Modele Autoregresive Scalare Station Are

Modele AR(p)Modele MA(q)

Econometrie avansat¼a

Gabriel Bobeic¼a

MBPM

8 aprilie 2011

Gabriel Bobeic¼a EA. Note de curs


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



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



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



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



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



White noiseAR(p)

Momentele unui proces AR(1):

Mediaµ � E [xt ] =

α

1� β.

Varianta

γ0 � Eh(xt � µ)2

i=

σ2

1� β2.



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.



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



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



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



White noiseAR(p)

Momentele unui proces AR(2):


α

1� β1 � β2.

Varianta

γ0 � Eh(xt � µ)2

i=1� β21+ β2

σ2

(1� β1 � β2) (1+ β1 � β2).



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.


ak � Corr [xt , xt�k jxt�1, . . . , xt�k+1] .a1 = ρ1; a2 = β2; ak = 0 pt. k > 2.



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



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



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



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 .



White noiseAR(p)

Momentele unui proces AR(p):


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) .



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 .


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.


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



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 .



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.


ak � Corr [xt , xt�k jxt�1, . . . , xt�k+1] ; a1 = ρ1; a2 = �ρ21

1� ρ21.



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



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



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


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.



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


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



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



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



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.



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 .


Modele Autoregresive Scalare Station Are

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