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609

This is to certify that the
dissertation entitled
KPSS AND LEYBOURNEchCABE AUTOCORRELATION
CORRECTIONS IN STATIONARITY TESTS
presented by

Yongsu Cho

has been accepted towards fulfillment
of the requirements for

Ph . D degree in ECODOEiCS

3&1»ka

Major professor

Date 8/22/02

 

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

 

LIBRARY
Michigan State
University

 

 

 

PLACE IN RETURN Box to remove this checkout from your record.
To AVOID FINES return on or before date due.
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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJClRC/DatoDuepGS—sz

 

KPSS AND LEYBOURNE-MCCABE AUTOCORRELATION
CORRECTIONS IN STATIONARITY TESTS

By

Yongsu Cho

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

2002

ABSTRACT

KPSS AND LEYBOURNE-MCCABE AUTOCORRELATION
CORRECTIONS IN STATIONARITY TESTS

By
Yongsu Cho

We consider tests of the null hypothesis that a time series is stationary that were
proposed by Kwiatkowski et al.(1992) and Leyboume and McCabe (1994, 1999). We
identify a problem with the Leyboume and McCabe (1999) test and suggest two
modifications of the test to solve it. We provide consistent model selection rules to pick
the number of lags used in the tests. Then we conduct simulations to compare the size
and power characteristics of the tests under different data generating processes and
different treatments of the number of lags. Generally speaking, the results are favorable
to the use of formal model selection rules, and they are unfavorable to the (unmodified)

Leyboume and McCabe (1999) test.

Dedicated to my parents,
And to my wife,

Eunj i

iii

Ft

would lil
and inter
grateful 1
Robert d
l
Institute
helping
their en
1
Slippon
Heeyeo

Support

ACKNOWLEDGEMENTS

For the completion of my dissertation, I am indebted to many people. First of all, I
would like to thank my dissertation supervisor, Professor Peter Schmidt for his excellent
and invaluable advice throughout the course of writing this dissertation. I am especially
grateful for his patience and intimate guidance on my dissertation. I also thank Professor
Robert de J ong and Professor Jeffrey Wooldridge for their helpful comments.

I gratefully acknowledge the financial support of the LG Economic Research
Institute for my study in the United States. I also would like to thank many friends for
helping me in different ways. I am grateful to Dr. Chirok Han and Jongbyung Jun for
their critical advice on computer simulations.

Finally, I would like to thank my parents and parents-in-law for their invaluable
support. My deepest gratitude goes to my wife, Eunji and my beloved two daughters,
Heeyeon and Heeseo, for their encouragement and patience. Without their love and

support, I could not have completed this dissertation.

iv

 

LIST (

C HAP
lNIRl

C HA

PERI
“Til

CPL:

PER
TES

CH,

PEF
\m

TABLE OF CONTENTS

LIST OF TABLES .................................................................................. vii

CHAPTER 1

INTRODUCTION .................................................................................... 1
1. Preliminaries .............................................................................. 1
2. The “local level model” and score tests .............................................. 2
3. LM99 test and its modifications ........................................................ 7
4. Short-run dynamics ..................................................................... 10
5. Organization of the thesis .............................................................. 16

CHAPTER 2

PERFORMANCE OF THE KPSS AND LEYBOURNE-McCABE TESTS

WITH A FD(ED NUMBER OF LAGS ........................................................... 18
1. Introduction ............................................................................. 18
2. Simulations .............................................................................. 19
3. Conclusions ............................................................................. 28
Appendix I ................................................................................... 31

CHAPTER 3

PERFORMANCE OF THE KPSS AND LEYBOURNE-MCCABE
TESTS WHEN THE NUMBER OF LAGS INCREASES WITH

THE SAMPLE SIZE ............................................................................... 51
1. Introduction ............................................................................ 51
2. Theoretical issues ..................................................................... 52
3. Simulations ............................................................................. 54
4. Conclusions ............................................................................ 60

CHAPTER 4

PERFORMANCE OF THE KPSS AND LEYBOURNE-McCABE TESTS

WITH MODEL SELECTION RULES .......................................................... 77
1. Introduction ............................................................................ 77
2. A consistent model selection rule for the Leyboume-McCabe tests. . . . . . ....78
3. A consistent model selection rule for the KPSS test 8O
4. Simulations .............................................................................. 82

C HA

CON

Bibli

 

5. Conclusions ............................................................................. 91

Appendix II .................................................................................. 93
CHAPTER 5
CONCLUDING REMARKS ..................................................................... 124
Bibliography ......................................................................................... 128

vi

 

 

 

 

CHAP'

Table I

Table I

Table j

Table

Table

Table

Table

Table

Table

Tab]:

Tablr

Tabll

Table

LIST OF TABLES

CHAPTER 2

Table 2.1: Size of KPSS Test with Fixed Number of Lags
(DGP: iid errors, no time trend) ....................................................... 33

Table 2.2: Power of KPSS Test with Fixed Number of Lags
(DGP: iid errors, no time trend) ....................................................... 34

Table 2.3: Actual Critical Values of KPSS Test with Fixed Number of Lags
(DGP: iid errors, no time trend) ....................................................... 35

Table 2.4: Size-Adjusted Power of KPSS Test with Fixed Number of Lags
(DGP: iid errors, no time trend) ....................................................... 36

Table 2.5: Size and Power of Leyboume-McCabe Tests with Fixed Number of Lags
(DGP: iid errors, no time trend) ....................................................... 37

Table 2.6: Actual Critical Values of Leyboume-McCabe Tests with Fixed Number of
Lags (DGP: iid errors, no time trend) ................................................ 39

Table 2.7: Size-Adjusted Powe of Leyboume-McCabe Tests with Fixed Number of
Lags (DGP: iid errors, no time trend) ................................................ 40

Table 2.8: Size and Power of KPSS and Leyboume-McCabe Tests with AR(1) Errors
(DGP. Yr: DYt-l'l' 8:, P=1/3) ............................................................ 42

Table 2.9: Actual Critical Values of KPSS and Leyboume-McCabe Tests with AR(1)
Errors (DGP: yt = PYt-1+ 8t, p=1/3) .................................................... 43

Table 2.10: Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with AR(1)
Errors (DGP: yt = py,_1+ 8,, p=1/3) ................................................... 44

Table 2.11: Size and Power of KPSS and Leyboume-McCabe Tests with MA(1) Errors
(DGP. Y! = aft-98H, 9:0.5) ............................................................ 45

Table 2.12: Actual Critical Values of KPSS and Leyboume-McCabe Tests with MA(1)
Errors (DGP: y. = 8t+98t-1, 9:0.5) .................................................... 46

Table 2.13: Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with MA(1)
Errors (DGP: yt= mesa, 9=O.5) .................................................... 47

vii

 

 

 

 

Table 2..

Table 2.

Table 2.

CRAFT

Table 3

Table 3

Table 3

Table 2

Table 1

Table

Table

Table

Table

Table

Table

Table 2.14: Size and Power of KPSS and Leyboume-McCabe Tests with ARMA(1, 1)

Errors (DGP: yt= pyt 1+ was. 1, p=1/3, 0=1/2) .................................... 48
Table 2.15: Actual Critical Values of KPSS and Leyboume-McCabe Tests

with ARMA(1, 1) Errors (DGP: yt= pyt- 1+ et+08t. 1, p=1/3, 0=1/2) ............. 49
Table 2.16: Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with

ARMA(1, 1) Errors (DGP: yt= pyt. 1+ 5t+93t-1, p=1/3, 0=1/2) ................... 50
CHAPTER 3
Table 3.1: Size and Power of KPSS and Leyboume-McCabe Tests when Number of

Lags Increases with Sample Size (DGP: iid errors, no time trend) .............. 63

Table 3.2: Actual Critical Values of KPSS and Leyboume-McCabe Tests when Number
of Lags Increases with Sample Size (DGP: iid errors, no time trend) ........... 65

Table 3.3: Size-Adjusted Power of KPSS and Leyboume-McCabe Tests when Number
of Lags Increases with Sample Size (DGP: iid errors, no time trend) ........... 66

Table 3.4: Size and Power of KPSS and Leyboume-McCabe Tests with AR(1) Errors
(DGP: y,= pyt-1+ 8t, p=1/3) ............................................................. 68

Table 3.5: Actual Critical Values of KPSS and Leyboume- -McCabe with AR(1) Errors
(DGP: yt= pyt-1+ st, p=1/3) ............................................................ 69

Table 3.6: Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with AR(1)
Errors (DGP: y. = pyt-1+ at, p=1/3) .................................................... 70

Table 3.7: Size and Power of KPSS and Leyboume-McCabe Tests with MA(1) Errors
(DGP: yt=81+981-1, 0=0. 5) ............................................................. 71

Table 3.8: Actual Critical Values of KPSS and Leyboume-McCabe Tests with MA(1)
Errors (DGP: yt= e.+03,- 1, 0=O. 5) ..................................................... 72

Table 3.9. Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with MA(1)
Errors (DGP: yt= 8t+98(-1, 0=O. 5) .................................................... 73

Table 3.10: Size and Power of KPSS and Leyboume-McCabe Tests with ARMA(1,1)
Errors (DGP: yt= pyt 1+ was..- 1, p=1/3, 0=1/2) ..................................... 74

Table 3.11: Actual Critical Values of KPSS and Leyboume-McCabe Tests with
ARMA(1,1) Errors (DGP: yt= pyH+ et+08.-1, p=1/3, 0=1/2) ................... 75

viii

 

 

 

 

C HAP'
Table -‘

Table .

Table

Table

Tablc

Tabl

Tab]

Tab

Tab

Tal

Tal

Ta‘

Table 3.12: Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with

ARMA(l,l) Errors (DGP: y,= py,-1+ St'l'GSH, p=1/3, 0=1/2)....................76

CHAPTER 4

Table 4.1:

Table 4.2:

Table 4.3:

Table 4.4:

Table 4.5:

Table 4.6:

Table 4.7:

Table 4.8:

Table 4.9:

Frequency of Lag Selection: KPSS Test Under the Null
(DGP: iid errors, no time trend)
(lmax=3, 10% significance level for pretest) ......................................... 96

Frequency of Lag Selection: KPSS Test Under the Alternative

(DGP: iid errors, no time trend)
(Im=3, 10% significance level for pretest) ......................................... 97

Size of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend) ........................................................ 98
Power of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend) ........................................................ 99
Actual Critical Values of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend) ..................................................... 100
Size-Adjusted Power of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend) ...................................................... 101

Frequency of Lag Selection: KPSS Test Under the Null
(DGP: iid errors, no time trend)
(1,....=3, c.v=(T/100)m) ............................................................... 102

Frequency of Lag Selection: KPSS Test Under the Alternative
(DGP: iid errors, no time trend) .
(1m..=3, c.v=(T/1OO)”4) ............................................................... 103

Size of KPSS Test with Model Selection Rule
(DGP: iid errors, no time trend) ..................................................... 105

Table 4.10: Power of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend) ..................................................... 106

Table 4.11: Actual Critical Values of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend) .................................................... 107

Table 4.12: Size-Adjusted Power of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend) ..................................................... 108

ix

 

Table 4

Table 4

Table -

Table

Table

Table

Table

Tabl:

Tabl

Tab]

Tabi

Tab

Tab

Tab

Tab

Table 4.13: Size and Power of Leyboume-McCabe Tests with Model Selection Rule

(Pmax=3) (DGP: iid errors, no time trend) ........................................ 109
Table 4.14: Actual Critical Values of Leyboume-McCabe Tests with Model Selection
Rule (pmax=3) (DGP: iid errors, no time trend) ................................... 110
Table 4.15: Size-Adjusted Power of Leyboume-McCabe Tests with Model Selection
Rule (pmx=3) (DGP: iid errors, no time trend) ................................... 111
Table 4.16: Size and Power of Leyboume-McCabe Tests with Model Selection
Rule (pmaX=3) (DGP: iid errors, no time trend) ................................... 112
Table 4.17: Actual Critical Values of Leyboume-McCabe Tests with Model Selection
Rule (pmax=3) (DGP: iid errors, no time trend) ................................... 113
Table 4.18: Size-Adjusted Power of Leyboume-McCabe Tests with Model Selection
Rule (pmax=3) (DGP: iid errors, no time trend) ................................... 114
Table 4.19: Size and Power of KPSS and Leyboume-McCabe Tests with AR(1) Errors
(DGP: Yr: Pyt4+ 8t, p=1/3) .......................................................... 115
Table 4.20: Actual Critical Values of KPSS and Leyboume-McCabe Tests with AR(1)
Errors (DGP: yt= pyt-1+ at, p=1/3) ................................................. 116

Table 4.21: Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with AR(1)
Errors (DGP: yt= pyt-1+ st, p=1/3) .................................................. 117

Table 4.22: Size and Power of KPSS and Leyboume-McCabe Tests with MA(1) Errors
(DGP: y, = 8:498“, 0=O.5) ............................................................ 118

Table 4.23: Actual Critical Values of KPSS and Leyboume-McCabe Tests with MA(1)
Errors (DGP: yt= ewes“, 0:0.5) ................................................... , 119

Table 4.24: Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with MA(1)
Errors (DGP: y, = ewes“, 0:0.5) .................................................. 120

Table 4.25: Size and Power of KPSS and Leyboume-McCabe Tests with ARMA(1,1)
Errors (DGP: yt= py,-1+ s,+03t.1, p=1/3, 0=1/2) .................................. 121

Table 4.26: Actual Critical Values of KPSS and Leyboume-McCabe Tests with
ARMA(1,1) Errors (DGP: yt= pyH-l- 8t+98t-|, p=l/3, 0=1/2) .................. 122

Table 4.27: Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with
ARMA(1,1) Errors (DGP: Yr: PYM+ ewes“, p=1/3, 0=l/2) ................. 123

 

 

 

1. Pre

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Static
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Chapter 1

Introduction

1. Preliminaries

From a statistical point of view, the correct treatment of the stationary or
nonstationary nature of time series data is quite crucial for valid statistical inference,
owing to the spurious regression phenomenon. However, standard unit root tests are not
necessarily very powerful against relevant alternatives. A unit root is typically the null
hypothesis being tested, and the null hypothesis is accepted unless there is strong enough
evidence against it.

Since the influential work of Nelson and Plosser (1982), which found that most
US. macroeconomic time series contain a unit root, it has been a well-established
empirical fact that standard unit root testing methods such as Dickey-Fuller tests, ADF
tests and Phillips-Perron tests do not clearly determine whether the observed time series
data contains a unit root or not. Dejong et al.(1989), Diebold and Rudebusch (1990),
Dejong and Whiteman (1991) and Phillips (1991) provide empirical evidence supporting
this argument.

These studies suggest that, in trying to decide whether macroeconomic data are
stationary or integrated, it would be useful to perform tests of the null hypothesis of
stationarity as well as tests of the null hypothesis of a unit root. Tanaka (1990),

Kwiatkowski, Phillips, Schmidt and Shin (1992), hereafter KPSS, Saikkonen and

 

 

Luukkon
score—ha:
of a unit
of the m

tests and

2. The “

par amel
nonstat;
the tim
and a s

66raI-1d0:

Value
I

Luukkonen (1993), and Leyboume and McCabe (1994), hereafier LM94, have proposed
score-based tests of the null hypothesis of stationarity against the alternative hypothesis
of a unit root. Leyboume and McCabe (1999), hereafter LM99, have also proposed a test
of the null hypothesis of stationarity. This thesis will propose some extensions of these

tests and will analyze their properties, mainly through a large number of simulations.

2. The “local level model” and score tests

The KPSS and Leyboume-McCabe stationarity tests were derived from a
parameterization which provides a plausible representation of both stationary and
nonstationary variables. The “local level model” is 3 components representation in which
the time series under study is written as the sum of a deterministic trend, a random walk,
and a stationary error. See, e.g., Harvey (1989, pp. 31-32), who also refers to this as the
“random walk plus noise” model. If y, is the observed series, we write it as follows:

y,=flt+,u,+u,. (1)

Here y, is a random walk: ,U, = ,U,_l + V, , where the v, are iid (0, 03,2) and the initial

value [10 is treated as fixed, and serves as an intercept. Also u, is iid (0, of); later, the iid

assumption will be relaxed. The term ,6! allows for deterministic linear trend.
Define It = 0'3 / 0'3 . Then the null hypothesis of stationarity corresponds to i=0

(hence of = 0 , so no random walk component exists). The unit root alternatives are

indexed by 2&0. Thus i=0 corresponds to stationarity around a constant level (if (i=0) or

 

 

 

 

 

around at

of a pure '

l) (I,2 is '

L
the rand
U1 test
(LBI't u
Le}bou

1}, be tl

(IE fine

LBI St;

The ]
deriv;

by T3

“’her

around a trend (if 0:0). Cases with 7t>0 have a unit root. As k—wo, we approach the case

of a pure random walk.

1) 6..2 is known

Under the further assumptions that the stationary error u, is normal white noise,
the random walk innovation v, is normal, and the variance of is known, the one-sided
LM test statistic for the stationarity hypothesis is the same as the locally best invariant
(LBI) test statistic. Nyblom (1986), Nabeya and Tanaka (1988), KPSS (1992), and
Leyboume and McCabe (1994) all consider a model equivalent to the model above. Let

:2, be the residual from an OLS regression of y, on the intercept and time trend. Then we

I
define the partial sum process of the residuals: S, =2 13,- , t= 1,2,. . .,T. Then the LM and

i =1
LBI statistic is

T
LM = ZS,2 /af. (2)

(=1
The LBI derivation is given by Nyblom and Nabeya and Tanaka, while the LM
derivation is given by KPSS. We will follow the notation of KPSS, with a normalization
by T'Z:

r

77, = T‘ZZS,2 /03, (3)

t=l

where the subscript “r” indicates that we have allowed for linear deterministic trend.

In the case that we wish to test the hypothesis of level stationarity (i.e., we impose

 

 

(i=0) inst:
intercept

N

Where tl

level st.

the Bro

Under

C:
H
‘a

Where

CI‘iti

166‘]

0:0) instead of trend stationarity, we define 12, as the residual from regression of y on an

intercept only (ti, = y, —- )7) instead of the above, and the rest of the test is unaltered.

Now we write
T 2

,7” = T‘ZZS, m}, (4)
(=1

where the subscript “it” indicates that we have extracted a mean but not a trend from y.
The asymptotics for the two tests are similar. First we will discuss the test for
level stationarity (77”). Let W(r) be a Wiener process (Brownian motion), and let V(r) be

the Brownian bridge:

V(r)=W(r)-rW(I), 0 S r S 1. (5)
Under the null hypothesis, y, = #0 + u, where ya is fixed and u, is iid. Then
:2, = y, — 7 = u, - r7 , and cumulations of the 1?, converge to a Brownian bridge:

T "WSW => 0..V(r), 0 s r 51, (6)

where [rT] denotes the integer part of rT and “:>” denotes weak convergence. Then it

follows that
l
77” :> IV(r)2dr (7)
0

Critical values based on this distribution have been widely tabulated; e. g., KPSS (1992, p.
166)

Now consider the alternative that DO. Let _Pl_’(r) be the demeaned Wiener process:

1(r)=W(r)—thb)db. (s)

Then KPSS

and cone.

 

 

andd

KPSS

note
LM

disr

no

 

Then KPSS (1992, p.168) show that (for 0.50)
T—3/ZS[rT] 2) av Il(S)dS (9)
0
and correspondingly
T 2 l r
7-2,” = r425, m: => (of /af)j(l_n_/(s)ds)2)dr., (10)
1:] 0 0

The analysis for the test of trend stationarity is very similar. Under the null, we

simply replace the Brownian bridge V(r) by the “second-level Brownian bridge”
1
V2(r) = W(r)+(2r—3r2 )W(1)+(6r2 -6r) [W(s)ds, as given by KPSS (1992, equation (16)).
0
Under the alternative, we replace the demeaned Wiener process E0) by the “demeaned
l l
and detrended Wiener process” W (r)=W(r)+(6r—4) IW(S)dS+ IS W(s)ds, as given by
0 0

KPSS (1992, equation (26)).

The essential point of this discussion is that 77,, (or 77,) is Op(1) under the null, but
Op(T2) under the unit root alternative. Thus these tests are consistent. It should also be
noted that the normality assumption for u, and V, was made to allow the derivation of the
LM or LBI test. However, the consistency of the tests and the validity of the asymptotic
distribution results given above do not depend on these normality assumptions. The tests
may have certain optimal properties under normality, but they are valid without the

normality assumption.

 

 

1I
chuu
\
the rand;
3.
0,1513

level-st

Th5-
static

CORE

mm.

2) 0,2 is unknown
Now we continue to assume that the stationary error u, is normal white noise, and

the random walk innovation v, is normal, but we relax the assumption that the variance
02,2 is known. Let a“: be an estimate of 0': that is consistent under the null. Then in the

level-stationary case we define the statistic
T 2
i7", =T‘ZZS, /&f. (11)
(=1
This differs from 77,, in (4) only because (if replaces 0'}. Similarly, in the trend-
stationary case, we define fir by replacing of in (3) by (if , an estimate of a: that is
consistent under the null. Replacing a: by a consistent estimate 6'3 does not alter the

distribution theory under the null.

For the case we are currently considering (iid u,, of unknown), both KPSS and

LM94 would suggest the following estimate of a: :
T
6-3 =T"Zu‘}. (12)
r=l
This is indeed a consistent estimate of 0': under the null. However, under the unit root

alternative, 63 is Op(T). Specifically, for the level-stationary case we have:
7' I
We: =T'ZZfiE = of lizards, (13)
1:1 0

where 22(3) is demeaned Wiener process of equation (8). As a result 17” is Op(T) under the

alternative (instead of Op(T2), as 27,, was, with known 0': ). Specifically,

 

 

T.
the deme

place of
change
alternat

place 0

3. L31

Static
null

“Sin

He

T l r 2 1
T4,)” = 21-423} /T"aj => Jl flaws] dr/ [[0de (14)
i=1 0 o 0

The analysis for the fit test is essentially the same. We just replace E0) by W‘(r),
the demeaned and detrended Wiener process. The essential point is still that using 6': in
place of of does not alter the asymptotic distribution theory under the null, but it does

change the distribution theory under the alternative. The test is Op(T2) under the

alternative with of known, but only Op(T) under the alternative when 6': is used in

place of 03.

3. LM99 test and its modifications

Leyboume and McCabe (1999) proposed a new version of the KPSS/LM94

stationarity test. The idea is to find an estimate a": that is consistent for a: under the
null of stationarity, and that is Op(1) under the unit root alternative. Then fill (or fi,)

using this estimate will be 0,,(T2), not 0,,(T), under the unit root alternative.
It is well known that the model (1) is second-order equivalent in moments to the
ARIMA(O,1,1) process:

(l—L)y,=,6+(1—6L)§,, 0<6<1. (15)
Here 4, is white noise with mean zero and variance of. The correspondence between the

parameterizations (15) and (1) is as follows:

ag=aj /6 (16A)

g=(,1+2-(,t2 +42)“2)/2 (16B)
where as before A = of / of. Here the null hypothesis of stationarity is of = 0 (or 2:0)

in (1), and corresponds to 0=1 in (15). It implies that y, is stationary. The alternative

of > 0 corresponds to OSG<1 and implies that ' y, has a unit root. It is important for later

development to stress that model (1) implies 03051 in (15); negative 0 are not consistent

with (1). Also the pure random walk corresponds to k=oo in (l ), or 0=0 in (15).

LM99 use the relationship (16A) to obtain their estimates of a": . Let

~ -_-,a§é (17)
where 6'; and 19 are the quasi-ML estimates of the ARIMA(O,1,1) process (15). “Quasi-

ML” refers to the fact that the form of the likelihood assumes normality, but the

consistency of the estimates does not depend on this assumption being correct. LM94
note that a"; and 9 are consistent under the null and Op(1) under the unit root alternative.
Therefore, 5: is also consistent under the null and Op(l) under the alternative. Ifwe use
6‘: as the denominator of the test statistic instead of 6': as in (11), we have the LM99

stationarity test 77),:

T‘ZZS . (18)

Obviously, f)", is 0,,(1) under the null hypothesis, and 0,,(T2) under the alternative. This
suggests that it may be more powerful than the KPSS/LM94 test 77” , which is only 0,,(T)

under the alternative.

We will now proceed to suggest two modifications of the LM99 test. These are

based on the following observation. The LM99 test, like KPSS and LM94, is an upper tail

test. However, the LM99 estimate 5: can be negative. Even though 0<0 is not consistent

with the local level model (1), é < 0 is possible, and 5": = a": 6’ is negative if B is
negative. In this case we will have 5” < 0 and the test will not reject. This will be a very
rare occurrence under the null (0=1), but it may not be rare under the alternative. Note

especially that in the pure random walk case (0=O) we will have é < 0 with a probability
that approaches 0.5 as T—-)oo, and the power of the LM99 test against pure random walk

alternatives will be close to 0.5, not 1.0, for large T. Our simulations will confirm this,

and will show that correspondingly the LM99 test will have poor power against unit root
alternatives that are close to random walks (i.e., for large values of 7c, correspondingly,

small values of 0). LM99 specifically assume 0>0, thus avoiding this problem in terms of
asymptotics, but still it is odd and not desirable to have a test whose power is low against
a random walk. This ought to be the easiest alternative to detect.

To avoid this problem, we propose two modifications of the LM99 test statistic.

The first, which we will call LMMl, uses the variance estimator E: = 6'2. This is a
consistent estimator of a: under the stationary null, since 0=1 under the null. Under the

alternative, it is not a consistent estimator of a": , but it is Op(1). Therefore, LMMl is
0,,(1) under the null and 0,,(T2) under the alternative. This modification of the LM99 test

may cost some power, because (3: > 06': when O<é <1, and we expect 0<é <1 when 9 is

not close to zero. However, we may gain power when 0 is close to zero since if can not

be negative.

We also propose another modification of the LM99 test statistic, which we will

call LMM2. This is based on the estimator 6': = '19

 

A 2 o . e
0'; , wh1ch 15 also consrstent under

the null and Op(1) under the alternative. For 0 close to one, we expect 9 > 0 with high
probability, and so a: should equal 5: with high probability. Thus we do not expect

substantial size distortions, and the power of LM99 and LMM2 should be similar when 0

is close to one (i.e., when power is low). However, for 0 close to zero (large A), we may
expect LMM2 to be more powerful than LM99, and LMMZ (unlike LM99) is consistent

against the pure random walk alternative.
4. Short-run dynamics

The time series data to which a stationary test is applied are typically highly
dependent over time, and so the iid error assumption under the null is unrealistic.
Empirically, it is important to allow the stationary errors u, to be correlated. The essential
assumption for the u, is that they satisfy a functional CLT, so that their cummulations

follow a Wiener process. That is, we assert

rT
T‘“2 u, => a'W(r) (19)

l=l
T 2
for O < a < oo . Here 02 = lim T “'E[Z u,) is the “long run variance”, and the assertion
ado

l=l

that it is finite is an assertion of “short memory” of the process 14,. Assumptions on u, that

10

guarantee (19) include the regularity conditions of Phillips-Perron (1988), which involve
mixing plus existence of certain moments, or the Phillips—Solo (1989) linear process

assumptions.

1) KPSS test

If (19) holds, then the numerator of the KPSS statistic follows:
T l

T‘ZZS} :> 0'2 lV(r)2dr (20)
r=l 0

Therefore KPSS use an estimate of 0'2 for the denominator of the statistic, to cancel the
0'2 in the numerator. A consistent estimator of the long—run variance 0'2 is constructed

from the residuals 12,:

T T
32(1) = T"Zii,2 + 2T"w(s,l)212,12,_5 , (21)
(=1

t=s+l
where w(s,1) is an optional weighting function that corresponds to the choice of a spectral
window. KPSS use the “Bartlett window”, which is 1- s/(I+1) as in Newey and West
(1987) to guarantee the nonnegativity of 32(1). For the consistency of 32(1), it is necessary
that the lag truncation number I—>oo but I / T —) 0 as T—->oo. The rate l=o(T”2) will usually
satisfactory under both the null and the alternative.

Let 7')” (I) be the KPSS statistic that uses 1 lags in estimation of the long run

variance. (Or, in the case of testing for trend stationarity, 71(1) is defined similarly).

Under the null it has the same asymptotic distribution as in the cases previously

considered:

11

T l
,7#(1)=T‘ZZ 5,2/s2(1)—> [V(rydr (22)
(=1 0

Under the alternative, the numerator is 0,,(T2) as before. However, KPSS (1992, p. 168)

show that 52(1) is 0,,(IT) under the unit root alternative. Therefore, under the unit root

alternative, I?” (l) is only Op(T/I). Recall that this compares to Op(T2) when the u, are

white noise and of is known, and to Op(T) when the u, are white noise but of is not

known. So we expect the allowance for autocorrelation of the u, to cause a loss of power.
A possibility that is not noted in the existing literature is that we can make the

KPSS test Op(T) under the alternative, under the assumption that the u, are MA(1), where l

is known, or where we have an upper bound for I that is “fixed” (does not depend on T).

Then the maximum non-zero autocorrelation is I, and we can estimate 02 consistently

using the ‘unweighted’ variance estimator
T
32(1) = T“ 2:23 + ZZT‘l 12,12”, (23)

where I is a fixed number. The unweighted variance estimator 32(1) is consistent under the
null hypothesis (32(0—902), and Op(T) under the alternative, with I fixed.

Note, however, that under the MA(1) assumption, with I fixed, the asymptotic
distribution of the KPSS statistic under the alternative does depend on I. The constant

K '=(1+21) would appear in the denominator of the expression for the distribution of

T "I?” (I). In that sense the KPSS statistic is still Op(T/l) ; but with I fixed, this does not

contradict the fact that it is 0,,(T).

We can summarize this discussion simply, in a way that relates to the remainder

of the thesis. We may let l—)oo as T—>oo, as in the original KPSS article. In this case the

12

 

 

 

Final];
select

tatist

aSSLm

Wher
aulor
Whit:
Whic

mod:

test is valid for very general forms of autocorrelation of the u,, and the statistic is Op(T/I)
under the alternative. This version of the test will be analyzed further in Chapter 3.
Alternatively, we may assume that u, is MA(1) with I known, in which case the statistic is
Op(T) under the alternative. This version of the test will be analyzed in Chapter 2.
Finally, we may assume that u, is MA(1) for finite but unknown 1, and use some model
selection procedure to choose I. If the model selection procedure is consistent, the
statistic is again 0,,(T) under the alternative. This version of the test will be considered in

Chapter 4.

2) LM94 test
Leyboume and McCabe (1994) based their test (which we call LM94) on the

assumption that u, is an autoregressive process with known order p. Their model is
<I>(L)y, = ,Bt + ,u, + a, , (24)

where ,u, is a random walk as in (1), <I>(L) = 1- ¢1L — ...... — (15pr is a pth order

autoregressive polynomial in the lag operator L with roots outside the unit circle, and s, is

white noise. Thus the stationary error in the solution for y, would be u, = <D"(L)£, ,

which is AR(p). Note that the equivalent of (15) above would be the ARIMA(p,1,1)

model:
¢(L)(1- L)y, = ,6 + (1 - 9104,- (25)
The LM94 test statistic is calculated as follows. First, we get ML estimates of ¢,~

(and Q from the ARIMA(p,1,1) model

13

Ayr = fl +i¢iAyr—i +41 —6;1—l ° (26)
i=1

Note that this model is estimated in first differences of the y:, so as to obtain consistent
estimates under both the null and alternative, and so to avoid low power problems1 under

the alternative hypothesis. Next, we construct the filtered series
yr. =y1 —:¢:y1—19 (27)
i=1

where ¢,.° are the ML estimates of ¢,-_ Then we calculate the residuals from the regression

of y: on an intercept (and time trend in the trend-stationary case). Finally, we construct
the LM94 statistic in the same way as the ii” (or fir) test was constructed from the
residuals 12,. That is, having filtered the data, we are back in the setting of white noise
error with unknown variance.

It follows that the LM94 test statistic, when the order (p) of the AR polynomial is
known, is 0,,(1) under the null and 0,,(T) under the alternative. Also the distribution of
the test statistic does not depend on the value of p. In Chapter 2, we will analyze this case
further. As with the KPSS test, there are other possibilities. In Chapter 3, we consider. the
case p—>oo as T—>oo. The asymptotic properties of LM94 with this method of choosing p
are unknown. Finally, we may use a model selection procedure to choose the relevant AR

order p. This will be discussed in Chapter 4.

 

' For simplicity, assume p=l, and a=B=O. In that case, if we regress y, on its lagged term y,.. to get residuals
that could be a basis for the test, then, under the unit root alternative, the parameter ,3. _) land the residuals

are approximately stationary, since y, and y,., are 1(1). If the stationarity test is based on these stationary
residuals, then the statistic will behave as if the null of stationarity is true. This will cause the test to lose
power under the unit root alternative. To avoid this problem, we estimate the parameters by ML estimation
on differenced y,. In this case, the parameter estimtes are consistent under both the null and the alternative,
and the residual is nonstationary under the alternative. See Leyboume and McCabe (1994) for details.

14

3) LM99 and LMMl, LMM2

The LM99 test (and its modifications LMMl, LMM2) handles short-run
dynamics in the same way as the LM94 test. Treating p as known, we estimate the
ARIMA(p,1,1) model (26) and filter the data as in (27). Having done so, we calculate the
LM99 statistic in the same way as was done with white noise errors. When p is known,
the LM99, LMMl and LMM2 test statistics are 0,,(T2) under the alternative and their
distribution does not depend on the value of p. We consider this case in Chapter 2. When
p is unknown, we can let p—)oo as T—>oo. The asymptotic properties of the test in this case
are unknown. We analyze this case further in Chapter 3. Finally, when p is finite but
unlmown, we can also use a model selection procedure to choose it. Leyboume and
McCabe (1999) suggest a consistent model selection rule for LM99 and they show that
the model selection rule does not affect the distribution of the test statistic. We consider

model selection rules further in Chapter 4.

4) Overfitting and near cancellation
The LM94 and LM99 test are based on estimation of the ARIMA(p,1,1) model

given in the equation (25) above. Here we rewrite this as:
<D(L)Ay. = [3 + ®(L)§, (28)
where @(L) =1— 6L . The asymptotic theory for the quasi-MLE assumes that there are no
cancellations in the lag polynomials (l>(L) and @(L). That is, if we factor <D(L) as
¢(L)=(1-¢IL)(1-¢2L)- - °(1'¢pL)2 (29)
it is assumed that 6|¢¢j for any j. There is generally no reason to expect such a

cancellation. However, it is worth noting that the tests may suffer from serious power loss

15

 

 

sir

UV

“2

ll€

(‘1

W:
M

f0:

seI

Sta

if there is a near cancellation. We will see examples of this phenomenon in our
simulations.

An important and practically relevant case in which this occurs is when we
overspecify p and the DGP has 3. = 0'3 / 0': large. For example, in the pure random

walk case we have k=oo, or 0=0; and in the near random walk case we have A large and 0
near zero. If we overspecify p, then one or more of the (1})- equals to zero, and we have a
cancellation or near cancellation of (1-¢,~L) and (1-0L). In this case, the test will be seen to
lose considerable power in finite samples. Thus we will see that all of the LM tests will

tend to perform poorly against alternatives with 1» large, unless the AR order p is known.

5. Organization of the thesis

In this chapter we have discussed the testing problem of interest and defined the
statistics that we will consider. Each of these depends on the choice of a parameter that
we will call the “number of lags” which is I for the KPSS test and p for the Leyboume-
McCabe tests. The remaining chapters analyze the size and power properties of the tests
for different methods of choosing the number of lags.

In Chapter 2, we will suppose that the number of lags is known and fixed to a pre-
selected value. We will investigate the size, power and size-adjusted power of the
stationarity tests using finite sample simulations. There will be three kinds of data
generating processes (DGP) used for our simulations. First, we will consider iid errors. In
this case, the true I or p is zero but the simulations will be performed assuming I and p

from zero to three. Secondly, we will consider MA(1) and AR(1) errors, and

16

correspondingly we will use the KPSS test with [=1 and the Leyboume-McCabe tests
with p=1. In this case, we might expect the KPSS test to work well for MA(1) errors and
the Leyboume-McCabe tests to work well for AR(1) errors. However, our interest in this
simulation is also on the opposite cases where the KPSS test is performed with AR(1)
errors and Leyboume-McCabe with MA(1) errors. This is to see the performance of each
stationarity test under incorrect specification. Similarly we will use ARMA(1,1) errors in
our simulations. In this case all of the tests are based on a misspecified model.

In Chapter 3, we will allow the number of lags in the KPSS and Leyboume-
McCabe tests to increase with the sample size. We will look at the size, power and size-
adjusted power of the various stationarity tests using same kinds of DGPs described in
Chapter 2.

In Chapter 4, we will consider model selection rules for the number of lags in the
KPSS and Leyboume-McCabe tests. Leyboume and McCabe (1999) proposed a model
selection rule to choose p in their tests, and we propose a rule to choose I in the KPSS
test. In both cases we also analyze the case that the critical levels of the pretest depend on
the sample size. This is done to avoid asymptotic overfitting (choosing I or p larger than
the true value). We investigate the size, power and size-adjusted power of each tests
using the same three DGPs as in the previous chapters.

Finally Chapter 5 gives our conclusions.

l7

Chapter 2

Performance of the KPSS and Leyboume-McCabe Stationarity Tests

with a Fixed Number of Lags

1. Introduction

In this chapter we consider the performance of the KPSS test and various versions
of the Leyboume-McCabe test (LM94, LM99, LMMl and LMMZ) when the number of
lags is fixed to some value chosen a priori. Here, as defined in Chapter 1, the “number of
lags” is the parameter “I”, the number of lagged terms in the long-run variance estimate,
for KPSS; and it is the parameter “p”, the assumed order of the AR polynomial, for the
various Leyboume-McCabe tests. In this chapter, the number of lags does not grow with
the sample size (as it does in Chapter 3) and is not determined by the information
contained in the observed data series (as it is in Chapter 4).

The use of a fixed number (I) of lags for KPSS reflects an assumption that. the
stationary error is MA(1), and therefore, in this chapter we use the “unweighted” long-run
variance estimator, as opposed to the “weighted” estimate of KPSS which used the

Bartlett window w(s,l) =1—s/(l +1) as in Newey-West (1987). Recall from Chapter 1

that with I fixed the KPSS statistic is 0,,(1) under the null hypothesis of stationarity and is
0,,(T) under the unit root alternative. However, the value of I does appear in the
asymptotic distribution under the alternative, so picking a needlessly large value of I

should be expected to cause a loss of power, even asymptotically.

18

All of the Leyboume-McCabe test variants are Op( 1) under the null hypothesis.
Under the unit root alternative, LM94 is Op(T) and LM99, LMMl and LMM2 are Op(T’).
See Chapter 1 for details. Furthermore, for all of the Leyboume-McCabe tests, the
limiting distribution is not affected by the value of p. Thus choosing a needlessly large
value of p may affect the power of the test in finite sample but it will not do so

asymptotically.

2. Simulations

In this section we provide some Monte Carlo evidence on the size and power of
the KPSS and Leyboume-McCabe tests in finite samples. The simulations were

performed using GAUSS 3.2.25 and the Maxlik optimization procedure. The DGP is
equation (1) of Chapter 1, with (i=0. Thus y, = ,U, + u, 9 l“, = #1,-) + V,, where the u,
are iid N(O, 0': ), the v, are iid N(0, a: ), and u and v are independent. The data contain no

deterministic trend and we consider only the tests that allow for level but not trend (e. g.,

KPSS 7')” but not 7?, , and similarly for the Leyboume-McCabe tests). The number of

replications is given below, but is generally 20,000 for KPSS and 10,000 for Leyboume-
McCabe tests.

We will first consider the case of white noise errors, with the number of lags for
the tests ranging from zero to three. White noise errors are probably not empirically
relevant. However, they do provide a fair comparison between the KPSS and Leyboume-
McCabe tests, in the following sense. The KPSS test with I fixed is based on an MA(1)

assumption, while the Letboume-McCabe tests with p fixed are based on an AR(p)

l9

assumption. The advantage of white noise is that it is both MA(1) with finite I and AR(p)
with finite p. Furthermore, with white noise errors we can easily investigate the power
loss from overspecifying I in the KPSS test or p in the Leyboume-McCabe tests.

We will also consider MA, AR, and ARMA errors. The primary point here will be
to see how the various tests perform when they are based on an incorrectly specified

model.

1) The KPSS test with iid errors

We first consider the size of the KPSS test in the presence of iid errors. The null
hypothesis is 03 = 0 (2:0) and then y, = u, , so y, is white noise. The test is set at the
5% nominal significance level, and the results are based on 20,000 replications.

Table 2.1 gives the size of the test with various sample sizes (T) and numbers of

lags (I) used to calculated 32(1). We consider T from 50 to 500 and I from zero to three.
The actual size of the KPSS test (0,.) as reported in Table 2.1 is 0.05 for all T when we

set the lag truncation number 1 equal to zero. Since the DGP is white noise, that is what
we expected. With the number of lags greater than zero, we found considerable size
distortions (overrejection) for relatively small sample size (T=50), and the size distortions
are worse as we increase 1. However, the size distortions disappear with larger sample
sizes such as T=200 even with I=3. Thus, for moderate sample sizes like T=200,
overspecifying I does not seem to cause much size distortion.

Now we turn to the power of the test. Here the main question is the extent to

which power is reduced when I is overspecified. Now 03 > 0 and our results depend on

20

A = of / of as well as T and I. The unit root component grows as 2 grows, and we expect

power to be higher when 2. or T is larger and when I is smaller.

Table 2.2 gives the poWer of the KPSS test as a function of T, 2., and I, for the
same values of T and l as in Table 2.1, and for 2. ranging fiom 0.001 to 10,000. As
expected, the power of the test increases with T and 2. for all values of I. Conversely,
given T and 2., power of the test decreases rapidly as we increase the number of lags (I).
(There are a few exceptions to this statement, for small T and 2., but these are cases of
significant size distortions, and these exceptions will disappear when we consider size-

adjusted power, in Table 2.4.)

The loss in power from choosing a needlessly large value of I can be substantial.
For example, for T=50 and 2t=0.1, compare power of 0.721 with I=0 (the “true value”
since the errors are white noise) to 0.432 with I=3. This loss of power seems to be less for
larger T, even though theoretically it does not disappear asymptotically. For example,
with T=500 and 2.=0.001, we have power of 0.788 with [=0 and 0.751 with I=3, a much
smaller power loss (at a comparable level of power with I=O) than with T=50 and 2.=0. 1.

We now proceed to consider size-adjusted power. As usual, the motivation is to
quantify the intrinsic ability of the test statistics to distinguish the null hypothesis and the
alternative. Table 2.3 gives the “actual critical values”, by which we mean the critical
values that would yield correct size (5%) in our simulations under the null hypothesis.

Table 2.4 provides the size-adjusted power of the KPSS test based on the actual
critical values in Table 2.3. The results are quite similar to those in Table 2.3. Power

increases with T and 2., and decreases with I. The main change is that we have now

21

removed the increase in power as 1 increases, for small values of T and 2.. We conclude

that this anomaly was due to size distortion, the effects of which have now been removed.

2) The Leyboume-McCabe tests (LM94, LM99, LMMl and LMM2) with iid errors

In this section we provide simulation results on the size and power of the various
Leyboume-McCabe tests in presence of iid errors. Our experimental design is very
similar to the design for the KPSS simulations just reported. We consider T=100, 200 and
500, values of 2. ranging from zero (the null) to 100, and number of lags (p) from zero to
three. To make the calculations simpler and faster, we used only 10,000 replications, and
we dropped T=50, because we encountered an annoyingly large number of failures of the
MLE algorithm when T=50 and p=2 or 3.

We first consider the size of the tests. This corresponds to the entries in Table 2.5
with 2t=0. Note that p=0 is the true value of p, since the errors are white noise, and that
for p=0, LM94 is the same as KPSS with I=0. For p=0, there are no substantial size
distortions though the LMMl test rejects too seldom. For larger values of p, the tests
reject too often under the null, and unsurprisingly these size distortions are larger when T
is smaller and p is larger. Most notably, the size distortions (overrejection) are worse for
all of the LM tests than they were for the KPSS test (with I for the KPSS test equal to p
for the Leyboume—McCabe tests), and they are worse for the LM99 and LMM2 tests than
for the LM94 and LMMl tests. For example, for T=100, compare size of 0.056 for KPSS
(with I=3), to 0.077 and 0.074 for LM94 and LMMl, respectively (with p=3), and to

0.100 for both LM99 and LMM2 (with p=3). Any power gains of the LM99 test or its

22

modifications would have to be weighed against its size problem when T is moderate and
p is overspecified.

We note in passing that LM99 and LMM2 are essentially identical in our
simulations, under the null hypothesis. The probability of obtaining a negative estimate of
0, when the true 0 equals one, is negligible.

Except for the LM99 test, the power of the tests increases with T and generally

with 2.. For the LM99 test, we have the disturbing feature that power increases with 2. for

small 2., but then decreases with further increases in 2. This is a reflection of the fact that,

for the pure random walk case of 2e=co, the probability of 19 < O approaches 0.5 as T—+oo.
Correspondingly the power of LM99 is close to 0.5, not 1.0, for large T and 21., and this is

what our simulations show. In our view, this is a serious defect of the LM99 test, but it is

easily solved by using the LMMZ test instead (that is, by taking the absolute value of (9 ).

For LM94, LMMl and LMMZ, power essentially always increases as we increase
T for a given 2. (and p), as we would expect, given the consistency of the tests. But, for
large 2., power does not necessarily increase with 2. for a given T (and p). This is a
reflection on the “near cancellation” problem discussed in Chapter 1. It does not occur
with the KPSS test.

Because we had some substantial size distortions, and these varied across tests
and the value of p, we will avoid a detailed comparison of power of the tests and the way
that it depends in p, and turn to a discussion of size-adjusted power.

Table 2.6 presents the “actual critical values”, as Table 2.3 did for KPSS. Then
Table 2.7 gives size-adjusted power (power using the actual critical values from Table

2.6) for the various LM tests.

23

We note first that the LM99 test has poor power (approximately 0.5) when T and
2. are large. This is the same phenomenon that was commented on above, and it is
probably the most striking result in either Table 2.5 or Table 2.7.

The other obvious and striking result in Table 2.7 is that, for given values of T, 2
and p, all of the various Leyboume-McCabe tests have very similar size adjusted power.
(The exception, as note, is the LM99 test, for large T and 2.) There is simply not much
difference between these tests. It is generally true, perhaps, that LMM2 has greater size-
adjusted power than LM94, but the differences are small. This is perhaps surprising, in
light of the fact that the LM94 statistic is only Op(T), while the others are Op(T2).

Size-adjusted power usually falls as p increases, for a given T and 2. This is as
expected since we are estimating needlessly many parameters. However, this decrease in
size-adjusted power is not terribly large. For example, for T=100, 2=0.01, for LM94 we
have size-adjusted power of 0.606, 0.579, 0.549 and 0.530 for p=0, 1, 2, 3, respectively.
For LMMZ, the size-adjusted powers follow a similar pattern: 0.621, 0.590, 0.556, 0.524.
It is revealing to compare these to the size-adjusted power of KPSS, from Table 4, where
for T=100, 2.=0.01, we find 0.590, 0.535, 0.504, 0.451, respectively, for [=0, 1, 2, 3.
Obviously overspecifying p in the Leyboume-McCabe tests does not cause as much ”of a
power loss as overspecifying I in the KPSS test. This is as suspected from the asymptotic
theory.

More generally, the KPSS test with [=0 is identical to LM94 with p=0, and it has
size-adjusted power that is very similar to that of the other Leyboume-McCabe tests with
p=0. However, KPSS with a positive number of lags “I” is generally less powerful than

the LM tests with p=[.

24

Empirically, one is unlikely to know the “correct” number of lags. Then the main
advantage of the KPSS test over the Leyboume-McCabe tests is that overspecifying the
number of lags causes less size-distortion for KPSS than for Leyboume-McCabe.
Conversely, the main advantage of the Leyboume-McCabe tests over the KPSS test is

that they are more powerful when the number of lags is overspecified.

3) The KPSS and Leyboume-McCabe tests with AR(1) errors

Here we perform simulations with AR(1) errors of the form: u,=pu,-1+e, , where s,
is normal white noise. We set the coefficient value p to be 1/3 to have the “long-run
variance” of the AR(1) series be equal to that of the MA(1) error series (which we will
consider in the following section) with coefficient 0=0.5. For details see Appendix I. We
consider the KPSS test with [=1 and the Leyboume-McCabe test with p=l. The
Leyboume-McCabe tests are based on a correctly specified model, while the KPSS test is
not (since AR(1) corresponds to the MA(oo)) and we want to see how much difference
this makes.

Table 2.8 gives the size and power of the various tests, for values of T and 2
similar to those considered previously. Table 2.9 gives the “actual critical values”, while
Table 2.10 gives size-adjusted power.

The KPSS test shows moderate size distortions (e.g., size=0.78 for T=500). This
should be expected since its long run variance calculation does not take into account the
correlations of order greater than one. (Presumably the size distortion would be larger for
larger values of p in the AR(1) DGP.) Its power compares favorably to the power of the

Leyboume-McCabe tests, but this is only due to the size distortion. From Table 2.10, the

25

size-adjusted power of KPSS is lower than that of the Leyourne-McCabe tests. This is
also as expected. We can also note that the LM99 test has low power compared to all of
the other tests when 2:1 as well as when 2=100. This is a reflection of a “near
cancellation” between the AR root of 1/3 and the MA root of 0.389, which causes the
estimates of p and 0 to be imprecise. Apparently this imprecision is sufficient to cause a
substantial number of negative estimates of 0. LMM2, which takes the absolute value,
does not suffer from this problem.

When we compare the various Leyboume-McCabe tests, we first notice that none
of them show any substantial size distortions. Power and size-adjusted power are
therefore more or less equivalent to compare. As in the previous section, the various
Leyboume-McCabe tests are all more or less equally powerful (except that, as before,
LM99 is very poor when 2 is large). LMM2 is a little more powerful than LM94, but the
difference is small.

The size and power characteristic of the Leyboume-McCabe tests with p=l are
very similar whether the DGP is AR(1) with p=1/3 (Table 2.8-2.10) or white noise (Table
2.5-2.7). Of course, white noise is AR(1) with p=0, so this is evidence supporting the
conjecture that, if p is correctly specified, the precise values of the AR parameters are not

too important.

4) The KPSS and Leyboume-McCabe Tests with MA(1) errors

Now we perform simulations with MA(1) errors of the form: y,=s,+08,-., where e,
is normal white noise. We pick 9=0.5 to make the long run variance equal to that of the

AR(1) process of the previous section. As in the previous section, we consider the KPSS

26

 

3-

 

 

 

 

test with [=1 and the LM tests with p=l. Now, however, the KPSS test is based on a
correctly specified model while the Leyboume-McCabe tests are not.

Our results are given in Tables 2.11-2.13. These have the same format as Tables
2.8-2.10 of the previous section.

Now the KPSS test has the correct size, whereas the Leyboume-McCabe tests
suffer from size distortions. The Leyboume-McCabe tests underreject under the null
hypothesis. This causes their power to be low. In terms of size-adjusted power, the
Leyboume-McCabe tests are roughly similar to each other (again, except for LM99 when
2 is large), and they generally, but not always, have slightly lower size-adjusted power
than the KPSS test.

The size-adjusted power of the KPSS test with [=1 is lower when the errors are
MA(1) with 0=0.5 than when the errors are white noise (i.e., MA(1) with 0=0). This is

most noticeable when power is relatively low.

5) The KPSS and Leyboume-McCabe tests with ARMA(1,1) errors

Here we perform simulations using ARMA(1,1) errors of the form: y,=py,-,+
s,+08,-,, where p=l/3 and 0:1/2. We choose these specific values of the AR and MA
parameters to equate the contributions of the AR and MA terms to the “long-run
variance” of the ARMA(1,1) error series. (See Appendix I for details.)

In doing so we are trying to ensure a fair comparison of the KPSS test with [=1
and the Leyboume-McCabe tests with p=1, none of which are based on a correctly
specified model. Our results are given in Tables 2.14-2.16, which have the same format

as the previous tables for the AR and MA cases.

27

All of the tests show considerable size distortions even for large sample sizes such
as T=500. The KPSS test overrejects while the Leyboume-McCabe tests underreject the
null. The power of the KPSS test is apparently greater than that of the Leyboume-
McCabe tests for small values of 2, but generally less for large values of 2 (except that,
as before, LM99 has low power for large 2). Given the size distortions of the tests, and
especially since these are in different directions for different tests, size-adjusted power is
a fairer comparison. The size-adjusted power of the KPSS test and Leyboume-McCabe
tests is similar when 2<1, but the power of the Leyboume-McCabe tests is greater than
that of the KPSS test when 221. The exception is still the LM99 test which is not
powerful when 2 is large.

An interesting detail is the very low power of the LM99 test when 2=1; e.g., size-
adjusted power is 0.002 for 2=1, T=500. This is again a reflection of a “near
cancellation” problem. The AR root of 1/3 nearly cancels the MA root of 0.389 implied
by 2=1, leaving the MA root of —0.5 from the ARMA process. Therefore the estimate of
0 is nearly always negative, and LM99 has virtually no power. LMMZ, which takes the

absolute value of the estimate of 0, does not suffer from this problem.

3. Conclusions

In this chapter we considered the KPSS and Leyboume-McCabe tests that use a
fixed number of lags. We investigated the size and power characteristics of the tests via

simulations. In these simulations our data generating processes included white noise,

28

'—

 

 

 

 

 

ARll

discus

laller
one 1
stair:

(list

5011
Ch:

COT

(10‘.

ha

P6

W11
ml.

the:

AR(1) errors, MA(1) errors, and ARMA(1,1) errors. We gave our conclusions as we
discussed the simulations, but we will repeat some of them here.

1. The LM99 test is not recommended. It has poor power for large values of 2
(alternatives close to a pure random walk). In fact, for 2=oo, as T—)oo power approaches
one half, not one. The LMM2 test, which simply uses the absolute values of the LM99
statistic, solves this power problem, and it does so without causing any noticeable size
distortions, since the probability of a negative test statistic under the null is negligible.

2. The LM94 test and the modified versions of the LMMl and LMM2
sometimes show a loss in power due to the “near cancellation” problem, identified in

Chapter 1, that occurs when the value of 0 is close to one of the AR roots. This

commonly occurs for large values of 2 when p is overspecified, so that 0 is close to zero
and one of the AR roots equals zero. In these circumstances the LM tests generally are
dominated by the KPSS test, which does not suffer from this problem.

3. There is not much difference in power between the LM94 test, on the one
hand, and the LM99 test or its modifications (LMMl, LMM2), on the other hand. This is
perhaps surprising because the LM statistic is Op(T) under the alternative while the others
are Op(T2). ‘

4. The white noise case was argued to be a fair setting for comparison of the
KPSS and Leyboume-McCabe tests, since it satisfies both the MA(1) and AR(p)
assumptions. With white noise errors, the KPSS test with [=0 is the same as the LM94
with p=0, and it performs similarly to the other LM tests with p=0. Ifthe errors are white
noise, but we overspecify [ (for the KPSS test) or p (for the Leyboume-McCabe tests),

there is a trade-off between size and power considerations. Overspecifying I in the KPSS

29

 

 

 

 

 

[8515 C
IESIS.

165111

are 5
KPS

ltSlE

tests causes smaller size distortions than overspecifying p in the Leyboume-McCabe
tests, but it also results in a greater loss of power for the KPSS test than for the
Leyboume-McCabe test.

5. Our simulations with AR(1), MA(1) and ARMA(1,1) errors show that there
are size distortions and loss of size-adjusted power if one underspecifies I or p. So, the
KPSS test with fixed [ does not do well if the DGP is AR(p), and the Leyboume-McCabe

tests with fixed p do not do well if the DGP is MA(I).

30

Appendix I

We wish to perform simulations with AR(1) errors and also with MA(1) errors.

We want to pick values for the AR parameter ‘ p and the MA parameter “0” that yield

equal values for the long-run variance of the process.

Here we define the long-run variance (1'2 as:

02=7o+271+272+273+ ...... =70+227j, (A1)

[=1

where Yj is the jth autocovariance.

1. MA(1) case: y, = u, = a, + 98H where a, is white noise. Then
70 = (1+ 02 )0’3, where 0': is variance ofe,
7, = 60’: , and
7)- = O forj>l.

Therefore, in the case of an MA(1) process, the long-run variance equals

02 = 70 +2yl = (1+6?2 +2600: = (1+6)zaf. (A2)

2. AR(1) case : y, = ,oy,_I + a, , where a, is white noise. Then

2

,0 2

1 2 '0 ——2—0'5......
(l-p )

2
= 0' , = = a", = =
(1_p2) 1: 71 p70 (l_p2) 72 p71

 

 

70

31

 

7—=,0}’~-= 5 .......
’ " (l-pz)

Therefore, long-run variance can be calculated as

02 =70+2y, +272+2y3+ ..... .

 

:[1+2p+2p2+ ...... ]0’f
G-pD O-p0 0-p0

 

 

= (1 _1p2)[1+2p+2p2 + ...... ]a52
= 12[(1+p+p2+ ...... )+(p+p2+ ...... )]rr52
(1-p)

 

1 [ 1 p ]
= + 05
(l-pz) (l-p) (l-p)

= 1 [(1+p)]52

(1-p2)(1-p) ‘

= 1 20.2. (A3)
(l—p)

 

 

For our MA(1) process with parameter 0 to have the same long mm variance as
our AR(1) process with parameter p, we require

1
(1—p)2'

 

(1+ 6)2 = (A4)

For 0=1/2, this is satisfied for p=1/3.

32

v—

Table 2.1

Size of KPSS Test with Fixed Number of Lags
(DGP: iid errors, no time trend)

5% significance level

 

 

T [=0 [=1 =2 [=3
50 0.050 0.050 0.061 0.077
100 0.049 0.051 0.052 0.056
200 0.051 0.051 0.051 0.050
500 0.050 0.047 0.050 0.050

 

Simulation results based on 20,000 replications.

33

Table 2.2

Power of KPSS Test with Fixed Number of Lags
(DGP: iid errors, no time trend)

5% significance level

 

 

 

 

 

T 2 [=0 L=l [=2 [=3
50 0.0001 0.051 0.057 0.067 0.075
0.001 0.075 0.081 0.084 0.091
0.01 0.287 0.261 0.234 0.203
0.1 0.721 0.615 0.524 0.432
1 0.924 0.741 0.607 0.502
100 0.958 0.762 0.625 0.509
10000 0.959 0.758 0.625 0.504
100 0.0001 0.063 0.062 0.066 0.066
0.001 0.168 0.161 0.153 0.151
0.01 0.587 0.543 0.51 1 0.473
0.1 0.927 0.845 0.757 0.681
1 0.989 0.909 0.809 0.723
100 0.994 0.921 0.812 0.723
10000 0.998 0.918 0.813 0.721
200 0.0001 0.097 0.095 0.097 0.099
0.001 0.399 0.280 0.373 0.370
0.01 0.846 0.814 0.782 0.746
0.1 0.990 0.963 0.919 0.872
1 0.999 0.980 0.942 0.891
100 1.000 0.983 0.944 0.896
10000 1.000 0.980 0.941 0.898
500 0.0001 0.307 0.305 0.305 0.298
0.001 0.788 0.774 0.764 0.751
0.01 0.997 0.979 0.971 0.955
0.1 1.000 0.998 0.993 0.984
1 1.000 0.979 0.995 0.985
100 1.000 0.999 0.996 0.987
10000 1.000 0.999 0.995 0.985

 

Simulation results based on 20,000 replications.

34

 

 

Table 2.3

Actual Critical Values of KPSS Test with Fixed Number of Lags
(DGP: iid errors, no time trend)

5% significance level

 

 

T =0 [=1 [=2 =3
50 0.4770 0.4937 0.5048 0.5365
100 0.4599 0.4733 0.4661 0.4877
200 0.4509 0.4753 0.4744 0.4586
500 0.4622 0.4585 0.4587 0.4646

 

Simulation results based on 10,000 replications

35

 

 

Table 2.4

Size-Adjusted Power of KPSS Test with Fixed Number of Lags
(DGP: iid errors, no time trend)

5% significance level

 

 

 

 

 

T 2 [=0 [=1 [=2 =3
50 0.0001 0.047 0.047 0.050 0.050
0.001 0.074 0.067 0.065 0.059
0.01 0.288 0.235 0.203 0.131
0.1 0.722 0.591 0.482 0.333
1 0.922 0.720 0.581 0.406
100 0.957 0.745 0.597 0.414
10000 0.960 0.736 0.590 0.418
100 0.0001 0.063 0.056 0.062 0.056
0.001 0.166 0.150 0.154 0.135
0.01 0.590 0.535 0.504 0.451
0.1 0.927 0.834 0.753 0.665
1 0.990 0.904 0.804 0.699
100 0.995 0.916 0.808 0.712
10000 0.994 0.915 0.817 0.709
200 0.0001 0.102 0.091 0.088 0.098
0.001 0.403 0.382 0.364 0.365
0.01 0.853 0.811 0.770 0.742
0.1 0.991 0.958 0.919 0.878
1 0.999 0.980 0.936 0.894
100 1.000 0.982 0.944 0.896
10000 1.000 0.982 0.940 0.896
500 0.0001 0.310 0.31 1 0.305 0.296 '
0.001 0.790 0.776 0.765 0.748
0.01 0.987 0.981 0.971 0.953
0.1 1.000 0.999 0.993 0.984
1 1.000 0.999 0.996 0.986
100 1.000 0.999 0.995 0.985
10000 1.000 0.999 0.996 0.986

 

Simulation results based on 20,000 replications.

36

 

 

Table

2.5

Size and Power of Leyboume-McCabe Tests with Fixed Number of Lags

(DGP: iid errors, no time trend)

5% significance level

 

 

 

 

 

 

 

 

 

 

p =0
T 2 LM94 LM99 LMMl LMM2
100 0 0.048 0.054 0.040 0.054
0.001 0.170 0.179 0.156 0.179
0.01 0.601 0.626 0.590 0.626
1 0.988 0.999 0.996 1.000
100 0.994 0.537 0.999 1.000
200 0 0.049 0.050 0.044 0.050
0.001 0.398 0.404 0.386 0.404
0.01 0.854 0.870 0.853 0.870
1 1.000 1.000 1.000 1.000
100 0.998 0.552 1.000 1.000
500 0 0.050 0.051 0.045 0.051
0.001 0.782 0.788 0.779 0.778
0.01 0.987 0.991 0.988 0.991
1 1 .000 1 .000 l .000 1 .000
100 1.000 0.582 1.000 1.000
p =1
T 2 LM94 LM99 LMMl LMM2
100 0 0.055 0.063 0.050 0.063
0.001 0.165 0.180 0.154 0.180
0.01 0.594 0.620 0.586 0.620 .
1 0.974 0.902 0.981 0.985
100 0.908 0.417 0.913 0.917
200 0 0.054 0.058 0.049 0.058
0.001 0.391 0.400 0.379 0.400
0.01 0.847 0.861 0.848 0.861
1 0.998 0.970 0.999 0.999
100 0.947 0.448 0.947 0.948
500 0 0.054 0.055 0.050 0.055
0.001 0.784 0.791 0.780 0.791
0.01 0.986 0.989 0.987 0.989
1 1.000 0.998 1.000 1.000
100 1.000 0.486 0.979 0.979

 

Simulation results based on 10,000 replications.

37

Table 2.5 (Continued)

Size and Power of Leyboume-McCabe Tests with Fixed Number of Lags
(DGP: iid errors, no time trend)

 

 

 

 

 

 

 

 

 

 

5% significance level
p =2
T 2. LM94 LM99 LMMl LMM2
100 0 0.070 0.085 0.064 0.086
0.001 0.187 0.205 0.180 0.205
0.01 0.588 0.618 0.584 0.618
1 0.931 0.705 0.937 0.941
100 0.915 0.433 0.922 0.926
200 0 0.060 0.065 0.055 0.065
0.001 0.400 0.412 0.390 0.412
0.01 0.839 0.855 0.839 0.855
1 0.982 0.805 0.983 0.984
100 0.952 0.456 0.952 0.953
500 0 0.050 0.051 0.047 0.051
0.001 0.780 0.787 0.778 0.787
0.01 0.986 0.989 0.987 0.989
1 1 .000 0.924 1 .000 1 .000
100 0.983 0.485 0.983 0.983
p =
T 2 LM94 LM99 LMMl LMMZ
100 0 0.077 0.100 0.074 0.100
0.001 0.194 0.217 0.188 0.218
0.01 0.576 0.608 0.576 0.609
1 0.914 0.685 0.920 0.926 '
100 0.919 0.432 0.927 0.930
200 0 0.062 0.072 0.058 0.072
0.001 0.405 0.420 0.399 0.420
0.01 0.832 0.847 0.834 0.847
1 0.975 0.800 0.976 0.976
100 0.985 0.469 0.959 0.959
500 0 0.057 0.058 0.054 0.058
0.001 0.781 0.789 0.780 0.789
0.01 0.985 0.988 0.987 0.988
1 0.999 0.950 0.999 0.999
100 0.983 0.498 0.983 0.983

 

Simulation results based on 10,000 replications.

38

Table 2.6

Actual Critical Values of Leyboume-McCabe Tests with Fixed Number of Lags
(DGP: iid errors, no time trend)

5% significance level

 

 

 

 

 

 

 

 

 

T LM94 LM99 LMMl LMMZ
p= 0

100 0.4575 0.4709 0.4323 0.4709

200 0.4578 0.4629 0.4371 0.4629

500 0.4618 0.4657 0.4476 0.4657
p=l

100 0.4870 0.5160 0.4621 0.5160

200 0.4805 0.4922 0.4584 0.4922

500 0.4756 0.4804 0.4613 0.4804
p=2

100 0.5285 0.5863 0.5158 0.5873

200 0.4946 0.5141 0.4767 0.5141

500 0.4631 0.4691 0.4491 0.4685
p=3

100 0.5501 0.6502 0.5461 0.6515

200 0.5040 0.5332 0.4897 0.5332

500 0.4885 0.4955 0.4774 0.4955

 

Simulation results based on 10,000 replications.

39

Table

2.7

Size-Adjusted Power of Leyboume-McCabe Tests with Fixed Number of Lags

(DGP: iid errors, no time trend)

 

 

 

 

 

 

 

 

 

 

5% significance level
p =0
T 2 LM94 LM99 LMMl LMMZ
100 0.001 0.172 0.175 0.171 0.175
0.01 0.606 0.621 0.610 0.621
1 0.988 1.000 0.997 1.000
100 0.994 0.537 1.000 1.000
200 0.001 0.401 0.405 0.404 0.404
0.01 0.855 0.870 0.863 0.870
1 1.000 1.000 1.000 1.000
100 0.998 0.552 1.000 1.000
500 0.001 0.783 0.787 0.785 0.787
0.01 0.987 0.991 0.989 0.991
1 l .000 1 .000 1 .000 l .000
100 1.000 0.582 1.000 1.000
p =1
T 2, LM94 LM99 LMMl LMMZ
100 0.001 0.153 0.153 0.154 0.153
0.01 0.579 0.590 0.586 0.590
1 0.972 0.901 0.981 0.984
100 0.905 0.413 0.913 0.914
200 0.001 0.380 0.381 0.383 0.381 -
0.01 0.841 0.854 0.850 0.854
1 0.997 0.970 0.999 0.999
100 0.945 0.447 0.948 0.947
500 0.001 0.777 0.782 0.781 0.782
0.01 0.985 0.988 0.987 0.988
1 1.000 0.998 1.000 1.000
100 0.979 0.486 0.979 0.979

 

40

Table 2.7 (Continued)

Size-Adjusted Power of Leyboume-McCabe Tests with Fixed Number of Lags
(DGP: iid errors, no time trend)

 

 

 

 

 

 

 

 

 

 

5% significance level
p =2
T 2, LM94 LM99 LMMl LMM2
100 0.001 0.156 0.152 0.153 0.152
0.01 0.549 0.556 0.554 0.556
1 0.923 0.699 0.933 0.935
100 0.908 0.425 0.918 0.919
200 0.001 0.377 0.378 0.391 0.378
0.01 0.827 0.838 0.833 0.838
1 0.981 0.804 0.983 0.983
100 0.950 0.454 0.952 0.951
500 0.001 0.780 0.784 0.783 0.784
0.01 0.986 0.989 0.988 0.989
1 1 .000 0.924 1 .000 1 .000
100 0.983 0.485 0.983 0.983
p =
T 2 LM94 LM99 LMMl LMM2
100 0.001 0.153 0.143 0.152 0.143
0.01 0.530 0.524 0.533 0.524
1 0.901 0.671 0.912 0.912
100 0.911 0.423 0.922 0.921
200 0.001 0.380 0.379 0.383 0.379 -
0.01 0.817 0.826 0.823 0.826
1 0.973 0.798 0.975 0.975
100 0.956 0.467 0.958 0.957
500 0.001 0.770 0.775 0.773 0.775
0.01 0.983 0.987 0.986 0.987
1 0.999 0.950 0.999 0.999
100 0.982 0.497 0.983 0.982

 

Simulation results based on 10,000 replications.

41

Table 2.8

Size and Power of KPSS and Leyboume-McCabe Tests with AR(1) Errors
(DGP: y1=pyt-l+819 p=1/3)

5% significance level

 

 

 

 

KPSS LM94 LM99 LMMl LMM2
T 2 [=1 p=1 p=1 p=l p=1
100 0 0.071 0.053 0.056 0.055 0.056
0.001 0.199 0.171 0.182 0.172 0.182
0.01 0.588 0.595 0.618 0.601 0.618
1 0.918 0.936 0.627 0.942 0.945
100 0.924 0.989 0.468 0.996 0.997
200 0 0.078 0.051 0.052 0.052 0.052
0.001 0.436 0.409 0.418 0.409 0.418
0.01 0.839 0.845 0.860 0.849 0.860
1 0.983 0.982 0.758 0.982 0.982
100 0.984 1.000 0.497 1.000 1.000
500 0 0.078 0.047 0.047 0.047 0.047
0.001 0.811 0.787 0.792 0.788 0.792
0.01 0.983 0.985 0.988 0.986 0.988
1 0.999 0.999 0.956 0.999 0.999
100 1.000 1.000 0.510 1.000 1.000

 

Simulation results based on 20,000 replications for the KPSS test, 10,000 replications for the Leybotune-

McCabe tests.

42

Table 2.9

Actual Critical Values of KPSS and Leyboume-McCabe Tests with AR(1) Errors

(DGP: YtszI-I'l'st, p=1/3)

5% significance level

 

 

KPSS LM94 LM99 LMMl LMM2

T [=1 p=l p=l p=1 p=l
100 0.5241 0.4710 0.4806 0.4780 0.4811
200 0.5501 0.4672 0.4711 0.4728 0.4711
500 0.5448 0.4519 0.4533 0.4514 0.4533

 

Simulation results based on 20,000 replications for the KPSS test, 10,000 replications for the Leyboume-

McCabe tests.

43

Table 2.10

Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with AR(1) Errors
(DGP: y,=py,-1+8,, p=l/3)

5% significance level

 

 

 

 

KPSS LM94 LM99 LMMl LMM2
T 2 [=1 p=l p=1 p=1 p=l

100 0.001 0.169 0.166 0.173 0.164 0.173
0.01 0.545 0.591 0.611 0.592 0.610

1 0.899 0.935 0.626 0.941 0.944

100 0.903 0.989 0.467 0.996 0.997

200 0.001 0.388 0.405 0.412 0.402 0.412
0.01 0.806 0.844 0.858 0.845 0.857

1 0.973 0.982 0.758 0.982 0.982

100 0.974 1.000 0.497 1.000 1.000

500 0.001 0.775 0.792 0.796 0.793 0.796
0.01 0.977 0.986 0.988 0.987 0.988

1 0.999 0.999 0.956 0.999 0.999

100 0.999 1.000 0.510 1.000 1.000

 

Simulation results based on 20,000 replications for the KPSS test, 10,000 replications for the Leyboume-

McCabe tests.

Table 2.11

Size and Power of KPSS and Leyboume-McCabe Tests with MA(1) Errors
(DGP. Y1: 8g + 98p], 9:0.5)

 

 

 

 

5% significance level
KPSS LM94 LM99 LMMl LMM2
T 7, [=1 p=1 p=1 p=1 p=1
100 0 0.051 0.029 0.028 0.032 0.039
0.001 0.095 0.054 0.058 0.051 0.058
0.01 0.384 0.314 0.325 0.309 0.325
1 0.900 0.877 0.848 0.883 0.892
100 0.918 0.910 0.419 0.916 0.920
200 0 0.053 0.025 0.025 0.026 0.030
0.001 0.242 0.171 0.174 0.166 0.174
0.01 0.685 0.630 0.642 0.629 0.642
1 0.980 0.971 0.969 0.972 0.974
100 0.981 0.947 0.451 0.947 0.949
500 0 0.052 0.022 0.024 0.022 0.025
0.001 0.622 0.546 0.548 0.543 0.548
0.01 0.946 0.932 0.937 0.932 0.937
1 0.999 1.000 1 .000 1.000 1 .000
100 0.999 0.980 0.496 0.980 0.980

 

Simulation results based on 20,000 replications for the KPSS test, 10,000 replications for the Leyboume-

McCabe tests.

45

Table 2.12

Actual Critical Values of KPSS and Leyboume-McCabe Tests with MA(1) Errors

(DGP: y,= 8, + 08,.,, 0=0.5)

5% significance level

 

T
100
200
500

KPSS LM94 LM99
[=1 p=l p=1
0.4460 0.3794 0.3704
0.4532 0.3647 0.3580
0.4503 0.3583 0.3591

LMMl LMM2
p=1 p=1
0.3846 0.4129
0.3589 0.3766
0.3499 0.3629

Simulation results based on 20,000 replications for the KPSS test, 10,000 replications for the Leyboume-

McCabe tests.

46

Table 2.13

Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with MA(1) Errors
(DGP: y,= 8, + 08,-1, 0=0.5)

 

 

 

 

5% significance level
KPSS LM94 LM99 LMMl LIVIMZ
T 2 [=1 p=l p=1 p=1 p=l
100 0.001 0.108 0.088 0.095 0.079 0.075
0.01 0.397 0.373 0.391 0.360 0.356
1 0.906 0.894 0.863 0.897 0.899
100 0.925 0.921 0.428 0.927 0.925
200 0.001 0.244 0.234 0.242 0.232 0.228
0.01 0.696 0.695 0.704 0.692 0.693
1 0.980 0.976 0.973 0.977 0.977
100 0.984 0.953 0.457 0.954 0.954
500 0.001 0.621 0.622 0.624 0.623 0.621
0.01 0.949 0.954 0.957 0.956 0.956
1 0.999 1.000 1 .000 l .000 1.000
100 0.999 0.982 0.499 0.983 0.982

 

Simulation results based on 20,000 replications for the KPSS test, 10,000 replications for the Leyboume-

McCabe tests.

47

Table 2.14

Size and Power of KPSS and Leyboume-McCabe Tests with ARMA (1,1) Errors
(DGP: y, = py,-1+ 8,+08,-,, p=1/3, 0=1/2)

5% significance level

 

 

 

 

KPSS LM94 LM99 LMMl LMM2
T 2 [=1 p=1 p=1 p=1 p=1
100 0 0.081 0.014 0.015 0.016 0.016
0.001 0.148 0.041 0.043 0.043 0.044
0.01 0.456 0.282 0.282 0.286 0.294
1 0.911 0.981 0.099 0.987 0.988
100 0.924 0.989 0.450 0.996 0.996
200 0 0.086 0.017 0.017 0.017 0.017
0.001 0.313 0.109 0.111 0.110 0.112
0.01 0.741 0.573 0.580 0.574 0.581
1 0.981 0.999 0.030 0.999 0.999
100 0.985 1.000 0.457 1.000 1.000
500 0 0.089 0.014 0.014 0.014 0.015
0.001 0.682 0.497 0.501 0.498 0.501
0.01 0.962 0.911 0.915 0.912 0.915
1 0.999 1 .000 0.002 1 .000 1.000
100 0.999 1.000 0.444 1.000 1.000

 

Simulation results based on 20,000 replications for the KPSS test, 10,000 replications for the Leyboume-

McCabe tests.

48

Table 2.15

Actual CriticalValues of KPSS and Leyboume-McCabe Tests with ARMA (1, 1) Errors
(DGP: yt= pyt-1+ 8t+98t-1, p=1/3, 0=1/2)

5% significance level

 

 

KPSS LM94 LM99 LMMl LMM2

T [=1 p=1 p=1 p=1 p=1
100 0.5643 0.3208 0.3210 0.3208 0.3236
200 0.5839 0.3188 0.3201 0.3218 0.3201
500 0.5844 0.3103 0.3125 0.3110 0.3125

 

Simulation results based on 20,000 replications for the KPSS test, 10,000 replications for the Leyboume-
McCabe tests.

49

Table 2.16

Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with ARMA (1, 1) Errors
(DGP: yt= pyH+ 8t+68t-1, p=1/3, 0=1/2)

5% significance level

 

 

 

 

KPSS LM94 LM99 LMMl LMMZ
T x [=1 p=1 p=1 p=1 p=1

100 0.001 0.104 0.100 0.101 0.103 0.100
0.01 0.389 0.396 0.391 0.399 0.401

1 0.870 0.988 0.101 0.989 0.989

100 0.888 0.995 0.450 0.997 0.997

200 0.001 0.236 0.189 0.189 0.189 0.189
0.01 0.678 0.664 0.668 0.661 0.669

1 0.963 0.999 0.030 0.999 0.999

100 0.969 1.000 0.457 1.000 1.000

500 0.001 0.617 0.617 0.617 0.616 0.617
0.01 0.941 0.949 0.950 0.949 0.950

1 0.998 1.000 0.002 1.000 1.000

100 0.998 1.000 0.444 1.000 1.000

 

Simulation results based on 20,000 replications for the KPSS test, 10,000 replications for the Leyboume-

McCabe tests.

50

Chapter 3

Performance of the KPSS and Leyboume—McCabe Tests when the

Number of Lags Increases with the Sample Size

1. Introduction

In this chapter, we consider the properties of the KPSS and Leyboume-McCabe
tests when we allow the number of lags to increase with the sample size. Here, as in the
previous chapters, the “number of lags” is the parameter “I”, the number of lagged terms
in the long-run variance estimate, for KPSS; and it is the parameter “p”, the assumed
order of the AR polynomial, for the various Leyboume-McCabe tests.

In the original KPSS (1992) paper, the number of lags l was required to satisfy

the requirements that, as T—>oo, I-—>oo, but l/T—>0. This ensured consistency of the long-
run variance estimator s2(l) , so long as certain regularity conditions are satisfied. The

Leyboume and McCabe papers (1994 and 1999) assumed a finite-order AR model. Our
treatment of the Leyboume-McCabe tests in this chapter lets p->oo, p/T—)0, as T—>oo, and
is analogous to the way that p is treated in the Said-Dickey (1984) ADF unit root tests.
Intuitively, we expect that an AR(p) model with large p can approximate any stationary
process, subject to some regularity conditions.

In this chapter we perform simulations to see how the size and power of the
various tests are affected when the number of lags grows with sample size. Our data

generating processes (DGPs) will be essentially the same as in the previous chapter. We

51

will follow Schwert (1989) and many subsequent papers and consider three rules for

choosing the number of lags: [0=0, l4=integer[4(T/100)]“4 and I 12=integer[12(T/ 100)] U 4.

2. Theoretical issues

In this section, we discuss the known properties of the KPSS and Leyboume-
McCabe tests when the number of lags is a function of the sample size.

KPSS (1992) have already addressed the distribution theory of the KPSS test

when l—>oo as T-—)oo. They use the ‘weighted’ long run variance estimator 32(1) to
construct the KPSS test, where 52(1) is defined in Chapter 1, equation (21). They show
that s2(l) is a consistent estimate of the long mm variance 02 when the lag truncation

number 1 satisfies the condition that l—>oo but [IT—>0, as T—>oo. Then, under the null

hypothesis of stationarity, the KPSS statistic rip is 0,,(1) and it has the asymptotic
distribution given in equation (22) of Chapter 1. Under the unit root alternative, I?” is

0,,(T/l). Thus the rate at which I grows affects the power of the test, even asymptotically.
Correspondingly we might expect that the power of the KPSS test will grow slowly
compared to other cases, where the lag truncation number I is assumed to be fixed (as in
Chapter 2), or determined by lag selection rules (as in Chapter 4).

The distribution theory for the Leyboume-McCabe tests when we allow the
number of lags to increase with the sample size is unknown. We can make an analogy, at
least at an intuitive level, to the augmented Dickey-Fuller (ADF) tests of Said and Dickey

(1984). They also let p—>oo, p/T—>O, as T—)oo, and they show that the ADF test is valid (in

52

the sense that it has the same asymptotic distribution as the standard DF test does with
white noise errors), provided that the errors satisfy some regularity conditions.
Specifically, they assume that the errors follow a finite order ARMA(p,q) model, but this
is a stronger assumption than necessary. Intuitively, the assumed AR(p) model
approximates any sufficiently regular stationary error very well, if p is large enough, and
the asymptotic distribution theory is unaffected by letting p—>oo so long as p does not
grow too quickly. We conjecture that similar results hold for the Leyboume-McCabe
tests; that is, that the Leyboume-McCabe tests are valid, as long as the errors satisfy some
regularity conditions and the number of lags increases to infinity but sufficiently slowly
relative to the sample size. Unfortunately we have no proof of this conjecture. The
difficulty, relative to the Said-Dickey analysis, is that the Leyboume-McCabe tests
depend on the results of a numerical optimization and thus cannot be written as an
explicit closed-form fimction of the data.

Under the alternative, the situation is even less clear. For the ADF test, Said and
Dickey do not provide any results under the alternative, and it is apparently not known
whether the power of the ADF test is different when p—mo than when a fixed value of p is
correct and is used. For the Leyboume-McCabe tests, we note simply that, for fixed p, the
asymptotic distribution under the alternative does not depend on p. Whether this carries

over to the case that p-)oo is not clear.

53

51;

88

Ill:

C0

M.

for

MC

3. Simulations

In this section we provide some Monte Carlo evidence on the size and power of
the KPSS and Leyboume-McCabe tests when the number of lags grows with the sample

size. The design of the simulations is very similar to that of Chapter 2. The DGP is

essentially equation (1) of Chapter 1, with (i=0. Thus y, = ,u, +u, , ,u, = ,u,_, + v,, where
the u, are N(0, a: ), the v, are N(0,o'v2 ), and u and v are independent. As in Chapter 2, we

consider the cases that the u, are iid (white noise), but also cases where they are AR(1),
MA(1) and ARMA(1,1) errors.
The data contain no deterministic trend and we consider only the tests that allow

for level but not trend (e.g., KPSS ii” but not 77. , and similarly for the Leyboume-

McCabe tests). The number of replications is given below, but is generally 20,000 for
KPSS and 10,000 for Leyboume-McCabe tests.

Simulations were performed using GAUSS 3.2.25 and the Maxlik optimization
procedure. We let the number of lags (I or p) follow the rules': [0=0,

l4=integer[4(T/100)m], 112=integer[12(T/100)”4].

 

' The number of lagged terms according to the above rule is as follows.

 

 

 

Lag truncation number or Order of AR polynomials
T =p=10=0 I=p=14 =p=112
50 0 3 10
100 0 4 12
200 0 4 14
500 0 5 l7

 

 

 

 

 

 

54

1) The KPSS and Leyboume-McCabe tests with iid errors

We first consider the size of the KPSS and Leyboume-McCabe tests in the
presence of iid errors. The null hypothesis is of = 0(7t=0) and then y, = u,, so y, is

white noise. The tests are set at the 5% nominal significance level, and the results are
based on 20,000 replications for the KPSS test and 10,000 replications for the
Leyboume-McCabe tests.

Table 3.1 gives the size and power of the KPSS and the Leyboume-McCabe tests

with various sample sizes (T) and values of A = of / 0'3. Size corresponds to the entries

for 7t=0. All of the tests have more or less correct size when l=p=10. For the case that the
number of lags is 14 or 112, there are size distortions in opposite directions: the KPSS test
rejects too seldom while the Leyboume-McCabe tests reject too often. For the KPSS test,
the size distortions disappear fairly rapidly as T increases, even for [=112. For the various
Leyboume-McCabe tests, the size distortions get smaller as T increases, which is
consistent with our conjecture that the Leyboume-McCabe tests are valid when p grows
with sample size. However, the decrease in the size distortions of the Leyboume-McCabe
tests as T grows is not very rapid. Correspondingly, the KPSS test with 1:112 hasvery
much smaller size distortions than the Leyboume-McCabe tests with p=112. For example,
for T=500, compare: KPSS, 0.046; LM94, 0.171; LM99, 0.190,

The power of the tests increases with T and generally with 7», except for the LM99
test. For l= =10 (=0), we have already discussed the results in Chapter 2. There is not
much difference between tests. It is obvious that increasing the number of lags to 14 or
112 costs power. However, it is hard to separate changes in power from changes in size

distortion in Table 3.1, so we will move on to a discussion of size-adjusted power. Table

55

3.2 gives the “actual” critical values, which would lead to size of 0.05 under the null in
our simulations, and Table 3.3 gives size-adjusted power (power using the “actual”
critical values).

Size-adjusted power increases with T and generally with k. In the cases of 14 or
112 lags, the KPSS test generally has greater size-adjusted power when A is small, while
the LM94, LM] and LMM2 tests have greater size-adjusted power for larger values of
A. The LM99 test still does poorly when A is large. The other Leyboume-McCabe tests
sometimes show evidence of the “near cancellation” problem discussed in Chapter 1;
power decreases as we move to the largest values of 7..

Increasing the number of lags from 10 to 14 to 112 causes size-adjusted power to
decrease, often substantially. That is, there is a loss in size-adjusted power from using too
many lags, as was also found in Chapter 2. There is no uniform comparison of tests in
terms of the power loss from increasing the number of lags. As we move from 10 to 14 to
l 12 lags, sometimes the loss in the size-adjusted power is larger for the KPSS test than for
the Leyboume-McCabe tests (e.g., T=500, i=1) and sometimes the reverse is true (e.g.,
T=200, 1:001). This is perhaps surprising, since in Chapter 2 it was more or less
unifome true that using too many lags affected the power of the KPSS test more than
the power of the Leyboume-McCabe tests. However, in Chapter 2 we never had more

than three lags, while now we have as many as 17 (for l or p=12, T=500).

2) The KPSS and Leyboume-McCabe tests with AR(1) errors

Here we perform simulations with AR(1) errors of the form : ut=put-1+e¢, where

at is normal white noise. We set p=1/3 as in Chapter 2. We consider the KPSS test with

56

[=14 and the LM tests with p=l4. The KPSS test should in principle be able to
accommodate AR(1) errors, if T is large enough, since the test is asymptotically valid
under the null and consistent under the alternative when l grows with T. However, we
presume that an AR error favors the Leyboume-McCabe tests, which are based on an AR
specification.

Table 3.4 gives the size and power of the various tests, for values of T and 71
similar to those considered previously. Table 3.5 gives the actual critical values, while
Table 3.6 gives size-adjusted power.

The KPSS test shows moderate size distortions. Interestingly, they do not
decrease noticeably as T increases (they should vanish asymptotically). The Leyboume-
McCabe tests do not show substantial size distortions, and this is perhaps surprising since
they did in the white noise case. The power of the KPSS test compares favorably to the
power of the LM tests (especially for small A), but this is only due to the size distortion.
When we look at size-adjusted power, the KPSS test is dominated by the Leyboume-
McCabe tests, all of which are fairly similar. (The exception to these statements is that
the LM99 test still does badly when k is large.) For example, for 7L=0.001, the power of
the KPSS test is 0.193, 0.422, and 0.785 for T=100, 200, and 500, respectively. However,
its size-adjusted power drops to 0.163, 0.379 and 0.746, and now is less than for its
Leyboume-McCabe counterparts. For example, for the LMM2 test, size-adjusted power
is 0.193, 0.396 and 0.777 for k=0.001 and T=100, 200 and 500; and the advantage of
LMMZ over KPSS is greater when 7. is larger. However, LM99 still does poorly for very

large k, and the other Leyboume-McCabe tests show modest decreases in power for very

large 7., due to the “near cancellation” problem.

57

Comparing the various Leyboume-McCabe tests, it seems that LMM2 test
generally has the largest size-adjusted power, but the differences are small. This is similar
to what was found in Chapter 2, with p fixed. With p fixed, this was a surprising result
since LM94 is Op(T) while the other Leyboume-McCabe tests are Op(T2). When p
increases with T, the asymptotic properties of the tests are unknown, so there is no theory
for our results to disagree with. However, at an intuitive level, the degree of similarity

between LM94 and the other Leyboume-McCabe tests is still surprising.

3) The KPSS and Leyboume-McCabe tests with MA(1) errors

Now we perform simulations with MA(1) errors of the form : y,=st+08.-1, where e,
is normal white noise. We pick 0=0.5 from the same reason we discussed in Chapter 2.
As in the previous section, we consider the KPSS test with [=14 and the Leyboume-
McCabe tests with p=l4. The KPSS test is asymptotically valid under the null, and
consistent under the alternative, while the asymptotic properties of the Leyboume-
McCabe tests are unknown. We presume that an MA error favors the KPSS test over the
Leyboume-McCabe tests, as explained in Chapter 2. Our results are given in Tables 3.7-
3.9, with the same format as before.

Consider first the size of the tests (results for i=0 in Table 7). The KPSS test has
a modest size distortion (rejection rate of 0.06 instead of 0.05), which does not clearly
diminish as T increases. The Leyboume-McCabe tests have more substantial size
distortions, but these do clearly diminish as T increases. For T=500, the degree of size
distortion is very minor for all of the tests (KPSS and all versions of Leyboume-

McCabe).

S8

In terms of size-adjusted power (Table 9), all of the Leyboume-McCabe tests are
quite similar to each other (except, again, LM99 when 7» is large). For T=500, the size-
adjusted power of the KPSS test is also quite similar. For smaller values of T, the KPSS
test is better than the LM tests when power is low, and worse when power is high. The
latter result is somewhat unexpected, since in Chapter 2 (with the number of lags fixed)
the KPSS test dominated the Leyboume-McCabe tests with MA(1) errors. The difference
may be that in Chapter 2 we used an unweighted long-run variance estimator, whereas
here we use the Newey-West weights. With MA(1) errors, only the first autocorrelation is
non—zero, and the Newey-West weights downweight this inappropriately except when 1 is

quite large.

4) The KPSS and Leyboume-McCabe tests with ARMA(1,1) errors

Here we perform simulations using ARMA(1,1) errors of the form: y,=pyH
+8t+981-1, where p=1/3 and 0=1/2. As before, we choose these specific values of the AR
and MA parameter to equate the contribution of the AR and MA terms to the “long-run
variance” of the ARMA(1,1) error series. We consider the KPSS test with [=14 and the
Leyboume-McCabe tests with p=l4. Our results are given in Tables 3.10—3.12, which
have the same format as the previous tables for the AR and MA cases.

The KPSS test shows moderate size distortion (rejection rate of about 0.08 instead
of 0.05), and this does not clearly decrease as T increases. This size distortion is only
very slightly smaller than was found in Chapter 2, with the same DGP, for the KPSS test
with [=1. The Leyboume-McCabe tests have slightly greater size distortions than the

KPSS test when T=100, but these size distortions do decrease as T increases, and have

59

disappeared for T=500. This results is quite different from what was found in Table 14 of
Chapter 2, where the Leyboume-McCabe tests with p=1 underrejected considerably
(rejection rate of about 0.015) and increasing T did not improve things. These results are
also more optimistic than the corresponding results in Table 3.1 for the case of white
noise errors. The latter result is surprising and deserves further study.

The size-adjusted power of the KPSS test with [=14 is less than that of the
Leyboume-McCabe tests with p=[4 (again, except LM99 when A. is large). This

difference in size-adjusted power is larger than was found in Table 2.16 of Chapter 2.

4. Conclusions

In this chapter we considered the KPSS and Leyboume-McCabe tests that use a
number of lags that increases with the sample size T. We investigated the size and power
characteristics of the tests via simulations. In the simulations our data-generating
processes included white noise, AR(1), MA(1), and ARMA(1,1) errors. Our main
conclusions are as follows.

1. The LM99 test is still not recommended, due to its poor power when A is
large. This case (LM99 test, 2. large) is an exception to the remaining conclusions for the
Leyboume-McCabe tests.

2. There is not much difference in power between the LM94 test and the LM99
test or its modifications (LMMl, LMMZ). This is similar to the results we have seen in

Chapter 2.

60

3. When p increases with T, the asymptotic properties of the Leyboume-McCabe
tests are unknown. However, our results seem to be consistent with the conj ecture that the
Leyboume-McCabe tests are asymptotically valid under the null and consistent under the
alternative.

4. Once again we can argue that the white noise case is a fair setting for
comparison of the KPSS test to the Leyboume-McCabe tests. All of the tests have size
distortions in the 14 or [12 cases - the KPSS test underrejects while the Leyboume-
McCabe tests overrej ect. The size distortions are much more severe for the Leyboume-
McCabe tests, however. In our opinion, they are severe enough to argue against the use of
the Leyboume-McCabe tests with the number of lags increasing with T (at least, at the
rate we consider, which is proportional to T1“). The Leyboume-McCabe tests are more
powerful but this is mostly due to the size distortions. Size-adjusted power favors the
Leyboume-McCabe tests in general, but not uniformly.

5. For all of the tests, an unnecessarily large number of lags costs power. This is
generally but not uniformly more true for the KPSS test than for the Leyboume-McCabe
tests.

6. The cases with autocorrelated errors are generally speaking more favorable to
the Leyboume-McCabe tests than the white noise case. For reasons that we do not
understand, the Leyboume-McCabe tests show smaller size distortions with
autocorrelated errors than with white noise errors. This is true even when the errors are
not AR. Power considerations favor the KPSS test in the MA case, but favor the
Leyboume-McCabe tests in the AR and ARMA cases. The ARMA cases, like the white

noise cases, were set up to be a fair setting for comparison of the KPSS and Leyboume-

61

McCabe tests, and the generally superior performance of the Leyboume-McCabe tests
balances its poor performance in the case of white noise errors. Clearly more work is

needed to determine which types of errors favor which tests.

62

Table 3.1

Size and Power of KPSS and Leyboume-McCabe Tests
when Number of Lags Increases with Sample Size

(DGP: iid errors, no time trend)

5% significance level

 

 

 

 

 

 

 

 

 

 

1=p=10=0

r 1. KPSS LM94 LM99 LMMl LMM2
100 0 0.049 0.048 0.054 0.040 0.054
0.001 0.168 0.170 0.179 0.156 0.179

0.01 0.587 0.601 0.626 0.590 0.626

1 0.989 0.988 0.999 0.996 1.000

100 0.994 0.994 0.539 0.999 1.000

200 0 0.051 0.049 0.050 0.044 0.050
0.001 0.399 0.398 0.404 0.386 0.404

0.01 0.846 0.854 0.870 0.853 0.870

1 0.999 1.000 1.000 1.000 1.000

100 1.000 0.998 0.552 1.000 1.000

500 0 0.050 0.050 0.051 0.045 0.051
0.001 0.788 0.782 0.788 0.779 0.778

0.01 0.997 0.987 0.991 0.988 0.991

1 1.000 1.000 1.000 1.000 1.000

100 1.000 1.000 0.582 1.000 1.000

[=p=[4=integer[4(T/100)]/4]

r 2. KPSS LM94 LM99 LMMl LMM2
100 0 0.043 0.103 0.129 0.104 0.131
0.001 0.147 0.215 0.243 0.211 0.244

0.01 0.508 0.572 0.597 0.573 0.601

1 0.818 0.912 0.669 0.919 0.924

100 0.826 0.917 0.433 0.923 0.927

200 0 0.049 0.074 0.084 0.071 0.084
0.001 0.372 0.402 0.418 0.397 0.418

0.01 0.776 0.833 0.846 0.833 0.846

1 0.943 0.976 0.800 0.976 0.977

100 0.945 0.964 0.464 0.965 0.966

500 0 0.048 0.062 0.065 0.059 0.065
0.001 0.757 0.779 0.786 0.777 0.786

0.01 0.962 0.983 0.986 0.984 0.986

1 0.992 0.999 0.970 0.999 0.999

100 0.992 0.985 0.500 0.985 0.985

 

63

Table 3.1 (Continued)

Size and Power of KPSS and Leyboume-McCabe Tests
when Number of Lags Increases with Sample Size

(DGP: iid errors, no time trend)

5% significance level

 

1=p=112=integer[12*(T/100)m]

 

 

 

 

T A KPSS LM94 LM99 LMMl LMM2
100 0 0.029 0.278 0.293 0.292 0.322
0.001 0.100 0.322 0.341 0.329 0.356
0.01 0.367 0.581 0.566 0.587 0.606
1 0.579 0.895 0.652 0.902 0.907
100 0.584 0.884 0.395 0.889 0.892
200 0 0.041 0.228 0.252 0.235 0.259
0.001 0.314 0.474 0.487 0.476 0.490
0.01 0.626 0.771 0.759 0.774 0.780
1 0.725 0.987 0.810 0.978 0.979
100 0.726 0.940 0.445 0.941 0.941
500 0 0.046 0.171 0.190 0.174 0.190
0.001 0.682 0.776 0.783 0.776 0.783
0.01 0.865 0.960 0.960 0.960 0.962
1 0.901 0.999 0.971 0.999 0.999
100 0.901 0.979 0.510 0.978 0.978

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

5% significance level

Table 3.2

(DGP: iid errors, no time trend)

Actual Critical Values of KPSS and Leyboume-McCabe Tests

when Number of Lags Increases with Sample Size

 

 

 

 

 

 

 

1=p=10=0

r KPSS LM94 LM99 LMM1 LMM2
100 0.459 0.457 0.471 0.432 0.471
200 0.451 0.457 0.462 0.437 0.463
500 0.462 0.462 0.465 0.447 0.465

[=p=14=integer[4(T/100)T4]

100 0.459 0.667 0.876 0.709 0.892
200 0.441 0.548 0.602 0.543 0.602
500 0.460 0.511 0.521 0.499 0.521

1=p=112=integer[12(T/100)”‘T

100 0.431 1.797 6.518 1.827 13.863
200 0.463 1.523 2.774 2.239 3.366
500 0.449 1.026 1.296 1.134 1.296

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

65

Table 3.3

Size-Adjusted Power of KPSS and Leyboume-McCabe Tests
when Number of Lags Increases with Sample Size
(DGP: iid errors, no time trend)

5% significance level

 

 

 

 

 

 

 

 

 

 

L=p=10=0

r 2. KPSS LM94 LM99 LMM1 LMM2
100 0.001 0.166 0.172 0.175 0.171 0.175
0.01 0.590 0.606 0.621 0.610 0.621

1 0.990 0.988 1.000 0.997 1.000

100 0.995 0.994 0.537 1.000 1.000

200 0.001 0.403 0.401 0.405 0.404 0.404
0.01 0.853 0.855 0.870 0.863 0.870

1 0.999 1.000 1.000 1.000 1.000

100 1.000 0.998 0.552 1.000 1.000

500 0.001 0.790 0.783 0.787 0.785 0.787
0.01 0.987 0.987 0.991 0.989 0.991

1 1.000 1.000 1.000 1.000 1.000

100 1.000 1.000 0.582 1.000 1.000

[=p=[4=integer[4(T/l00)m]

T 7. KPSS LM94 LM99 LMM1 LMM2
100 0.001 0.143 0.130 0.118 0.123 0.116
0.01 0.511 0.478 0.451 0.474 0.449

1 0.818 0.881 0.643 0.899 0.898

100 0.830 0.890 0.414 0.909 0.907

200 0.001 0.475 0.351 0.341 0.349 0.341
0.01 0.849 0.804 0.808 0.809 0.808

1 0.974 0.973 0.797 0.975 0.975

100 0.975 0.961 0.461 0.963 0.963

500 0.001 0.761 0.758 0.762 0.761 0.762
0.01 0.963 0.980 0.982 0.981 0.982

1 0.991 0.999 0.970 0.999 0.999

100 0.992 0.984 0.499 0.984 0.984

 

66

Table 3.3 (Continued)

Size-Adjusted Power of KPSS and Leyboume-McCabe Tests

when Number of Lags Increases with Sample Size
(DGP: iid errors, no time trend)

5% significance level

 

l=p=112=integer[12(T/100)m]

 

 

 

 

T 7L KPSS LM94 LM99 LMM1 LMMZ

100 0.001 0.139 0.088 0.056 0.058 0.043
0.01 0.425 0.298 0.194 0.225 0.166

1 0.618 0.659 0.475 0.691 0.647

100 0.623 0.668 0.275 0.747 0.708

200 0.001 0.340 0.208 0.154 0.167 0.135
0.01 0.642 0.581 0.514 0.547 0.498

1 0.747 0.920 0.772 0.941 0.934

100 0.744 0.875 0.395 0.892 0.884

500 0.001 0.693 0.616 0.579 0.599 0.580
0.01 0.870 0.921 0.916 0.921 0.917

1 0.903 0.999 0.971 0.999 0.999

100 0.903 0.969 0.499 0.969 0.967

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

67

Table 3.4

Size and Power of KPSS and Leyboume-McCabe Tests with AR(1) Errors
(DGP: y,= py..1+ at, p=1/3)

 

 

 

 

5% significance level
KPSS LM94 LM99 LMMl LMMZ
T 7. [=14 p= l4 p= [4 p= I4 p= [4
100 0 0.067 0.058 0.062 0.061 0.062
0.001 0.193 0.200 0.211 0.201 0.211
0.01 0.544 0.571 0.589 0.577 0.593
1 0.825 0.943 0.617 0.949 0.953
100 0.835 0.936 0.371 0.942 0.944
200 0 0.071 0.054 0.056 0.055 0.056
0.001 0.422 0.397 0.405 0.398 0.405
0.01 0.804 0.813 0.827 0.817 0.827
1 0.946 0.986 0.761 0.987 0.987
100 0.950 0.968 0.399 0.969 0.969
500 0 0.070 0.054 0.054 0.053 0.054
0.001 0.785 0.777 0.781 0.777 0.781
0.01 0.967 0.978 0.981 0.979 0.987
1 0.991 1.000 0.963 1.000 1.000
100 0.993 0.987 0.402 0.987 0.988

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for LM tests.

68

Table 3.5

Actual Critical Values of KPSS and Leyboume-McCabe Tests with AR(1) Errors

(DGP: y1= 9344+ 86 p=1/3)

5% significance level

 

T

KPSS LM94 LM99
1:14 p= 14 p= 14

LMM1 LMM2
p= 14 p= 14

 

100
200
500

0.5033 0.4909 0.5084
0.5204 0.4739 0.4810
0.5360 0.4745 0.4749

0.4997 0.5084
0.4792 0.4810
0.4741 0.4749

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

69

Table 3.6

Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with AR(1) Errors
(DGP: y1= 1m:+ at, p=1/3)

 

 

 

 

5% significance level
KPSS LM94 LM99 LMM1 LMM2
T )1, [=14 p= [4 p= 14 p= I4 p= [4
100 0.001 0.163 0.187 0.192 0.185 0.193
0.01 0.513 0.558 0.571 0.562 0.575
1 0.796 0.939 0.615 0.949 0.950
100 0.806 0.933 0.369 0.939 0.942
200 0.001 0.379 0.390 0.396 0.389 0.396
0.01 0.775 0.809 0.822 0.811 0.822
1 0.930 0.985 0.760 0.987 0.987
100 0.937 0.968 0.398 0.969 0.969
500 0.001 0.746 0.771 0.777 0.773 0.777
0.01 0.955 0.977 0.980 0.979 0.980
1 0.986 1.000 0.963 1.000 1.000
100 0.987 0.987 0.402 0.987 0.988

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

70

Table 3.7

Size and Power of KPSS and Leyboume-McCabe Tests with MA(1) Errors
(DGP: Y: = 8t+98t-1, 9:0.5)

5% significance level

 

 

 

 

KPSS LM94 LM99 LMM1 LMM2
T 7. [=14 p= [4 p= [4 p= [4 p= 14
100 0 0.056 0.129 0.149 0.141 0.166
0.001 0.109 0.169 0.193 0.171 0.196
0.01 0.388 0.443 0.468 0.446 0.473
1 0.813 0.924 0.638 0.931 0.934
100 0.824 0.920 0.429 0.926 0.929
200 0 0.060 0.086 0.098 0.087 0.101
0.001 0.253 0.282 0.294 0.278 0.294
0.01 0.677 0.713 0.728 0.713 0.728
1 0.944 0.977 0.740 0.978 0.979
100 0.944 0.962 0.455 0.962 0.963
500 0 0.059 0.063 0.066 0.062 0.067
0.001 0.626 0.617 0.622 0.615 0.622
0.01 0.933 0.943 0.947 0.944 0.947
1 0.991 1.000 0.981 1.000 1.000
100 0.992 0.984 0.491 0.984 0.984

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

71

Table 3.8

Actual Critical Values of KPSS and Leyboume-McCabe Tests with MA (1) Errors
(DGP: y. = 8¢+98t-1, 0=0.5)

5% significance level

 

 

KPSS LM94 LM99 LMM1 LMM2

T [=14 p= [4 p= [4 p= [4 p= I4
100 0.4721 0.7883 1.1373 0.9951 1.6230
200 0.4896 0.5884 0.6457 0.6024 0.6626
500 0.4908 0.5112 0.5198 0.5075 0.5225

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

72

Table 3.9

Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with MA (1) Errors
(DGP: y,= 8(+98t-1, 0=0.5)

5% significance level

 

 

 

 

KPSS LM94 LM99 LMMl LMM2
T )1, [=14 p= [4 p= [4 p= [4 p= [4
100 0.001 0.103 0.073 0.062 0.058 0.038
0.01 0.386 0.305 0.269 0.269 0.207
1 0.805 0.880 0.608 0.897 0.886
100 0.817 0.878 0.401 0.897 0.886
200 0.001 0.238 0.212 0.202 0.203 0.196
0.01 0.663 0.659 0.657 0.656 0.652
1 0.937 0.974 0.737 0.976 0.976
100 0.938 0.956 0.448 0.956 0.957
500 0.001 0.604 0.590 0.592 0.590 0.591
0.01 0.921 0.936 0.940 0.938 0.939
1 0.989 1.000 0.981 1.000 1.000
100 0.990 0.983 0.490 0.983 0.983

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

73

Table 3.10

Size and Power of KPSS and Leyboume-McCabe Tests with ARMA (1,1) Errors
(DGP: yt= pyM+ st+08,-1, p=1/3, 0=1/2)

5% significance level

 

 

 

 

KPSS LM94 LM99 LMMl LMM2
T 7. =14 p= [4 p= 14 p= 14 p= [4
100 0 0.073 0.082 0.089 0.087 0.092
0.001 0.138 0.164 0.174 0.168 0.178
0.01 0.425 0.483 0.484 0.495 0.511
1 0.818 0.949 0.555 0.956 0.960
100 0.830 0.938 0.372 0.943 0.945
200 0 0.081 0.068 0.072 0.070 0.072
0.001 0.300 0.280 0.289 0.282 0.289
0.01 0.706 0.727 0.733 0.730 0.738
1 0.942 0.989 0.704 0.990 0.991
100 0.946 0.966 0.396 0.967 0.968
500 0 0.079 0.047 0.048 0.047 0.047
0.001 0.655 0.609 0.615 0.610 0.615
0.01 0.937 0.940 0.944 0.941 0.944
1 0.992 1.000 0.961 1.000 1.000
100 0.992 0.988 0.393 0.988 0.988

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

74

Table 3.11

Actual Critical Values of KPSS and Leyboume-McCabe Tests

 

 

with ARMA (1,1) Errors
(DGP: y,= pyt-1+ ect-08H, p=1/3, 0=1/2)
5% significance level
KPSS LM94 LM99 LMM1 LMMZ
T [=14 p= [4 p= [4 p= [4 p= [4
100 0.5357 0.6233 0.7115 0.6713 0.7407
200 0.5530 0.5294 0.5501 0.5377 0.5505
500 0.5484 0.4483 0.4534 0.4516 0.4528

 

Simulation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

75

Table 3.12

Size-Adj usted Power of KPSS and Leyboume-McCabe Tests
with ARMA (1,1) Errors
(DGP: y,= pyM+ et+08t-1, p=1/3, 0=1/2)

 

 

 

 

5% significance level
KPSS LM94 LM99 LMM1 LMM2
T A, =14 p= I4 p= [4 p= I4 p= 14
100 0.001 0.094 0.119 0.117 0.117 0.116
0.01 0.374 0.405 0.395 0.410 0.415
1 0.768 0.933 0.547 0.947 0.951
100 0.785 0.920 0.362 0.932 0.933
200 0.001 0.239 0.245 0.245 0.243 0.245
0.01 0.650 0.699 0.703 0.701 0.708
1 0.916 0.987 0.703 0.989 0.989
100 0.924 0.963 0.393 0.964 0.965
500 0.001 0.603 0.619 0.622 0.618 0.622
0.01 0.919 0.942 0.944 0.943 0.945
1 0.986 1.000 0.961 1.000 1.000
100 0.986 0.988 0.393 0.988 0.988

 

Sirmrlation results based on 20,000 replications for the KPSS tests, 10,000 replications for the LM tests.

76

Chapter 4

Performance of the KPSS and Leyboume-McCabe Tests

with Model Selection Rules

1. Introduction

In this chapter we consider the performance of the KPSS and Leyboume-McCabe
tests when the number of lags is determined by a model selection rule. In the previous
chapters, we either assumed that the number of lags is finite and known a priori (Chapter
2), or we let the number of lags be a function of the sample size (Chapter 3). In this
chapter we assume that the true number of lags is unknown, but we have a finite upper
bound for it. The number of lags to be used is then the outcome of a general to specific
(G-S) testing procedure.

In the next section, we consider the Leyboume-McCabe tests, for which the
“number of lags” is the order “p” of the autoregressive model for Ay,. Leyboume and
McCabe (1999) suggested a model selection procedure that is based on a G-S sequential
testing of the AR coefficients. Their approach is analogous to those of Hall (1994) and
Ng and Perron (1995), which also used a G-S testing procedure to select the AR lag order
in the augmented Dickey-Fuller regression. The LM99 procedure is “consistent” in the
sense that, as T—)oo, the probability approaches one that the number of lags chosen is at
least as large as the true number. However, the probability of overfitting does not go to

zero. We suggest the possibility of making the critical value for the pretest depend on the

77

sample size, so that the model selection procedure is consistent in the stronger sense that
it picks the true number of lags with probability one asymptotically.

In the following section, we propose a model (lag) selection procedure for the
KPSS test. This is based on a 6-8 sequential testing of the correlations between first-
differenced residuals. Finally, we provide some Monte Carlo evidence on the finite-
sample properties of the KPSS and Leyboume-McCabe tests when these model selection
procedures are used to pick the number of lags. We do this for a variety of different

DGP’s, similar to those considered in Chapter 2 and 3.
2. A consistent model selection rule for the Leyboume-McCabe tests

We first discuss the model selection rule suggested by Leyboume and McCabe

(1999). To do so, we consider the ARIMA(p,l,1) model

Ayr = fl+i¢iAyt-i +91 _6C1—l (1)
i=1
as previously given in equation (26) of Chapter 1. We assume that this model holds for
some “true value” of p, say pa, with ¢p. ¢ 0. While p0 is unknown, there is a known

finite upper bound p"m (pm 2 p0).

Selecting the order of the AR component is done sequentially, using a general-to-
specific (G-S) strategy. We start by estimating the model (1) with p= pm, and testing the
null hypothesis that 90,, = 0. The test statistic for this pretest is defined as
Z (p) = T ” zépé -) N(0,l), where 03p and 61 are the quasi-ML estimates from the

ARIMA(pm,1,l) model. If we reject the null, we pick pm as the number of lags for the

78

Leyboume-McCabe tests. If we do not reject the null, we reduce p by one and repeat the

test until we can reject the null. Thus the number of lags used will be the largest value of

p such that we can reject the null that 05’, = 0 . If the null is never rejected, we set p=0.

Leyboume and McCabe (1999, p.267) show that this pretest is “consistent” both
under the stationary null and unit root alternative. In addition, Leyboume and McCabe
show that the LM99 test is asymptotically not affected by the pretest. This implies that,
with the number of lags p chosen by their model selection rule, we can proceed in large
samples as if the chosen lag p were equal to the true order.

It is instructive to note what it means for the pretest to be “consistent”. For p=po,
Leyboume-McCabe note that Z(p) is 0,,(Tm) under both the null of stationarity and the
alternative of unit root. Thus asymptotically the probability of picking a model that is
false, in the sense that p<po, is zero asymptotically. However, there is a non-zero
probability of overfitting (picking p>po). For example, if po=l and pm=5, and if the
pretests are at the 10% a-level, the probability of picking p>l is (asymptotically) equal to
1-(0.9)“=0.344‘.

However, it is not hard to modify this model selection procedure so that it picks
p=p0 with probability one (asymptotically). We simply need to use critical values for the

pretest that depend on the sample size. If the critical value is C, as T—>oo we need to
require that C—>oo but C / JT —) 0. Since Z(p) is Op(1) under the hypothesis that p>p0 (¢p

=0), the requirement that C—>oo will ensure that the probability of rejecting the hypothesis

 

' This calculation is justified by Theorem 3 of Leyboume and McCabe (1999), which shows the asymptotic
independence of the statistics Z(p) for different p> po,

79

that Q, =0 will go to zero as T—>oo. However, since Z(p) is Op(T”2) under the hypothesis

that p=p0 (so 05,, $0), the requirement that C/J—f —> 0 will ensure that the probability of
rejecting the hypothesis that 15,, =0 will go to one as T->oo.

In our simulations we will consider critical values of the form C=kT“4 for some
k>0. These satisfy the conditions of the previous paragraph for consistent model
selection.

In the theory of statistical tests, the notion that the size of the test should approach
zero as sample size approaches infinity is implicit. Our approach of letting the critical
values grow with sample size accomplished this. However, to ensure a consistent model
selection rule it is much more convenient to specify the rate at which the critical value
changes with the sample size than it would be to specify the rate at which the size of the
test changes with sample size. We know the order in probability of the test statistic under
the alternative, and the only important consideration is how this compares to the rate of

growth of the critical value.

3. A consistent model selection rule for the KPSS test

In this section we propose a model selection rule for the KPSS test. For the KPSS

test we assume the model
y,=131+#1+“1 (2)
as previously given in equation (1) of Chapter 1. Here ,u, is a random walk and u, is a

short-memory process. We now make the parametric assumption that u, is MA(1) for

80

some “true value” of 1, say [0. While 10 is unknown, there is a known finite upper bound
[max (1max 2 [0).

Our model selection procedure is based on the autocorrelations of Ay,. Clearly

Ay, = ,6 + w, , where w, = v, + Au, , (3)
and where v, = A14. Now, if we suppose the error term u, follows MA(1) process, the

maximum number of non-zero autocorrelation of the series w, is (1+1). So we can do

model selection based on significance tests of the correlations of the w,. Once again we

adopt a 6—8 strategy. Let 7]. be the jth autocovariance of w,. So, we start with [=1m, and

correspondingly we test the hypothesis that 71+: = 0. If we reject this hypothesis, we pick

[max as the number of lags. If we do not reject the null, we reduce [ by one and repeat the
procedure until we can reject the null. Thus the chosen value of I is one less than the
order of the largest significant autocorrelation of Ay,. If no autocorrelations of order j22
are significant, we pick [=0.

To carry out the pretest of the hypothesis that y, = 0 , for any value of 1: 2.2, we

first define the residuals 13/, = Ay, — ,0 = Ay, —X)7 , where Z; is an estimate of [3 that is

consistent under the null 03 = 0 or the alternative of > 0. Then we use the simple test

of Barttlet (1946):
T‘” ‘, —) N(0,l), (4A)
7' T 2
where p. = 2w.-. /2 . . (48)

Now we consider the consistency of our model selection rule. Consider first the

case of a fixed critical value (equivalently, fixed a-level). If 10 is the true lag length, so

81

the 7,0,1: 0, we will reject the hypothesis that 7,0,1: 0 with probability one,

asymptotically. Therefore, our rule is “consistent” in the sense that asymptotically it will
choose a value of 1 at least as large as [0. However, even asymptotically there is a positive
probability of overfitting (picking [> 10).

As in the previous section, we can also consider critical values that depend on the
sample size. If the critical value (C) satisfies C—)90, C/JT —+ 0 as T-—>oo, our selection
rule will be consistent in the strong sense that it will pick [=10 with a probability that goes
to one asymptotically. An example of a possible choice is C=kT”4 for some k>0. See

Appendix II for further details.

4. Simulations

In this section, we provide some Monte Carlo evidence on the size and power of
the KPSS and Leyboume-McCabe tests with model selection rules. Simulations were

performed using GAUSS 3.2.25 and the Maxlik optimization procedure. The DGP is
equation (1) of Chapter 1, with B=0. Thus y, = ,u, +u, , p, = ,u,_l + v,, where the u, are
iid N(0,a: ), the v, are iid N(0,0'3 ), and u and v are independent. The data contain no

deterministic trend and we consider only the tests that allow for level but not trend (e.g.,
KPSS 77,1 but not ii, , and similarly for the Leyboume-McCabe tests). As described in
previous chapters, white noise errors are used for a fair comparison of the KPSS and
Leyboume-McCabe tests. We will also consider MA, AR and ARMA errors. Again the
primary point here will be to see how the various tests perform when they are based on an

incorrectly specified model.

82

For our model selection procedures, we will use “fixed” critical values for our
pretest, with critical values of i165, corresponding to a nominal significance level of

10%. We will also consider “data dependent” critical values of the form C=(T/100)m,

which corresponds to C=kT"4 with k=100‘“ 4=1 / M =0.3162.

1) The KPSS test with iid errors - Fixed critical values

We first consider the KPSS test in the presence of white noise errors. We first
report the fi'equency distribution of lags chosen by model selection rule, under the null
hypothesis and the unit root alternative respectively. The results are from 10,000
replications. We use the 10% significance level (i.e., critical value=l.65) for the pretests
and the upper bound of lags 1max is set to three.

Tables 4.1 and 4.2 give the simulation results. The model selection rule works

well for large values of ,1 = of / of and T. For example, for A.=l0,000, and T=500, the

frequency of lag selection is (0.7341, 0.0804, 0.0930, 0.0925) for [=0, l, 2, 3,
respectively. This agrees quite closely with the frequencies (0.729, 0.081, 0.090, 0.10)
predicted by asymptotic theory. However, the model selection rule does not work. well
under the null hypothesis (i=0) or generally for small values of 71 (say M1). The
frequency of choosing the upper bound lag (1m=3) is greater than 0.1, and it shows no
sign of approaching 0.1 as T—->oo. We do not understand this result.

Table 4.3 gives the size of the KPSS test. We use various values of the upper
bound 1m, namely 3, 5, and 10. There are size distortions for all values of 1m, and as
expected, the size distortions are greater for larger 1m. However, these size distortions

disappear quite rapidly as we increase the sample size (T) for all values of [my

83

 

Table 4.4 gives the power of the KPSS test. Power increases with T and 3., but
decreases with [m. This is also as expected. When we compare these results to the power
of the KPSS test with the true number of lags ([=0), as in Table 2.2 of Chapter 2, the
power of the test with the model selection rule is clearly less. For example, for T=200 and
A=100, the power of the test is l for [=0, 0.984 for 1m=3, 0.933 for 1m=5, and 0.851 for
1m=10. Clearly this power loss reflects the positive probability of overspecifying 1. This
is the cost of using a fixed critical value and it motivates our consideration of data-
dependent critical values for the pretest.

We now proceed to consider size-adjusted power. Table 4.6 provides the size-
adjusted power of the KPSS test based on the actual critical value in Table 4.5. The
results are quite similar to those in Table 4.4. Power increases with T and 7., and
decreases with [map The only substantial difference is that size-adjusted power (Table 4.6)

is less than power (Table 4.4) for small values of T.

2) The KPSS test with iid errors -Data dependent critical values

Now we consider the KPSS test when the model selection rule is performed with
the “data dependent” critical values that increase with the sample size T. The critical
values are C=(T/100)m. We note that these critical values are of the form C=kT'/4, and
our choice of k is essentially arbitrary. It yields a critical value of 1.65 (as used in the
previous section) for T=741 (approximately). For small values of T, therefore, we will
have smaller critical values than 1.65, so we will choose larger values of 1 than in the
previous section. Conversely, for T greater than 741, we will choose smaller values of 1.

As T->oo, we will choose 1% (the true value) with a probability that approaches one.

84

 

Our simulation results are given in Tables 4.7-4.12, which have the same format
as the previous tables for the case of fixed critical values. We first consider the frequency
distribution of lags chosen with 1m=3. These results are given in Table 4.7 and 4.8.
Under both the null of stationarity and the unit root alternative, the probability of
choosing lags which are greater than the true lag converges to zero and the probability of
choosing the true lag ([=0) converges to 1. That is, the results are as expected given the
consistency of the model choice rule.

Table 4.9 gives the size of the test. There are substantial size distortions for small
sample sizes and large 1m. However, the size distortions disappear fairly rapidly as T
increases. We can note that the results in Table 4.9 are for T5500. For T in this range, the
pretest critical values are less than 1.65, and correspondingly the size distortions in Table
4.9 are larger than in Table 4.3 (where the critical value was fixed at 1.65, for a nominal
10% a-level). However, the size distortions in Table 4.9 are not very severe for T2200
and [m not unreasonably large. For T larger than 741 , the data-dependent critical values
will be greater than 1.65 and we presume that the size distortions of the stationarity test
will be smaller than before.

Now we turn to the power of the test. This is given in Table 4.10. The power
increases with T and A and decreases with 1",”, as expected. For T, in the range
considered here (T5500), power is lower in Table 4.10 than in Table 4.4 because now the
pretest critical values are smaller and we pick larger values of I. This comparison would
reverse for larger T.

Size-adjusted power is given in Table 4.12, using the actual critical values from

Table 4.11. We will not discuss these results separately.

85

3) The Leyboume-McCabe tests with iid errors — Fixed critical values

Now we provide simulation results on the size and power of the various
Leyboume-McCabe tests in presence of iid errors. The DGP is same as in the KPSS case
and we first consider the model selection rule with a fixed critical value of 1.65,
corresponding to the 10% significance level. The upper bound pmax is set to three and the
simulation results are based on 10,000 replications.

Tables 4.13-4.15 give the simulation results. The size and power of the tests are in
Table 4.13. The tests have moderate size distortions, and these size distortions do not
decrease as rapidly as for the KPSS test in Table 4.3. The power of the tests increases
with T and generally with 1. (except LM99). As in the previous chapters, the various
Leyboume-McCabe tests are all more or less equally powerful (again except LM99).
LMMZ is a little more powerful than LM94, but the difference is small.

We can compare the power of the Leyboume-McCabe tests with the model
selection rule to the power with p=0, as in Table 2.5 of Chapter 2. There is a
considerable power loss for larger values of 71.. For example, for T=100 and i=1, for
LM94 and LMM2 we have power of 0.988 and l with p=0, but the power of the tests
with model selection is only 0.917 and 0.925. Clearly this power loss is due to the fact
that the model selection rule overspecifies p with positive probability; this is true even for
large T.

We can also see in Table 4.13 that, for fixed T, the power of the LM tests with
model selection decreases as we move to the largest value of A. (2=100). This is a
reflection of the hear-cancellation” problem discussed in Chapter 1. This problem does

not occur for the KPSS test.

86

 

Table 4.15 gives the size-adjusted power of the Leyboume-McCabe tests, based
on the actual critical values given in Table 4.14. The results are fairly similar to those in
Table 4.13.

We can compare the size-adjusted power of the Leyboume-McCabe tests with
pm=3 (Table 4.15) with the size-adjusted power of the KPSS test with 1m=3 (Table
4.12). They are not too different. The results favor the LM tests over the KPSS test,

except for large 7., when the KPSS test is preferred.

4) The Leyboume-McCabe tests with iid errors - Data dependent critical values

Here we perform simulations on the size and power of the Leyboume-McCabe
tests when the pretest is performed with the data-dependent critical values we discussed
in previous sections. That is, now the critical value equals (T/100)”4. The DGP is same as
before, and the upper bound pm is set to three. Our simulation results are based on
10,000 replications. The main interest of this simulation is to see if there is any difference
in the performance of the Leyboume-McCabe tests according to the choice of critical
value for pretest.

Tables 4.16-4.18 give our simulation results. The tests have moderate size
distortions, but these decrease fairly rapidly as T increases. Power in Table 4.16 is rather
similar to power in Table 4.13 (with fixed critical values) and this is also true of size-
adjusted power (Table 4.18 versus Table 4.15). Larger sample sizes than we considered
would presumably be necessary to find power gains fi'om the use of data-dependent

critical values.

87

Power or size-adjusted power for the Leyboume-McCabe tests is not too different
from power or size-adjusted power for the KPSS test with a data-dependent critical value
for the pretest and with [mx=3. Once again the results seem to favor the Leyboume-

McCabe tests except for large values of A.

5) The KPSS and Leyboume-McCabe tests with AR(1) errors

Now we perform simulations with AR(1) errors of the form: ut=pum+et , where e;
is normal white noise. We set the coefficient value p to be 1/3 for the same reason we
discussed in previous chapters. We consider the KPSS test with 1m=3 and the LM test
with pm=3 and use fixed critical values (1.65, for a 10% nominal significance level) for
the pretest.

Our simulation results are provided in Tables 4.19-4.21. Table 4.19 gives the size and
power of the various tests, for various T and 7.. The KPSS test has large size distortions
and these do not disappear for T=500 (size=0.080). This should be expected since its long
run variance calculation does not take account correlations of order greater than three.
The Leyboume-McCabe tests have smaller size distortions and these disappear quickly as
T grows. The power of all of the tests increases with T and 7L. The power of the KPSS test
compares favorably to the power of the Leyboume-McCabe tests, but this is only due to
the size distortion. From Table 4.21, the size-adjusted power of the KPSS test is lower
than that of the Leyboume-McCabe tests.

For the Leyboume-McCabe tests, it is interesting to compare the present results to
the earlier results with iid errors. Comparing Table 4.13 (iid errors) to Table 4.19 (AR(1)

errors), we see that the size distortions are actually smaller in the AR(1) case. Size-

88

adjusted power (Table 4.21 versus Table 4.15) is similar. We can also compare the
present results to the results in Chapter 2 for the AR(1) DGP and with p set equal to one
(the true value). See Table 2.8 of Chapter 2 for these results. In terms of size or power,
there seems to be little cost to model selection in the AR(1) case, in the sense that the
results for the Leyboume-McCabe tests are not very different in Table 4.19 than in Table
2.8 of Chapter 2. It seems that the Leyboume-McCabe tests with model selection perform

quite well, given our AR(1) DGP.

6) The KPSS test and Leyboume-McCabe tests with MA(1) errors

Here we perform simulations with MA(1) errors of the form: y,=st+08t-1, where at
is normal white noise. We pick 0=0.5 for the same reason we discussed in previous
chapters. We consider the KPSS test with 1m=3 and the LM test with pm=3 as in the
previous section. We use fixed critical values (1.65, for a 10% nominal significance
level) for the pretest.

Our simulation results are given in Tables 4.22-4.24. Now the KPSS test has
correct size, whereas the Leyboume-McCabe tests suffer from size distortions. This is as
expected. However, comparing Table 4.19 and 4.22, the size distortions of the
Leyboume-McCabe tests when the DGP is MA(1) are less serious than the size distortion
of the KPSS test when the DGP is AR(1), and they go away more quickly as T increases.

The power of the Leyboume-McCabe test is comparable to that of the KPSS test,
but this is largely due to the size distortion. The KPSS test is generally superior in terms
of size-adjusted power (Table 4.24), especially when T is small. For T=500, there is not

much difference, except when 7. is large; then KPSS is again better.

89

 

It is interesting to compare the present results for the KPSS test to the earlier
results with iid errors. Comparing Table 4.3 and 4.4 (iid errors) to Table 4.22 (MA(1)
errors), there are no substantial size distortions in either case, but power is less in the
MA(1) case. We can also compare the present results to the results in Chapter 2 for the
MA(1) case and with I set equal to one. See Table 2.11 of Chapter 2 for these results.
Once again, in terms of size or power, there seems to be little cost to model selection. The

KPSS test with model selection seems to work quite well, given our MA(1) DGP.

7) The KPSS test and Leyboume-McCabe tests with ARMA(1,1) errors

Here we perform simulations using ARMA(1,1) errors of the form:

y, = pyH +5, +9£H , where p=1/3 and 0=l/2. As before, we choose these specific

values of the AR and MA parameters to equate the contribution of the AR and MA terms
to the “long-run variance” of the error series. Our results are given in Tables 4.25427,
which have same format as the previous tables for the AR and MA cases.

The Leyboume-McCabe tests have modest size distortions for T=100, but these
have essentially disappeared for T2200. The KPSS test has larger size distortions for
T=100, and they decrease more slowly as T increases. The Leyboume-McCabe tests also
typically have greater size-adjusted power than the KPSS test, except when 7. is large.
Overall, the Leyboume-McCabe test seem to do better than the KPSS test when the DGP

is ARMA(1,1).

90

 

5. Conclusions

In this chapter we considered the KPSS and Leyboume-McCabe tests when the
number of lags is determined by a model selection rule. LM99 proposed a model
selection rule to pick the AR order (p) in their test. We proposed a similar model

selection rule to pick the MA order (1) for the KPSS test. This rule is based on testing the

significance of correlations of the first differenced series (Ay, ). We also proposed

consistent model selection rules for the Leyboume-McCabe and KPSS tests. To obtain
consistency, we let the critical values for the pretests go to infinity with T, but not too fast
(e.g., critical value=kTm).

We proposed a consistent model selection rule that can be applied to the KPSS

test. The model selection procedure is based on tests of correlations of residuals from the

regression of the first differenced series Ay, on an intercept. For consistency of the

model selection rule, we need a critical value that grows with sample size T, but not too
quickly (i.e., cv=(T/100)”4). We also discussed the distribution theory of the KPSS test
with a model selection rule. The KPSS test with a consistent model selection rule is
Op(T), not Op(T/[), under the alternative.

Finally, we investigated the size and power characteristics of the tests via
simulations. In these simulations, our data generating processes included white noise,
AR(1) errors, MA(1) errors and ARMA(1,1) errors. Our conclusions are as follows.

1. Our consistent model selection rules work, in the sense that the probability of

choosing the correct number of lags goes to one as T increases.

91

 

2. There is a power loss from model selection, as compared to knowing the
correct number of lags, in finite samples. This loss seems to be less for the Leyboume-
McCabe tests than for the KPSS test.

3. The LM99 test is still not recommended, due to its very poor power when 7. is
large. The other Leyboume-McCabe tests are quite similar to each other.

4. With autocorrelated errors, the KPSS test with model selection does not do well
if the DGP is AR(p) and the LM tests with model selection do not do well if the DGP is
MA(1). When we compare different tests for different types of DGP, the Leyboume-
McCabe tests can be argued to be more robust in finite samples than the KPSS test, in the
sense that the size distortions of the KPSS test with AR(1) errors are greater than those of
the Leyboume-McCabe tests with MA(1) errors. Similarly the LM tests do better than the

KPSS tests under ARMA( 1,1) errors.

92

 

Appendix II: Asymptotic Effects of Model Selection Rules on the KPSS Test

Let [0 be the true value of [ (the order of the MA process for u, ), and [m 2 10 be
the specified upper bound. We called a model selection rule “consistent” if it picked an
[e [10, [max] with probability one, asymptotically. This is the same notion of “consistency”
as in LM99. F

A “consistent” model selection rule does not affect the distribution of the KPSS

test under the null. So long as the unweighted long run variance estimate is used,

plimSz(m)=-'O'2 for all me[[.,, 1max ], and therefore

 

axlfi.(m)—fi,.(l.)l—>0 (A1)

where the max is over me [[0, 1max ].
However, unless 1m=[o, a “consistent” model selection rule does affect the
distribution of the KPSS test under the unit root alternative. For example, consider the

simplest case in which [0=1 and m= 1m=2. Then

T T
T—Isz (1) = r4 203+ 27 4212,11“ —> 303 I Z(syds. (A2)
(=1 .

—+ 502 [1(5de (A3)

93

T“[fi..(2) - 27,. (1)] = Tris} (7452(2)? .. 7‘2 S; {T432 (1)}"

1:1
1 1 l a 2 l
—>[—-—]af J1!_W_’(s)ds) da/af J-Ej(s)2ds
5 3 0 0 0

2 l a 2 I 2
= [_ Em ojmsws] da/ ojms) ds :4 0 (A4)

More generally, let 1 represent the true lag and m=1+q represent one of the lags possibly

chosen (i.e., chosen with positive probability asymptotically), with ISqSUM- 0). Then
7 1 r
T"82 (1) = T“2 212,2 + 22[T’2 Za,a,_,]
1:1 3:1 t=s+l

T T T
= r2212; + {742.251, +....+ r2 2,2,]
1 (=2

1: r=l+l

1
—-) (1+ 2003 IZ(S)2 dS , (A5)
0

T"sz(m) = Tail}? +2i[T-2 £81814]
(=1 t=s+l

s=l

T T T
_ -2::A2 —2::AA —22:AA
(=1 [=2

t=m+l

_.[1+2<I+q)laf 1110241. (A6)

Therefore,

94

 

{TH/ASN.9] da
2‘1 0 0

T‘ 14.0")- 77,101» ‘ (1+ 200 + 20 + q»

. ¢ 0. (A7)
[msyds
0

The above results imply that, under the unit root alternative, the KPSS test with a

“consistent” model selection rule with a positive probability of overspecification will lose

 

1
power asymptotically. Obviously, this would not be the case if the probability of
overspecification (as well as underspecification) is asymptotically zero. That is, the ,
asymptotic distribution will be unaffected if the model selection rule is consistent in the
sense that the probability of choosing [=1o equals one, asymptotically. This is the point of 5'

our “data-dependent” critical values, say C, that satisfy the requirements that C—)oo,

C/JT—90, as T—-)oo.

95

Table 4.1

Frequency of Lag Selection: KPSS Test Under the Null

(DGP: iid errors, no time trend)

(Im=3, 10% significance level for pretest)

 

 

 

Lag (Selected by the Rule)
T 0 1 2 3
50 0.6772 0.0883 0.0929 0.1416
100 0.6536 0.0862 0.0977 0.1625
200 0.6374 0.0898 0.1015 0.1713
500 0.6341 0.0896 0.1017 0.1746
1,000 0.6349 0.0851 0.1031 0.1769

 

Simulation results based on 10,000 replications.

96

Table 4.2

Frequency of Lag Selection: KPSS Test Under the Alternative

(DGP: iid errors, no time trend)
([m=3, 10% significance level for pretest)

 

 

 

 

 

 

 

 

 

Lag (Selected by the Rule)

4 T 0 1 2 3
0.0001 50 0.6628 0.0887 0.0942 0.1543
100 0.6490 0.0917 0.1019 0.1574
200 0.6258 0.0962 0.1060 0.1720
500 0.6336 0.0894 0.1032 0.1738
0.001 50 0.6647 0.0915 0.0916 0.1522
100 0.6528 0.0912 0.0963 0.1597
200 0.6373 0.0937 0.1022 0.1668
500 0.6341 0.0962 0.1019 0.1678
0.01 50 0.6657 0.0905 0.0915 0.1523
100 0.6457 0.0917 0.0983 0.1643
200 0.6393 0.0908 0.0985 0.1714
500 0.6326 0.0982 0.1025 0.1667
0.1 50 0.6776 0.0895 0.0930 0.1399
100 0.6554 0.0881 0.0974 0.1591
200 0.6371 0.0955 0.1013 0.1661
500 0.6525 0.0891 0.0969 0.1615
1 50 0.7231 0.0821 0.0838 0.1110
100 0.7108 0.0825 0.0836 0.1231
200 0.6991 0.0825 0.0890 0.1294
500 0.6889 0.0854 0.0927 0.1330
100 50 0.7662 0.0742 0.0731 0.0865
100 0.7468 0.0718 0.0863 0.0951
200 0.7360 0.0812 0.0891 0.0937
500 0.7291 0.0799 0.0928 0.0982
10000 50 0.7652 0.0729 0.0729 0.0854
100 0.7488 0.0722 0.0884 0.0906
200 0.7383 0.0807 0.0835 0.0975
500 0.7341 0.0804 0.0930 0.0925

 

Simulation results based on 10,000 replications.

97

 

Table 4.3

Size of KPSS Test with Model Selection Rule
(DGP: iid errors, no time trend)

10% significance level for pretest, 5% significance level for stationarity test

 

 

r 1AM=3 1...,=5 [m=10
50 0.065 0.084 0.097
100 0.053 0.059 0.076
200 0.049 0.051 0.059
500 0.051 0.049 0.051

 

Simulation results based on 20,000 replications.

98

Table 4.4

Power of KPSS Test with Model Selection Rule
(DGP: iid errors, no time trend)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

 

 

T 7. 1m,=3 [MES [,m=10
50 0.0001 0.064 0.076 0.100
0.001 0.092 0.093 0.109
0.01 0.278 0.258 0.210
0.1 0.652 0.575 0.385
1 0.843 0.748 0.514
100 0.886 0.809 0.586
10000 0.886 0.801 0.590
100 0.0001 0.062 0.073 0.085
0.001 0.159 0.160 0.151
0.01 0.561 0.526 0.414
0.1 0.859 0.800 0.617
1 0.930 0.872 0.690
100 0.949 0.900 0.734
10000 0.952 0.898 0.738
200 0.0001 0.097 0.097 0.102
0.001 0.389 0.385 0.346
0.01 0.817 0.785 0.684
0.1 0.963 0.921 0.785
1 0.979 0.944 0.831
100 0.984 0.955 0.851
10000 0.984 0.954 0.847
500 0.0001 0.311 0.301 0.292
0.001 0.780 0.763 0.730
0.01 0.980 0.967 0.913
0.1 0.996 0.988 0.939
1 0.998 0.992 0.947
100 0.998 0.993 0.953
10000 0.999 0.992 0.95 8

 

Simulation results based on 20,000 replications.

99

 

Table 4.5

Actual Critical value of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend)

10% significance level for pretest, 5% significance level for stationarity test

 

 

T [max=3 [MES [max=10
50 0.5089 0.5426 0.6546
100 0.4774 0.4890 0.5502
200 0.4760 0.4643 0.4846
500 0.4556 0.4622 0.4710

 

Simulation results based on 10,000 replications.

100

 

Size-Adjusted Power of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

 

 

T 7. [max=3 [max=5 Imm=10
50 0.0001 0.050 0.050 0.049
0.001 0.066 0.063 0.051
0.01 0.212 0.173 0.093
0.1 0.542 0.422 0.226
1 0.673 0.542 0.330
100 0.701 0.587 0.374
10000 0.695 0.596 0.392
100 0.0001 0.058 0.057 0.054
0.001 0.152 0.147 0.099
0.01 0.520 0.491 0.315
0.1 0.806 0.744 0.478
1 0.870 0.820 0.559
100 0.889 0.836 0.616
10000 0.889 0.842 0.614
200 0.0001 0.094 0.096 0.088
0.001 0.373 0.378 0.331
0.01 0.789 0.764 0.667
0.1 0.942 0.91 1 0.777
1 0.967 0.931 0.812
100 0.971 0.941 0.835
10000 0.970 0.944 0.835
500 0.0001 0.306 0.300 0.283
0.001 0.774 0.761 0.715
0.01 0.976 0.965 0.906
0.1 0.996 0.988 0.937
1 0.997 0.990 0.942
100 0.997 0.992 0.953
10000 0.998 0.991 0.954

 

Sirmrlation results based on 20,000 replications.

 

Table 4.7

Frequency of Lag Selection: KPSS Test Under the Null

(DGP: iid errors, no time trend)
(1...,=3, c.v.=(r/100)"‘)

 

Lag (Selected by the Rule)

 

 

T 0 l 2 3
50 0.2167 0.1173 0.2013 0.4647
100 0.2879 0.1262 0.1836 0.4023
200 0.4007 0.1286 0.1541 0.3166
500 0.5622 0.1023 0.1185 0.2170
1,000 0.6853 0.0817 0.0893 0.1437
2,000 0.8085 0.0517 0.0544 0.0854
5,000 0.9275 0.0235 0.0202 0.0288
10,000 0.9763 0.0061 0.0077 0.0099

 

Simulation results based on 10,000 replications

102

 

Table 4.8

Frequency of Lag Selection: KPSS Test Under the Alternative
(DGP: iid errors, no time trend)

(1m=3, c.v.=(T/100)"‘)

 

 

 

 

 

 

Lag (Selected by the Rule)

7. T 0 1 2 3
0.0001 50 0.2111 0.1251 0.1978 0.4660
100 0.2880 0.1270 0.1849 0.4001
200 0.3945 0.1 198 0.1585 0.3272
500 0.5613 0.0993 0.1192 0.2202
1,000 0.6936 0.0777 0.0857 0.1430
2,000 0.8152 0.0500 0.0525 0.0823
5,000 0.9274 0.0206 0.0217 0.0303
10,000 0.9744 0.0077 0.0099 0.0080
0.001 50 0.2183 0.1 178 0.2020 0.4619
100 0.2683 0.1264 0.1892 0.3981
200 0.3927 0.1223 0.1660 0.3190
500 0.5617 0.0961 0.1263 0.2159
1,000 0.6915 0.0802 0.0835 0.1448
2,000 0.8071 0.0528 0.0543 0.0859
5,000 0.9284 0.0206 0.0230 0.0280
10,000 0.9758 0.0073 0.0074 0.0095
0.01 50 0.2147 0.1221 0.1983 0.4649
100 0.2963 0.1250 0.1818 0.3969
200 0.3975 0.1210 0.1602 0.3213
500 0.5491 0.1017 0.1234 0.2258
1,000 0.6882 0.0830 0.0896 0.1392
2,000 0.8125 0.0508 0.0533 0.0834
5,000 0.9305 0.0200 0.0240 0.0255
10,000 0.9759 0.0073 0.0074 0.0094
0.1 50 0.2148 0.1217 0.2060 0.4575
100 0.3048 0.1282 0.1848 0.3822
200 0.4018 0.1248 0.1557 0.3177
500 0.5585 0.1078 0.1219 0.2118
1,000 0.7017 0.0738 0.0825 0.1420
2,000 0.8109 0.0526 0.0546 0.0819
5,000 0.9346 0.0203 0.0191 0.0260
10,000 0.9776 0.0059 0.0072 0.0093

 

103

 

Table 4.8 (Continued)

Frequency of Lag Selection: KPSS Test Under the Alternative

(DGP: iid errors, no time trend)
(1,..=3, c.v.=(r/100)"‘)

 

 

 

 

 

Lag Selected by the Rule

1 r 0 1 2 3
50 0.2347 0.1388 0.2037 0.4228
100 0.3176 0.1380 0.1911 0.3533
200 0.4319 0.1273 0.1599 0.2809
500 0.6118 0.0968 0.1160 0.1736
1,000 0.7384 0.0740 0.0798 0.1078
2,000 0.8630 0.0396 0.0425 0.0549
5,000 0.9573 0.0134 0.0148 0.0145
10,000 0.9866 0.0043 0.0045 0.0046
100 50 0.2322 0.1513 0.2305 0.3860
100 0.3255 0.1500 0.2154 0.3091
200 0.4490 0.1278 0.1848 0.2384
500 0.6541 0.0979 0.1151 0.1329
1,000 0.7908 0.0658 0.0681 0.0753
2,000 0.9047 0.0299 0.0329 0.0325
5,000 0.9762 0.0075 0.0073 0.0090
10,000 0.9957 0.0013 0.0020 0.0010
10000 50 0.2280 0.1507 0.2390 0.3823
100 0.3253 0.1484 0.2206 0.3057
200 0.4619 0.1328 0.1756 0.2297
500 0.6449 0.1013 0.1227 0.1331
1,000 0.7904 0.0651 0.0656 0.0789
2,000 0.9109 0.0302 0.0348 0.0331
5,000 0.9774 0.0079 0.0070 0.0077
10,000 0.9958 0.001 1 0.0016 0.0015

 

Simulation results based on 10,000 replications.

104

 

Table 4.9

Size of KPSS Test with Model Selection Rule
(DGP: iid errors, no time trend)

c.v.=(T/100)m for pretest, 5% significance level for stationarity test

 

 

T 1w=3 I,,.,,=5 lw=10
50 0.072 0.093 0.135
100 0.057 0.065 0.096
200 0.049 0.054 0.061
500 0.050 0.050 0.050

 

Simulation results based on 20,000 replications.

 

105

Table 4.10

Power of KPSS Test with Model Selection Rule
(DGP: iid errors, no time trend)
1/4

c.v.=(T/ 100) for pretest, 5% significance level for stationarity test

 

 

 

 

 

T 7t [max=3 [MES [,m=10
50 0.0001 0.074 0.092 0.138
0.001 0.096 0.108 0.135
0.01 0.240 0.188 0.122
0.1 0.535 0.352 0.080
1 0.652 0.431 0.065
100 0.684 0.448 0.066
10000 0.683 0.444 0.067
100 0.0001 0.065 0.074 0.098
0.001 0.163 0.155 0.133
0.01 0.517 0.467 0.286
0.1 0.784 0.684 0.397
1 0.852 0.728 0.431
100 0.863 0.739 0.441
10000 0.865 0.740 0.434
200 0.0001 0.097 0.094 0.099
0.001 0.381 0.374 0.324
0.01 0.799 0.748 0.620
0.1 0.939 0.873 0.706
1 0.956 0.891 0.719
100 0.962 0.900 0.733
10000 0.965 0.905 0.731
500 0.0001 0.305 0.307 0.292
0.001 0.772 0.760 0.714
0.01 0.978 0.962 0.899
0.1 0.995 0.984 0.929
1 0.997 0.988 0.937
100 0.998 0.989 0.942
10000 0.997 0.989 0.943

 

Simulation results based on 20,000 replications.

106

Table 4.11

Actual Critical Values of KPSS Test with Model Selection Rule
(DGP: iid errors, no time trend)

c.v.=(T/100)m for pretest, 5% significance level for stationarity test

 

KPSS with Lag Selection Rule

 

 

T [m=3 Imm=5 [max=10
50 0.5235 0.6543 0.9447
100 0.4869 0.5017 0.6075
200 0.4614 0.4819 0.5064
500 0.4591 0.4437 0.4705

 

Simulation results based on 10000 simulations.

107

Table 4.12

Size-Adjusted Power of KPSS Test with Model Selection Rule

(DGP: iid errors, no time trend)

c.v.=(T/100)"‘ for pretest, 5% significance level for stationarity test

 

 

 

 

 

T 7. [max=3 [,m=5 Lm=10
50 0.0001 0.054 0.044 0.048
0.001 0.064 0.047 0.044
0.01 0.191 0.068 0.030
0.1 0.471 0.138 0.017
1 0.607 0.184 0.020
100 0.639 0.199 0.018
10000 0.634 0.203 0.017
100 0.0001 0.055 0.059 0.053
0.001 0.148 0.128 0.061
0.01 0.511 0.438 0.110
0.1 0.779 0.651 0.157
1 0.840 0.705 0.183
100 0.849 0.718 0.190
10000 0.855 0.712 0.192
200 0.0001 0.098 0.087 0.082
0.001 0.384 0.354 0.289
0.01 0.801 0.735 0.588
0.1 0.937 0.861 0.678
1 0.959 0.885 0.696
100 0.966 0.895 0.707
10000 0.964 0.896 0.696
500 0.0001 0.311 0.317 0.287
0.001 0.780 0.772 0.708
0.01 0.977 0.966 0.898
0.1 0.995 0.986 0.924
1 0.997 0.990 0.936
100 0.998 0.991 0.940
10000 0.998 0.991 0.937

 

Simulation results based on 20,000 replications.

108

Size and Power of Leyboume-McCabe Test with Model Selection Rule (pmx=3)

Table 4.13

(DGP: iid errors, no time trend)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

 

T 7. LM94 LM99 LMM1 LMMZ
100 0 0.062 0.072 0.056 0.073
0.001 0.172 0.183 0.160 0.183
0.01 0.572 0.596 0.564 0.596
1 0.917 0.828 0.921 0.925
100 0.895 0.423 0.900 0.903
200 0 0.052 0.056 0.047 0.056
0.001 0.442 0.502 0.483 0.502
0.01 0.842 0.856 0.841 0.856
1 0.972 0.876 0.973 0.974
100 0.940 0.547 0.940 0.942
500 0 0.057 0.057 0.052 0.057
0.001 0.787 0.794 0.785 0.794
0.01 0.987 0.990 0.988 0.990
1 0.998 0.975 0.998 0.998
100 0.976 0.502 0.975 0.975

 

Simulation results based on 10,000 replications.

109

 

Table 4.14

Actual Critical Values of Leyboume-McCabe Test with Model Selection Rule (pmax=3)
(DGP: iid errors, no time trend)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

 

T LM94 LM99 LMM1 LMM2
100 0.5041 0.5417 0.4816 0.5437 T
200 0.4680 0.4824 0.4509 0.4824
500 0.4828 0.4888 0.4694 0.4888
Simulation results based on 10,000 replications. ..
b,

110

Size-Adjusted Power of Leyboume-McCabe Test with Model Selection Rule (pmax=3)

Table 4.15

(DGP: iid errors, no time trend)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

 

T 7. LM94 LM99 LMM1 LMM2

100 0.001 0.152 0.150 0.151 0.149
0.01 0.547 0.555 0.552 0.554

1 0.912 0.823 0.919 0.920

100 0.889 0.417 0.898 0.896

200 0.001 0.439 0.490 0.491 0.490
0.01 0.840 0.850 0.845 0.850

1 0.972 0.876 0.973 0.973

100 0.939 0.455 0.941 0.940

500 0.001 0.778 0.782 0.782 0.782
0.01 0.986 0.989 0.988 0.989

1 0.998 0.974 0.998 0.998

100 0.975 0.501 0.975 0.975

 

Simulation results based on 10,000 replications.

111

 

Size and Power of Leyboume-McCabe Test with Model Selection Rule (pmx=3)

Table 4.16

(DGP: iid errors, no time trend)

”4

c.v.=(T/ 100) for pretest, 5% significance level for stationarity test

 

 

 

 

T 7. LM94 LM99 LMM1 LMM2
100 0 0.071 0.087 0.067 0.088
0.001 0.184 0.205 0.177 0.205
0.01 0.572 0.600 0.569 0.600
1 0.909 0.745 0.915 0.920
100 0.885 0.394 0.890 0.893
200 0 0.061 0.068 0.055 0.068
0.001 0.385 0.399 0.377 0.399
0.01 0.836 0.851 0.836 0.851
1 0.969 0.852 0.969 0.970
100 0.935 0.451 0.936 0.938
500 0 0.048 0.050 0.046 0.050
0.001 0.786 0.793 0.782 0.793
0.01 0.986 0.989 0.987 0.989
1 0.998 0.968 0.998 0.998
100 0.977 0.494 0.977 0.977

 

Simulation results based on 10,000 replications.

112

 

Table 4.17

Actual Critical Values of Leyboume-McCabe Test with Model Selection Rule (pm=3)
(DGP: iid errors, no time trend)

c.v.=(T/100)"‘ for pretest, 5% significance level for stationarity test

 

 

T LM94 LM99 LMM1 LMMZ

100 0.5337 0.5932 0.5244 0.5951

200 0.4991 0.5225 0.4805 0.5225 1'3:
500 0.4569 0.4637 0.4465 0.4637

 

Simulation results based on 10,000 replications.

 

if

113

Size-Adjusted Power of Leyboume-McCabe Test with Model Selection Rule (pmax=3)

Table 4.18

(DGP: iid errors, no time trend)

c.v.=(T/100)m for pretest, 5% significance level for stationarity test

 

 

 

 

T 71 LM94 LM99 LMM1 LMMZ

100 0.001 0.150 0.151 0.151 0.150
0.01 0.534 0.541 0.538 0.539

1 0.900 0.737 0.910 0.911

100 0.875 0.385 0.884 0.883

200 0.001 0.364 0.364 0.368 0.364
0.01 0.822 0.832 0.829 0.832

1 0.968 0.850 0.970 0.969

100 0.932 0.447 0.935 0.933

500 0.001 0.790 0.793 0.791 0.793
0.01 0.986 0.989 0.987 0.989

1 0.998 0.968 0.998 0.998

100 0.977 0.494 0.977 0.977

 

Simulation results based on 10,000 replications.

114

 

10% significance level for pretest, 5% significance level for stationarity test

Table 4.19

(DGP: yt= py.-1+e., p=1/3)

Size and Power of KPSS and Leyboume-McCabe Tests with AR(1) Errors

 

 

 

 

 

KPSS LM94 LM99 LMM1 LMM2
T 7. [max=3 pmax=3 pw= 3 pmx= 3 pmx= 3
100 0 0.134 0.055 0.05 8 0.057 0.058
0.001 0.282 0.178 0.188 0.179 0.188
0.01 0.650 0.583 0.602 0.587 0.603
1 0.947 0.933 0.624 0.937 0.941
100 0.937 0.927 0.385 0.934 0.936
200 0 0.1 15 0.053 0.054 0.053 0.054
0.001 0.479 0.390 0.399 0.390 0.399
0.01 0.852 0.831 0.844 0.835 0.844
1 0.984 0.977 0.762 0.977 0.978
100 0.977 0.958 0.408 0.958 0.959
500 0 0.080 0.052 0.052 0.052 0.052
0.001 0.804 0.782 0.787 0.784 0.787
0.01 0.980 0.983 0.987 0.985 0.987
1 0.998 0.999 0.957 0.999 0.999
100 0.997 0.986 0.428 0.986 0.986

 

Simulation results based on 20,000 replications for KPSS test, 10,000 replications for Leybom’ne-McCabe

tests.

115

- .-

Table 4.20

Actual Critical Values of KPSS and Leyboume-McCabe Tests with AR(1) Errors
(DGP: y.= pyt-1+e., p=1/3)

10% significance level for pretest, 5% significance level for stationarity test

 

 

KPSS LM94 LM99 LMM1 LMM2

T [max=3 pmx=3 pmax= 3 pmx= 3 pmx= 3
100 0.7268 0.4855 0.5045 0.4934 0.5045
200 0.6628 0.4781 0.4843 0.4812 0.4837
500 0.5861 0.4679 0.4695 0.4690 0.4695

 

Simulation results based on 20,000 replications for KPSS test, 10,000 replications for Leyboume-McCabe
tests.

 

116

Table 4.21

Size-Adj usted Power of KPSS and Leyboume-McCabe test with AR(1) Errors
(DGP: yt= py.-1+st, p=1/3)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

 

KPSS LM94 LM99 LMM1 LMM2

T 7L [max=3 pmax=3 pmax= 3 pmax= 3 pmx= 3
100 0.001 0.148 0.164 0.168 0.164 0.168
0.01 0.503 0.571 0.584 0.570 0.584
1 0.903 0.930 0.622 0.935 0.939
100 0.875 0.925 0.382 0.932 0.934
200 0.001 0.359 0.379 0.385 0.380 0.385
0.01 0.776 0.825 0.838 0.829 0.838
1 0.964 0.976 0.761 0.977 0.978
100 0.950 0.957 0.407 0.957 0.958
500 0.001 0.746 0.780 0.785 0.781 0.785
0.01 0.966 0.983 0.986 0.984 0.986
1 0.996 0.999 0.957 0.999 0.999
100 0.994 0.986 0.428 0.986 0.986

 

Simulation results based on 20,000 replications for KPSS test, 10,000 replications for Leybomne-McCabe

tests.

117

 

Table 4.22

Size and Power of KPSS and Leyboume-McCabe Tests with MA(1) Errors
(DGP. Y! = €t+981-|, 9:0.5)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

 

KPSS LM94 LM99 LMM1 LMM2
T k Imm=3 pmx=3 pmax= 3 pmx= 3 pmx= 3
100 0 0.047 0.072 0.082 0.073 0.087
0.001 0.100 0.120 0.137 0.116 0.137
0.01 0.387 0.407 0.427 0.404 0.427
1 0.894 0.868 0.778 0.872 0.880
100 0.949 0.901 0.414 0.905 0.909
200 0 0.050 0.061 0.067 0.059 0.067
0.001 0.235 0.250 0.260 0.246 0.260
0.01 0.674 0.693 0.706 0.693 0.706
1 0.967 0.957 0.894 0.958 0.960
100 0.984 0.937 0.442 0.938 0.939
500 0 0.048 0.048 0.049 0.046 0.049
0.001 0.611 0.607 0.613 0.604 0.613
0.01 0.941 0.948 0.953 0.949 0.953
1 0.997 0.998 0.977 0.998 0.998
100 0.998 0.977 0.485 0.977 0.977

 

Simulation results based on 20,000 replications for KPSS test, 10,000 replications for Leyboume-McCabe

tests.

118

 

Table 4.23

Actual Critical Values of KPSS and Leyboume-McCabe Tests with MA(1) Errors
(DGP: yt = St'l'GSH, 0=0.5)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

KPSS LM94 LM99 LMMl LMMZ
T Imax=3 pmx=3 max= 3 w= 3 pmax= 3 P
100 0.4610 0.5585 0.6104 0.5661 0.6343
200 0.4522 0.5061 0.5227 0.4942 0.5239
500 0.4529 0.4541 0.4601 0.4496 0.4061
Simulation results based on 20,000 replications for KPSS test, 10,000 replications for Leyboume-McCabe
tests. 1

 

119

Table 4.24

Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with MA(1) Errors
(DGP. yt = 8t+98t.|, 9:0.5)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

 

 

KPSS LM94 LM99 LMM1 LMM2

T )1 Imax=3 pmx=3 pmax= 3 pmax= 3 pmax= 3
100 0.001 0.098 0.088 0.088 0.084 0.083
0.01 0.385 0.357 0.359 0.353 0.348
1 0.896 0.850 0.762 0.857 0.861
100 0.949 0.888 0.403 0.896 0.896
200 0.001 0.242 0.224 0.225 0.228 0.225
0.01 0.677 0.672 0.681 0.678 0.680
1 0.968 0.954 0.891 0.956 0.957
100 0.986 0.933 0.438 0.935 0.935
500 0.001 0.621 0.613 0.616 0.612 0.616
0.01 0.942 0.950 0.953 0.952 0.953
1 0.998 0.998 0.977 0.998 0.998
100 0.998 0.977 0.485 0.977 0.978

 

Simulation results based on 20,000 replications for KPSS test, 10,000 replications for Leyboume-McCabe

tests.

120

Table 4.25

Size and Power of KPSS and Leyboume-McCabe Tests with ARMA(1,1) Errors
(DGP: yt = pyt-1+et+081-1, p=1/3, 0=1/2)

10% significance level for pretest, 5% significance level for stationarity test

 

 

 

 

 

KPSS LM94 LM99 LMMl LMM2
T 7t Imm=3 pmax=3 pmx= 3 max= 3 pmx= 3
100 0 0.073 0.058 0.062 0.060 0.063
0.001 0.135 0.126 0.135 0.128 0.135
0.01 0.431 0.416 0.431 0.421 0.432
1 0.915 0.948 0.212 0.952 0.954
100 0.937 0.933 0.379 0.939 0.941
200 0 0.071 0.048 0.052 0.051 0.052
0.001 0.277 0.241 0.248 0.243 0.248
0.01 0.694 0.677 0.686 0.678 0.686
1 0.970 0.988 0.187 0.988 0.988
100 0.975 0.957 0.390 0.957 0.958
500 0 0.062 0.046 0.048 0.046 0.046
0.001 0.625 0.559 0.603 0.559 0.603
0.01 0.937 0.935 0.939 0.936 0.939
1 0.997 1.000 0.107 1.000 1.000
100 0.997 0.987 0.407 0.986 0.987

 

Simulation results based on 20,000 replications for KPSS test, 10,000 replications for Leyboume-McCabe

tests.

121

“3;...

Table 4.26

Actual Critical Values of KPSS and Leyboume-McCabe Tests with ARMA(1,1) Errors
(DGP: yt= pyt-1+81+981-1, p=1/3, 0=1/2)

10% significance level for pretest, 5% significance level for stationarity test

 

 

KPSS LM94 LM99 LMM1 LMM2
T [max=3 pmax=3 pm= 3 pmx= 3 pmx= 3
100 0.5283 0.4887 0.5102 0.5004 0.5106
200 0.5191 0.4592 0.4720 0.4673 0.4695
500 0.5042 0.4476 0.4559 0.4508 0.4502

 

Simulation results based on 20,000 replications for KPSS test, 10,000 replications for Leyboume-McCabe
tests.

 

122

Table 4.27

Size-Adjusted Power of KPSS and Leyboume-McCabe Tests with ARMA(1,1) Errors
(DGP: yt= pyt-1+81+081-1, p=1/3, 0=1/2)

10% significance level for pretest, 5% significance level for stationary test

 

 

 

 

KPSS LM94 LM99 LMM1 LMM2
T 7L Imm=3 pmx==3 pmx= 3 pmx= 3 pmx= 3
100 0.001 0.098 0.117 0.119 0.116 0.120 =
0.01 0.379 0.402 0.409 0.400 0.410 I”.
1 0.894 0.945 0.210 0.951 0.953
100 0.922 0.931 0.377 0.936 0.939
200 0.001 0.236 0.244 0.242 0.241 0.244 7‘
0.01 0.667 0.679 0.683 0.676 0.684 ”J
1 0.961 0.988 0.187 0.988 0.988
100 0.971 0.957 0.390 0.957 0.957
500 0.001 0.61 1 0.607 0.607 0.605 0.610
0.01 0.929 0.937 0.940 0.938 0.941
1 0.996 1.000 0.107 1.000 1.000
100 0.997 0.987 0.470 0.987 0.987

 

Simulation results based on 20,000 replications for KPSS test, 10,000 replications for Leyboume-McCabe

tests.

123

 

Chapter 5

Concluding Remarks

This thesis has considered three different types of stationarity test: Kwiatkowski,
Phillips, Schmidt, and Shin (1992), or KPSS; Leyboume and McCabe (1994), or LM94;
and Leyboume and McCabe (1999), or LM99. The tests are all similar, but they differ in
the way that they estimate the variance of the process and accommodate short-run
dynamics. The KPSS test estimates a long-run variance and lets the number of lags go to
infinity with sample size. The LM94 test whitens the data by fitting an ARMA(pJJ)
model, where the correct number of lags (p) is assumed known. The LM99 test also fits
an ARIMA(p,1,1) model, but it uses a different variance estimate, and it chooses p
through model selection.

One aim of this thesis is to separate the issue of “how is variance estimated” from
the issue of “how is the number of lag is chosen”. Thus, for each of the three tests, we
consider the possibilities that the correct number of lags is known, or that the number of
lags goes to infinity with sample size, or that a model selection procedure is used to pick
the number of lags.

The thesis does not contain any substantial theorems, but it contains some original
theoretical contributions. First, we show that the LM99 test has very poor power
properties when the alternative is close to a random walk (K=oo, or 0=0). In fact, the
asymptotic theory of LM99 excludes this case, because in the pure random walk case

power approaches 0.5, not 1, as T—)oo. The pure random walk case is not one that we

124

 

 

should want to exclude, and based on this observation and our simulations, the LM99 test
is definitely not recommended. We propose two modifications of the LM99 test that
avoid this difficulty. The simplest, LMM2, just takes the absolute value of the LM99
statistic.

Second, we note that the LM94 and LM99 tests (including our modifications of
LM99) may have poor power against near random walk alternatives (large X) when the
AR order (p) is overspecified. This is due to a near-cancellation of one of the AR roots
(which is zero) with the MA root (which is near zero). This is a fundamental problem
with no obvious solution, but it explains why in our simulations KPSS tends to be
preferred when A is large.

Third, we provide a consistent model selection procedure to pick the order of the
MA process in the KPSS test. We also show how we can remove the possibility of
overfitting, either in the AR case or the MA case, by using critical values that grow with
T at an appropriate rate.

The main contribution of the thesis is to investigate the size and power
characteristics of the various stationarity tests via large numbers of Monte Carlo
simulations. We do this for three different treatments of the ntunber of lags.

We first consider the case that the number of lags is fixed. This includes cases
where the true number of lags is known, but also cases where we only have an upper
bound so that we overspecify the model, and also cases where an incorrect model is used.
We consider white noise, AR(1), MA(1) and ARMA( 1,1) errors. None of the tests does
well if it is based on an incorrect (e.g., underspecified) model. The white noise case is

interesting because both the AR and the MA specifications are correct, though possibly

125

 

 

overspecified. Overspecifying l in the KPSS test causes smaller size distortions but
greater power loss than overspecifying p in the Leyboume-McCabe tests, so there is a
trade off between size and power. Except for the fact that the LM99 test is poor when k is
large, all of the Leyboume-McCabe tests are quite similar. This is surprising because
their rates of divergence under the alternative are different.

Second, we consider the case that the number of lags increases with sample size.
In this case there are no known asymptotic properties for the Leyboume-McCabe tests,
but the results are consistent with the conjecture that the Leyboume-McCabe tests are
asymptotically valid under the null and consistent under the alternative. With white noise
errors, we are massively overspecifying I and p. The KPSS test underrejects while the
Leyboume-McCabe tests overreject. The Leyboume-McCabe test appear to be more
powerful, but this is mostly fiom size distortions. Size-adjusted power still favors the
Leyboume-McCabe tests in general. The case with autocorrelated errors are generally
also favorable to the Leyboume-McCabe tests, but threre are still substantial size
distortions. More work is needed to determine which types of errors favor which tests.

Finally, we consider the tests with model selection rules. These model selection
rules work reasonably well, but there is definitely a loss in power from not knowing the
true number of lags. With white noise errors, the KPSS test generally has smaller size
distortions but also less power than the Leyboume-McCabe tests. With autocorrelated
errors, the KPSS test does better with MA errors, while the Leyboume-McCabe tests do
better with AR errors and generally with ARMA errors.

Speaking generally, the LM99 test is the biggest loser in these simulations,

because of its poor power when the data contain a strong random walk component. There

126

 

 

is no clean cut winner among the remaining tests, and one of the main results of these
simulations is that this is so despite the fact that asymptotic theory appears to favor LM99
or its modifications over the LM94 and KPSS tests. Model selection procedures seem to
be useful, and another of our general conclusions is that this is so for the LM94 and

KPSS tests as well as for the LM99 test and its modifications.

127

 

wEul—I' F' If 1.

 

BIBLIOGRAPHY

1. Dejong, D.N., J. C. Nankervis, N. E. Savin, C. H. Whiteman (1992), “Integration
versus trend stationarity in time series”, Econometrica, 60, 423-433.

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

3. Hall, A. (1994), “Testing for a unit root in time series with pretest data-based model
selection, Journal of Business and Economic Statistics, 12, 461 -470.

4. Kwiatkowski, D.P, 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.

 

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