carlos3 cs
TRANSCRIPT
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 117
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 217
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 317
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 517
- 2 0 E + 1 1
- 1 5 E + 1 1
- 1 0 E + 1 1
- 5 0 E + 1 0
0 0 E + 0 0
5 0 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B D
- 1 2 - 8 - 4 0 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B G D P R
5 0 E + 1 0
1 0 E + 1 1
1 5 E + 1 1
2 0 E + 1 1
2 5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C G V
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
6 E + 1 1
7 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C P V
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E E
2 2 0 0 0 0 0
2 4 0 0 0 0 0
2 6 0 0 0 0 0
2 8 0 0 0 0 0
3 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E S
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
E T
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E T_ N A
0 8
1 2
1 6
2 0
2 4
2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H E B
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
F D D V
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
1 4 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
G D P V D
4 0 E + 1 0
8 0 E + 1 0
1 2 E + 1 1
1 6 E + 1 1
2 0 E + 1 1
2 4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I B G V
0 5 1 0
1 5
2 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
I R L
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I R S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T I S K V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T V
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
L F
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
M G S V D
- 2 E + 1 0
- 1 E + 1 0
0 E + 0 0
1 E + 1 0
2 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
N T R D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P C G
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P C P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P F D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P G D P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T I S K
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M G S
0 2
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P M G S X
0 0
0 5
1 0
1 5
2 0
2 5
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M N W
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P T D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S X
0 4
0 8
1 2
1 6
2 0
2 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X N W
0 8
0 9
1 0
1 1
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
R P M G S
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
R P X G S
0 1 6
0 2 0
0 2 4
0 2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
S H T G S V D
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T D D V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
2 0 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V D
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
U N R
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
X G S V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S V D
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X M K T
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X P E R F
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 717
-02
-01
00
01
02
03
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
GDP growth ndash 2 SE
Root Mean Squared Error 0005725Mean Absolute Error 0003868Mean Abs Percent Error 1738783Theil Inequality Coefficient 0415270 Bias Proportion 0154562 Variance Proportion 0353654 Covariance Proportion 0491785
n
X t q f it nminus et
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 817
f t
et
X t = λ(L)f t + et
Γ(L)f t = ηt
λi(L) X t
et
E (et η
tminusk) = 0 forallk
E (eitejs = 0)foralls
i = j
et ηt
n
rtimes1 (f
t f
tminus p)
F t
ntimesr (λ0λ p) Λ
λi
ntimesq
i Γ(L)
X t = ΛF t + et
A(L)F t = Gηt
A(L) 1 0 Γ(L) G 1 0
r le pq
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 917
n
X t
limnrarrinfinnminus1Λ
Λ = DΛ
maxeval(
e
) le c lt infin forallm
DΛ r times r maxeval
e
et
W
W
Wn = I
F t = nminus1W
X t
limnrarrinfinnminus1W
Λ = H rtimesr H
ˆF t = nminus1W ΛF t + nminus1W et rarr p HF tas n rarr infin
nminus1W
et rarr p 0
F t Λt H H r times r r2
nminus1Λ
Λ = I r r(r + 1)2
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 217
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 317
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 517
- 2 0 E + 1 1
- 1 5 E + 1 1
- 1 0 E + 1 1
- 5 0 E + 1 0
0 0 E + 0 0
5 0 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B D
- 1 2 - 8 - 4 0 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B G D P R
5 0 E + 1 0
1 0 E + 1 1
1 5 E + 1 1
2 0 E + 1 1
2 5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C G V
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
6 E + 1 1
7 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C P V
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E E
2 2 0 0 0 0 0
2 4 0 0 0 0 0
2 6 0 0 0 0 0
2 8 0 0 0 0 0
3 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E S
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
E T
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E T_ N A
0 8
1 2
1 6
2 0
2 4
2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H E B
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
F D D V
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
1 4 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
G D P V D
4 0 E + 1 0
8 0 E + 1 0
1 2 E + 1 1
1 6 E + 1 1
2 0 E + 1 1
2 4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I B G V
0 5 1 0
1 5
2 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
I R L
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I R S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T I S K V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T V
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
L F
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
M G S V D
- 2 E + 1 0
- 1 E + 1 0
0 E + 0 0
1 E + 1 0
2 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
N T R D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P C G
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P C P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P F D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P G D P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T I S K
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M G S
0 2
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P M G S X
0 0
0 5
1 0
1 5
2 0
2 5
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M N W
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P T D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S X
0 4
0 8
1 2
1 6
2 0
2 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X N W
0 8
0 9
1 0
1 1
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
R P M G S
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
R P X G S
0 1 6
0 2 0
0 2 4
0 2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
S H T G S V D
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T D D V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
2 0 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V D
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
U N R
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
X G S V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S V D
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X M K T
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X P E R F
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 717
-02
-01
00
01
02
03
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
GDP growth ndash 2 SE
Root Mean Squared Error 0005725Mean Absolute Error 0003868Mean Abs Percent Error 1738783Theil Inequality Coefficient 0415270 Bias Proportion 0154562 Variance Proportion 0353654 Covariance Proportion 0491785
n
X t q f it nminus et
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 817
f t
et
X t = λ(L)f t + et
Γ(L)f t = ηt
λi(L) X t
et
E (et η
tminusk) = 0 forallk
E (eitejs = 0)foralls
i = j
et ηt
n
rtimes1 (f
t f
tminus p)
F t
ntimesr (λ0λ p) Λ
λi
ntimesq
i Γ(L)
X t = ΛF t + et
A(L)F t = Gηt
A(L) 1 0 Γ(L) G 1 0
r le pq
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 917
n
X t
limnrarrinfinnminus1Λ
Λ = DΛ
maxeval(
e
) le c lt infin forallm
DΛ r times r maxeval
e
et
W
W
Wn = I
F t = nminus1W
X t
limnrarrinfinnminus1W
Λ = H rtimesr H
ˆF t = nminus1W ΛF t + nminus1W et rarr p HF tas n rarr infin
nminus1W
et rarr p 0
F t Λt H H r times r r2
nminus1Λ
Λ = I r r(r + 1)2
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 317
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 517
- 2 0 E + 1 1
- 1 5 E + 1 1
- 1 0 E + 1 1
- 5 0 E + 1 0
0 0 E + 0 0
5 0 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B D
- 1 2 - 8 - 4 0 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B G D P R
5 0 E + 1 0
1 0 E + 1 1
1 5 E + 1 1
2 0 E + 1 1
2 5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C G V
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
6 E + 1 1
7 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C P V
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E E
2 2 0 0 0 0 0
2 4 0 0 0 0 0
2 6 0 0 0 0 0
2 8 0 0 0 0 0
3 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E S
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
E T
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E T_ N A
0 8
1 2
1 6
2 0
2 4
2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H E B
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
F D D V
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
1 4 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
G D P V D
4 0 E + 1 0
8 0 E + 1 0
1 2 E + 1 1
1 6 E + 1 1
2 0 E + 1 1
2 4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I B G V
0 5 1 0
1 5
2 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
I R L
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I R S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T I S K V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T V
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
L F
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
M G S V D
- 2 E + 1 0
- 1 E + 1 0
0 E + 0 0
1 E + 1 0
2 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
N T R D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P C G
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P C P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P F D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P G D P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T I S K
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M G S
0 2
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P M G S X
0 0
0 5
1 0
1 5
2 0
2 5
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M N W
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P T D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S X
0 4
0 8
1 2
1 6
2 0
2 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X N W
0 8
0 9
1 0
1 1
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
R P M G S
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
R P X G S
0 1 6
0 2 0
0 2 4
0 2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
S H T G S V D
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T D D V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
2 0 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V D
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
U N R
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
X G S V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S V D
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X M K T
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X P E R F
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 717
-02
-01
00
01
02
03
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
GDP growth ndash 2 SE
Root Mean Squared Error 0005725Mean Absolute Error 0003868Mean Abs Percent Error 1738783Theil Inequality Coefficient 0415270 Bias Proportion 0154562 Variance Proportion 0353654 Covariance Proportion 0491785
n
X t q f it nminus et
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 817
f t
et
X t = λ(L)f t + et
Γ(L)f t = ηt
λi(L) X t
et
E (et η
tminusk) = 0 forallk
E (eitejs = 0)foralls
i = j
et ηt
n
rtimes1 (f
t f
tminus p)
F t
ntimesr (λ0λ p) Λ
λi
ntimesq
i Γ(L)
X t = ΛF t + et
A(L)F t = Gηt
A(L) 1 0 Γ(L) G 1 0
r le pq
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 917
n
X t
limnrarrinfinnminus1Λ
Λ = DΛ
maxeval(
e
) le c lt infin forallm
DΛ r times r maxeval
e
et
W
W
Wn = I
F t = nminus1W
X t
limnrarrinfinnminus1W
Λ = H rtimesr H
ˆF t = nminus1W ΛF t + nminus1W et rarr p HF tas n rarr infin
nminus1W
et rarr p 0
F t Λt H H r times r r2
nminus1Λ
Λ = I r r(r + 1)2
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 517
- 2 0 E + 1 1
- 1 5 E + 1 1
- 1 0 E + 1 1
- 5 0 E + 1 0
0 0 E + 0 0
5 0 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B D
- 1 2 - 8 - 4 0 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B G D P R
5 0 E + 1 0
1 0 E + 1 1
1 5 E + 1 1
2 0 E + 1 1
2 5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C G V
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
6 E + 1 1
7 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C P V
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E E
2 2 0 0 0 0 0
2 4 0 0 0 0 0
2 6 0 0 0 0 0
2 8 0 0 0 0 0
3 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E S
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
E T
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E T_ N A
0 8
1 2
1 6
2 0
2 4
2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H E B
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
F D D V
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
1 4 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
G D P V D
4 0 E + 1 0
8 0 E + 1 0
1 2 E + 1 1
1 6 E + 1 1
2 0 E + 1 1
2 4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I B G V
0 5 1 0
1 5
2 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
I R L
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I R S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T I S K V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T V
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
L F
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
M G S V D
- 2 E + 1 0
- 1 E + 1 0
0 E + 0 0
1 E + 1 0
2 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
N T R D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P C G
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P C P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P F D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P G D P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T I S K
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M G S
0 2
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P M G S X
0 0
0 5
1 0
1 5
2 0
2 5
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M N W
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P T D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S X
0 4
0 8
1 2
1 6
2 0
2 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X N W
0 8
0 9
1 0
1 1
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
R P M G S
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
R P X G S
0 1 6
0 2 0
0 2 4
0 2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
S H T G S V D
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T D D V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
2 0 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V D
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
U N R
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
X G S V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S V D
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X M K T
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X P E R F
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 717
-02
-01
00
01
02
03
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
GDP growth ndash 2 SE
Root Mean Squared Error 0005725Mean Absolute Error 0003868Mean Abs Percent Error 1738783Theil Inequality Coefficient 0415270 Bias Proportion 0154562 Variance Proportion 0353654 Covariance Proportion 0491785
n
X t q f it nminus et
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 817
f t
et
X t = λ(L)f t + et
Γ(L)f t = ηt
λi(L) X t
et
E (et η
tminusk) = 0 forallk
E (eitejs = 0)foralls
i = j
et ηt
n
rtimes1 (f
t f
tminus p)
F t
ntimesr (λ0λ p) Λ
λi
ntimesq
i Γ(L)
X t = ΛF t + et
A(L)F t = Gηt
A(L) 1 0 Γ(L) G 1 0
r le pq
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 917
n
X t
limnrarrinfinnminus1Λ
Λ = DΛ
maxeval(
e
) le c lt infin forallm
DΛ r times r maxeval
e
et
W
W
Wn = I
F t = nminus1W
X t
limnrarrinfinnminus1W
Λ = H rtimesr H
ˆF t = nminus1W ΛF t + nminus1W et rarr p HF tas n rarr infin
nminus1W
et rarr p 0
F t Λt H H r times r r2
nminus1Λ
Λ = I r r(r + 1)2
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 517
- 2 0 E + 1 1
- 1 5 E + 1 1
- 1 0 E + 1 1
- 5 0 E + 1 0
0 0 E + 0 0
5 0 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B D
- 1 2 - 8 - 4 0 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C B G D P R
5 0 E + 1 0
1 0 E + 1 1
1 5 E + 1 1
2 0 E + 1 1
2 5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C G V
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
6 E + 1 1
7 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
C P V
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E E
2 2 0 0 0 0 0
2 4 0 0 0 0 0
2 6 0 0 0 0 0
2 8 0 0 0 0 0
3 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E S
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
E T
8 0 0 0 0 0 0
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E T_ N A
0 8
1 2
1 6
2 0
2 4
2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
E X C H E B
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
F D D V
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
1 4 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
G D P V D
4 0 E + 1 0
8 0 E + 1 0
1 2 E + 1 1
1 6 E + 1 1
2 0 E + 1 1
2 4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I B G V
0 5 1 0
1 5
2 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
I R L
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I R S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T I S K V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
I T V
1 2 0 0 0 0 0 0
1 6 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 4 0 0 0 0 0 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
L F
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
5 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
M G S V D
- 2 E + 1 0
- 1 E + 1 0
0 E + 0 0
1 E + 1 0
2 E + 1 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
N T R D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P C G
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P C P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P F D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P G D P
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P I T I S K
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M G S
0 2
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
P M G S X
0 0
0 5
1 0
1 5
2 0
2 5
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P M N W
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P T D D
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S
0 0
0 4
0 8
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X G S X
0 4
0 8
1 2
1 6
2 0
2 4
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
P X N W
0 8
0 9
1 0
1 1
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
R P M G S
0 4
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
R P X G S
0 1 6
0 2 0
0 2 4
0 2 8
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
S H T G S V D
4 0 E + 1 1
6 0 E + 1 1
8 0 E + 1 1
1 0 E + 1 2
1 2 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T D D V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V
4 0 E + 1 1
8 0 E + 1 1
1 2 E + 1 2
1 6 E + 1 2
2 0 E + 1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
T E V D
0 1 0
2 0
3 0
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
U N R
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0 1
5
X G S V
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X G S V D
0 E + 0 0
1 E + 1 1
2 E + 1 1
3 E + 1 1
4 E + 1 1
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X M K T
0 6
0 8
1 0
1 2
7 0
7 5
8 0
8 5
9 0
9 5
0 0
0 5
1 0
1 5
X P E R F
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 717
-02
-01
00
01
02
03
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
GDP growth ndash 2 SE
Root Mean Squared Error 0005725Mean Absolute Error 0003868Mean Abs Percent Error 1738783Theil Inequality Coefficient 0415270 Bias Proportion 0154562 Variance Proportion 0353654 Covariance Proportion 0491785
n
X t q f it nminus et
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 817
f t
et
X t = λ(L)f t + et
Γ(L)f t = ηt
λi(L) X t
et
E (et η
tminusk) = 0 forallk
E (eitejs = 0)foralls
i = j
et ηt
n
rtimes1 (f
t f
tminus p)
F t
ntimesr (λ0λ p) Λ
λi
ntimesq
i Γ(L)
X t = ΛF t + et
A(L)F t = Gηt
A(L) 1 0 Γ(L) G 1 0
r le pq
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 917
n
X t
limnrarrinfinnminus1Λ
Λ = DΛ
maxeval(
e
) le c lt infin forallm
DΛ r times r maxeval
e
et
W
W
Wn = I
F t = nminus1W
X t
limnrarrinfinnminus1W
Λ = H rtimesr H
ˆF t = nminus1W ΛF t + nminus1W et rarr p HF tas n rarr infin
nminus1W
et rarr p 0
F t Λt H H r times r r2
nminus1Λ
Λ = I r r(r + 1)2
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 717
-02
-01
00
01
02
03
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
GDP growth ndash 2 SE
Root Mean Squared Error 0005725Mean Absolute Error 0003868Mean Abs Percent Error 1738783Theil Inequality Coefficient 0415270 Bias Proportion 0154562 Variance Proportion 0353654 Covariance Proportion 0491785
n
X t q f it nminus et
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 817
f t
et
X t = λ(L)f t + et
Γ(L)f t = ηt
λi(L) X t
et
E (et η
tminusk) = 0 forallk
E (eitejs = 0)foralls
i = j
et ηt
n
rtimes1 (f
t f
tminus p)
F t
ntimesr (λ0λ p) Λ
λi
ntimesq
i Γ(L)
X t = ΛF t + et
A(L)F t = Gηt
A(L) 1 0 Γ(L) G 1 0
r le pq
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 917
n
X t
limnrarrinfinnminus1Λ
Λ = DΛ
maxeval(
e
) le c lt infin forallm
DΛ r times r maxeval
e
et
W
W
Wn = I
F t = nminus1W
X t
limnrarrinfinnminus1W
Λ = H rtimesr H
ˆF t = nminus1W ΛF t + nminus1W et rarr p HF tas n rarr infin
nminus1W
et rarr p 0
F t Λt H H r times r r2
nminus1Λ
Λ = I r r(r + 1)2
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 717
-02
-01
00
01
02
03
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
GDP growth ndash 2 SE
Root Mean Squared Error 0005725Mean Absolute Error 0003868Mean Abs Percent Error 1738783Theil Inequality Coefficient 0415270 Bias Proportion 0154562 Variance Proportion 0353654 Covariance Proportion 0491785
n
X t q f it nminus et
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 817
f t
et
X t = λ(L)f t + et
Γ(L)f t = ηt
λi(L) X t
et
E (et η
tminusk) = 0 forallk
E (eitejs = 0)foralls
i = j
et ηt
n
rtimes1 (f
t f
tminus p)
F t
ntimesr (λ0λ p) Λ
λi
ntimesq
i Γ(L)
X t = ΛF t + et
A(L)F t = Gηt
A(L) 1 0 Γ(L) G 1 0
r le pq
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 917
n
X t
limnrarrinfinnminus1Λ
Λ = DΛ
maxeval(
e
) le c lt infin forallm
DΛ r times r maxeval
e
et
W
W
Wn = I
F t = nminus1W
X t
limnrarrinfinnminus1W
Λ = H rtimesr H
ˆF t = nminus1W ΛF t + nminus1W et rarr p HF tas n rarr infin
nminus1W
et rarr p 0
F t Λt H H r times r r2
nminus1Λ
Λ = I r r(r + 1)2
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 817
f t
et
X t = λ(L)f t + et
Γ(L)f t = ηt
λi(L) X t
et
E (et η
tminusk) = 0 forallk
E (eitejs = 0)foralls
i = j
et ηt
n
rtimes1 (f
t f
tminus p)
F t
ntimesr (λ0λ p) Λ
λi
ntimesq
i Γ(L)
X t = ΛF t + et
A(L)F t = Gηt
A(L) 1 0 Γ(L) G 1 0
r le pq
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 917
n
X t
limnrarrinfinnminus1Λ
Λ = DΛ
maxeval(
e
) le c lt infin forallm
DΛ r times r maxeval
e
et
W
W
Wn = I
F t = nminus1W
X t
limnrarrinfinnminus1W
Λ = H rtimesr H
ˆF t = nminus1W ΛF t + nminus1W et rarr p HF tas n rarr infin
nminus1W
et rarr p 0
F t Λt H H r times r r2
nminus1Λ
Λ = I r r(r + 1)2
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 917
n
X t
limnrarrinfinnminus1Λ
Λ = DΛ
maxeval(
e
) le c lt infin forallm
DΛ r times r maxeval
e
et
W
W
Wn = I
F t = nminus1W
X t
limnrarrinfinnminus1W
Λ = H rtimesr H
ˆF t = nminus1W ΛF t + nminus1W et rarr p HF tas n rarr infin
nminus1W
et rarr p 0
F t Λt H H r times r r2
nminus1Λ
Λ = I r r(r + 1)2
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1017
r(r minus 1)2 F
F F = (F
1F
T )
W W
X t k
r
minF 1F T ΛS k
S k = (nT )minus1T
t=1(X t minus ΛF t)
(X t minus ΛF t)
nminus1Λ
Λ = I r F
F
F t
F t
Λ
F (Λ) = (Λ
Λ)minus1Λ
X t
minΛT minus1
X T t=1X
t [I minus (Λ
Λ)minus1Λ
]X t
maxΛtr(Λ
Λ)minus12
Λ
(T minus1T
t=1
X tX
t)Λ(Λ
Λ)minus12
maxΛΛ
T
t=1
X tX
tΛ
nminus1Λ
Λ = I k
Λ
X = X
tX t k
ΛΛ = nI r
ˆF t = nminus1 ˆ
Λ
X t
r
k k min[N T ]
t
radic N
i
radic T
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1117
et
P CP (k) = S k + kσ2g(n T )
IC k = ln(S k) + kg(n T )
g(n T )
S k
macrσ2 = S kmax
kmax
k
k g(n T )rarr
0 [min(n T )]g(n T )rarr infin
n T rarr infin
g(n T ) = n+T nT ln[min(n T )]
L1 minus norm ||β ||
λ||β || λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1217
minβ
1
n(Y minus X β )(Y minus X β ) + λ
k
i=1
||β i||
0 le λ lt infin
λ 0
λ
ˆyt+h = c+Π(L)F t+Φ(L)yt
Π(L) Φ(L)
λ
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
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8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1317
λ
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1417
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1517
983085983088983086983091983088
983085983088983086983090983088
983085983088983086983089983088
983088983086983088983088
983088983086983089983088
983088983086983090983088
983088983086983091983088
983088983086983092983088
983090983088983089983091983153983089 983090983088983089983091983153983090 983090983088983089983091983153983091 983090983088983089983091983153983092
983110983137983139983156983151983154 983117983151983140983141983148 983089 983116983105983123983123983119 983092 983107983151983149 983138983145 983150983137983156 983145983151983150
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1617
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717
8132019 Carlos3 CS
httpslidepdfcomreaderfullcarlos3-cs 1717