1,; BD(a)BD(a)'da.
Since l'l Z’F, w(j,l) —> 1/2 as I —> oo , the result for CASE A directly follows.

Also, we may obtain the results for CASE B and CASE C by a Similar argument. 9

71

CHAPTER 4

CONVERGENCE CLUBS AMONG OECD COUNTRIES

4.1 Introduction

The Solow-type neoclassical growth model and the new growth models (see Romer
(1986) and Lucas (1988), for example) give different predictions on how output
discrepancies across countries evolve. The convergence of international outputs is
considered to be evidence supporting the neoclassical growth model while it is
considered to refute the new growth theories. Accordingly, a large number of papers
have attempted to test the hypothesis of convergence empirically. However, there is
more than one possible definition of convergence and it seems that different definitions
of convergence lead to different conclusions. For example, studies that employed a
cross-section method (say, convergence as "catching-up") tend to favor international
output convergence (see Barro and Sala-i-Martin (1995)) while tests on the basis of time
series analysis find little evidence of convergence even among relatively similar
countries, such as the fifteen OECD countries (see Bernard and Dulauf (1995)).

In most of the studies of convergence (and in the cross-section approach especially),
an "all or nothing" hypothesis is considered: either the whole world is convergent or it is
not. Baumol (1986) has suggested that the world might be divided into several
"convergence clubs": within each club, countries converge to each other; but convergence

does not occur across clubs. This feature is implied by growth models that exhibit

72

multiple locally stable steady state equilibria. Examples are Azariadis and Drazen (1990)
and Galor (1996). A similar idea can also be found in the literature on "world income
distribution dynamics" -- see Quah (1996), Bianchi (1997) and Jones (1997), for
example. Recently, empirical work has been done on finding convergence clubs.
Durlauf and Johnson (1995) use a cross-section approach and a regression tree procedure
to classify countries into different convergence groups. Hobijn and F ranses (1997, 1998)
use a time series approach and an algorithm based on test statistics to cluster countries
into several convergence clubs. The empirical results of these papers seem to suggest
that convergence might not be a universal phenomenon; instead, they find a number of
relatively small clubs. As in Hobijn and Franses (1998), we consider three kinds of
convergence: perfect convergence in output, relative convergence in output, and
convergence in growth rate. The first two types of convergence correspond to
stationarity of output differences, and we test this hypothesis using a test of joint
stationarity developed in Chapter 3. Others such as Bernard and Durlauf (1995) have
used joint unit root tests to test the hypothesis of no convergence, and have failed to
reject this hypothesis, so it is natural to see whether we can reject the hypothesis of joint
convergence. Hobijn and F ranses (1997) applied the KPSS test to countries in a pairwise
fashion, and the motivation for this chapter was to see what difference it made to use a
joint test. Subsequently, Hobijn and Franses (1998) developed and applied a joint
stationarity test that is essentially the same as that of Chapter 3, so that this chapter is
broadly similar to Hobijn and Franses (1998). It differs in several regards, however.

First, in addition to testing stationarity of output differences, we use a modified

Hotelling's T 2 statistic to test for grth rate convergence. Second, in addition to the

73

 

OECD data set of Bernard and Durlauf(1995), we study another data set of the same
OECD countries. Third, we examine the robustness (or sensitivity) of our findings to
choice of data and to various econometric details in ways that differ from Hobijn and
Franses. Correspondingly, we obtain different results. However, our findings support
their main empirical result. There is evidence of convergence in grth rates, but
convergence in output levels does not seem to occur except within convergence clubs that
are quite small.

The rest of this chapter is structured as follows. In section 4.2, we discuss the
concepts of convergence. In section 4.3, we formulate the corresponding convergence
measures. Then, based on these measures, we establish a procedure to form convergence
clubs in section 4.4. In section 4.5, we describe the data sets we use. In section 4.6, we
report our empirical findings and some relevant implications. Finally, in section 4.7, we

give some concluding remarks.

4.2 Convergence Hypotheses

Let y" be the log per capita real GDP for country i (i=1,. . .,N) in period t
(t=1,...,T). We consider the following representation:
y" = d” + s” ,
where d,, and s" are the deterministic and stochastic parts respectively. For the
deterministic part, we assume a linear trend: d" = a, + flit . We assume that the
stochastic part s” is [(1). Following Hobijn and Franses (1997, 1998), two definitions
of convergence in output are considered.

74

Definition I Countries i and j (i¢j) are "convergent perfectly in output" if ( y” — y j.,) is

zero—mean stationary.

Definition 11 Countries i and j (iij) are "convergent relatively in output" if ( y” - y 1,) is

B‘s

level stationary.

Actually, in terms of what is ultimately tested, it would be more proper to say "short

memory" instead of stationary. However, we follow the terminology of Hobijn and

 

Franses.
Let g” = Ayn be the growth rate of country i. Then g” = ,6, + 8,., where 8,., = A5,,

is stationary. A definition of convergence in growth rate is given as follows.

Definition 111 Countries i and j (i¢j) are "convergent in growth rate" if the mean of

(g,, - g”) is zero.

According to Bernard and Durlauf (1995), Definition I implies that in the long run
(or steady-state) the output gap between countries i and j disappears. It requires a, = aj. ,
,6, = ,6 . , and that s” and s], be cointegrated with a cointegrating vector [1.-l].

Definition 11 implies that in the long run the output gap between countries i and j settles

on some (non-zero) constant (see Hobijn and Frances (1998)). Definition 11 requires

,3, = ,6]. and that s” and S], be cointegrated as in Definition 1, but it allows a,- ¢ a j.

75

Definition III implies that the grth rate difference between countries i and j tends to be

zero in the long run. Definition 111 requires only that ,6, = ,6. ; it puts no restriction of

cointegration on the stochastic parts. Thus, if countries i and j are convergent perfectly in
output, they also converge to each other relatively as well as in growth rate. And, if
countries i and j are convergent relatively in output, they also converge to each other in
grth rate.

Extensions of the convergence definitions to cover a group of (more than two)
countries can be done in two ways: pairwise and multivariate. Hobijn and Franses (1997)

considered pairwise convergence while Hobijn and Franses (1998) considered

 

multivariate convergence.

Definition I(A) (Perfect Convergence in Multivariate Output) A group of k countries
converge in multivariate output if ( yz, — y“ , y3, — y“ ,..., y,“ — y,, )' is zero-mean

stationary.

We note that in Definition I(A) it does not matter which country in a group is

chosen as the base country. Thus, it does not matter in what order the countries are

numbered.

Definition [(8) (Perfect Convergence in Pairwise Output) A group of k countries

converge in pairwise output if any pair of countries follows Definition 1.

76

Qefinition [I(A) (Relative Convergence in Multivariate Output) A group of k countries

converge in multivariate output if ( yz, - y], , y3, — y“ ,..., y,“ — y,, )' is level stationary.

Definition 11(8) (Relative Convergence in Pairwise Output) A group of k countries

converge relatively in pairwise output if any pair of countries follows Definition 11.

 

1?"
Definition ”I(A) (Convergence in Multivariate Growth Rate) A group of countries k
countries converge in multivariate growth rate if ( g” , gz‘, , ..., g,” ) have the same L
F
mean. L:
E

 

Definition 111(8) (Convergence in Pairwise Growth Rate) A group of k countries

converge in pairwise growth rate if any pair of countries follows Definition 11].

4.3 Convergence Measures

In this section we consider statistical tests of the convergence hypotheses listed in
the previous section. In each case we will test the null hypothesis of convergence against
the alternative of non-convergence. We continue to assume the representation of the last
section: y,, = a, + fig + 5,, where s” is I(l) so that g,, = ,8, + As,, is stationary.

We test the hypotheses of perfect and relative convergence in output using the zero-
mean and level-corrected MKPSS statistics respectively. Let 5,

2y" Ty}! bethe

1.1).:

output gap between countries i and j at period t and let Supt) = T‘I 2,115“ I be its

1)

77

sample mean. Define E: (E, ,E2,...,ET) where E, =(5(2,l),,,6(3.,)‘,,...,6(,,‘,)’,)' for the
zero-mean version of the test, and where

E, =(5(2.I).t —6(2‘l),6(3,,)‘, — (3J),...,6(,‘J)J —5(k,m)' for the mean-corrected version. Let

(1(1) be the Newey and West (1987) estimate of the long-run covariance of E , with I

 

lags:
(“2(1) = (‘20 + jam j,1)(f2 j + (‘21!)
where w(i,l)=1—j/(I+I) with l—)oo as T—m and l/T—)0; and f2, = T" 2,11,, (i3, EH').
Let AT be the TxT matrix with the (i,j)th entry equal to max(T-i+1, T-j+1). E

Then the MKPSS statistic is:
P = *Tl—z[vec(E')‘(§A2(I)-I ® AT)vec(E')].

See Chapter 3 for more detail. We note that when k=2, the MKPSS statistic is the
univariate KPSS statistic. We also note that this statistic is the same as the statistic
proposed and used in Hobijn and F ranses (1998) with one minor difference. We use the
original data (for the zero-mean version) or the demeaned data (for the level-corrected
version) in the calculation of Newey-West estimator, while Hobijn and Franses use the
demeaned and detrended data for both cases. This makes no difference asymptotically
but may matter in finite samples. Interestingly, our derivation of the MKPSS statistic is
different from theirs. The derivation in Chapter 3 follows the lines of KPSS, by deriving
the LM statistic for the hypothesis that the variance of the random-walk component of the
data is zero. Hobijn and Franses (1998) simply present the statistic as an algebraic

generalization of KPSS.

78

is?

Proposition 1 Let F“) and 1“” be the zero-mean and the level-corrected MKPSS
statistics, repectively. Then
1. f0) and Fm are invariant to the choice of base country.
2. When the group is convergent perfectly in output (Definition I(A)),
F") :> 1““) = 2:1“; W,2 (r)dr] , where W,(r) is the standard Brownian motion; when

the group is convergent relatively in output (Definition II(A)),

1‘”) => r”) = {51 1;, B} (r)dr], where B, (r) is the Brownian bridge.

 

3. When the group is not convergent in the sense of Definition I(A): F”) -> 00; when

the group is not convergent in the sense of Definition II(A): 11(2) —) 00.

Proof: For part 1, the result follows Hobijn and Franses (1998, Appendix A). A proof of
part 2 can be found in Chapter 3 (Theorem 1). Chapter 3 (Theorem 2) provides a proof
for part 3 when the non-convergence is caused by non-cointegration of the stochastic part
of the outputs. For the case that non-convergence is caused by the deterministic part

(different levels or trends), see Hobijn and Franses (1998, Appendix A). 0

By Proposition 1(1), we see that the statistic, F , is independent of the choice of

base country (or, equivalently, it is independent of the ordering of the countries). By

Proposition 1(2) and 1(3), we see that the test based on F is consistent as a test of

(perfect/relative) convergence in output againt non-convergent alternatives.

79

When we turn our interest to "convergence in growth rate", our primary concern
changes, since we assume that growth rates are stationary. Now we simply wish to test
whether mean growth rates are the same, which is a hypothesis about the mean in a

multivariate setting. The standard test in this case is based on Hotelling's T2 statistic.

We note that the Hotelling's T2 statistic usually assumes random sampling. In our

case, the independence condition for the growth rate of a country over time can hardly be is!

Ffi‘hln

presumed. Below, we introduce a modified Hotelling's T2 statistic. Let the growth rate

difference at periodtbe Q, = (Am),,i(,,),,,...,/1(,,,,,,) with )1

(1.1).: = gm ”8),; and the

 

12‘
t

average growth rate discrepancy be O = (T -l)'12,7:2 Q, . We consider a modified
Hotelling's T2 statistic as

H = (T -1)§f2(1)"§'
where 52(1) is the Newey-West estimator of Q , the long run variance matrix of Q, . We
need to specify some regularity conditions on the growth rates to ensure that a central

limit theorem holds for E and that (2(1) is consistent for Q. Basically, we need to

assert weak dependence (or "short memory"), which we can do with a number of standard

conditions such as Phillips and Durlauf (1986, p. 475).

Proposition 2 H has the following properties.
1. H is invariant to the choice of base country.

2. When the countries are convergent in growth rate (Definition III(A)), we have
2
H => 10H).
3. When the countries are not convergent in the sense of Definition III(A), H —> oo.

80

Proof: The proof of part 1 essentially follows Hobijn and F ranses (1998, Appendix A).

For the second part, since T" 2? :> MN (0,9" 2) (MN: multivariate normal distribution)
and (2(1) —>p Q , we have H :> 13“). For part 3, it is easy to show that H is oforder

O(T), so it goes to infinity as T goes to infinity. 9

4.4 Finding Convergence Clubs

In the previous section, several convergence statistics have been introduced. In this
section, we apply them to cluster countries into several disjoint "convergence clubs" and
leave all the countries that belong to no club as "isolated countries".

The club formation procedure is based on a "bottom-up" cluster algorithm. In the
first place, we put each country into a single country club (that is, k clubs for k countries).
Secondly, we compute a specified convergence measure (defined below) between every
pair of clubs and find the minimum measure. If it is larger than a given critical value, we
stop the procedure and conclude that no non-trivial convergence club exists. Otherwise,
we cluster the two clubs with minimun convergence measure into a new bigger club -- so
we have one convergence club with two countries and all other clubs with only one
country. Then, we compute the same convergence measure between every pair of clubs
(now we have only k-l clubs), and find the minimum one. Then we either stop the
procedure (because the convergence measure for every pair exceeds the critical value) or
we form a new club by combining the pair of clubs which had the minimal convergence

measure. We continue until no new clubs is formed. Thus, ultimately we classify all

81

 

countries in two big categories: convergence or isolation. Furthermore, countries in the
convergence category are clustered into several convergence clubs.
The convergence measure we use is the "p—value" for the statistic that is used to test

the hypothesis of convergence for the combined club. For example, for the case of

perfect convergence, the test statistic is F“) , and the corresponding random variable is
F“) , whose asymptotic distribution is given in Proposition 1. Then the p—value
corresponding to a value F“) is Pr(F(') S P (1)) , where F“) is treated as fixed and the

probability is evaluated from the asymptotic distribution of 1‘“). Our critical value is

chosen as 0.95. Thus we stop forming clubs when any newly formed clubs would
generate a test statistic for the convergence hypothesis that is significant at 5% level; that
is, any newly formed clubs would be rejected by the data at 5% level. Otherwise we
form the clubs whose p-value is smallest; that is, the clubs for which convergence is least
strongly contradicted by the data. We use p-values rather than values of the test statistic
because in the course of our procedure we consider combinations involving different
numbers of countries, and larger clubs will tend to generate larger statistics but not
necesasarily larger p—values.

A formal description of our cluster procedure is as follows.

Cluster Procedure

Let k. be the number of clusters, and K, (i=1,. . ., k.) be the set of countries in

cluster i.

1. Set k. =kand set K,={i} for all i=1,2,...,k.

82

 

2. Define m,.="statistical measure on (K ,UK J. )" and m' = min(m,j) for
i. j

1 _<_ i, j S k. but iag'. Let i‘ and j i be the indices of the two groups corresponding

to m‘. If m‘ is greater than c (=O.95), stop. Otherwise, proceed to the next step.

3. Reset K| =K ,.UK ,. and reset all groups other than i‘ and j. into K2 through
I 1

K,,_,. Let k. = k. —1. Return to Step 2.

v ...". b.1121 ,L'fl‘Aiz—y
r
F

Two points are worth noting here. The first is the reason we adopt a "bottom-up"

procedure (from small to big) instead of a "top-down" procedure (from big to small). As ..

 

mentioned in Chapter 3, the MKPSS test may suffer low power in some cases in which
only some of the series have unit roots or in cases of cointegration in which the series
share several common trends. Power may actually be lower when the number of non-
convergent countries is larger than when it is smaller. Thus, we worry that a "top-down"
procedure may lead us to find some spurious (too big) clubs. Certainly, a bottom-up
procedure is not totally immune to this problem, but the chance of finding spurious clubs
is lessened, because there are more hurdles to be passed before we cluster too many
countries into a big convergence club.

Second, our cluster procedure differs from the procedure used by Hobj in and
Franses (1998) since we do not condition our procedure for finding relative convergence
clubs on our results for perfect convergence clubs. Rather the two analyses are done

independently.

4.5 Data Descriptions and Preliminary Results

83

In this chapter, we use two data sets. The first is the data set of Bernard and
Durlauf (1995), consisting of annual real per capita GDP for some 15 OECD countries
ranging from 1900 to 1987. The second data set is from Maddison (1995). For the
purpose of comparison, we choose the same 15 OECD countries, but these data run from
1885 to 1994. B

The major difference between these two data sets is how they convert the individual

countries' GDP into a common unit. For the Bernard and Durlauf data, based on the

 

International Comparison Project V (ICP V) of the United Nations, a 1980 PPP-adjusted E
dollar is used as a common unit. For the Maddison data, a 1990 Geary-Khamis PPP-
adjusted dollar is considered. Comparisons in the ICP V are done on a binary basis while
the Geary-Khamis approach is based on a multilateral comparison.
The conventional wisdom for data such as these is that output levels are I(l) while
grth rates are stationary. In Table 4-1 we present results of the KPSS test of

stationarity applied to the levels and growth rates for each country (KPSS fifl test). We
choose the number of lags (I) in long-run variance estimation in each of two ways. First

we consider 1 =14 = int[4(T/10mm] . Second, we use a modified version of the

selection procedure of Newey and West (1994). See the Appendix for details. We
denote this procedure by "I=auto-sel."

In Table 4-1, "Data Set 1" refers to the Bernard-Durlauf data; "Data Set 2(1)" refers
to the Maddison data; and "Data Set 2(II)" refer to the Maddison data restricted to the

same time period as the Bernard-Durlauf data, 1900-1987.

84

The first part of Table 4-1 shows that, in all three data sets, the data for all countries
but one reject the null hypothesis of (linear) trend-stationarity at the 5% level. For the
US the null is typically rejected at the 10% level. Paradoxically, despite the fact that per
capita real GDP of each country exhibits nonstationarity individually, the MKPSS test of
the fifteen countries as a whole does not always reject the null of multivariate trend-
stationarity. For all three data sets, the null of joint trend stationarity is rejected at the 5% h
level when the Newey-West automatic selection method is used, but not when 1 =14.

In the second part of Table 4-1, we see that we cannot reject the null of level

stationarity of the growth rates at the 5% level. (A minor exception is that we do reject at

 

the 5% level for Australia, for Data Set 1, when I is chosen by the automatic selection
rule.) Thus, broadly speaking, the results in Table 4-1 confirm the conventional wisdom
that output levels are 1(1) and growth rates are 1(0).

In Table 4-2 we report the results of tests of convergence between every pair of
countries in the sample. It shows that out of 105 distinct combinations, 9%~17% show
evidence of perfect convergence in output, 15%~36% show relative convergence in
ouput, and virtually all show convergence in growth rate. Thus, as it should be, the

weaker the convergence hypothesis we use, the larger the number of convergent pairs we

find.

4.6 Empirical Results on Finding Convergence Clubs

In this section we apply the clustering procedure described in section 4.4 to the data

sets discussed in section 4.5. Our basic results are presented in Table 4-3. The clustering

85

results are presented for three different definitions of convergence, for three different data
sets, and for two methods of choosing the number of lags in estimation of the long-run
variance.

We will first discuss the results for clustering based on multivariate perfect
convergence, which are given in Table 4-3. 1. We certainly do not find evidence of
convergence among the entire set of 15 OECD countries. Rather we find the existence of
four or five small convergence clubs, with two to four members each, and a few isolated
countries.

The choice of lag length makes some difference but not too much. Choosing 1 =14
instead of l=auto-sel increases the number of isolated countries for two of the three data
sets, but does not otherwise change the composition of the convergence clubs. The
choice of data set makes more difference, but there is some regularity. Some of the
regularity seems intuitively reasonable. For example, Japan is always an isolated
country. Belguim, Denmark and The Netherlands are always in the same club. Except in
one case, Australia and the UK are always in the same club. Some of the findings are
very hard to understand. For example, in Data Set 1, France and Norway are a club; so
are Finland and German. In either version of Data Set 2, Italy and Norway are a club.
Probably it is too much to expect for all of the clusters from any algorithm to appear
reasonable, and the main conclusion from Table 4-3.1 is that we have a moderate number
of fairly small clubs.

Table 4-3.2 gives the results for clustering based on multivariate relative
convergence. Since this is a weaker requirement than perfect convergence, we should

expect larger clubs, and hence perhaps fewer clubs, and less isolated countries. This

86

 

happens only partially: we do have fewer isolated countries, and we usually have
somewhat larger clubs, but we do not get less clubs. More interestingly, there is
surprisingly little correspondence between the clubs defined in terms of perfect
convergence (Table 4-3. 1) and those defined in terms of relative convergence (Table 4-
4.1). The latter are definitely not just combinations or augmentations of the former. This
frankly calls into question the reliability of forming convergence clubs by the type of ':
clustering algorithm we use.

Table 4-3.3 gives the results for clustering based on convergence in grth rates.

Now we clearly do find fewer clubs and larger clubs. In fact, in each case we find

 

exactly two clubs, with the number of countries in the larger club between 10 and 14.
The clubs do not seem to have any simple or obvious interpretation.

In Table 4-4, the convergence measures of Hobijn and Franses (1998) are
considered. As mentioned in Section 3, the only difference between the MKPSS statistic
and the Hobijn-Franses measure is in the calculation of the Newey-West estimator. For
the Hobijn-Franses statistic we use the demeaned and detrended residuals in the
calculation of the Newey-West estimator while for the MKPSS statistic we used either
the levels (zero mean case) or deviations from means (level-stationary case). We will use
the 5% critical level and our methods of choosing I (1 =14 or l=auto-sel) so our results are
not the same as those in Hoijn and F ranses (1998). This one minor difference in
econometric detail makes a surprisingly large difference. We now have more isolated

countries and smaller clubs. In addition there are some shifts in membership.

87

In order to be able to compare the compositions of the convergence clubs in Table
4-3 and Table 4-4, we also use a descriptive statistic proposed by Hobijn and F ranses

(1998), ra'b :

 

 

b
r _ l—127=16:j5r,j
a,b _ 1) °
(2E2: 7¢r5§jX2§Lt ind-,1)
Here a,b represent two clustering procedures, and 6:, =1 if countries i and j are in the

same convergence club for procedure "x", otherwise, 6,3. = 0. Obviously this is a kind of

sensitivity measure. The higher raj, , the less sensitive the results are to the choice of

clustering procedure.

Table 4-5 gives the results of this analysis. In Table 4-5.1 we present the results for
different choices of the truncation lag (keeping other things fixed). These correlations are
reasonably high, in excess of 0.7, except for perfect convergence and Data Set 1. In
Table 4-5.2 we present the correlations for the choice between Data Set 1 and Data Set
2(II), so that we are comparing the Bemard-Durlauf data with the Maddison data over the
same time period. These correlations are fairly low for the case of perfect convergence
but fairly high for the case of growth rate convergence. In Table 4-5.3 we present the
results for the choice between Data Set 2(1) and Data Set (11); that is, between the entire
Maddison data set and a subset of it. Finally, in Table 4-5.4 we present the correlations
for the choice between the MKPSS measure and the Hoijn-Franses measure. These are

also fairly high, except perhaps for perfect convergence and Data Set 1.

4.6 Conclusion

88

In this chapter we investigate whether a group of 15 OECD countries exhibit
convergence in output levels or in growth rates. Convergence in output levels is defined
in terms of the joint stationarity of cross—country output differences, and this hypothesis is
tested using a multivariate version of the KPSS test that was developed in Chapter 3 and

Hobijn and Franses (1998). Convergence in growth rates is defined as equality of mean

growth rates, and this hypothesis is tested using a modification of Hotelling's T 2 test.
We investigate the sensitivity (or robustness) of our results by considering some minor
variations in econometric detail related to estimation of long-run variances.

We consistently reject the hypothesis of convergence for the entire set of 15 OECD
countries. This is true for all three definitions of convergence and is clearest conclusion
of our study. We then use a clustering algorithm to creat "convergence clubs" that are
characterized by within-clubs convergence. For convergence in output levels, we find
four or five clubs with two to four members each, plus a few isolated firms. Some of
these clubs seem to make intuitive sense and some do not. For convergence in growth
rate, we find that the 15 OECD countries can be separated into two convergence clubs,
where the larger clubs typically has about 10 members.

The composition of the convergence clubs is moderately sensitive to the choice of
data and to questions of econometric detail. Thus we don't have the same degree of belief
in the composition of our convergence clubs that we do in our results on the number of

clubs.

89

 

..-i...

52.8% N “em 28 n E N .8 28 .AamEmm: N am 28 u e N am 98

.8 saw 825 .vs ”503:: 8:5 .95 $85 £02 ”£202 .52 ”88.55% ..3 ”can...
.5. is. .mmo “:38 {E ”seen. .2: ”use: .28 season .25 n85 ...mm ”2298 .22 ”£3. .w2 a__e....=< e8. 0. 9:8

33 23 H2...... .23: $9.: $9 32 ”as: are: 3.22
3.9 53 3.: 83 “can; mwnx 3.0: ONE .2.va Ed ”685 3.2

w. .olgml.) .835

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

83 32 32 EN 83 82 :2 e 2 82 :3 £2 E: 3:
N86 $0.0 80.0 $2 NSC $0.0 Ntd mNS $3 83 5.0 8:. w:
52 38 83 23 CNS 83 55 83 go 82 a3 23 v5
5N6 53 £8 £3 83 83 ES 83 83 82 88 22 £5
88 $3 $8 $8 82 Ed Es 38 83 3N6 23 88 «.02
Ned N80 83 9.3 No; 32 88 83 83 :3 $3 83 $2
$2 $3 23 SS 32 $3 83 33 83 NeNd 8:. :3 as
83 N26 $8 83 $3 83 we; 38 0:3 53 23 83 E
83 Ne; NSC 82 NS 3.3 $8 83 88 83 $8 :3 «mo
83 9.3 33 a; $8 $3 88 $2 8% $3 a; 83 S:
83 82 9.3 $8 33 None ”Ned 23 22 $3 5.0 :3 z;
88 83 2 3 N03 83 No; 8:. NeNd 23 83 $3 $2 2mg
35.0 80.0 $8 3.8 $2 22 58 N80 :2 53 88 53 25
88 N80 32 $8 58 98 go 83 82 23 ES 33 mm
83 $3 $3 83 8; ES 33 $8 $2. 82 $2 8N6 :2
a; :2 NNS 82 88 82 83 83 8g 38 82 2N6 £2
E 3833 2a e833 an 3833 41 e833 E Caesar ..I 3833 558
g N am 28 e N 3 28 = 5m 23 g N am 98 e N am «no F 3 28
sex 2590 .93 598

3582.8 .2... .e .8» use. 535 5:238 25 s :3 ".25.. 598.

 

90

 

m: 23. 2E .5< ”83:80 35.8.

as. 2E .5< ”33:30 35.8.

 

 

 

 

 

 

 

 

 

 

 

vs m2 e 8 2: m2 .2.
«oz 5. .N «oz 5. ..., a N .8 28
23. ”mo 5: .N 25 $0 «E .N
52 28 28 48 .F 52 28 25 ,_wm .2
n3 z: :2 ”8258 e228 .3 z: :2 ”82:80 828.
8 v5 8... .e 8 es mi 2.
25m $2 3: .m 2% Eu 5: .m s N 8 28
$2 28 25 mm .N 52 28 25 ..8 .N
«02 ,E .2 «Oz 3. .F
8 226 .3 28 22 ”85:80 828 8 E ”8258 828
$2 28 a $2 28 28 .8
v5 25 .e vs 25 22 .e
«we 2: .m «.8 z: .N F .8 28
<2 :2 .N m3 5. S... .N
«oz 5: .2 202 s: .F
2 1 8.2.... 1 a... , 28
N2 2: 2: N2 N2 2: 28 5290 g 8:83.80
“N 8 e N e N... 52.5 585928 828
z e z e m 2 =28 5 8592.8 52.8
:2 28282 22 3.232 E 3838 889280
a N .8 28 a N .8 28 F 28 28

 

 

 

 

91

 

@202. 82.80 828.
.5 «oz .3 2.. 83 .N
8 m2... 52 5. ”mo 5... 28 25 .8 5... F

52 25. <._._ mum (”E 2i zwo z<o ..mm .S< 932 .N

w: 8.....80 828. a. N .8 28
258 .62

 

3202. H82.2500 350$
«.02 23.. 2E .N
w:
v5 95 .52 (.2 «$0 (mu. zmo z<o ...wm :2 Q7. ._.

w: m>>w 102 23. <: mmo «an. 2:. zmo z<o ._.:< .N

.sz2. 8......8 828. s N .8 28
.5 mi .8 82 .F

 

@202. 8......8 82......
m: 2.8 ..02 3 5. mm... 5.. 2.. 28 25 5.. .N
.5 hi ..8 8.... F

m>>m mOz $2 2.4.. <._._ mmw (an. 2:. zmo z<o ._.:< .N
.F

@202. 82.58 8.29.
8: F .8 28

x: ..mm m3...

 

 

2n.

 

.863... u. mm. . .28

 

 

 

 

«Oz 28 8.5.80 8.28. $202. 8......8 828.
258 (F. .m
8 52 .e 2.8 ”.02 ...
..mm. s... z<o 5.. .N 8 52 5. $0 5... 28 25 S... .N 2. N .8 28
3 2.. .N 3 z: .N
v... .8 8... F .5 .8 8.. F
”.02 3 2.. 8.....80 8.28. @202. 8......8 828.
8.. 28 ..., ..02 3 2.. 8
$2 .8 .e m: 5... 2.
.5 8:... ..., v... 52 .8 83 .N e N .8 28
«we s... 3. 5.. N .80 5... 28 2.... .N
2...... (F. F 258 E. F
28 z... .8 8......8 828. z... 8......8 828.
<2. s... 3. 5.. m. S. s... 28 25 5.. ...,
8 $2 2. m: E .8 ...
3 .82 .N 3 .82 ..N F .8 28
2.5 «02 .N 258 ”.02 .N
.5 8.... .F .5 8.. .F
2 u. 88.8 u. 8. . 28

 

 

 

92

 

m3 n_<... z_n_ ...mm .5... .85500 356$

w: as. 27. .mm .5... ”83:30 356$

 

 

 

82.8 <2. .8. < 82.8 <2. 88 e
v... 8..< .8 v... 82< .8 .5 N .8 28
82 28 2<o .N 82 28 2<o .N
.02 <2 .F .02 <2 .F
8.. 3 2.. S< 8......88 8.28 8.. 3 2.. S< 8.588 8.28.
82.8 2.8 <2. .8 82.8 88 <... .8
2.. 8..< 8 V... 8..< 2.
28 25 .8 28 2<o .8 .2 N .8 28
82 .8 .N 82 .8 .N
..02 <2 .F .02 <2 .F
8.. v... 8.. 82.8 3 88 2.. 28 .8 8.....88 8.28.
8.28 3 88 2.. 2<8 .8 8..< 8......88 8.28. 2.. 8..< ..
52 mo .8 .82 28 .8 F .8 28
<2 S< .N <2 S< .N
.02 <2. .F .02 <2. .
2 u. 828 n. 8.. . 28

 

 

 

93

 

 

 

 

 

 

 

N88... 8:... 888... .2 N .8 28

88.... 8.8... 8.... s N .8 28

83.0 .88... 8.8.0 . 8.80
828982200 8.8.5390 828982200 82.282 828982200 .8288. 8.80

 

 

 

 

2.. 28 .8 8.880 8.28. 202 8.888 8.28.
3 280 .8 3 2.. .8
8.. .82 8 8.. 82 2.< 8

v... 8..< .8 2.. .8 8..< .8 2.. N .8 28
<2 <2. 3. 2.< .N 280 <2. 28 2<0 .N
8.28 202 .. 22.8 <2 ..
202 3 2.. 28 .8 8.880 8.28. 202 8......80 8.28
3 2.. .8
8.. 5< .8

2.. 8..< 2. 2.. 8..< .. .2 N .8 28
280 <2. 2<0 S< .8 220 <2. 28 2<0 .8
8.. .82 .N 52 .8 .N
22.8 <2 .. 8.2.8 <2 ..
202 3 2.. 5< 8.....80 8.28 2.. 28 8.....80 8.28
8.. 28 .8 3 288 .8
52 .8 2. 8.. .82 .8 2.

2.. 8..< .8 2.. 8..< .8 . .8 28
280 <2. 2<0 .N <2 <2. 2<0 S< .N
82.8 <2 2 82.8 202 ..
.2 u. .828 u. 88. .28

 

 

 

94

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

88:. 8N8... .5 N .8 28
8.8... .88... .2 N .8 28 2".
000.. 808.0 . 8.80
8k... 888.8 2.. N .8 28
288.8 .88... e N .8 28 .828...
.880 E0 _ 8.80
828082200 8.2.8.82 880.8228 58.288. 8.80 08..
882.0 888.0 880.0 8. n.
88.0 58.0 000. F 88.0.2.8".
8280.82.00 8.8.2.390 802808.600 8.2.282 828082200 .8288. 08..
000. F 828.0 080 ...u.
080 N820 828.0 88.0.28”.
828982200 8.9-5.220 880.9600 8.2.282 8080.228 .8288. 08..

 

 

 

 

95

APPENDIX

(Automatic Lag Selection Procedure)

The following procedure is based on Newey and West (1994). Since it is not
invariant to which country is chosen to be the base country, we suggest that, for a given
group of countries, first we search for the number of lags by the method of Newey and
West (1994) according to all possible choices of the base country. Then, among all lags

we choose the minimum one as our data-driven (or automatically-selected) truncation lag.

Let éb = Zfrfiléfi} where £131} is the ith column of E with the bth country as the base.

Let fl, (i) = 2,1,.“ (2,”??? . We calculate "I=auto-sel" through the following steps.

Step 1. Choose bandwidth: m = int[4(T/100)2/9] .

Step 2. Let b=1 (the index of the base country).

Step 3. Calculate: 8,” = 8(0) + 2% 5,0) and 55,21 = 8(0) + 23"; mi) .
i=1 i=l

“I 1 2
Step 4. Calculate 1,, = int[l.447(s’/,m) T ]
Sb

Step 5. Let b=b+l, return Step 3 if b S k (k: group size); otherwise, proceed to the next

“3

step.

Step 6. Choose 1 = Imbink(l,,) and stop.
S S

96

LIST OF REFERENCES

Andrews, D.W.K. and W. Ploberger, 1994, Optimal tests when a nuisance parameter is
present only under the alternative, Econometrica 62, 1383-1414.

Azariadis, C. and A. Drazen, 1990, Threshold extemalities in economic development,

Quarterly Journal of Economics 105, 501-526.

Barro, R.J. and X. Sala-i-Martin, 1992, Convergence, Journal of Political Economy 100, R
223-251.

Baumol, W.J., 1986, Productivity growth, convergence and welfare: what the long-run
data show, American Economic Review 76, 1072-1085.

Bernard, AB. and S. Durlauf, 1995, Convergence in international output, Journal of
Applied Econometrics 10, 97-108.

 

Bernard, AB. and S. Durlauf, 1996, Interpreting tests of the convergence hypothesis,
Journal of Econometrics 71, 161-73.

Bianchi, M., 1997, Testing for convergence: evidence from non-parametric
multimodality tests, Journal of Applied Econometrics 12, 393-409.

Bhargava, A., 1986, On the theory of testing for unit root in observed time series, Review
of Economic Studies 52, 369-384.

Chung, C.F., 1996, Sample variance, sample covariances, and linear regression of
stationary multivariate long memory processes, working paper, Michigan State
University.

Choi, I. and BC. Ahn, 1995, Testing for cointegration in a system of equations,
Econometric Theory 11, 912-951.

Choi, I. and BC. Ahn, 1999, Testing the null of stationarity for multiple time series,
Journal of Econometrics 62, 1383-1414.

Dickey, DA. and WA. Fuller, 1979, Distribution of the estimators for autoregressive time
series with a unit root, Journal of the American Statistical Association 74, 427-431.

Diebold, F.X. and GD. Rudebusch, 1991 , On the power of Dickey-Fuller tests against
fractional alternatives, Economic Letters 35, 155-160.

Granger, C.W.J., 1980, Long memory relationships and the aggregation of dynamic
models, Journal of Econometrics 14, 227-238.

97

Granger, C.W.J. and R. Joyeux, 1980, An introduction to long memory time series models
and fiactional differencing, Journal of Time Series Analysis 1, 15-29.

DeJong, R. and J. Davidson, 1997, The functional central limit theorem and weak
convergence to stochastic integrals: results for weakly dependent and fractionally
integrated processes, working paper, Michigan State University.

Durlauf, S. and RA. Johnson, 1995, Multiple regimes and cross-country growth behavior,
Journal of Applied Econometrics 10, 365-384.

Galor, O., 1996, Convergence? Inferences from theoretical models, The Economic
Journal 106, 1056-1069.

Harman, E.J., 1970, Multiple Time Series, John Wiley, New York.

Hansen, BE, 1992, Consistent covariance matrix estimation for dependent
heterogeneous processes, Econometrica 60, 967-972.

 

Hobijn, B. and PH. F ranses, 1997, Asymptotically perfect and relative convergence of
productivity, working paper, New York University.

Hobijn, B. and PH. Franses, 1998, Asymptotically perfect and relative convergence of
productivity, working paper, New York University.

Hosking, J ., 1981, Fractional differencing, Biometrika 68, 165-176.

Hosking, J ., 1996, Asymptotic distributions of the sample mean, autocovariances, and
autocorrelations of long-memory time series, Journal of Econometrics 73, 261-284.

Jones, CL 1997, On the evolution of the world income distribution, Journal of Economic
Perspectives 11:3, 19-36.

King, M.L. and P.X. Wu, 1997, Locally optimal one-sided tests for multiparameter
hypotheses, Econometric Reviews 16, 131-156.

King, M.L. and TS. Shively, 1993, Locally optimal testing when a nuisance parameter is
present only under the alternative, Review of Economics and Statistics 75, 1-7.

Kwiatowski, 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?, Journal of Econometrics 54, 159-178.

Lee, HS, 1995, Analysis of economic time series with long memory, unpublished Ph.D.
dissertation, Michigan State University.

98

Lee, D. and P. Schmidt, 1996, On the power of the KPSS test of stationarity against
fractionally-integrated alternatives, Journal of Econometrics 73, 285-302.

Leyboume, SJ. and B.P.M. McCabe, 1994, A consistent test for a unit root, Journal of
Business and Economic Statistics 12, 157-166.

Lo, A.W., 1991, Long-term memory in stock market prices, Econometrica 59, 1279-1313.

Lucas, R.E., On the mechanics of economic development, Journal of Monetary Economics
22, 342.

Nelson, CR. and CI. Plosser, 1982, Trends versus random walks in macroeconomic time
series: some evidence and implications, Journal of Monetary Economics 10, 139-162.

Maddison, A., 1995, Monitoring the world economy 1820-1992, Development Center of i
the Organization for Economic Co-Operation and Development.

 

“'1.“ . _-'V‘ a

Newey, W.K. and K.D. West, 1987, A simple, positive semi-definite, heteroskedasticity
and autocorrelation consistent covariance matrix, Econometrica 55, 703-708.

Newey, W.K. and K.D. West, 1994, Automatic lag selection in covariance matrix
estimation, Review of Economic Studies 61, 631-653.

Park, J .Y. and PCB. Phillips, 1988, Statistical inference in regression with integrated
processes: Part I, Econometric Theory 4, 468-498.

Park, J .Y. and PCB. Phillips, 1989, Statistical inference in regression with integrated
processes: Part II, Econometric Theory 5, 95-131.

Phillips, PCB. and S. Durlauf, 1986, Multiple time series regression with integrated
processes, Review of Economic Studies LIII, 473-495.

Phillips, PCB. and P. Perron, 1988, Testing for a unit in time series regression, Biometrka
75, 335-346.

Phillips, PCB. and V. Solo, 1992, Asymptotics for linear processes, Annals of Statistics
20, 971-1001.

Potscher, B.M., 1991, Non-invertibility and psudo-maximun likelihood estimation of mis-
specified ARMA models, Econometric Theory 7, 435-499.

Quah, D.T., 1996, Twin peaks: growth and convergence in models of distribution
dynamics, The Economic Journal 106, 1045-1055.

Romer, P.M., Increasing returns and long-run growth, Journal of Political Economy,
1002-1037.

99

Said, SE. and DA. Dickey, 1984, Testing for unit roots in autoregressive-moving average
models of unknown order, Biometrika 71, 599-608.

Saikkonen, P. and R. Luukkonen, 1993, Testing for a moving average unit root in
autoregressive integrated moving average models, Journal of the American Statistical
Association 88, 596-601.

Schmidt, P. and PCB. Phillips, 1992, LM tests for a unit root in the presence of
deterministic trend, Oxford Bulletin of Economic and Statistics 54, 257-287.

Schmidt, P., 1993, Some results on testing for stationarity using data detrended in
differences, Economic Letters 41,1-6.

Sim, C. A., J .11. Stock, M.W. Watson, 1990, Inference in linear time series models with
some unit roots, Econometrica 58, 113-144.

Sowell, F., 1989, Maximum likelihood estimation of fractionally integrated time series
models, working paper, Carnegie Mellon University.

Sowell, F ., 1990, The fractional unit root distribution, Econometrica 58, 495-50.

Tsay, W.J. and C-F. Chung, 1995, The spurious regression of fractionally integrated
process, working paper, Michigan State University.

100

 

"111111111111111“