Stock, J.H. (1987) “Asymptotic Properties of Least Squares Estimators of Cointegrating Vectors,” Econometrica, Vol.55, pp.1035 – 1056.

Stock, J.H. (1987) “Asymptotic Properties of Least Squares Estimators of Cointegrating Vectors,” Econometrica, Vol.55, pp.1035 – 1056.

Proposition:

Let y

be a scalar, y

be a k × 1 vector, and (y

, y

)

be a g × 1 vector, where g = k + 1.

Consider the following model:

y

= α + γ

y

+ z

∆ y

= u

( z

u

)

= Ψ

(L)

is a g × 1 i.i.d. vector with E(

) = 0 and E(

) = PP

. OLSE is given by:

( α ˆ γ ˆ )

=

( T ∑

y

∑ y

∑ y

y

)

( ∑ y

∑ y

y

)

.

Define λ

, which is a g × 1 vector, and Λ

, which is a k × g matrix, as follows:

Ψ

(1) P = ( λ

Λ

)

.

Then, we have the following results:

( T

( ˆ α − α ) T ( ˆ γ − γ )

)

−→

 



1

( Λ

∫

W(r)dr )

Λ

∫

W(r)dr Λ

(∫

(W(r)) (W(r))

dr )

Λ

 



−1

( h

h

)

, where

( h

h

)

=

 



λ

W(1) Λ

(∫

W(r) (dW(r))

)

λ

+ ∑

τ=0

E(u

z

)

 

 .

W(r) denotes a g-dimensional standard Brownian motion.

1) OLSE of the cointegrating vector is consistent even though u

is serially

correlated.

2) The consistency of OLSE implies that T

∑

ˆu

−→ σ

. 3) Because T

∑

(y

− y

)

goes to infinity, a coe ffi cient of determination,

R

, goes to one.

3.4 Testing Cointegration

3.4.1 Engle-Granger Test y

∼ I(1)

y

= α + γ

y

+ u

• u

∼ I(0) = ⇒ Cointegration

• u

∼ I(1) = ⇒ Spurious Regression

Estimate y

= α + γ

y

+ u

by OLS, and obtain ˆu

.

Estimate ˆu

= ρ ˆu

+ δ

∆ ˆu

+ δ

∆ ˆu

+ · · · + δ

∆ ˆu

+ e

by OLS.

ADF Test:

• H

: ρ = 1 (Sprious Regression)

• H

: ρ < 1 (Cointegration)

= ⇒ Engle-Granger Test

For example, see Engle and Granger (1987), Phillips and Ouliaris (1990) and Hansen

(1992).

Asymmptotic Distribution of Residual-Based ADF Test for Cointegration

# of Refressors, (a) Regressors have no drift (b) Some regressors have drift

excluding constant 1% 2.5% 5% 10% 1% 2.5% 5% 10%

1 − 3 . 96 − 3 . 64 − 3 . 37 − 3 . 07 − 3 . 96 − 3 . 67 − 3 . 41 − 3 . 13

2 − 4 . 31 − 4 . 02 − 3 . 77 − 3 . 45 − 4 . 36 − 4 . 07 − 3 . 80 − 3 . 52

3 − 4 . 73 − 4 . 37 − 4 . 11 − 3 . 83 − 4 . 65 − 4 . 39 − 4 . 16 − 3 . 84

4 − 5 . 07 − 4 . 71 − 4 . 45 − 4 . 16 − 5 . 04 − 4 . 77 − 4 . 49 − 4 . 20

5 − 5 . 28 − 4 . 98 − 4 . 71 − 4 . 43 − 5 . 36 − 5 . 02 − 4 . 74 − 4 . 46

J.D. Hamilton (1994), Time Series Analysis, p.766.

3.4.2 Error Correction Representation

VAR(p) model:

y

= α + φ

y

+ φ

y

+ · · · + φ

y

+

,

where y

, α and

indicate g × 1 vectors for t = 1 , 2 , · · · , T , and φ

is a g × g matrix

for s = 1 , 2 , · · · , p.

Rewrite:

y

= α + ρ y

+ δ

∆ y

+ δ

∆ y

+ · · · + +δ

∆ y

+

, where

ρ = φ

+ φ

+ · · · + φ

,

δ

= − ( φ

+ δ

+ · · · + φ

) , for s = 1 , 2 , · · · , p − 1.

Again, rewrite:

∆ y

= α + δ

y

+ δ

∆ y

+ δ

∆ y

+ · · · + +δ

∆ y

+

, where

δ

= ρ − I

= −φ (1) ,

for φ (L) = I

− δ

L − δ

L

− · · · − δ

L

.

If y

has h cointegrating relations, we have the following error correction represen- tation:

∆ y

= α − BA

y

+ δ

∆ y

+ δ

∆ y

+ · · · + +δ

∆ y

+

,

where A

y

is a stationary h × 1 vector (i.e., h I(0) processes), and B and A are g × h matrices.

Note that φ (1) = BA

for φ (L) = I

− δ

L − δ

L

− · · · − δ

L

.

Each row of A

denotes the cointegrating vector, i.e., A

consists of h cointegrating

vectors.

Suppose that

∼ N(0 , Σ ). The log-likelihood function is:

log l( α, δ

, · · · , δ

, B | A)

= − T g

2 log(2 π ) − T

2 log |Σ|

− 1 2

∑

( ∆ y

− α + BA

y

− δ

∆ y

− · · · − δ

∆ y

)

Σ

× ( ∆ y

− α + BA

y

− δ

∆ y

− · · · − δ

∆ y

) Given A and h, maximize log l with respect to α, δ

, · · · , δ

, B.

Then, given h, how do we estimate A? = ⇒ Johansen (1988, 1991)

(*) Canonical Correlatoion (正準相関)

x

= (x

, x

, · · · , x

) and y

= (y

, y

, · · · , y

), where n ≤ m.

u = a

x = a

x

+ a

x

+ · · · + a

x

, v = b

y = b

y

+ b

y

+ · · · + b

y

, where V(u) = V(v) = 1 and E(x) = E(y) = 0 for simplicity.

Define:

V(x) = Σ

, E(xy

) = Σ

, V(y) = Σ

, E(yx

) = Σ

= Σ

.

The correlation coe ffi cient between u and v, denoted by ρ , is:

ρ = Cov(u , v)

√ V(u) √

V(v) = a

Σ

b , where V(u) = a

Σ

a = 1 and V(v) = b

Σ

b = 1.

Maximize ρ = a

Σ

b subject to a

Σ

a = 1 and b

Σ

b = 1.

The Lagrangian is:

L = a

Σ

b − 1

2 λ (a

Σ

a − 1) − 1

2 µ (b

Σ

b − 1) .

Take a derivative with respect to a and b.

∂ L

∂ a = Σ

b − λΣ

a = 0 ,

∂ L

∂ b = Σ

a − µΣ

b = 0 . Using a

Σ

a = 1 and b

Σ

b = 1, we obtain:

λ = µ = a

Σ

b . From the first equation, we obtain:

a = 1

λ Σ

Σ

b ,

which is substituted into the second equation as follows:

1

λ Σ

Σ

Σ

b − λΣ

b = 0 , i.e.,

( Σ

Σ

Σ

Σ

− λ

I

)b = 0 , i.e.,

|Σ

Σ

Σ

Σ

− λ

I

| = 0 .

The solution of λ

is given by the maximum eigen value of Σ

Σ

Σ

Σ

, and b is

the corresponding eigen vector.

Back to the Cointegration:

Estimate the following two regressions:

∆ y

= b

+ b

∆ y

+ b

∆ y

+ · · · + b

∆ y

+ u

y

= b

+ b

∆ y

+ b

∆ y

+ · · · + b

∆ y

+ u

Obtain ˆu

for i = 1 , 2 and t = 1 , 2 , · · · , T , and compute as follow:

Σ ˆ

= 1 T

∑

ˆu

ˆu

, Σ ˆ

= 1 T

∑

ˆu

ˆu

, Σ ˆ

= 1

T

∑

ˆu

ˆu

, Σ ˆ

= Σ ˆ

.

From ˆ Σ

Σ ˆ

Σ ˆ

Σ ˆ

, compute h biggest eigenvalues, denoted by ˆ λ

, ˆ λ

, · · · , ˆ λ

, and the corresponding eigen vectors, denoted by ˆa

, ˆa

, · · · , ˆa

, where ˆ λ

> λ ˆ

> · · · >

λ ˆ

,

The estimate of A, ˆ A, is given by ˆ A = (ˆa

, ˆa

, · · · , ˆa

).

How do we obtain h?

3.5 Testing the Number of Cointegrating Vectors

Trace Test (

):

H

: λ

= 0 and H

: λ

> 0.

2(log l

− log l

) = − T

∑

i=h+1

log(1 − λ ˆ

) −→ tr(Q) , where

Q = (∫

0

W(r)dW(r)

)

(∫

0

W(r)W(r)

dr

)

(∫

W(r)dW(r)

)

.

Trace Test for # of Cointegrating Relations

# of Random (a) Regressors have no drift (b) Some regressors have drift

Walks (g − h) 1% 2.5% 5% 10% 1% 2.5% 5% 10%

1 11.576 9.658 8.083 6.691 6.936 5.332 3.962 2.816

2 21.962 19.611 17.844 15.583 19.310 17.299 15.197 13.338

3 37.291 34.062 31.256 28.436 35.397 32.313 29.509 26.791

4 55.551 51.801 48.419 45.248 53.792 50.424 47.181 43.964

5 77.911 73.031 69.977 65.956 76.955 72.140 68.905 65.063

J.D. Hamilton (1994), Time Series Analysis, p.767.

Largest Eigenvalue Test (最大固有値検定):

H

: λ

= 0 and H

: λ

> 0.

2(log l

− log l

) = − T log(1 − λ ˆ

) −→ maxmum eigen value of Q ,

Maximum Eigenvalue Test for # of Cointegrating Relations

# of Random (a) Regressors have no drift (b) Some regressors have drift

Walks (g − h) 1% 2.5% 5% 10% 1% 2.5% 5% 10%

1 11.576 9.658 8.083 6.691 6.936 5.332 3.962 2.816

2 18.782 16.403 14.595 12.783 17.936 15.810 14.036 12.099

3 26.154 23.362 21.279 18.959 25.521 23.002 20.778 18.697

4 32.616 29.599 27.341 24.917 31.943 29.335 27.169 24.712

5 38.858 35.700 33.262 30.818 38.341 35.546 33.178 30.774

J.D. Hamilton (1994), Time Series Analysis, p.768.

4 GMM (Generalized Mothod of Moments,

)

1. Method of Moments (

):

Regression Model: y

= x

β +

From the assumption, E(x

) = 0.

The sample mean is given by:

1 T

∑

x

= 1 T

∑

x

(y

− x

β ) = 0 .

Therefore,

β

=

 

 1 T

∑

x

x

 



−1





 1 T

∑

x

y

 

 ,

which is equivalent to OLS.

2. Generalized Mothod of Moments (GMM,

):

E (h( θ ; w

)) = 0 θ is a k × 1 parameter vector to be estimated.

w

is an observed vector w

= (y

, x

).

h( θ ; w

) is a r × 1 vector function, where r ≥ k.

Define g( θ ; W

) as follows:

g( θ ; W

) = 1 T

∑

h( θ ; w

) , where W

= { w

, w

, · · · , w

} .

Compute:

min

( g( θ ; W

) )

S

(

g( θ ; W

) )

The solution of θ , denoted by ˆ θ

, corresponds to the GMM estimator, where

S is defined as follows:

S = lim

T→∞

1 T

∑

∑

E (

(h( θ ; w

)) (h( θ ; w

))

) .

In empirical studies, S is replaced by its estimate, i.e., ˆ S

.

When h( θ ; w

), t = 1 , 2 , · · · , T , are not serially correlated, the following ˆ S

is consistent, i.e.,

S ˆ

= 1 T

∑

( h(ˆ θ

; w

) ) (

h(ˆ θ

; w

) )

−→ S .

When h( θ ; w

), t = 1 , 2 , · · · , T , are serially correlated, S ˆ

= Γ ˆ (0) +

∑

k( τ

q + 1 )( ˆ Γ ( τ ) + Γ ˆ ( τ )

) , where ˆ Γ ( τ ) = 1

T

∑

h(ˆ θ

; w

)h(ˆ θ

; w

)

.

k(x) = 1 − x = ⇒ Bartlett kernel (Newwey-west estimator), k(x) = ⇒ Parzen kernel, and etc.

Then, we obtain:

√ T (ˆ θ

− θ ) −→ N (

0 , (DS

D

)

)

,

where

D = ∂ g( θ ; W

)

∂θ

. Note that D is a r × k matrix.

Let ˆ D

be an estimate of D.

The variance estimator of ˆ θ

is given by:

D ˆ

= ∂ g(ˆ θ

; W

)

∂θ

.