WikiDer > VAR ning umumiy matritsali yozuvi (p)

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Ushbu sahifada $ a $ ning turli xil matritsa yozuvlari uchun tafsilotlar ko'rsatilgan vektor avtoregressiyasi bilan ishlash k o'zgaruvchilar.

Var (p)

${ displaystyle y_ {t} = c + A_ {1} y_ {t-1} + A_ {2} y_ {t-2} + cdots + A_ {p} y_ {tp} + e_ {t}, ,}$

har birida ${ displaystyle y_ {i}}$ uzunlik vektori k va har biri ${ displaystyle A_ {i}}$ a k × k matritsa.

Shovqin haqida qanday taxminlar mavjud?

Katta matritsali yozuv

${ displaystyle { begin {bmatrix} y_ {1, t} y_ {2, t} vdots y_ {k, t} end {bmatrix}} = { begin {bmatrix} c_ { 1} c_ {2} vdots c_ {k} end {bmatrix}} + { begin {bmatrix} a_ {1,1} ^ {1} & a_ {1,2} ^ {1 } & cdots & a_ {1, k} ^ {1} a_ {2,1} ^ {1} & a_ {2,2} ^ {1} & cdots & a_ {2, k} ^ {1} vdots & vdots & ddots & vdots a_ {k, 1} ^ {1} & a_ {k, 2} ^ {1} & cdots & a_ {k, k} ^ {1} end { bmatrix}} { begin {bmatrix} y_ {1, t-1} y_ {2, t-1} vdots y_ {k, t-1} end {bmatrix}} + cdots + { begin {bmatrix} a_ {1,1} ^ {p} & a_ {1,2} ^ {p} & cdots & a_ {1, k} ^ {p} a_ {2,1} ^ { p} & a_ {2,2} ^ {p} & cdots & a_ {2, k} ^ {p} vdots & vdots & ddots & vdots a_ {k, 1} ^ {p} & a_ {k, 2} ^ {p} & cdots & a_ {k, k} ^ {p} end {bmatrix}} { begin {bmatrix} y_ {1, tp} y_ {2, tp} vdots y_ {k, tp} end {bmatrix}} + { begin {bmatrix} e_ {1, t} e_ {2, t} vdots e_ {k, t} end {bmatrix}}}$

Regressiya belgisi bilan tenglama

Qayta yozish y o'zgaruvchilar birma-bir beradi:

${ displaystyle y_ {1, t} = c_ {1} + a_ {1,1} ^ {1} y_ {1, t-1} + a_ {1,2} ^ {1} y_ {2, t- 1} + cdots + a_ {1, k} ^ {1} y_ {k, t-1} + cdots + a_ {1,1} ^ {p} y_ {1, tp} + a_ {1,2 } ^ {p} y_ {2, tp} + cdots + a_ {1, k} ^ {p} y_ {k, tp} + e_ {1, t} ,}$

${ displaystyle y_ {2, t} = c_ {2} + a_ {2,1} ^ {1} y_ {1, t-1} + a_ {2,2} ^ {1} y_ {2, t- 1} + cdots + a_ {2, k} ^ {1} y_ {k, t-1} + cdots + a_ {2,1} ^ {p} y_ {1, tp} + a_ {2,2 } ^ {p} y_ {2, tp} + cdots + a_ {2, k} ^ {p} y_ {k, tp} + e_ {2, t} ,}$

${ displaystyle qquad vdots}$

${ displaystyle y_ {k, t} = c_ {k} + a_ {k, 1} ^ {1} y_ {1, t-1} + a_ {k, 2} ^ {1} y_ {2, t- 1} + cdots + a_ {k, k} ^ {1} y_ {k, t-1} + cdots + a_ {k, 1} ^ {p} y_ {1, tp} + a_ {k, 2 } ^ {p} y_ {2, tp} + cdots + a_ {k, k} ^ {p} y_ {k, tp} + e_ {k, t} ,}$

Qisqacha matritsali yozuv

VAR-ni qayta yozish mumkin (p) bilan k o'z ichiga olgan umumiy usulda o'zgaruvchilar T + 1 kuzatishlar ${ displaystyle y_ {p}}$ orqali ${ displaystyle y_ {T}}$

${ displaystyle Y = BZ + U ,}$

qaerda:

${ displaystyle Y = { begin {bmatrix} y_ {p} & y_ {p + 1} & cdots & y_ {T} end {bmatrix}} = { begin {bmatrix} y_ {1, p} & y_ {1 , p + 1} & cdots & y_ {1, T} y_ {2, p} & y_ {2, p + 1} & cdots & y_ {2, T} vdots & vdots & vdots & vdots y_ {k, p} & y_ {k, p + 1} & cdots & y_ {k, T} end {bmatrix}}}$

${ displaystyle B = { begin {bmatrix} c & A_ {1} & A_ {2} & cdots & A_ {p} end {bmatrix}} = { begin {bmatrix} c_ {1} & a_ {1,1} ^ {1} & a_ {1,2} ^ {1} & cdots & a_ {1, k} ^ {1} & cdots & a_ {1,1} ^ {p} & a_ {1,2} ^ {p} & cdots & a_ {1, k} ^ {p} c_ {2} & a_ {2,1} ^ {1} & a_ {2,2} ^ {1} & cdots & a_ {2, k} ^ {1 } & cdots & a_ {2,1} ^ {p} & a_ {2,2} ^ {p} & cdots & a_ {2, k} ^ {p} vdots & vdots & vdots & ddots & vdots & cdots & vdots & vdots & ddots & vdots c_ {k} & a_ {k, 1} ^ {1} & a_ {k, 2} ^ {1} & cdots & a_ {k , k} ^ {1} & cdots & a_ {k, 1} ^ {p} & a_ {k, 2} ^ {p} & cdots & a_ {k, k} ^ {p} end {bmatrix}}}$

${ displaystyle Z = { begin {bmatrix} 1 & 1 & cdots & 1 y_ {p-1} & y_ {p} & cdots & y_ {T-1} y_ {p-2} & y_ {p-1} & cdots & y_ {T-2} vdots & vdots & ddots & vdots y_ {0} & y_ {1} & cdots & y_ {Tp} end {bmatrix}} = { begin { bmatrix} 1 & 1 & cdots & 1 y_ {1, p-1} & y_ {1, p} & cdots & y_ {1, T-1} y_ {2, p-1} & y_ {2, p} & cdots & y_ {2, T-1} vdots & vdots & ddots & vdots y_ {k, p-1} & y_ {k, p} & cdots & y_ {k, T-1} y_ {1, p-2} va y_ {1, p-1} & cdots & y_ {1, T-2} y_ {2, p-2} & y_ {2, p-1} & cdots & y_ {2, T-2} vdots & vdots & ddots & vdots y_ {k, p-2} & y_ {k, p-1} & cdots & y_ {k, T-2} vdots & vdots & ddots & vdots y_ {1,0} & y_ {1,1} & cdots & y_ {1, Tp} y_ {2,0} & y_ {2,1} & cdots & y_ {2, Tp} vdots & vdots & ddots & vdots y_ {k, 0} & y_ {k, 1} & cdots & y_ {k, Tp} end {bmatrix} }}$

va

${ displaystyle U = { begin {bmatrix} e_ {p} & e_ {p + 1} & cdots & e_ {T} end {bmatrix}} = { begin {bmatrix} e_ {1, p} & e_ {1 , p + 1} & cdots & e_ {1, T} e_ {2, p} & e_ {2, p + 1} & cdots & e_ {2, T} vdots & vdots & ddots & vdots e_ {k, p} & e_ {k, p + 1} & cdots & e_ {k, T} end {bmatrix}}.}$

Keyinchalik koeffitsient matritsasini echish mumkin B (masalan, oddiy kichkina kvadratchalar taxmin qilish ${ displaystyle Y taxminan BZ}$).

Adabiyotlar

• Lyutkepol, Helmut (2005). Ko'p sonli seriyalarni tahlil qilish uchun yangi kirish. Berlin: Springer. ISBN 3540401725.