Serially Correlated Errors (Time Series Data):

(1)

Serially Correlated Errors (Time Series Data):

• Suppose that u

₁

, u

₂

, · · · , u

_n

are serially correlated.

Consider the case where the subscript represents time.

Remember that β

GM M

∼ N (

β, σ

²

(X

⁰

Z(Z

⁰

Ω Z)

⁻¹

Z

⁰

X)

⁻¹

) , We need to consider evaluation of σ

²

Z

⁰

Ω Z = V(u

^∗

), i.e.,

V(u

^∗

) = V(Z

⁰

u) = V(

∑

n i=1

z

⁰_i

u

i

) = V(

∑

n i=1

v

i

)

= E ( (

∑

n i=1

v

i

)(

∑

n i=1

v

i

)

⁰

)

= E ( (

∑

n i=1

v

i

)(

∑

n j=1

v

j

)

⁰

)

= E(

∑

n i=1

∑

n j=1

v

_i

v

⁰_j

) =

∑

n i=1

∑

n j=1

E(v

_i

v

⁰_j

) where v

_i

= z

⁰_i

u

_i

is a r × 1 vector.

145

(2)

Define Γ

_τ

= E(v

_i

v

⁰_i−τ

).

Γ

0

= E(v

_i

v

⁰_i

) represents the r × r variance-covariance matrix of v

_i

. Γ

_−τ

= E(v

_i_−τ

v

⁰_i

) = E((v

_i

v

⁰_i−τ

)

⁰

) = (

E(v

_i

v

⁰_i−τ

) )

₀

= Γ

⁰_τ

. V(u

^∗

) =

∑

n i=1

∑

n j=1

E(v

_i

v

⁰_j

)

= E(v

₁

v

⁰₁

) + E(v

₁

v

⁰₂

) + E(v

₁

v

⁰₃

) + · · · + E(v

₁

v

⁰_n

) + E(v

₂

v

⁰₁

) + E(v

₂

v

⁰₂

) + E(v

₂

v

⁰₃

) + · · · + E(v

₂

v

⁰_n

) + E(v

₃

v

⁰₁

) + E(v

₃

v

⁰₂

) + E(v

₃

v

⁰₃

) + · · · + E(v

₃

v

⁰_n

)

...

+ E(v

_n

v

⁰₁

) + E(v

_n

v

⁰₂

) + E(v

_n

v

⁰₃

) + · · · + E(v

_n

v

⁰_n

)

= Γ

0

+ Γ

−1

+ Γ

−2

+ · · · + Γ

1−n

+ Γ

1

+ Γ

0

+ Γ

−1

+ · · · + Γ

2−n

146

(3)

+ Γ

2

+ Γ

1

+ Γ

0

+ · · · + Γ

3−n

...

+ Γ

n−1

+ Γ

n−2

+ Γ

n−3

+ · · · + Γ

0

= Γ

0

+ Γ

⁰₁

+ Γ

⁰₂

+ · · · + Γ

⁰_n−1

+ Γ

1

+ Γ

0

+ Γ

⁰₁

+ · · · + Γ

⁰_n−2

+ Γ

2

+ Γ

1

+ Γ

0

+ · · · + Γ

⁰_n−3

...

+ Γ

n−1

+ Γ

n−2

+ Γ

n−3

+ · · · + Γ

0

= n Γ

0

+ (n − 1)( Γ

1

+ Γ

⁰1

) + (n − 2)( Γ

2

+ Γ

⁰2

) + · · · + ( Γ

n−1

+ Γ

⁰n−1

)

= n Γ

0

+

n−1

∑

i=1

(n − i)( Γ

i

+ Γ

⁰_i

)

147

(4)

= n ( Γ

0

+

n−1

∑

i=1

(1 − i

n )( Γ

i

+ Γ

⁰_i

) )

≈ n ( Γ

0

+

∑

q

i=1

(1 − i

q + 1 )( Γ

i

+ Γ

⁰i

) ) .

In the last line, n − 1 is replaced by q, where q < n − 1.

We need to estimate Γ

_τ

as: Γ ˆ

_τ

= 1 n

∑

n i=τ+1

ˆ

v

_i

v ˆ

⁰_i_−τ

, where ˆ v

_i

= z

⁰_i

u ˆ

_i

for ˆ u

_i

= y

_i

− x

_i

β

GM M

. As τ is large, ˆ Γ

τ

is unstable.

Therefore, we choose the q which is less than n − 1.

148