On the indeterminacy of determinacy and indeterminacy

EUROPEAN

CENTRAL

BANK

E

C

B

E

Z

B

E

K

T

B

C

E

E

K

P

W O R K I N G PA P E R S E R I E S

WORKING PAPER NO. 277 ON THE INDETERMINACY OF DETERMINACY AND INDETERMINACY ANDREAS BEYER, ROGER E. A. FARMER SEPTEMBER 2003

EUROPEAN

CENTRAL

BANK

W O R K I N G PA P E R S E R I E S

WORKING PAPER NO. 277 ON THE INDETERMINACY OF DETERMINACY AND INDETERMINACY1 ANDREAS BEYER2, ROGER E. A. FARMER3 SEPTEMBER 2003

1

2 3

This note has benefitted from conversations with Harold Cole,Thomas Lubik, Lee Ohanian and Frank Schorfheide.The authors wish to thank two anonymous referees for the ECB working paper series for helpful suggestions.This note was completed in the summer of 2003 while Farmer was visiting the Directorate General Research as part of the European Central Bank’s Visitor Programme. He wishes to thank members of DG-Research for their hospitality. The views expressed in this note are those of the authors and do not necessarily represent those of the ECB.This paper can be downloaded without charge from http://www.ecb.int or from the Social Science Research Network electronic library at http://ssrn.com/abstract_id=487417. European Central Bank, Postfach 16 03 19, D-60066, Frankfurt am Main, [email protected] UCLA, Dept. of Economics, 8283 Bunche Hall, Box 951477, Los Angeles, CA 90095-1477, [email protected]

©

European Central Bank, 2003 Address

Kaiserstrasse 29 D-60311 Frankfurt am Main Germany

Postal address

Postfach 16 03 19 D-60066 Frankfurt am Main Germany

Telephone

+49 69 1344 0

Internet

http://www.ecb.int

Fax

+49 69 1344 6000

Telex

411 144 ecb d

All rights reserved. Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged. The views expressed in this paper do not necessarily reflect those of the European Central Bank. ISSN 1561-0810 (print) ISSN 1725-2806 (online)

Table of Contents

Abstract

4

Non technical summary

5

1

Introduction

6

2

Placing our work in context

7

3

Two equivalent models 3.1 Model 1 3.2 Model 2

7 8 9

4

Models 1 and 2 compared

11

5

Conclusion

11

Appendix

13

References

17

European Central Bank working paper series

19

ECB • Working Paper No 277 • September 2003

3

Devwudfw D qxpehu ri dxwkruv kdyh dwwhpswhg wr whvw zkhwkhu wkh X1V1 hfrqrp| lv lq d ghwhuplqdwh ru dq lqghwhuplqdwh htxloleulxp1 Zh dujxh wkdw wr dqvzhu wklv txhvwlrq/ rqh pxvw lpsrvh d sulrul uhvwulfwlrqv rq odj ohqjwk wkdw fdqqrw eh whvwhg1 Zh surylgh h{dpsohv ri wzr hfrqrplf prghov1 Prgho 4 glvsod|v dq lqghwhuplqdwh htxloleulxp/ gulyhq e| vxqvsrwv1 Prgho 5 glvsod|v d ghwhu0 plqdwh htxloleulxp gulyhq e| ixqgdphqwdov1 Jlyhq dvvxpswlrqv derxw wkh vkrfn glvwulexwlrq ri prgho 5/ lw lv srvvleoh wr qg d glvwulexwlrq ri vxqvsrw vkrfnv wkdw gulyh prgho 4 vxfk wkdw wkh wzr prghov duh revhuydwlrqdoo| htxly0 dohqw1 MHO0Fodvvlfdwlrq= F6 lw lv lqghwhuplqdwh li pdq| rwkhu htxloleuld duh duelwudulo| forvh wr wkh uvw1 Li htxloleuld duh lqghwhuplqdwh/ qrq0ixqgdphqwdo vkrfnv pd| frqwulexwh wr wkh yduldqfh ri hfrqrplf xfwxdwlrqv dqg/ li djhqwv duh ulvn dyhuvh/ wkhvh xfwxdwlrqv zloo uhgxfh zhoiduh1 Lq wkh uhfhqw olwhudwxuh uhvhdufkhuv kdyh dwwhpswhg wr whvw hpslulfdoo| zkhwkhu wkh X1V1 hfrqrp| lv lq d ghwhuplqdwh ru dq lqghwhuplqdwh htxloleulxp1 Zh dujxh wkdw wr dqvzhu wklv txhvwlrq/ rqh pxvw lpsrvh d sulrul uhvwulfwlrqv rq odj ohqjwk wkdw fdq0 qrw eh whvwhg1 Lq wklv qrwh zh srlqw rxw/ e| phdqv ri d vlpsoh h{dpsoh/ wkdw lw lv qrw srvvleoh wr ghflgh zkhwkhu uhdo zruog gdwd lv jhqhudwhg e| d ghwhuplqdwh ru dq lqghwhuplqdwh surfhvv1 Zh frqvwuxfw wzr prghov wkdw jhqhudwh wkh vdph olnholkrrg ixqfwlrq dqg khqfh duh revhuydwlrqdoo| htxly0 dohqw1 Rqh prgho glvsod|v dq lqghwhuplqdwh htxloleulxp gulyhq sxuho| e| qrq0ixqgdphqwdo +vxqvsrw, vkrfnv1 Wkh rwkhu prgho glvsod|v d ghwhuplqdwh htxloleulxp gulyhq sxuho| e| ixqgdphqwdo vkrfnv1 Rxu uhvxow lv glvfrxudjlqj iru wkh srvvlelolw| ri glvwlqjxlvklqj ehwzhhq jrrg dqg edg hfrqrplf srolflhv vlqfh lw lpsolhv wkdw/ dw d yhu| ixqgdphqwdo ohyho/ ghwhuplqdwh dqg lqghwhu0 plqdwh prghov fdqqrw eh glvhqwdqjohg1


5

4

Lqwurgxfwlrq

Wkh htxloleulxp ri d udwlrqdo h{shfwdwlrqv prgho lv ghwhuplqdwh li lw lv orfdoo| xqltxh> lw lv lqghwhuplqdwh li pdq| rwkhu htxloleuld duh duelwudulo| forvh wr wkh uvw1 Li htxloleuld duh lqghwhuplqdwh/ qrq0ixqgdphqwdo vkrfnv pd| frqwulexwh wr wkh yduldqfh ri hfrqrplf xfwxdwlrqv dqg/ li djhqwv duh ulvn dyhuvh/ wkhvh xfwxdwlrqv zloo uhgxfh zhoiduh1 Khqfh/ lw lv ri vrph lpsruwdqfh wr d srolf| pdnhu wr hqvxuh wkdw klv dfwlrqv gr qrw lqgxfh lqghwhuplqdf|1 Lq dq lq xhqwldo duwlfoh/ Fodulgd Jdol dqg Jhuwohu ^8` kdyh dujxhg wkdw X1V1 prqhwdu| srolf| ohg wr dq lqghwhuplqdwh htxloleulxp lq wkh shulrg iurp 4 f1 Lq rxu h{dpsoh wkhuh duh qr ixqgdphqwdo vkrfnv dqg wt uhsuhvhqwv dq lqghshqghqw qrq0ixqgdphqwdo vkrfn1 Lq jhqhudo/ wkh uhgxfhg irup ri wkh v|vwhp lv irxqg e| vroylqj wkh frp0 sdqlrq irup h{solflwo| dqg holplqdwlqj wkh lq xhqfh ri wkh xqvwdeoh urrwv1 Lq rxu h{dpsoh wkh pdwul{ A lv lqyhuwleoh dqg rqh fdq frpsxwh wkh urrwv ri

8


A3 B e| kdqg=4 Wkh| duh htxdo wr f dqg zkhuh a3 : Wkh uhgxfhg irup lv jlyhq e| wkh h{suhvvlrq 5 7

pt Et dptn o

6

5

8'7

f f

65 87

pt3 Et3 dpt o

6

5

8n7

6 8 wt :

+6,

Uhzulwlqj wklv htxdwlrq zh rewdlq wkh iroorzlqj h{suhvvlrqv iru pt dqg Et dptn o dv ixqfwlrqv ri wkh revhuydeoh yduldeoh pt3 dqg wkh vxqvsrw vkrfn wt

pt ' pt3 n wt ; Et dptn o ' 2 pt3 n wt :

615

+7, +8,

Prgho 5

Iru wkh fdvh ri Prgho 5 zh dvvxph djdlq wkdw wkhuh lv d vlqjoh vwuxfwxudo htxdwlrq jlyhq e| wkh h{suhvvlrq pt ' aEt dptn o n bpt3 n vt :

+9,

Htxdwlrq +9, glhuv iurp +4, lq wkuhh uhvshfwv1 Iluvw/ wkh odjjhg vwdwh yduldeoh pt3 hqwhuv wkh htxdwlrq/ vhfrqg/ wkhuh lv d ixqgdphqwdo vkrfn/ vt dqg wklug/ zh fkrrvh a dqg b vxfk wkdw wkh htxloleulxp ri wkh prgho lv ghwhuplqdwh1 Wkh frpsdqlrq irup ri Htxdwlrq +9, lv uhsuhvhqwhg ehorz/ 5

65

Yt

6

5

B

65

3

5 [w 6 6 5 [v 6 a pt b f pt3 f 7 87 87 8'7 8n7 8vt n 7 8wt : +:, f f f Et dptn o Et3 dpt o A

Yt

Prgho 5 kdv wzr vkrfnv> vt lv d ixqgdphqwdo vkrfn dqg wt lv d qrq0ixqgdphqwdo vkrfn1 wt lv ghqhg lq wkh vhfrqg urz ri Htxdwlrq +:, wr eh wkh glhuhqfh 4

Fkulv Vlpv ^:` surylghv frgh lq pdwode wr frpsxwh wkh uhgxfhg irup ri d olqhdu prgho

ri wklv nlqg lq zklfk wkh glphqvlrq ri wkh v|vwhp lv duelwudu| dqg wkh pdwulfhv A dqg B pd| eh vlqjxodu1


9

ehwzhhq pt dqg lwv gdwh t h{shfwdwlrq1 Vlqfh zh fkrrvh sdudphwhuv vxfk wkdw wkhuh lv d xqltxh htxloleulxp/ wkh qrq0ixqgdphqwdo vkrfn wt zloo eh ghwhuplqhg hqgrjhqrxvo| dv d ixqfwlrq ri vt 1 Suhpxowlso|lqj htxdwlrq +:, e| A3 ohdgv wr wkh h{suhvvlrq/ 5 7

pt Et dptn o

6

5

8'7

f

ab

a

65 87

6

pt3 Et3 dpt o

5

8n7

f a

6

5

8 vt n 7

6 8 wt :

+;,

a

Lw lv frqyhqlhqw iru wkh iroorzlqj dqdo|vlv wr uhsdudphwhul}h wkh prgho lq whupv ri wkh wzr urrwv ri

5 A3 B ' 7

zklfk zh fdoo dqg 1

f

b a

6 8

a

Wkh sdudphwhuv a dqg b duh jlyhq e| wkh h{suhv0

vlrqv/ a ' = E n dqg b ' = E n 1 Li wkh htxloleulxp lv xqltxh/ wkhuh pxvw eh rqh xqvwdeoh urrw wkdw doorzv rqh wr slq grzq wkh qrq0suhghwhuplqhg yduldeoh Et dptn o dv d ixqfwlrq ri wkh odjjhg vwdwh yduldeoh pt3 dqg wkh ixqgd0 phqwdo vkrfn vt 1 Zlwkrxw orvv ri jhqhudolw| zh dvvxph wkdw lv wkh xqvwdeoh urrw vxfk wkdw mm > ;

mm < :

Lq Dsshqgl{ D zh vkrz krz wr vroyh h{solflwo| iru wkh uhgxfhg irup/ zklfk fdq eh zulwwhq dv iroorzv=

E n vt ; E n Et dptn o ' 2 pt3 n vt : pt ' pt3 n

10

+ exw lq wkh devhqfh ri lqghshqghqw lqirupdwlrq rq wkh wuxh yduldqfh ri wt ru vt wkhuh lv qr zd| wr glvwlqjxlvk wt iurp EE n = vt : Vxssrvh wkdw Prgho 5 lv wkh gdwd0jhqhudwlqj surfhvv dqg wkdw vt kdv glvwulex0 wlrq Dv zlwk phdq f dqg vwdqgdug ghyldwlrq £v 1 Wkhq wkhuh h{lvwv d vxqvsrw huuru zlwk glvwulexwlrq Dw dqg vwdqgdug ghyldwlrq £w ; zkhuh £w '

E n £v

vxfk wkdw wkh olnholkrrg ixqfwlrqv ri prghov 4 dqg 5 duh lghqwlfdo1 Li Dv lv qrupdo +dv lv riwhq dvvxphg, wkhq Dw lv dovr qrupdo1 Zh kdyh surylghg dq h{dpsoh ri d ghwhuplqdwh prgho dqg dq lqghwhuplqdwh prgho wkdw duh revhuydwlrqdoo| htxlydohqw1

8

Frqfoxvlrq

Wkhuh lv qr uhdvrq wr wklqn wkdw rxu h{dpsoh lv vshfldo dqg lq fxuuhqw uhvhdufk ^5` zh duh h{sorulqj pruh jhqhudo h{dpsohv zlwk pxowlsoh htxdwlrqv wkdw duh ghulyhg iurp vwuxfwxudo prghov wkdw duh zlgho| xvhg lq wkh olwhudwxuh1 Rxu uhvxow lv glvfrxudjlqj iru wkh srvvlelolw| ri glvwlqjxlvklqj ehwzhhq jrrg dqg edg hfrqrplf srolflhv vlqfh lw lpsolhv wkdw/ dw d yhu| ixqgdphqwdo ohyho/ ghwhuplqdwh dqg lqghwhuplqdwh prghov fdqqrw eh glvhqwdqjohg1


11

Li rxu uhvxow lv fruuhfw wkhq krz duh Oxeln dqg Vfkruikhlgh deoh wr glvwlq0 jxlvk ghwhuplqdwh dqg lqghwhuplqdwh uhjlrqv ri wkh sdudphwhu vsdfh lq X1V1 gdwdB Zh wklqn wkdw wkhlu uhvxow klqjhv rq sulru uhvwulfwlrqv ryhu odj ohqjwk wkdw h{foxgh fhuwdlq prghov iurp frqvlghudwlrq1 Wr vhh krz wklv pljkw zrun/ vxssrvh wkdw d Ed|hvldq zhuh wr eh frqiurqwhg zlwk gdwd jhqhudwhg e| Prgho 5 lq zklfk wkh htxloleulxp zdv ghwhuplqdwh1 Ohw wkh Ed|hvldq fkrrvh d sulru suredelolw| glvwulexwlrq ryhu sdudphwhuv wkdw sodfhv }hur zhljkw rq wkh srv0 vlelolw| wkdw b 9' f/ khqfh/ qr dprxqw ri hylghqfh zloo doorz klp wr uhylvh wklv sulru lq idyru ri d prgho zlwk b 9' f: Wklv lqglylgxdo zrxog frqfoxgh/ lqfruuhfwo|/ wkdw wkh gdwd zdv jhqhudwhg e| Prgho 4 zlwk dq lqghwhuplqdwh htxloleulxp1 Dv Shvdudq srlqwhg rxw lq klv 4 5 A3 B ' 7

f

65

E n n

87

n

f

f

6

5

f

8'7

n

6 8:

Wkh hljhqyhfwruv ri A3 B dvvrfldwhg zlwk wkh urrwv dqg duh jlyhq e| wkh h{suhvvlrqv

5 $ 7

6

5

8

dqg $ 7

6 8;

dqg khqfh A3 B fdq eh gldjrqdol}hg dv 5 A3 B ' Q\Q3 ' 7

Q

65 87

\

f f

3

Q

65

87 3

3 3

6

3 3 8 ; 3

zkhuh wkh froxpqv ri Q duh hljhqyhfwruv1 Zh qrz zulwh wkh v|vwhp dv d sdlu ri vfdodu htxdwlrqv e| lqwurgxflqj wkh iroorzlqj ghqlwlrqv= 5 Zt ' 7 5 t ' 7 5 t ' 7

14

zt z2t t 2t t 2t

6

5

8 ' Q3 7 6

pt Et dptn o 5

8 ' Q3 A3 7 6

6

5

8 ' Q3 A3 7

f f

8;

6 8 vt ; 6 8 wt :


Xvlqj wkh ghqlwlrqv ri Q3 ; A3 dqg Q3 A3 / 5

6

3 3 8 ; 3

Q3 ' 7 3 5 A3 ' 7 5 Q3 A3 ' 7

3 3

f

6

E n E n 6 n

3

n 3

8;

3 3 8 ; 3

zh fdq zulwh wkh h{suhvvlrqv iru zt ; z2t ; t ; 2t ; t dqg 2t lq whupv ri wkh sdudphwhuv dqg = zt ' 3 p 3 Et dptn o ; t z2t ' 33 pt n 3 Et dptn o ; t ' n 3 vt ; 2t ' n 3 vt ;

t ' 33 wt ; 2t ' 3 wt :

< A A A A A A A A A A A @ A A A A A A A A A A A >

+D6,

Xvlqj wkhvh ghqlwlrqv/ wkh v|vwhp fdq eh ghfrpsrvhg lqwr wkh iroorzlqj sdlu ri vfdodu glhuhqfh htxdwlrqv= zt ' zt3 n t n t ;

+D7,

z2t ' z2t3 n 2t n 2t :

+D8,

Vlqfh > ; zh pxvw vhw zt ' zt3 ' f; wr holplqdwh wkh lq xhqfh ri wkh h{sorvlyh urrw1 Lw iroorzv iurp +D7, wkdw t n t ' f;


+D9,

15

l1h1 wkh vxp ri ixqgdphqwdo dqg qrq0ixqgdphqwdo huuruv pxvw dgg xs wr }hur1 >Iurp +D6, lw dovr iroorzv wkdw pt Et dptn o ' f; dqg khqfh Et dptn o ' pt :

+D:,

Xvlqj +D:, dqg wkh ghqlwlrq ri z2t iurp +D6, |lhogv z2t ' pt ;

+D;,

dqg xvlqj wkh h{suhvvlrq +D9, dqg wkh ghqlwlrqv ri t dqg t iurp +D6, lw iroorzv wkdw wt '

n vt :

+D