ASemiparametricPanelModelforUnbalancedDatawithApplicationtoClimateChangeintheUnitedKingdom Atak,AlevandLinton,OliverB.andXiao,Zhijie MunichPersonalRePEcArchive

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Munich Personal RePEc Archive

A Semiparametric Panel Model for Unbalanced Data with Application to Climate Change in the United Kingdom

Atak, Alev and Linton, Oliver B. and Xiao, Zhijie

Queen Mary, University of London, London School of Economics and Political Science, Boston College

15 March 2010

Online at https://mpra.ub.uni-muenchen.de/22079/

MPRA Paper No. 22079, posted 13 Apr 2010 14:06 UTC

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T

s i∈I

s

y is − α i − β ^⊤ _i D s − η K h ((t − s)/T ),

η = g θ (t/T ) = T ⁻ ⁿ _i ^T _{s t}

_i

y is − α _i − β ^⊤ _i D _s K h ((t − s)/T ) T ⁻ ⁿ _i ^T _{s t}

i

K _h ((t − s)/T )

= T ⁻ ^T _s K _h ((t − s)/T ) ⁿ _i

^s

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i

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ε _jt (θ) = y _jt − α _j − β ^⊤ _j D _t − g _θ (t/T )

∂ε _jt (θ)

∂α i

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if j = i

− ^∂g

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else

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if j = i

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else i = 1, . . . , n,

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1 T

T

s t

i

K h ((t − s)/T ) → − _m

_t

, i ≤ m t

#, i > m t

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∂g θ (t/T )

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1 T

T

s t

i

K h ((t − s)/T )D s → − m

t

, i ≤ m t

# , i > m t

T → ∞

- + 0 3 θ

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d

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T ⁻ ⁿ _i ^T _{s t}

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0 , X ^⊤ Xθ = X ^⊤ y, , X ^⊤ X -

: 0 q ^⊤ = (1, . . . , 1, 0, . . . , 0), q ^⊤ θ = 0. + 0

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, R, k × (k − 1) , k = n(d + 1), (q, R) R ^⊤ q = 0, 1'CD( § '-92. @

R = R R

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; R = I _nd×nd ,

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{ (ξ _t , F t ), −∞ < t < ∞} β4 , 1 2 , / β(n)

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| P (A | F −∞ ^t ) − P (A) |

E n → ∞ - β4 ,

7 : 1'CC92 , -

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^⊤

. + θ

-

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'- i, ε _it β β _it lim sup _t b ^t max _≤i≤n β _it <

∞ b > 1 ^∞ _h _−∞ E (ε it ε it h ) = ω _i s _i = ^∞ _k _−∞ E (ε it ε i,t k ) 0 < ω ≤ min ≤i≤n ω i ≤ max ≤i≤n ω i ≤ ω < ∞ -

)- g : [ , ] → τ ≥ p-

8- K [ − 1, 1] K (u)du = 1

u ^j K(u)du = 0, j = 1, . . . , p − 1, u ^p K(u)du = 0 1 _p (K) = u ^p K(u)du

|| K || = K (z)dz.

9- !

1 2 " T → ∞ h → 0 T h → ∞ T h ^p → 0

1 2 h = c T T ⁻ ^{/ p} 0 < lim inf

T →∞ c T ≤ lim sup

T →∞

c T < ∞ .

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- +

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, 3 >

4T 3 7

-

+ t ≤ t ≤ ≤ t n - @

t ≤ t ≤ ≤ t n 0

4 F

F F - . t i → ∞

t _i = ⌊ r _i T ⌋ r _i ∈ (0, 1) 1(2

i = 1, . . . , n, 1 r _n = 12 F -

+ - a kj = ⁿ _{s j} (r s − r s ) /s ^k k = 1, 2, 3, 4

δ i = (1 − r i − 2a i + a i ) f i = (n + 2)a ,i − 2a ,i − na ,i λ i = (n a ,i − 4na ,i + 4a ,i )

n = δ ω , . . . , δ i ω _i , . . . , δ n ω _n S n = δ s , . . . , δ i s _i , . . . , δ n s _n

∆ n = diag { 1, . . . , 1 − r i , . . . , 1 − r n } .

D

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@ A _n n × n , (i, j )4 [A n ] _i,j = λ i i−

j ω _j + ⁿ _{j i} λ j ω _j i = j

f i ω _j + ω _i + λ i l j,l<i ω _l + _l>i λ l ω _l j < i ,

G n =



 

 

(a − 2a + ⁿ _l a l ) . . . ia i − 2a i + ⁿ _{l i} a l . . . (na ,n − 2a ,n ) ia i − 2a i + ⁿ _{l i} a l ia i − 2a i + ⁿ _{l i} a l (na ,n − 2a ,n ) (na _,n − 2a _,n ) (na _,n − 2a _,n ) (na _n − 2a _n )



 

  .

+ 0 >

Q = ∆ n + G n (∆ n + G n ) ⊗

(∆ n + G n ) ⊗ ^⊤ ∆ n ⊗ I + G n ⊗ J , 1A2

= ⁿ + A n [ n + A n ] ⊗ ^⊤

[ n + A n ] ⊗ ^⊤ S n ⊗ I + A n ⊗ J , 1B2

i 11 × 1 J = i i ^⊤ 11 × 11 ,

∗ = b

b ⊗ b = b , . . . , b i , . . . , b n

⊤

b i = 1

p! 1 _p (K)



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n

l

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r

l

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

r

i

g ^p (s) ds

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



δ(s) [0, 1], δ(s) = 1/j r j < s < r j j = 1, 2, . . . , n. . E -

T 1 - # " "$ "% &

& '() T → ∞ ,

√ T R ^⊤ θ − R ^⊤ θ + h ^p R ^⊤ QR ⁻ R ^{⊤ ∗} ⇒ N 0, R ^⊤ QR ⁻ R ^⊤ R R ^⊤ QR ⁻ -

R 1. + 0

F -

R 2. +

4 %1)**D2 -

0 0 $ 4

C

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3 4 - @ K

0 -

R 3. ? i t F

- + F J ω _i s _i -

@ t = 1 r i = 0

i = 1, . . . , n r n = 1 δ(s) = 1/n 0 < s < 1 j = 1, 2, . . . , n. 3 b i = 1

p! 1 _p (K )





n

l



 δ(s)g ^p (s) ds



 −



 g ^p (s) ds







 = 0.

+ -

+ 0 0

-

Q = Σ X − 1

n Σ ^∗ _X , 1D2

Σ X = I n I n ⊗ ^⊤

I n ⊗ I n

Σ ^∗ _X = J n J n ⊗ J n ⊗ ^⊤ J n

,

0 1B2

n = 1 − n ω , . . . , 1 − n ω _i , . . . , 1 − n ω _n S n = 1 − _n s , . . . , 1 − _n s _i , . . . , 1 − _n s _n ,

(i, j)4 A n

[A n ] _i,j = ⁿ ^{j i} ω _j i = j

− _n 1 − _n ω _j + ω _i + _n _{l j,i} ω _l j < i .

C 1. # " "$ "%

T → ∞ ,

√ T R ^⊤ θ − R ^⊤ θ ⇒ N 0, R ^⊤ QR ⁻ R ^⊤ R R ^⊤ QR ⁻ -

@ ε it E σ n = S n =

1 − _n σ I n I n 4 , (i, j)4 A n

[A n ] _i,j = ⁿ 1 − _n σ i = j

− _n σ j = i .

'*

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. , E - +

E + ) ,-

T 2 - # " "$ "% &

& '() T → ∞ ,

√ T h [g(u) − g(u) − h ^p b(u)] ⇒ N 0, 1

m ω _m || K || u ∈ [r m , r m ) m = 1, . . . , n − 1,

√ T h [g(u) − g(u) − h ^p b(u)] ⇒ N 0, 1

n ω || K || u > r n - b(u) = _p g ^p (u)1 _p (K), ω _m = m ⁻ ^m _i ω _i ω = n ⁻ ⁿ _i ω _i -

@ -

C 2. # " "$ "% & t = 1

T → ∞ ,

√ T h [g(u) − g(u) − h ^p b(u)] ⇒ N 0, 1

n ω || K || . 1C2

R 4- @ , 4

+ -

ε t = (ε t , . . . , ε nt ) ^⊤ = Ξ(t/T ) ^/ η _t , η _t = (η _t , . . . , η _nt ) ^⊤

β4 , ' Ξ(u) 0

, - Ψ(s) = Eη _t η ^⊤ _{t s} Ψ ∞ = ^∞ _s _−∞ Ψ(s)- + 1C2 || K || i ^⊤ Ξ(u) ^/ Ψ ∞ Ξ(u) ^/ i/n, i = (1, 1, . . . , 1) ^⊤ . ?

θ -

R 5- ! , + ) n → ∞ .

@ g(u) 1/ √

T mh, u > r n 1/ √ T nh.

+ 3 r , r , . . . [0, 1].

? n - +

θ -

( ) *

@ 1)2- @

q4 - - y i,T q T - !

y i,T q 0 q q → ∞

, q- + ,

1 2

''

(13)

4 - +

? 1)***2 6: - @

@ 1)**C2 ,

-

=

y _{i,T q} = α _i + β ^⊤ _i D _{T q} + g(1 + q/T ) + ε _{i,T q} .

+ y i,T q ,

α i β _i g(1 + q/T ) i = 1, . . . , n t ≤ T -

α i β _i g(1 + q/T )

y _{i,T q} g(1 + q/T )- .

y _{T q} = ⁿ _i y i,T q /n

y _{T q} = β ^⊤ D T q + g(1 + q/T ) + ε t , 1'*2 β ^⊤ = ⁿ _i β _i /n, ε _{T q} = ⁿ _i ε _{i,T q} /n.

. 0 { ε it } t F 3 -

g(1 + q/T )

E T y i,T q = α i + β ^⊤ _i D T q + g(1 + q/T ),

E T , -

.

'K i, ε it * E (ε _it ) = σ _i 0 < σ ≤ min ≤i≤n σ i ≤

max ≤i≤n σ i ≤ σ < ∞ -

)K g : [ , + ǫ] → ǫ > 0, τ ≥ p-

( K 1 2 K K ^′ [ − 1, 0]7 1 2 1 ^∗ ( K ) > 0

1 ^∗ ( K )1 ^∗ ( K ) − 1 ^∗ ( K ) > 0 1 ^∗ _j ( K ) = ₋ u ^j K (u)du-

A h "%' ) h h/h → 0 T → ∞

. g(1 + q/T )- = g ( )

)K7 T → ∞ q/T → 0 + , g( )

u = 1 τ 4 1τ = p − 12 g(1 + q/T ) =

τ

k

1 k! g ^k (1) q T

k

+ o q T

τ

=

τ

k

γ _k q T

k

+ o q T

τ

.

')

(14)

T F T - .

y t = n ⁻

n

i

(y it − α i − β ^⊤ _i D t ) = y _t − β ^⊤ D t ,

t n ≤ t ≤ T. K ( ) 4 0 (

T >

T

t

K T − t

T h y _t −

τ

k

γ _k t − T T

k

. 1''2

h A-

. E 1''2

+ -

B( K ) = 1

(τ + 1)! g ^τ (1)



 



1 ^∗ _τ ( K ) 1 ^∗ _τ ( K )

. . . 1 ^∗ _τ ( K )



 



M ( K ) =



 



1 ^∗ ( K ) 1 ^∗ ( K ) . . . 1 ^∗ _τ ( K ) 1 ^∗ ( K ) 1 ^∗ ( K ) 1 ^∗ _τ ( K )

. . . . . . . . . 1 ^∗ _τ ( K ) 1 ^∗ _τ ( K ) . . . 1 ^∗ _τ ( K )



 

 , V ( K ) =



 



ν ^∗ ( K ) ν ^∗ ( K ) . . . ν ^∗ _τ ( K ) ν ^∗ ( K ) ν ^∗ ( K ) ν ^∗ _τ ( K )

. . . . . . . . . ν ^∗ _τ ( K ) . . . ν ^∗ _τ ( K )



 

 ,

1 ^∗ _k ( K ) = ₋ K (u) u ^k du ν ^∗ _j ( K ) = ₋ u ^j K (u)du. D h = diag (1, h, . . . , h ^τ ) .

T 3. # " "$ "+ ^′ ", "% "( "- T → ∞ ,

√ T hD h γ − γ − h ^τ M ( K ) ⁻ B( K ) ⇒ N 0, 1

n σ M( K ) ⁻ V ( K )M ( K ) ⁻ , σ = n ⁻ ⁿ _i σ _i .

+ F (γ , γ , . . . , γ _τ )

h ^τ D h M ( K ) ⁻ B( K ), F

ω D ⁻ _h M ( K ) ⁻ V ( K )M ( K ) ⁻ D _h ⁻ /nT h. + g(1 + q/T ) g(1 + q/T ) =

τ

k

γ _k q T

k

, y _{i,T q}

y _{i,T q} = α _i + β ^⊤ _i D _{T q} + g(1 + q/T ). 1')2

'8

(15)

+ $ y _{T q} = β

⊤

D T q + g(1 + q/T ), 1'82

β = n ⁻ ⁿ _i β ^⊤ _i -

+ - P ^τ = (1, (q/T h), . . . , (q/T h) ^τ ) ^⊤ .

E _T ^∗ , -

T 4. # " "$ "+ ^′ ", "% "( T → ∞

y _{i,T q} &

E ^∗ _T [y i,T q − y i,T q ] = b g = h ^τ "

P τ ^⊤ M ( K ) ⁻ B( K ) + o(1) #

& y _{i,T q} &

E _T ^∗ "

(y i,T q − E T y i,T q ) #

= σ _i + 1 T nh

"

P τ ^⊤ M ( K ) ⁻ V ( K )M ( K ) ⁻ P ^τ + o(1) # σ ,

σ , & y _{T q}

y i,T q & & & y _{T q}

&

E _T ^∗

$

y _{T q} − E _T ^∗ y _{T q}

%

= 1

n 1 + 1 T h

"

P τ ^⊤ M ( K ) ⁻ V ( K )M( K ) ⁻ P ^τ + o(1) # σ .

+ + 8 9 y i,T q

g(1 + q/T )- @

0 q 0 b g : h ^τ B B 0

1τ + 124 M ( K ) ⁻ B( K )- + σ _i + V σ /T nh,

V 1' '24 , M ( K ) ⁻ V ( K )M( K ) ⁻ - y _{T q} - +

q/T h → 0-

@ q → ∞ $

h q/T - @ y i,T q q/T h → 0

0 b g : h ^τ B

σ _i + V σ /T nh B V 0 - @ q/T h → δ ∈ (0, ∞ ),

F b _g : h ^τ ∆ ^⊤ _τ M ( K ) ⁻ B ( K ), ∆ _τ = (1, δ, . . . , δ ^τ ) ^⊤ - + : σ _i + ∆ ^⊤ _τ M ( K ) ⁻ V ( K )M ( K ) ⁻ ∆ τ σ /T nh- @ q/T h → ∞ ,

-

R 4. @ { ε it } t

E T y i,T q = α i + β ^⊤ _i D T q + g(1 + q/T ) + E T ε i,T q ,

'9

(16)

E _T , - ' E _T ε _{i,T q} = 0

1 E T ε i,T q → 0 q → ∞ 2- + E T ε i,T q 0 1

< & , ; 2 ,

- @ y i,t

α i β _i g(t/T ) - -

ε i,t = y i,t − α i − β ^⊤ _i D t − g(t/T )

0 ε i,t <

ε i,T q E T ε i,T q - y i,T q g(1 + q/T )

- - y i,T q = α i + β ^⊤ _i D T q + g(1 + q/T ) + E T ε i,T q .

@ <1'2 ε i,t = ρε i,t− +η _it η _it E T ε i,T q = ρ ^q ε i,T -

< 3

& , ; -

y _{i,T q} y _{T q} 1')2 1'82- 3

ε _i,t 4 1 q → ∞ 2

/ 4 E -

+ ''

! , (T M AX )

(T M IN ) F ,

(T RAN GE)

- +

/ , - ⁾ + 0 'D(8

!, 'D(D -

)

+ >LL - / - - L L L L

'(

(17)

5 '-

@

3 - +

- +

6 < ? F - + F /

- ,

- . - + F -

0 / 0

- <

0 F 0 / - +

- + 0

, - !

0 - . 0 -

! 3

- ",

$ - + -

. , - @ + ' )

θ + % + @=- +

'A

(18)

0 - +

-

. 5 ) 8 - + $

-

3 , - ! F 6

? 1)**A2 , 0 - @ 5

9 ( - +

- @ 5 A B $

- "

- - 'CC(- @

+ % + @= - + 6

K 1 h ≃ 0.05).

, F - +

+ % + @=

-9BABC -9DA) 0 -

. , - .

- . 4 p =

1, 2, . . . , 12, - + 5 D C

-

MMMM5 + ? MMM

, -

@

- + 0

- +

0 0

-

'B

(19)

. '' $

. " " !

P T 1- + 0 15! 2 θ

∂ L (θ)

∂α i

= −

j i T

t t

j

y _jt − α _j − β ^⊤ _j D _t − g _θ (t/T ) ∂g _θ (t/T )

∂α i

−

T

t t

i

y it − α i − β ^⊤ _i D t − g θ (t/T ) 1 + ∂g θ (t/T )

∂α i

= 0

∂ L (θ)

∂β _i = −

j i T

t t

j

y jt − α j − β ^⊤ _j D t − g θ (t/T ) ∂g θ (t/T )

∂β _i

−

T

t t

i

y it − α i − β ^⊤ _i D t − g θ (t/T ) D t + ∂g θ (t/T )

∂β _i = 0,

>

∂g θ (t/T )

∂α _i = − 1 m _t

1 T

T

s t

i

K h ((t − s)/T ) → − _m

_t

, i ≤ m t

#, i > m t

∂g θ (t/T )

∂β _i = − 1 m _t

1 T

T

s t

i

K h ((t − s)/T )D s → − _m

_t

, i ≤ m t

# , i > m t

. + i = 1, . . . , n,

l i T

t t

l



y _lt − α l − β ^⊤ _l D t − 1 m t

1 T

n

j T

s t

j

y js − α j − β ^⊤ _j D s K h ((t − s)/T )



 ∂g θ (t/T )

∂α i

+

T

t t

i



y it − α i − β ^⊤ _i D t − 1 m _t

1 T

n

j T

s t

j

y js − α j − β ^⊤ _j D s K h ((t − s)/T )



 1 + ∂g θ (t/T )

∂α _i = 0

l i T

t t

l



y _lt − α l − β ^⊤ _l D t − 1 m t

1 T

n

j T

s t

j

y js − α j − β ^⊤ _j D s K h ((t − s)/T )



 ∂g θ (t/T )

∂β _i

T

t t

i



y it − α i − β ^⊤ _i D t − 1 m _t

1 T

n

j T

s t

j

y js − α j − β ^⊤ _j D s K h ((t − s)/T )



 D t + ∂g θ (t/T )

∂β _i = 0 y it = α i + β ^⊤ _i D t + g(t/T ) + ε it 5!

y it − α i − β ^⊤ _i D t = ε it + g(t/T ) − (α i − α i ) − β ^⊤ _i − β ^⊤ _i D t ,

'D

(20)

i = 1, . . . , n 5! - - - α _i

l i T

t t

l

∂g θ (t/T )

∂α i

(α l − α l ) +

l i T

t t

l

∂g θ (t/T )

∂α i

D t β ^⊤ _l − β ^⊤ _l

−

j i



 1 T l i

T

t t

l

1 m t





T

s t

j

K h ((t − s)/T ) ∂g θ (t/T )

∂α i







 α _j − α j

− 1 T l i

T

t t

l

1 m t

T

s t

i

K h ((t − s)/T ) ∂g θ (t/T )

∂α i

(α _i − α i )

−

j i

β ^⊤ _j − β ^⊤ _j





l i T

t t

l

1 m t



 1 T

T

s t

j

D s K h ((t − s)/T )



 ∂g θ (t/T )

∂α i





− β ^⊤ _i − β ^⊤ _i

l i T

t t

l

1 m t

1 T

T

s t

i

D s K h ((t − s)/T ) ∂g θ (t/T )

∂α i

+ (α _i − α i )

T

t t

i

1 − 1 m t

1 T

T

s t

i

K h ((t − s)/T ) 1 + ∂g θ (t/T )

∂α i

+ β ^⊤ _i − β ^⊤ _i

T

t t

i

D t − 1 m _t

1 T

T

s t

i

D s K h ((t − s)/T ) 1 + ∂g θ (t/T )

∂α _i

−

n

j i,j

α _j − α j T

t t

i



 1 m _t

1 T

T

s t

j

K h ((t − s)/T )



 1 + ∂g θ (t/T )

∂α _i

−

n

j i,j

β ^⊤ _j − β ^⊤ _j

T

t t

i



 1 m t

1 T

T

s t

j

D s K h ((t − s)/T )



 1 + ∂g θ (t/T )

∂α i

=

l i T

t t

l



ε lt − 1 m t

1 T

n

j T

s t

j

ε js K h ((t − s)/T )



 ∂g θ (t/T )

∂α i

+

l i T

t t

l



g(t/T ) − 1 m t

1 T

n

j T

s t

j

g(s/T )K h ((t − s)/T )



 ∂g θ (t/T )

∂α i

+

T

t t

i

ε it − 1 m t

1 T

T

s t

i

ε is K h ((t − s)/T ) 1 + ∂g θ (t/T )

∂α i

+

T

t t

i



g(t/T ) − 1 m t

1 T

n

j T

s t

j

g(s/T )K h ((t − s)/T )



 1 + ∂g θ (t/T )

∂α i

−

T

t t

i



 1 m t

1 T

n

j i,j T

s t

j

ε _js K _h ((t − s)/T )



 1 + ∂g _θ (t/T )

∂α i

'C

(21)

5! - - - β _i

l i T

t t

l

∂g θ (t/T )

∂β _i (α _l − α l ) +

l i T

t t

l

∂g θ (t/T )

∂β _i D t β ^⊤ _l − β ^⊤ _l

−

j i



 1 T l i

T

t t

l

1 m _t





T

s t

j

K h ((t − s)/T ) ∂g θ (t/T )

∂β _i







 α _j − α j

− 1 T l i

T

t t

l

1 m t

T

s t

i

K _h ((t − s)/T ) ∂g θ (t/T )

∂β _i (α _i − α _i )

−

j i

β ^⊤ _j − β ^⊤ _j





l i T

t t

l

1 m t



 1 T

T

s t

j

D s K h ((t − s)/T )



 ∂g θ (t/T )

∂β _i





− β ^⊤ _i − β ^⊤ _i

l i T

t t

l

1 m t

1 T

T

s t

i

D _s K _h ((t − s)/T ) ∂g _θ (t/T )

∂β _i + (α i − α i )

T

t t

i

1 − 1 m _t

1 T

T

s t

i

K h ((t − s)/T ) D t + ∂g θ (t/T )

∂β _i + β ^⊤ _i − β ^⊤ _i

T

t t

i

D t − 1 m t

1 T

T

s t

i

D s K h ((t − s)/T ) D t + ∂g θ (t/T )

∂β _i

−

n

j i,j

(α j − α j )

T

t t

i



 1 m t

1 T

T

s t

j

K h ((t − s)/T )



 D t + ∂g θ (t/T )

∂β _i

−

n

j i,j

β ^⊤ _j − β ^⊤ _j

T

t t

i



 1 m t

1 T

T

s t

j

D s K h ((t − s)/T )



 D t + ∂g θ (t/T )

∂β _i

=

l i T

t t

l



ε _lt − 1 m t

1 T

n

j T

s t

j

ε js K h ((t − s)/T )



 ∂g θ (t/T )

∂β _i +

l i T

t t

l



g(t/T ) − 1 m t

1 T

n

j T

s t

j

g(s/T )K h ((t − s)/T )



 ∂g θ (t/T )

∂β _i +

T

t t

i

ε _it − 1 m t

1 T

T

s t

i

ε _is K _h ((t − s)/T ) D _t + ∂g θ (t/T )

∂β _i +

T

t t

i



g(t/T ) − 1 m t

1 T

n

j T

s t

j

g(s/T )K h ((t − s)/T )



 D t + ∂g θ (t/T )

∂β _i

−

T

t t

i



 1 m t

1 T

n

j i,j T

s t

j

ε js K h ((t − s)/T )



 D t + ∂g θ (t/T )

∂β _i .

)*

(22)

@

C T,a =



 



C a, . . . C a, n

. . . . . . C a,n . . . C a,nn



 

 , C T,b =



 



C b, . . . C b, n

. . . . . . C b,n . . . C b,nn



 



C _T,A =



 



C _A, . . . C _{A, n} . . . . . . C A,n . . . C A,nn



 

 , C _T,B =



 



C _B, . . . C _{B, n} . . . . . . C B,n . . . C B,nn



 



d a =



 

 d a,

. . . d a,n



 

 d A =



 

 d A,

. . . d A,n



 

 e a =



 

 e a,

. . . e a,n



 

 e A



 

 e A,

. . . e A,n



 



C _a,ii = 1 T

 



T

t t

i

1 − _m

_t

_T ^T s t

i

K h ((t − s)/T ) 1 + ^∂g

^θ

_∂α ^t/T

i

− _T l i T t t

l

m

t

T

s t

i

K _h ((t − s)/T ) ^∂g

^θ

_∂α ^t/T

i

 

 C a,ij = 1

T





T t t

j

∂g

θ

t/T

∂α

i

− T l i

T t t

l

m

t

T

s t

j

K h ((t − s)/T ) ^∂g

^θ

_∂α ^t/T

i

− ^T t t

i

m

t

T T

s t

j

K h ((t − s)/T ) 1 + ^∂g

^θ

_∂α ^t/T

i





C b,ii = 1 T

 



T

t t

i

D _t ^⊤ − _m

_t

_T ^T s t

i

D _s ^⊤ K h ((t − s)/T ) 1 + ^∂g

^θ

_∂α ^t/T

i

− l i T

t t

l

m

t

T T

s t

i

D ^⊤ _s K h ((t − s)/T ) ^∂g

^θ

_∂α ^t/T

i

 

 C b,ij = 1

T





T

t t

j

D ^⊤ _t ^∂g

^θ

_∂α ^t/T

i

− T l i

T t t

l

m

t

T

s t

j

D ^⊤ _s K h ((t − s)/T ) ^∂g

^θ

_∂α ^t/T

i

− ^T t t

i

m

t

T T

s t

j

D s K h ((t − s)/T ) 1 + ^∂g

^θ

_∂α ^t/T

i





d a,i = 1

√ T _{l i}

T

t t

j



g(t/T ) − 1 m t

1 T

n

j T

s t

j

g(s/T )K h ((t − s)/T )



 ∂g _θ (t/T )

∂α i

+

T

t t

i



g(t/T ) − 1 m _t

1 T

n

j T

s t

j

g(s/T )K h ((t − s)/T )



 1 + ∂g θ (t/T )

∂α _i

)'

(23)

e a,i = 1

√ T

T

t t

i

ε it − 1 m t

1 T

T

s t

i

ε is K h ((t − s)/T ) 1 + ∂g θ (t/T )

∂α i

− 1

√ T

T

s t

i

l i T

t t

l

1 m t

1 T K _h ((t − s)/T ) ∂g θ (t/T )

∂α i

ε _is

+ 1

√ T _{j i}





T

t t

j

∂g θ (t/T )

∂α i



 ε jt −

n

j i,j

1 T

T

s t

j

l i T

t t

l

1 m t

K h ((t − s)/T ) ∂g θ (t/T )

∂α i

ε js

− 1

√ T

n

j i,j

1 T

T

s t

j

T

t t

i

1 m t

K h ((t − s)/T ) 1 + ∂g θ (t/T )

∂α i

ε js

C A,ii = 1 T

 



T

t t

i

1 − m

t

T T

s t

i

K h ((t − s)/T ) D t + ^∂g

^θ

_∂β ^t/T

i

− T l i T t t

l

m

t

T

s t

i

K h ((t − s)/T ) ^∂g

^θ

_∂β ^t/T

i

 

 C A,ij = 1

T





T t t

j

∂g

θ

t/T

∂β

i

− _T l i

T t t

l

m

t

T

s t

j

K h ((t − s)/T ) ^∂g

^θ

_∂β ^t/T

i

− ^T t t

i

m

t

T T

s t

j

K h ((t − s)/T ) D t + ^∂g

^θ

_∂β ^t/T

i





C _B,ii = 1 T

 



T

t t

i

D _t ^⊤ − _m

_t

_T ^T s t

i

D _s ^⊤ K h ((t − s)/T ) D t + ^∂g

^θ

_∂β ^t/T

i

− l i T

t t

l

m

t

T T

s t

i

D _s ^⊤ K h ((t − s)/T ) ^∂g

^θ

_∂β ^t/T

i

 

 C B,ij = 1

T





T

t t

j

D ^⊤ _t ∂g θ (t/T )

∂β _i − 1 T l i

T

t t

l

1 m t





T

s t

j

D _s ^⊤ K h ((t − s)/T ) ∂g θ (t/T )

∂β _i









− 1 T





T

t t

i



 1 m t

1 T

T

s t

j

D _s K _h ((t − s)/T )



 D _t + ∂g θ (t/T )

∂β _i





d _A,i = 1

√ T _{l i}

T

t t

j



g(t/T ) − 1 m t

1 T

n

j T

s t

j

g(s/T )K _h ((t − s)/T )



 ∂g θ (t/T )

∂β _i +

T

t t

i



g(t/T ) − 1 m t

1 T

n

j T

s t

j

g(s/T )K h ((t − s)/T )



 D t + ∂g θ (t/T )

∂β _i

))

(24)

e A,i = 1

√ T

T

t t

i

ε it − 1 m t

1 T

T

s t

i

ε is K h ((t − s)/T ) D t + ∂g θ (t/T )

∂β _i

− 1

√ T

T

s t

i

l i T

t t

l

1 m t

1 T K _h ((t − s)/T ) ∂g θ (t/T )

∂α i

ε _is

+ 1

√ T _{j i}





T

t t

j

∂g θ (t/T )

∂β _i



 ε jt −

n

j i,j

1 T

T

s t

j

l i T

t t

l

1 m t

K h ((t − s)/T ) ∂g θ (t/T )

∂β _i ε js

− 1

√ T

n

j i,j

1 T

T

s t

j

T

t t

i

1 m t

K h ((t − s)/T ) D t + ∂g θ (t/T )

∂β _i ε js ,

C T,a C T,b

C T,A C T,B





√ T (α − α)

√ T β − β



 = d a

d A

+ e a

e A

.

C T = C T,a C T,b

C T,A C T,B

d T = d a

d A

e T = e a

e A

,

5! >

C T

√ T θ − θ = d T + e T . 1'92

+ 0 $ q ^⊤ θ = 0 0

√ T θ − θ = R R ^⊤ C _T R ⁻ R ^⊤ d _T + R R ^⊤ C _T R ⁻ R ^⊤ e _T ,

R K × (K − 1) E q.

& ' ) T → ∞ :

C T,a ⇒



 

 

c . . . c i . . . c n

c i c ii c in

c n c ni c nn



 

 

= ∆ n + G n = C n ,

)8

(25)

C T,b → C n ⊗ 1 12

⊤

=



 

 

c . . . c i . . . c n

c i c ii c in

c n c ni c nn



 

 

⊗ 1 12

⊤ = (∆ n + G n ) ⊗ 1 12

⊤ ,

C T,A → C n ⊗ 1 12

=



 

 

c c i c n

c i c ii c in

c n c ni c nn



 

 

⊗ 1

12 = (∆ n + G n ) ⊗ 1 12 ,

C _T,B → ∆ _n ⊗ 1

12 I + G _n ⊗ 1 12

⊤ . +

C T → Q = ∆ n + G n (∆ n + G n ) ⊗

(∆ n + G n ) ⊗ ^⊤ ∆ n ⊗ I + G n ⊗ ^⊤ .

& 8

d a

d A

= − √

T h ^p b

b ⊗ + o( √ T h ^p )

b = b , . . . , b i , . . . , b n

⊤

b i = 1

p! 1 _p (K)





l i





r

l

δ(s)g ^p (s) ds



 −





r

i

w(s)g ^p (s) ds







 ,

w(s) δ(s) N* 'O :

δ(s) = 1

j r j < s < r j j = 1, 2, . . . , n.

w(s) = 1 − δ(s) = 1 − 1

j r j < s < r j j = 1, 2, . . . , n.

)9

(26)

& 9 e T

,

= = ⁿ + A n [ n + A n ] ⊗ ^⊤

[ n + A n ] ⊗ ^⊤ S n ⊗ I + A n ⊗ J .

P T 2 - 5

g P (u) =

T ⁻ ⁿ _i ^T _{s t}

_i

y is − α i − β ^⊤ _i D s K h (u − s/T ) T ⁻ ⁿ _i ^T _{s t}

_i

K h (u − s/T )

@ ^t _T

^m

< u < ^t

^m

_T , ⁿ _i ^T _{t t}

_i

K h (u − t/T )/T = ^m _i ^T _{t t}

_i

K([u − t/T ] /h)/T h = m. + g _P (u) = 1

T m

n

i T

t t

i

K _h (u − t/T ) y _it − α _i − β ^⊤ _i D _t

= 1

T mh

m

i T

t t

i

K(u − t/T ) y it − α i − β ^⊤ _i D t

= 1

T mh

m

i T

t t

i

K([u − t/T ] /h) y it − α i − β ^⊤ _i D t − (α i − α i ) − β ^⊤ _i − β ^⊤ _i D t

= 1

T mh

m

i T

t t

i

K([u − t/T ] /h) g(t/T ) + ε it − (α i − α i ) − β ^⊤ _i − β ^⊤ _i D t

= 1

T mh

m

i T

t t

i

K([u − t/T ] /h)g(t/T ) + 1 T mh

m

i T

t t

i

K([u − t/T ] /h)ε it

− 1 T mh

m

i T

t t

i

K ([u − t/T ] /h) (α i − α i )

− 1 T mh

m

i T

t t

i

K ([u − t/T ] /h) β ^⊤ _i − β ^⊤ _i D t

5 0

1 T mh

m

i T

t t

i

K([u − t/T ] /h)ε it = 1 m

m

i

1 T h

T

t t

i

K([u − t/T ] /h)ε it

,

i _t K([u − t/T ] /h)ε it

+

√ 1 T h

T

t t

i

K([u − t/T ] /h)ε _it ⇒ ω _i || K || ^/ ξ _i .

)(

(27)

+ g(u) 1

T mh

m

i T

t t

i

K([u − t/T ] /h)g(t/T )

= 1 m

m

i

1 T h

T

t t

i

K ([u − t/T ] /h)g(t/T )

= 1 m

m

i

1 T h

T

t t

i

K ([u − t/T ] /h) g(u) +

p

j

1 j! h ^j u − t/T h

j

g ^j (u) ,

+ o(h ^p )

= 1 m

m

i

g(u) + 1

p! h ^p g ^p (u) z ^p K(z)dz + o(h ^p ).

= g(u) + 1

p! h ^p g ^p (u) z ^p K(z)dz + o(h ^p ).

5 1 T mh

m

i T

t t

i

K([u − t/T ] /h) (α i − α i ) = o p

√ 1 T h , 1

T mh

m

i T

t t

i

K([u − t/T ] /h) β ^⊤ _i − β ^⊤ _i D t = o p

√ 1 T h ,

θ F 0 -

+ t m /T < u < t m /T m = 1, . . . , n − 1

√ T h [g ^∗ (u) − g(u) − h ^p b(u)] ⇒ N 0, 1 m

1 m

m

i

ω _i || K || - 5 u > t n /T

√ T h [g ^∗ (u) − g(u) − h ^p b(u)] ⇒ N 0, 1 n

1 n

n

i

ω _i || K || .

P T 3 4. = q/T → 0 T → ∞

) ^′ + ,

g(1 + q/T ) =

τ

k

1 k! g ^k (1) q T

k

+ o q T

τ

=

τ

k

γ _k q T

k

+ o q T

τ

.

+ T >

T

t

K T − t

T h y _t − γ ^⊤ x t , γ =



 

 γ

--- γ _τ



 

 x t =



 

 1

---

t−T T

τ



 

 .

)A

(28)

+

γ = γ +

T

t

K T − t

T h x t x ^⊤ _t

− T

t

K T − t T h x t ε t

−

T

t

K T − t

T h x _t x ^⊤ _t

− T

t

K T − t

T h x _t β ^⊤ − β ^⊤ D _t +

T

t

K T − t

T h x t x ^⊤ _t

−

1 (τ + 1)! g ^τ (1)

T

t

K T − t T h x t

t − T T

τ

.

& %1)**D2

γ = γ +

T

t

K T − t

T h x t x ^⊤ _t

− T

t

K T − t T h x t ε t

+

T

t

K T − t

T h x _t x ^⊤ _t

− h ^τ

(τ + 1)! g ^τ (1)

T

t

K T − t

T h x _t t − T T h

τ

+ o _p ((T h) ⁻ ^/ + h ^τ )

= (

1 T h

T

t

K t − T T h

t − T T h

k

→

− K (u) u ^k du = 1 ^∗ _k ( K ),

1 T h

T

t

K T − t

T h x t x ^⊤ _t →



 



1 ^∗ ( K ) 1 ^∗ ( K ) . . . 1 ^∗ _τ ( K ) 1 ^∗ ( K ) 1 ^∗ ( K ) 1 ^∗ _τ ( K )

. . . . . . . . . 1 ^∗ _τ ( K ) 1 ^∗ _τ ( K ) . . . 1 ^∗ _τ ( K )



 

 = M ( K ).

= T

T i

√ 1 T h

T

t

K T − t T h

t − T T h

k 1

n

i

ε it ⇒ 1 n

n

i

N 0, ω _i ν ^∗ _k ( K ) = N 0, ν ^∗ _k ( K ) n

n

i

ω _i ,

√ 1 T h

T

t

K t − T

T h x t ε t ⇒ N



 

 0,

n i ω _i

n



 



ν ^∗ ( K ) ν ^∗ ( K ) . . . ν ^∗ _τ ( K ) ν ^∗ ( K ) ν ^∗ ( K ) ν ^∗ _τ ( K )

. . . . . . . . . ν ^∗ _τ ( K ) . . . ν ^∗ _τ ( K )



 





 

 = N 0, 1

n ω V ( K ) ,

)B

(29)

ω = ⁿ _i ω _i /n,

" 1

√ T h

T

t

K T − t T h

t − T T h

k

ε _it 1

√ T h

T

s

K T − s T h

s − T T h

l

ε _is

→

∞

j −∞

γ _ε

_i

(j ) K (u) u ^{l k} du = ω _i ν ^∗ _{l k} ( K ).

+ 1 T h

T

t

K T − t

T h x t x ^⊤ _t

−

√ 1 T h

T

t

K T − t

T h x t ε t ⇒ M ( K ) ⁻ N 0, 1

n ω V ( K )

= N 0, 1

n ω M( K ) ⁻ V ( K )M( K ) ⁻ .

1 (τ + 1)! g ^τ (1) 1 T h

T

t

K T − t

T h x _t t − T T h

τ

→ 1

(τ + 1)! g ^τ (1)



 



1 ^∗ _τ ( K ) 1 ^∗ _τ ( K )

. . . 1 ^∗ _τ ( K )



 

 = B( K ).

+ √

T h γ − γ − h ^τ M( K ) ⁻ B( K ) ⇒ N 0, 1

n ω M( K ) ⁻ V ( K )M( K ) ⁻ .

=

γ _k − γ _k = h ^τ−k B k + 1

√ T h ^k ^/ U k , g(1 + q/T)

g(1 + q/T ) =

τ

k

γ _k q T

k . +

g(1 + q/T ) − g(1 + q/T )

=

τ

k

h ^τ−k q T

k

B k + o q T

τ

+

τ

k

√ 1

T h ^k ^/ q T

k

U k + o q T

τ

. +

b g =

τ

k

h ^τ−k q T

k

B k = h ^τ

τ

k

q T h

k

B k

v g =

τ

k

√ 1

T h ^k ^/ q T

k

U k = 1

√ T h

τ

k

q T h

k

U k .

)D

(30)

$ h q/T - @

y i,T q − y i,T q = ε i,T q − (α i − α i ) − β ^⊤ _i − β ^⊤ _i D T q − [g(1 + q/T ) − g(1 + q/T )] , 0, q

ASemiparametricPanelModelforUnbalancedDatawithApplicationtoClimateChangeintheUnitedKingdom Atak,AlevandLinton,OliverB.andXiao,Zhijie MunichPersonalRePEcArchive

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