Modele AR(p)Modele MA(q)
Econometrie avansat¼a
Gabriel Bobeic¼a
MBPM
8 aprilie 2011
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
White noise: εt � iidN�0, σ2
�;E [εt ] = 0;Var [εt ] = σ2;Cov [εt , εs ] = 0 pt. s 6= t.Traiectorie simulat¼a pentru un proces White noise
4
3
2
1
0
1
2
3
50 100 150 200 250 300 350 400 450 500
σ2 = 1
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Autocorelatie estimat¼a pentru un proces White noise
1
0.75
0.5
0.25
0
0.25
0.5
0.75
1
1 6 11 16 21 26 31 36 41 46
σ2 = 1
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Autocorelatie partial¼a estimat¼a pentru un proces White noise
1
0.75
0.5
0.25
0
0.25
0.5
0.75
1
1 6 11 16 21 26 31 36 41 46
σ2 = 1
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Propriet¼atile statistice estimate pentru un proces White noise simulat
0
10
20
30
40
50
60
3 2 1 0 1 2 3
Medie 0.06Medianã 0.05Maximum 2.92Minimum 3.10Std. Dev. 1.04Skewness 0.02Kurtosis 2.81
JarqueBera 0.75Prob. 0.69
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
AR(1): xt = α+ β � xt�1 + εt .
Traiectorie simulat¼a pentru un proces AR(1)
0
4
8
12
16
20
50 100 150 200 250 300 350 400 450 500
beta = 0,5 beta = 0,9
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Momentele unui proces AR(1):
Mediaµ � E [xt ] =
α
1� β.
Varianta
γ0 � Eh(xt � µ)2
i=
σ2
1� β2.
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Momentele unui proces AR(1) 2:
(Auto)-covarianta
γ1 � E [(xt � µ) (xt�1 � µ)] = β � γ0.
γk � E [(xt � µ) (xt�k � µ)] ;γk = βk � γ0.
(Auto)-corelatia
ρ1 � γ1γ0= β.
ρk � γkγ0; ρk = βk .
Autocorelatia partial¼a
ak � Corr [xt , xt�k jxt�1, . . . , xt�k+1] .a1 = β; ak = 0 pt. k > 1.
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Autocorelatie estimat¼a pentru un proces AR(1)
0.2
0
0.2
0.4
0.6
0.8
1
1 6 11 16 21 26 31 36 41 46
α = 5; β = 0, 8
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Autocorelatie partial¼a estimat¼a pentru un proces AR(1)
0.2
0
0.2
0.4
0.6
0.8
1
1 6 11 16 21 26 31 36 41 46
α = 5; β = 0, 8
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
AR(2): xt = α+ β1 � xt�1 + β2 � xt�2 + εt .
Traiectorie simulat¼a pentru un proces AR(2)
40
44
48
52
56
60
50 100 150 200 250 300 350 400 450 500
α = 5; β1 = 0, 6; β2 = 0, 3
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Momentele unui proces AR(2):
Mediaµ � E [xt ] =
α
1� β1 � β2.
Varianta
γ0 � Eh(xt � µ)2
i=1� β21+ β2
σ2
(1� β1 � β2) (1+ β1 � β2).
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Momentele unui proces AR(2) 2:
(Auto)-covarianta
γ1 � E [(xt � µ) (xt�1 � µ)] =β1
1� β2γ0.
γk � E [(xt � µ) (xt�k � µ)] ;γk = β1 � γk�1 + β2 � γk�2.
(Auto)-corelatia
ρ1 � γ1γ0=
β11� β2
.
ρk � γkγ0; ρk = β1 � ρk�1 + β2 � ρk�2.
Autocorelatia partial¼a
ak � Corr [xt , xt�k jxt�1, . . . , xt�k+1] .a1 = ρ1; a2 = β2; ak = 0 pt. k > 2.
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Autocorelatie estimat¼a pentru un proces AR(2)
0.2
0
0.2
0.4
0.6
0.8
1
1 6 11 16 21 26 31 36 41 46
α = 5; β1 = 0, 6; β2 = 0, 3
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Autocorelatie partial¼a estimat¼a pentru un proces AR(2)
0.2
0
0.2
0.4
0.6
0.8
1
1 6 11 16 21 26 31 36 41 46
α = 5; β1 = 0, 6; β2 = 0, 3
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
AR(p)
xt = α+ β1 � xt�1+ β2 � xt�2+ . . .+ βp � xt�p + εt ; εt � iid N�0, σ2
�.�
1� β1L� β2L2 � . . .� βpL
2�
| {z }B (L)
xt = α+ εt .
0BBBBB@xtxt�1xt�2...
xt�p+1
1CCCCCA| {z }
yt
=
0BBBBB@α00...0
1CCCCCA| {z }
+
a
0BBBBB@β1 � � � βp1 � � � 00 � � � 0...
. . ....
0 � � � 0
1CCCCCA| {z }
b
�
0BBBBB@xt�1xt�2xt�3...
xt�p
1CCCCCA| {z }
yt�1
+
0BBBBB@εt00...0
1CCCCCA| {z }
ut
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Conditii de stabilitate pentru un proces AR(p)
λp � β1λp�1 � . . .� β1λ� βp = 0; jλj < 1.
Scrierea MA(∞) a unui proces AR(p) stabil
xt = B (1)�1 � α+ B (L)�1 � εt
=�1� β1 � . . .� βp
��1� α+
∞
∑k=0
ψk � εt�k ;Ψ (L)B (L) = 1.
yt = (I � b)�1 � a+∞
∑k=0
bk � ut�k .
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Momentele unui proces AR(p):
Mediaµ � E [xt ] =
α
1� β1 � β2 � . . .� βp.
Varianta
γ0 � Eh(xt � µ)2
i= σ2.
∞
∑k=0
ψ2k
Γ0 � E�(yt � E [yt ])
�y 0t � E
�y 0t���
Γ0 = b � Γ0 � b0 + Σ;Σ = E�ut � u0t
�.
vec (Γ0) = (I � b b)�1 � vec (Σ) .
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q)
White noiseAR(p)
Momentele unui proces AR(p) 2:(Auto)-covarianta
γk � E [(xt � µ) (xt�k � µ)] ;γk =
∞
∑j=0
ψj � ψj+k
!� σ2.
Γ1 � E�(yt � E [yt ])
�y 0t�1 � E
�y 0t�1
���; Γ1 = b � Γ0.
Γk � E�(yt � E [yt ])
�y 0t�k � E
�y 0t�k
���; Γk = bk � Γ0.
(Auto)-corelatia
ρk � γkγ0.
$k � Γ�10 � Γk .
Autocorelatia partial¼a
ak � Corr [xt , xt�k jxt�1, . . . , xt�k+1] .a1 = ρ1; a2 6= 0; . . . ; ap�1 6= 0; ap = βp ; ak = 0 pt. k > p.
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
MA(1): xt = εt + θ1 � εt�1; εt � iid N�0, σ2
�.
Traiectorie simulat¼a pentru un proces MA(1)
4
3
2
1
0
1
2
3
4
50 100 150 200 250 300 350 400 450 500
θ1 = 0, 7
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
MA(1)xt = εt + θ1 � εt�1; εt � iid N
�0, σ2
�.
xt = (1+ θ1L)| {z }Θ(L)
εt .
xt =�1 θ1
�| {z }θ0
��
εtεt�1
�| {z }
ut
Conditii de inversabilitate pentru un proces MA(1): jθ1j < 1.Scrierea AR(∞) a unui proces MA(p) inversabil
Θ (L)�1 xt = εt
xt = �∞
∑k=1
(�θ1)k xt�k + εt .
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
Momentele unui proces MA(1):Media
µ � E [xt ] = 0.Varianta
γ0 � E�x2t�= σ2
�1+ θ21
�.
γ0 = σ2 � θ0 � θ.
(Auto)-covarianta
γ1 � E [xt � xt�1] = θ1 �σ2;γk � E [xt � xt�k ] ;γk = 0, pt. k > 1.(Auto)-corelatia
ρ1 �γ1γ0=
θ1
1+ θ21; ρk � 0, pt. k > 1.
Autocorelatia partial¼a
ak � Corr [xt , xt�k jxt�1, . . . , xt�k+1] ; a1 = ρ1; a2 = �ρ21
1� ρ21.
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
Autocorelatie estimat¼a pentru un proces MA(1)
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
1 6 11 16 21 26 31 36 41 46
σ2 = 1; θ1 = 0, 7
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
Autocorelatie partial¼a estimat¼a pentru un proces MA(1)
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
1 6 11 16 21 26 31 36 41 46
σ2 = 1; θ1 = 0, 7
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
MA(2): xt = εt + θ1 � εt�1 + θ2 � εt�2; εt � iid N�0, σ2
�.
Traiectorie simulat¼a pentru un proces MA(2)
8
6
4
2
0
2
4
6
8
50 100 150 200 250 300 350 400 450 500
θ1 = �1, 7; θ2 = 0, 72Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
MA(2)
xt = εt + θ1 � εt�1 + θ2 � εt�2; εt � iidN�0, σ2
�.
xt =�1+ θ1L+ θ2L2
�| {z }Θ(L)
εt .
xt =�1 θ1 θ2
�| {z }θ0
�
0@ εtεt�1εt�2
1A| {z }
ut
Conditii de inversabilitate pentru un proces MA(2):jλj < 1;λ2 + θ1λ+ θ2 = 0.Scrierea AR(∞) a unui proces MA(p) inversabil
Θ (L)�1 xt = εt
xt = �∞
∑k=1
ψkxt�k + εt ;Ψ (L) �Θ (L) = 1.
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
Momentele unui proces MA(2):Media
µ � E [xt ] = 0.Varianta
γ0 � E�x2t�= σ2
�1+ θ21 + θ22
�;γ0 = σ2 � θ0 � θ.
(Auto)-covarianta
γ1 � E [xt � xt�1] = σ2 � θ0 �
0@ 0 0 01 0 00 1 0
1A � θ;
γ2 = σ2 � θ0 �
0@ 0 0 00 0 01 0 0
1A � θ;
γk � E [xt � xt�k ] ;γk = 0, pt. k > 2.(Auto)-corelatia: ρk �
γkγ0.
Autocorelatia partial¼a: ak .Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
Autocorelatie estimat¼a pentru un proces MA(2)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
1 6 11 16 21 26
σ2 = 1; θ1 = �1, 7; θ2 = 0, 72
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
Autocorelatie partial¼a estimat¼a pentru un proces MA(2)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
1 6 11 16 21 26
σ2 = 1; θ1 = �1, 7; θ2 = 0, 72
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
MA(q)
xt = εt + θ1 � εt�1 + θ2 � εt�2 + . . .+ θq � εt�q ; εt � iid N�0, σ2
�.
xt =�1+ θ1L+ θ2L2 + . . .+ θqLq
�| {z }Θ(L)
εt .
xt =�1 θ1 θ2 . . . θq
�| {z }θ0
�
0BBBBB@εt
εt�1εt�2...
εt�q
1CCCCCA| {z }
ut
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
Conditii de inversabilitate pentru un proces MA(q)
jλj < 1;λq + θ1λq�1 + . . .+ θ1λ� θq = 0.
Scrierea AR(∞) a unui proces MA(q) inversabil
Θ (L)�1 xt = εt
xt = �∞
∑k=1
ψkxt�k + εt ;Ψ (L) �Θ (L) = 1.
Gabriel Bobeic¼a EA. Note de curs
Modele AR(p)Modele MA(q) MA(q)
Momentele unui proces MA(q):
Mediaµ � E [xt ] = 0
Varianta
γ0 � E�x2t�= σ2
�1+ θ21 + θ22 + � � �+ θ2q
�;γ0 = σ2 � θ0 � θ.
(Auto)-covarianta
γk � E [xt � xt�k ] =(
σ2 �∑q�kj=1 θj � θj+k ; k � q0; k > q
.
(Auto)-corelatia: ρk �γkγ0.
Autocorelatia partial¼a: ak .
Gabriel Bobeic¼a EA. Note de curs