Appendix Research OKUI, Ryo auit appendix v2

Appendix to “Asymptotically unbiased estimation of

autocovariances and autocorrelations for panel data with

incidental trends”

Ryo Okui

^∗

February 3, 2011

1 Proof of Theorem 1

Let xt= (1, t)^′, βi= (ηi, δi)^′and ˆβi= (ˆηi, ˆδi)^′. Then we can write ˜yit= yit_−x^′_tβ^ˆi = x^′_t(βi_{− ˆ}βi)+wit, and:

ˆ

γk = ¹

N (T − k)

N

∑

i=1 T

∑

t=k+1

˜ yity˜i,t−k

= ¹

N (T − k)

N

∑

i=1 T

∑

t=k+1

witwi,t−k+ ¹ N (T − k)

N

∑

i=1 T

∑

t=k+1

x^′_t(βi− ˆ^βi)wi,t−k

+ ¹

N (T − k)

N

∑

i=1 T

∑

t=k+1

x^′_t−k(βi− ˆ^βi)wit+ ¹ N (T − k)

N

∑

i=1 T

∑

t=k+1

(βi− ˆ^βi)^′xtx^′_t−k(βi− ˆ^βi).

Lemma 1 in Okui (2010) gives: 1 N (T − k)

N

∑

i=1 T

∑

t=k+1

witwi,t−k→pγk

and

√N T

( 1

N (T − k)

N

∑

i=1 T

∑

t=k+1

witwi,t−k_{− γ}k

)

→^dN



0,

∞

∑

j=−∞

{γ_j²+ γk+jγk−j+ cum(0, −k, j, j − k)^}



.

Next, we consider the second term in the expansion of ˆγk. We have: 1

N (T − k)

N

∑

i=1 T

∑

t=k+1

x^′_t(βi− ˆβi)wi,t−k _{= −}

1 N

N

∑

i=1

1 T − k

( _T

∑

t=k+1

xtwi,t−k

)^′( _T

∑

t=1

xtx^′_t

)⁻¹( _T

∑

t=1

xtwi,t

) .

∗Institute of Economic Research, Kyoto University, Yoshida-Hommachi, Sakyo, Kyoto, Kyoto, 606-8501, Japan. Email: okui@kier.kyoto-u.ac.jp

1

Simple matrix algebra shows that: ( _T

∑

t=k+1

xtwi,t−k

)^′( _T

∑

t=1

xtx^′_t

)⁻¹( _T

∑

t=1

xtwi,t

)

= ⁴ T

( _T

∑

t=1

wit

) (_{T −k}

∑

t=1

wit

)

−_{T (T + 1)}⁶ ( _T

∑

t=1

twit

) (_{T −k}

∑

t=1

wit

)

−_{T (T + 1)}⁶ ( _T

∑

t=1

wit

) (_{T −k}

∑

t=1

twit

)

+ ¹²

T (T + 1)(T − 1) ( _T

∑

t=1

twit

) (_{T −k}

∑

t=1

twit

) .

Let ¯wi =^∑T_t=1wit/T and ¯zi =^∑T_t=1twit/T². Then, because ¯wi = Op(1/^√T ), ¯zi = Op(1/^√T ) and wit’s are i.i.d. over individuals and have finite fourth moments, we have that:

1 N

N

∑

i=1

1 T − k

( _T

∑

t=k+1

xtwi,t−k

)^′( _T

∑

t=1

xtx^′_t

)⁻¹( _T

∑

t=1

xtwi,t

)

= ¹ N

N

∑

i=1

T T − k^{(4 ¯w}

2

i − 12 ¯^wiz¯i+ 12¯z_i²) + O ( 1

T^3/2 )

.

We now note that:

E( ¯w²_i) = γ0+ ¹ T

T −1

∑

j=0

2^{T − j} T ^γj=

VT

T ^,

E( ¯wiz¯i) = ¹ T³

T

∑

t=1

t

T

∑

j=1

γ|t−j| =¹ 2

VT

T ^{+ o} ( 1

T )

,

E(¯z_i²) = ¹ T⁴

T

∑

t=1

t

T

∑

j=1

jγ|t−j| =¹ 3

VT

T ^{+ o} ( 1

T )

.

Therefore, we have that:

E





 1 N

N

∑

i=1

1 T − k

( _T

∑

t=k+1

xtwi,t−k

)^′( _T

∑

t=1

xtx^′_t

)⁻¹( _T

∑

t=1

xtwi,t

)^





= 4^VT T ^{− 12}

1 2

VT

T ^{+ 12} 1 3

VT

T ^{+ o} ( 1

T )

= 2^VT T ^{+ o}

( 1 T

) .

Moreover, because wit have finite fourth moments by assumption, we have var( ¯w²_i) = O⁽_T¹2) as well as var( ¯wi¯zi) = O⁽_T¹2) and var(¯z_i²) = O⁽_T¹2). It therefore follows that:

var





 1 N

N

∑

i=1

1 T − k

( _T

∑

t=k+1

xtwi,t−k

)^′( _T

∑

t=1

xtx^′_t

)⁻¹( _T

∑

t=1

xtwi,t

)^





= o ( ₁

N T )

.

Therefore, we have that: 1

N

N

∑

i=1

1 T − k

( _T

∑

t=k+1

xtwi,t−k

)^′( _T

∑

t=1

xtx^′_t

)⁻¹( _T

∑

t=1

xtwi,t

)

= ²

T^{VT + op} ( ₁

√N T )

,

and

1 N (T − k)

N

∑

i=1 T

∑

t=k+1

x^′_t(βi_{− ˆ}βi)wi,t−k_{= −}²

T^{VT+ op} ( 1

√N T )

,

2

when N/T → c. Similarly, we have:

1 N (T − k)

N

∑

i=1 T

∑

t=k+1

x^′_t−k(βi_{− ˆ}βi)wit_{= −}²

T^{VT + o} ( 1

√N T )

,

and

1 N (T − k)

N

∑

i=1 T

∑

t=k+1

(βi− ˆ^βi)^′xtx^′_t−k(βi− ˆ^βi) = ² T^{VT + o}

( ₁

√N T )

.

To sum up, we get:

√N T (

ˆ

γk− γk+ ² T^VT

)

→dN



0,

∞

∑

j=−∞

{γ²_j+ γk+jγk−j+ cum(0, −k, j, j − k)^}



.

References

Okui, R. (2010). Asymptotically unbiased estimation of autocovariances and autocorrelations with long panel data, Econometric Theory 26: 1263–1304.

3