262 Y
. Arai, T. Yamamoto Economics Letters 67 2000 261 –271
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 s 5 m 2 r 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: y 5 AL y
1 ´ , t 5 1,2, . . . ,T
1
t t 21
t p
i 21
where y 5 y , . . . , y 9, AL 5 o
A L , the A are m 3 m coefficient matrices, L is the
t 1t
mt i 51
i i
backward shift operator, Ly 5 y , and ´ is iid with zero mean, variance matrix S . 0, and finite
t t 21
t ´
fourth cumulants. The system 1 is initialized at t 5 2 p 1 1, . . . ,0 and these initial values can be any random vectors including constants. Here we set the initial conditions so that the I0 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:
p 21
Dy 5 Py 1
O
G Dy 1
´ , t 5 1,2, . . . ,T 2
t t 21
i t 2i
t i 51
p
where P 5 A1 2 I , G 5 2 o
A , I is the identity matrix with dimension m and for later
m i
j 5i 11 j
m p 21
convenience we define G 5 I 2 o
G . To focus on the problem dealt in this paper, we
m i 51
i
conventionally assume the following
Assumption 2.1.
a I 2 Azz 5 0 implies z 5 1 or uzu . 1.
u u
m
b P 5 ab 9 where a and b are m 3 r matrices of full column rank r. Without loss of generality, we assume that b is orthonormal.
9
c a Gb has full rank, where a
and b are m 3 m 2 r matrices of full column rank that are
orthogonal to a and b, respectively. Again without loss of generality we assume that b is
orthonormal. Next we consider the companion form of the system 1 to express the impulse responses explicitly:
Y . Arai, T. Yamamoto Economics Letters 67 2000 261 –271
263
Y 5 CY 1
J 3
t t 21
t
where:
9 9
J 5 [´ ,0, . . . ,0]
t t
9 9
9
Y 5 [ y , . . . , y ]
t t
t 2p 11
A ? ? ?
A A
1 p 21
p
I ? ? ?
m
C 5 ?
: ?
: :
3 4
? ? ? ?
I
m
In this case, up to the initial conditions and deterministic components, the moving average MA representation of the system can be written as:
t 21 i
Y 5
O
C J
t t 2i
i 51
or:
t 21
y 5
O
Q ´ 4
t i
t 2i t 50
where:
i
Q 5 M9C M 5
i
M9 5 [I ,0, . . . ,0] 6
m
We usually interpret the elements of the Q as the system’s impulse responses.
i
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 5 PP9
´ 21
and define n 5 P ´ . Then we have:
t t
t 21
y 5
O
C n 7
t i t 2i
t 50
where C 5 Q P. The elements of the C are usually interpreted as orthogonalized impulse responses of
i i
i
the system.
264 Y
. Arai, T. Yamamoto Economics Letters 67 2000 261 –271
3. Alternative representation for the asymptotic distribution of impulse responses estimated by RRR
