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. Arai, T. Yamamoto Economics Letters 67 2000 261 –271
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 i 21
j 21
9 9
V 5 N V N , N 5 S
Q M9C K
i i a
i i
j 50 i 212j
where K is defined in Appendix A and V is defined in Theorem 3.1. It is clear that N must be
a i
convergent for the convergence of V as i →
`. Unfortunately, this is not the case because neither Q
i 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 s 5 m 2 r unit roots in the companion matrix C causes this. As an illustration, consider the following model:
y 5 y 1
´
1t 1,t 21
1t
y 5 by 1
´
2t 1,t 21
2t
In this model, the impulse responses are given by:
i
1 1
Q 5 5
, for any i,
5 A say
F G F G
i
b b
Then, N in Theorem 2.9 of Phillips 1998 are in this case:
i
N 5 I 1 i 2 1A A9
i 2
Since A is a finite matrix, it implies that N diverges as i →
`. Consequently, Theorem 2.9 of Phillips
i
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 be the impulse responses estimated by the RRR:
i
i For fixed i we have:
p d
1 2
ˆ ˆ
Q →
Q , n Q 2 Q
→ N0,V
8
i i
i i
i
as n →
`, where:
9
V 5 N V N
i i a
i i 21
k 9
N 5
O
Q Q9L9G9D G
i i 212k
z k 50
21
V 5 S S
a ´
j j
Y . Arai, T. Yamamoto Economics Letters 67 2000 261 –271
265
9
S 5 Ej j
j j t
t
where: b 0
G 5 ,
Q9 5 [b,H,0, . . . ,0], L9 5 [0,I
]
F G
z m p 21 1r
I
m p 21
b b 0
G 5 I H 5
F G
p
I H
p 21
I 1 ab 9 G
G
m 1
p 21
ab9 G
G
1 p 21
D 5 0 I
m
3 4
? ?
I
m
9 9
9
j 5 [y b,Dy
, . . . ,Dy ]9
t t 21
t 21 t 2p
ii If i →
` as n →
` with either i 5 fn or i n →
0 where f . 0 is a fixed fraction of the sample, we have:
p d
1 2
ˆ ˆ
¯ ¯
¯ ¯
Q →
Q, n Q 2 Q
→ N0,V
9
i
¯ as n
→ `, where Q is the limit of impulse response defined in A.11:
¯ V 5 NV N9
a
I
r 21
¯
9
N 5 Q Q9I 2 E
F G
22
I H9
p 21
where E is the submatrix of the companion matrix E given in
A.10.
22
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 Assumption 2.1 hold and let C be the orthogonalized impulse responses estimated
i
by a RRR: i For fixed i we have:
p d
1 2
ˆ ˆ
9 9
C →
C , n C 2C
→ N0,J V J 1 J V J
10
i i
i i
1i a 1i
2i s 2i
as n →
`, where:
266 Y
. Arai, T. Yamamoto Economics Letters 67 2000 261 –271
J 5 I P9N
1i m
i
J 5 Q I W
2i i
m 1
19
V 5 D var´ ´ D
s t
t 1
21
9 9
D 5 D D D
m m
m 21
9 9
W 5 L hL I 1 K I PL j
2
m m
m mm
m 2
L is the mm 1 1 2 3 m elimination matrix defined such that, for any m 3 m matrix F,
m
v echF 5 L vecF , and the commutation matrix K
is defined such that, for any m 3 n matrix
m mn
2
G, K vecG 5 vecG9, and the m 3 mm 1 1 2 duplication matrix D is defined such that
mn m
D vechF 5 vecF for a symmetric m 3 m 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 i 5 fn or i n →
0 where f . 0 is a fixed fraction of the sample, we have:
p d
1 2
ˆ ˆ
¯ ¯
¯
9 9
C →
C, n C 2 C
→ N0,J V J 1 J V J
11
i 1 a
1 2 s
2
as n →
`, where: ¯
¯ C 5 QP
J 5 I P9N
1 m
¯ J 5 Q I W
2 m
¨ Proof. See the proof of the Proposition 1 in Lutkepohl 1990.
Theorem 3.1 indicates that N →
N when i →
` as n →
` with either i 5 fn or i n →
0. It implies
i
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,
Y . Arai, T. Yamamoto Economics Letters 67 2000 261 –271
267
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
Kategori