Seasonal Cointegration Rank Tests for Daily Data

(1)

(2)

n Xt

X_t=

Σ

_{i = 1}

k

Π_iX_t−i+ Φ D_t+ _t

t Nn(0, Σ )

X

0

, X

₋₁

, , X

_{− k +1}

D

_t

(n n )

A (z ) = I_n

− Σ

i = 1 k

Π_izⁱ

X_t

|A(z)| =0

z1= 1, z2= e^{2 πi/5}, z3= e⁻^{2π i/5}, z4= e^4πi/5, z5= e⁻^4πi/5)

2π/5 − 2π/5 4π/5 − 4π/5

p (L )X_t= _{m = 1}

Σ

5

α_m _mX_t^{(m )}+

Σ

_{j = 1}

k−5

Γ_jp (L )X_t₋_j+ ΦD_t+ _t

α

m

m n r_m r_m

α

m

= α

mR

+ iα

mI

m

=

mR

+ i

m I

α

_{m m} _m

=

_R

− i

I

(3)

X_t^{(m )}= pm(L )L pm(zm)zm

X_t

p

j

(z ) = Π

m≠ j 5

(1 − z

m

z ) = p (z ) 1 − z

j

z

p (z ) =

Π

m = 1 5

(1

−

z _mz ) = (1

−

z⁵) = (1

−

z )(1 + z + z²+ z³+ z⁴)

z

_m

z

m

Xt^{(m )}

X_t^{(1 )}= 1

5(L + L²+ L³+ L⁴+ L⁵)X_t X_t^{(2 )}= 1

5 [ cos (2

5π )L + cos (4

5π )L²+ cos (4

5π )L³+ cos (2

5π )L⁴+ L⁵]X_t +1

5i [− sin (2

5π )L− si n (4

5π )L²+ sin (4

5π )L³+ sin (2

5π )L⁴]X_t X_t^{(3 )}= 1

5 [ cos (2

5π )L + cos (4

5π )L²+ cos (4

5π )L³+ cos (2

5π )L⁴+ L⁵]X_t +1

5i [ sin (2

5π )L + sin (4

5π )L²− sin (4

5π )L³− sin (2

5π )L⁴]X_t X_t^{(4 )}= 1

5 [ cos (4

5π )L + cos (2

5π )L²+ cos (2

5π )L³+ cos (4

5π )L⁴+ L⁵]X_t +1

5i [ − sin (4

5π )L + s in (2

5π )L²− si n (2

5π )L³+ sin (4

5π )L⁴]X_t X_t^{(5 )}= 1

5 [ cos (4

5π )L + cos (2

5π )L²+ cos (2

5π )L³+ cos (4

5π )L⁴+ L⁵]X_t +1

5i [ sin (4

5π )L− sin (2

5π )L²+ sin ( 2

5π )L³− sin (4

5π )L⁴]X_t

Xt^{(2 )}= Xt^{(2R )}+ iXt^{(2I )} Xt^{(3 )}= Xt^{(2R )}− iXt^{(2R )}

X_t^{(4 )}= X_t^{(4R )}+ iX_t^{(4I )} X_t^{(5 )}= X_t^{(4R )}− iXt^{(4I )}

X_t^{(2R )}= 1 5 [ cos (2

5π )L + cos (4

5π )L²+ cos (4

5π )L³+ cos (2

5π )L⁴+ L⁵]X_t X_t^{(2I )}= 1

5[− sin (2

5π )L − sin ( 4

5π )L²+ sin (4

5π )L³+ sin (2

5π )L⁴]X_t

(4)

X_t^{(4R )}= 1 5 [ cos (4

5π )L + cos (2

5π )L²+ cos (2

5π )L³+ cos (4

5π )L⁴+ L⁵]X_t X_t^{(4I )}= 1

5[ − si n (4

5π )L + sin (2

5π )L²− sin (2

5π )L³+ sin ( 4

5π )L⁴]X_t

p (L )X_t= α₁ ₁X_t^{(1 )}+ 2 (α_{2 R} _2R+ α_{2I 2I})X_t^{(2R )}+ 2 (α_2R _2I

−

α_{2I 2R})X_t^{(2I )} + 2 (α_4R _4R+ α_{4I 4I})X_t^{(4R )}+ 2 (α_4R _4I

−

α_{4I 4R})X_t^{(4I )}

+

Σ

_{j = 1}

k− 5

Γ_jp (L )X_t_{− j}+ ΦD_t+ _t

(α

_2R _2I

− α

_{2I 2 R}

) = 0

(α_4R _4I

−

α_{4I 4 R}) = 0

r r = n

Φ

m

ΦD_t _{m = 1}

Σ

s˜

Φ_mz ^t_m _{m = 2}

Σ

s˜

Φ_mz ^t_m

Φ

₁

m = 1

Σ

s

Φmz ^tm= _{m = 1}

Σ

s

αmρmz ^tm

ρ

m

(1 r

m

)

(5)

p (L )X

t

= Σ

m = 1 5

α

m

 

 

 

m

ρ

_m

 

 

 

X

t^{(m )}

z

^tm

+ Σ

j = 1 k− 5

Γ

j

p (L )X

t− j

+ Σ

m = 6 s˜

Φ

m

z

m t

+

t

p (L )X

t

= α

1 1

X

t^{(1 )}

+ Σ

m = 2 5

α

m

 

 

 

m

ρ

_m

 

 

 

X

t^{(m )}

z

^t_m

+ Σ

j = 1 k− 5

Γ

j

p (L )X

t− j

+ Φ

1

+ Σ

m = 6 s˜

Φ

m

z

m t

+

t

D

_t

= 0

k

∆₅X_t= _t (t = 1 , 2, ..., T ) _t

∼i.i.d. N(0,I

k

)

k = 1 , 2, , 5

(n− r) = 1, 2, , 5

∆

₅

X

_t

= 2 1

_k

+

_t

1

_k k

r n

(n− r )

(6)

(n− r ) (n− r )

(n− r )

π

(n− r )

(7)

n−r T θ 2π/ 5 4π/ 5 2π/ 5 4π/ 5 2π/ 5 4π/ 5 2π/ 5 4π/ 5

2π/ 5 4π/ 5 2π/ 5 4π/ 5 2π/ 5 4π/ 5 2π/ 5 4π/ 5

(8)

n−r T θ 2π/ 5 4π/ 5 2π/ 5 4π/ 5 2π/ 5 4π/ 5 2π/ 5 4π/ 5

2π/ 5 4π/ 5 2π/ 5 4π/ 5 2π/ 5 4π/ 5 2π/ 5 4π/ 5

(9)