18 C
.-F. Chung Economics Letters 71 2001 17 –25
hyperbolic rates differ from the geometric rates of the ARMA model. The hyperbolic impulse responses of the ARFIMA model imply similar hyperbolic patterns in its autocorrelations, which is
usually referred to as long memory as opposed to the short memory of the ARMA model. Because of the sharp distinction in the impulse response patterns between the ARFIMA model and the ARMA
model, the impulse response analysis has become an important tool in the study of long memory economic series. See, for example, Diebold and Rudebusch’s 1989, 1991 study on aggregate
income; Diebold et al. 1991 and Cheung’s 1993 work on exchange rates; Baillie et al.’s 1996a,b paper on inflation rates; and Baillie et al.’s 1996a,b and Chung’s 2000 study on volatility
conditional variances. The results in this note should facilitate the impulse response analysis in this growing literature.
2. Calculating impulse responses
Let us start with the general stationary and invertible vector ARFIMA VARFIMA model of the
m-variate process x :
t d
FB D Bx 2 m 5 QB ´
t t
where B is the lag operator,
m is the mean vector, ´ is white noise with E´ 5 0 and Var´ 5 RR9.
t t
t p
j q
j
The ARMA operators FB 5 F 2 o
F B and QB 5 Q 1
o Q B
are m 3 m matrix
j 51 j
j 51 j
polynomials in B. The coefficients of polynomials FB and QB are assumed to satisfy the standard stationarity and invertibility conditions and are nonlinear functions of some estimable parameters after
d
normalization and identification restrictions are imposed. The operator D B is an m 3 m diagonal matrix characterized by the m parameters d , d , . . . ,d
as follows:
1 2
m d
1
1 2 B ? ? ?
d
2
1 2 B ? ? ?
d
D B ;
? :
: ?
: ?
3 4
d
m
? ? ? 1 2 B
d
i
where d , d , . . . ,d , can be fractional numbers. The term 1 2 B is defined by the following power
1 2
m
expansion:
` 2d
d j
d
i i
i
1 2 B 5
O
c B
; c B
j j 50
d
i
where c 5 Gd 1 j [G j 1 1 Gd ], c
5 1, and c 5 0, for j ± 1. Consequently, we have:
j i
i j
d
1
c B
? ? ?
` d
2
c B
? ? ?
d 21
d d
j
[D B ] 5
; C B 5 I 1
O
C B 1
m j
? :
: ?
:
j 51
?
3 4
d
m
? ? ? c
B
d d
i
where I is the identity matrix of order m and each C is an m 3 m diagonal matrix with c
as the
m j
j
C .-F. Chung Economics Letters 71 2001 17 –25
19
ith diagonal element. To stress the central role of the fractional differencing parameters d , d , . . . ,d
1 2
m
in our analysis, we call x the VARFIMAd , . . . ,d process.
t 1
m
If 2 0.5 , d , 0.5, for all i, then the VARFIMAd , . . . ,d model is both stationary and invertible
i 1
m
Hosking, 1981, 1996; Chung, 2001 so that it has the infinite moving-average representation:
21 `
j
x 5 m 1 AB u , where u 5 R ´ , and Az
; I 1 o A z . The moving-average coefficients A
t t
t t
m j 51
j j
are referred to as impulse responses and each of them reflects the change in x caused by a shock in
t
previous u , for j 5 1,2, . . . . We note Varu 5 I for all t. Throughout our analysis, we concentrate
t 2j t
m n
j
on the following truncated version of Az: A z 5 I 1
o
A z , for a finite n. We will show that
n m
j 51 j
the impulse responses analysis can be substantially simplified if we follow a simple rule to express them as products of finite-order power series. Let’s first define such a rule of operation.
, j
Given two arbitrary power series in m 3 m matrices of different orders Pz 5 I 1
o
P z and
m j 51
j n
h j
Qz 5 I 1
o
Q z , we use a special notation } to denote the operation of truncating the series
m j 51
j n
n j
that result from power series multiplication: M z}PzQz, where M z 5 I 1
o
M z is power
n n
m j 51
j j
series with the n matrix coefficients being defined by M 5
o
P Q , j 5 1, . . . ,n. Here, we set
j k 50
k j 2k
n
P 5 Q 5 I , P 5 0 for k . ,, and Q 5 0 for k . h. The basic properties of the operation } are
m k
k
included in Lemma 1 in Appendix A. We can also define the truncated series that result from the
n 21
n j
power series inversion by N z}Qz Pz, where N z 5 I 1
o
N z is power series with the n
n n
m j 51
j s
matrix coefficients being defined through recursion: N 5 P 1
o
Q N , j 5 1, . . . ,n. Here, we set
j j
k 51 k
j 2k
P 5 0 for k . , and s ; minhn, jj. The basic properties of N z are included in Lemma 2 in
k n
Appendix A. The key idea of our formulation is that any n impulse responses in a VARFIMAd , . . . ,d model
1 m
can be exactly calculated from the n coefficients of a finite-order power series resulting from truncated power series multiplication and inversion:
n n
j d
21
A z 5 A 1
O
A z }C z Fz
Qz R 2
n j
n j 51
where
n n
d d
j d
21
C z 5 I 1
O
C z }[D B] 3
n m
j j 51
d
is the truncated version of the infinite series C z in 1. We call 2 the impulse response
n
generating function for the VARFIMAd , . . . ,d model. The operator } is used simply to represent
1 m
n
the truncation rule when only finitely many impulse responses A are desired. We note } does not
j
mean the n impulse responses A are approximated. They are all computed exactly. Also, if all d are
j i
d
equal to zero so that C z 5 I , then 2 gives the impulse response generating function for the
n m
vector ARMA model. It is also common to consider the n-horizon cumulative impulse responses, or n-horizon dynamic
]
n
multipliers, which are defined as A 5 I 1
o
A 5 A 1, for all n . 0. To construct the
n m
j 51 j
n
generating function for cumulative impulse responses, we need to first define the nth order power
n 21
n j
21
expansion of I 2 I z which is L z 5 I 1
o
I z }I 2 I z . With L z, the cumulative
m m
n m
j 51 m
m m
n
impulse response generating function can then be expressed as:
20 C
.-F. Chung Economics Letters 71 2001 17 –25
n n
n
] ]
]
j d
21
A z 5A 1
O
A z }L z A z}L z C z Fz
Qz R 4
n j
n n
n n
j 51
The series L z is a convenient tool as indicated in the following proposition where the relationship
n d
between L z and the generating functions C z in 3 is examined. This proposition can be easily
n n
d 1k
proved by induction. Let’s first define C z to be the operator which is exactly the same as
n d
C z except that the value of each of the m parameters d is increased by k. Also note that
n i
21
L z 5 I 2 I z.
n m
m n
k d
d 1k
Proposition 1. For any integer k 5 0, 61, 62, . . . , we have [L z] C z}C
z. Also,
n n
n d
C z 5 L z if d 5 1 for all i 5 1, . . . ,m.
n n
i
Corollary 1. The impulse response generating function of the VARFIMAd 1 k, . . . ,d 1 k model
1 m
n d 1k
21
with an
arbitrary integer
k can
be computed
from C
z Fz
Qz R}
n k
d 21
[L z] C z Fz
Qz R
n n
So by repeatedly multiplying the impulse response generating function of the VARFIMAd , . . . ,d
1 m
21
model by L z or L z , we may obtain the impulse response generating function of the
n n
VARFIMAd 1 k, . . . ,d 1 k model. Hence, impulse responses can always be computed even
1 m
though the VARFIMAd 1 k, . . . ,d 1 k model may not be stationary or invertible. Also note that
1 m
the cumulative impulse response generating function of the VARFIMAd , . . . ,d model, defined in
1 m
4, is simply the impulse response generating function of the VARFIMAd 1 1, . . . ,d 1 1 model.
1 m
Let l be a vector that includes all estimable structural parameters in F , F , . . . ,F , Q z,
1 p
Q z, . . . ,Q z, and R, as well as the m fractional differencing parameters d , d , . . . ,d . Given the
1 q
1 2
m 2
vectorization operator ‘vec ? ’, let’s also stack vecA , vecA , . . . , and vecA into an nm 3 1
1 n
vector a. We note that impulse responses are nonlinear functions of the estimable parameters l, which
ˆ are indirectly defined by 2, and can be written generally as
a 5 al. Given an estimator l of l, ˆ
impulse responses a can be estimated by al .
Many estimators of l have been proposed in the literature for the univariate ARFIMA model, such
as the maximum likelihood estimators, both in the frequency-domain Fox and Taqqu, 1986 and in the time-domain Sowell, 1992, the conditional sum of squares estimator Chung and Baillie, 1993;
Baillie et al., 1996a,b, and the minimum distance estimator Tieslau et al., 1996, Chung and Schmidt, 1999. Moreover, under stationarity and invertibility conditions, all of these estimators have normal
limiting distributions. Let S
l be the corresponding asymptotic variance–covariance matrix, then the ˆ
asymptotic variance–covariance matrix for
al is Dl Sl Dl9, where Dl 5 ≠al ≠l. Since the operators of vectorization and Kronecker product are linear, we find D
l is exactly not just approximately equal to the stack of the coefficients of the following matrix generating function:
d
≠vec[A z] ≠vec[C
z]
n n
n 21
]]]]
9
]]]]] }
[R9Qz9 Fz I ]
m
≠ l
≠ l
≠vec[Qz] ≠vecR
d 21
d 21
]]]] ]]]
1 [R9 C
z Fz ]
1 [I C
z Fz Qz]
n m
n
≠ l
≠ l
≠vec[Fz]
21 d
21
9
]]]] 2 [R9Qz9 Fz
C z Fz ] 5
n
≠ l
C .-F. Chung Economics Letters 71 2001 17 –25
21
The procedure for computing the derivatives of the cumulative impulse responses is quite similar. All it requires is to multiply 5 by I L z.
m n
While the computation involved in 4 and 5 for the vector ARMA case i.e. d 5 ? ? ? 5 d 5 0
1 m
is identical to Mittnik and Zadrozny’s 1993 recursive forms [their equations 3 and 5], it seems easier to write computer programs based on our formulation. The advantage of the expressions 4 and
5 is that, by using the notation of generating functions, we can avoid the messy expressions of sums of products and recursion as well as some unusual large matrices such as the permutation matrix. In
the terminology of computer programming, what we do here essentially is using products and inverse of generating functions to symbolize the computer programs that calculate the sums of products and
recursion. We also note that inversion is always with respect to the same series Fz. With such uniformity, the resulting computer programs will be highly structural and efficient. Because 5
includes all parameters, it may in fact be viewed as a kind of closed-form expression that resembles Mittnik and Zadrozny’s equation 8.
Note that in the univariate case with m 5 1, all the m 3 m matrices reduce to scalars and the VARFIMAd , . . . ,d model scales down to the usual univariate ARFIMA p, d, q model. In such a
1 m
case the vector l contains 1 1 p 1 q scalar parameters d, F , . . . ,F , Q , . . . ,Q . The 1 1 p 1 q
1 p
1 q
columns of D l are simply the coefficients of the power series ≠A z ≠d, ≠A z ≠F , j 5 1, . . . , p,
n n
j
and ≠A z ≠Q , j 5 1, . . . ,q, respectively, while these power series can be easily computed as
n j
follows:
j 21 d
n
≠A z ≠
C z
n n
n n
21 21
d 21
j
]] ]]]
} Fz
Qz }
Fz Qz
O
C
O
d 1 k z
6
HF G J
j
≠d ≠d
j 51 k 50
≠A z ≠
Fz
n n
n d
22 21
j
]] ]]
} 2
C z Fz Qz
} Fz
A z z , j 5 1,2, . . . , p
7
n n
≠ F
≠ F
j j
≠A z ≠
Qz
n n
n d
21 d
21 j
]] ]]
} C z Fz
} C z Fz
z , j 5 1,2, . . . ,q
8
n n
≠ Q
≠ Q
j j
j
The last results in 7 and 8 are based on the facts that 2 ≠Fz ≠F 5 ≠Qz ≠Q 5 z , while the
j j
d d
d
last result in 6 is due to the equalities ≠C ≠d 5 C
[≠ ln Gd 1 j ≠d 2 ≠ ln Gd ≠d] 5 C
j j
j j 21
21
o d 1 k
. See Gradshteyn and Ryzhik 1980, Eq. 8.365.3.
k 50
3. Analyzing impulse responses
