Appunti Statistica - parte 1

, = 1, ...,

y t T

t

Statistics 1

T t

C t T C t

S t u t

Statistics 2

y t

= + +

y T C S u

t t t t

= × ×

y T C S u

t t t t

y t

= × ×

y T C S u

t t t t

) = ) + ) + )

ln(y ln(T C ln(S ln(u

t t t t

y t , , ...,

y y y

1 2 T (Ω, )

A, P

Statistics 3

Z Z} R ∞

{Y : ∈ : Ω ⟶

t

t ω ∈Ω

ω Y 3

ω ω 1

(… , (ω ), (ω ), (ω ), … )

y y y

−1 1 0 1 1 1

(ω)

ω y t

Z R

R +

Y t

Ω

A u t

Statistics 4

2

σ

Z

2 2

{Y } ∼ (0, ) ⟺ ∼ ), ∀t ∈

W N σ Y i.i.d.(0, σ

t t Z

2 2

{Y } ∼ (0, ) ⟺ ∼ (0, ), ∀t ∈

N W N σ Y i.i.d. N σ

t t

2

σ μ

2

σ Z

2 2

{Y } ∼ (μ, ) ⟺ ∼ ), ∀t ∈

W N σ Y i.i.d.(μ, σ

t t Z

2

= + , where ∼ (0, ), ∀t ∈

Y μ u u W N σ

t t t

Statistics 5

μ μ μ

∘

= 20 C

μ u t

∘ ∘ ∘

20.2 C 19.8 C 20.1 C Z,

{Y } ∼ ⟺ = + , ∀t ∈ where ∼

RW Y Y u u

t t t−1 t t

2

(0, ) i.i.d.

W N σ

u t Y 0

t

= + ∑

Y Y u

0

t i

i=1

Statistics 6

δ

{Y } ∼ + ⟺ = + + , ∀t ∈

RW drif t Y δ Y u

t t t−1 t

Z, 2

where ∼ (0, ) i.i.d.

u W N σ

t δ

u t Y 0

t

= + ⋅ + ∑

Y Y δ t u

0

t i

i=1

⋅ ∑

δ t u i

Statistics 7

Y t−1

u δ

t

u t δ

1 ⋅ Y t−1 α ω

{Y } ∼ ⟺ = + + , ∀t ∈

AR(1) Y ω αY u

t t t−1 t

Z, 2

where ∼ (0, ) i.i.d.

u W N σ

t k

Y α

t

∣α∣ < 1

=1 =0

α ω

Statistics 8

μ

u u

t t−1

{Y } ∼ ⟺ = + + , ∀t ∈

M A(1) Y μ β u u

1

t t t−1 t

Z, 2

where ∼ (0, ) i.i.d.

u W N σ

t Y t−1 u t

β 1

μ

β 1 Z,

{Y } ∼ ⟺ = + + , ∀t ∈ where ∼

LT Y β β t u u

0 1

t t t t

2

(0, ) i.i.d.

W N σ

Statistics 9

β β

0 1

+

β β t

0 1

β 0

β 1 1, 2, 3, …

β β

0 1 u