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1 1 1 1 t h t t t t t N C σ σ _ϕ ϕ C N C C u _ _ + _ _ +

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271-$&D"("G$=(&h !"#0,1$ G-$(18&+=*D-$5&=(#"%*VC-&)"#8,I#&,*)+<GDC8J)8CHA&`G&#-O,+1"&#"(=*VQ& D"("G$=(<&h/&=8G&+-;)-$J&)"&7$C81J$&2*+D*7"(+=#&7*G*#8CH&#D%8#"&"2($2"=*#"&,*)+<GDCJ"&1&D*b D(1$7)-$2*&*,($+<A&F*7"=)-&D"("G$=(&! #81)"C1"&($;"=8#)N&"#$(+JO&7*&(818,"A&e("1&1&J$2*&#1(*b +=$G&G-O718*,($+*#"&";*,"CJ"&,*)+<GDCJ-&J$+=&G)-$J&#("P;-#"&)"&1G-")8&($";)$J&+=*D8&D(*C$)b =*#$JA&B&,*;$-&*7#(*=)*VQ&D"("G$=(<&% > 0K*7D*#-$7)-,&$;"+=8C1)*VC-&p(-+CH"M&G*P)"&-)=$(D($b =*#"Q&J",*&1G-")O&-;*VC-&D("C8&V#-"7C1*)$J&D(1$1&2*+D*7"(+=#*&7*G*#$&)"&1G-")O&($";)$J&D%"C8& D(18&+="%$J&($;"CJ-&R-$PNC$J&,*)+<GDCJ-&7*&D*1-*G<&*7)-$+-$)-"&#8)-,"JNC$2*&1&=(#"%*VC-&)"#8,I#& K=JA&278&Ct/

(

Ct–1

)

h=const.MA

d$D($1$)="=8#)$& 2*+D*7"(+=#*& 7*G*#$& G",+8G";-1<J$& 178+,*)=*#")N& +<GO& <P8=$C1)*VC-& R-$PNC$J&-kD(18+1%$J&K*C1$,-#")$JM/&C*&*D-+<J$&#1I(9

&& & & &&&&&&&& = =t

(

t t– t

)

t t t N Ct t

U

E

β

u

C

,

C

,

N

max

_,

∑

∞_τ τ ₁

& & & &&&&&&&K>M 271-$& β∈

(

0,1

)

& J$+=& +=")7"(7*#8G& #+DI%C18))-,-$G& 78+,*)="& <P8=$C1)*VC-& ,*)+<GDCJ-/& "&E_t&& U&*D$("=*($G&("CJ*)";)8CH&*C1$,-#"5&4*(G<%*#")8CH&#&CH#-;-&tA

o2(")-C1$)-$&R<7P$=*#$&1"D-+<J$G8&#&D*+="C-&(I#)*VC-9 &&& & & & &&

P

_t

C

_t +

B

_t =

(

1

+

i

_t_₁

)

B

_t_₁+

W

_t

N

_t

& & & & &&&&&&K:M 271-$9

&B_t&U&1"+IR&*+1C1O7)*VC-&2*+D*7"(+=#"&7*G*#$2*/&

i_t&i&)*G-)";)"&+=*D"&D(*C$)=*#"/&

W_t&i&)*G-)";)$&#8)"2(*71$)-$&K)"&J$7)*+=,O&D("C8MA

^NC1)"& ,*)+<GDCJ"& ($D($1$)="=8#)$2*& 2*+D*7"(+=#"& 7*G*#$2*&C_t& J$+=& #8)-,-$G& "2($2"CJ-& ,*)+<GDCJ-&1(IP)-C*#")8CH&7IR(&C_t

( )

i _{)"&,=I($&D*D8=&,+1="%=*#")8&J$+=&#&#"(<),"CH&,*),<($)b}

CJ-& G*)*D*;-+=8C1)$JA& j2($2"CJ"& #8,*(18+=<J$& C1O+=*& D(18JG*#")8& #k G*7$;"CH& F30L& "2($2"=*(&& F-q-="&-&3=-2;-=1"/&RO7NC8&4<),CJN&=8D<&\L3&K*&+="%$J&$;"+=8C1)*VC-&+<R+=8=<CJ-MZ₉

&& & & & &&&&&& ≡

∫

_ _

C

_t

C

_t

( )

i

1 1ε ε ε 0 1

di

1& & & & &&&&&&K[M j;*,"CJ"& #87"=,I#& )"& D*+1C1$2I;)$& 7*R("& *D-$("& +-O& )"& 1"+"71-$& G",+8G";-1"CJ-& %NC1)$J& ,*)+<GDCJ-&C_t& D(18& 7")8CH& C$)"CH& 7IR(&P_t

( )

i _{-& <+=";*)8G& D*1-*G-$& )*G-)";)8CH& #87"=,I#& )"&}

#+18+=,-$& 7*R("& 1

( ) ( )

0 di i C i P

∫

t t A& !(*#"71-& =*& 7*& D*#+=")-"& <,%"7<& (I#)"5& D*D8=<& )"& 7*R("& KD*(A&

0";-&>??l/&+Ak[>M9

&& & & & &&&&&&&& ₌

1

_

( ) _t

[

_{t t}

]

ε

t i C P i P

C

(

)

/

& & & & &&&&&&KZM

271-$&D"("G$=(&& > 1&*1)"C1"&$;"+=8C1)*VQ&C$)*#N&D*D8=<&)"&D*J$78)C1$&7*R(*/&"&P_t&"2($2"=*#8& -)7$,+&C$)&7IR(9

&& & & & &&&&&&&&&

≡

∫

( )

1 1 1 0 1 dj j P Pt t

__ε __ε & & & & &&&&&&

KEM & B&(*1#-N1")-"&1"2"7)-$)-"&G",+8G";-1"CJ-&<P8=$C1)*VC-&K>M&D(18&*2(")-C1$)-<&R<7P$=*#8GkK:M& *7D*#-$7)-*&#12;O7$G9&R-$PNC$J&,*)+<GDCJ-&-&D("C8&*("1&R-$PNC$J&-&*C1$,-#")$J&,*)+<GDCJ-/&G*b P$G8&#8D(*#"71-Q&#"(<),-&,*)-$C1)$&*D=8G";-1"CJ-&1"CH*#"5&2*+D*7"(+=#&7*G*#8CH&K+,*(18b +=")*&(I#)-$P&1&#"(<),<& =

(

)

t t C C /&#8)-,"JNC$2*&1&4",=</&P$&2*+D*7"(+=#"&7*G*#$&+N&($D($1$)="b =8#)$ME_9&

&& & & &&&&&&&&&

(

)

=

(

)

t t t t t c w p h c n 1 1 _ _ _ + _ σ σ σ

φ && & & &&&&&&KTM

&& & &&&&&&&&

_[

₍

₎

_]

₌

₍

₎

t t t t t t t t Ec h c Ep p i c h 1 1 1 1 1 _ _ _ _ _{+ +} + + + _

σ σ σ σ & & &&&&&&&KlM

&

271-$& G"%8G-& ;-=$("G-& *1)"C1*)*& D(*C$)=*#$& *7CH8;$)-"& *(82-)";)8CH& 1G-$))8CH& *7& #"(=*VC-&& #&+=")-$&<+=";*)8G/&)DA _≡

(

ss

)

t t N N n ln / A

JD(K9?#/;"%7".9;:)+

!(1$7+-OR-*(+=#"&+N&J$7)*(*7)$&D*7&#12;O7$G&=$CH)*;*2--&D(*7<,CJ-&K+=*+<JN&J$7)*C18))-,*#N&D*b =O2*#N&4<),CJO&D(*7<,CJ-M&*("1&D("C8&1<P8#")$J&#&D(*C$+-$&D(*7<,C8J)8GA&_"P7$&D(1$7+-OR-*(+=#*& D(*7<,<J$&J$7$)&(*71"J&7*R("/&,=I($&+D(1$7"J$&)"&(8),<&*&G*)*D*;-+=8C1)$J&+=(<,=<(1$&,*),<($)CJ-A

e8RI(& D(1$7+-OR-*(+=#"& #& 1"+"71-$& *2(")-C1"& +-O& 7*& <+=";")-"& C$)& 1R8=<& +#*-CH& D(*7<,=I#&

(

*

)

t

p /&D(18&C18G&#8(IP)-"&+-O&7#-$&2(<D8&D(1$7+-OR-*(+=#/&1";$P)-$&*7&+D*+*R<&<+=";")-"&C$)&KD*(A& 0";-&-&0$(=;$(&@SSSMA&r$7)$&1&4-(G/&J",&#&D*7+="#*#8G&+CH$G"C-$&\";g*&KD*(A&)DA&0";-&>??l/&+A&[[&& -& )"+=AM/& 7*,*)<JN& #8R*(<& C$)/& G",+8G";-1<JNC& 178+,*)=*#")$& *C1$,-#")$& 18+,-& #& C1"+-$& *R*b #-N18#")-"&)*#*&<+=";")$J&C$)8A&]71-"%&=8CH&4-(G/&*,($V;")8CH&G-")$G&1*(-$)=*#")8CH&)"&D(18b +1%*VQ&K")2A&forward lookingM/&#8)*+-&(1 – ")A&]+=";")$&D(1$1&)-$&C$)8

(

)

(

)

_ f t p &&G*P)"&#81)"C18Q&)"& D*7+="#-$&(I#)")-"T₉ E_{& &W$=*78&(*1#-N18#")-"&78)"G-C1)8CH&G*7$;-&(I#)*#"2-&*2I;)$J&D(1$7+="#-"JN&)DA&F$r*)2&-&F"g$&K>??TM&*("1&0";-&} K>??lMA T_{& &&!*&;*2b;-)$"(81"CJ-&#"(<),<&G-O718*,($+*#$J&*D=8G";-1"CJ-&178+,*)=*#")8CH&18+,I#&D(1$1&D(1$7+-OR-*(+=#"A}

&& & & && =

(

p

_t

)

(

)

_ +

β

π

θ

βθ

t

p

(

1

)

_k=0

(

)

t

(

_{t+k t+k}

)

k f t

E

mc

p

p

∑

∞ * b

p

p

&& & & &&&&&&KSM 271-$& D"("G$=(&θ ∈ 0,1

* p

& )"18#"G8& +=*D)-$G& +1=8#)*VC-& C$)& \";g*/& "& 1G-$))"&mc_t& *1)"C1"& D(*C$)=*#$& *7CH8;$)-$& ($";)$2*& J$7)*+=,*#$2*& ,*+1=<& ,("5C*#$2*& D(*7<,CJ-& *7& J$2*& #"(=*VC-&& #&+=")-$&<+=";*)8GA

]71-"%&D*1*+="%8CH&4-(G&#8)*+-&"A&6-$&7*,*)<JN&*)$&G-O718*,($+*#$J&*D=8G";-1"CJ-&18+,I#/& ;$C1& J$78)-$& )"V;"7<JN& ,*),<($)=I#A& B$& #12;O7<& )"& +D*+IR& D*7$JG*#")-"& 7$C81J-& *,($V;")$& +N& G-")$G& 1*(-$)=*#")8CH& )"& D(1$+1%*VQ& K")2A& backward lookingMA& 3=*+<JN& D(*+=$& ($2<%8& -)7$,+"CJ-&& -&<+=";"JN&C$)8&pt

b

/&,*(18+="JNC&J$78)-$&1&-)4*(G"CJ-&7*+=OD)8CH&#&*,($+-$&t – 1/&#&=8G&1k-)4*(G"CJ-& *&D*1-*G-$&=8CH&C$)/&,=I($&1G-$)-%8&+-O&#&D*D(1$7)-G&*,($+-$&p*t19&

&& & & & &&&&&&&&&&& =

= _ _ + _ β π θ _k t

∑

1 * 1 t π t b t p p p p

& & & & & &&&&&K@?M 271-$ = = = _ _ _ + + π θ θ 1 1 t π_t t t p p 1 t t p p

(

1

)

p p &J$+=&-)4;"CJN&#&*,($+-$&tA 6-$1";$P)-$&*7&D(18JO=$2*&+D*+*R<&<+=";")-"&C$)&4-(G8&)"D*=8,"JN&#&+#*-CH&7$C81J"CH&*2("b )-C1$)-"&7*=8C1NC$&G*P;-#*VC-&1G-")8&C$)8&#$7%<2&D*D<;"()$2*&#&)$*,$8)$+*#+,-CH&G*7$;"CH& +CH$G"=<&+1=8#)*VC-&C$)&\";g*A&B2*7)-$&1&=8G&+CH$G"=$G&=8;,*&C1OVQ&4-(G&G*P$&+#*R*7)-$&1G-$b )-Q&C$)O&#&R-$PNC8G&*,($+-$&tA&!(1$7+-OR-*(+=#"&1G-$)-"JN&C$)8&)"&D*7+="#-$&($2<%&*,($V;*)8CH& (I#)")-$G&KSM&;<R&K@?M&i&#&1";$P)*VC-&*7&2(<D8/&7*&,=I($J&)";$PNA&e8RI(&4-(G/&,=I($&G*2N&1G-$)-Q& C$)8/&*7R8#"&+-O&#&,"P78G&*,($+-$&;*+*#*&-&)-$1";$P)-$&*7&#8R*(<&7*,*)")$2*&#&D*D(1$7)-CH& *,($+"CHA&31")+"&)"&1G-")O&C$)&#&,"P78G&,*;$J)8G&*,($+-$&#8)*+-&t – #A&!*1*+="%$&4-(G8&K1*(-$)b =*#")$&1"(I#)*&)"&D(18+1%*VQ/&J",&-&)"&D(1$+1%*VQM/&D*1*+="#-"JN&C$)8&)"&D*1-*G-$&1&*,($+<&t – 1A& e&=",-G&<JOC-<&"2($2"=*#8&D*1-*G&C$)&p_tJ$+=&7")8&(I#)")-$G9

&& & & & &&&&&&&&& ₌

* p b f p p 1_ _ + θ θ ω ω

(

)

* 1 _t t t p p

p & & & & &&&&K@@M

271-$& D*1-*G& )*#*& <+=";")8CH& C$)&

(

*

)

t

p & J$+=& #8D"7,*#N& #8R*(I#& C$)& D(1$1& 4-(G8& backward&& -&forward looking9

&&& & & & &&&&&&& *₌

t p

(

)

f t b t p p 1 f f p E p p _ _ + + βθ θ θ ω ω t t t

& & & & &&&&&K@>M e&#8)-,<&;*2b;-)$"(81"CJ-&(I#)")-"&KSM&#&*=*C1$)-<&+=")<&<+=";*)$2*&*&1$(*#$J&-)4;"CJ-&*=(18b G"G8&(I#)")-$&*D-+<JNC$&#12;O7)N&1G-")O&C$)&=8CH&D(1$7+-OR-*(+=#&forward looking/&,=I($&)-$& )"D*=,"%8&R"(-$(8&+1=8#)*VC-&)*G-)";)$J&=",&J",&#&D(18D"7,<&+=")7"(7*#$2*&G*7$;<&R$1&-)7$,b +"CJ-&K1*RA&0";-&>??l/&+A&[TM9

&&& & & ₌

t pt

(

1

)

pt

(

)

(

t

) (

)

t t f t t t f t p E p p mc p _ _{1 1}_ ₊

1

_ + + + _ βθ βθ βθ ω ω ω

π & & & &&&&&K@:M

!*&#+="#-$)-<&#8("P$)-"&K@:M&7*&(I#)"5&*,($V;"JNC8CH&"2($2"=*#8&D*1-*G&C$)&K@@M&-&K@>M&-&)"b +=OD)-$&(*1#-N1")-<&#*,I%&D<),=<&(I#)*#"2-&*&1$(*#$J&-)4;"CJ-&*=(18G<J$G8&(I#)")-$&*D-+<JNC$& -)4;"CJO&K1*RA&0";-/&0$(=;$(&@SSSM9

&& & &&&&&&&& =

(

)

[

]

[

₍

₎

]

(

)(

)(

)

(

)

[

]

t t t t t

mc

E

₊

1

1

1

1

1

_ _ 1 _ _ 1 _ + + + + + + _

β

β

1

_

β

1

_

β

β

θ

θ

θ

θ

1

_

θ

θ

1

_

θ

θ

θ

ω

ω

ω

ω

ω

π

π

π

&& & & & & & & & & & & &&&&&K@[M

3,*(18+="G8&=$("1&1&4",=</&P$&)"&#&D$%)-&,*),<($)C8J)8G&(8),<&D("C8&J$7)*+=,*#8&,*+1=&D("C8&J$+=& (I#)8&($;"CJ-&D%"C8&($";)$J&7*&,("5C*#$J&#87"J)*VC-&D("C8/&=JA9&mc_t=

(

w_t p_t

)

ln

(

Y_t N_t

)

1 α α ∂ ∂ _ _ _ σ A&e+="#-"b JNC&7*&=$2*&#8("P$)-"&#$#)N=(1*,($+*#8&#"(<)$,&,*)-$C1)8&G",+8G";-1"CJ-&7;"&2*+D*7"(+=#&7*b G*#8CH&KTM&&*=(18G<J$G8&)"+=OD<JNCN&1";$P)*VQ&D*G-O718&,*+1="G-&,("5C*#8G-&"&D(*7<,=$Gl₉

&&& & & &&&&&&&&

=

(

)

(

1

)

1 1 1 t t t

x

h x mc + + _ α α α _ _ _ _ ϕ σ σ

&& & & & & & & & & & & &&&& &K@ZM &&

271-$&x_t=*& ;<,"& D(*7<,C8J)"/& RO7NC"& D(*C$)=*#8G& *7CH8;$)-$G&y_t*7& J$2*& D*1-*G<& *,($V;*)$2*& #k#"(<),"CH&2-O=,-CH&C$)A

LD(M./#0(6(!.;:+N(;:91=:19+0$+

!(1$,+1="%C"JNC&G-O718*,($+*#8&#"(<)$,&,*)-$C1)8&7;"&,*)+<G$)="&KlM&-&D*7+="#-"JNC&7*&)-$2*& =*P+"G*VQYt =Ct/&*=(18G<J$G8&(I#)")-$&;<,-&D(*7<,C8J)$J&K=1#A&78)"G-C1)N&,(18#N&`3&1&=(#"b

%*VC-N&)"#8,I#M9

&&& & & &&& = = + + + _ _ _

(

) (

)

(

)

x t t t t t t t t E x x i E v x _{+ – +} – 1 1 1

~

1 x t x t x x t v v 1 ,

(

)

X x t

~

IID0

,

γ γ σ ξ ξ π ∑

ρ

& & & &&&&&K@EM

271-$vtx($D($1$)=<J$&1"<=*,*($;*#")8&+1*,A

B&,*;$-&1$+="#-$)-$&(I#)"5&K@[M&-&K@ZM&*D-+<JNC8CH&1"CH*#")-$&4-(G&D(*#"71-&7*&+4*(G<%*#"b )-"&(I#)")-"&-)4;"CJ-&K=1#A&H8R(87*#$J&)$*,$8)$+*#+,-$J&,(18#$J&!H-;-D+"/&6s!\M9

&&& & & &&& = = +β + + + β ₊+ – – –

(

t

)

b t t t t t f t

E

1 1 0x 1x 1 v t t t v v 1+ξ , ξt ~ IID

(

0

,

)

π π π π π π π π π π ∑

κ

κ

ρ

& & & &&&&& K@TM 271-$vt π ($D($1$)=<J$&1"<=*,*($;*#")8&+1*,A !"("G$=(8&D*+="C-&+=(<,=<(";)$J&D*1*+="JN&#&)"+=OD<JNC8CH&)-$;-)-*#8CH&($;"CJ"CH&1&D"("G$b =("G-&<,(8=8G-9

l& &_{F;"& 4<),CJ-& D(*7<,CJ-& =8D<& \*RR"bF*<2;"+"& =}

t

t n

y (1 α_ ) A& _*(18+="G8& (I#)-$P& 1& 1"%*P$)-"/& P$& 2*+D*7"(,"& J$+=& 1"G,)-O="/&R$1&<71-"%<&D"5+=#"&-&-)#$+=8CJ-&#&,"D-="%&(1$C1*#8/&1&C1$2*&#8)-,"/&P$9&yt=ctA

&& & & &&& = = = = = =

(

1

)

h 1 ~

(

)

[

1 1

]

f b

(

) (

)(

)(

)

(

)

[

]

1 1 1 1 0

(

)(

)(

)(

)

(

)

[

]

1 1 h γ – + σ σ σ

(

1

)

h – + + σ σ σ σ σ β β β _ _ _ _ _ _ βθ β 1_β θ 1_β 1_ β θ θ 1_θ 1_θ θ 1_θ θ θ ω ω θ

(

)

[

1 1

]

+ + + + + β _ _ θ ω ω 1_ω ω ω θ α α 1_ _ α ϕ

κ

κ

]1<D$%)-$)-$G& G*7$;<& J$+=& (I#)")-$& *D-+<JNC$& +D*+IR& D(*#"71$)-"& D*;-=8,-& D-$)-OP)$JA& !(18JG<J$G8/&P$&R"),&C$)=(";)8&,+1="%=<J$&)*G-)";)N&+=*DO&D(*C$)=*#N&12*7)-$&1&($2<%N&f"8;*("& K@SS:M/&($"2<JNC&)"&R-$PNCN&-)4;"CJO&-&;<,O&D(*7<,C8J)NA&e&*R<&D(18D"7,"CH&($",CJ"&="&J$+=&7*7"=)-"&& -&1"D$#)-"&D*#(I=&2*+D*7"(,-&7*&7%<2**,($+*#$J&(I#)*#"2-&i&+=")<&<+=";*)$2*A&]#12;O7)-"JNC&& #&G*7$;<&)-$CHOQ&R"),<&C$)=(";)$2*&7*&7<P8CH&J$7)*("1*#8CH&1G-")&+=ID&D(*C$)=*#8CH/&#&($b 2<;$&#D(*#"71"G8&G$CH")-1G&#82%"71")-"&+=ID&D(*C$)=*#8CH9

&& & & &&&&&&&&&&& ₌

₍

₎₍

₎

i t t x t t t i x i ₁ 1

(

i

)

IID0

,

~

+ + + _ λ λ φ φ ξ i t ξ _ ππ ∑

& & & & & & & & & & & & &&&&&K@lM

271-$λ∈ 0,1 _{J$+=& D"("G$=($G& #82%"71")-"& +=ID& D(*C$)=*#8CH/& "} i

_~

t

ξ ($D($1$)=<J$& +1*,& D*;-=8,-& D-$)-OP)$JA

F*7"=,*#*& (*1D"=(18G8& C1OVC-*#*& ")=8C8D"C8J)N& Kforward-lookingM& ($2<%O& f"8;*("/& #& ,=I($J& R"),&C$)=(";)8&($"2<J$&)"&-)4;"CJO&1&*,($+<&t + 1/&#8)-,"JNCN&1&("CJ*)";)8CH&*C1$,-#"5/&*("1&)"& R-$PNCN&;<,O&D(*7<,C8J)N&K\;"(-7"/&0";-/&0$(=;$(&@SSlMA

&& & & &&&&&&&&&&

= t

(

)(

t t x t

)

t i E x i 1 1 +1

(

)

IID

~

+ + + _ λ λ i t ξ i t ξ _ _φ π π φ i

,

0 ∑

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