Appunti Statistica - parte 2

Y t

C(Y V(Y

, ) = ) =

Y γ 0

t t t γ 0 ρ h

C(Y ,Y ) Z

γ γ

, ) = = = , ∀t ∈

=

t t+h

ρ(Y Y ρ

h h

t t+h h

γ γ γ

V(Y )V(Y ) 0 0 0

t t+h

γ h ρ h

ρ h

γ

= = 1

0

ρ 0 γ 0 γ h

V(Y )

t

V(Y ) γ 0

t+h 02 =

⋅ =

γ γ γ γ

0 0 0

ρ h

Statistics 12

C(Y C(Y

= , ) = , ) = ⟹ =

γ Y Y γ ρ ρ

−h −h

t t−h t−h t h h

C(A, C(B,

=

B) A)

Ω , ...

W W

1 2

A

P Ω ∈Ω

W

Statistics 13

Z

2 2

{Y } ∼ (0, ) ⟺ ∼ ), ∀t ∈

W N σ Y i.i.d.(0, σ

t t

E(Y Z

) = 0, ∀t ∈

t E(Y )

t

Z

2

∼ ), ∀t ∈

Y i.i.d.(0, σ

t

Z

∈

t

V(Y Z

2

) = , ∀t ∈

σ

t

V(Y )

t 2

∼ )

Y i.i.d.(0, σ

t

2

σ 2

σ

=

0

h

C(Y , ) = 0

Y

t t+h

Statistics 14

Y Y

t t+h =

0

h C(A,

Se due variabili casuali e sono indipendenti ⟹ = 0

A B B)

= +

Y μ u

t t

E(Y ) = μ

t

V(Y 2

, ) = =

0

variance σ h

t =

0

h

2

{Y } ∼ (0, )

W N σ

t

Statistics 15

∼

Z, 2

{Y } ∼ ⟺ = + , ∀t ∈ where ∼ (0, )

RW Y Y u u W N σ

t t t−1 t t

E

E(Y E(Y E(Y E(u E(Y

) = + ) = ) + ) = ) = ⋯ ⟹

u

t t−1 t t−1 t t−1

mean stationarity

E(Y E(Y

) = )

0

t

u t

E(u ) = 0

t V(u ) =

t

2

σ

Statistics 16

V(A V(A) V(B) C(A,

+ = + + 2

B) B)

V(Y V(Y V(Y V(u V(Y 2

) = + ) = ) + ) + 2C(Y , ) = ) +

u u σ

t t−1 t t−1 t t−1 t t−1

u Y

t t−1

C(Y , ) = 0

u

t−1 t

E(u E(Y E(Y

) = 0 ⋅ ) = ) ⋅

u

t t−1 t t−1

C(Y

0=0 , ) = 0

u

t−1 t

2 > 0

σ

V(Y V(Y

2

) + > ) ⟹ NO variance stationarity

σ

t−1 t−1

E

E(u )

t

Statistics 17

V(u 2

) = σ

t 1

Y t−1

−1

t u t

δ

δ Z,

{Y } ∼ + ⟺ = + + , ∀t ∈ where ∼

RW drif t Y δ Y u u