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Testing for the Cointegrating Rank of a VAR Process with Level Shift at Unknown Time

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f[b!Z^_n+Z^nH \‰d®w\Hb_wRlT_fcl\b!nHw@ZT…nRSRlOi^_\HRS`_f[†Dzgb!Z^b_@«pfkw@ZcS\Hb!šj1š>clRSR¨`_clfwTfffnHRlR

fZn_Rl_\H][kZclfX\Ho¥Z^ŽfZf†

(12)

u fnHRSRœt\bZqrnHkRS\=@«cl[b›šjnHf›][kov\k[`±\=@«_RS\H®Z^_4§…b_ceZ®fnHRS®_k\][kcSfŽ\Ho

\T_1_\pw[`_T_[f%nHb_`´Z\3w[\_nH{^_*gš \ _knHY `…n*fZcSn\f

Ù

Ž nb_`µàŽ

†±sX^_=fcS1T_R ncS\bf

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œ

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š>^_[

L = `_cn˜

œ]\

CTjkkkrj

\ 3 Ÿ nHb`UX nH

œ ® \ ® Ÿ nHb_`

œ ® \ œ[Z

? ®

ŸŸ {n~Zclw[ffk][w[cld[RlOH† u f

f^\„š>b® O›s¥\`…n~^_cSf¨gOp] \Ho…_k\w[ff{nOŽ jk˜nHk`_`nHf£nVwnHb\b_cSwnHRov\ ov\H¥clbdfZcl˜nZclb_˜>Z^_

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

u+ž

œ Ÿ _\pw[ffkf>\Ho¨cSb[fk

wnb± ®\ ZnHcSb`!ov\H

œ

†l Ÿ O3RSclb_nHVZnHb_fkov\HnZcl\b_fš>^cSw}^ªRSn‰dH®fT_w}^ªZ[fkf›clbdHncnbP†

5

^…ndVT_fk`4gšj\‚ZZ^_k@‚£nHb`4ov\HT_‚¦`_cS%b_fkcS\b_nHR_k\w[fkff¤ov\-fkcST_RSnZcl\b_f£nHb_`%\¤f\%X\HoZ^_

[fT_ReZf-ov\H Z^>^_[@‚¦`_cS%b_fkcS\b_nHR_‡Ž

m

f-cSb4\Hr`_@}nHclR_^_[> ][wnHTf>Z^_@O4nH>k_kf[b}ncld

ov\\HZ^_[DwnHf[fŽnHfš RlR©† 5 1š>clRSR¥nHb…nRlOp¬._\pw[ff[f+š>cl^!w\clbZ˜HZncSb_˜ZnHb_Y

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®

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®

†!z¦o

®

=J

L = \

CŽš>ceZ^_^

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CC^

Œ

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¾)CTj¥¾

ò Ÿ [_[f[bPf*Z^_tcSb_f}nHbZnHb_\HT_f{`_[][b_`_[b_w[O ]@gšj[b²Z^™fkZnZcl\b…nHO°nHb_`

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 uDž

\H`_[

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nHb_`B _knHYª`…n~Z

Ž

£^_fk1[fT_ReZfn

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Ù

Ž nHb_`}àŽ š>^_cSw}^±cSb

ZTbi{nOLn0&Œ[w[>Z^_Žfk{nHRlRŒfnHRS_k\][kcSfX\o£Z^ŽfZfXov\rZ^›w[\cSb˜nZclb_˜®ZnHbY†

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ov\pwT_f\b.^_jkfTRlZf¨ov\.{—= †¥sX^_jbPT_ \Ho…k_RlcSwn~ZcS\Hb_f¨cSf¤‰†sX^PT_f£^_jf}nHb`…nH`1k\

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Ÿ ø

‰HpHH†þ˜_†lg*~ih~%j= /k؊†

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

u+ž \k`_kf



¨nHRl^_\T_˜H^!Z^_T_4\k`_clf

 = ¾Z\=@«_Rl\^_®cS%…nHw@\Ho¤Z^cSfº PT…nHbZcegO3\b

(13)

s£n _RStLnHb_`´µcS˜T_kf[‚Riw\b}ncSb³kfkT_Rlf1ov\{ = ‰tnHb_`°nif^_ceo¡`…n

Ž = H¥Z^…n~cSf[

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Z\tnHRlR]\Hffcl _RS4 _knHY3\clbPf ]@gšj[bB^_~Z^³nHb_`ªZ^_Ã^±\H _f[kdnZcl\b¾† ž [fT_ReZf.nH4fk^_\‰š>b

ov\r`c,&][brdnHRST_[fr\Ho¨Z^+f^_ceo¡X…nHnH%[Z[ft‚=

œ

#jZ#j¥‚Wk

Ÿ d_nHRlb_nZced

 u+ž

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Ù

Ž

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Ž

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5

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\

CX= #kšoX = œ

#j}

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nHb_`‚=

œ

/j}#j|

Ÿ dSÂàŽ

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š>^_[nfZ^_RSnZced4ovr pTb_w@O´\Ho

Ù

Ž = HtcSf1Šnm

œ

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œœ

¾)C¥j¥¾

ò Ÿ = œ

/kš2@jZ#k؋ ŸŸ

œ

w\H…n

µ£cl˜T_®

Ÿ †

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œ

fk

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w\H[R ncS\b

œ

¾)Cœ=5#k¡2_@¾

ò

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Ž

w\H…n`!\{Z^®]\‰š \ ZnHcSb`™š>ceZ^

Ù

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nH _fkb

œ

¾)C’=Þ¾

ò = Ÿ œ

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fZcl{ncS\b4%[^_\``_\1b\Hr`_c+&Œ[¤T_w}^{ovk\ Z^_nH__k\nHw}^{š>^cSw}^*clb_wRlT_`_fjnHb{clT_RSfkV`T_.O

cSb.nH`_`clZcl\b

œ Ù

Ž Ÿ † Q b_w[Hclb›^_j_kfkbZReO.w[\b_fkcS`_[`1wnHfk š>^_[

 = ~cloZ^_ w\clbZ˜HZncSb_˜>Znb_Y

\HoZ^_V_k\w[ff¢cSf-^_>\b_RlO PT…nHbZcegO4\HoŒclbZkfpT_fclb_˜1à

Ž

nO®b_\H¤RlnH`4Z\Žfcl˜b_ce§…wnHb¢`_clfkZ\HkZcl\b_f[†

(14)

cSb˜ ž nb_Y

® = HF¾)C<= ¾ ò

=0

\

C›= #kš¢nH%_Rl*cl¬ã{ = -hjknHY

m

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Ž = H!nHb_`

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

uDž T_

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fk^_clo¡>`_nZ[fkZcl{n~Z\<à

ŽBœ

š>ceZ^_\HT>cS%_T_Rlf+`_T%cS[f Ÿ

Œ

†š2 †š2PH †¡2PH †¡2Š †šH‹ †l‹2

=5 †Ø †Ø †Ï †e †š‹2 †³

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® = †ÏŠH‹ †lH †l †e †e‰Š †e‰ †Ø

® = †Ï †Ø‰‹ †Ø‹ †ÏH †Ï †Ï‰‹ †Ø

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Œ

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_

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®  †Ï †ØH †Ø †Ï †Ï †Ï †ØH

fk^_clo¡>`_nZŽfZcl{n\

Ù

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Ù

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Œ

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® = †Ï‹ †l‹ †l †e„ †e‰ †Ï †Ø

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®  †Ï †ØH †Ø †Ï †Ï †Ï †ØH2

udsG=

œ

#jZ#j Ÿ d

(15)

Without Impulse Dummies With Impulse Dummies θ 1 =0.4 θ 2 =0.8 rejection frequencies of rank tests θ 1 =θ 2 =0

break date estimates break date estimates rejection frequencies of rank tests

0.0

0.2

0.4

0.6 0.8

1.0 0 1 3 5 7 10 δ 3

<50 50 >50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 3 5 7 10 δ 3

r0=0 r0=1 r0=2 0.0 0.2 0.4 0.6 0.8 1.0 0 1 3 5 7 10 δ 3

<50 50 >50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 3 5 7 10 δ 3

r0=0 r0=1 r0=2

0.0

0.2

0.4

0.6

0.8

1.0 0 1 3 5 7 10 δ 3

<50 50 >50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 3 5 7 10 δ 3

r0=0 r0=1 r0=2 0.0 0.2 0.4 0.6 0.8 1.0 0 1 3 5 7 10 δ 3

<50 50 >50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 3 5 7 10 δ 3

r0=0 r0=1 r0=2

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(16)

Without Impulse Dummies With Impulse Dummies θ 1 =0.4 θ 2 =0.8 rejection frequencies of rank tests θ 1 =θ 2 =0

break date estimates break date estimates rejection frequencies of rank tests

0.0

0.2

0.4

0.6

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r0=0 r0=1 r0=2

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0.2

0.4

0.6

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1.0 0 1 3 5 7 10 δ 3

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r0=0 r0=1 r0=2 0.0 0.2 0.4 0.6 0.8 1.0 0 1 3 5 7 10 δ 3

<48 48/49 50 51/52 >52 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 3 5 7 10 δ 3

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(17)

Without Impulse Dummies With Impulse Dummies rejection frequencies of rank tests break date estimates p=1 p=3 break date estimates rejection frequencies of rank tests

0.0

0.2

0.4

0.6 0.8

1.0 0 1 3 5 7 10 δ 3

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r0=0 r0=1 r0=2 0.0 0.2 0.4 0.6 0.8 1.0 0 1 3 5 7 10 δ 3

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r0=0 r0=1 r0=2

0.0

0.2

0.4

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0.8

1.0 0 1 3 5 7 10 δ 3

<50 50 >50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 3 5 7 10 δ 3

r0=0 r0=1 r0=2 0.0 0.2 0.4 0.6 0.8 1.0 0 1 3 5 7 10 δ 3

<50 50 >50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 3 5 7 10 δ 3

r0=0 r0=1 r0=2

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cpi t -3.0 -1.0 1.0 3.0 5.0 7.0

19 70 -1

19 72 -1

19 74 -1

19 76 -1

19 78 -1

19 80 -1

19 82 -1

19 84 -1

19 86 -1

19 88 -1

19 90 -1

19 92 -1

19 94 -1

19 96 -1

19 98

-1 ex t 2.0 0.0 -2.0 -4.0 -6.0

19 70 -1

19 72 -1

19 74 -1

19 76 -1

19 78 -1

19 80 -1

19 82 -1

19 84 -1

19 86 -1

19 88 -1

19 90 -1

19 92 -1

19 94 -1

19 96 -1

19 98 -1

w t -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

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19 72 -1

19 74 -1

19 76 -1

19 78 -1

19 80 -1

19 82 -1

19 84 -1

19 86 -1

19 88 -1

19 90 -1

19 92 -1

19 94 -1

19 96 -1

19 98 -1

lp t -10.1 -9.9 -9.7 -9.5 -9.3 -9.1 -8.9 -8.7 -8.5

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19 72 -1

19 74 -1

19 76 -1

19 78 -1

19 80 -1

19 82 -1

19 84 -1

19 86 -1

19 88 -1

19 90 -1

19 92 -1

19 94 -1

19 96 -1

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epop t -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0

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19 74 -1

19 76 -1

19 78 -1

19 80 -1

19 82 -1

19 84 -1

19 86 -1

19 88 -1

19 90 -1

19 92 -1

19 94 -1

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