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Essays on nonlinear transformations of nonstationary time
series
By

Chien-Ho Wang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

2003

Copyright © by
Chien—Ho Wang

2003

ABSTRACT

Essays on nonlinear transformations of nonstationary time

series
By

Chien—Ho Wang

My dissertation consists of four essays on nonlinear transformations of nonsta—
tionary time series. The dissertation has five chapters. The first chapter gives the
motivation for each of the four essays on nonlinear transformations of nonstationary
time series.

In Chapter 2, we consider periodic transformations of nonstationary time series.
We will use the scaled 1(1) process n"°‘:rt, where a E (0,1/4) It is shown that a

result of de Jong (2001) can be extended to

n+1” Erma) — u) i» M0, V)

t=l
where u = (27r)‘l ff” T(:L‘)d:z:, V is the covariance matrix, and mt is a so—called unit
root process.

In Chapter 3, we extend the asymptotic results for nonlinear transformations of
integrated time series of Park and Phillips (1999). We use less restrictions than Park

and Phillips to derive the improved results for integrable functions and asymptot-

ically homogeneous functions. In addition to the improved results, we propose a

new asymptotic result for non-integrable functions. This new result can extend the
original Park and Phillips results to some functions that are not locally integrable.
In Chapter 4, we investigate the question as to what happens to Dickey-Fuller
tests when the data under consideration is a trigonometric transformation of an 1(1)
process. We use analytical tools provided by de Jong (2001) to establish that for the

Dickey-Fuller t—test, we have
n-1/2ip L (ECOS(€t) _ 1)(1_ (Ecos(5t))2)'1/2

where f“ is Dickey-Fuller t-test under trigonometric transformations of 1(1) processes
with intercept. The above result implies that the periodic transformation of integrated
process will asymptotically indicate stationarity.

In Chapter 5, we consider a different approach for threshold unit root model. We

consider the Dickey-Fuller unit root test of the threshold unit root model

Ayt = 5t if lyt—ll S C ,
u+cpy¢_1+€, if |y¢_1|> C
where —2 < «p < 0. we will relax the assumption that threshold value,C,is known.
We derive the asymptotic results that can be used to establish the asymptotic dis-
tribution of the Dickey-Fuller unit root test in a regression of Ag; on a constant and

yt_11(|y¢_1| > C) that has been optimized over the parameter C' that is unidentified

under the null hypothesis.

For my father, Pei-Chen Wang and my mother, Shu-Yz'ng Wang

ACKNOWLEDGMENTS

First, I want to thank my dissertation advisor, Professor Robert de Jong. He
taught me a great details about time series methodology. His enthusiasm and knowl-
edge inspired me to pursue my studies of theoretical econometrics. Without Professor
de J ong’s valuable help, it would have been impossible for me to complete this disser-
tation. I am deeply appreciative of the guidance and encouragement which I received
from him over this period at Michigan State University.

I am also specially grateful to professor Richard Baillie, Professor Peter Schmidt
and Professor Jeffrey Wooldridge for their kindness and valuable help for my disser-
tation. I am extremely fortunate to have the opportunity to associate with such a
top group of econometricians. Their intelligence, diligence and success have inspired
me so much.

Many of my fellow graduate students have also contributed to the completion of
this study through their encouragement and friendship. My special thanks go out to
Mao-Sheng Chen, Wen-Hao Chen, Daiji Kawaguchi, Rehim Kilic, Jing-I Lu, Jen-Je
Su, and Bei Zhou. I feel grateful for all they did for me.

Finally, I keep special thanks to my family. I thank my parents who provide me
a total support for my long time research. Their loves are greatly appreciated. I also
want to thank my younger brother, and his wife. They give me a lot of help when I

process to write my dissertation. I dedicate my great thanks to them.

vi

TABLE OF CONTENTS

1 Introduction 1
1.1 Basic properties of 1(1) processes ...................... 1
1.1.1 Concepts of unit root models ....................... 1
1.1.2 The concept of cointegration ........................ 2
1.1.3 Scope of this dissertation ......................... 3
1.2 Thesis Structure ............................... 4
2 Asymptotics for scaled periodic transformations of integrated time
series 8
2.1 Introduction .................................. 8
2.2 Assumptions and main result ........................ 9
2.3 Conclusions and possible extensions ..................... 13
2.4 Mathematical Appendix ........................... 13
3 Further results on the asymptotics for nonlinear transformations of
integrated time series 41
3.1 Introduction .................................. 41
3.2 Assumptions and result for integrable functions .............. 43
3.3 Asymptotically homogeneous functions ................... 46
3.4 Nonintegrable functions ........................... 48
4 Unit root tests when the data are a trigonometric transformation of
an integrated process 62
4.1 Introduction .................................. 62
4.2 Assumption and main results ........................ 64
4.3 Conclusion ................................... 66
4.4 Mathematical Appendix ........................... 67
5 Some results on the asymptotics for threshold unit root test 71
5.1 Introduction .................................. 71
5.2 Main results .................................. 73
5.3 Applications .................................. 74
5.4 Conclusion and further research ....................... 75
APPENDICES 81

vii

A Introduction to local time 81

A.1 Definition and properties ........................... 81
A2 The Tanaka formula ............................. 83
A.3 The occupation times formula ........................ 84
BIBLIOGRAPHY 87

viii

CHAPTER 1

Introduction

This thesis consists on four essays of nonlinear transformations on nonstationary
time series. Before we discuss nonlinear transformation on nonstationary time se-
ries, we will first introduce some basic concepts about nonstationary time series and

provide a limited overview of the relevant literature.

1.1 Basic properties of 1(1) processes

In this section, we will introduce the concepts about nonstationary time series regres-

sion. I will introduce the cointegration and unit root models separately.

1.1.1 Concepts of unit root models

First, we introduce the basic linear time series regression model,
xt=psct_1+w¢ t=l,2,...,T (1.1)

where wt is a stationary process. In general, we need that the coefficient, p, must
satisfy the condition that l pl < 1 to ensure stationarity. In two breakthrough articles
by Dickey and F‘uller (1979, 1981), the case of p = 1 was first investigated. They

find that the t-test will converge to a non-standard distribution under p = 1. They

also simulate this unit root distribution. After Dickey and Fuller’s articles, Said and
Dickey (1984) and Phillips and Perron (1988) extended the unit root limit theory to
serially correlated errors. The limit theory of linear unit root models is by now well

developed.

1.1.2 The concept of cointegration

In international economics and macroeconomics, there exist long run relationships
between nonstationary variables. One example is the Purchasing Power parity (PPP)
research in international finance. PPP states that in the long run the exchange rate

adjusted price levels in two countries should be the same. The empirical models about

PPP is
Pi = PM + W + “a“

where p,- (pj) is the price level in country i (j ), 73,- is exchange rate between country i
and j, and uij is a stationary series. Because 1),- and 13,- are 1(1) processes, we cannot
use the traditional ordinary least squares method to obtain the limit properties of the
estimated coefficients. In a breakthrough paper, Engle and Granger (1987) developed

the linear cointegration regression model. They considered the time series regression

yt = 231% + Ut (12)

where wt and yt are two different I(1) processes and at is a white noise process. We can
investigate long run relationships between some economic and financial time series
using the cointegration concept. Since cointegration was proposed, it has become

mainstream in econometric research.

1.1.3 Scope of this dissertation

There are a lot of nonlinear relationships between economic variables in economic
theory. Using linear times series models for all economic times series has a lot of
restrictions. However, if we only transform the 1(1) processes and directly use trans-
formed variables to regress two transformed variables for cointegration or unit root
models, we will have some problems. The main problem about nonlinear transfor-
mations of nonstationary time series is that transformed 1(1) series may not keep
the same nonstationary properties as before. Granger and Hallman (1989, 1991) first
discussed these possible problems. They used nine kinds of functional forms to in-
vestigate whether the transformed 1(1) series still keep the nonstationary properties.
They found whether the integrated process keep its nonstationary characteristic after
transformed will depend on functional forms. The other problem about nonlinear
transformations of nonstationary time series is the use of unit root tests. Because
the properties of transformed 1(1) series change, we may misjudge the properties of
transformed 1(1) series when we use the Dickey-Fuller unit root tests. In Granger and
Hallman’s research, they simulated Dickey-Fuller test critical values under different
transformations. They found the critical values will change depending on functional
forms. Granger and Hallman’s papers investigated about these problems in detail,
but they only used simulations to investigate these problems. They did not derive any
formal limit theory for transformed 1(1) series. After Granger and Hallman’s papers,
Park and Phillips (1999) extended the existing limit theory for integrated processes
to nonlinear models. They considered three classes of functional forms: integrable
functions, asymptotically homogeneous functions and explosive functions. They use
the concept of local time to derive asymptotic results for nonlinear regression models.
Although Park and Phillips’ results are remarkable, some functional forms cannot

be considered in their results, and their results are still restrictive in terms of the

necessary conditions. In this dissertation, we will investigate these questions.

1.2 Thesis Structure

In Chapter 2, we consider periodic transformations of nonstationary time series.
In Park and Phillips’ paper ( Park and Phillips (1999)), the authors derive asymptotic
properties of nonlinear transformations of 1(1) series for three classes of functional
forms: integrable functions, asymptotically homogeneous functions, and explosive
functions. The key element here is that the I(1) process was not rescaled by the
square root of sample size. After Park and Phillips’ work, de Jong (2002) established
the asymptotics for periodic transformations of 1(1) processes. In that paper, it is

proven that for periodic function T(.) and for I( 1) processes :rt,
71“” 2mm -— u) —d—» No, 02)
t=1

where u = (27r)‘1 f; T(:r)d:1:. De Jong derived this result for periodic functions of
1(1) processes that have not been scaled. In Chapter 2, we will use the scaled 1(1)
process n‘axt instead of :13, in de Jong’s paper, where a E (0, 1/4). It is shown that

de Jong’s original result can be extended to

n-a'l/2 En:(T(n'°;rt) - ,u.) —d—> N(0, 2o-2 :03_j-2(a.32 + b§))
t=1 i=1
where p = (27r)‘1f:rT(:r)dx, aj = r‘lffwcos(ja:t)T(:r)dzr and (31 =
r‘lffw sin(j:rt)T(a:)d:r. When a = 0, this new result will specialize to Theorem
1 of de Jong (2001). In this chapter, we therefore extend the results for periodic
transformations of I(1) processes in de Jong’s original paper.

In Chapter 3, we extend the asymptotic results for nonlinear transformations of

integrated time series of Park and Phillips (1999). In Park and Phillips, they prove

that for 1(1) processes xt and integrable function T(.),

n—l/2ZT(CEt) —"—» (/ T(s)ds)L(1,0)
t=1 '00
where L(t,s) is a two-parameter stochastic process called ”Brownian local time”.
They established the above result for 1(1) processes that have not been scaled by

n‘axt. In this chapter, we use the scaled 1(1) processes n‘axt instead of It. We will

use the results of de Jong (2001) to extend their result for integrable functions to

co

THU?” ZTUL'Oxt) —d-> (/_ T(s)ds)L(1,0),

where a E [0,1/2) The asymptotics for integrable functions is derived under less
strict conditions than in Park and Phillips (1999).
The asymptotically homogeneous functions as defined in Park and Phillips ( 1999)

are assumed to satisfy
T(Ax) = u()\)H(:c) + R(:r:, A).

For the remainder function R(., .), Park and Phillips ensured asymptotically neglibil-

ity of

71
71-1: R(:1:t,n1/2).
t=1

Their result for asymptotically homogeneous functions is then

u(n1/2)_ln—IZT($t) L/o H(oW(r))dr,

where 02 = lim,,_.o,D n'lErrfl. We use the scaled 1(1) processes 12-02:; instead of 23¢,
as in Park and Phillips (1999). We generalized the original definition of Park and
Phillips of asymptotically homogeneous functions. For functions H () and u(.), we

assume that for all K > 0,

K
/ |V()\)_1T()\$) — H(:r)|d:1: —> 0,
—K

when /\ —> 00. Under regularity conditions, we derive the result

71 1
V 72—1/2” "In"1 71—“, —d—> o f r r
< > gm 1:) [0m ”()ch

where a E [0, 1 / 2). This result extends the limit theory for the asymptotically horne-
geneous functions as derived by Park and Phillips.

In Chapter 4, we investigate the question as to what happens to Dickey-Fuller
tests when the data under consideration is a trigonometric transformation of an 1(1)
process. Granger and Hallman (1991) investigated this question by simulations. They
concluded that the 1(1) series will change its properties after periodic transformations.
We use analytical tools provided by de Jong (2001) to establish that for the Dickey-

Fuller t-test, we have
n‘l/gffl —p—-» (E cos(s,) — 1)(l — (Ecos(e¢))2)'l/2

where t), is Dickey-Fuller t-test under trigonometric transformations of 1(1) processes

with intercept. Otherwise, for the coefficient ,6 we show that
Til/20f) — Ecos(et)) —d-+ N(0, V)

where V = (3/8)E(cos(et) — Ecos(et))2 + (1/8)E(sin(et))2. Because Dickey-Fuller t-
tests diverge at rate JR. The above result implies that the periodic transformation of
integrated process will asymptotically indicate stationarity. These theoretical results
are supported by the simulations in Granger and Hallman (1991).

In Chapter 5, we consider a different approach for threshold unit root model.
In Gonzalez and Gonzalo’s paper (Gonzalez and Gonzalo (1997)), they used the

threshold unit root model:

Solyt—l + 5t if ”yr—1 S C

31¢
$023/t—1 + 5t if yt—l > C

They derive the asymptotic properties of Dickey-Fuller unit root tests that the null
hypothesis of unit root exists in at least one regime against stationary threshold au-
toregressive model. But their model has a main drawback. In Gonzalez and Gonzalo’s
TUR model, they only allow the case that all regimes are stationary in alternative hy—
pothesis. In this chapter, we consider the Dickey-filler unit root test of the threshold

unit root model

Ayt = at if Iyt._1| S C ,
prt_1 + at if lyt_1| > C
where —2 < (p < 0. We will relax the assumption that threshold value,C,is known.
We derive the asymptotic results that can be used to establish the limit distribution
of the Dickey-Fuller t-test for H0 : 4,0 = 0 against the alternative of H1 : —2 < (,0 < 0

that has been optimized over the parameter C that is unidentified under the null

hypothesis.

CHAPTER 2

Asymptotics for scaled periodic
transformations of integrated time

series

2.1 Introduction

Nonstationary time series have been attractive for recent research in econometrics.
The applications of nonlinear transformation of nonstationary time series have been
of major interest in international economics and macroeconomics. Although a lot of
macroeconomic models had used nonlinear transformations for some time series data,
the transformed data properties do not totally understand by econometricians. The
first paper to investigate this question was Granger and Hallman (1991). Granger
and Hallman (1991) used the Monte Carlo method to investigate the relationship
in nonlinear transformations of nonstationary time series. They concluded that the
stationarity of nonlinear transformation depends on functional forms. After Granger
and Hallman’s breakthrough research, Ermini and Granger (1993) investigate the

variances, covariances and high monent conditions under transformed data series with

Gaussianity, but they did not build the limit theory under nonlinear transformations
of I(1) processes. In recent paper, Park and Phillips(1999) established the limit

distribution of the form.

a, Z T(rt)

where :rt = 1130 + 23:15], 1:0 is an arbitrary random variable that is independent
of all other at, the ej satisfy a weak dependence condition, It 6 IR, an is a proper
scaling factor such that an —> 0 as n —> co, and T(.) is a transformation of the inte—
grated process 2:; that is allowed to be within one of three function classes: integrable
functions, the asymptotically homogeneous functions and the explosive functions. Af-
ter Park and Phillips, de Jong (2000) extended Park and Phillips original results to
periodic transformations on nonstationary time series. De Jong considered continu-
ously differentiable periodic functions and concluded that the periodic nature of the
trigonometric functions effectively ”reduces” the dependence in the integrated process
to a point at which a central limit theorem holds.

In this chapter, we extend the result in de J ong (2001). We use a scaled integrated
process n’o‘z't instead of :13, in de Jong. We use a martingale approximation and a
Fourier series expansion result to obtain the main theorem about periodic transfor-
mations for scaled integrated process. Compared with the main results in de Jong
(2001), we can find that Theorem 1 of de Jong (2001) is a special case of our general

results.

2.2 Assumptions and main result

We consider a time series art generated by

3% =33t-1'l'5t (2-1)

in which at is a sequence of independent and identical distributed random variables
with mean zero and variance 02. B = 9(51, et_1, . . . ,51, x0) is a sigma field including

the information until time period t. Other assumptions will be made throughout this

paper.

Assumption 2.1 at has a symmetric distribution with E(et) = 0 and Var(et) = 02.

Assumption 2.2 Elastl5 < 00.
Using these assumptions, we can obtain the useful lemmas as below.

Lemma 2.1 For the process (L‘t defined before, if at satisfies Assumption 2.1 and 2.2
with 0 < a <1/4 and for any C E RC 7f 0, then

 

n’a’l/zz sin(n”°’(:rt) — (2(‘20'2)n°‘"1/2: (sin(n_°‘(a:t) — E(sin(n-°(:ct)|F¢-1))
t=1 t=l

 

= 0,,(1).

and

 

n-a-l/tzcosrn-acx.) — (wanna-“2: (case-arr.) — E(.os(n—oa.)lp.-.))
t=l

t=l

 

= 0,,(1).
Lemma 2.2 For the process :rt defined before, if at satisfies Assumption 2.] and 2.2
with 0 < 01 <1/4 andfor any (,7 E R,(,’7 75 0, then for)! =C
nth-Ii E{[sin(n‘°(a:t) - E(sin(n—°'C.’Et)lFi—1)l2lFt-ll—E’U/QNCUV,
t=1
and for 7 31$ C
”20-12": E{[sin(n'°'y:ct) — E(sin(n-°'y:rt)|Ft_1)]
t=1

x [sin(n’°‘(:rt) — E(sin(n’°(a:t) Ft_1)]|F}_1}—p—+0.

 

10

With the same method, we can also obtain a lemma about the cosine function.

Lemma 2.3 For the process at as defined before, if at satisfies Assumption 2.1 and

2.2 with 0 < a <1/4 and for any (,7 E R,C,'y 75 0, then fory = C

730—1: E{[cos(n‘0‘(a:t) — E(cos(n"°§:rt)|F,_1)]2|F,_1}i>(1/2)(Co)2,

t=l

and for ’7 79 C
n20— 1:E{[cos(n 0/33,) —E(cos(n‘“7xt)lFt—1)l

[cos(n—°(:rt) — E(cos(n'°‘(:rt)|Ft_1)]IE_I}-L>0.

From these three lemmas, we can build the limit distribution of the periodic trans-

formation of rescaled integrated process.

Theorem 2.1 For the process :rt defined before, if at satisfies Assumption 2.1, 2.2

and 0 < a <1/4, then

n 71

n
. CC _ :17 _ _ :1:
(71-0-3 (Clot),~-c, —a—;§ :SIII(—'— (ma t) a— ;E :COS(n—a" (1 t)
n
i=1

t=l t=l

       

‘ y it N(0, A)

         

where A is a 2m x 2m matrix with diagonal elements

(2((1o)'2,...2(Cmo)‘2,2(Clo)‘2...2(Cmo)‘2). The other elements are zero.

From Theorem 2.1, we obtain two main results. First, we can find that the limit
variances depend on the square of rescaled parameter Q. When (3- is large, the limit
variance is small. Second, when a = 0, the result of Theorem 2.1 will be equal with
Theorem 1 of de Jong (2001). From our result, we can find Theorem 2.1 extends
the result obtained from de Jong. From Theorem 2.1, we can obtain the following

corollary.

11

Corollary 2.1 For the process :rt defined before, if at satisfies Assumption 2.1 and
2.2 and 0 < a <1/4, then

n'o‘ l(223mm “(xt)—d—+N(0, 2((0) 2) and

71—0—1/2 Z cos(n‘°(a:t)—d-+N(0, 2((o)'2).

t=l

Horn these results, we can find periodic transformations decrease the dependence
in sealed nonstationary time series. In fact, this result support Granger and Hallman’s
conclusion that periodic transformation of 1(1) process is stationary series. But the
variance forms of the limit distribution will depend on scaled factors C,. Using these
results with Fourier series concepts, we can extend our results to more general result

about periodic transformations of scaled integrated process.

Theorem 2.2 For the process 2;, defined before, assuming that at is an i.i.d. se-
quence of random variables satisfied Assumption 2.1, 2.2, and assuming that T(.) is

continuously differentiable and periodic on [—7r, 7r] and 0 S a < 1/4, we have

n+1” Z(T(n'°xt) — u)——d—>N(0,2o'2i(j"2)(a§ + b3». 3' 2 0

i=1

u = 27r‘2/ T(sr)d:r aj = n-I/ cos(ja:¢)T(:c)d:r and

71' -7l'

bi = n‘lf sin(j:rt)T(:r)d:r.

An possible extension of the above results is to the case of asymptotic distribution
of 5;. But from the present proof, it is far from clear how to go about to estabilish

such the results.

12

2.3 Conclusions and possible extensions

In this chapter, we established the limit distribution for summations of continu-
ously differentiable periodic functions of scaled integrated process. We use scaled I(1)
processes n‘o‘xt instead of at, in de Jong (2000). We can obtain more general result for
limit theory of periodic transformations of integrated time series under 0 < a < 1/4
Even though we obtain more general result, but these results still depend on 5; must
be an i.i.d. and symmetric distribution. From these results, we can build the limit
behavior of regression under periodic transformations of I(1) processes. For example,
we can establish the behavior of least squares estimator b without intercept in the

model
yt = bT(n"’a:t) + ut.

Where at is a martingale difference sequence of random variable with respect to the

sigma field and T(.) is a periodic function. The least square estimator b is equal to

b = (Z T(n’o‘xt)2)‘l(z T(n"°’:rt)yt) = b + (ZT(n‘°:rt)2)—1(Z T(n’ocrt)ut).

For the denominator of the least square estimator is the periodic function. We can use
the theorem we developed to build the asymptotic properties. About the numerator
of least square estimator, we need to analyze the property of 21;, T(n‘amt)ut. This
term is a summation of martingale difference equation. If E(uf|F¢-1) = Euf, the
asymptotic normality holds for \/7_i(b — b). This result enlarge the original result from

de Jong.

2.4 Mathematical Appendix

For the proofs of Lemma 2.1 and 2.2, we need the following lemmas.

13

|/\

|/\

Lemma 2.4 For the process 3:, as defined above, under Assumption 2.1 and 2.2,
IE (sin (n"°<:v¢) lFt—l) _ (1‘ (1/2)n“2°‘((o)2)sin(n'°‘(:rt_1)| S (1/24)C4El5tl4nw4a-
and

IE (cos (Went) lFH) — (1 — (1/2)n‘2‘*(<a)2)cos(n‘am-1)l s (1/24)<“Els.l4n"“°.

Proof of Lemma 2.4:

From conditional expectation definition and the identity sin(:r + y) = sin(a:) cos(y) +

cos(rr) sin(y),
|E (sin (n-ng.) IFH) — (1 — (1 /2)n'2°‘((o)2) sin(n-acx,-1)|
= |sin(n'°(:rt_1)E cos(n-acs.) + cos(n’oCrrt_1)Esin(n"°‘(et)

-(1 - (1/2ln’2°(<0)2) sin(n“’Cxt—1)l
|sin(n-acx._1)(E cos(n'o‘Cet) — 1 + (1/2)n—'~’“(ca)2) + cos(n-acx,-1)Esin(n-0(e,)|
lsin(n'°(a:t_1)(Ecos(n‘o‘Cet) — 1 + (1 /2)n-2“(Ca)2)l + lcos(n'°€rct_1)Esin(n’°‘<€t)|
|sin(n'°C:vt—1)IIEcos(n"“<6z) - 1+(1/2)n"2°(<€t)2| + |cos(n‘04xt—1llEsin(n‘“C€t)I.

By the inequality |cos(:r) — 1 — x2| S (1/24);r:4 and |sin(:r) — x] _<_ (1/6):I:3 with

Assumption 2.1, it follows that

|Ecos(n*aqe,) — (1 — (1 /2)n'2a((o)2)l g (1 /24)§“E|s.|4n—4a (2.2)
Because 0 < sin(n’°‘(:rt) < 1 and 0 < cos(n‘OCxt) < 1, it follows that
IE (sin (n‘OCxt) IFt_1) — (1 — (1/2)n"2°((o)2) sin(n'°(:r¢_1)|

S lEcos(n_°Cet) — l + (1/2)n_2°(Co)2| + lEsin(n‘a(5,)| S (1/24lC4E lEtl4 n—4a_

14

where the last inequality uses the following Equation (2.2). Using the same terminol-

ogy, we can obtain another result.
|E (cos (n‘o‘Catt) |Ft_1) — (1 — (1/2)n'2°‘((o)2) cos(n'a($t_1)|
S IEcos(-n"°‘(et) - 1+ (1/2)n’2°(Ca)2l + |Esin(n'“(et)| S (1/24)(4E|etl4n_4a.
El

Lemma 2.5 Let 5; satisfy Assumption 2.1 and 2.2 with 0 < a < 1/4 Then

 

 

n 2
E 722042 (sin(n_°'y:1:t) sin(n’a(:rt) — E(sin(n-a’r$t) sin(n-0C$t)lFt—1))
t=1
= 0(1)
and
n 2
E nza’lz (cosh—0733;) cos(n_"C:1:t) - E(cos(n'°"y:rt)cos(n-OCI1)IF¢_1))

 

 

t=l

= 0(1).

Proof of Lemma 2.5:

First, we note that by the martingale difference property of the summands,

n 2

E nib-1: (shim-“73:0 sin(n’°(:rt) — E(sin(n-°7xt) Sin(n—GC$t)lFt-1))

t==l

= n4a"2ZE[(sin(n"°7xt) sin(n—°C:rt) — E(sin(n—°'y:r¢) sin(n“’(:rt)|F¢_1))]2. (2.3)

t=l
Because sin(n""ya:t) sin(n"’(:rt) and E(sin(n‘°7:rt) sin(n“°C;r¢)|Ft_1) take their val-

ues in [0,1], it follows that

|sin(n‘°"y:rt) sin(n‘°(:rt) — E(sin(n”°'y.rt) sin(n'°§a:t)|F,_1)| g 2. (2.4)

15

Horn Equation (2.3) and (2.4), it now follows that.

71

”20-1: (sin(n'°‘7:c¢) sin(n'°C;rt) - E(sin(n“°7:rt) sin(n"°‘C;rt)|Ft_1))

t=l

E

 

 

g rims-1. (2.5)

From the assumption 0 < a < 1/4 and Equation (2.5), when n —> co, the result

follows. Similarly,

it 2

n2a’lZ (cos('n"°7:rt) cos(n‘O‘Crt) — E(cos(n‘°’7.rt) COS(n_oCJIt)|Ft—1))

t=l

E

—>0.

 

 

C]

Lemma 2.6 Under Assumption 2.1 and 2.2, if0 < a < 1/4 and for any C,’7 E

R,C,'y;é0, thenfor7=C 7,C¢0

(1/n)Zsin2(n'aC:rt)ii(1/2) and (1/n)thos2(n "’)Ca:¢ p(1/2).

i=1 ,1:

andf0r7#C7,C#0

(1/n)Zsin(n—°‘7:rt)sin(n"°C:c,)—p+0 and ()l/n Zcosm 071:; )cos(n "——+C:rt) p 0.

i=1

Proof of Lemma 2.6:

1. First, from Lemma 2.5, it follows that

 

 

n 2
E nta—IZeinan-ux.)—E<sin?v(- mm 1)) =00).
t=1
implying that:
”2.42 [sin (nom-(smiin’aCn)IE_1)ll = ope). (2.6)
t=l

 

16

Second, from the definition of an, we can write sin2(n’°‘C;rt) as below:
sin2(n’°‘C:rt) = (sin(n’°‘C:rt_1)cos(n"" 5;) + cos(n‘a'Csr:t_1)sin(n'C'Ce,))2
= sin2(n_°C;rt_‘1)cos2(n“aCst) + cos2(n‘°C:rt_1)sin2(n'°Cet)

+2 sin(n"'C;rt_1)cos(rfaCxt_1)sin(n_°Cet)cos(n_°Ce¢). (2.7)

Under Assumption 2.1, using independence and the symmetry of the
distribution of at, the conditional expectation from Equation (2.7),

E(Sin2(n_acxt)lFt—1)a i5

E(sin2(n’aCz‘t)

 

Ft—l)
= si112(n'°C:rt_1)Ecos2(n‘°Cet) + cos2(n’°‘C:rt_1)Esin2(n'°Cet). (2.8)

Substituting (2.8) into (2.6), we can obtain result as below:

n

In?“—1 2 sin2(n_°C:rt)

t=l

—n2""1 2": (sin2(n—OC:C¢_1)E cosQ(n'°C5¢) + cos2(n_aCa:¢_1)Esin2(n_°Cet)) |
t=1
= 0,,(1). (2.9)
Also note that
cos2(n—°C:rt) = 1 — sin2(n"’C1:t). (2.10)
Combining (2.9) and (2.10), we can obtain the equation

71
[n20'lZsin2(n'aC;r¢)
t=1

17

n

n20- 1:“ ((ECOS2( (n OrCa)— Esi112(n‘°‘C€t)) SIII2(Tl—GCIEt_1) + Esin2(n-°‘C€t))l
i=1
=op(l). (2.11)

From Equation (2.11), it implies
n

WG—1: sin2(n_°‘C:rt)
t=1

= 11204:: ((Ecosg(n‘°‘Cet) — Esin2(n_“Cet)) sin2(n_°‘C:1:t_1) + Esin2(n_°C€i))

t=1

+op(1). (2.12)
Also, we have equality:

Z sin2(n"°‘C:r¢_1) = Z sin2(n‘oCxt)+sin2(n"°Ca:0)—sin2(n’°Ca:,,).(2.13)
£21

£21

and
Esin2(n‘°Cet) + Ecosz(n'°‘Cet) = 1. (2.14)

Substituting (2.13) and (2.14) into (2.12), we can rewrite Equation (2.12) as

below:

[2Esin2(n “C Ce n2“ 1Zsin2 (n C):rt= n2“ (Esin2(n"°Cet)) + 0,,(1).
By Taylor expansion, we can obtain the inequality

|n2aEsin2(n_aCet) - 2o2| S (1/3)C4E letl4 n'2".

By this inequality, it suffices to show

[2(C2o2 + O(n"2"))]n‘l Z sin2(n_aC:1:t) = C202 + O(n‘2°) + 0,,(1).

£21

18

After some algebra, we can obtain

n_IZsin2(n_aC$t)i>1/2.

t=1

-F0r7#C%C#0

First, from Lemma 2.5, we obtain that

 

 

 

 

E n2a'li [sin(n“"ya:t) sin(n'°Ca:t) — E(sin(n_°‘7:rt) sin(n'°C:rt)|Ft-1)] 2
i=1
= 0(1).
It follows that
rah—Ii [sin(n—°7:1:¢)sin(n—°C;rt) — E(sin(n"°‘7:rt) sin(n'°C:I:t)|Ft_1)]
t:1
= 0p(1)_ (2.15)

Second, from the definition of 23,, we can write sin(n““7:rt) sin(n‘°C:rt) as below:

sin(n'°'y.’rt) sin(n_°C:rt)

= Sln(n-a’)’l‘(—1) sin(n—OC.T:¢_1)COS(Tl--O’7€t) COS(n-OC5t)

+ cos(n’C'VCBt—flSin(n—OC$t—1)Sin(n_a’75t) cos(n-00%)

+ sin(n—°7xt_l) cos(n—“Czrt-1)cos(n 075‘) sin(n—°Cet)
+ cos(n—ayxt-1)cos(n-“Cast_1)sin(n"'7et)sin(n'°Cet). (2.16)

Under Assumption 2.1, the conditional from Equation (2.16) is,

E(sin(n'°'yatt) sin(n-“Ca:t)|Ft_1)

19

= sin(n'°‘7;rt_1)sin(n—°C:rt_1)E cos(n‘ayet) cos(n—(’Cet)
+ c0s(n"°7;r,-1) cos(n'oCrt_1)E sin(n“"yet) sin(n'°Cet). (2.17)

With the same method as first proof, we can obtain the equality as below:

n
”20-1: sin(n'°7;rg) sin(n”"C;rt)
t=1

n
= E cos(n—“150cos(n—“Ce,)n2°'1Zsin(n"°7.’r¢_1)sin(n’°Cart_1)
t=l

+E sin(n'°‘75t) sin(n”°Ce,)n2a"IZ cos(n‘ayxt_1)cos(n—“Ca:¢_1)+ 0p(1).(2. 18)

t=l

We move first item of the right side in Equation (2.18) to the left side. We
obtain the following Equation:
nib—1: sin(n_°7:r¢) sin(n—“C:1:¢)
t=1

—E cos(n—“7'et) cos(rz‘aCet)n20-IZ sin(n’°7:ct_1)sin(n"°C:1:t_1)
i=1

2 E sin(n‘°"yet)sin(n'aCet)n2°"1Ewan—“711.14)cos(n—“C33,-1). (2.19)
i=1

Also, we have equalities:
n
E sin(n‘°"y:rt_1)sin(n_°C;rt_1)
t=l
n

= Z sin(n_“7:1:t) sin(n"°C:rt) -l- sin(n—°‘7.r0) sin(n'°C;r0)

t=l

— sin(n"a'y;rn) sin(n‘°C:c,,). (2.20)

20

71.
Z cos(n'a711:¢_1)cos(n—“Ca:,_1)

t=l
= Z cos(n'o7rt) cos(n—"Cam + cos(n—07.130) cos(n‘aCato)
t=l

— cos(n-071:") cos(n-”Catn). (2.21)

Substitute (2.20) and (2.21) into (2.19), we can rewrite Equation (2.19) as below:

fl.

[1 — Ecos(n'°7€t) cos(rz""Cet)]n2‘l'"'1 Z sin(n'°7:1:t) sin(n’°C:1:t)

i=1
=[Esin( n “7et)sin(n "¢)]Ce n20 1::cos( (n “7:1:t)cos(n O'Catt)+op(1). (2.22)
t: 1

By Taylor expansion, we have two inequalities

]n2°_1[1 — Ecos(n’°7et) cos(n—“Cam — (72 + C2)02] S (l/4)(7C)2E]e4]n"2°
and

]712°_1[Esin(na 7e,)sin(n e)]— 7C02] <( (1/6) )E]7C( 72 +C2) 53hr?“

By these inequalities, they suffice to show:

n

[(72 +6172 + (Dru-2011112“: sine-07x.) sine-“cm

t=l

[=7Co2 +O(n ‘20 )]112° lZcos( n “7:1:t)cos(n “C1170 +op(1) (2.23)

t: 1

Using the same trick, we can obtain another equation about

n
112""1 Z cos(n—07m) cos(n”“C:1:t)
t=1

[(72 +C2 )02 +O(n ‘2“) n2“ 1Zcos(n“7a't)cos(n"C;1:¢)
£21

21

n

='2[7Co +O(n ‘2") n2" lZsin(n"7:r¢ )si11(n"C:1:t)+op(1). (2.24)

t=l

Solving Equations (2.23) and (2.24) simultaneously, we can obtain

11

n Zsin(n "7:11) )sin(n "Crt)—>0 and n 1Z:cos(n"7:1:t)cos(n"C:1:t)-—-»1D-—+0.

i=1

[3
Proof of Lemma 2.1:
First, we note that the following identity holds:
n"‘1/2 Zsin(n-"C:rt) = n"'1/2Z (sin(n-"C:1:¢) — E(sin(n_"C:rt)|Ft_1))
t=l t=l
+n"‘1/2 Z E(sin(n'"C:1:t)|Ft_1). (2.25)

i=1

By Lemma 2.4, we can rewrite Equation (2.25) as below:

11

71‘“ 1/2:Sin(ri—"Cx) - n“ ”22 (sine-xx.)—E<sin(n"°c:vt>lH-1>)

t=1

Tl

+n"—1/2Z sin(n‘"C$t_1)E cos(n—"C50.

t=1
We move the second item on the right hand side of the equality to the left and leave

the summation of differences on the left side. We obtain,

11

n
n"‘” X sine-arr.) — n‘H/iZ sin<n-°<x.-.>Escanner)

t=l t=l

Tl

= n"’1/QZ (sin(n—"C:1:¢) — E(sin(n""C:rt)|F¢_1)) (2.26)

t=1
Also we have following equality:
gsim n"_C:1:¢ 1) =Zsin(n’"C:rt) + sin(n'"C:1:0) — sin(n-"C1:,,). (2.27)
t: 1

22

Now plugging in the result of Equation (2.27) into Equation (2.26) gives
n"—1/2Z sin(n‘"C1:t) - n"_1/2E cos(n—"Cet) Z sin(n_"C;rt)
t=1 i=1

_na—l/2(E cos(n—0451))(Sln(n—OCZE0) — 8111077043311»

71.

= n"”1/2Z (sin(n_"Ca:t) — E(sin(n’"Ca:t)|F,_1)). (2.28)

i=1
After combination the left side of Equation (2.28), we can obtain

11

n"'1/2(1 — E cos(n—"Cam Z sin(n—°CI1)

t=l

_na—l/‘zwcos(n-age,))(sin(n‘0cxo) — sin(n‘°an))

fl

= n"—1/2Z (sin(n_"Ca:t) — E(sin(n""C:rt)|Ft_1)). (2.29)

t=1
We move the last item on the left side of Equation (2.29) to the right side. After

some algebra, we obtain.

n"‘1/2(1 — Ec0s(n_"Cet))Z sin(n_"C:1:t)
i=1

= n"‘1/2i (sin(n'"C:1:,) — E(sin(n_"C:1:t)]F¢_1))
t=1
+n"_1/2(E cos(n—"Cam(sin(n—"C:1:0) — sin(n—"C.r,,)).
By Taylor expansion, we can obtain the following inequality.
|n2°(1— Ewan-area) — (1/2>(<o>2| s (lawman—2°.

By this inequality, it suffices to show.

71

((1/2)(Co)2 + 0<n-2“>1n-0-”2Z sine-0cm.)

i=1

Tl

= n"’1/ZZ(Sin(n-OC$t) — E(sin(n_"C$i)|Ft-1))

t=l

23

+n"‘1/2(1 — (1/2)n"2"(C0)2 + 0(n’4"))(sin(n—"C:r0) - sin(n”"C:r,,)).
By rearranging terms and boundedness of summands,
n—"'1/2Z sin(n—"C230 : (2(Co)‘2)n""1/2 Z (sin(n'"C:1:t) — E(sin(n‘"C:r¢)|F¢-1))
t=l t=1

+op(1).

Proof of Lemma 2.2:

1. First, by the law of iterated expectation, it follows that

n2"_lz E{[sin(n_"C:rt) — (E sin(n—"Czrt)]Ft-1)]2|Ft-1}
t=1

- n2""lZ{E(sin2(n""C:1:t)|F,_1)— E(sir1(n‘"Ca:t)]E-1)2}.
t=1

From the definitions and assumptions of wt, this statistic can be rewritten as

”20— IZ{E( sinz Wm, |F,_1)— (ESIn(n_aC$t)lE—l)2}

n“""“1 i E[sin2(n‘"C:1:t_1)cos2(n—"Ce¢) + 0082(n’"C:rt-1)sin2(n""Ce¢)
+2 sin(n"C:1:t_1) cos(n"-C:1:t 1) sin(n "Cet) cos(n e)-|Ft 1]
—[sin2(n’"C:1:t_1)(E cos(n—"Cat”2 + cos2(n'"C:1:t-1)(Esin(n—"Cat”2
+2 sin(n'"C:rt-1) cos(n—"Cxt-1)Esin(n-"CEJE cos(n-"C50”. (2.30)
The conditional expectation of Equation (2.30) is
n2"'12n:{E(sin2(n'"C:rt)|Ft_1) — (E sin(n'"Cxt)|Ft-1)2}
t=l

24

n

= ”2"“: {[sin2(n‘"C;r,_1)EcosZ(n’"Cet) + cosz(n‘"C:rt-1)E si112(n'"Cet)

t=1

+2 sin(n'"C:rt-1) cos(n—"Cxt-1)Esin(n—"C50 cos(n'"Cet)]
—[sin2(n—"C:r¢-1)(E cos(n—"Cet))2 + cos2(n_"C:1:t-1)(E sin(n-"Cet))2
+2 sin(n—"CIt-l) cos(n—"Crt_1)Esin(n—"Cet)E cos(n—"Cet)]}.

From Assumption 2.1 and 2.2, the odd moments of 51 are equal to zero. We can

rewrite original equation as below:

”—201 IZ{E(Sin2 ‘00“) ]F,_1)-— (E(sin(n’aca:1)lFt—1)2}

- n2"'lz cos2(n'"C:1:t-1)Esin2(n'"Cet) + 0,,(1)

t=l

= 71—12 cos2(n“"C:1:,-1)(C2or2 + O(n"2")) + 0,,(1)

i=1
= n_l(Co)QZ cosz(n’"C:rt_1)+ 0,,(1).
t=l
By Lemma 2.6,

n2"'IZ {E(sin2(n‘"CxtllFt—1) — (E(Sin(n-OC$t)lFt-1)2}

n

= n"1(Ca)2Z C082('n—0C$t-l) + 012(1) - (1/2)(C0)2 + 012(1)-

t=1

2. For77éC

First, by the law of iterated expectation, it follows that

”201—12 {lsin(n‘°7r1) sin(n-"C100 _ E( Si11(n—07$t)lFt-llE(Sin(n—a<$t)lFt—l)llFt—ll

t=l

25

= n2""1:{E(sin(n_"7:rt)sin(n‘"C:1:t)|Ft-1)

t=l

—E(Sin(n—a’lxt)lFt—l)E(Sin(n-ocxt)lFt—l)}
From the definition of act, this statistic can be rewritten as

712"‘1 Z{E(sin(n""7:r,) sin(n'"Cart)|Ft-1)

i=1
—E(sin(n’"7:r.¢)]F¢_1)E(sin(n_"Ca:t) |F¢-1)}

= n2""li sin(n—"7xt-1)sin(n—"Cxt-1)E cos(n—"7a) cos(n—"Cad
t=l

+ si11(n""7:1:t-1) cos(n—"Catt-1)E cos(n-"7a) sin(n-"C50

+ cos(n—"7:1:t_1) sin(n'"C:rt-1)E sin(n-"7a) cos(n—"C50

+ cos(n-"72:34) cos(n-"Cxt-1)E sin(n-"7a) sin(n-"Cad
—[sin(n_"7:rt_1)sin(n—"Cxt-1)E cos(n—"7et)E cos(n‘"Cet)

+ sin(n’"7:1:t-1)cos(n'"C:rt-1)E cos(n-"750E sin(n—"Cad

+ cos(n'"7:1:t_1) sir1(n""C:rt-1)E sin('n‘"7et)E cos(n-"Cad

+ cos(n—"7:1:t-1)cos(ri‘"C:1:t-1)Esin(n‘"75,)E sin(n—"Cet)]}

From Assumption 2.1 and 2.2, the odd moments of e, are equal to zero. We can

obtain:

THO-1: {E(Sln(n"0’)’l‘t)si11(n’°CIt)|Ft_l) __ E( SIII(TI_O’7'115¢)]F1_1)E(SlIl(n—OC$¢)]Ft-1)}

t=l

26

= WWI: {cos(n—"71m-1)cos(n""C:1:t-1)E sin(n—"750E sin(n—"Cet)} + 0,,(1)
t=1

Because —1 S cos(n‘"7:ct_1)cos(n-"Cxt-l) S 1, it implies
n?“1irEeintn-va.)sin(n-04min-»

t=l

—E(sin(n""7:1:,)|Ft-1)E(sin(n‘"C:rt)|Ft_1)}
= (C02 + O(n‘2"))n_li {cos(n—"7:1:t-1) cos(n—"CLEt-fl} + 0,,(1).

t=l

By the Lemma 2.6, we know

n'li cos(n—"7:1:t-1)cos(n—"Czrt-1) = 0 under 7 71$ C and 7,C Z 0

t=1

We can obtain the result as below:
712"‘1 i{[sin(n'"7:rt) sin(n""C:rt)

t=1

—E(sin(n'"7xt)lFt_1)E(sin(n'"C:1:t)[Ft-1)]|Ft-1}
= (C02 + 0(n'2"))n’li {cos(n-"7:1:¢_1) cos(n—"Cxt_1)} + 0,,(1) = 0.

t=l

Proof of Lemma 2.3:

1. Using the law of iterated expectation and Assumption 2.1, we can obtain the

equation as below:

 

n2"_12 E{[cos(n""C:rt) — (E cos(n—"C:1:¢)]l~"t-1)]2 [ft-1}

t=l

27

n2" 1 sin2 (n "Crt- 1)Esin2 (n "(€t)+0p(1)
i=1

= n-‘Zsintrn-°<xt_.>((<a>2 + (1(n-2°» + 0.11)

“((0)22 Sin2(n_"Cxt_1) + 0,,(1).

t=l

By Lemma 2.6,

n2a_1;{E(Cos2(n'aC$tllFt-1) - (E(cos(n’"Ca:t)|F¢_1)2}

-1(Co)2Zsin2(n"C:rt-1)+op(1)
= (1/2)(C0)2 +0100)-

479C

By the law of iterated expectation and Assumption 2.1, we can use similarly

way as proof of Lemma 2.2 to obtain the equation.

n2"_1 Z{[cos(n'"7a:t) cos(n-"Cxa

t=l

-E(cos(n_"7:1:t)|Ft-1)E(cos(n‘"C;rt)|Ft-1)]|Ft-1}

= (C02 + 0(n'2"))n’lz {sin(n-"7:1:t_1) sin(n-"Cxt-1)} + 0,,(1).

t=l

By the Lemma 2.6, we know

n‘lz sin(n-"7.174) sin(n—"Crt-l) = 0 under 7 # C and 7,C Z 0

t=l

28

We can obtain the result as below:

it
n2"—l Z{[cos(n"a7xt) cos(n—“Csrfl

1:1

—E(COS("_07~Tt)lFt—1 )E(COS('n-aCxt)lFt—1)llFt—1}

= (C02 + 0(n'2“))n_1::{sin(n—“7x,_1)sin(n—“CI¢_1)} + 013(1) = 0.

i=1

For proof Theorem 2.1, we need another lemma as below.

Lemma 2.7 For the process It defined before, ifet satisfies Assumption 2.1, 2.2 with

0 < a <1/4 for any Cm E R, then

(J 211% >- (lawns-Em —E(sin(<——1:::1)1F ) l

 

 

\” ”’° iicos s<:::*>— (;%>ssa-si(cos<%fl>—E<css<‘——'":é“IF-1)}

Proof of Lemma 2.7:

By Lemma 2.1 and Taylor expansion, we can obtain the following equalities.

n—a- ézsin(i:_‘_ _(<2_202_)n°'%z (sin(%) — E(Sin(%)lFt—l))

t=1 t=l

MO. 200%; (Cit) _<_2_)na—%Z (004%) — E(cos(%)IFt_1))’ = 0,,(1)

(C202

= 012(1)

 

 

So they suffice to show.

/ n-a"%zsin<9—“)-<§s>na-s (mm—Esmeflnao l

no

”_0_%§Sin(%-fl) — (%)7i0‘%; (sin(§;‘—f‘) — E(sin(—C"‘:;‘I)IE_1) ( )
i r? = 0P 1
n-a-sz cos(gg—ii) — (%)n“‘%2 (cos(%§—‘) — E<cos<fli+l>IFH>

 

 

 

Proof of theorem 2.1:

By Lemma 2.1 and 2.7, VC 6 R

”WSW qn_it)_ 2 )na-i (sin(%)—E(sin(%)|fl_l))l=0p(1).

 

 

 

—(C202 t=l
2 1 "
WZZcoq §:_‘)_( ((272)710-52 (cos(%) — E(cos(-C;:—t)|Ft_1))l = 0,,(1).
t=1
So these suffice to show that:
2 1 n t 2
(gags—s; (sin(%) — s(si no” 1m 11):; NO( ’32?
2 1 n t 2
@7107; (cos(cnia) ‘_ E(COS(C—axt)lFlt_1))—LN(O,'CT‘2). VC E R

By the martingale difference central limit theorem (Hamilton (1994) p.193-195) and

Lemma 2.2 and 2.4, it follows that.

(n’a‘i :sin(%),,n . .. ”a 528m (cmft), 71-0—5 QZCOS (Git),
t=l

 

 

 

(“szcos (me‘n ‘1 N(0,AI)

. . . . - ‘ 2 2 2 2
where A 18 a 2m x 2m matrix that dlagonal elements is (215?, . . . W, a? . . . W).
The other elements are zero. Cl

30

Proof of Theorem 2.2:

First we know that if for all k, Xnk—p—rX as n ——> 00 and

lim;H00 lim supn_,00 Ean — Xnkl = 0, then Xn—p—sX. This is , for all C,

31.“; E exp(iC X11) = 330,113; [E eprCXnk) - E eprCanl + 3371123013 eprCXnk)
= Eexp(iCX) + 0(1)

Because lim lim |Eexp(iCXnk)— Eexp(iCXn)| g |C|klim lim Ean;c — an.

kn—ooo n—soo

Second, we have Fourier series

m =()aO/2 +2] ajcosm.) )+b sin(n»)
j=l

where aj = n‘lffflcos(ja:t)T(:r)dx and bj = n‘lfffl sin(jxt)T(:L')dLr for j Z 0. Noting
that ,u = 27r'1f:r T(zr)d:1: = 0.0/2,

n
n'a—1/22(T( (n amt) =Z[a- n’“ 1/2Z:cos( (n O‘jzt) )+bj n‘“ 1/2::sin( (n “jxt)]
t=1

Now set

N
A
A
H.
II M 75--
H
K).

”(‘6 + bib—l 214% + (EDI/2

71

k n
x Zan-a-W Z cos(n-“jx,) + bjn'a'l/2 Z sin(n—“jay”

j=l t=1 i=1

and

TI 71

k
X”: Z( ajn ‘0‘ l/2Zcos(n"‘j113t)-+-bj’n " 1/2Z:sin(n_°‘jart))

j=1 t=1 t=1

From Corollary 2.1, we know.

at _ 00 ._
Xnk—smo, 2a 2 Z (a 2><a§ + b?»-

1:1

31

Also by the Fatou lemma (see Chung (2001, p.45)) and the inequality ((1 + b)2 <

2a2 + 2192,

lim lim sup(E|Xn _Xnk|)2

k—’°° n—ooo

=lim lim supE (En a 1/22( aj cos(n_°j$1) + bj SiIl("-aj$1)))2

k—*OO _§
" °° jzk t=1

S 2klim limsupiiaja En'l 20Z‘chofl (n 0‘th )ncos( "01%)

—100 _.
"°° '=1cz=1c 1—13=1

+2 klim lim sup :3 20.: bjb, E1144" 2 2 sin( najxt )sin(n-“1%)

—'OO _,
" °° j=k 1:1: t: 13:1

k.)

We will only consider the first term, since the second can be dealt with analogously.

First, we consider the case that for t < s and j # 1.
IE cos('n“aj:r¢) cos(n-01$,”
= I(1/4)E(exp(n’aijxt) + exp(—n“°‘ij:rt))(exp(n‘°il:c3) + exp(—n'“il$t)|

g (1/4) Eexp(n_°i(j — l):1:0)EH exp(n_°‘i(j — l)ep) H exp(—n‘°il€q)

p=l q=t+l

 

+(1/4) Eexp(n"ai(j + l)xO)EH exp(n"°i(j + l)5p) H exp(n'°11c

q=t+l

 

+(1/4) Eexp(n’°i(t — j)x0)EH exp(n i(l — j)e p)H exp(— °ileq)

12:1 q: —t+l

 

+(1/4) Eexp(n’ai(—j — [)170)EH exp(n’°i(—j — 05,) H exp(—n‘°ileq) .

p=l q=t+1

 

 

(2.31)
Because E exp(n‘°i(j + l):1:0) is bounded by 1, We can rewrite equation (2.31):

lEcos(n ajxt)cos(n O’l:1:s)l <EI:[exp(n ai(j+l) up )H exp(n “is! q)(2. 32)

q: —t+1

 

For Equation (2.32), we separate four cases to discuss.

32

1. O < n“°‘(j +1) 3 (1/2) and 0 < 11""! S (1/2)

By Taylor expansion and de Jong ((2001) P.6-7), we can obtain the inequality

as below:

IEexp(n"ai(j + l)5,,)| S 1 — (1/6)n"2a((j + [)0)2 (2.33)
Using the same method, we can obtain another inequality:

|Eexp(n-°115q)| g 1—(1/6)n-20'(la)2 (2.34)

We can rewrite Equation (2.32) as below:

 

t s
EHexpm-w +z>ep1 H expo-“215.11
p=l

q=t+l

 

t s

H (1 — (1/6)n‘2°‘((1' + 110)?) H (1— (won-2000)?)

<

 

 

exp(ZIog(1— (1/61n-20((j + 11012)) exp( 2 logu — (won-200012))

p=1 q=t+l

 

 

t s

s exp(§j (—<1/6>n-20(<j + no)?» exp( 2 (—(1/6)n-2“(zo)2>)

pzl q=t+l

= exp(-(1/6)n’2at((j + lW2) €XP(-(1/6)n_2a(8 -t)(10)2) (2-35)

and the same inequality as (2.35) will also hold for t Z s. For j = l, assuming

again that t S 3,
IE cos(n-“33131) cos(n—“jig”

= (1/4)|E(exp(n"°z'ja:t) + exp(—n’a'ij:rt))(exp(n'°ij:cs) + exp(—n'°"ij:1:3))|

33

S (1/4)IE(GXP(”_GU($t + 335))I +(1/4)IE(exp(n—°1'j(:ct — 1‘s)”
+(1/)4IE(exp(n“°z'j(—:rt+:1:s)I+()1/4IE((expnaij(-$1—$s))l
s exp<—<1/6>n-2at<2j>202) exp<—<1/6>n-20(s 411202)

and again the same inequalities holds for t Z 5. Therefore,

«3 oo
- - -2a—l -0
klirnonlir’roio sup 2 E ajazEn in in cos( n OEjam) c)os(n 1:123)

j=k 1:1: t: 1 3:1
00 00
3 lim limZZ lajnad
k—roon—wo
j=k l=k

71’2" 1: Zexp(— (1/6) 20‘t(j + [)202) exp(—(1/6)n"2o‘(s — t)(la)2)

t==lsl

(2.36)

Because the last item of Equation (2.36) is independent of t, under s > t we

can rewrite Equation (2.36) as below:

lem limsupZZaJ-alEn—20 1:200“ 11 “just )cos(n O'l:1:3)

"—‘°° j=kl=k 1::131

00 00
< | lim limsupZZajaz

kH°° "“°° =k 1:1:

K)

Mn: exp<—<1/6>n-20't<j +1>2021exp(—<1/6>n*20<s — woof)!

X71720”: exp(-(1/6)n"2°t(j + 0202) Z eXp(-(1/6)N'2°‘(é>‘ - t)(10)2)

t=l s—tzl

34

(2.37)

The last item of Equation (2.37) can be calculate as following:

"11120 2 ex1)(-—(1/6)%(—t)(10)2)

s— t: 1

= :3 exp(— (>n1/6 °s—( 4111012)
s—t=1
=€XP(—(1/6)”_2 (10) )(1-exp(-(1/6‘)n"""(10)2))-1
= (exp((1/6)11’2°‘(l0)2) — 1)”1 (2.38)
Under 0 < n’al S 1/2, we have the following inequality by Taylor expansion.

exp(( ((1/6)n 2"(210) ) — 1 2 (1 /6)n-2°(1a)'~’

From this inequality, we can obtain the following relationship by inverse this

inequality.
(exp((1/6)11—2“(la)2) — 1)‘1 S 6112000)“2 (2.39)

Combining (2.38) with (2.39), we can obtain the result as below:

311.130 2: exp(— (11/)6 )1'2°(s — t)(la)2)
8— i=1
= (exp((1/6)11—2°(l0)2) — 1)‘1 S (1 + (1/6)11_2"(la)2 —1)'1 S (3112"(10)'2
(2.40)
Using the same method, we can obtain the other inequality:

limooZexpE (1)/6)11’2°‘t((j + [)0 )2 )3 6112°((j + ()0)’2 (2.41)

35

We substitute (2.40) and (2.41) into (2.37). The Equation (2.37) can be rewrit-
ten as below:

71

(2.37) S 3111307111130 supzz lajllalln‘l(6(10)‘2)z exp(—(1/6)11‘2“t(j + 0202)

j=k (:1: t2]

M8
M8

S lim lim
bacon—100

lajllazl'n‘l(6(101‘2)(6'112°((j+l)0)‘2)

KI.
II
a.
N
ll
3'

M8
M8

=li1n lim IajIlazIn20‘1(6(10)‘2)(6((j + 001—2)
k—ooon—ooojzk (:1:
< um limZ aim 2° 1(6(ja)‘2)(6((2j)0)‘2)
j: —k

+£33.31): Z 111111111120“(6001-2110+1101”)

j=k l=k,l;£j
oo 00
. 2 2 .
S CHEESE; aj + (Z Iajl) ) for some constant C1
J:

11‘°(j + 1) > (1/2) and 11‘01 > (1/2)

According to Theorem 2.1.4 of Lukacs (1970, p18) Eexp(n'“i(j +1)5p) < 1 if

11. 0‘(j + 1) E R\{0} and Eexp(n‘°ilsq) < 1 if 11-01 E IR\{0}. We assume
a = max{ sup IEexp(11‘°‘1'(j+1)€p)|, sup lEexp(11‘°‘11€q)|} < 1.

(n-°(j+1)|>1/2 |n“"l|>l/2
We can rewrite Equation (2.31) as below:

SH E exp(1j+1)£p) H E exp(11‘°1'15q)

q=t+1

 

IEcos(11‘°ja:t)cos(11 ”1:13)
_<_ atas—t S amax(t,s) (242)

and the same inequality as (2.41) will also hold for t Z s. For 3' = 1, assuming

again that t S s,

IEcos(n ajfllt) @8010]st

36

= (1/4)IE(exp(11‘°ij:1:t) + exp(—11‘“1'j:1:t))(exp(11“’1'ja:s) + exp(—11‘aijxs))I
S (1/4)|E(exp(n‘°ij(rv1 + 151))I + (1/4)|E(exp(n‘“ij(xt - Is))l
+(1/4)IE(exp(11‘°ij(—:1:t + 13))I +(1/4)IE(€Xp(n‘01j(—:1:t— 1133))I S amaX(t,s)

and again the same inequalities holds for t 2 3. Therefore,

lirgolimsupZZaJ-alEn‘ 2"“ 12:2:cos(1“"j;1:t)cos(11“"1:1:s)

"“°° 3': _k1: Ic t: 1 s: 1
oo 00
< lirn lim E E 103‘1101171- 20‘ IE" En amm”)
k—won—wo
j=k 12k t: 1 3:1
00 (X) n CXD
_<_ lim lim (12 ‘112"‘ IE 2 as + lirn lim E E Iajllalln‘2a‘12 E as
kdoon—‘oo k—aooTI—‘OO
j: _k t: 1 s: 1 j:k1:k,1¢j 1:1 5:1
00
_<_ lim lim (1211‘20‘ 1((1 — a)‘111a)
k—won—wo k
j:

00 oo
o o . _20—1 _ _1
+Iclggnlgno2 E |a1||a1|11 ((1 a) na)

j=k l=k.l;£j
00 (X)

S C211220(Z a? + (Z la.J-I)2). for some constant 02
J-

.0 < 11“”(j + 1) S 1/2 and 11—01 > 1/2

First, we assume a = sup IEexp(11“’1(j+l)sq)| < 1. We can rewrite
1711"’(J"H)|>1/2

Equation (2.32) with Equation (2.34) as below:

 

 

t s
EHeXp(11‘°‘1(j +1)e,,) H (mm-0115,)
=1

 

_<. H <1 — (116111-2011 + 11012) H Eexpwaz‘lsq)

 

37

exp Zlog 1(-(1/6)11 ‘2"( ((j+l)0 2))liI Eexp(11[€q)

p=1 q=t+l

t

s “ma—(1161114111 11110121112“

10:1

= exp(- (1/6)n ‘2°( (1 +110)2)03“ (2-43)

and the same inequality as (2.43) will also hold for t Z s. For j = 1, assuming

again that t S 3,
lE cos(n‘ajxt)cos(11‘°jxs)|
= (1/4)|E(exp(11‘°1ja:t) + exp(-—11“’1'j:1:t))(exp(11“'1j:1:3) + exp(—'n‘°1'j:1:s))|
s (1/41IE<exp(n‘°z'1(a:1 + rs))l + (1/4)|E<exp(n'“11(xt— maul
+()1/4IE((exp11“’1'.j(—:1:t+;1:3)))l+(1)/4)(|E exp(11‘°1j(—;1:t —1:s))|
_<_ exp(-(1/6)N‘2“t(2102)a””‘

and again the same inequalities holds for t 2 3. Therefore,

00 oo
lim lim supE E a.J-alE11‘2O‘IEn En cos( 11‘ 0.1.1:, )cos(n ‘°l;rs)
k—ooon—wo

3:11 1:1: 1: 1 3:1

gklirrgo'leZZIaJ-Hagln‘ ”2“ IZGXPC- (1/)6 )11‘(2°t( ((j+l)0 )2)SZas

j=k l: 1: 1:1
00 Tl 00

< - - 2 —2a—l _ —2a

_ £112,113ng aJn 2 exp(— (1/6)1 t( (230)2 )21H
1: = 8- =

+lim 001nm: Z IaJllalln‘Qa‘ 1tZ;cz)<1)(— (1/)611‘2”t(( 0)2)a"‘{:

j:kl= kl¢j s—tzl

38

3132015130211 n (6((21'1a 1 21((1—a1—‘a1

j: —k
+1ggogim2 2 11.111112 +101 21((1—a1-1a1
j: _=kl kl¢j

oo 00
S C3k11m(2;aj + (2'01”?)- for some constant C3
J= J

4. 11‘°‘(j +1) > 1/2 and O < 11‘“! S 1/2

First, we assume a = sup |Eexp(11“’1(j+l)5q)| < 1. We can rewrite
ln“‘(j+l)l>1/2

Equation (2.32) with Equation (2.33) as below:

8

1
EHexp(11‘“1(j+l)5p) H exp(n“"1l€q)
p=l

 

 

q=t+l

8

s HEexp(n‘°z‘(1° +1151) II (1—(1/6111‘2010121
p=l

q=t+1

 

 

8

S a‘exp( Z (—(1/6)11‘°(10)2))

q=t+l

(2 log( ((1— (1/6) 111-211(10) 11

q: t+l

 

           

= at exp(—(1/6)11‘2“(s — t)(l0)‘2) (2.44)

and the same inequality as (2.44) will also hold for t 2 s. For j = l, assuming

again that t S s,
lE cos(n‘ajxt)cos(11‘aj;rs)|
= (1/4)|E(exp(11‘°1j:rt) + exp(—11‘“17j:r¢))(exp(11‘°1j:rs) + exp(—11“’1j:1:s))|
S (1/4)|E(exp(11‘°1j(xt + 13))I+(1/4)|E(exp(11‘°1'j(:rt— 18))l

+(1)/4 IE(exp(11‘°1j(—Lrt +13) )|+( 1/4) lE(exp(11‘a-1j(-:rt —;r.s))|

39

Sa‘exp( (1/6) ‘2“(8 -t)(10)2)

and again the same inequalities holds for t 2 .9. Therefore,

(x3 00 n n
Alim lim sup E E (zJ-alE11‘2C"1 E E cos(n‘ajzrt)cos(11‘°l:rs)
‘—+oon—+oo

3:1: 1:1: 1:1 3:1

I(1)-11111111 2012012: BXP(- (1/6) )n‘2“(S-t)(10)2)

M8

00
S lim lim
k—vocn—wo Z

j=k 1:}: 3— =1
00
< - - --2a 1 _ 2
_ klmgonlgnoa k ;a :exp(— ()111/6 20(5 t)(10) )
J=‘ 8-

+1111;O 11m: 2: |(1J-||a1|11‘ 211— 12a 2 exp(— ()111/6 ‘2“(s—t)(l0)2)

j===klk.l;éj t=l s—=tl

S 11111 lim a311‘1((1 - a)‘la)(6(l0)‘2)

+lim 00hm: Z laJ-llalln‘ ((1—a)‘1a)(6(10)_2)

j=kl= kl¢j

(X)
S C4klim (2;) a- + (2k IaJ-l) 2.) for some constant C4
1= 1

00 00
From four cases before, if it can be shown that Z IaJI < 00 and Z a; < oo,

j=0 j=0

this proof will be complete. These conditions hold because 2 (112+ b?) < oo

1:0

and Z (laJI + Ile) < 00 by the assumptions on T(..) (See Apostol (1971) p340.)
j=0

E]

40

CHAPTER 3

Further results on the asymptotics
for nonlinear transformations of

integrated time series

3. 1 Introduction

This chapter proves three results about functions of integrated processes. Our first
result is an extension of a result in Park and Phillips (1999), where it is proven that

for integrable functions T(.) and for I(1) processes :1},

n-W 21m) A» ([0 T(s)ds)L(1,0), (3.11

where L(t, s) is a two-parameter stochastic process called (Brownian) local time. The
remarkable thing about this result is that it establishes limit theory for a function of
an I(1) process that has not been rescaled by 11‘1/2. Park and Phillips establish the
above result under some regularity conditions on the I(1) process wt and the integrable
function T(..) In this paper, we show that Park and Phillips’ regularity conditions for

the above result can be relaxed and also that their result can be extended to yield,

41

forOSa<1/2,

oo

11‘1/2‘0 iTM‘Ol‘t) 41—1 ([- T(s)ds)L(1,0). (3.2)

A central tool for the proof of this first result is a lemma that was recently established
in de Jong (2001). Also in Park and Phillips ( 1999), it is shown that for functions

T(.) that satisfy
T(/\;1:) = V()\)H(;r) + R(;r, A) (3.3)
under conditions on R(., .) that basically serve to ensure asymptotic negligibility of

1/(111/2)‘111‘1 Z R(.1:¢,111/2), (3.4)

1:1
we have

n

1/(111/2)‘111‘IZT(:1:t)—d—+/0 H(0l/V(1‘))d'r, (3.5)

t=l

where 02 = limnaoo 11‘1E1r,2,. Again the interesting aspect of the above result is
the fact that it considers integrated processes that have not been rescaled by 11“”.
Functions T(.) that satisfy the appropriate condition are coined asymptotically homo-
geneous by Park and Phillips. The asymptotically homogeneous condition is trivially
satisfied for T (:r) = lxl“ for a Z 0, but is general enough to also deal with functions
such as T(zr) = Ier“ log Ier for all a 2 0. In this paper, we show the more general

result that whenever for functions H ( ) and V(.) we have
V(/\)‘1T(A:r) —> H(.r) as A —> 00 (3.6)
in L1 sense, we have for 0 S a < 1 / 2, under regularity conditions,
11(111/2‘a)‘l11‘1iT(11‘°:rt) —d+ folH(0W(1))d1‘. (3.7)
1:1

Therefore, we show that Park and Phillips’ class of asymptotically homogeneous func-

tions can be extended, and we consider 11‘0er for O S a < 1 / 2 instead of Jr; as the

42

argument for T ()

A third result that is proven in this chapter concerns averages of the type

11‘ 12:;[11‘1/211 l‘ mI(1 ‘12/ It > C") (3.8)

and

n

11*Z1n-1/21,|-"11(1n*1/21.| > on), (3.91

where m > 1. While it has been shown in de Jong (2001) and Potscher (2001) that

under regularity conditions for locally integrable functions T(.) we have

11‘1 iTM‘l/Zrt) -—d—> /1T(0l/V(1))d1‘, (3.10)
1:1 0

it is yet unknown what happens to functions T(.) that are not integrable. Using a
“clipping device” involving a deterministic sequence on that converges to 0 with 11, it

will be proven that for 111 > 1,
(1n —1)c:,‘m11‘1Z[0‘111‘1/2zrtl‘m1(0‘111‘1/2:rt > on) —d1 L(1,0), (3.11)
and also that

(1/2)(m- c}‘ ‘m’ 11‘1210‘ 11‘ Wxtl‘mlfla‘ln‘lflxtl > C”) —d—> L(1,0).(3.12)

3.2 Assumptions and result for integrable func-
tions

Identically to Park and Phillips (1999), linear process conditions for 51:1 are assumed
$1 ‘-‘ 131—1 + wt: (3.13)
where 1111 is generated according to

wt = 291511514 (3.14)
11:0

43

where at is assumed to be a sequence of i.i.d. random variables with mean zero, and
where it is assumed that 2:10 (1'11 51$ 0. In addition, we will assume that .120 is an
arbitrary random variable that is independent of all w, t Z 1. The main assumptions

used in this paper are Assumption 2.1 and 2.2 from Park and Phillips (1999):

Assumption 3.1 2::0 [cl/21160.1) < 00 and E5? < oo.

Assumption 3.2
(a) Ziokl‘rbkl < 00 and Eletl” < 00 for some p > 2.

(b) The distribution of e, is absolutely continuous with respect to the Lebesgue measure
and has characteristic function 1/1(s) for which limsnoo s"1,b(s) = 0 for some

17>0.

Assumption 3.1 guarantees that n‘1/211:[m] :> 0W(r) where “=>” denotes weak con-
vergence in C'[0,1], i.e. the space of functions that are continuous on [0,1], while
Assumption 3.2 in addition also guarantees a convergence rate for a Skorokhod repre-
sentation of n‘l/ 2131",]. Several of the manipulations in the proofs of the results in this

paper require the use of local time L(., ..) Local time is a random function satisfying
1

L(t,s) =lirr(1)(2e)‘1/ I(|W(r) — s| < e)dr. (3.15)
5“” 0

See Park and Phillips (1999, p. 271-272) and Chung and Williams (1990, Ch. 7) for
more details regarding local time.
Park and Phillips (1999) establish the following result for integrable functions of

integrated random variables:

Theorem 3.1 Suppose that T(.) is integrable and Assumption 3.2 holds with p > 4.

If T(.) is square integrable and satisfies the Lipschitz condition
|T(JI) - T(yll S Old? - 11!1 (3-16)

44

over its support for some constants c and l > 6/(p - 2), then

n-l/2iT($t)—d1(/Oo T(s)ds)L(1,0). (3.17)

—00

For differentiable functions T(.), we need to set l = 1, implying that we need p > 8
in order for the theorem to work. In order to improve the above result, we needed

the following useful lemma, that was established in de Jong (2001):

Lemma 3.1 Under Assumption 3.2, for all y E R, 6 > 0, and 11 Z M for some value

of A4,
P(y s 114/21,. s y + 6) 3 Cd. (3.18)
where C and M do not depend on y, (5, or 11.

Using this lemma, we were able to improve Park and Phillips’ result and show the

following quite general result:

Theorem 3.2 Suppose Assumption 3.2 holds. Also assume that lT(:1:)| S R(:r),
and assume that R(.) is integrable, continuous on R, and monotone on (0,00) and

(—oo,0). If T() is continuous, then for 0 S a < 1/2,

n‘l/2‘O‘ :T(n‘°a:t) —d-> ([00 T(s)ds)L(1,0). (3.19)

Compared to Park and Phillips’ theorem, we have completely removed their Lipschitz—
continuity condition and weakened it to continuity, and in addition, their requirement
on p has been removed. Also, weights n‘° for 0 S a < 1 / 2 are allowed for. While no
H(.) function such as present in Theorem 3.2 is explicitly used in their Theorem 3.1,
from Park and Phillips’ proof it is clear that existence of such a function is implied.

Therefore, Theorem 3.2 is a “clean” improvement to Park and Phillips’ Theorem 3.1.

45

3.3 Asymptotically homogeneous functions

In this section, we improve Park and Phillips’ (1999) result for asymptotically homo-

geneous functions. Park and Phillips assume that
T(Aa‘) = V(A)H(:1:) + H(.r, A) (3.20)

and they show that

n

1/(n1/2)‘1n‘1 277.1,) —d—1/0 H(0W(r))dr (3.21)

i=1

if either

a. |R(;r, A)| S a(A)P(:r), where lim supxqoo a(A)/1/(A) = 0 and P is locally integrable,

01'

b. |R(:1:,A)| S b(A)Q(Aa:), where lim supka b(A)/1/(A) < 00 and Q is locally inte-

grable and vanishes at infinity, i.e. Q(:1:) —1 0 as |a‘| —> 00.

In this paper, we redefine their notion of an asymptotically homogeneous function,

as follows:

Definition 3.1 A function T(.) is called asymptotically homogeneous if for all K >

0 and some function H (.),
K
[\lim / |1/(A)‘1T(A.1:)- H(x)|d:r = 0. (3.22)
_m _K
Obviously from the dominated convergence theorem it follows that if for some 1/(.)
and H ( .), pointwise in :r,
V(A)‘1T(A;1:) —> H(ar) as A —> oo (3.23)

and |1/(A)‘1T(A:r)| S C(r) for a locally integrable function G(.), then T(.) is asymp-

totically homogeneous. Below, we will call a function monotone regular if for some

46

01,...,a , T . is monotone on a-,a-+1 for j = 0, ...,q setting 00 = —00 and
q .7 J
aq+1 — 00).

The main result of this section is the following:

Theorem 3.3 Suppose Assumption 3.1 holds. Also assume that T () is asymptoti—
cally homogeneous. In addition, assume that H(.) is continuous and T(.) is monotone

regular. Then, for 0 S a < 1/2,

1/(111/2“")‘1n‘l ZTM‘O‘xt) —d—+/0 H(0W(r))dr = [00 H(0s)L(1,s)ds.(3.24)

It is also possible to show that our definition of an asymptotically homogeneous

function is more general than Park and Phillips’. Under Assumption a. above,

K K
/ |1/(A)‘1T(A;r) — H(.r)|d;1: = V(A)‘1/ IR(1:,A)|d;r
-K —K
S a(A)1/(A)‘l /K P(1r)d:r —1 0 (3.25)
-K

as A ——+ 00 if P(.) is locally integrable. Under Assumption b. above,

K K
/ |1/(A)‘1T(A:I:) — H(r)|d:c = 1/(A)‘1/ |R(1:, A)|d:1:
—K —K
S b(A)1/(A)‘1/K Q(A;r)d:1: —1 0 (3.26)
—K

as A -—> 00, because lim sup)H00 b(A)1/(A)‘l < 00 and ling...” [_KK Q(A:1:)d:1: = 0 by
boundedness of Q(.) (which is also assumed in Park and Phillips (1999)). Therefore,
obviously the set of functions that is “asymptotically homogeneous” in this paper is
wider than in Park and Phillips ( 1999). But clearly, most functions that one may
expect to be useful for applications should be expected to already be in Park and
Phillips’ class of asymptotically homogeneous functions, and the main achievement
of our analysis is the redefinition of the class of asymptotically homogeneous functions

to as large as possible a collection of functions. It appears to us that the above result

47

should be close to the limits of what should be possible in this setting, and for
the authors of this paper, it is hard to see how the above definition of the class of
asymptotically homogeneous functions can be relaxed further to yield an even larger

function class that generates similar behavior.

3.4 Nonintegrable functions

In de Jong (2001) and Potscher (2001) it is proven that under regularity conditions,

in spite of possible poles in T(.), as long as f_KK |T(:r)|d:1: < 00 for all K > 0, we have

n‘lZT(n‘l/2xt) 15-1/0 T(0W(r))dr. (3.27)

These results raise the question as to what will happen if a nonintegrable function of

an integrated process is used for T(.) in statistics of the form

11‘1 2 T(n‘lflxt). (3.28)
1:1

This issue appears to have never been tackled before in either the statistics or the

econometrics literature. This section explores this issue for functions

T(.r) = la:|""1(1: > 0) (3.29)
and

T(zc) = latl‘m, (3.30)

for m > 1. As it turns out and is perhaps to be expected, the observations “close
to zero” take over the limit behavior of the statistic in this case. We will need a
“clipping device” and we construct statistics similar to those constructed in Park and

Phillips (1999) for integrable functions. Our first result is the following:

48

Theorem 3.4 Let on = n‘(2p+1)/3p+’7 for some 1) > 0 such that —(2p+ 1)/3p+17 < 0.

In addition, assume that
T(zr) = Irrl‘m (3.31)
for some m > 1. Let d" = f; T(zr)d;1:. Then under Assumption 3.2,

dgln‘lZT(0‘111‘1/2271)I(0‘111‘1/2:rt > en) —d1 L(1,0). (3.32)
1:1

Clearly, in the above theorem (1,, = (111 — 1)‘1(c},‘"’ — 1), but we choose the above
formulation to bring out better where our rescaling factor (1,, originates from.
The proof of the following “two—sided” version of the above theorem is analogous and

therefore omitted:

Theorem 3.5 Let cn = n‘(2”+1)/3”+" for some 1) > 0 such that —(2p+ 1)/3p+17 < 0.

Assume that
T(zr) = |$|‘m (3.33)
for some 111 > 1. Let d, = 2 f6: T(x)d:r. Then under Assumption 3.2,

d,‘,1n‘l zT(0‘1n‘1/211)1(|0"n"/2;rt| > C") i1 L(1,0). (3.34)
1—1

The above theorems leave the issue wide open to what function class the above theo-
rem can be extended. The line of proof employed in the Appendix may allow for some
generalization, but it is not clear to the authors what the outer limits are for which a
result as the above might hold. Furthermore, the clipping device is intriguing, and one
could conjecture that for the above definitions the theorem will remain true if cn in the
theorem and in the definition of (1,, were to be replaced by minlstg n‘l/ 21:11 (.131 > 0)

and minlggn n‘l/ 2|1‘g' respectively.

49

Proofs

Throughout this section, to improve readability, we will assume for every proof that
02 = 1.
Below we use the following definitions, which are identically to Park and Phillips

(1999):
l n
Nn(1/,,; a,b) = / I(a S unn‘1/21:[m] S b)dr = 11‘1 Z [(0 S unn‘l/zart S b),(3.35)
0 1:1
and

1
N(1/,,;a,b) 2/ [(0 S 11,,11‘1/2l/V(r) S b)dr. (3.36)
0

In the proofs below, M and C are the constants from Lemma 3.1. The following

lemma from Park and Phillips ( 1999) was needed in order to prove our results.
Lemma 3.2 Under Assumption 3.2, as 11 —1 oo,

E(N,,(V,,; 0, 6) — Nn(u,,; M, (k +1)(5))2 S c(6n‘11/;1)(1 + k62nlog(n)u,:2) (3.37)
and

Nn(1/,,;0,1r,,) = N(1/,,;0,1rn)+ op(n‘(2p")/3”+E) (3.38)

or 11,, 2 V,,11‘2(p+1)/3P and an e > 0.
y

Proof:

See Park and Phillips (1999). C]

We are now in a position to prove the main theorems of this paper.

50

Proof of Theorem 3.2:

Define TK(LII) = T(sr)I(|.r| S K), T,’{(:r) = T(a‘)l(:1: > K), and Tflr) = T(.r)I(.1: <
—K). We will show that

im lim sup Eln‘m‘“ Z T,’{(n‘°:rt)| = 0 (3.39)

l
KHOO ""°° 1:1

and the same argument, mutatis mutandis, will hold for ”7112—02; Tfln‘aart).

Then, we will show that for all K > 0,
K

n‘1/2‘a :TKM‘OQ) —d-> ([KT(s)ds)L(1,0), (3.40)

and the result then follows (for a formal proof that this is sufficient, see for example
the start of the proof of Theorem 1 of de Jong (2001)). To show the result of Equation

(3.39), note that for all K > 0,

M
|n‘1/2‘°ZT(n‘°a:t)I(n‘°att > K)| S M11‘1/2R(K) —1 0 (3.41)

t=1

asn——>oo,and

Eln‘l/2‘or Z T(n‘°x¢)I(11‘O:L‘t > K)|

t=lU+l

= E|Zn‘1/2‘° Z 1(n-0.1.1I(Kj < n‘asu s K(1‘+1))|

j=l t=l\11+1

g EZn-l/H Z R(Kj)I(Kjt‘1/2n" <1—1/2x. g K(j+1)t‘1/211")

j=l 1:M+1

3 2 11‘1/2 X R(Kj)CKt“/2
3:1 1:1

3 C(sup n‘1/2 214/2111}: R(Kj)
"31 1:1 3:1
= 0' / H(KU])(1(K1')
1

51

oo 2K 00

= C /K R(K[;1:/K])d:r = C /K R(K[a‘/K])d;r + C AK R(K[a:/K])da:

S C'(KR(K) + /°° R(:L‘)d;r) -—1 0 (3.42)
K

as K —> 00, where C’ = Csup,,21n'1/22;’=Jt‘l/2, and KR(K) -—> 0 under the
assumptions of the theorem because
2K 00

R(2K)K S [K R(:1:)da: S /K R(:r)d:r —> 0 (3.43)
as K —1 00. The first inequality follows from the assumed boundedness of |T(.)|
by H(.) and the assumed monotonicity of H(.), and the second is an application of
Lemma 3.1. This completes the proof of the result of Equation (3.39). The remainder
of the proof follows the line of proof of Park and Phillips (1999, proof of Theorem
5.1), but some modifications will be made. In order to show the result of Equation
(3.40) and thereby make the proof of Theorem 3.2 complete, define for 6 > 0

K/6 1
T6(:1:) =/ T(j6)1(j6__< 11‘ O‘attS (j +1)6)dj, (3.44)
—K/6

K-/6 l

and note that for all K > 0, f_ M6

10 5 S n‘°r1< _ (1' +1)5)d1= 1011‘“le < K)

and therefore

11

Eln‘1/2‘0 201111-01.) — T"(11‘°.1:,))|

t=l

K/ 6— l n
— —E| /_ ”2 “Z ZT< (1'61— 1(n-01111111 s 0°11: (1' +1161d1'l

K/6 ,_1
K/6—1 n
s sup sup 11(11— 1(n/11E / W Z 1(16 s m 3 (1+ 1111.11
xE[—K.K]I’6[—K,K]:|x-x’|S6 —K/6 t=1
= sup sup |T(.1:) — T(a:')|11‘1/2‘° Z P(—n"t‘1/2K S t‘l/2att S nat‘1/2K)
xE[—K.K]:r’E[—K,K]:|.t—:1:’|S6
n
S sup sup |T(a:) — T(zr')|11‘1/2 Z 2CKt‘1/2

:1:€[—K,K]x’E[-K,K]:|J:-:r’|S6 t=l

52

S 2C’K sup sup |T(;r.) — T(a")| —> 0 (3.45)
x€[—K,K]:r’e[—K,I\'j:1.r—x’|S6

as 6 —> 0 by continuity of T(.), where the second inequality is Lemma 3.1. Therefore,

we can consider n‘l/Z‘O‘ 2;, T"(11‘°‘;1:t) instead of n‘l/z‘“ 2;, TAN-0‘11). Now

11‘1/2‘0‘ Z T"(11‘01‘t)
(=1

K/6—l n
= / roam-12:210.; s 110.113 <1+1161d1
K

’/6 1:1
K/6—1
= Z T(161n1/2"°Nn(n”2‘°;16, (1' + 1111, (346)
—K/6

and

K/6—1
| f «11 11111111101111.1121, (1' + 116111

K/6-1
—/ T(j6)dj11l/2‘0Nn(nl/2‘a;0,5)I = 0,,(1) (3.47)
—K/6

because by the Cauchy-Schwartz inequality,

K/6—l K/é-l
E(/ T(1'61n1/2'°Nn(n”2-°;1'6,(1 + 1111111 — / T<161d1n1/2‘°Nn(n”2‘°; 0,1112
—K/6 —K/6

K/6 K/6
3 111—20 / 1211112111 / E(N..(n‘/2‘°;1'6, (1' + 116) — Nu(n”"°;0,6))’dj
-K/6 —K/6

K/6 - K/6 -,
g 111-20 / 110112111] C(6n-3/2+“1(1+ ljlozlog(n)112")dj
—K/6 —K/6

K
S 11‘1/2‘°(1/5)(‘/—K R(s)2ds)c2K(1+ K6n2° log(n)) = 0(1), (3.48)

where the second inequality is Lemma 3.2. Therefore, it suffices to consider

K/6—l K—6
/ T(j6)djn’/2‘°Nn(11’/2‘°;0,5) 2 6‘1/ T(s)dsnl/2‘°N,,(111/2‘“;0, (5).
—K/6 —K

Now note that
Inl/2‘ON,,(11l/2‘°;0,5) _ n1/2-aA1(nl/2—-a; 0, (”I : Op(nl/2-an-(2p—l)/3p)

53

= 010("(1-11/21/(3111) = 0,,(1) (3.49)

by the second part of Lemma 3.2. Therefore,

K-6 K—6
|/ T(s)dsn1/2‘0Nn(nl/2“’;0,6)—/ T(s)dsn1/2‘0N(nl/2‘°;0,6)| = op(1),(3.50)

—K -—K

implying that it suffices to analyze
K-6
(/ T(s)ds)(6‘1n1/2‘°N(n’/2‘°;0,6)). (3.51)
—K

As 11 —+ 00,

6‘1n1/2‘“N(nl/2“'; 0,6) ——> L(1,0) almost surely, (3.52)

as explained in the text following Lemma 2.5 of Park and Phillips (1999). In addition,

as 6 —1 0, by continuity of T(.),

[:4 T(s)ds —> /_: T(s)ds. (3.53)

Therefore,

K

n‘1/2‘° ZTKUl-OIQ) —d> (/_KT(s)ds)L(1,0), (3.54)

implying that the condition of Equation (3.40) is now verified. This completes the

proof. 1:]

For the proof of Theorem 3.3, we need the following lemma:
Lemma 3.3 Under Assumption 3.1, for any K > 0,
n 1
n‘1 2 1(n—“2:1. g :11 : / 1(n/(r1 5 11111, (3.551
1:1 0

where “:>” denotes weak convergence in D]—K, K] (i.e. the space of functions that

are continuous on [0,1] except for a finite number of discontinuities )

54

Proof of Lemma 3.3:

Pointwise in .13, the result follows from Theorem 3.2 of Park and Phillips ( 1999), and
therefore it suffices to show stochastic equicontinuity of 11‘1 2;, I (n‘l/ 2.1:, S 11:). By
the Skorokhod representation, we can assume that SUE-510,1] |n'1/ 211m] — W(r)| 333+ 0.
Then for 11 large enough, SUPre[o,1] ln‘l/ 2513],,” — W(r)| S 6 almost surely, implying

that for 11 large enough

11

sup sup In‘1 Z(I(n‘l/21:t S :r.) — I(n‘l/th S :r’))|
|:1:|SK 12’21:<J:’<:r+6 t=l

n
S sup 11‘1 21(13 S n‘l/za‘t S 1+ 6)
iIISK i=1

1
S sup/ I(1—6SW(1)S.1:+26)dr
0

IIISK
x+26
= sup / L(1,s)ds S 36 sup IL(1,s)] (3.56)
IIISK z—6 ISISK

where the equality follows from the occupation times formula (see Park and Phillips
(1999, Lemma 2.4)) and because 5‘1qung IL(1,s)] is a well-defined random vari-
able. The above chain of inequalities establishes stochastic equicontinuity of

11‘1 22;, 1 (11‘1/ 22:, S :13), which completes the proof. Cl

Proof of Theorem 3.3:

Because 911313191 n‘1/2lrtl = Op(1), it now suffices to show that for any K > 0,

1/(n1/2‘C’)‘111‘l ZT(n‘°xt)I(]n‘1/2:rt] S K) i1 /0 H(lV(r))I(]W(r)| S K)d1‘

i=1

K
= / H(s)L(1, s)ds. (3.57)
—K

55

Now, by Lemma 3.3, n‘IZI’=,I(n‘l/2xt S :11) => f011(W(r) S :r)dr. By the
Skorokhod Representation Theorem, we can assume without loss of generality that

11‘1 n: I n‘1/21‘, S a: - 1 I W(r S :1:)dr = c,, is: 0. Now for all 6 > 0, let
1 1 0

fl

31116 = 8111 = anl/2_a)—1"l_lZT(n—O$1)I(In_l/2$tl _<_ K)

t=l

K/a—i n
= p(nl/Q‘O)‘1/ 11‘1 :T(n‘°:rt)l(j6 S n‘l/zxt S (j+1)6)(lj,(3.58)

“/5 1:1
K/6—1 n
82,, : 11(111/2-01—1 / T(n1/2‘aj6)n": I(j6 g 114/21. s (j+1)6)dj,(3.59)
4‘75 1:1
K/a—i 1
83116 = V(n1/2‘“)"/ T(nm‘aflilf [(15 S W0) S (1+115ldrd11(3-60)
—K/6 0
K/6—1
5...... = val/“1“ / T(n1/2-°1616L(1,161dj
—K/6
K—6
= u(n1/2‘°‘)‘1/ T(nl/2‘03)L(1,s)ds, (3.61)
_I"
K 1
55m; = S5 =/ H(s)L(1,s)ds =/ H(l’V(1‘))I(|l4/(r)| S K)dr. (3.62)
—K 0
We will show that lim,5_1()limsup,,__.00 iSjné—Sj+l,n6] = 0 almost surely forj = 1, . . . ,4.

By the monotone regular condition, we can act as if T(.) is monotone without loss of

generality. For |S1 — 82mg] we then have

lim sup |Sl — S2,,6]

n—+oo

K/6—1 n
S lim sup V(n’/2‘“)"/ 11‘1 Z [T(n‘aa‘t) — T('111/2‘°j6)]1(j6 S n‘l/zxt S (j + 1)15).

71—400 [‘76 t=l

K/6—1 n
S lim sup V(n’/2“’)‘1/ 11‘1 Z |T(n’/2“’(j + 1)15) — T(111/2‘°j6)]1(j6 S n‘l/2:1:¢ S (j +1
n—ooo _K/6 1:1

K/6—1
S lim sup/ ]1/(n1/2‘°)"T(n1/2‘°(j+ 1)15)
n—wo -K/6

56

_,(,,112-.)-1T(,,.12-.J,)_ 1111+ 1161 + 1106111

+/K/6—1]H((j+1)6) — H(j6)]dj = /1(_6 |H(:1: + 6) — H(1)]d:1:, (3.63)
-K/6 —K

and as 6 —-> 0, the last term disappears because of continuity of H (), the second
inequality follows from monotonicity of T(.), and the third by our definition of an
asymptotically homogeneous function. To show that lim5_.0 lim sup,,_,00 ]Sgn5—S3n6] =
0 almost surely, note that

K/6—1 n
]1/(111/2‘°‘)"/ T(nl/2‘0j6)(11‘121(j6 S n‘1/2$¢S(j + 1)15)

K/6 t=1

_/0 [(36 g W(r) S (1+ 1)5)d7”)dj|

K/6—1
S 2cn1/(111/2‘C’)‘1/ |T(111/2‘aj6)|dj
—K/6
K K
S2c,,6‘1/ |1/(n’/2‘°)‘1T(n’/2‘°:1:)—H(a:)ldx+2cn6"/ |H(;1:)|d:1:=o(1)(3.64)
-K —K

almost surely under our assumptions and by the definition of on. For [53,,5 — 54mg] we

have
]S3n6 - 84nd]

K/6 1
s val/2'11“ / 1.111111101111114 / 1(16 s Mr) 3 (1+ 116111 — L(11611d1‘
—K/6 0

K/6 1
S u(n1/2‘°)‘1/ 6]T(n1/2“’j6)|dj sup |6"/ I(a: S W(r) S :1:+6)dr—L(1,:r)|.(3.65)
—K/6 lrlSK 0

By the earlier argument,

K/6
s11psup1/(nl/2‘“)‘1/ 6|T(n1/2‘0j6)]dj < 00, (3.66)
1121 6>0 —K/6

and therefore it suffices to show that as (5 -—> 0,

1
sup I6"/ [(13 S W(r) S :1: + 6)dr — L(1,:1:)| —+ 0. (3.67)
IIISK 0

57

By the occupation times formula, the above expression satisfies

sup [6‘ [W5 L(,l s)ds— L,(1 1)]: sup |6‘/IH(L L(,11))ds|

IrISK |1|<I\

S sup sup |L(1,s)—L(1,;r)|—+0 as6—->0 (3.68)

IxISK s€[:r,:1:+6]

           

by uniform continuity of L(1, .) on [—K, K]. Finally, for |S4n5 — 've
K
lim I (11(111/2‘0)‘ 1T(n 1/2‘ 0‘ s)—H(s))L-(1,s)ds|
11—ooo —K
K
S sup IL(1,s)] lim / ]1/(111/2‘°)‘1T(111/2‘0s)— H(s)]ds = 0 (3.69)
lslSK ""00 —K

by the definition of an asymptotically homogeneous function, which completes the

proof. CI

The following lemma is needed for the proof of Theorem 3.4.

Lemma 3.4 For any sequence bn such that c,1 = 0(bn), under the assumptions of

Theorem 3.4,
(151113) lim T(j6c,,)I((j+1)6c,, > cn)I(j6c,, S b,,)d;’6c,, = 1. (3.70)
j: —0

Proof of Lemma 3.4:

This result follows because

ZT(j6c.,,)1((j+1)6c,, > 1.11011, 3 11.111351,
:/ T([j]6c,,)1((]j] +1)6c.. > Cn)1(ijl60n S b.1)d.‘.’50ndj
g/ T((j —- 1151,11((j + 1151:... > c..)I((j — 116C" 5 bn)d;‘6cnd1
1

58

=/ T(x)I(:r + 26c" > cn)1(a: S bn)d,‘,ldx

=0

1 11..
= (/ T(x)d;r)‘1/ T(cr)d:1:. (3.71)
I=Cn .z:=c,.(l—26)
Now because T(x) = |:r|‘"‘1(a: > 0), the last expression equals
(m —11(cl.""—11“(m—11“(<cn(1— 21111-11— bli'")
= (Cl—"’ —1)_’((Cn(1- 2501”" - 1)1—m), (3-72)

71

and because m > 1 and c" = o(b,,), the result now follows. A similar argument will

hold for a lower bound, which then completes the proof of the lemma. 1:]

Proof of Theorem 3.4:

1—1/m—a

Note that, for bn = CH for some a > 0 small enough that bn —1 0 and

d;1T(bn) -—1 0 as 11 —> 00,

n
dgln‘1 Z T(‘n‘1/21,)1(n"/2;1:t > c,,)
1:1
11
= dgln" :T(n‘1/2;1:t)1(n‘1/2:rt > cn)I(n"/2;1:t S bn)
1:1

+d,‘,1n‘l i T(n‘1/211)1(11"/2:r¢ > b,,), (3.73)
1:1
and the second term is 0,,(1) because
dgln‘liT(n‘1/2:rt)l(n‘l/2It > bn)
i=1
3 d;’T(b,,) —+ 0 (3.74)
by assumption. Now note that trivially, for all 6 > 0, defining Wn(1‘) = n"/ 2513111111

d;111‘1ZT(11‘1/21:t)1(n‘1/2;rt > cn)I(11‘l/21:t S bn)
1-1

59

oo 1
= 2d,:1/ T(W,,(r))I(W,,('r) > 0,.)1(Wn(r) S bn)I(j6c,, S Wn(r) < (j+1)6c,,)dr.(3.75)
i=0 0
An upper bound for the last term is

00 1
2110514111: A 1(w,(1~1 > cn11(w,,(1~1 3 1,1106% 5 Wn(r) < (j + 11151.,1111

g ZT(jdcn)I((j + 116e,. > c..)I(j<5cn S M121]0 I(jécn S Wn(1‘) < (1+1)6cn)dr

i=0

oo

= [macaw +1111, > 1.110612. 3 bn)d;1N,,(1;j6c,,, (1' + 1111,). (3.761

1=0

Similarly, a lower bound is

00

ZT((1+1)5C11)1(16611 > Cn)1((j+1)66n S bnldiianU-ijdcm (1+ 115%)(3-77)

i=0
We will only consider the upper bound and determine its limit, but the argument for

the lower bound is identical and renders the same limit. By Lemma 3.2,

EZT(j6c,.)I((j + 1161. > (2.11060. 5 b..1d.:lan(1;1‘6cm (1‘ + 1161:.) — Nn(1;0,6cn)l

i=0

3 :71”ch (0' + 1150+ > 1.111111, 3 bn1d;1<c<6c,./n1(1+(1(6c..12nlog(n1111‘/2

s (11:161. 211111.11111 + 1161:. > 1.111111, 3 1.11

1:0

X5‘IC;’(C(5Cn/n)(1 + ((1)11/(6011000021110s(11))))’/2- (3-78)
Now, by Lemma 3.4,

alsimlim sup d,‘,’6anT(j6cn)I((j + 1)6cn > en)I(j6c,, S bn) = 1, (3.79)

n-aoo j:0

and therefore the expression of Equation (3.78) converges to zero in probability if

052((611/11) + (cu/n)((1)11/(011))(61112'1’1108110)) —* 0- (3-80)

60

First, note that by assumption cgln“1 —> 0, and that the second part of the above

expression is
0(1),, log(n)) = 0(1) (3.81)

by assumption. Therefore, it suffices to consider

00

Z T(j5cn)1((j + U501: > c,,)1(j6c,, S b11)d;16Cn(Nn(1; Dim/(5611))- (3-82)

i=0

Now by the comment following Lemma 2.5 in Park and Phillips (1999),
Nn(1; 0, (Sag/(60,.) = L(1,0) + 0,,(1) (3.83)

if 60,, 2 n’(2p‘1)/3P+” for some n > 0, which is the case by assumption for n large
enough. Therefore, we only need consider

L(1,0)d;16cn fT<j6q.)I<(j + me. > calms. 5 b.). (3.84)

j=0
Now by Lemma 3.4, it follows that by choosing 6 arbitrarily small, the limit distribu-

tion will be arbitrarily close to L(1,0); and noting that the same argument will work

for the lower bound, this suffices to prove the result. I]

61

CHAPTER 4

Unit root tests when the data are a
trigonometric transformation of an

integrated process

4. 1 Introduction

Unit root tests were first studied by Dickey and Fuller (1979) with proof and simu-
lations. me this beginning paper, unit root testing became mainstream research in
time series econometrics. It is a widely believed that many time series in macroeco-
nomics are I(1) processes, as argued by Nelson and Plosser (1982). Economists have
concentrated on how to test for a possible unit root in data series. After the Dickey-
Fhller unit root test, Said and Dickey (1984) and Phillips and Perron (1988) proposed
revised unit root tests to take into account the possible autoregressive-moving average
in errors. Their papers corrected the drawbacks of the Dickey-Fuller unit root test.
For the development of unit root tests in econometric time series, see Phillips and
Xiao (1998).

In international finance and macroeconomics, there are a lot of nonlinear models

62

for time series, for example used for modelling the real exchange rate. For empiri-
cal reasons, researchers often use nonlinear transformations to transform integrated
time series. One important question is whether the unit root phenomena still exist
after transformation. The first paper to discuss this question is Granger and Hall-
man (1988, 1991). They used simulation to analyze the characteristics of unit root
tests when the data is a function of an integrated process. After Granger and Hall-
man’s paper, Ermini and Granger (1993) established some asymptotic properties for
transformations of I( 1) processes under normality assumptions. Following these three
papers, Franses and Koop (1998), Franses and McAleer (1998), and Kobayashi and
McAleer (1999) analyzed unit root tests when the data are functions of an integrated
process. They find that Dickey-Fuller tests are sensitive to nonlinear transforma-
tions; for example, it can happen that a variable is found to be nonstationary in
level, but stationary after transformations. They consider the logarithm transforma-
tion of integrated time series and propose the revision for sensitive problem when we
use augment Dickey-Fuller unit root tests under transformed integrated process. But
all these papers only study the logarithm transformation. About other functional
forms, they do not establish theoretical results.

This chapter establishes analytically what the asymptotic behavior of the Dickey-
Fuller unit root tests will be when the true data-generating process is a trigonometric
function of an integrated process. For example, the data could be generated as sin(:ct),
where :13, is an integrated series. This problem has been analyzed mainly through
simulations in Granger and Hallman (1991), and this chapter gives the mathematical
underpinning for their conclusions. Another paper that is related is Ermini and
Granger ( 1993); in that paper, various moments and covariances are calculated that
involve functions of integrated processes. Ermini and Granger’s (1993) results are

obtained by strongly relying on a normality assumption. In this chapter, we try

63

to relax the normality assumptions under Ermini and Granger. We only keep the
symmetric distribution of residual item,5t, and obtain the asymptotical distribution
of Dickey—Fuller unit root tests under periodic transformation of integrated process.

One important tool for the analysis of this chapter is provided in de Jong (2001).
In that paper, it is established that for functions T(.) that are periodic on [—7r, 7r],

for an integrated process x; that satisfies some regularity conditions,

1r

n—l/QZ(T(:I:,) — (27T)_1/ T(zr)da:) —d—+ N(0,02), (4.1)

~77

where ” —‘—i—>” denotes convergence in distribution. This paper extends the tools devel-
oped in de Jong (2001) somewhat in order to arrive at a complete asymptotic analysis

of the problem under consideration.

4.2 Assumption and main results
We consider a time series :5, generated by
as, = 513,4 + at (4.2)

where at is a sequence of independent and identical distributed mean zero random
variables with a continuous distribution, a mean of zero, and a variance 02. ft =
0(5t, 5,4, . . . ,51, 1:0) is the sigma field that includes all the information in 5} until time
period t. Below, let ,6 denote the regression coefficient resulting from a regression of
yt on yt..1, and let ,6” denote the regression coefficient resulting from a regression of
yt on yt_1 and a constant. In all results below, we will allow for both yt = sin(:ct) and
yt = cos(:ct), but as intuition suggests, for both choices of y, the asymptotic results
are identical.

For the convergence behavior of ,6 and ,5“, the following result can be established:

64

 

IF

Theorem 4.1 For the process :13, as defined before, for y, = sin(xt),
,6 —p—* Ecos(et) and [3,, l—r Ecos(s¢), (4.3)
and similarly for y, = cos(ltt),

f) —p—> Ecos(st) and )5), L Ecos(e)). (4.4)

. . p .
In the theorem above and elsewhere in this chapter, ” ——> ” denotes convergence in

probability. All proofs for this chapter are deferred to the Mathematical Appendix.
For the regression coefficients [3 and 15;“ the following theorem establishes root-n
consistency and asymptotic normality under the additional assumption that the dis-

tribution of at is symmetric:
Theorem 4.2 For the processes :5, defined before, if at has a symmetric distribution,
for y; = sin(:rt),

n1/2(,b — Ecos(5t)) —d’ N(O, V) and n1/2(p‘p — Ecos(et)) L N(O, V), (4.5)
and similarly for yt = COS(1,‘¢),

711/29: — Ecos(st)) —". N(0, V) and 721/203,, — Ecos(et)) —"—> N(0, V), (4.6)
where

V = (3/8)E(cos(et) — Ecos(et))2 + (1/8)E(sin(st))2. (4.7)

The above theorem implies that the Dickey-Fuller coefficient tests will go off to —00
at rate n - the same rate as would apply for stationary processes y; - and therefore
the Dickey-Fuller coefficient tests will asymptotically indicate stationarity. Finally,
we establish the asymptotic behavior of the Dickey-Fuller t-statistics f and t1, for
the coefficients of yt_1 resulting respectively from a regression of Ag) on 311-1 and a

regression of Ag, on yt_1 and a constant:

65

Theorem 4.3 For the processes :rt defined before, for y, = sin(xt), defining

c = (Ecos(st) — 1)(1— (E cos(et))2)_l/2, (4.8)
we have
n'l/Qf _‘L, c and n’l/pr —p+ c, (4.9)

and similarly for y) = cos(:rt),

n—l/2t 1» c and n'1/2f# —p—> c. (4.10)

Unlike Theorem 4.2, the result of Theorem 4.3 does not rely on a symmetry assump-
tion for the distribution of st.

From our results, it is clear that the asymptotic behavior of the Dickey-Fuller
coefficient and t-tests in terms of convergence rates is identical to that of the case of
stationary random variables, and that the t-test will asymptotically indicate station-
arity. This conclusion was also obtained through simulation in Granger and Hallman

(1991).

4.3 Conclusion

In this chapter, we introduced the unit root test under trigonometric transforma-
tions. As is shown in the preceding theorems, the trigonometric transformation of an
integrated process will result in a stationary process. When we use the Dickey-Fuller
unit root test under trigonometric transformation, the test will diverge to —00. These
results support the Monte Carlo simulation of Granger and Hallman. Compared with
the Ermini and Granger paper, we only keep symmetric distribution and relax all nor-
mality assumptions. From our proof, we can obtain more generalized results about

trigonometric transformation of an integrated process.

66

4.4 Mathematical Appendix

Proof of Theorem 4.1:

We will consider y, = sin(a‘t); the case yt = cos(;rt) is analogous. By the law of large

numbers for bounded martingale difference sequences,
n“l Z(sin(:rt) sin(x,_1) — E(sin(:rt) sin(xt_1)l.7-"t_1)) —p—-> O, (4.11)
t=2

and therefore in order to find the probability limit of (n — 1)‘l 221:2 ytyt_1, it suffices

to consider

”—1 Z E(sin(a:t)sin(:rt_1)|ft_1)

t=2

Tl

= n"1 Z(sin(:rt_1)2Ecos(et) + sin(:r¢_1) cos(n—1)1": Sifl(5t))

t=2

11 Tl

= Ecos(st)n_1 Z sin(;rt_1)2 + Esin(a€¢)n’l Z sin(:r¢_1)cos(;rt_1). (4.12)

t=2 t=2
From Theorem 2 of de Jong (2001), i.e. the result of Equation (4.1), it follows that
n’1 :sin(:rt_1)2 —p—> 1/2 and n"1 :sin(;rt_1) cos($t_1) 1» O, (4.13)
t=2 t=2
and therefore
n'1 2 E(sin(a:t) sin(:rt_1)|ft_1) i» (1/2)Ecos(st). (4.14)
t=2

It now follows that

n'12?=2 ytyt_1 p (1/2)E cos(et)

 

 

 

)6 = _ n ——+ = Ecos(st). (4.15)
n 18:2 313.1 (1/2)
For )6”, the same result follows by noting that
- _ "-1 Z?=2(yt — g)(yt—l — 3?)
pH _ _1 n , — 2 i (416)
n Zt=2(yt—1 — y)
and by noting that again by the result of Equation (4.1), 3] —p—> 0. C]

67

Proof of Theorem 4.2:

Again, we will consider yt = sin(:rt), and note that the case y, = cos(:rt) is analogous.
For such yt,

n-W :z‘zxsina.) sin(n-1) — sin2<xt—1>Ecos<€t>>

"-1 211:2 sin2(a:t_1)

= 72'”2 Z?=2(sin(:rt_1) cos(:rt_1) sin(et) + sin(:Iti-il2 (308(5t) " Sin($t—1)2E 903930)

"—1 221:2 sin2 (27;)

Now, noting that the denominator converges in probability to 1 / 2 as before, and in

 

n1/2(/3 — Ecos(et)) =

 

.(4.17)

addition note that the summands gt in the numerator are martingale differences with
respect to .7} by symmetry of the distribution of at, which implies that E sin(st) = 0.
We now apply the martingale difference central limit theorem; see e.g. Theorem 3.2 of
Hall and Heyde (1980). To verify the conditions of this theorem, it now only remains

to be shown that 71‘1 2:2 9,2 L V 6 (0,00). This will be true because

11
-l 2
n 2 :9t
t=2

fl

= n—1 2 sin(xt—l)2(Sin($t—l)(COS(5t) _ ECOS(5t)) + cos($t_1) Sin(€‘))2
t=2

n

= n_1 2 sin(xt—1)4(COS(€t) - ECOS(5‘)l2

t=2

n

+1771: sin(:13t_1)2 (308(flvt—1)2 sin(ft)2
t=2

11

+7771 2: 2 sin(x,_,)3 cos(:ct_1)(cos(et) — E cos(st)) sin(et). (4.18)

t=2

By the martingale difference law of large numbers, the last expression equals

0P(1) + 1f]: sin(:1:t_1)4E(cos(et) — ECOS(5t))2
i=1

+n‘l Z sin(:I:t_1)2 COS($t-1)2E(Sin(5t))2

t=1

68

+n'l Z 2 sin(:r,_1)3 cos(;17,_1)E((cos(st) — Ecos(et)) sin(et)).
By Theorem 2 of de Jong (2001) as quoted in Equation (4.1), we know that

n—1 Zsin(;1:t_1)4 L(27r)_1/ sin 411:( )d:1:= 3/8,

t=l 7'

n—l Zsin(:1:,_1)2 cos(:rt-1)2 —p—1 (27r)"1/ sin2(:r) cos2(:r)d:1: = 1/8,
t=l ‘

11'

and

n—1 : 2sin(17t-1)3 cos(;rt_1) —p—2 (27r)—1/ 2si11(:1:)3 cos($)da: = 0.

t=1 1r

Therefore, it follows that

”-1 ZQEL 3(/)8 E(cos(€t) — Ecos(st))2 + (1/8)E(sin(5,))2 : V.
For p,“ note that

|nl/2(p — p)|_ < 0p(1) + 2711/2y2=0p(n'1/2),

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)

(4.24)

implying that the same limit as for n1/2(p— Ecos(e )) results for n1/2(pp — Ecos(et))

as well, and this observation completes the proof of Theorem 4.2.

Proof of Theorem 4.3:

C]

First note that, for both the cases yt = sin(;rt) and yt = cos(:rt), using the results

obtained in the proof of Theorem 4.1,

n

.2 = (n —1>-‘Z<y.— 44.-.)?

t=2

n 71 n
""1 Z 31? - 2/3'1’12 yty¢_1 + (32""1 Z 313
t=2 t=2 t=2

69

—p—»(1/2)—2Ecos(e(C,)(1/2)Ecos(s)+(Ecos(e¢))2(1/2)=(1/2)-(1/2)(Ecos(et))2.(4.25)

Therefore, it now follows that

 

 

 

n‘1/2t2 "-1/2 (71 “ 1)1/2W“ 1)
(32/(( n)_1 2t: 23/1— 1))”2
: _1/2 (n—1)1/2(p—Ecos(C)) +n'1/2 (n—1)1/2(Ecos(et)— 1)
(2/((n"1)21-2’y¢1))”2 (2/(01-‘1 21- 231:- 1))”2
—2 (E cos(et) — 1)(1 — (Ecos(5¢))2)'1/2. (4.26)

For n'l/Ql”, the same result holds, because the 3] that would appear in the expression
for t), converges to 0 in probability, and therefore the difference between n"1/2f and

n’l/gffl converges to zero in probability asymptotically. Cl

70

CHAPTER 5

Some results on the asymptotics

for threshold unit root test

5. 1 Introduction

Economic time series data often show some sudden changes, as a result of an exter-
nal shock. It is generally believed that linear time series models cannot capture such
a structural change. One statistical model that attempts to capture such a sudden
structural change in different regimes is the threshold autoregressive model devel-
oped by Tong (1990). The threshold autoregressive model captures regime switching
based on the lagged values of the variables. This is a very attractive property for
economists, but the threshold autoregressive model still has some drawbacks. One
of main drawback is that for inference in threshold autoregressive models, there is
limited theory for testing null hypotheses that imply a unit root. The first paper to
investigate unit root structure in threshold autoregressive model is Gonzalez and Con-
zalo (1997). They present a threshold unit root (TUR) model that has either stable
roots existing in all regimes or unit roots in at least one regime. In the context of their

threshold unit root model, they derive the asymptotic distribution of a Dickey-Fuller

71

t-test. However, their analysis has some problems. First, in Gonzalez and Gonzalo’s
threshold unit root model, they consider the threshold value to be known and fixed.
But generally in economics time series, threshold values can be unknown. Second,
the threshold unit root test in Gonzalez and Gonzalo consider unit root exists in one
regime threshold autoregressive model. They test one of all regimes existing unit root
against alternative hypothesis that threshold autoregressive model does not have unit
root in any regimes. But their model does not consider the case that unit root exists
in one regime of TUR model in advance and test the null hypothesis of a pure I(1)
process against the alternative hypothesis of a TUR model that has one regime with
unit root. For improvement of these drawbacks, we establish an asymptotic result
that can be used for testing the null of a unit root ((p = 0) against the alternative of

a threshold unit root model:

Ay. = u + 5t if lyt—ll S C (5.1)
p+<pyH+et if Iyt..1| >C,
where -2 < p < 0. We will relax the assumption that threshold value, C, is known,
as in Gonzalez and Gonzalo’s TUR model, and we consider tests that have been
optimized over the unidentified parameter,C.

This chapter is organized as follow. In Section 5.2, we will derive the appropriate
asymptotic results. With results, we can establish the asymptotic distributions of
Dickey-filler t-test in regression Ayt on constant and y¢_11(yt-1 > C), optimized
over a set of possible value of C. In section 5.3, we will explore the possible further

extension with the asymptotics. The conclusion will be found in Section 5.4. All

proofs are in Mathematical Appendix.

72

5.2 Main results

For developing the asymptotic distribution of the threshold unit root test, we will
use results involving the Brownian local time and a result by Perkins (1982) involving

convergence to Brownian local time. In Perkins’ Theorem 1.1, it is shown that

p.-

(1/2)Ln(1,7F) = |n_l/2yi - 7rI(1('n_1/2yi—1S 7r) — 1014/2311 S 70)
t

ll
H

n-l
= Z In"1/2yt — 7r|(I(n"1/2yt_1 S 7r)I(n'1/2yt > 7r) + I(n'l/Qyt_.1 > 7r)I(n_1/2yt S 1r))
t=1

n—l
= Z (n‘1/2yt — 7r|1(min(n'l/2yt,”nil/2M4) S 7r S max(n—l/2yt,n’l/zyt_1))
i=1

=> (1/2)L(1,7r), (5.2)
where L(t, s) is the two-parameter stochastic process called ”Brownian local time”
and ” =>” denote weak convergence. In order to establish our results, we need the

following assumption for at.

Assumption 5.1 51 is an i.i.d. sequence random variables with mean zero, variance
02 and E l52l4 < 00. The distribution of st is absolutely continuous with respect to the
Lebesgue measure and has characteristic function ib(s) for which limsnoo s"i,b(s) = 0

for some 77 > 0.

Assumption 5.1 implies Assumptions 1 and 2 of Park and Phillips (1999) and the
assumptions of Theorem 1.2 from Perkins (1982), implying that we can combine

results from both papers here. The following results now follow relatively easily from

Perkins (1982):

Theorem 5.1 Assume 5, satisfies Assumption 5.1, and assume that Ayt = 5; and

310 = 0.771671,
71 l
n'1/225t1(n‘1/2yt_1 S it) :> o/ I(0‘W(r) S 7r)dW(r). (5.3)
t=1 0

73

 

Theorem 5.2 Assume 5, satisfies Assumption 5.1, and assume that Ayt = 5t and

yo = 0. Then
it“1 Zetyt_11(n'1/2yt_1 S 7r) 2) 02/0 (W(r))I(oW(r) S 7r)dW(r). (5.4)

The proofs of Theorems 5.1 and 5.2 can be found in the l\/Iathematical Appendix.

5.3 Applications

Consider the threshold unit root model

Ayt = II + 5t if lyt—ll S C (5.5)
u+<pyt_1+et if lyt_1| >C.

From Chan, Petrucelli, Tong and Woolford (1985), it is known that under regularity
conditions, if at is an i.i.d. error and —2 < (,0 < 0, then yt will be ergodic, and the
usual law of large numbers will be hold for y, and yf. With the results we establish in
Section 5.2, we can construct tests for the unit root hypothesis H0 : (p = 0 against the
alternative of an ergodic TUR model, i.e. —2 < (p < 0. If the threshold value were
known, we could obtain an estimator e of (p by a regression of Ayt on constant and
yt_11(yt_1 > C). However, if the threshold value is a priori unknown, the problem
arises that under H0, the threshold value is unidentified. One solution for this problem
is to use the smallest possible t-value over the space of relevant values for threshold

value as our test statistics. Assuming C = Til/2

7r is given, define (0“,, as the least square
regression coefficient from a regression of Ayt on 313.1] (yt_1 > n1/27r) with intercept,

and similarity define 5:220 as the usual t-test for H0 : (p = 0 from the regression with

intercept. Under the null hypothesis of cp = 0 and assuming that yo = 0 , the t":
(a 0

statistics can be written as
(1/81)(Zyi_11(3/1—1 > "mfill-IWZ 5191—11fyt—1 > 711/270)» (5-6)

i=1 t=l

74

 

where sf is the usual error variance estimator. For numerator of (5.6), we can use the

results that we establish in this chapter to obtain

71

'n-125tyt_11(n—1/2y,_1 S 7r) :> 02/0 (W(r))I(oW(r) S 7r)dW(r). (5.7)

t=l

For the denominator, Park and Phillips (2001) established that for a compact subset
of II of R,

71

1
n‘1 :(n"1/2yt)2l(yt_1 > nlflir) => 02/ W(r)21(o|14/(r)| > 7r)dr. (5.8)
t=1 0

Combining these two results, we can conjecture the possible asymptotic distribution

of the statistic under the null to be,

 

inf i“ _“L. inf f01W(r)I(|W(r)| > 7r/o)dW(r) — W(1) fO‘W(r)1((W(r)| > 7r/0)dr.
«en “”0 «en ( f01 W(r)21(|W(r)| > 7r/o)dr— (f01W(r)I(|W(r)| > 7r/o)dr)2)1/2

(5.9)

The problem with this conjecture is that the denominator equals zero for 7r >
”SUE-ems] |W(T)l, and therefore the above result does not follow straightforwardly
from the continuous mapping theorem. The application of our theorems towards the

problem of testing for a threshold unit root will be part of the future research.

5.4 Conclusion and further research

In this chapter, we derived two theorems involving the product of an error and an
indictor function. With the two results we established, we can consider Dickey-Fuller
t-tests that detect the null hypothesis of a unit root against alternative of a threshold
unit root model. With regard to further research, we can derive the Dickey-Fuller
unit root test for TUR model under 7r 6 II with our asymptotic results. In addition to
obtain asymptotics of the Dickey-Fuller unit root tests under our TUR model, we can

relax the assumptions about residuals, at. We can use the stationary ARMA processes

75

instead of white noises in residual series of TUR model. For this improvement, the

asymptotic properties of augment Dickey-Fuller unit root tests can be derived.

Mathematical Appendix

Proof of Theorem 5.1:

First note that,

"—1/2ZEtU’vI /2yt— 1377)

n

= if”2 2(3)) — nl/zr)1(n_l/ 2_y, 1 S 7r) — n ”2 :(y _1— nl/ 7r)1(n"1/2y¢_1 S 7r)
t=1
n—l

= "71/2 20/: -'n1/27r)1(n_1/2yt—1 S 7r) + ”"1/2(yn- n1/2W)I(n'l/2yn—1 S 7r)
t=l

_n-1/2 2:0” _ nl/27r)l(n‘1/2yt S 7r) _ ”_1/2(y0 _ nl/27r)1(n-l/2yo S 71')

- n"1/2(yn — 72,1/27r)1(n 12/ yn_1 S 7r) — n l”(yo — n1/27r)1(n_1/2y0 S 7r)

n

+n-l/2 Zak _ 711/27?)1(n-1/2yt_1 S 71') _ n-l/2 2(3/1 _ n1/27r)1(n—1/2yi S 7r)
t=1 i=1

= n'1/2(yn - nl/2r)1(n’”2yn—1 S 71) - n”/2010 - (11/270101 "Wye- < 7r)
n

+n'1/2 :(y — 711/27r)[1(n‘1/2yt_1 S 7r) — I(n‘lflyt S 7r)]
i=1

n—1/2(yn _ Til/2r)1(n—l/2yn_1 S 7r) _ n —1/2(yo __ n1/27T)1(n—1/2y0 S 7r)

n
+n'1/2 2(yt — 711/27r)1(min(n‘1/2yt, n’1/2yt_1) S 7r S max(n'1/2yt, n'l/2yt_1)).
t=l

(5.10)

For the first term of the last formula, we have
(Tl—”23111 _ 7r)I(n—l/2y‘n—l S 71') = (n—l/2yn—l _ 77 + n-l/an)1(n—l/2yn—l S 71’).

76

By Chebyshev’s inequality,

Esreig In’l/2st|1(n_l/2yn_1 S 7r) S n”l/2E|s¢| —> 0
as n —> 00. Also,

I(n'l/Qyn_1)1(n"1/2yn_1 S 7r)| => (oW(1) — 7r)I(|oW(1)| S 7r). (5.11)
For the second term, we can obtain

(Tl—Wye — 7r)1('n—1/2y0 _<_ 77) => (‘77)“0 S 7T)- (5-12)

For the third term, from Perkins ( 1982), it follows that

"-1/2 2%: - nl/er)1(min(n‘l/2yz,n‘l/2yi41) S r S maX(n"l/2yun'l/2yt-I))
i=1
:4 (1/2)L(1,7r). (5-13)

The results of Equation (5.11), (5.12) and (5.13) now imply that the statistic of

Equation (5.10) converges weakly to

(oW(1)— 7r)1(|aW(1)l S r) — (-7r)1(0 3 7r) + (1/2)L(1,7r). (5.14)
The Tanaka Formula (see Perkins (1982) p437-p439)

max(VV(1) — 7r,0) = max(—7r,0) + A11(W(r) > 7r)dW(r) + (1/2)L(1,7r)

now implies that the process of Equation (5.14) can be written as 0f011(0l/V(7‘) S

7r)dW(r). We can conclude that

n 1
n‘1/2Zetl(n_l/2y¢_1 S 7r) => 0/ I(0W(r) S 7r)dW(r).
t=1 0

77

 

Proof of Theorem 5.2:

Fl‘om the definition of y,_1, we know

Czyz—1=(1/2){y2- yt2.1- 52}

The pointwise convergence in distribution of the statistic follows from Park and
Phillips (2001). Therefore, we only need to show stochastic equicontinuity to complete

the proof. To Show stochastic equicontinuity note that

n
"221 thyt—11(n-l/2yt—l S 7r)
=()1/2” lle/i2 _ yt2-1 - €2)1(n’1/2y1_1 S 77)

=()nl/2 ‘12:“ y — n1/27r)2 —(yt_1 — n1/27r)2 + 2n1/27ret— e2]1(n (2yt_1 S 7r)

n

=(1/2>n-‘Z(y.—nl/2 )I( y. .< 7r)

t=l

Tl

—(1/2)n—‘ 2a-] - n1/2,)2,(,,—u2y,_1 s 7r)

t=l

n

+(1/2)n’1 2(2711/27ret — €2)I(n—1/2yt_1 S 7r)

t=l

n~1

= (1/2)Tl—12(yt — nl/27r)21(n—l/2yt_1 S 77)
t=l

+(1/2)n_1(yn — n1/27r)21(n"1/2yn_1 S 7r)

n.—

—(1/2)n‘l (yz-rzl/2r)2n’1(2y Sr)

-(1/2)n‘1(yo - n1/27r)21(7l’”2yo S 7r)

g—n

+(1/2)n_1 2(2711/27re, — 5,2)I(n_1/2yt_1 S 7r)
t=l

78

= (1/2)n’1(yn - n1/27r)21(n"”2yn-1 S 7?)

-(1/2)'n‘1(yo - Til/2702101 ”/2110. < 7r)

H

+(1/2ln—l (y -nl/27T)2l1(n /2y1—1S7r)-1(n/2y SW1]

+()711/2 "12(2n1/27ret— C2)171( *1/2yt_1< 77)
Now note that
91(7) = (yt - 711/271)2(1(n“/2yt_1 S 7r) - [(71 *1/23), S 71))
—e,21(n’1/2y¢-1 S 77)

is continuous in 7r, and also note that gt(7r)is differentiable in 77 and that its first

derivative is
9“”) = ‘2"122ly1 “ 711/27r)(1(n12/ 3/1— 1 S 7T) _ 1(n/2.311 < 71))

Define

t=1
and
02(7) = 7771:9371)
t=l
Now

IGn(7r) - Gn(7”r)| S |7r - 7~rl SUP |G1.(7r)|
KER

11
= Ir — 1144231414... — rllltn‘WyH s 7r) — 1(n-”21. s «)1
7r _

=14 — vita/2511141144141.

«ER

79

 

Therefore, it follows that 0,,(77) is stochastically equicontinuous. The proof of stochas-
tic equicontinuity of 71—1/2 212:1 etyt_11(n'l/2yt_1 S 77) is therefore complete if we can

Show that.
n
27rn'1/2 Z st1(n'l/2y,_1 S 77)
t=l

is stochastically equicontinuous, which follows from Theorem 1 and the continuity of

g(7r) = 77. D

80

APPENDICES

81

APPENDIX A

Introduction to local time

A.1 Definition and properties

In my dissertation, we use the concept of local time. I will give a simple intro-
duction for local time in this appendix. Local time is a continuous two-parameter
stochastic process that characterizes a continuous time martingale process. When
this continuous time martingale process is Brownian motion, the associated local
time function is called the Brownian local time, which we well denote by L(t, 3). Like
Brownian motion, local time is a random function that has a well-defined distribution
for any given value of the argument; the finite-dimensional distributions of L(., .) a
spatial density, i.e. L(., .) are not normally distributed, however. The intuitive inter-
pretation of local time is that it is a spatial density, i.e. L(., .) provides information
about how much time a Brownian motion process spends in the neighborhood of a
given point 3.

To get to the standard definition of local time, we first need to define the occu-
pation time H. Let [M] denote the quadratic variation process of M, where M is a

continuous time semimartingale process. Then the occupation time of 111(7), for any

82

Borel measurable set A, is given by

H(A, t) = /t1(1\rl(r) e A)d[A»I](/r).

For the special case M = B, [B](7) = r, and the reason for naming H (., .) “occupation
time” is clear for that case. In the more general case, we can think of the amount of
time spent by M () in the set A as being measured in units of quadratic variation.

The following theorem now defines the local time function L M(t, 3):

Theorem A.1 For a continuous time semimartingale process M (), there exists a

continuous function L(., .) such that

H((—oo,.i:],t) = /2 LM(t,s)d[1\/1](s).

—00

Proof of Theorem A.1:

See Chung and Williams (1990).

The above theorem implies that
t
LM(t,s) =lin(13(2e)_1/ 1(l111(r) — s| S e)d[1\/1](r).
5" 0

Trotter (1958) was the first to show the result of Theorem A.1 for the special case of
Brownian motion, i.e. M = B. For the case of the Brownian local time, the above

theorem implies that

H((—oo,:1:],t) = /2 L(t,s)ds

—00

and that

(d/ds)'/0 1(B(r) S s)dr = L(t,.s).

We note that in the article by Park and Phillips (1999), that seemed to have started

interest of the econometrics profession in local time, the order of the arguments of the

83

L(., .) function see to be reversed, compared to what is convention in the statistics
literature. Here, we will follow Park and Phillips’ notation.

In order to get some idea of how the local time function behaves, it may be
worthwhile here to realize that since suprew] B (r) and infreml B (r) are well-defined
random variables, fot 1(B(r) S s)dr = 0 for s < infremy] B(r). Therefore, for such 5,
L(t,s) = 0 as well. Similarly, for all s > suprem] B(r), fut 1(B(r) S s)dr = 1 and

therefore, L(t, s) = 0 for s > SUPrem] B(r). Also,

/: L(t,s)ds = [1:(03/(15) /0t1(3(.,~) S s)dr]:_00 :1.

These facts together complete the picture that we should have in mind for L(., .:) as a
function of s, L(., ) is a function with bounded (yet random) support that integrates

to one.

A.2 The Tanaka formula

The It formula states that, if (d2/dx2)F(:r) = (d/dzr)f(:1:) = f’(2:) and f’(;r) is con-

tinuous, we have

F<B<t>>—F<B<0)>= / 7(B<r>>dB(r>+<1/2)/ 77301144

The Tanaka formula now states that a form of the It formula holds for f (W) = 1 (W S
s) as well. The local time L(t, s) will appear in this formula, as a replacement for the
lt correction term. Basically, for this choice of f (), the Tanaka formula justifies that

one can consider

—<d/ds> / rainwater = —L<t.s>

instead of the undefined
t
/ (d/dW)HW>IW=B(.)4r
0

84

in the It formula. For a heuristic application of the It formula along these lines, to
make F () continuous and have f (W) = 1( W S s) as its derivative at any point except
W = s, we should choose F (W) = (W — 5)] (W S s). This heuristic implication of

the Tanaka formula is then

(B(t) - 8)1(B(t) S 8) - (—S)1(0 S 8) = /0t I(B(r) S 8)dB(T) - (1/2)L(t98),
which can be rewritten as

max(s — B(t),0) + ma.x(s,0) = /0t1(B(r) S s)dB(r) + (1/2)L(t,s). (A.1)
However, the most cited form of the Tanaka formula is as follows:

Theorem A.2 Tanaka formula
t
L(t, s) = [B(t) — s| — Isl —/ sgn(B(r) — s)dB(r). (A.2)
0

Proof of Theorem A.2

See McKean (1969).

This second form of the Tanaka formula easily results from our conjecture of

Equation(A.1) by noting that sgn(W — s) = 1 — 21(VV S s).

A.3 The occupation times formula

Because of the interpretation of local time as a spatial density, we may expect a
relationship between integrals over a function of Brownian motion and an expression

involving local time. Specifically, for a continuous function T(.), we may expect that

/01T(B(r))dr

85

can be approximated, for small 5 > 0, by

l. ZTeroe < B(r) s (j + 1).).14.

For small 5 > 0, we should now have that

1
5’1/ I(je < B(r) S (j+1)€)d7‘ z L(Ljf),
0

suggesting that

1 1
f0 71(B(7.))dr z/O ZT(js)I(j5 < B(r) S (j+1)e)dr

co

m ZT<j€)E—1L(l,j€) z/ T(s)L(1,s)ds.

—00

This can be formalized in the following theorem:

Theorem A.3 Let T : IR —2 IR be a locally integrable function. Then

[012(Blflldr = [0 T(5)L(1,3)ds.

00

Proof of Theorem A.3:

See Chung and Williams (1991).

86

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87

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