Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol67.Issue3.Jun2000:

www.elsevier.com / locate / econbase

Alternative representation for asymptotic distributions of

impulse responses in cointegrated VAR systems

*

Yoichi Arai, Taku Yamamoto

Department of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan Received 12 July 1999; accepted 21 October 1999

Abstract

We show an alternative representation for the asymptotic distributions of impulse responses in cointegrated VAR systems. Our representation has the advantage that the asymptotic variances are convergent at long horizons.  2000 Elsevier Science S.A. All rights reserved.

Keywords: Cointegration; Impulse response; Reduced rank regression

JEL classification: C32

1. Introduction

Impulse responses have been important tools for analyzing the interrelationships among variables. ¨

For stationary systems, Lutkepohl (1990) provides the simple representations for the asymptotic distributions of the impulse responses estimated by an OLS regression. For cointegrated systems,

¨

Lutkepohl and Reimers (1992) derive the asymptotic distributions of the impulse responses estimated by an OLS regression for a fixed finite lead time i. Phillips (1998, Theorem 2.3) shows that the impulse response matrices estimated by an OLS regression converge in distribution to random matrices and are inconsistent when i→`. Phillips (1998, Theorem 2.9) also shows that the impulse responses estimated by a reduced rank regression (RRR) are consistent even when i→`, and derived the asymptotic distributions for cointegrated VAR systems. The asymptotic distributions have an undesirable property in that the asymptotic variances diverge as the lead time goes to infinity. This is rather puzzling in view of the fact that the true impulse response matrices converge to a fixed finite matrix as i→` and the impulse responses estimated by a reduced rank regression (RRR) are consistent.

*Corresponding author. Tel.: 181-42-580-8793; fax:181-42-580-8793.

E-mail address: yamamoto@econ.hit-u.ac.jp (T. Yamamoto)

0165-1765 / 00 / $ – see front matter  2000 Elsevier Science S.A. All rights reserved. P I I : S 0 1 6 5 - 1 7 6 5 ( 9 9 ) 0 0 2 7 8 - 5

In this paper we present alternative representations of the asymptotic distributions of the RRR estimates of impulse responses with the convergent asymptotic variances for cointegrated VAR systems. The derivation closely follows that of Phillips (1998) except one important difference concerning the treatment of unit roots. Our derivation explicitly utilizes the fact that s5m2r unit

roots are not estimated by a RRR, where m is the dimension of VAR system and r is the cointegration rank. This difference leads to the asymptotic distributions with the convergent asymptotic variances even if the lead time goes to infinity.

In Section 2, we will give a brief description of the model. In Section 3, we will show alternative representations for the asymptotic distributions of impulse responses estimated by a RRR. Section 4 concludes the paper.

2. The model

We consider an m-dimensional vector autoregressive (VAR) model of the form:

yt5A(L )yt211´t, t51,2, . . . ,T (1)

p i21

where yt5( y , . . . , y )1t mt 9, A(L )5oi51 A Li , the A are mi 3m coefficient matrices, L is the

backward shift operator, Ly_t5y_t21, and ´_t is iid with zero mean, variance matrix S_´.0, and finite fourth cumulants. The system (1) is initialized at t5 2p11, . . . ,0 and these initial values can be any random vectors including constants. Here we set the initial conditions so that the I(0) component of (1) is stationary. Our conclusions do not depend on the presence of deterministic components in (1), so we assume for simplicity that they are absent.

It will be convenient to write the system (1) in an error correction form:

p21

Dy_t5Py_t211

O

G_iDy_t2i1´_t, t51,2, . . . ,T (2)

i51

p

where P5A(1)2I ,_m G_i5 2o_j5i11 A , I_{j m} is the identity matrix with dimension m and for later

p21

convenience we define G5I_m2o_i51 G_i. To focus on the problem dealt in this paper, we conventionally assume the following

Assumption 2.1.

(a) Iu m2A(z)zu50 implies z51 or uzu.1.

(b)P5ab9 where a andb are m3r matrices of full column rank r. Without loss of generality,

we assume that b is orthonormal.

9

(c)a Gb' ' has full rank, wherea' and b' are m3(m2r) matrices of full column rank that are

orthogonal to a and b, respectively. Again without loss of generality we assume that b' is orthonormal.

Yt5CYt211Jt (3)

where:

9

9

J_t 5[´_t,0, . . . ,0]

9

9

9

Y_t 5[ y , . . . , y_{t t2p11}]

A1 ? ? ? Ap21 Ap

I_m ? ? ? 0 0

C5 _?

: ? : :

3

?

4

0 ? ? ? I_m 0

In this case, up to the initial conditions and deterministic components, the moving average (MA) representation of the system can be written as:

t21 i

Y_t5

O

CJ_t2i

i51

or:

t21

y_t5

O

Q ´_{i t2i} (4)

t50

where:

i

Q_i5M9C M (5)

M9 5[I ,0, . . . ,0]m (6)

We usually interpret the elements of the Q_i as the system’s impulse responses.

We also consider orthogonalized impulse responses because they are preferred in applied researches. Suppose a lower triangular matrix P with positive diagonal elements such that S_´5PP9

21

and define nt5P ´t. Then we have:

t21

y_t5

O

C n_{i t2i} (7)

t50

whereCi5QiP. The elements of theCiare usually interpreted as orthogonalized impulse responses of the system.

3. Alternative representation for the asymptotic distribution of impulse responses estimated by RRR

First of all, let us see that the asymptotic variances V (in his notation) given in Theorem 2.9 of_i Phillips (1998) generally diverge as the lead time i goes to infinity. Phillips (1998) provides the following representation of V :_i

i21 j 21

9

9

V5N V N , N 5S Q ^_{M9C K}

i i a i i j50 i212j

where K is defined in Appendix A and V is defined in Theorem 3.1. It is clear that N must bea i

convergent for the convergence of V as i_i →`. Unfortunately, this is not the case because neitherQ_j

j 21

9

nor M9C K generally converges to zero as j→`. It will be explained in the proof of our Theorem 3.1 that the presence of s5m2r unit roots in the companion matrix C causes this. As an illustration,

consider the following model:

y1t5y1,t211´1t

y_2t5by_1,t211´_2t

In this model, the impulse responses are given by:

i

1 0 1 0

Qi5

F G F G

_{b 0} 5 _{b 0} , for any i, 5A (say)

Then, N in Theorem 2.9 of Phillips (1998) are in this case:i

N_i5(I₂1(i21)A)^_A9

Since A is a finite matrix, it implies that N diverges as ii →`. Consequently, Theorem 2.9 of Phillips (1998) indicates that the asymptotic variances diverge as i→`.

We will now present the alternative asymptotic distributions of the RRR estimates of impulse responses with the convergent asymptotic variances for cointegrated VAR systems.

ˆ

Theorem 3.1. Let Assumption 2.1 hold and let Q_i be the impulse responses estimated by the RRR:

(i ) For fixed i we have:

p _{1 / 2} d

ˆ ˆ

Q_i→Q_i, n (Q_i2Q_i)→N(0,V )_i (8)

as n→`, where:

9

V_i5N V N_{i a i}

i21

k9

N 5

O

Q ^_{Q9L9G9D G}

i i212k z k50

21

V 5S ^_S

9

Sj j5E(j jt t)

where:

b 0

G_z5

F

_{0 I}

G

, Q9 5[b,H,0, . . . ,0], L9 5[0,I_{m( p21 )1r}]

m( p21 )

b' b 0

G5I_p^_H5

F

G

0 0 I_p21^_H

I_m1ab9 G₁ G_p21 ab9 G₁ G_p21

D5 _{0 I 0}

m

3

? ?

4

0 I_m 0

9

9

9

j_t5[ y_t21b,Dy_t21, . . . ,Dy_t2p]9

(ii ) If i→`as n→` with either i5fn or i /n→0 where f.0 is a fixed fraction of the sample,

we have:

p _{1 / 2 ˆ} d

ˆ ¯ ¯ ¯ ¯

Q_i→Q, n (Q2Q)→N(0,V ) (9)

¯

as n→`, where Q is the limit of impulse response defined in (A.11):

¯

V5NV N_a 9

I_r 0

21

¯

₉

N5Q^_{Q9(I2E )}

F

G

22 _{0 I} ^_H9

p21

where E₂₂ is the submatrix of the companion matrix E given in (A.10). Proof. See Appendix A.

The asymptotic distributions of the RRR estimates of orthogonalized impulse responses can be ¨

easily derived in the same manner as the proof of the Proposition 1 in Lutkepohl (1990). We present them for the convenience of applied researchers.

ˆ

Corollary 3.2. Let Assumption2.1hold and let C_ibe the orthogonalized impulse responses estimated by a RRR:

(i ) For fixed i we have:

p _{1 / 2} d

ˆ ˆ

₉

₉

Ci→Ci, n (Ci2Ci)→N(0,J V J1i a 1i1J V J )2i s 2i (10)

J1i5(Im^_P9)Ni

J_2i5(Q_i^I )W_m

1 19

V 5D (var(´ ^´))D

s t t

1

₉

21

₉

D 5(D D )m m Dm

21

9

9

W5L_mhL (I_{m m}21K_mm)(I^_P)Lmj

2

L_m is the (m(m11) / 23m ) elimination matrix defined such that, for any (m3m) matrix F,

vech(F )5L vec(F ), and the commutation matrix K is defined such that, for any (m3n) matrix

m mn

2

G, K vec(G)5vec(G9), and the (m 3m(m11) / 2) duplication matrix D is defined such that

mn m

D vech(F )5vec(F ) for a symmetric (m3m) matrix F. N and V are the same as those defined in

m i a

Theorem 3.1.

(ii ) If i→`as n→` with either i5fn or i /n→0 where f.0 is a fixed fraction of the sample,

we have:

p _{1 / 2 ˆ} d

ˆ ¯ ¯ ¯

₉

₉

C_i→C, n (C2C)→N(0,J V J_{1 a 1}1J V J )_{2 s 2} (11)

as n→`, where:

¯ ¯

C5QP

J 5(I ^_P9)N

1 m

¯

J₂5(Q^I )W_m

¨

Proof. See the proof of the Proposition 1 in Lutkepohl (1990).

Theorem 3.1 indicates that Ni→N when i→` as n→` with either i5fn or i /n→0. It implies that the RRR estimates of impulse responses have the convergent asymptotic variances. It may be noted that (ii) of the above theorem is the same as (ii) of Theorem 2.9 in Phillips (1998), while (i) is different from his (i). Specifically, the expression of N in (i) is different, where Phillips (1998)_i presents the same formula as given for the OLS estimates of impulse responses in his Theorem 2.3. In order to derive (i) of the theorem, it is critical that we remove the effects of unit roots which are not estimated by a RRR. This leads to the asymptotic distributions with convergent variances even when the lead time goes to infinity. The asymptotic distributions with this property is accessible and suitable for the inference at long horizons.

The procedure for applying Theorem 3.1 or Corollary 3.2 to empirical research is as follows. First of all, we need to estimate the cointegrating rank consistently. It is enough for us to use the consistent method like Johansen (1988, 1991) with appropriate modifications to ensure that the size of the test goes to zero as the sample size goes to infinity. This is necessary for a RRR. Secondly we estimate parameters by a RRR. Since the inconsistency of the OLS estimates were shown in Phillips (1998,

Theorem 2.3), the estimation method should be a RRR. These procedures enable us to apply Theorem 3.1 or Corollary 3.2.

4. Conclusion

In this paper we have considered impulse responses in cointegrated VAR systems. We have derived the asymptotic distributions of the RRR estimates of impulse responses with the convergent asymptotic variances.

Acknowledgements

Arai’s research was partially supported by the Japan Society for the Promotion of Science and Yamamoto’s research was partially supported by the Ministry of Education, Science and Culture under Grants-in-Aid No. 10630021.

Appendix A

Proof of Theorem 3.1

Since our derivations heavily depend upon several results in Phillips (1998, Appendix), they will be a great help for understanding the proof of Theorem 3.1. By estimating (2) by a RRR, we can write estimated impulse responses as in (A.4) of Phillips (1998, Appendix), namely:

i 21

ˆ ˆ

Qi5M9KD K M (A.1)

ˆ

where D is the estimated companion matrix associated with an error correction form (2)

ˆ ˆ

ˆ ˆ

Im1ab9 G1 Gp21



_abˆ ˆ_{9 G}ˆ _Gˆ



1 p21

ˆ D5

0 I 0



m



? ?

0 I 0



m



I 0 ? ? ? 0

I 2I ? ? ? 0

K_{5 ? ?}

3

_{? ? ? ?}

4

I 2I ? ? ? 2I

ˆ ˆ

ˆ

Hereaand bare the RRR estimates ofa andb, respectively. Moreover we assume that b is suitably

ˆ ˆ

ˆ ˆ

normalized for identification. Under these conditions it is well known that a, b and G₁, . . . ,G_p21 are

ˆ ˆ

ˆ ¨

consistent — see Johansen (1991) for_p a and b, and Lutkepohl and Reimers (1992) for G’s. This

ˆ

means D→D, giving the required result.

ˆ ˆ

ˆ

terms of E that is the estimated companion matrix associated with the I(1) /I(0) VAR representation — see Phillips (1998, Appendix A.1) for the I(1) /I(0) VAR representation.

i 21 i 21

ˆ ˆ ˆ ˆ ˆ

Q_i5M9KD K M5M9KG E G9K M (A.2)

where:

ˆ ˆ ˆ ˆ ˆ

G5I ^_{H, H5[b ,b]}

p '

and

ˆ ˆ

¯ ¯

ˆ

₉

_ˆ

I_s b a' G₁ ? ? ? G_p21





ˆ ˆ

0 I_r1b9a



_ˆ

₉

_ˆ ˆ_{¯ ¯}ˆ



0 b a' G1 ? ? ? Gp21

ˆ ˆ ˆ ˆ

E5G9DG5





(A.3)

ˆ ˆ

0 b9a

0 0 I_m 0



_{? ? ? ?}



? ?

0 ? 0 I 0



m



In stationary systems, the derivation will be completed by taking differentials of this estimated impulse response. In cointegrated systems, however, we must pay attention to the structure of the

ˆ

estimated companion matrix E. Although there are s5m2r unit roots in the system, they are not

estimated in a RRR. It means that the first (mp3s) block in E is not estimated. Then we need to

i

ˆ ˆ

discard this block when taking differentials. By partitioning E as [E ,E ] where E_{1 2 1} is the first (mp3s) block in E, we have:

i 21 21

ˆ ˆ ˆ ˆ ˆ ˆ ˆ

Q_i5M9KGE G9K M5M9KG [E ,E ]G_{1 2} 9K M ˆ ˆ

₉

ˆ ˆ ˆ

5M9KGE₁b'1M9KGE Q₂

21

ˆ ˆ ˆ

by the fact that G9K M5[b',Q9]9. Then:

i

ˆ ˆ ˆ

9

ˆ ˆ ˆ ˆ ˆ

9

ˆ ˆ ˆ

Q_i5b b' '1M9KG E Q2 5b b' '1M9KG E LQ (A.4)

1 / 2

ˆ ˆ ˆ

9

where L5[0,I_{m( p21 )1r}]. In this representation, when we normalize Q_i2Q_i with n , b b' ' can be

21

ˆ

9

replaced by the true value b b' ' because b'2b' is O (n_p ). Then we should be concerned only

21

ˆ ˆ

with the second term of the right-hand side of (A.4). Since G2G and Q2Q are also O (n_p ), we behave as they are true. The differentials of estimated impulse responses based upon (A.4) become:

i21

i i212k k

ˆ ˆ ˆ ˆ ˆ

dQ_i5M9KG dE LQ5M9KG

O

E dE E LQ (A.5)

k50

Im

ˆ ˆ ˆ ˆ

Im 0 d(b a

9

_'ˆ) dG¯ ? ? ?dG¯ ₂₁ 0 d(b a

9

ˆ) dG¯ ? ? ?dG¯

1 p21 ' 1 p21

ˆ

dE5 ₀

F

G F

5K M

G

ˆ ˆ

0 d(b9a) 0 d(b9a)

3 4

?

0

ˆ ˆ ˆ ˆ

¯ ¯ ¯ ¯

9

ˆ

9

ˆ

0 d(b a) dG ? ? ?dG 0 d(b a) dG ? ? ?dG

21 ' 1 p21 21 ˆ ' 1 p21

5K G9GM

F

G

5K G9MH

F

G

ˆ ˆ

0 d(b9a) 0 d(b9a)

21 _{ˆ ˆ ˆ} 21 _{ˆ ˆ ˆ}

5K G9M[dP,dG₁,? ? ?,dG_p21]G5G9K M[dP,dG₁,? ? ?,dG_p21]G

Here:

ˆ _ˆ

dP5dab9

Then:

21 _{ˆ ˆ}

ˆ _ˆ

dE 5G9K M[(da)b9,dG1, . . . ,dGp21]G

21 _{ˆ ˆ}

ˆ

9

5G9K M[da,dG1, . . . ,dGp21]G Gz

ˆ ˆ

Thus substituting this dE into the Eq. (A.5) yields the representations of dQ_i as:

i21

i212k k

ˆ ˆ ˆ ˆ

dQi 5M9KG

O

E dE E LQ

k50 i21

i212k 21 _{ˆ ˆ} k

ˆ _ˆ

₉

ˆ _(A.6)

5M9KG

O

E G9K M [da,dG₁, . . . ,dG_p21]G G E LQ_z

k50 i21

k

ˆ ˆ

ˆ _ˆ

₉

ˆ

5

O

Qi212k[da,dG1, . . . ,dGp21]G GE LQz k50

Ahn and Reinsel (1990) show that:

d

1 / 2 _{ˆ ˆ} 21

ˆ

n [a2a,G 2G, . . . ,G 2G ]→N(0,S ^_{S ) (A.7)}

1 1 p21 p21 ´ j j

Then we have:

d 1 / 2 _ˆ

n (Q_i2Q_i)→N(0,V )_i

where:

9

V_i5N V N_{i a i}

21

V_a5S_´^_{Sj j}

i21 i21

k9 i212k k9

N 5

O

Q ^_{Q9L9E G9G 5}

O

_Q ^_{Q9L9G9D G (A.8)}

i i212k z z

k50 k50

b' G12

G5

F

G

0 G₂₂

we have:

k21

b9 0 b' G12 E (I12 1E221 ? ? ? 1E22 )

k

9

G GE LQz 5

F

_{0 I}

GF GF

_{0 G E}i

G

Q

m( p21 ) 22 22

k21

0 b9G₁₂ E (I₁₂ 1E₂₂1 ? ? ? 1E₂₂ ) b9G_{12 k}

5

F

GF

i

G F G

Q5 E Q22

0 G₂₂ E₂₂ G₂₂

I_r 0 _k

5

F

G

E Q₂₂

0 I ^_H

p21

Thus N can be written in terms of E :i 22 i21 _{I 0}

r k9

N_i5

O

Q_i212k^Q9E₂₂

F

_{0 I ^H9}

G

(A.9)

p21 k50

1

Since E₂₂ corresponds to the coefficient matrix for the stationary components , this representation implies the convergent property of N . This proves part (i) of the theorem._i

In the case that i→` as n→` with either i5fn or i /n→0, remember the Eq. (A.3)

ˆ ˆ

¯ ¯

ˆ

₉

_ˆ

Is b a' G1 ? ? ? Gp21





ˆ ˆ

0 Ir1b9a



_{0 b aˆ}

₉

_{ˆ G}ˆ_{¯ ? ? ? G¯}ˆ



_ˆ

I E

' 1 p21 s 12

ˆ

E5





5

F G

(say) (A.10)

ˆ

ˆ _{ˆ 0 E}

0 b9a 22

0 0 I_m 0



_?



? ? ? _{? ?}

0 ? 0 I 0



m



Then:

2 i21 21

ˆ ˆ ˆ ˆ

I E (I 1E 1E 1 ? ? ? 1E ) p I E (I2E )

i s 12 m( p21 )1r 22 22 22 s 12 22

ˆ

E 5

F

_i

G F

→

G

ˆ

0 E22 0 0

Thus, we have:

21 p

i ₂1 I_s E (I₁₂ 2E )₂₂ 21 21

ˆ ˆ ˆ ˆ

₉

Q_i5M9KGE G9K M→M9KG

F

G

G9K 5b b' '1b'E (I₁₂ 2E )₂₂ Q

0 0

¯

5Q (say) (A.11)

giving the required results of first statement of part (ii). 1

ˆ¯

With regard to deriving the asymptotic distributions of Q, it is enough to show that Ni→N as i→`. From Eq. (A.9):

i21 _{I 0 I 0}

r r

i9 _¯ 21

Ni5

O

Qi212k^_Q9E22

F

_{0 I ^H9}

G

→Q^_Q9(I2E )22

F

_{0 I ^H9}

G

p21 p21

k50

This completes the proof of part (ii) of the theorem.

References

Ahn, S.K., Reinsel, G.C., 1990. Estimation for partially nonstationary multivariate autoregressive models. Journal of the American Statistical Association 85, 813–823.

Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231–254. Johansen, S., 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models.

Econometrica 59, 1551–1580.

¨

Lutkepohl, H., 1990. Asymptotic distributions of impulse response functions and forecast error variance decompositions of vector autoregressive models. Review of Economics and Statistics 72, 116–125.

¨

Lutkepohl, H., Reimers, H.E., 1992. Impulse response analysis of cointegrated systems. Journal of Economic Dynamics and Control 16, 53–78.

Phillips, P.C.B., 1998. Impulse response and forecast error variance asymptotics in nonstationary VARs. Journal of Econometrics 83, 21–56.

J1i5(Im^P9)Ni J_2i5(Q_i^I )W_m

1 19

V 5D (var(´ ^´))D

s t t

1

₉

21

₉

D 5(D D )m m Dm

21

9

9

W5L_mhL (I_{m m}21K_mm)(I^_P)Lmj

2

L_m is the (m(m11) / 23m ) elimination matrix defined such that, for any (m3m) matrix F, vech(F )5L vec(F ), and the commutation matrix K is defined such that, for any (m3n) matrix

m mn

2

G, K vec(G)5vec(G9), and the (m 3m(m11) / 2) duplication matrix D is defined such that

mn m

D vech(F )5vec(F ) for a symmetric (m3m) matrix F. N and V are the same as those defined in

m i a

Theorem 3.1.

(ii ) If i→`as n→` with either i5fn or i /n→0 where f.0 is a fixed fraction of the sample,

we have:

p _{1 / 2 ˆ} d

ˆ ¯ ¯ ¯

₉

₉

C_i→C, n (C2C)→N(0,J V J_{1 a 1}1J V J )_{2 s 2} (11)

as n→`, where:

¯ ¯

C5QP J 5(I ^_P9)N

1 m

¯

J₂5(Q^I )W_m

¨

Proof. See the proof of the Proposition 1 in Lutkepohl (1990).

Theorem 3.1 indicates that Ni→N when i→` as n→` with either i5fn or i /n→0. It implies that the RRR estimates of impulse responses have the convergent asymptotic variances. It may be noted that (ii) of the above theorem is the same as (ii) of Theorem 2.9 in Phillips (1998), while (i) is different from his (i). Specifically, the expression of N in (i) is different, where Phillips (1998)_i presents the same formula as given for the OLS estimates of impulse responses in his Theorem 2.3. In order to derive (i) of the theorem, it is critical that we remove the effects of unit roots which are not estimated by a RRR. This leads to the asymptotic distributions with convergent variances even when the lead time goes to infinity. The asymptotic distributions with this property is accessible and suitable for the inference at long horizons.

The procedure for applying Theorem 3.1 or Corollary 3.2 to empirical research is as follows. First of all, we need to estimate the cointegrating rank consistently. It is enough for us to use the consistent method like Johansen (1988, 1991) with appropriate modifications to ensure that the size of the test goes to zero as the sample size goes to infinity. This is necessary for a RRR. Secondly we estimate parameters by a RRR. Since the inconsistency of the OLS estimates were shown in Phillips (1998,

Theorem 2.3), the estimation method should be a RRR. These procedures enable us to apply Theorem 3.1 or Corollary 3.2.

4. Conclusion

In this paper we have considered impulse responses in cointegrated VAR systems. We have derived the asymptotic distributions of the RRR estimates of impulse responses with the convergent asymptotic variances.

Acknowledgements

Arai’s research was partially supported by the Japan Society for the Promotion of Science and Yamamoto’s research was partially supported by the Ministry of Education, Science and Culture under Grants-in-Aid No. 10630021.

Appendix A

Proof of Theorem 3.1

Since our derivations heavily depend upon several results in Phillips (1998, Appendix), they will be a great help for understanding the proof of Theorem 3.1. By estimating (2) by a RRR, we can write estimated impulse responses as in (A.4) of Phillips (1998, Appendix), namely:

i 21

ˆ ˆ

Qi5M9KD K M (A.1)

ˆ

where D is the estimated companion matrix associated with an error correction form (2)

ˆ ˆ

ˆ ˆ

Im1ab9 G1 Gp21



_abˆ ˆ_{9 G}ˆ _Gˆ



1 p21 ˆ

D5

0 I 0



m



? ?

0 I 0



m



I 0 ? ? ? 0

I 2I ? ? ? 0

K_{5 ? ?}

3

_{? ? ? ?}

4

I 2I ? ? ? 2I

ˆ ˆ

ˆ

Hereaand bare the RRR estimates ofa andb, respectively. Moreover we assume that b is suitably

ˆ ˆ

ˆ ˆ

normalized for identification. Under these conditions it is well known that a, b and G₁, . . . ,G_p21 are

ˆ ˆ

ˆ ¨

consistent — see Johansen (1991) for_p a and b, and Lutkepohl and Reimers (1992) for G’s. This

ˆ

means D→D, giving the required result.

ˆ ˆ

ˆ

terms of E that is the estimated companion matrix associated with the I(1) /I(0) VAR representation — see Phillips (1998, Appendix A.1) for the I(1) /I(0) VAR representation.

i 21 i 21

ˆ ˆ ˆ ˆ ˆ

Q_i5M9KD K M5M9KG E G9K M (A.2)

where:

ˆ ˆ ˆ ˆ ˆ

G5I ^_{H, H5[b ,b]}

p '

and

ˆ ˆ

¯ ¯

ˆ

₉

_ˆ

I_s b a' G₁ ? ? ? G_p21





ˆ ˆ

0 I_r1b9a



_ˆ

₉

_ˆ ˆ_{¯ ¯}ˆ



0 b a' G1 ? ? ? Gp21 ˆ ˆ ˆ ˆ

E5G9DG5





(A.3)

ˆ ˆ

0 b9a

0 0 I_m 0



_{? ? ? ?}



? ?

0 ? 0 I 0



m



In stationary systems, the derivation will be completed by taking differentials of this estimated impulse response. In cointegrated systems, however, we must pay attention to the structure of the

ˆ

estimated companion matrix E. Although there are s5m2r unit roots in the system, they are not

estimated in a RRR. It means that the first (mp3s) block in E is not estimated. Then we need to i

ˆ ˆ

discard this block when taking differentials. By partitioning E as [E ,E ] where E_{1 2 1} is the first (mp3s) block in E, we have:

i 21 21

ˆ ˆ ˆ ˆ ˆ ˆ ˆ

Q_i5M9KGE G9K M5M9KG [E ,E ]G9K_{1 2} M

ˆ ˆ

₉

ˆ ˆ ˆ

5M9KGE₁b'1M9KGE Q₂

21

ˆ ˆ ˆ

by the fact that G9K M5[b',Q9]9. Then:

i

ˆ ˆ ˆ

9

ˆ ˆ ˆ ˆ ˆ

9

ˆ ˆ ˆ

Q_i5b b' '1M9KG E Q2 5b b' '1M9KG E LQ (A.4)

1 / 2

ˆ ˆ ˆ

9

where L5[0,I_{m( p21 )1r}]. In this representation, when we normalize Q_i2Q_i with n , b b' ' can be 21

ˆ

9

replaced by the true value b b' ' because b'2b' is O (n_p ). Then we should be concerned only 21

ˆ ˆ

with the second term of the right-hand side of (A.4). Since G2G and Q2Q are also O (n_p ), we behave as they are true. The differentials of estimated impulse responses based upon (A.4) become:

i21

i i212k k

ˆ ˆ ˆ ˆ ˆ

dQ_i5M9KG dE LQ5M9KG

O

E dE E LQ (A.5)

k50

Im

ˆ ˆ ˆ ˆ

Im 0 d(b a

9

_'ˆ) dG¯ ? ? ?dG¯ ₂₁ 0 d(b a

9

ˆ) dG¯ ? ? ?dG¯

1 p21 ' 1 p21

ˆ

dE5 ₀

F

G F

5K M

G

ˆ ˆ

0 d(b9a) 0 d(b9a)

3 4

?

0

ˆ ˆ ˆ ˆ

¯ ¯ ¯ ¯

9

ˆ

9

ˆ

0 d(b a) dG ? ? ?dG 0 d(b a) dG ? ? ?dG

21 ' 1 p21 21 ˆ ' 1 p21

5K G9GM

F

G

5K G9MH

F

G

ˆ ˆ

0 d(b9a) 0 d(b9a)

21 _{ˆ ˆ ˆ} 21 _{ˆ ˆ ˆ}

5K G9M[dP,dG₁,? ? ?,dG_p21]G5G9K M[dP,dG₁,? ? ?,dG_p21]G Here:

ˆ _ˆ

dP5dab9

Then:

21 _{ˆ ˆ}

ˆ _ˆ

dE 5G9K M[(da)b9,dG1, . . . ,dGp21]G

21 _{ˆ ˆ}

ˆ

9

5G9K M[da,dG1, . . . ,dGp21]G Gz

ˆ ˆ

Thus substituting this dE into the Eq. (A.5) yields the representations of dQ_i as:

i21

i212k k

ˆ ˆ ˆ ˆ

dQi 5M9KG

O

E dE E LQ

k50 i21

i212k 21 _{ˆ ˆ} k

ˆ _ˆ

₉

ˆ _(A.6)

5M9KG

O

E G9K M [da,dG₁, . . . ,dG_p21]G G E LQ_z

k50 i21

k

ˆ ˆ

ˆ _ˆ

₉

ˆ

5

O

Qi212k[da,dG1, . . . ,dGp21]G GE LQz k50

Ahn and Reinsel (1990) show that:

d

1 / 2 _{ˆ ˆ} 21

ˆ

n [a2a,G 2G, . . . ,G 2G ]→N(0,S ^_{S ) (A.7)}

1 1 p21 p21 ´ j j

Then we have:

d 1 / 2 _ˆ

n (Q_i2Q_i)→N(0,V )_i

where:

9

V_i5N V N_{i a i}

21 V_a5S_´^_{Sj j}

i21 i21

k9 i212k k9

N 5

O

Q ^_{Q9L9E G9G 5}

O

_Q ^_{Q9L9G9D G (A.8)}

i i212k z z

k50 k50

b' G12

G5

F

G

0 G₂₂

we have:

k21

b9 0 b' G12 E (I12 1E221 ? ? ? 1E22 )

k

9

G GE LQz 5

F

_{0 I}

GF GF

_{0 G E}i

G

Q

m( p21 ) 22 22

k21

0 b9G₁₂ E (I₁₂ 1E₂₂1 ? ? ? 1E₂₂ ) b9G_{12 k}

5

F

GF

i

G F G

Q5 E Q22

0 G₂₂ E₂₂ G₂₂

I_r 0 _k

5

F

G

E Q₂₂

0 I ^_H

p21

Thus N can be written in terms of E :i 22

i21 _{I 0}

r k9

N_i5

O

Q_i212k^Q9E₂₂

F

_{0 I ^H9}

G

(A.9)

p21 k50

1

Since E₂₂ corresponds to the coefficient matrix for the stationary components , this representation implies the convergent property of N . This proves part (i) of the theorem._i

In the case that i→` as n→` with either i5fn or i /n→0, remember the Eq. (A.3)

ˆ ˆ

¯ ¯

ˆ

₉

_ˆ

Is b a' G1 ? ? ? Gp21





ˆ ˆ

0 Ir1b9a



_{0 b aˆ}

₉

_{ˆ G}ˆ_{¯ ? ? ? G¯}ˆ



_ˆ

I E

' 1 p21 s 12

ˆ

E5





5

F G

(say) (A.10)

ˆ

ˆ _{ˆ 0 E}

0 b9a 22

0 0 I_m 0



_?



? ? ? _{? ?}

0 ? 0 I 0



m



Then:

2 i21 21

ˆ ˆ ˆ ˆ

I E (I 1E 1E 1 ? ? ? 1E ) p I E (I2E )

i s 12 m( p21 )1r 22 22 22 s 12 22

ˆ

E 5

F

_i

G F

→

G

ˆ

0 E22 0 0

Thus, we have:

21 p

i ₂1 I_s E (I₁₂ 2E )₂₂ 21 21

ˆ ˆ ˆ ˆ

₉

Q_i5M9KGE G9K M→M9KG

F

G

G9K 5b b' '1b'E (I₁₂ 2E )₂₂ Q

0 0

¯

5Q (say) (A.11)

giving the required results of first statement of part (ii). 1

ˆ¯

With regard to deriving the asymptotic distributions of Q, it is enough to show that Ni→N as i→`. From Eq. (A.9):

i21 _{I 0 I 0}

r r

i9 _¯ 21

Ni5

O

Qi212k^Q9E22

F

_{0 I ^H9}

G

→Q^Q9(I2E )22

F

_{0 I ^H9}

G

p21 p21

k50

This completes the proof of part (ii) of the theorem.

References

Ahn, S.K., Reinsel, G.C., 1990. Estimation for partially nonstationary multivariate autoregressive models. Journal of the American Statistical Association 85, 813–823.

Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231–254. Johansen, S., 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models.

Econometrica 59, 1551–1580.

¨

Lutkepohl, H., 1990. Asymptotic distributions of impulse response functions and forecast error variance decompositions of vector autoregressive models. Review of Economics and Statistics 72, 116–125.

¨

Lutkepohl, H., Reimers, H.E., 1992. Impulse response analysis of cointegrated systems. Journal of Economic Dynamics and Control 16, 53–78.

Phillips, P.C.B., 1998. Impulse response and forecast error variance asymptotics in nonstationary VARs. Journal of Econometrics 83, 21–56.