Testing covariance stationarity

(1)

!" #$$%

& ' ' " ( ( (

)* ' ( ' + , ,

- . ' ' ( ' + / ,

- 0 /' ( ' ' '

+ ( 1 2 3 & ' +

" ' ' ' ( + )*

4 2 3 & ' "

' + ( ' ' ' ( '

/ 4 2 3 ' '

' + - 5

'' + 6 7 89 /

/ + (

' " + , '

-* 2!:;#3" +

+ (

' ' - "

+ ' + ' + 2 3

' ' !<

-& " ( ' )*

" ' ' )( = ( = " * ' " " 2!::#3 /

+

-0 2!::%3 ( > + )*

+ '' ( '

5 2k3 ' - 2#$$!3 ' ' /

? ' ) >,

(2)

' + - 0 " 2#$$@3 ' '

-5 ' '' "

( ' ' (

- + (

+ ' '

2 2!::!33- ( = 2 "

" * ) 2!::#33 ' ( (

- & ( " (

' + 2 3 ' ''

-5 ' + +

(

-' + #$

-+ = A " 0 . 2!:B%3

= - "

+ ' ( - 5 '

* ( 2!::$ 3" * (,

2!::$ 3" * ' 2!::C3 0 = 2!:::3- &

" + D ,

' - 5 ' + 2 3

+ D E ' - & ' +

2 3 " ( + ,

+ D ' - & " '

' ' ' +

- &+ ' " +

+ ' 2* ( "

!::$ 3- F ' " 5 A 5 A

'' ' +

2 * ' "

!::C3-A ( " ( + ''

" G ( ,

- & ' ' " ( ' ' ( +

' + 2 3 + ' ' ,

' - ' )* ' " ' '

(3)

H I ' (

( " ' (

' ' - ' ' '

- 0 ' '

' ( + ' ' ( ,

" ( ( ' (

- 0 " ( ' ,

( ' +

- . ' ' '' '' +

' ' - & ' " (

+ 6 7 89 / "

' - (

* ( 2!::$ 3" ( ' (

+ , ' ( " + "

' ' '

+ ' +

,' - F (

' + + '

-+ ' ' + ( - # E

( + '' (

=-@ + - 0 /'

%- < ' ' ''

C

-( ( = ⇒ ,

' →- & ( '

W(s)ds ( W, ' W ( ,

- 5 ' ' = ' n→ ∞,

/ ' (

(4)

y ( +

d ' u1

y =d +u , t= 1, ..., n. 2!3

d ' = ( ' ' E

d = γ′_{x ,} ₍ _γ _{= (}_{γ , ...., γ} ₎′ ₊ _G _x

+ = ( + " - -"x = (1, t, ..., t )′_. ₊

' 2 3 x = 1, 2 3

x = (1, t)′_{. u} _' ₊_y

-6 ' H , u E

' ' - + ' " ( u

+ ( ' 2* ' " !::#3 u =C(L)ε ,

( ε ( ' +

C(L) = ∞ c L , C(1) = 0" ( G + ,

( u ' '

- & ' " ( + ( '

-A A: _ε ₍ ₍ _E ₊ _.

A B: ∞ _{jc <}_∞_.

' + + ' ,

- & + + ( ' '' +

+ * ' 2!::#3- "

" / " ( ' ,

' - ' + '

+ " +

> + " " > D

-& '' " ( ' + (

- E + H : '

-' H 1 (

2 3 - . + ( + ,

(5)

-. ( / + ' ' ' ,

-)( = ( = " * ' " " 2!::#3 ' ' +

' + - )*

E + (

KP SS= 1

(ωn) (u ) 2#3

( ω ' + u

-!"

2#$$@3 ' ' + ( '

+

V/S= 1

(ωn) 

 u −_n1 u

 

V /S ,(

V /S=n− V ar(S , S , ..., S )

ω 2@3

( S = (u ) ' + V ar(S , S , ..., S )

' + '

-#

)* J , 0 + ? ,

X . 2#$$!3 ' ' +

) >, > + ? "

1

(6)

KS= 0 /

≤ ≤

1

√ n

1

ω u −

k

n u .

#

$

%

&

& " ( ' ' + =

? E ( ' ' +

( / ?

-+ ' ' + ( 1 &+ u ,

" E ( - A ( "

' ( + ' ' "

' ( (

' + - (

' + ' 2 ' ( " ,

" , 3 = ? E

( '

-&+ ( u v , - - v =u −σ "

z = u

v

H 5 ' 5 " z E

' ' n− _z _⇒ _B₍_r_{) = (}_B ₍_r₎_{, B} ₍_r₎₎′ ₌_BM_{( )}_, ₍ _B₍_r₎

( ( + ( /

= ω ω

ω ω

( ω ω , + ' {u} {v }"

' - ' ω , +{u} {v}.

&+u ( " ( + ( 6 60

C = max

≤ ≤

− _√1

n z

( + " '' ' +

-6 ' ' "

C ⇒sup W(r)

(7)

− 1

√

n z ⇒W(r) =

W (r) W (r)

( W(r) #, (

-A ( " u ' d = γ′_x

= ( - 2 3 ( d = γ "

2 3 ( d = γ +γ t = γ′_x_{" (} _x _{= (1} _t₎′ _. _&

H ,( u 2 y )E

2 3 - 5

/D Dx →X(r) n→ ∞- / ' " +x

p, ' "D=diag[1, n, , n ] X(r) = (1, r, , r )

-. y " " , D

z = u

v

u =y −γ′x"v =u −σ " ( σ = 1

n u .

. + ( z"

C = max

≤ ≤

− _√1

n z .

' ' ' + ' '

!-T 1: _n_{→ ∞}_,

C = max

≤ ≤

− 1

√

n z ⇒sup W(r)

(

W(r) =W(r)−



 dW (s)X(s)

′ _X₍_s₎_X₍_s₎′_ds −

W (1) ---0×



 X(s)ds.

' ' ,

" ' +C + + '

-+ ' " +sup W(r)

(8)

- & ' (

' 2 - -1 x = 13

# ( 1

W(r) =W(r)−rW(1) = W (r)−rW (1) W (r)−rW (1) .

& ( ' x =

(1, t)′_,

W(r) = W (r)−rW (1) + 6r(1−r) W (1)− W (s)ds W (r)−rW (1)

+ " ( ' " " 9 1 +

x= (x , , x ) :

x = x + +x .

+ ( !$"$$$

+ !"$$$- & ' !K" <K !$K +

-!1

!K !-;% !-CC

<K !-C@ !-%C

!$K !-<! !-@!

& + + ' " (

+ , /, ..

+ ( = 2 " - -" * ' " !::<31

ω =

−

k h

q γ (h), ω =

−

k h

q γ (h), ω =

−

k h

q γ (h), 2%3

( ' - &

2%3" D γ (h), γ (h), γ (h) ' E

n− ′

u u "n− ′

v v , n− ′

u v ( ′ E

1≤t, t+h≤n"k(.) ( ( E [−1,1]( k(0) = 1,

(9)

q ( ' + ' ' q→ ∞ q/n→0

-= + + / 2 - - A " !:B$4

* " !:;!3- / ' " ( ( k(x) = 1− |x|,(

"

ω =

−

(1−|h| q )γ(h)

6 ' + " ω , ω , ω ,

+ω , ω , ω . " ( u " ( ' ω , ω ,

ω ω , ω , ω " ( u v ( E

-'

%

& " ( /' ' +

+ ( - . ( + + (

1

y =γ′x+u

( 1 u =αu− +e " γ= 0

-+ ( +α / /' 1 α= 0.0,0.5

1- e ,( 27 *!31 e ∼N(0,(1+c∗t))( c= 0.0,

0.002" 0.005" 0.05" 5 t = 1,2, ..., n ( n '

-( |α|<1 c= 0.& +|α|<1

c= 0,

- & + a= 1, c= 0, '

' - + " ' ' (

( α= 1 c= 0.

0 " DGP1 ( "

E ' ' '

* ' 2!::C3- & /' " (

( = 1 27 *#31 e ∼N(0,1) +t <(τ∗n)

e ∼N(0,1 +c) +t ≥(τ∗n)( τ = 0.5 c= 0.0, $-<" !" # @- .

τ = 0.8" τ = 0.5 " + " (

' - " 7 *# ' + - &

(10)

' " '( + ( =

'

-. ' ( + <K )* " 8 " ) ' ' C

- 5 ' u =

y − y .. <$$$ ( n= 300 500.5 +

( ' 2D3" ( ' , ' (

2#$$%3" 1

q= min{q , B(n)},

( B(n) = [8∗(n/100) ]

q = 3n

2 .

2ρ 1−ρ

( [ ] '

-'

(

$

9 ' ( # @- ,

|α|<1 c= 0- + " ' # @

D - . α= 0.0, ' '

-& " ( '

' + )* " 8 )

-. α = 0.5, C + /

- & " ' + ( + "

+ ' + " n= 300 n= 500

-' +C " )* " ) 8 "

( ) - / ' " + n = 500

α= 0.5" ' + )* " 8 " ) C $-$C<" $-$C$"

$-$%; $-$C!" ' - + ' +

" ( ' + ,

- '' H ' "

)* " 8 " ) C ' E

(11)

'

$

' + +

2 3 + , ,

- + 7 *! 7 *# ' # @"

' - . E ' ( +

' ' + - 5 /' "

' ( ( n

' + - A ( " ' +

( c= 0, '' +

-# @ ( )* " 8 ) ' (

( |α|<1 c= 0. ' ( + n c.

/ ' " ( n= 500 7 *! ( c= 5, ' (

+ $-$C +α= 0 $-$B +α= 0.5.

2' ( 3 + α = 0 c= 0

-'' ( 7 *# - )* "

8 ) D ' +

+ ,

-6 = + " C ? ,

E ( ' - + " + +

/ " ( /' (

( ' + + '

- E 0 ,

1 + 27 *!3

/ '( + 27 *#3" # @

C ' ( + ' . ' (

( n (

+

-" ( ' + α= 1 c= 0"

( C ' ( - ''

C + ? E '

" = ' ( +

- A " ?

α = 1 ? c = 0

' ( - F "

(12)

' ( 2 3

+ ' ( C (

α= 1 c= 0.

& " ' 1

2 3 6 ' " ' ' '

)* 4 2 3 & ' +

" ' ' ' ( + )* " 8 ) 4

2 3 & ' + " ' +

( ' ' ' ( ' / 4

2 3 & ' + , "

' ' ' ( + - &

/ " ( '' + ' ' C

, +

(13)

#1 * ( + , 7 *!

n=300 n=500

α c _α c

KPSS KPSS

0.0 0.002 0.005 0.05 5 0.0 0.002 0.005 0.05 5 0.0 0.043 0.045 0.044 0.047 0.047 0.0 0.045 0.046 0.045 0.047 0.048

0.5 0.065 0.063 0.065 0.068 0.066 0.5 0.065 0.066 0.067 0.064 0.065

1.0 0.858 0.858 0.856 0.860 0.856 1.0 0.920 0.922 0.920 0.920 0.920

V/S V/S

0.0 0.002 0.005 0.05 5 0.0 0.002 0.005 0.05 5 0.0 0.044 0.045 0.043 0.044 0.044 0.0 0.046 0.045 0.044 0.043 0.042

0.5 0.064 0.064 0.066 0.064 0.065 0.5 0.060 0.062 0.062 0.062 0.063

1.0 0.910 0.910 0.911 0.915 0.915 1.0 0.957 0.957 0.956 0.960 0.960

KS KS

0.0 0.002 0.005 0.05 5 0.0 0.002 0.005 0.05 5 0.0 0.034 0.037 0.040 0.050 0.054 0.0 0.039 0.040 0.042 0.053 0.056

0.5 0.046 0.047 0.051 0.061 0.066 0.5 0.048 0.050 0.055 0.068 0.070

1.0 0.812 0.810 0.810 0.811 0.816 1.0 0.894 0.896 0.895 0.892 0.894 C C

0.0 0.002 0.005 0.05 5 0.0 0.002 0.005 0.05 5 0.0 0.043 0.201 0.640 1 1 0.0 0.037 0.653 0.992 1 1

0.5 0.069 0.185 0.480 0.984 0.999 0.5 0.061 0.484 0.898 0.999 1

1.0 0.879 0.891 0.900 0.954 0.994 1.0 0.942 0.948 0.958 0.986 0.999

(14)

@1 * ( + , 7 *#

n=300 n=500

α c α c

KPSS KPSS

0.0 0.5 1 2 3 0.0 0.5 1 2 3

0.0 0.043 0.044 0.045 0.043 0.042 0.0 0.045 0.044 0.046 0.047 0.044

0.5 0.065 0.065 0.065 0.067 0.067 0.5 0.065 0.066 0.065 0.066 0.067

1.0 0.858 0.857 0.854 0.853 0.853 1.0 0.920 0.924 0.925 0.923 0.920

V/S V/S

0.0 0.5 1 2 3 0.0 0.5 1 2 3

0.0 0.044 0.044 0.042 0.043 0.042 0.0 0.046 0.047 0.047 0.045 0.044

0.5 0.064 0.067 0.067 0.067 0.067 0.5 0.06 0.062 0.061 0.063 0.065

1.0 0.910 0.909 0.910 0.910 0.908 1.0 0.957 0.958 0.958 0.957 0.955

KS KS

0.0 0.5 1 2 3 0.0 0.5 1 2 3

0.0 0.034 0.038 0.041 0.046 0.056 0.0 0.039 0.039 0.041 0.051 0.057

0.5 0.046 0.049 0.052 0.061 0.069 0.5 0.048 0.051 0.055 0.064 0.071

1.0 0.812 0.808 0.805 0.813 0.820 1.0 0.894 0.899 0.902 0.898 0.895 C C

0.0 0.5 1 2 3 0.0 0.5 1 2 3

0.0 0.043 0.433 0.925 1 1 0.0 0.037 0.700 0.997 1 1

0.5 0.069 0.323 0.735 0.982 0.999 0.5 0.060 0.501 0.928 0.999 1

1.0 0.880 0.892 0.898 0.925 0.943 1.0 0.942 0.948 0.954 0.999 0.999

)

**

+

,

' + + D ' ',

' ( = - " +

E + ' + ,

- & " ( +

' + E - .

y = log(E /E− ) ( E 6 7 89 /

+ $!8$%8!::: !#8@!8#$$@" ( !$$% - y

- ! ( +y - F

' {y} / "

- A ( " +

+ - '

+ " * (,

(15)

2!::$ 3 = +

+ + " ' '

0 2!:C@3- & ( " + u > ( y

"

/(t) =t− _u

+ t- #

' ' + /(t) . ' - &

E " #$$ "

D - 5+ "

<@$ - 5 ' * ' 2!::C3" ,

/' '

+ + = ' ( +

' ( + - " ' "

<@! "

( + ,

' - & " + ( y ( "

( ' " (

' ,' ( (

" + " '

' ' + '

+ ,' - " ' '

' ' '+ 1 + ,' (

-% / + - . ',

' )* " 8 " ) C u ..

+ ' + , '

( <@! - , '

' ' ' # (

'' - . ( =

( ' " % ( , + '

)* " 8 ) " ' y

- A ( " ' " +y

" ,

(16)

-3 -2 -1 0 1 2 3

2 5 0 5 0 0 7 5 0 1 0 0 0

!1 6 7 89 9/

. 1 . 2 . 3 . 4 . 5 . 6 . 7

2 5 0 5 0 0 7 5 0 1 0 0 0

#1 9 + 6

(17)

= - '

% ( )* " 8 )

- . ( ''

( ' " ( ' '

<K + E " ' y ,

- & " ( ( = ' ( 6

7 89 /

" ' +y ( '

-+ ' "

( '' = ,

L . ( D

+ ( ' ,

' ' ' - & ( " ( E = # +

( '' " ,

" ( '' ( ( - *

( 2!::$ 3 ' + (

' = + '' ( ,

' + - (

%- 7 > + ( ( + ' "

' +

, ' " !$K- "

( ' + (

( ' E , ' !

-%1 5 +

. ' , '

)* $-C< $-%$

8 $-!! $-$B

) !-#! !-$<

C !-;@MM !-%@

! _{0 =} _2!:::3 _' ₊ _,

5 A ' ' ( 5 A ( '

(18)

-

%

' ' ' + ' (

( ,

- & ' '' " ( ( 6 7 ,

89 / - F

+ ( ( '

' +

- ' E E ( = * (

2!::$ 3 * ' 2!::C3 + ,

(19)

.

**

* + + !

1

√

n z =

1 √ n u v = 1 √ n

u −(γ−γ)′_x

u −σ

= √1_n u −(γ−γ)

′_x

v −(σ −σ ) +o (1)

= √1 n

u

v −

(γ−γ)′

(σ −σ ) ---0 1

√

n x +o (1)

= √1 n

u

v −

(γ−γ)′

(σ −σ ) ---0 D √ n − 1 √ n D √ nx

+o (1)

⇒ B(r)−



 dB (s)X(s)

′ _X₍_s₎_X₍_s₎′_ds −

B (1) ---0 0



 X(s)ds

=

 

W(r)− 

 dW (s)X(s)

′ _X₍_s₎_X₍_s₎′_ds −

W (1) ---0 0



 X(s)ds

  

+ " A ' 5 " n→ ∞,

C = max

≤ ≤

− _√1

n z ⇒sup W(r) .

$

N!O P " 0- - 0 = >" #$$@- & /

? " 5'' 9 !@"

;BB,;;@-N#O " - *-" - ' ) = = - " #$$@- ,

+ " Q +

9 " !!#"

#C<,#:%-N@O A " 9-Q-" !:B$" 0 ' 2 ( R =1 .

(20)

N%O A " 7 ,5 " - - 0 " 7-.-. " !:B%- F *

+ =, = ' " Q + 5 5 ,

C:"

!$;,!!@-N<O )( = ( = " 7-" *- - - * ' " *- " R- " !::#- ,

' + + "

Q + 9 " <%"

!<:,!B;-NCO " - Q- - *- 0- 0 " !::%- 5 +

- Q + 9 !#"

!<B,CC-NBO " - - - " #$$<- 5 , ' E ,

- + Q +

-N;O " 5-" !::!" , 0 = 0 = * " "

<:"

!#B:,!@!@-N:O " 0- *- - - * ' "

!::C-+ , 1 5 ( + ( ''

E " Q + 9 ' !"

#!!,#%;-N!$O 0 " " !:C@" + ' ' "

Q + @C"

@:%,%!:-N!!O " - -&- * " !:;#- ( =

1 ' " Q +

0 9 !$"

!@:,!C#-N!#O * " 5- - -.- ( " !::$ - +

= = " 9 @@"

!C<,!B$-N!@O * " 5- - -.- ( " !::$ - 5 +

= " Q + 9 %<"

#CB,#:$-N!%O * " .- .- ) " !::#- 6 60 ( F "

9 C$"

#B!,#;<-N!<O * ' " *- - -" !::<- E D ,

- 9 C@"

(21)

N!CO * ' " *- - - - " !::#- 5 ' + ' - 5

+ #$"

:B!,!$$!-N!BO * " 0- -" !:;!" ' 2 (

R =15 *

3-N!;O " - - 0 = " !:::- + E

" ' 5 A " 7 ' +

0 " 6 + " "

(

-N!:O . " - -"!:C!- , +,E - = %;" !$:,

!!%-N#$O " -" #$$!- ' + ,

- Q + 5 " ##2!3"