VECTORES AUTORREGRESIVOS

yt
E(yt ) = µ
yt
I(d)
d=1
∆yt = yt − yt−1
yt
Yt
Yt = (zt , xt , vt )
p
p
p
1
1
zt = c1 + φ1
11 zt−1 + ... + φ11 zt−p + φ12 xt−1 + ... + φ12 xt−p + φ13 vt−1 + ... + φ13 vt−p + a1t
p
p
p
1
1
xt = c 2 + φ1
z
+
...
+
φ
z
+
φ
x
+
...
+
φ
x
+
φ
t−1
t−p
t−1
t−p
21
22
23 vt−1 + ... + φ23 vt−p + a2t
21
22
p
p
p
1
1
vt = c3 + φ1
z
+
...
+
φ
z
+
φ
x
+
...
+
φ
x
+
φ
v
+
...
+
φ
v
+ a3t
31 t−1
32 t−1
33 t−1
31 t−p
32 t−p
33 t−p
Yt = C + Φ1 Yt−1 + ... + Φp Yt−p + at
E(at ) = 0 E(at a′s ) = Ω
t=s 0
p
Φn
⎡
(k × k)
k
⎤
φn11 φn12 φn13
⎢ n
Φn = ⎣ φ21 φn22 φn23 ⎥
⎦
φn31 φn32 φn33
Yt−n
(1 × k)
⎡
⎤
zt−n
⎢
⎥
= ⎣ xt−n ⎦
vt−n
Yt − Φ1 Yt−1 + ... − Φp Yt−p = C + at ,
L
Yt (1 − Φ1 L + ... − Φp Lp ) = Φ(L)Yt = C + at ,
Yt−n
(k × k)
φ(L)
Φ(L)
ij
L
[δij − φ1ij L... − φpij Lp ]
δij = 1
i ̸= j
i=j
|Ik − Φ1 L + ... − Φp Lp | = 0
Yt = µ + at + Ψ1 at−1 + Ψ2 at−2 + ... = µ + Ψ(L)at
µ = (Ik − Φ1 − Φ2 − ... − Φp )−1 c Ψ(L) = (Ik + Ψ1 L + Ψ2 L + ...)
Ψ0 = I
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Yt
Yt−1
Yt−2
Yt−p−1
⎤
⎡
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥=⎢
⎥
⎢
⎥
⎢
⎦
⎣
C
0
0
0
⎤
⎡
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥+B⎢
⎥
⎢
⎥
⎢
⎦
⎣
Yt−1
Yt−2
Yt−3
Yt−p
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥+⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
at
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
B
0 Ik
Φ
Y = BZ + A
B
Φ
B̂ = (Z ′ Z)−1 Z ′ Y
k
T +p
p+1
p
Yt
f (YT , YT −1 , ..., Y1 |Y0 , Y1 , ..., Y−p+1 ; θ)
θ
yt
C, Φ1 , ..., Φp
y
t−1
Ω
C + Φ1 yt−1 + Φ2 yt−2 + ... + Φp yt−p
at ∼ N (0, Ω)
yt |yt−1 , ..., yt−p ∼ N (C + Φ1 yt−1 + Φ2 yt−2 + ... + Φp yt−p , Ω)
Π′ = [CΦ1 Φ2 ...Φp ]
Xt′ = [1Yt−1 Yt−2 ...Yt−p ]
Yt = Π′ Xt + at
yt |yt−1 , ..., yt−p ∼ N (Π′ Xt , Ω)
yt |yt−1 , ..., yt−p
fyt |yt−1 ,...,yt−p (yt |yt−1 , ..., yt−p , θ) =
= (2π)−k/2 |Ω−1 |−1/2 exp[(−1/2)(yt − Π′ Xt )′ Ω−1 (yt − Π′ Xt )]
1 t
y0 , y1 , ..., y−p+1
fyt yt−1 ,...,y1 |y0 ,y1 ,...,y−p+1 (yt yt−1 , ..., y1 |y0 , y1 , ..., y−p+1 ; θ) =
= fyt−1 ,...,y1 |y0 ,y1 ,...,y−p+1 (yt−1 , ..., y1 |y0 , y1 , ..., y−p+1 ; θ)
×fyt |yt−1 ,yt−2 ,...,y−p+1 (yt |yt−1 , yt−2 , ..., y−p+1 ; θ)
yT , yT −1 , ..., yT −p
y0 , y1 , ..., y−p+1
fyT ,yT −1 ,...,y1 |y0 ,y1 ,...,y−p+1 (yT , yT −1 , ..., y1 |y0 , y1 , ..., y−p+1 ; θ) =
=
T
'
t=1
fyt |yt−1 ,yt−2 ,...,y−p+1 (yt |yt−1 , yt−2 , ..., y−p+1 ; θ)
L(θ) =
t
(
t=1
logf (Yt |past; θ)
L(θ) = −(T k/2)log(2π) + (T /2)log|Ω−1 |
−(1/2)
T )
(
t=1
(Yt − Π′ Xt )′ Ω−1 (Yt − Π′ Xt )
*
Yt
Π̂k×(kp×1)
Zt
Ω̂k×k
Π̂ =
+
T
(
t=1
Xt′ Xt
,−1 +
Ω̂ = (1/T )
T
(
T
(
Xt′ Yt
t=1
,
ât â′t
t=1
ât = Yt − Π̂′ Xt
p
Π̂ Ω̂.
L(Π̂, Ω̂) = −(T k/2)log(2π) + (T /2)log|Ω̂−1 | − (1/2)
T )
(
â′t Ω̂.1 ât
t=1
*
T n/2
L(Π̂, Ω̂) = −(T k/2)log(2π) + (T /2)log|Ω̂−1 | − (T n/2)
p1
p0
L0
p1 > p 0
Ω̂0
Ω̂1
L1
2(L1 − L0 ) = T {log|Ω̂0 | − log|Ω̂1 |}
k
χ2 (k 2 (p1 −p0 )) p1 −p0
k(p1 − p0 )
k 2 (p1 − p0 ).
L1
p1
p0
L0
p1 = 4
k = 2)
p0 = 3
T = 46
Ω̂0 =
+
2.0 1.0
1.0 2.5
,
Ω̂1 =
+
1.8 0.9
0.9 2.2
,
2(L1 − L0 ) = 46(1.386 −
2
χ (4)
Π̂
j
yt−p−1
Π
22 (4 − 3) = 4
10.99 > 9.49
j
yt−p
h
yt−p
Yt = µ + at + Ψ1 at−1 + Ψ2 at−2 + ... = µ + Ψ(L)at
Ψ̂
Π̂
i
yti = γi +
∞
(
y
j
ψijp at−p
t=0
ψijp
j
i
p
ytj
ψijp
E(at a′t ) = Ω
Ω
aP IB,t
ainf l,t
ψ
C(L) = Ψ(L)Ω1/2
ηt = Ω−1/2 at
Yt = µ + Ψ(L)Ω1/2 Ω−1/2 at = µ + C(L)ηt
i
yti = µi +
∞ (
n
(
j
cpij ηt−p
p=0 j=1
yti
cpij
j
ηt−p
cpij
ηt
c2P IB,inf l
yt
xt
yt
xt
xt ,
yt
s>0
(xt , xt−1 , ...)
xt+s
xt+s
(xt , xt−1 , ...)
(yt , yt−1 , ...)
M EC[E(xt+s |xt , xt−1 , ...)] = M EC[E(xt+s |xt , xt−1 , ..., yt , yt−1 , ...)]
yt
Φj
xt
j
+
+
+
xt
yt
,
φ211 0
φ221 φ222
+
=
,+
c1
c2
,
xt−2
yt−2
+
,
φ111 0
φ121 φ122
+ ... +
+
yt
+
+
a1t
a2t
+
,+
xt−1
yt−1
φp11 0
φp21 φp22
,
,+
+
xt−p
yt−p
,
+
,
xt
xt
xt = c1 + φ111 xt−1 + φ211 xt−2 + ... + φp11 xt−p
+φ112 yt−1 + φ212 yt−2 + ... + φp12 yt−p + a1t
H0 : φ112 = φ212 = ... = φp12 = 0
yt
yt
-T
xt
RSS1 =
2
t=1 â1t
RSS0 =
yt
-T
2
t=1 ê1t
S1 =
(RSS0 − RSS1 )/p
RSS1 (T − 2p − 1)
5
2p − 1)
yt
F (p, T −
xt
j
yti = µi +
∞ (
n
(
j
cpij ηt−p
p=0 j=1
Yt = µ + Ψ(L)Ω1/2 Ω−1/2 at = µ + C(L)ηt
C(L) = Ψ(L)Ω1/2
ηt = Ω−1/2 at
C(L)
Ψ(L)
Φj
1/2
C(L) = Ψ(L)Ω
Ĉ(L) = Ψ̂(L)Ω̂1/2
Ω̂ =
Φj
1
T
-T
i=1
âit â′it
Ω1/2
WW′ = Ω
W = Ω1/2
⎡
⎣
w11
w21
w12
w22
...
...
w1m
w2m
wm1
wm2
...
wmm
⎤⎡
⎦⎣
Ω1/2
w11
w12
w21
w22
...
...
wm1
wm2
w1m
w2m
...
wmm
⎤ ⎡
⎦=⎣
σ11
σ21
σ12
σ22
...
...
σ1m
σ2m
σm1
σm2
...
σmm
m2
m2
⎤
⎦.
m2
Ω
Ω
i
σij =
m
(
wik wjk =
k=1
m2 −
m2
m
(
j
j
i
wjk wik = σji
k=1
m2 − m
m(m + 1)
=
2
2
w11 , w12 , ..wmm )
(m2 − m)/2
Ω1/2
W
wij = 0
W
⎡
0
⎢
⎢ w21
W =⎢
⎢
⎣
i<j
⎤
0
0
... 0
... 0 ⎥
⎥
wm1 wm2
... 0
⎥
⎥
⎦
Ω
ηt Ω−1/2 at = W −1 at
W η t = at
ayt 3
ayt 1 = w11 ηt1
= w12 ηt1 + w22 ηt2
= w13 ηt1 + w23 ηt2 + w33 ηt3
ayt 2
ayt 1
ηt1
ηt2 = E[ayt 2 |ayt 1 ) = E(w12 ηt1 + w22 ηt2 |ayt 1 ) = w12 E(ηt1 |ayt 1 ) + w22 E(ηt2 |ayt 2 ) =
= w12 w111 ayt 1 + w22 0
E(ηt2 |ayt 2 ) = Cov(ηt2 , ayt 1 ) = w11 Cov(ηt1 , ηt2 ) = 0.
ηt2 =
w12 y1
a
w11 t
w22 ηt2 = ayt 2 − w12 ηt1 = ayt 2 − E(ayt 2 |ayt 1 ).
ηtj =
*
1 ) yj
y
y
at − E(at j |ayt 1 , ayt 2 , ayt 3 ..., at j−1 ) .
wjj
ηt1
ηt2
ηt1
W
W =
+
a b
0 c
,
ηt1 = ηtd
+
inf lacion
output
,
= Ψ(L)
+
Ψ0 = I
1/2
Ω
Ω
1/2
,+
ηtd
ηt0
,
Co = Ψo Ω1/2 =
= W
C0
ηt
a b
0 c
ηt2 = ηto
Y
C(L)
1
Lnyt1 − Lnyt−1
+
1
Lnyt1 − Lnyt−1
2
yt
at = W
+
+
ηtd
ηto
1
Lnyt1 − Lnyt−1
2
yt
Lnyt1
yt2
−
1
Lnyt−1
,
,
,
=γ+
∞
(
p=0
, cov(ηtd , ηto ) = 0
=γ+
= γ1 +
∞
(
C
p=0
∞
(
+
d
ηt−p
o
ηt−p
,
p=0
p=0 [(
-∞
τ
τ =0 c11
τ =0
p
d
o
(cp11 ηt−p
+ cp12 ηt−p
)
-k
∞
(
Ψp at−p
p
C =
+
-∞
τ
d
τ =0 c11 )ηt−p
=0
√ , (
∞
0
√ √ =
Ψp Ω1/2
τ =0
Lnyt1
o
+ ( τ =0 cτ12 )ηt−p
]
-∞
w11
∞
(
p
ψ11
+ w21
p=0
∞
(
p
ψ12
=0
p=0
W
p
p
p
1
1
1
qt = βpt + β11
qt−1 + ... + β11 qt−p + β12
pt−1 + ... + β12 pt−p + β13
wt−1 + ... + β13 wt−p + ad
t
p
p
p
1
1
1
pt = γqt + hwt + β21
qt−1 + ... + β21 qt−p + β22
pt−1 + ... + β22 pt−p + β23
wt−1 + ... + β23 wt−p + as
t
p
1
w
wt = β33 wt−1 + ... + β33 wt−p + at
qt
pt
wt
B0 yt = B1 yt−1 + B2 yt−2 + ... + Bp yt−p + at
⎡
⎤
1 −β 0
⎥
B0 = ⎢
⎣ 1 −γ −h ⎦
0 0
1
yt = B0−1 B1 yt−1 + B0−1 B2 yt−2 + ... + B0−1 Bp yt−p + B0−1 at
M A(∞)
Yt = µ + Ψ(L)B0−1 at = µ + C(L)ηt
C(L) = Ψ(L)B0−1
ηt = B0−1 at
B0
W
B0
′
′
D = E(ηt ηt′ ) = E(B0−1 at a′t B0−1 ) = B0−1 ΩB0−1
D = E(ηt ηt′ )
Ω = B0 DB0′
M A(∞)
yt = µ + at + Ψ1 at−1 + Ψ2 at−2 + ...
yt+s
yt+s − ŷt+s|t = at+s + Ψ1 at+s−1 + ... + Ψs−1 at+1
s
M EC(ŷt+s|t ) = E[(yt+s − ŷt+s|t )(yt+s − ŷt+s|t )] =
= Ω + Ψ1 ΩΨ′1 + Ψ2 ΩΨ′2 + ... + Ψs−1 ΩΨ′s−1
E(at a′t ) = Ω
at = Ω1/2 ηt
ηt
ηt
Ω = E(at a′t ) = a1 a′1 V ar(η1t )+a2 a′2 V ar(η2t )+...+am a′m V ar(ηmt )
jj
E(ηt ηt′ ) = D.
M EC(ŷt+s|t ) =
m
(
j=1
{V ar(ujt )[aj a′j + Ψ1 aj a′j Ψ1 + ...
+Ψs−1 aj a′j Ψs−1 ]}
V ar(ηjt )
ηj
s
j