貨幣、匯率與動態均衡之學術前沿研究-子計畫七：匯率預測：估計風險之角色(IV)

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行政院國家科學委員會專題研究計畫成果報告

貨幣、匯率與動態均衡之學術前沿研究--子計畫七：匯率

預測：估計風險之角色(4/4)

研究成果報告(完整版)

計畫類別：整合型

計畫編號： NSC 98-2752-H-004-001-PAE

執行期間： 98 年 04 月 01 日至 99 年 12 月 31 日

執行單位：國立政治大學國際貿易學系

計畫主持人：郭炳伸

報告附件：國外研究心得報告

出席國際會議研究心得報告及發表論文

公開資訊：本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中華民國 101 年 05 月 03 日

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中文摘要：

中文關鍵詞：

英文摘要： This paper develops a simple long-difference

transformation for

estimation and inference in general AR(1) models. As

in Phillips and

Han (2008), a Gaussian limit theory with a

convergence rate of

$\sqrt{T}$ is available, whether or not a unit root

is present in

the process. Yet, the novelties of our limit results

are that the

same weak convergence applies to the models with or

without trend,

and that the asymptotic distribution is characterized

by a constant

variance of value 2. The merits promise usefulness of

the

long-difference transformation in applications to

dynamic panels.

英文關鍵詞： unit root, AR model, long difference, first

difference

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1 I. COVER

Program for Promoting Academic Excellence of Universities

ȐPhase IIȑ

Midterm Report

༊౗ႣෳǺ՗ी॥ᓀϐفՅ (4/4)

Understanding Exchange Rate Predictability: The Role of Estimation Risk

(4/4)

Serial No.: 98-2752-H-004-001-PAE

Overall Duration: Month 04 Year 06

- Month 03 Year 10

Report Duration: Month 04 Year 09

- Month 03 Year 10

National Chengchi University

01/10/10

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2

II. (FORM1) BASICINFORMATION OF THE PROGRAM/Sub-project

Program/Sub-project Title: ༊౗ႣෳǺ՗ी॥ᓀϐفՅ(4/4)

Understanding Exchange Rate Predictability: The Role of Estimation Risk (4/4)

Serial No.:98-2752-H-004-001-PAE Affiliation

Princi

pal Investigator

Name ೾ࣂ՜ Biing-Shen Kuo

Program C

o

ordinator

Name

Tel: (02) 29393091 ext 81029 Tel:

Fax: (02) 29387699 Fax:

E-mail [email protected] E-mail

Expenditures1 (in NT$1,000) Manpower2:Full time/Part time(Person-Months)

Projected Actual Projected Actual

FY 95 1052 1052 3 3

FY 96 1052 1052 3 3

FY 97 1052 1052 3 3

FY98 1052 483 3 3

Overall 4208 3639 12 12

Serial No. _{Project Title} Principal

Investigator Title Affiliation

Serial No. (95) 95-2752-H-004 -002 -PAE ೾ࣂ՜

Biing-Shen Kuo ௲௤ Professor ࡹݯεᏢ୯ຩس National Chengchi Univ.

Serial No. (96) 96-2752-H-004-002-PAE ೾ࣂ՜

Serial No. (97) 97-2752-H-004-002-PAE ೾ࣂ՜

Serial No. (98) 98-2752-H-004-001-PAE ೾ࣂ՜

Notes:1,2Please explain large differences between projected and actual figures.

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6

III. (FORM2) LIST OF WORKS, EXPENDITURES, MANPOWER, AND MATCHINGSUPPORTS FROM THE PARTICIPATING INSTITUTESȐREALITYȑ.

Serial No.: (95) Program/Sub-project Title:

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Research Item (Include sub projects) Major tasks and objectives

Expenditures (in NT$1,000) Manpower (person-month)

Matching Supports from the Participating Institutes (in English & Chinese) Salary Seminar/ Conference-re lated expenses Project- related expenses Cost for Hardware & Software Total Principal Investigators Consultants Research/ Teaching Personnel Supporting Staff Total Sub-project 10 Develop of new estimators and explore its application 660 186 63 36 945 1 0 2 0 3 0 SUM 660 186 63 36 945 1 0 2 0 3 0

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Matching Supports from the Participating Institutes (in English & Chinese) Salary Seminar/ Conference-re lated expenses Project- related expenses Cost for Hardware & Software Total Principal Investigators Consultants Research/ Teaching Personnel Supporting Staff Total Sub-project 10 Develop of new estimators and explore its applications 680 140 85 30 935 1 0 2 0 3 0 SUM 680 140 85 30 935 1 0 2 0 3 0

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7

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Matching Supports from the Participating Institutes (in English & Chinese) Salary Seminar/ Conference-re lated expenses Project- related expenses Cost for Hardware & Software Total Principal Investigators Consultants Research/ Teaching Personnel Supporting Staff Total Sub-project 10 Develop of new estimators and explore its application 700 207 21 18 946 1 0 2 0 3 0 SUM 700 207 21 18 1 0 2 0 3 0

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Matching Supports from the Participating Institutes (in English & Chinese) Salary Seminar/ Conference-re lated expenses Project- related expenses Cost for Hardware & Software Total Principal Investigators Consultants Research/ Teaching Personnel Supporting Staff Total Sub-project 10 Develop of new estimators and explore its application 432 96 41 0 569 1 0 2 0 3 0 SUM 432 96 41 0 569 1 0 2 0 3 0

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8

IV. (FORM3) STATISTICS ON RESEARCHOUTCOME OF THIS PROGRAM/Sub-project

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LISTING TOTAL DOMESTIC INTERNATIONAL SIGNIFICANT1 CITATIONS2 TECHNOLOGYTRANSFER

PUBLISHED ARTICLES JOURNALS ! ! ! ! ! CONFERENCES ! 1 ! ! ! TECHNOLOGY REPORTS ! ! ! ! ! ! PATENTS PENDING ! ! ! ! - ! ! GRANTED ! ! ! ! - ! ! COPYRIGHTED INVENTIONS ITEM ! ! ! ! ! !

WORKSHOPS/CONFERENCES3 ITEM 1 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! PARTICIPANTS Around 20 ! ! TRAINING COURSES ȐWORKSHOPS/CONFERENCESȑ HOURS ! PARTICIPANTS ! PERSONAL ACHIEVEMENTS HONORS/ AWARDS4 ! KEYNOTES GIVEN BY PIS ! EDITOR FOR JOURNALS !

TECHNOLOGY TRANSFERS ITEM ! ! ! LICENSING FEE ROYALTY

INDUSTRY STANDARDS5 ITEM ! ! !

TECHNOLOGICAL SERVICES6

ITEM ! ! ! - - -

SERVICEFEE ! ! ! - - -

1_{Indicate the number of items that are significant. The criterion for “significant” is defined by the PIs of the program. For example, it may refer to Top journals (i.e., those with impact factors in the upper 15%) in the area of research, or}

conferences that are very selective in accepting submitted papers (i.e., at an acceptance rate no greater than 30%). Please specify the criteria in Appendix IV.

2_{Indicate the number of citations. The criterion for “citations” refers to citations by other research teams, i.e., exclude self-citations.} 3_{Refers to the workshop and conferences hosted by the program.}

4_{Includes Laureate of Nobel Prize, Member of Academia Sinica or equivalent, fellow of major international academic societies, etc.} 5

Refers to industry standards approved by national or international standardization parties that are proposed by PIs of the program.

6_{Refers to research outcomes used to provide technological services, including research and educational programs, to other ministries of the government or professional societies.}

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9

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PUBLISHED ARTICLES JOURNALS ! ! ! ! ! CONFERENCES ! ! ! ! TECHNOLOGY REPORTS ! ! ! ! ! ! PATENTS PENDING ! ! ! ! - ! ! GRANTED ! ! ! ! - ! ! COPYRIGHTED INVENTIONS ITEM ! ! ! ! ! !

WORKSHOPS/CONFERENCES3 ITEM 3! 1 2! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! PARTICIPANTS Around 60 Around 20 Around 40! !

TRAINING COURSES ȐWORKSHOPS/CONFERENCESȑ HOURS ! PARTICIPANTS ! PERSONAL ACHIEVEMENTS HONORS/ AWARDS4 ! KEYNOTES GIVEN BY PIS ! EDITOR FOR JOURNALS !

ITEM ! ! ! - - -

SERVICEFEE ! ! ! - - -

2_{Indicate the number of citations. The criterion for “citations” refers to citations by other research teams, i.e., exclude self-citations.} 3

Refers to the workshop and conferences hosted by the program.

4_{Includes Laureate of Nobel Prize, Member of Academia Sinica or equivalent, fellow of major international academic societies, etc.} 5_{Refers to industry standards approved by national or international standardization parties that are proposed by PIs of the program.}

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10

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WORKSHOPS/CONFERENCES3 ITEM 3! 1 2! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! PARTICIPANTS Around 80 Around 40 Around 40! !

TRAINING COURSES ȐWORKSHOPS/CONFERENCESȑ HOURS ! PARTICIPANTS ! PERSONAL ACHIEVEMENTS HONORS/ AWARDS4 ! KEYNOTES GIVEN BY PIS ! EDITOR FOR JOURNALS !

ITEM ! ! ! - - -

SERVICEFEE ! ! ! - - -

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11

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WORKSHOPS/CONFERENCES3 ITEM 1 1 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! PARTICIPANTS 80 80 ! TRAINING COURSES ȐWORKSHOPS/CONFERENCESȑ HOURS ! PARTICIPANTS ! PERSONAL ACHIEVEMENTS HONORS/ AWARDS4 ! KEYNOTES GIVEN BY PIS ! EDITOR FOR JOURNALS !

ITEM ! ! ! - - -

SERVICEFEE ! ! ! - - -

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12 V. (F ORM 4) E X E CUTIVE S UMMARY ON R ESEARCH O UTCOMES OF THIS P ROGRAM /Sub-project (Please state t h e followi ngs c oncisely and clearly) 1. Gene ral Descri ption of t h e P rogram /Sub-projec t: Including Objectives of t h e P rogram The rese arc h attem pts to offer ec onome tric expl anati ons to the ne ar random -wal k exc hange rates. I t argues

that previous empirical

evidence for or a g ainst predictabilit y in ex cha n g e rate movements m ig ht have been consider abl

y flawed by the existence

of estimation ri sk due to the strong per sistence in fundamentals. T h e primar y goal of the project is in a pursuit of a more relia

ble inference pro

cedure fo r the predicta bility both i n

-sample and out-of--sample by appropriat

ely controlling the estimati

on risk. To ac hieve th e goal, an ave ragin g es ti mator th at c o mb ines infor mation optim a ll y both fr om the univariate

time series under study

and from cross-sec tional time ser ies is devel ope d. Another g oal of the pro je c t is to ex pl or e useful applic ati ons of the id ea of the ave ra g in g es timator . While the project s tarts with an attem pt to e

stablish robust infere

nce pr ocedure s for the e x chan g e rate predictabilit y usi n g the aver a g in g estimator

, the estimator itself, thr

ou g h combinati on, offers alernative wa y s to m a ke be tter u se o f the inform ation

from the data. T

h

is is impor

tant because efficienc

y or power g ain s could be antic ipate d when the inform ation fr om the data is better e x ploi ted. 2. Breakthroughs and Major Ac hievem ents Evidence for the e x change rate predic tability in the past literature

has been mixed. In c

o ntrast, the current project, after controlling

the estimation risks, is able to establish

a mo

re uni

form evidence

for the predictabilit

y , whether the forecasting horiz ons are short or l o ng. Thi s is s o me how remarkable because to our knowled g e, little evidence

for the exc

hange r a te predictabi lity in the sh ort h o riz ons w a s

found in the literature.

In addition to establishing evidence

for the predictability, tw o ma jor conclusi ons emer g in g fr om the rese arch so fa r can be su mma r ized : 1) th e mag n it u d e of t h e est ima tio n risk s is so hi g h th at the exch an g e rate predictability can

be masked even when

i t exists in the data; 2) T h e i n fo

rmation about exchange rate movements

from cros s-se ction is valuable in the w a ys that it can reduce the e

stimation risk and thus im

prove the te stin g power for predictability, i f i t could be exploi

ted effectively as the

averag

ing estimator does.

To extend the idea of the ave raging es timator further, we also de velope d a new differ ence-ba sed estimator. O f significance is that a Gaussian limit is fou nd to be available for the es timator , whe ther a uni t r oot presents in the data gen erate d fr om sim p le autoregres sions. 3. Categorize d Summary of Rese arch Outcom es. T h e criteri a for top c o nfere nces and journal s should be given and introduce d briefly in t h e begi nni ng of t h is section. In e ach researc h a rea, please gi ve a brief s u mm ar y on the researc h

outcomes associated with t

h e a rea. Note t h at the summaries should be

consistent with the statistics give

n in Form 3. Please list and num ber eac h re searc h outcom es in sort ed or der in Appe ndi x II, an

d list all the publications in

top c o nfe renc es and journals in Appe ndi x II I. 3. 1 Devel opm ent of an avar agin g e stim a to r (1/ 4) : An averaging estim ator that is to co ntrol potenti al e stimation risk s associ ate d w ith the predic tive regres sion is developed i n the first-ye ar study. The so urces of the estimation ris ks comes fr om hi g h per sistence of predic tive regress ors, an d the de pende nt variable be ing the overlapping s u ms of shor t-h oriz o n chan g e in lo g exchan g e rate. The former c reates bias in small-s ample

s and the latter brin

g

s

forth remarkable estimation variabilit

y in long-horizon predictions. The considere d aver aging es timator optim ally c o mbines two alternati ve estimators that differ in their bias and

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13 precision characteristics. B

y

constr

uc

ti

on, the sugge

sted estim

ator for the sl

ope c o efficients utiliz es informati on from cross-se ctions in a si

milar way that the

panel-based estim ators do. The im plicit ass umption underl y in g the use of inform ati on fr om cr oss-secti ons for our es timator

, however, is very muc

h differ ent fr om that for the panel -b ased e stimator s. I n con tr ast, the panel-b ased estimator s ar e built on th e ass ump tion th at the slope coefficients ar e all the s a m e for all the cross -secti onal cou n tri e s. O n the o ther h a nd , the aver agin g estim a to r allows for separate sl ope estimate fo r e a ch cross-sec tion country as

the OLS estim

ator

does

, but makes use

of the cross -secti onal inform ati on that the OLS estimator doe

s not. Thus both the aver

aging estim

ator and the pooled

estimator are the s a me to re duce the es ti mati on err ors , but di ffer in the way h o w the cross -secti onal infor matio n is proces sed. Yet, our averaging estim ator has the advantages of pr oducing m o re re asonable sl ope estimates. 3. 2 Risk re duc tion : simul ati on an alysis (1/ 4 )

We examined to what exte

nt the pr

op

osed estimator can improve over the tr

aditi o nal esti mator in ter ms o f risk reduc tion thr ou gh simu lations. Un der the setup th at mimics the r e ality, we doc umente d that the avera g in g estimator e m pirically dom inates the le ast-s qu are (LS ) estim ator, r e g ar dless of

which simulation sce

nari o is considere d. Virtually the ri sk reduc tions

using the averaging es

timator can be as large

as between 10\% and 35\% , compar ed with the L S estim ator. More importantl y, the risk improvement by the averaging es timator is embodie d fur

ther into power gains in testing. Our

simulati

on s

h

ows that the power gains from

usin g the a v e rag in g est ima to r, ag a in rela tiv e to t h e LS est imat o r , is 10 % t o 3 0 % o r mo re in ma ny ca ses. A n si g nificant implication

of the finding is simply that the predictabil

ity

alternati

ve

can now be better de

tected from the data when the tes t statistics are ba sed on the averagin g e stim a tor. 3.3 A re-examinati on of the exchange rate predictability (2/4) We re-investiga ted the empirica l va lidity of the ex chang e rate predictability apply

ing the averaging

estimator. The testing strategy basically

fo

llows

that utilized in the l

itera

ture where these studi

es all base their inference on the boots tr ap approach in order to contr ol for sm all-sample bi as

for which the

as y mp totic appr oximation ge nerally fai ls to correc t. We accesse d the rel ative forec ast acc ur acy of the two competing models with

Theil's U and DM statistics. It

should be noted that the

problem with

estimating the long-run variance precise

ly when c a lculatin g the DM

statistic often leads to

spuri ous inference . Impor tant me ssages emer gi ng fr om the empirical exerc ises include:

1) There is now much more significant evidence

presented for the

do minance of the mone tary model over the random walk when predic ting, after accounting for es timation risks using the considered es timator. With onl y

few exceptions, the

p-value s ass ociated with the aver aging es

timator for both statistics are

smaller, relative to those associated with the L S estim ator. 2) It

stands out from

the

results that

controlling over the risks uncovers

mo

re favor

able evidence in suppor

ts

of

the m

o

netary model, while there is essentially no e

v

idence for s

o

when leaving

the risks unattended. Man

y more ins tance s of this ar e found fr om the repor ted T h eil’s U statist ic. Particularl y, at alm o st all horiz o ns, the mo neta ry model is fo und to be superi o r to

the random walk in terms of predictabilit

y for Ge rman y and Japan .

This contrasts shar

ply with

the

previ

o

us

findings where little evidence fo

r predictability is repor ted. Consideri n g the Theil's U

statistic is more robust,

this ev

idence lends quit

e a goo d dea l o f credence to the predicta bility a t both s h or t- and long-horiz ons. 3.4 Asymptotic the or y of the averaging estimator (3/ 4) The use of the aver a g in g e stimator in te stin g f o r ex c h a n g e rate predictabilit y bri n g s for th som e econometric interesting questions . T h is entails the de velopment of an as y mptotic the or y of the aver a g in g estim ator. We inv oke a lo ca l-to -unity framewo rk to build the a sy m ptotic the or y based on the obser vation w ith inherent hi g h persistence in the data. We are now able to deri ve the as y mptotic distributions of the aver a g in g estimator unde r

the simplified assumpti

ons where regress ion err ors are uncorrelate d with predicti n g variables. The as y mptotic distribution derived is a mixture nor m al. T h e mi xture norm al collapses into

the limit distributi

on

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14 least-square e

stimator

, or

that

of the panel estimator,

when either r eceives z ero weights in form ing the avera g in g estimator . 3.5 Devel opm ent of a new differenced-bas e estim ator (4/ 4) Our diffe rence-based e stimator poss esses a num ber of intere sting pr oper ties. Firstl y, the transform ed second-di ffere nce es timator of the auto regressive coe ffic ient has a Gaussian as y mptotic distributi on that applies to both the

unit root cas

e and c

o

nve

ntional ca

ses. This implies

that the normality limit stands for the

local-to-u

nity cases as

well. Thus, the

limi t distributi on is con tinu ous as the autoregressive co effic ient passes thr ough uni

ty. Our simul

ati ons further re veals that the estimator displa y s ne g li g

ible bias for ver

y small sam ples, as op pose d to the conve nti o n a l leas t s q uares estim a tor. More over , the limit dis tributi on of our estim ator in

simple AR(1) models without time trend exhibi

ts a constant variance of valu e 2. This is in contrast to the compe ting esti mator, which is a line ar incr easing func tion of the autor egressive coe fficient. P a rticularly,

the limit variance

for

our estimator is

smaller for that

of the

competing counterpart

for any positi

v e autore gressi ve coe fficients. When a

time trend in the

model is ente rtaine d, the c o rrespondin g limit variance o f our estim ator is muc h less affec ted by

the underlying true

c o efficients th an that of the c o mpeting and Han estimator . T h e estim ati on e fficiency m a y

turn into power gains

for te sts built on the estim ator in d y na mic pa nel contexts. 4. International C o op eration Act ivities (Optional)

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15 VI. A PPENDIX I: M INUTES FROM P ROGRAM D ISCU SSION M EETINGS VII. A PPENDIX II: 1. P UBLICAT ION L IST Ȑ C ONFERENCES ,J OURNALS ,B OOKS ,B OO K C HAPTERS , etc. ȑ 2. P ATENT L IS T 3. I NVENT ION L IST 4. L IST OF W O R KSHOPS /C ONFERENCES H OSTED BY THE P ROGR AM 5. L IST OF P ERSONAL A CHIEVEMENTS OF THE PI S 6. L IST OF T ECHNOLOGY T RANSFER S 7. L IST OF T ECHNOLOGY S ERVICE S P UBLICAT ION L IST

W

orking paper:

first year

and second year:

Doing Justice to Fundamentals in Exchange

Rate Forecasting: A Control over Estimation

Risks (under review

)

third year:

Averaging to Improve Efficiency in Time Series Regressions

fourth year:

Gaussian Infer

ence in General

AR

(1) Models Based on Long Differ

ence

VIII. A PPENDIX III: L IST OF P UBLICATIONS IN “T OP ”J OURNALS AND C ONFERENCES P UBLICAT IONS :

first year:

N

/A

second year:

N

/A

third year:

N

/A

fourth year

N

/A

IX . A PPENDIX IV: S LIDES ON S CIENCE AND T ECHNOLOGY B REAKTHROUGHS (

TWO SLIDES FOR EACH BREAKTHROUGH

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16 X. APPENDIX V: MIDTERM/FINAL SELF-ASSESSMENT

P

ROGRA M

T

IT LE

:

Gaussian Infer

ence in General

AR(1) Models Based on Long Differ

ence

A SSESSMENT S UBJECT S CORE Ȑ 1~5 , L OW T O H IGH ȑ

PROGRAM’SCONTENTS & PERFORMANCE

Importance & In

novation

of th

e

Program’s Major Tasks

4 Clarit y and Pres enta tion of th e R eport 4 Viability of th e Program’s Appr oaches & Metho dologies 4 Principal Invest ig ator’s Com petence for L ead ing the Program 3 Interface & Integration between

Overall & Sub-Project(s)

3

Interface & Integration

among A ll Sub-Projects 2 Manpower & Ex penditur es 4 PROGRAM’S RESULTS Contribution in Enhancing

the Institute’s International Academic Standing

4 Impact on Advancing Teaching or on Techno log y D evelopment 2 Total Score 30

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17 R EVIEWER ’ S C OMMENTS &S UGGESTION : 1. This question is important, because it is closely related to the uni t root tes ting and c a n be g en eraliz ed to the panel

unit root literature

. 2. Th e author s the n focu s on the c a se

2 k

, i.e., the second-or der difference oper ator . 3. The authors

propose a second-order difference operator,

2

(1

)

L

, as c o mpared to

the usual first-or

der difference ope rator,

(1

)

L

, used in Phillips and Han

(2008). 4 . Th e s e co n d -o rd er d if fe ren cin g es timator is as y mptoticall y norm al under suitable re g ularit y condi tions . The asymptotic variance

of the second-order differenci

ng es timator does not de pend on the value of

U

. Further more , the pr opose d estimator is mo re e fficient than the first-or der differe ncin g estimator of Phillips and Han (2008) when

0 U

!

. This result is inte

resting becaus e most m a cr oeconomic tim e series belon g to this cate gory. 5. P. 2, line 12?, why 2 1

()

t tp

yO

U

'

when

1 U

!

? Any pr oof or re ference ? 6. Section 3 c o nsiders the c a se where the re is a linear trend in the data-g ener atin g process (DGP). Phillips and Han (2008) pr op ose a d o ubl e FD oper ator to deal with

this case. The

double FD e stimator for the model in (1) is show n t o be as y mptotic norm

al. One interestin

g

question is that the FD esti

mator of Phillips and Han (2008) can be m o difi ed to d e al with the m o del w ith a linear tre nd with ou t usi ng the double FD es timator. 7. The auth or s mi g ht c o nsi der the relati ve e fficienc y of the double F D es

timator with the

m o dified sin g le FD estimator under the m o del i n (1) when

1 U

d

. This might pr ovide an alter nati ve es

timator for the d

y namic panel data m o del. 8. The s a me

argument also applied to the single

second-orde

r differencin

g

estim

ator and the

double second-order differencing e sti mator unde r the same se t-up.

9. What is the order

of magnitude of 1, 1 1 t

y

'

when

1 U

!

? Wh at is the or der of magnitu de of 2, 2 1 t

y

'

when

1 U

!

? This is related to the l o cal behavi or of the estim ator around

1 U

.

10. What would happen if

there are lagge d depe ndent variables 2

,...,

tt k

y

at the RHS?

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18 P RINCIPLE I NVESTIG AT O R ’ S F EEDBACK :( A VAILABLE ) 1. Com ments 1-4 are s u mm ar

ies of the pape

r. 2. Com ment 5: Take the l o ng difference on AR(1), and re cursively subs tituting the lag de pende nt variables backw

ards, the desired result will be established.

3. Com ment 6: The as ymptoti cs for the double FD estim ator indeed are ver y m u ch invol ved for the models with time trend. After the c o

nference, we thought hard abou

t the c o m ment by the r eferee, and di d find an othe r muc h simpler tr ans form ation th at c a n yield the nor mality as the d o u b le F D estim a tor d o es . We sh all repor t the res ults in the revi sion of the paper. 4. Comment 7:

We will investigate the possi

bility. 5. Comment 8: A suggesti on

that will be well-taken.

6. Com ment 9: 2 cases need to be considered when

1 U

!

, t h e explo siv e AR (1) sy st em a n d t h e midly ex plo siv e AR(1) s y st

em. The paper

does not inves

tiage the for mer case as i ts as ymptotics is non-st andar d again. But the c o nsidere d system falls into the latter case, the no rmality results applies as we found for the s imple models without trend in the paper. 7. Comment 10: This is a hard questi on. We have tried to consider AR(2) models. But it turned out that the calculations were ver y m u ch involved and

the derivations did not lead

to results of si gnific ance. Program Re vi ewer’s Signature: ጰ Ў ᅼ T sa i, W e n -J en

(18)

Sub-pro

ject

#

7 U

nderstanding

Exc

h

ange

Rate

Predictabilit

y:

The

R

ole

of

Estimation

Risk

4/4

1 In

tro

duction

and

Summary

Diﬀerence-t

y

p

e

transformation

h

as

b

een

one

o

f

commonly

emplo

y

ed

p

ractices

in

time

series

researc

h.

In

studies

w

ith

d

ynamic

panel

d

ata,

diﬀerencing

eliminates

the

need

to

estimate

ﬁxed-eﬀect

parameters.

In

cases

where

the

data

is

generated

b

y

a

unit

ro

ot

se-ries,

b

y

doing

so,

the

stationarit

y

of

the

series

under

transformation

can

b

e

ac

hiev

ed.

The

transformations,

suc

h

a

s

P

rais-Winsten

estimator

a

nd

Co

ch

rane-Orcutt

estimator,

are

a

lso

useful

in

reducing

the

d

egrees

of

auto

correlations

in

regression

errors.

E

ﬃciency

o

f

the

re-sulting

estimates

can

b

e

made

close

to

the

b

o

und

attained

b

y

the

Gauss-Mark

o

v

theorem.

Researc

h

in

the

y

ears

o

f

the

pro

ject

suggests

that

a

com

bination

b

et

w

een

the

dif-ference

estimator

a

nd

the

O

LS

one

leads

to

eﬃciency

impro

v

emen

ts

in

time

series

con

texts.

F

u

rthermore,

P

hillips

and

H

an

(2008)

as

w

ell

as

P

a

paro

ditis

and

P

olitis

(2000)

sho

w

that

the

diﬀerence-based

metho

d

leads

to

the

standard

Gaussian

asymptotic

theory

for

the

AR(1)

series,

irresp

ectiv

e

o

f

w

hether

the

series

is

a

unit

ro

ot

pro

cess.

The

clev

er

transformation

o

f

P

hillips

and

H

an

(2008)

and

P

aparo

ditis

and

P

olitis

(2000)

comes

from

the

observ

a

tion

that

the

a

utoregressiv

e

co

eﬃcien

t

o

f

concern

is

in

a

one-to-one

linear

relation

to

the

ﬁ

rst-order

a

uto

correlation

co

eﬃcien

t

for

the

diﬀerenced

series.

Rather

than

o

n

the

lev

el

regressions,

the

a

utoregressiv

e

co

eﬃcien

t

can

no

w

b

e

reco

v

ered

b

y

w

o

rking

on

the

transformed

diﬀerenced

regressions.

Because

o

f

the

stationarit

y

nature

of

the

diﬀerenced

series,

the

u

n

u

sual

limit

normalit

y

is

a

ble

to

b

e

obtained

for

all

v

alues

o

f

the

autoregressiv

e

co

eﬃcien

t,

including

the

case

o

f

a

unit

ro

ot.

It

th

u

s

a

v

o

ids

the

discon

tin

uit

y

problem

in

the

limit

distributions

asso

ciated

with

the

lev

el

regressions

(Phillips

and

H

an,

2008).

This

pap

er

in

v

estigates

whether

the

aforemen

tioned

transformation

is

unique

to

the

con-sidered

A

R(1)

mo

dels.

O

f

p

articular

concern

is

if

a

n

y

linear

relations

that

could

yield

stan-dard

limit

results

remain

to

b

e

found,

when

“higher-order

diﬀerence”

tec

hnique

applies.

Our

inquiry

in

to

the

question

a

rises

from

the

p

oten

tial

uses

of

the

diﬀerence-based

estimator.

Virtually

,

regardless

of

whic

h

o

rder

is

tak

en

o

n

A

R(1)

pro

cess,

to

obtain

consistency

for

the

p

arameter

en

tails

the

estimation

of

the

m

etho

d

o

f

m

omen

ts.

While

the

m

etho

d

o

f

m

o-1

(19)

men

ts

estimator

is

u

seful

to

a

chiev

e

the

consistency

in

large

samples,

it

is

found

to

suﬀer

small-sample

bias,

sp

eciﬁcally

in

the

con

text

with

panel

d

ata.

Hahn

et

al.

(2007)

adv

o

cate

the

long-diﬀerence

metho

d

of

Grilic

hes

a

nd

Hausman

(1986),

a

n

extreme

v

ersion

o

f

higher-order

diﬀerence,

to

reduce

the

small-sample

problem

in

this

situation.

Nev

ertheless,

Han

a

nd

Phillips

(2009)

sho

w

that

their

diﬀerence-based

estimator,

when

applying

to

dynamic

panel

data

mo

dels,

b

asically

incur

no

bias,

ev

en

in

the

cases

where

time

dimension

o

f

the

data

is

v

ery

short.

More

than

this,

their

estimator

is

imm

une

to

the

w

eak

instrumen

t

problem

that

o

ccurs

to

some

of

the

w

idely

used

metho

d

of

momen

ts

estimator

for

the

cases

where

the

autoregressiv

e

co

eﬃcien

t

is

close

of

unit

y.

The

P

hillips

and

H

an

estimator

is

a

ctually

based

on

the

n

otion

o

f

p

ro

cessing

the

ﬁ

rst-diﬀerence

time

series

under

study

.

S

eeing

these

practical

adv

a

n

tages,

whether

the

notion

with

pro

cessing

data

is

applicable

to

higher-order

diﬀerence

time

series

equally

deserv

es

careful

in

v

estigations.

W

e

demonstrate

in

this

part

of

the

sub-pro

ject

that

for

second-diﬀerence

AR(1)

series,

another

transformation

w

hic

h

g

iv

es

rise

to

some

linear

relationship

b

et

w

een

the

A

R

co

eﬃcien

t

and

the

auto

correlation

co

eﬃcien

t

d

o

es

exist.

T

he

transformation

is

distinct

from

that

for

ﬁrst-diﬀerence

series

unco

v

ered

b

y

Phillips

and

H

an

(2008)

and

P

aparo

ditis

and

P

olitis

(2000).

There,

ho

w

ev

er,

is

essen

tially

no

suc

h

linear

transformations

a

v

ailable

for

a

n

y

higher-order-diﬀerence

series.

Our

diﬀerence-based

estimator

p

ossesses

a

n

um

b

er

o

f

in

teresting

prop

erties.

Lik

e

that

of

Phillips

and

H

an

(2008),

the

transformed

second-diﬀerence

estimator

o

f

the

autoregressiv

e

co

eﬃcien

t

h

as

a

G

aussian

asymptotic

distribution

that

a

pplies

to

b

oth

the

unit

ro

ot

case

and

con

v

en

tional

cases.

This

implies

that

the

normalit

y

limit

stands

for

the

lo

cal-to-unit

y

cases

as

w

ell.

Th

us,

the

limit

distribution

is

con

tin

u

ous

a

s

the

autoregressiv

e

co

eﬃcien

t

p

asses

through

unit

y.

Our

sim

ulations

further

rev

eals

that

the

estimator

displa

ys

negligible

bias

for

v

ery

small

samples,

as

opp

o

sed

to

the

con

v

en

tional

least

squares

estimator.

Chen

and

K

uo

(2009)

in

fact

sho

w

analytically

that

the

ﬁnite-sample

bias

for

b

o

th

Phillips

and

H

an

estimator

and

o

urs

shares

the

same

appro

x

imate

magnitudes.

Moreo

v

er,

the

limit

distribution

o

f

o

ur

estimator

in

simple

AR(1)

m

o

d

els

w

ithout

time

trend

exhibits

a

constan

t

v

a

riance

of

v

a

lue

2. This

is

in

con

trast

to

that

of

Phillips

and

H

an

estimator,

a

linear

increasing

function

of

the

autoregressiv

e

co

eﬃcien

t.

P

a

rticularly

,

the

limit

v

a

riance

for

o

ur

estimator

is

smaller

for

that

of

Phillips

and

H

an

estimator

for

an

y

p

ositiv

e

a

utoregressiv

e

co

eﬃcien

ts,

typical

situations

encoun

tered

in

p

ractice.

When

a

time

trend

in

the

m

o

d

el

is

en

tertained,

the

corresp

onding

2

(20)

limit

v

a

riance

of

our

estimator

is

m

uc

h

less

aﬀected

b

y

the

underlying

true

co

eﬃcien

ts

than

that

of

Phillips

and

H

an

estimator.

The

estimation

eﬃciency

ma

y

turn

in

to

p

o

w

er

gains

for

tests

built

on

the

estimator

in

d

ynamic

panel

con

texts.

2 M

o

d

el

and

the

T

r

ansformation

Consider

a

simple

autoregressiv

e

mo

del

w

here

y

t

=

α

+

u

t

_,u

t

=

ρu

t− 1

+

ε

t

_,

where

ρ

is

the

autoregressiv

e

co

eﬃcien

t

o

f

in

terest,

ε

t

∼

iid

(0

,σ

2

),

and

ρ

∈

(−

1 ,1].

T

he

structural

AR(1)

mo

del

a

b

o

v

e

corresp

onds

to

a

reduced

form

written

a

s

y

t

=(

1 −

ρ)

α

+

ρy

t− 1

+

ε

t

(1)

from

whic

h

the

data

generated

is

a

simple

unit

ro

ot

pro

cess

for

the

b

o

undary

case,

or

a

stationarit

y

pro

cess

when

|ρ|

<

1. No

w

consider

a

long-diﬀerence-t

y

p

e

transformation

o

f

(1):

k

y

t

=

ρ

k

y

t− 1

+

k

ε

t

_,

(2)

where

k

=(

1 −L

k

)=(

1 −L

)(1

+

L

+

L

2

+

···

+

L

k− 1

),

with

L

b

eing

the

lag

op

erator.

Note

that

the

o

rthogonalit

y

condition

for

the

transformed

mo

del

d

o

es

n

ot

generally

hold,

as

E[

k

y

t− 1

k

ε

t

_]=E

[

k

y

t− 1

(

k

y

t

₋

ρ

k

y

t− 1

_)]

=

−

ρ

k− 1

σ

2

.

A

s

a

result,

inconsisten

t

estimates

of

the

co

eﬃcien

t

are

w

ell

exp

ected

when

the

least

squares

p

rinciple

applies

to

(2).

P

hillips

and

H

an

(2008)

deriv

ed

a

simple

transformation

o

f

the

ﬁrst

diﬀerenced

equation

that

can

yield

consisten

t

estimates

u

sing

least

squares.

W

e

in

tend

to

in

v

estigate

if

a

n

y

transformations

of

the

k

ind

of

Phillips

and

H

an

(2008)

exist

for

the

long-diﬀerenced

equations.

As

in

Phillips

and

H

an

(2008),

the

transformed

regression

tak

es

the

form:

k

y

t

=(

ρ−

φ

)

k

y

t− 1

+(

k

ε

t

₊

φ

k

y

t− 1

₎₌

θ

k

y

t− 1

+

η

t

_.

(3)

where

θ

=

ρ−

φ

,

η

t

=

k

ε

t

₊

φ

k

y

t− 1

_,a

n

d

φ

is

the

co

eﬃcien

t

to

b

e

determined

from

the

transformation.

The

d

esired

transformation

is

to

re-establish

the

orthogonalit

y

condition.

Th

us,

w

e

exp

ect

the

follo

wing

condition

E

(

k

y

t− 1

_η

t

_)=E

[

k

y

t− 1

(

k

ε

t

+

φ

k

y

t− 1

_)]

=

0 to

hold

for

the

transformed

regression.

W

e

can

th

us

solv

e

for

φ

=

ρ_k − 1 (1 − ρ₂ ) 2(1 − ρ_k )

,

w

hic

h

o

nly

is

a

function

of

the

a

utoregressiv

e

co

eﬃcien

t,

giv

en

a

n

y

k

.

T

aking

adv

a

n

tage

of

the

d

eriv

ed

equiv

alence

relations

a

b

o

v

e,

the

estimated

a

utoregressiv

e

co

eﬃcien

t

can

no

w

b

e

reco

v

ered

from

the

relation

that

θ

=

ρ−

ρ_k − 1 (1 − ρ₂ ) 2(1 − ρ_k )

.

3

貨幣、匯率與動態均衡之學術前沿研究-子計畫七：匯率預測：估計風險之角色(IV)

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