Full text of DOI:10.1080/000368400322813

Applied Economics
ISSN: 0003-6846 (Print) 1466-4283 (Online) Journal homepage: http://www.tandfonline.com/loi/raec20
A general model for short-term interest rates
Ching-Fan Chung & Mao-Wei Hung
To cite this article: Ching-Fan Chung & Mao-Wei Hung (2000) A general model for short-term
interest rates, Applied Economics, 32:2, 111-121, DOI: 10.1080/000368400322813
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32
Applied Economics, 2000, , 111±121
A general model for short-term interest
rates
CHING-FAN CHUNG* and MAO-WEI HUNG
The Institute of Economics, Academia Sinica, Taipei, Taiwan
A general one-factor model for short-term interest rates is proposed. Besides the long
memory fractionally integrated mean process, the model also consists of a power
function of the interest rate as well as the GARCH e€ ect in the conditional variance.
The estimation results show that, while there is no evidence for fractional integration
in the mean beyond the well-known martingale property, both the power function of
the interest rate and the GARCH e€ ect but not the ARCH e€ ect are highly sig-
(
)
ni®cant in the formation of the conditional variance. Test results also con®rm a
structure change in October 1979 due to the shift in the Federal Reserve monetary
policy.
I. INTRODUCTION
by, for example, Mankiw and Miron 1986 and Mishkin
(
)
1992 . Backus and Zin demonstrate theoretically that long
(
)
The behaviour of the short-term interest rate has been the
focus of extensive studies in economics and ®nance. Many
researchers have investigated the relationship between
macroeconomic variables and the term structure of interest
memory may exist in the short-term interest rates and ®nd
some empirical support for their argument. Such long
memory in interest rates can have profound implications
on asset pricing and thus deserves further investigation.
In this paper, we propose a generalized one-factor model
for the short-term interest rate process. In the continuous-
time version of the model we propose Fractional Brownian
Motion, instead of the standard Brownian Motion, for the
innovations to capture the potential long memory in the
mean of the short-term interest rate. Besides this long
memory mean process, our model also includes the
GARCH e€ ect and a power function of the interest rate
in the conditional variance. We believe the GARCH e€ ect
rates. For instance, Estrella and Hardouvelis 1991 ex-
(
)
amine the relationship between output and the term struc-
ture of interest rates. In ®nance, pricing interest rate
options and designing hedging strategies all depend on
the dynamic behaviour of the term structure of interest
rate. A great number of models have been suggested in
the literature to explain such dynamics. See, for example,
Brennan and Schwartz 1977, 1979, 1980 , Vasicek 1977 ,
(
)
(
)
and Cox et al. 1981, 1985 .
(
)
In a recent paper Chan et al. 1992 propose a continu-
(
)
is a natural extension of Chan et al.’s 1992 model of the
( )
ous-time model for the short-term interest rate that nests
most of the previous di€ usion models. In their model both
mean and conditional variance of the interest rate are func-
tions of the level of interest rate. Their main empirical
result is that incorporating a power function of interest
rate into the conditional variance substantially improves
the ®t of the model.
conditional variance. With a long memory mean as well as
the power function of the interest rate and the newly added
GARCH e€ ect in the conditional variance, our framework
permits much greater ¯ exibility than all previous models.
Our empirical results show that, in addition to the power
function of the interest rate, the GARCH e€ ect but not
(
ARCH e€ ect is signi®cant in determining the conditional
)
More recently, along a di€ erent line of research Backus
variance of the short-term interest rate. We also ®nd that
there is no empirical evidence for the long memory in the
mean beyond the well-known martingale property. It is
and Zin 1993 challenges the well-known belief that the
(
)
short-term interest rate process is a martingale, suggested
*To whom all correspondence should be addressed; e-mail: chungcf@ccms.ntu.edu.tw
#
Applied Economics ISSN 0003±6846 print/ISSN 1466±4283 online
2000 Taylor & Francis Ltd
111

112
C.-F. Chung and M.-W . Hung
µ
suggested that the long memory detected by Backus and
cant improvement over those models with being ®xed at a
Zin 1993 may result from an overly simpli®ed constant
conditional variance in their model.
Although it is widely believed that a structure break in
the interest rate process occurs around October 1979 due to
given value such as 0 Vasicek, 1977 , 1 Cox et al., 1985 ,
( ) ( )
(
)
or 2 Brennan and Schwartz, 1977 . In view of this inter-
(
)
esting result it is natural to ask the following question:
how is such a model of volatility related to the popular
changes in Federal Reserve policy, Chan et al. 1992 were
(
)
GARCH generalized autoregressive conditionally hetero-
(
not able to ®nd such a structure break in their empirical
investigation. They attribute their results to the more ¯ ex-
ible treatment of the conditional variance. However, being
equipped with an even more general model, we obtain evi-
dence contrary to Chan et al.’s ®nding of no structure
break. Instead, our estimation con®rms the earlier result
that a structure break in the short-term interest rates did
exist in October 1979. It seems that the model of Chan et
al., although more ¯ exible than its predecessors, may still
su€ er from misspeci®cation bias due to the omission of the
GARCH e€ ect. In terms of the testing of structural breaks,
the incorporation of the additional GARCH e€ ect appears
to be critically important.
scedastic speci®cation? It is particularly interesting to
)
know whether rµt =2 is alternative or complementary to the
GARCH e€ ects. An obvious way to examine this issue is
to consider a model which includes both rµt =2 and the
GARCH e€ ects.
Once the GARCH e€ ects are incorporated into the vola-
tility process, the maximum likelihood estimation MLE
(
)
seems to be more straightforward than the GMM pro-
cedure. So in this paper we employ the approximate
MLE which is standard in the GARCH literature. We
note one prerequisite for the MLE is the distributional
assumption. Under an appropriate distributional assump-
tion the MLE will be more e cient than its GMM counter-
part. The price we pay for such e ciency gain is of course
the potential biases in MLE when the assumed distribution
is not supported by the data. However, recent literature
The remainder of the paper is organized as follows. The
proposed one-factor model for short-term interest rate is
described in Section II. The estimation method and the
estimation results are presented in Section III and Section
IV, respectively. The testing results for the structure break
are given in Section V. Section VI discusses the estimation
and testing results.
e.g. Bollersley and Wooldridge, 1992 indicates that the
(
)
MLE under the normality assumption may still be robust
to certain distributional misspeci®cations.
In this paper we also investigate the e€ ects of the large
kurtosis of the interest rate distribution through the t-dis-
tribution with its degree of freedom being treated as a free
parameter. A small estimate of the degree of freedom
implies a kurtosis that may be larger than the one from
the normal distribution. Procedures based on t-distribu-
II. THE MODEL
Chan et al. 1992 combine several well-known continuous-
(
)
tions are common in the GARCH literature Bollerslev,
time models for the riskless interest rate and propose the
following speci®cation:
(
1986, 1987 .
)
In addition to the distributional assumption, an even
more important condition for using MLE is that the data
must be stationary. In this regard, we note it is widely
believed that the short-term interest rate, especially before
October 1979, was a martingale. See for example, Mankiw
rµt =2dWt
ˆ
…
†
rt dt
…
†
1
‡
‡
¼
¬
-
drt
where rt is the spot rate at time t and Wt is the standard
Brownian motion. The key feature of their model is that
the volatility of interest rate changes depends on a power
function of the level of the interest rate. The discrete-time
approximation to Equation 1 is
and Miron 1986 and Mishkin 1992 . As a result, apply-
(
)
(
)
ing the OLS estimation which is MLE under the normality
(
ˆ
µ
assumption and the assumption that
where the dependent variable rt‡
0 to Equation 2
)
µ=2
ˆ
…
†
2
¡
‡
‡
¼
¬
-
"
‡
t
rt‡
rt
rt
rt
1
1
¡
rt is martingale di€ er-
1
2
…
†
ˆ
…
†
ˆ
ence while the independent variable rt is non-stationary, we
"
"
where E
0 and E
1. Using the generalized
‡
‡
t
1
t
1
will inevitably get a small estimate of the coe cient.1
-
method of moments GMM estimation, Chan et al. 1992
)
(
)
(
®nd that the term rµt =2 plays a very important role in the
These previous empirical results indicate that the usual
stochastic di€ erential Equation 1 may not be a suitable
model for the short-term interest rate.2
formation of volatility. More speci®cally, they demonstrate
µ
that allowing to be a free parameter constitutes a signi®-
1 For example, using Mishkin’s 1990, 1992 monthly data from February 1964 to October 1979 see Section IV , we obtain the following
(
)
(
)
^
^
ˆ
…
†
ˆ
:
¡
… ¡
†
¬
:
:
-
:
:
OLS estiamtes t-statistics in parentheses . On one-month interest rate, we have
0 2428 1 87222,
0 0399 1 7098 ,
(
)
2
2
2
^
^
¼
^
¬
^
¼
ˆ
ˆ
ˆ
…
†
ˆ
¡
… ¡
†
ˆ
ˆ
:
:
:
:
-
:
:
:
0 502, and R
0 016. On two-month rate, we have
0 2212 1 7717 ,
0 0347 1 5900 ,
0 473, and R
0 014.
2
2
^
^
^
¼
ˆ
…
†
ˆ
¡
… ¡
†
ˆ
ˆ
¬
:
:
-
:
:
:
:
-
On three-month rate, we have
0 2227 1 7513 ,
0 0339 1 5634 ,
0 473, and R
0 013. None of the estimates is
signi®cant and R2 are all small.
„
t
0
2 Given the stochastic di€ erential Equation 1, it is the di€ erence rt
¡
…
†
rs ds, instead of the interest rate rt itself, that is a
‡
¬
-
martingale. See éksendal 1992 .
(
)

General model for short-term interest rates
113
One way to modify model in Equation 1 is to replace the
standard Brownian motion Wt by the increasingly more
popular fractional Brownian motion so that more ¯ exible
non-integer orders of integration may be considered. The
main di€ erence between the standard Brownian motion
and the fractional version is that the latter permits persis-
Simple algebra yields an alternative expression for
Equation 6:
d
…
†
…
Š
†
…
†
ˆ
…
†
7
‰
¡
ˆ
‡
¡
¡
rt‡
1
-
¬
¼
"
‡ ‡
1 t
1
1 L 1
L
t
1
¬
¬=-
where
. We note this model reduces to Equation 2
ˆ
if d 0. Since as mentioned earlier there is plenty of
empirical evidence for the martingale property of the
short-term interest rate, we believe it is the ®rst-di€ erence
tent autocorrelations long memory . We thus propose the
(
)
following two-equation continuous-time model for the
short-term interest rate rt:
ˆ
…
†
ˆ
rt‡
1
¡
¡
rt, instead of the level rt, that
rt‡
1
L rt‡
1
1
is stationary. In terms of rt‡ the Equation 7 becomes
d
1
ˆ
…
†
…
†
3
‡
‡
¬
-
¼
drt
rt dt
tdWt
d¡ 1
…
†
…
Š
†
ˆ
…
†
8
‰
¡
‡
¡
1 L 1 L
-
¼
"
‡ ‡
t 1 t
1
rt‡
1
1
p

2t dWt
2
µ
rt
2
ˆ
…
†
dt
t
…
†
4
¡
‡
¼
®
½¼
¯¼
d
2
t
¬
where the
term vanishes. It should be pointed out that
the fractionally integrated model is particularly useful here
since not only does it allow for a more ¯ exible order of
where Wt is a standard Brownian motion with a unit vari-
ance and Wtd is a fractional Brownian motion that is inde-
pendent of Wt and is de®ned by3
¡
integration so that both orders d and d 1 are equally
natural, it also nests the unit root or the martingale
(
)
…
t
1
‡
0
d
Wtd
t
x dWt
ˆ
case with d 1. Hence, with Equation 8 we are able to
test the signi®cance of unit root against the fractional alter-
natives.
…
†
t
ˆ
…
†
…
†
x
¡
…
†
1
¡
d
0
0
Here, d is an arbitrary real number, Wt is an ordinary
The fractionally integrated process Equation 5 was ®rst
(
)
Brownian motion which has a unit variance and is inde-
introduced by Granger and Joyeux 1980 , Granger 1980
)
(
)
(
…
†
is the gamma function.
¡
pendent of Wt, and
To specify the discrete-time counterparts of Equation 3,
we note that Avram and Taqqu 1987, Theorem 2 have
and Hosking 1981 . In contrast to the unit root process
which exhibits perfect persistence with the impulse
(
)
(
)
responses approaching 1, the fractionally integrated pro-
f
g
shown that if the stochastic process Zt follows a fraction-
<
cess with d 1 is mean reverting and its impulse responses
ally integrated process
converge to 0 at a slow hyperbolic rate. We note that the
impulse responses of the conventional ARMA model con-
verge at fast exponential rates. While the ARMA model is
said to have short memory, the fractionally integrated pro-
d Zt
t
1
T
…
†
ˆ
ˆ
…
†
5
¡
"
; . . . ;
1
L
t
<
:
"
t
for d 0 5, where L is the usual lag operator and are
independently and identically distributed i.i.d. with ®nite
(
)
cess has long memory. See Chung 1995 for a more com-
(
)
second moments, then
plete analysis of the impulse responses of various time
series models and Baillie 1996 for a survey of the applica-
(
)
‰
Š
Ts
X
1
!
d
!
d
tions of the long memory process in ®nance and economics.
The speci®cations Equations 7 and 8 belong to the class of
fractionally integrated autoregressive moving-average
models, or the ARFIMA models, and can be denoted as
! 1
as T
Zt Ws
T
ˆ
t
1
d
2
!
!
where
denotes convergence in distribution,
1 Zt and x is the largest integer that is smaller
Var
T
P
T
…
†
‰
Š
ˆ
t
†
…
†
;
¡
;
;
;
ARFIMA 1 d 0 and ARFIMA 1 d 1 0 , respectively.
(
than or equal to x. This is a functional central limit the-
Finally, we note Backus and Zin 1993 also propose the
(
)
orem for the fractionally integrated process Equation 5 .4
(
)
ARFIMA models for the interest rates to explain certain
properties of the term structure of interest rates. By ®tting
a number of ARFIMA models, Backkus and Zin suggest
that fractional integration, although modest, exists in the
interest rates.
It helps justify the following discrete-time approximation
to Equation 3:
¡ d
ˆ
…
†
…
†
6
¡
‡
‡
¡
1 L
¬
-
¼
"
‡
t
rt‡
rt
rt
‡
1
t
1
1
…
† ˆ
"
"
t
where
are assumed to be i.i.d. with E 0 and
† ˆ
1. Note that we also rely on the standard func-
t
Let’s turn to the conditional variance process in Equation
4. To implement it empirically, we simply note Bollerslev
2
t
…
"
E
tional central limit theorem to justify the discrete-time
model in Equation 2 as an approximation to the continu-
et al. 1994 have shown its discrete-time approximation is
(
)
2
µ
rt
2
¯"2t
ˆ
…
†
9
‡
‡
¼
®
¶¼
‡
t
1
t
ous-time model Equation 1 .
(
)
‡ :
d 0 5
d
3 The basic properties of the fractional Brownian motion Wtd are as follows: 1 E W
0. 2 Both W d and W¬d
† ˆ
have the same
2
d
…
=¬
t
( )
( )
t
t
d
… ¡
†
: ; :
¡
ˆ
2
0. 3 If d
( )
j
j
¬ >
®nite-dimensional distributions for any
0 5 0 5 , then Wt has stationary increments and E Wt Wsd
‡
1. The standard Brownian motion is a special case of the fractional Brownian motion with d 0. See Mandelbrot and
2d
2
¡
ˆ
j
j
s
¼
t
Van Ness 1968 and Jonas 1983 for an overview of the fractional Brownian motion.
(
)
(
)
4 A slightly more restricted version can also be found in Davydov 1970 .
(
)

114
C.-F. Chung and M.-W . Hung
2
1
ˆ
d¡ 1
¡
¡
. That is, the conditional variance
2
¶
½
¯
¼
where
1
t
ˆ
ˆ
…
…
†
…
Š
†
L
‰
‰
¡
¡
‡
¡
1 L 1
"
-
1
1
rt‡
‡
t
1
1
2
¼
¼
depends on a power function of rt and follows a
GARCH 1,1 process. Thus the empirical model we esti-
mate consists of Equations 8 and 9. To guarantee that the
model is stationary and the conditional variances are
positive, we impose the following restrictions on the
‡
t
1
1
( )
t¡ 1
X
1
j
†
Š
‡
-
º
1 L
jL rt‡
1
2
‡
t
ˆ
j
0
ˆ
…
†
1
…
†
…
†
Š
‰
¡
‡
¡
‡
º0
ºj
¡
= ¡
¡
j
j
1
with
while
1 and
j
d
1
d
j 1 ,
¡
‡
<
:
-
<
®
¶ <
parameters: d
1
0 5,
1, 0
, 0
1,
2
¼
‡
t
are computed recursively from Equation 9.
¯ <
and 0
1. Note that the model 2 of Chan et al. is a
( )
1
Here we note the parameter
is not involved.5
ˆ
ˆ
ˆ
¯
0,
¬
¶
special case of Equations 7 and 9 with d 0 and
while Equation 7 is an alternative to Equation 8.
IV. THE ESTIMATION RESULTS
We use Mishkin’s 1990, 1992 monthly data on one-
month, two-month, and three-month rates from February
1964 to December 1986, which are a well-known data set
and largely overlap with those used by Chan et al. Hence,
the ®ndings reported in this paper can be readily compared
with those earlier results.
(
)
III. THE ESTIMATION METHOD
Given the proposed model Equations 8 and 9, the
ˆ
" ;
t
; . . . ;
1
assumption that the innovations
t
T , are i.i.d.
standard normal random variables, then the approximate
maximum likelihood estimation maximizes the following
function:
We ®rst ®t the level data to model Equations 7 and 9 and
®nd that, unsurprisingly, the fractional di€ erencing par-
ameter d is uniformly greater than 0.5 which renders the
model non-stationary. We thus switch to the model
Equations 8 and 9 for the ®rst-di€ erenced data. The esti-
mation is conducted under both normal and t-distribu-
tions. In the latter case there is an additional parameter,
the degree of freedom, that characterizes the kurtosis. The
estimation results are presented in Table 1.
¡
2
T
1
ˆ
… ¬ ; - ; ; ®; µ; ¶; ¯†
º
¡
¡
L
d
log 2
T ¡ 1
T ¡ 1
X
X
2
2
1
2
1
2
…
†
10
¡
¼
"
‡
t
log
‡
t
1
1
ˆ
ˆ
t
1
t
1
Apart from the Jacobian of the transformation from rt‡ 1 to
¡
‡
1 are quite small
-
The estimates of both d 1 and
"
which is close to one and asymptotically negligible,
‡
t
1
and statistically insigni®cant. These ®ndings strongly sug-
gest that the short-term interest rates are martingales and
Equation 10 is the same as the true log-likelihood function.
;
;
; . . .
If the initial observations r0 r¡ 1 r¡ 2
are assumed to be
0, then maximizing
thus verify what Mankiw and Miron 1986 and Mishkin
ˆ
ˆ
"
¡ 2
ˆ
(
)
"
"
zero or, equivalently,
0
¡ 1
1992 have suggested. As a result, the best predictors of all
(
)
Equation 10 is asymptotically equivalent to the MLE. As
Bollerslev 1986 suggests, it is also straightforward to write
the future short-term interest rates, conditional on all avail-
able information, are the current level: Et rt‡
(
)
…
† ˆ
rt, for all
down the approximate log-likelihood function when the
j
ˆ
ˆ
; ; . . .
1 2
j
In contrast to such relatively simple dynamics
";
; . . . ;
1
innovations
t
T , are assumed to have the stan-
in the level of the interest rate, the formation of volatility
dardized t-distribution with the degree of freedom being
treated as a free parameter. The asymptotic theory for
the approximate MLE in the ARFIMA models is consid-
appears to be much more complex. While the estimates of
¯
the ARCH parameter are small, almost all the estimates
¶
µ
of the GARCH parameter and the power parameter
ered by Li and McLeod 1986 and Chung and Baillie
(
)
are highly signi®cant. These results seem to imply that the
GARCH e€ ect, as opposed to the ARCH e€ ect, is com-
plementary to the term rtµ . that is, the GARCH e€ ect and
rµt may represent di€ erent aspects of volatility formation,
while the ARCH e€ ect may be o€ set by the rµt term.6
1993 . The approximate MLE has been applied to in¯ a-
(
)
tion rate data by Baillie et al. 1996 .
(
)
Given the model Equation 8 for the ®rst-di€ erenced
(
process rt‡ 1, we compute
tion as follows:
)
"
in the log-likelihood func-
‡
t
1
5
"
If the estimation were based on the level data as in Equation 7, then the innovations
would be
1
‡
t
¡
t
1
X
1
1
d
2
j
‰
Š
‰
Š
†
1 L
ˆ
¡
…
†
…
¡
†
…
¡
†
ˆ
¡
…
… ¡
jL rt‡
1
†
¬
‡
‡
"
-
¬
-
º~
1
1 L 1
L
rt‡
1
‡
t
1
1
2
2
¼
¼
‡
‡
t
1
t
1
ˆ
j
0
‰
Š
†
:
1
ˆ
…
¡
†
…
† = ¡… ¡
¡
‡
º~
º~
¡
with
1 and
j
d
d
j
j
6 Like 0the majority of the previous studies of GARCH models, our analysis concentrates on the GARCH 1, 1 model. We do try to ®t
(
)
other GARCH speci®cations of higher orders and then use the standard model selection criteria such as AIC and SIC to pick the best
one. It is found that none of those GARCH models of higher orders survives this screening process.

General model for short-term interest rates
115
Table 1. Maximum likelihood estimates
¡
‡
-
®
µ
¶
¯
d
1
1
d.f.
One-month interest rate
Normal distribution
¡
¡
:
0 0816
0.0612
0.5193
0.7290
2.9775
16.1586
0.3719
0.0787
†
†
:
0 9059
3.6299
2.2552
1.2734
(
(
(
(
(
)
)
(
)
)
(
)
)
(
)
)
)
)
¡
:
t-distribution
¡ 0 0912
0.0875
0.7573
0.5759
3.1278
0.5449
2.9982
0.0635
5.5013
2.2467
:
1 0352
2.3993
10.1021
0.7661
(
(
(
(
(
)
)
)
Two-month interest rate
Normal distribution
¡
(
:
0 1134
0.2074
1.5955
0.7696
3.7134
2.7897
15.0875
0.0000
0.0000
0.1121
1.9846
¡
†
:
2 2093
(
(
)
)
(
)
)
(
)
)
(
(
)
)
(
)
)
:
t-distribution
0 0114
0.0989
0.8750
0.2931
0.9242
6.8739
0.5651
4.2070
0.1746
4.0439
2.5870
(
0.1338
2.3012
1.6276
(
)
(
(
(
Three-month interest rate
Normal distribution
:
0 0267
0.0602
0.4415
0.2473
4.1310
3.2628
0.5392
6.1178
0.2060
3.5113
0.3219
10.9247
(
)
)
(
(
)
)
(
)
)
(
)
)
(
(
)
)
(
)
)
:
t-distribution
0 0383
0.1179
1.0606
0.2615
2.1384
3.1514
6.7683
0.6120
5.0113
0.1834
1.6665
3.4458
2.8764
(
0.4386
(
(
(
(
Notes: This table presents the approximate MLE of the model Equations 8 and 9:
¡
d
1
‰
Š
…
1
¡
…
†
¡
†
ˆ
‡
-
¼
"
‡
1 t
1
1 L
L
rt‡
‡
1
t
1
µ
rt
2
2
2
ˆ
‡
‡
¼
®
¶¼
¯"
‡
t
1
t
t
under the normal and t-distributional assumptions. The estimation is based on 275 monthly data on three types of short-term interest
µ
®
rates from February 1964 to December 1986. Numbers in parentheses are t-statistics. The value of rt in the rt term is divided by 10 to
stabilize the iterative estimation. As a result, the estimates are magni®ed by the factor of 10µ .
®
ˆ
…
…
†
†
¼
"
‡
t
rt‡
11
‡
As to the degree of freedom of the t-distribution, we ®nd
its estimates range from 3.44 to 5.50 so that the corre-
sponding kurtosis is larger than that of the normal distri-
bution. One of the most striking results in Table 1 is that
under the t-distribution assumption we obtain very similar
estimation results for all three types of rates. We note the
two most important parameters and have similar esti-
mates around 3 and 0.6, respectively. For the one-month
and three-month rates, the parameter estimates are also
quite similar under the normality assumption. However,
in the case of the two-month rate the di€ erent distribu-
tional assumptions appear to have a stronger e€ ect on
the estimates, especially those of the ARCH and
GARCH parameters. Because obviously all three interest
rate distributions have larger kurtosis, the estimation
results based on the t-distribution seem to be more
reliable.7
1
t
1
1
2
µ
rt
2
ˆ
‡
¶¼
¼
®
12
‡
t
1
t
This is a preferred martingale model in which volatility
contains the GARCH e€ ect and the power term rµt . The
estimation results are presented in Table 2. There we ®nd a
highly signi®cant e€ ect of a power function of the interest
rate and a moderate GARCH e€ ect.
The main conclusion we draw from Tables 1 and 2 is that
the level of the short-term interest rate follows a simple
martingale process while volatility is determined by a
more sophisticated dynamic mechanism.
We now show how misspeci®cation in volatility might
a€ ect the estimation of the d 1 and
Table 3 we report the estimation results for a model where
the GARCH e€ ect is suppressed, while in Table 4 we
further constrain the value of to be 1 so that we e€ ec-
µ
¶
¡
‡
1 parameters. In
-
µ
Following the estimation results in Table 1, especially
those from the t-distribution, we consider a reduced speci-
®cation where the values of the three insigni®cant par-
tively have an Ornstein±Uhlenbeck model. These increas-
ingly simpli®ed speci®cations for volatility seem to move
¡
‡
-
the estimates of d 1 and
1 away from zero. In par-
¡
ˆ
ˆ
ˆ
¯
0.
¡
‡
1
-
ameters are restricted as follows: d
1
ticular we ®nd the estimate of d 1 for the one-month rate
The restricted model consists of two equations:
becomes a signi®cant negative value. That is, if the volati-
7 Even though theoretically the estimation under the normality assumption is more robust to the distributional assumptions, the par-
ameter estimates for the two-month rate under the normality assumption appear quite di€ erent from the rest of Table 1. These aberrant
results may indicate problems with the normality assumption.

116
C.-F. Chung and M.-W . Hung
Table 2. Maximum likelihood estiamtes
®
µ
¶
d.f.
One-month interest rate
Normal distribution
0.7522
3.1141
19.9312
0.4724
4.2964
3.7374
(
)
)
(
)
)
(
)
t-distribution
0.5979
2.7092
3.2324
12.0498
0.6112
4.6698
5.7232
2.3281
(
(
(
)
(
)
)
)
Two-month interest rate
Normal distribution
0.7101
4.7010
2.8268
17.2830
0.2229
1.3515
(
)
)
(
)
)
(
)
)
t-distribution
0.7125
2.2687
2.8185
10.4621
0.2826
1.0383
4.3351
2.4575
(
(
(
(
Three-month interest rate
Normal distribution
0.5746
4.4239
2.8168
18.7928
0.3106
1.9860
(
)
)
(
)
(
)
)
t-distribution
0.5466
1.9486
2.9432
9.6628
0.4972
2.3945
3.3082
3.2029
(
(
(
)
(
Notes: This table presents the approximate MLE of the model Equations 11 and 12:
ˆ
¼
"
‡
t
rt‡
‡
1
t
1
1
µ
rt
2
2
ˆ
‡
¶¼
¼
®
‡
t
1
t
under the normal and t-distributional assumptions. The estimation is based on 275 monthly data on three types of short-term interest
µ
®
rates from February 1964 to December 1986. Numbers in parentheses are t-statistics. The value of rt in the rt term is divided by 10 to
stabilize the iterative estimation. As a result, the estimates are magni®ed by the factor of 10µ .
®
Table 3. Maximum likelihood estimates
¡
‡
-
®
µ
d
1
1
d.f.
One-month interest rate
Normal distribution
¡
:
¡ 0 1500
0.1569
1.6416
1.4044
8.8818
3.0046
18.4650
:
1 7786
(
)
)
(
)
)
(
)
)
(
)
)
¡
:
t-distribution
0 1681
0.1718
1.8057
1.4688
4.9357
3.0508
12.2612
6.5717
1.9495
¡
:
2 0320
(
(
(
(
(
)
)
)
Two-month interest rate
Normal distribution
¡
:
¡ 0 0885
0.1532
1.9889
0.9023
2.7830
17.7075
:
1 1682
10.2997
(
)
)
(
)
)
(
(
)
)
(
)
)
¡
:
t-distribution
¡ 0 0854
0.1944
2.0817
0.9652
2.7718
10.3324
4.3448
2.6582
(
:
1 0972
3.9445
(
(
(
(
Three-month interest rate
Normal distribution
¡
:
0 0824
0.1510
2.1316
0.8243
10.7931
2.7566
19.3166
¡
:
1 1492
(
)
(
)
)
)
)
(
)
)
¡
:
t-distribution
¡ 0 0719
0.2192
0.9753
2.8541
3.5367
3.4541
(
:
0 9167
2.4173
(
3.1869
(
10.2758
(
)
(
This table presents the approximate MLE of the following models:
p

µ=
rt
¡
2
d
1
‰
Š
…
1
¡
…
†
¡
†
ˆ
1
‡
-
®
"
‡
t
1
1 L
L
rt‡
1
under the normal and t-distributional assumptions. The estimation is based on 275 monthly data on three types of short-term interest
µ
®
rates from February 1964 to December 1986. Numbers in parentheses are t-statistics. The value of rt in the rt term is divided by 10 to
stabilize the iterative estimation. As a result, the estimates are magni®ed by the factor of 10µ .
®

General model for short-term interest rates
117
Table 4. Maximum likelihood estimates
¡
‡
-
®
d
1
1
d.f.
One-month interest rate
Normal distribution
¡
¡
:
0 1826
0.1729
2.0796
0.7398
20.0671
:
0 5961
(
(
)
)
(
)
)
(
)
)
¡
:
t-distribution
¡ 0 1551
0.1499
0.9996
1.9496
2.7846
:
2 2185
1.8816
4.1882
(
(
(
)
)
)
Two-month interest rate
Normal distribution
¡
:
0 1411
0.2186
3.0314
0.5411
19.3219
¡
:
2 0028
(
¡
)
)
(
)
)
(
)
)
:
t-distribution
¡ 0 0492
0.1601
0.7936
1.6349
2.6475
4.3363
(
:
0 6756
1.8573
(
(
(
Three-month interest rate
Normal distribution
¡
:
0 1377
0.2286
3.2065
0.5111
19.0018
¡
:
1 8376
(
¡
¡
)
)
(
)
)
(
)
)
:
t-distribution
0 0459
0.2029
2.3415
0.7507
1.6443
2.6066
4.6905
(
:
0 6006
(
(
(
Notes: This table presents the approximate MLE of the following model:
p

¡
d
1
‰
Š
…
1
¡
…
†
¡
†
ˆ
1
‡
-
®"
1
1 L
L
rt‡
‡
t 1
under the normal and t-distributional assumptions. The estimation is based on 275 monthly data on three types of short-term interest
µ
®
rates from February 1964 to December 1986. Numbers in parentheses are t-statistics. The value of rt in the rt term is divided by 10 to
stabilize the iterative estimation. As a result, the estimates are magni®ed by the factor of 10µ .
®
It should be noted that the t-distributions are not consistent with the well-known theory by Cox et al. 1985 : the short-term interest
(
)
2
À
rates which follow the present model should have non-centra2l
distributions. While the normal distribution results can be considered
À
large-sample approximations to the case of the non-central
distributions, the testing results for the t-distribution cases reported here
are simply for the reference purpose only.
lity of this short-term interest rate is not properly speci®ed,
then its level can turn into a mean-reverting fractionally
integrated process which is quite di€ erent from a martin-
gale. Other than this long memory result, the signi®cant
We note that, by restricting the order of integration and
applying the GMM estimation to the one-month rate,
µ
Chan et al. obtained a signi®cant estimate 2.9998 for
‡
-
and an insigni®cant but large estimate 0.4079 for
1.
‡
-
1 estimates also indicate a form of short-run dynamics
These results are more or less compatible with our esti-
mates on the one-month rate in Table 3.8 Here, we should
note the methodological di€ erence between the GMM
estimation and the MLE. The GMM estimation procedure
does not require distributional assumption so it is robust
to the distributional assumptions.9 Also, whether data are
stationary or not does not seem to matter in the GMM
estimation so that Chan et al. do not consider the unit
root problem in the interest rate data. In contrast, the
MLE procedure requires us to make a great deal of
e€ ort to take each of these issues explicitly into con-
sideration. Because of this, we believe the resulting MLE
should achieve greater e ciency than the GMM method
does, provided that the distributional assumptions, the t-
in the interest rate level which contradicts the martingale
property. So from Tables 3 and 4 we conclude that the ®rst
and the second conditional moments of the short-term
interest rates cannot be separately estimated. If the volati-
lity process is misspeci®ed, then the estimation of the ®rst
moments can also be spurious. We note our estimation
results from Tables 1±4 not only refute Backus and Zin’s
1993 theoretical proposition of the fractional integration
(
)
in the interest rate but also explain why empirical support
for their theory could have been `found’ when an overly
simpli®ed constant conditional variance is assumed.
The model speci®ed in Table 3 is similar to the model 2
( )
of Chan et al., but with a more ¯ exible order of integration.
8 We have tried to estimate the model of Chan et al. using the approximate MLE. That is, we estimate the model Equation 7 with d
0
ˆ
‡
-
µ
and no GARCH e€ ects. The parameter estimates t-statistics for and
1 are 1.0072 66.4082 and 2.9987 18.5858 , respectively,
( ) ( )
(
)
under the normality assumption. The estimation results under the t-distribution are almost identical. Here, we note that the estimate of
‡
-
1 is larger than but insigni®cantly di€ erent from 1. So the estimated model essentially contains a unit root, which is not inconsistent
with the martingale result we reported earlier.
9Chan et al. claim the GMM estimation is also robust to the GARCH e€ ects without explanation. We will provide some evidence about
how the GARCH e€ ect might bias the GMM estimation and the ensuing hypothesis testing.

118
C.-F. Chung and M.-W . Hung
distribution in particular, are not too far away from the
truth.
In Table 7 we present the log-likelihood values. In the
column under the title `without structural break’ are the
ones corresponding to the estimates for the entire sample
that were reported in Table 2. Those in the next column
under the title `with structural break’ are the sums of two
log-likelihood values obtained from the two sub-period
estimations. The entries in the last column are two times
the di€ erences between the previous two columns and are
the likelihood ratio test statistics for testing the hypothesis
V. STRUCTURAL CHANGES IN OCTOBER
1979
It is widely believed that the short-term interest rate experi-
ences a structural change in October 1979 due to the shift
in Federal Reserve monetary regimes. See, for example,
of no structural break. The degree of freedom of the 2 test
À
is 3 for the normal distribution cases. The critical values at
the 95% level and the 99% level are 7.82 and 9.35, respect-
ively. The degree of freedom is 4 for the t-distribution cases
and the critical values at the 95% level and the 99% level
are 9.49 and 11.14, respectively. From Table 7 we conclude
that the hypothesis of no structural break is strongly
rejected for the two-month and three-month rates, while
it is marginally rejected for the one-month rate.
Conseqently, our estimation results appear to be in con¯ ict
with the conclusion of Chan et al. on one-month rate but
consistent with many earlier results obtained by other
researchers. Since our model Equations 11 and 12 di€ er
from the model of Chan et al. mainly in the additional
GARCH e€ ect, we may argue that their ®nding of no
structural break is due to their insu cient consideration
of conditional heteroscedasticity, which, ironically, was
precisely the same criticism they made on previous works
in the literature.
Huizinga and Mishkin 1984 and Clarida and Friedman
(
)
1984 . However, Chan et al. 1992 report that they could
(
)
(
)
not ®nd evidence for such a structural change based on
their GMM estimation of model 2 . They interpret their
( )
[
]
results as follows: ` . . . their interest rate models may be
rich enough to capture the change in interest rate behav-
iour evident in the post-1979 period. These results also
raise the possibility that previous tests for structural breaks
may be misspeci®ed because of their failure to model the
conditional heteroscedasticity in interest rate changes cor-
rectly’ p. 1222 . We also examine this issue by estimating
(
)
model Equations 11 and 12 separately with data before and
after October 1979. The estimation results are presented in
Tables 5 and 6. It is surprising to see that the estimates of
µ
¶
both and for each of the two sub-periods are much
larger than those from the whole period in Table 2. We also
µ
note the t-statistics for the
estimates are smaller,
obviously due to smaller sample sizes in each sub-period.
Table 5. The ML E for the ®rst subsample from February 1964 to October 1979
®
µ
¶
d.f.
One-month interest rate
Normal distribution
0.3586
3.3630
0.7845
11.5879
3.6063
9.0404
(
)
)
(
)
)
(
)
)
t-distribution
0.4267
(
3.5470
0.7876
9.2487
4.9627
1.9547
5.8980
1.9221
(
(
(
)
)
)
Two-month interest rate
Normal distribution
0.2724
4.3126
3.7624
8.2375
0.8334
18.8842
(
)
)
(
)
)
(
)
)
t-distribution
1.6443
(
3.0277
0.9203
2.0450
16.2446
(
0.3634
3.4780
20.2536
(
(
Three-month interest rate
Normal distribution
0.2542
4.7966
4.3650
7.4079
0.8697
32.8638
(
)
)
(
)
)
(
)
)
t-distribution
6.3153
0.3720
3.9679
4.9215
0.9355
29.0881
2.0136
55.2955
(
(
(
(
Notes: This table presents the approximate MLE of the model Equations 11 and 12:
ˆ
¼
"
‡
t
rt‡
‡
1
t
1
1
µ
rt
2
2
ˆ
‡
¶¼
¼
®
‡
t
1
t
under the normal and t-distributional assumptions. The estimation is based on 187 monthly data on three types of short-term interest
µ
®
rates from February 1964 to December 1986. Numbers in parentheses are t-statistics. The value of rt in the rt term is divided by 10 to
stabilize the iterative estimation. As a result, the estimates are magni®ed by the factor of 10µ .
®

General model for short-term interest rates
119
Table 6. The ML E for the second subsample from November 1979 to December 1986
®
µ
¶
d.f.
One-month interest rate
Normal distribution
0.5108
2.7863
4.4869
8.4381
0.6091
4.9833
(
)
)
(
)
)
(
)
)
t-distribution
0.4618
(
4.4393
0.6411
12.1273
2.1782
6.9563
4.1517
0.4841
(
(
(
)
)
)
Two-month interest rate
Normal distribution
0.3997
2.1521
4.9023
9.6723
0.4075
1.6779
(
)
)
(
)
)
(
)
)
t-distribution
0.3487
(
4.9771
0.4730
14.3891
1.9832
7.9035
2.1024
0.4201
(
(
(
Three-month interest rate
Normal distribution
0.2075
2.0683
4.9738
9.4855
0.6176
3.8909
(
)
)
(
)
)
(
)
)
t-distribution
0.1506
1.6297
5.2170
6.6641
0.7043
4.5802
(
8.3116
0.7436
(
(
(
Notes: This table presents the approximate MLE of the model Equations 11 and 12:
ˆ
¼
"
‡
t
rt‡
‡
1
t
1
1
µ
rt
2
2
ˆ
‡
¶¼
¼
®
‡
t
1
t
under the normal and t-distributional assumptions. The estimation is based on 88 monthly data on three types of short-term interest rates
µ
®
from November 1979 to December 1986. Numbers in parentheses are t-statistics. The value of rt in the rt term is divided by 10 to
stabilize the iterative estimation. As a result, the estimates are magni®ed by the factor of 10µ .
®
Table 7. L og-likelihood values
Without structural break
With structural break
Likelihood-ratio test statistics
12.6144*
One-month interest rate
¡
¡
¡
¡
:
:
Normal distribution
t-distribution
231 5513
225 2441
:
:
{
11.0020
226 7645
221 2635
Two-month interest rate
¡
¡
¡
¡
:
:
Normal distribution
t-distribution
202 2414
189 4505
25.5818*
23.7338*
:
:
195 2152
183 3483
Three-month interest rate
¡
¡
:
:
:
:
Normal distribution
t-distribution
¡ 200 8301
¡ 186 7173
28.2256*
29.9554*
188 8084
173 8307
Notes: This table present the log-likelihood values under the normal and t-distributional assumptions. The second column gives the log-
likelihood values based on 275 monthly data on three types of short-term interest rates from February 1964 to December 1986. The
(
corresponding parameter estimates are in Table 2. The entries in the third column are the sums of two log-likelihood values of the two
)
subsample estimations: one from February 1964 to October 1979 and the other from November 1979 to December 1986. The corre-
sponding parameter estimates are in Tables 5 and 6, respectively. So the log-likelihood values in the third column are from estimations
(
)
that allow for structural break. The models under consideration are Equations 11 and 12:
ˆ
¼
"
‡
t
rt‡
‡
1
t
1
1
µ
rt
2
2
ˆ
‡
¶¼
¼
®
‡
t
1
t
The fourth column gives the likelihood-ratio test statistics. The degree of freedom of the 2 test is 3 for the normal distribution cases. The
critical values at the 95% level and the 99% level are 7.82 and 9.35, respectively. The degree of freedom is 4 for the t-distribution cases
and the critical values at the 95% level and the 99% level are 9.49 and 11.14, respectively. The test statistics with * are signi®cant at 99%
À
{
level. The one with is insigni®cant at 99% level but signi®cant at 95% level.

120
C.-F. Chung and M.-W . Hung
Table 8. L og-likelihood values
Without structural break
With structural break
Likelihood-ratio test statistics
One-month interest rate
¡
¡
¡
¡
:
:
{
{
Normal distribution
t-distribution
233 6993
231 0451
5.3084
4.8794
:
:
230 6871
228 2474
Two-month interest rate
¡
¡
:
:
:
:
Normal distribution
t-distribution
¡ 202 6113
¡ 192 6287
19.9652*
16.3986*
194 1166
185 9173
Three-month interest rate
¡
¡
¡
¡
:
:
Normal distribution
t-distribution
201 7283
192 8603
17.7360*
11.4808*
:
:
186 6077
180 8673
Notes: This table presents the log-likelihood values under the normal and t-distributional assumptions. The second column gives the log-
likelihood values based on 275 monthly data on three types of short-term interest rates from February 1964 to December 1986. The
entries in the third column are the sums of two log-likelihood values of the two subsample estimations: one from February 1964 to
October 1979 and the other from November 1979 to December 1986. So the log-likelihood values in the third column are from
estimations that allow for structural break. The model under consideration is Equation 13:
p

µ=
2
‰
Š
¡
…
†
ˆ
1
‡
-
®
"
‡
t
1
1 L rt‡
rt
1
which is essentially the same as Cehn et al.’s model based on the ®rst-di€ erenced data. The fourth column gives the likelihood-ratio test
statistics. The degree of freedom of the 2 test is 3 for the normal distribution cases. The critical values at the 95% level and the 99% level
À
are 7.82 and 9.35, respectively. The degree of freedom is 4 for the t-distribution cases and the critical values at the 95% level and the 99%
level are 9.49 and 11.14, respectively. The test statistics with * are signi®cant at 99% level. The one with is insigni®cant at 99% level but
{
signi®cant at 95% level.
To rea rm that the di€ erent testing results Chan et al.
get is caused by the omission of the statistically signi®cant
GARCH term in their model of volatility, we conduct
further testing based on the model of Chan et al. for the
®rst-di€ erenced data:
rates. A signi®cant GARCH e€ ect is also present, but it is
less apparent in the two-month rate than the others. In the
absence of the GARCH e€ ect as in the model of Chan et
µ
al., the unambiguously signi®cant estimate of
such as
2.9998 indicates striking sensitivity of the interest rate vola-
tility with respect to the level changes in the interest rate.
However, the presence of the positive GARCH e€ ect in our
p

µ=2
…
†
ˆ
…
†
13
‰
Š
¡
‡
1 L rt‡
-
®
"
‡
t
1
rt
1
1
The testing procedure is the same as that given in Table 7
and the results are shown in Table 8. Unsurprisingly, for
the one-month rate the hypothesis of no structural break
cannot be rejected at 95% level, just like what Chan et al.
have reported. This result clearly suggests the inference of
no structural break does have something to do with the
omission of the GARCH term in their speci®cation of vola-
tility, even though they have correctly included the rµt term.
Incidentally, from Table 8 we also note, contrary to the
case of the one-month rate, that the structural changes
are again present in the two-month and three-month
rates, just like the earlier result in Table 7 based on
model Equations 11 and 12.
µ
model lessens the in¯ uence of the estimate on the e€ ect of
the level changes in the interest rate. The implication of an
additional GARCH e€ ect is that a change in the interest
rate volatility in the next period results not only from the
change in the current level of the interest rate but also from
the persistence in volatility itself. Consequently, the e€ ects
of the level changes in the interest rate on its volatility is
µ
not ®xed at the value of the estimate, but may in fact
µ
¯ uctuate around some values below the estimates ranging
from 2.8 to 3.2. Obviously, it is this extra ¯ exibility that
makes the detection of the structural breaks possible. From
Tables 5 and 6, we notice for all three types of interest rates
µ
¶
the estimates from the ®rst subsample and the estimates
from the second subsample are consistently lower than
their respective counterparts from the other subsamples.
It appears that after 1979 the interest rate volatility
becomes more sensitive to the level of the interest rate
and less persistent.
VI. DISCUSSION
The main conclusion drawn from the estimates of our pre-
ferred model Equations 11 and 12 in Tables 2, 5 and 6 is as
follows. Over the entire sampled period from February
1964 to December 1986, the power function of the interest
rate is highly signi®cant in the determination of the con-
ditional variances of all three types of short-term interest
Also note that the estimates in Tables 5 and 6 di€ er from
those in Table 2 in a systematic way: the subsample esti-
µ
¶
mates of are higher while the subsample estimates of
are lower in comparison with those of the whole sample. In
µ
¶
other words, the estimates of the and parameters from

General model for short-term interest rates
121
Brennan, M. and Schwartz, E. 1979 A continuous time
(
)
the whole sample are not averages of those from the two
subsamples. The reason for this somewhat counter-intui-
tive result may lie in the possible e€ ect cancellation
approach to the pricing of bonds, Journal of Banking and
3
Finance, , 133±55.
Brennan, M. and Schwartz, E. 1980 Analyzing convertible
(
)
µ
¶
between the
and
estimates when the whole sample
15
bonds, Journal of Financial and Quantitative Analysis,
,
period is considered. It should also be pointed out that
such e€ ect cancellation may result in less ¯ uctuating esti-
mates of the e€ ect of the level changes in the interest rate.
That is, estimations based on the longer sampled period
produce smoother e€ ects of the level of interest rate on
volatility than those in the subsamples.
As a concluding remark, we believe our ®ndings in this
paper are substantive contributions to the literature of the
interest rate modelling and our results can be useful in the
valuation of interest rate contingent claim and optimal
hedging strategies. We note, in particular, valuation of
itnerest rate contingent claim is very sensitive to the vola-
tility of the interest rates, while ¯ exible modelling of the
volatility is the centrepiece of our analysis. An interesting
future research subject concerns how the valuation of inter-
est rate contingent claim can take into account both the
power function of interest rate and the GARCH e€ ect.
907±29.
Chan, K., Karolyi, A., Longsta€ , F. and Sanders, A. 1992 An
(
)
empirical comparison of alternative models of the short-term
47
interest rate, Journal of Finance, , 1209±27.
Chung, C.-F. 1995 Calculating and analyzing impulse responses
(
)
and their asymptotic distributions for the ARFIMA and
VARMA models, Econometrics and Economic Theory
Paper No. 9402, Michigan State University.
Chung, C.-F. and Baillie, R. T. 1993 Small sample bias in con-
(
)
ditional sum of squares estimators of fractionally integrated
18
ARMA models, Empirical Economics, , 791±806.
Clarida, R. H. and Friedman, B. M. 1984 The behavior of U.S.
(
)
short term interest rates since October 1979, Journal of
39
Finance, , 671±84.
Cox, J. C., Ingersoll, J. E., Jr. and Ross, S. A. 1981 A re-ex-
(
)
amination of traditional hypotheses about the term structure
36
of interest rates, Journal of Finance, , 769±99.
Cox, J. C., Ingersoll, J. E., Jr. and Ross, S. A. 1985 A theory of
(
)
53
the term structure of interest rates, Econometrica, , 385±
407.
Davydov, Y. A. 1970 The invariance principle for stationary
(
)
15
processes, Theory of Probability and Its Applications,
,
ACKNOWLEDGEMENTS
487±9.
Granger, C. W. J. 1980 Long memory relationships and the
(
)
We are very grateful to a referee for his valuable sugges-
tions.
aggregation of dynamic models, Journal of Econometrics,
14
, 227±38.
Granger, C. W. J. and Joyeux, R. 1980 An introduction to long
(
)
memory time series models and fractional di€ erencing,
REFERENCES
1
Journal of Time Series Analysis, , 15±39.
Avram, F. and Taqqu, M. S. 1987 Noncentral limit theorems
(
)
68
Hosking, J. R. M. 1981 Fractional di€ erencing, Biometrika,
,
(
)
15
and Appell polynomials, Annals of Probability, , 767±75.
165±76.
Backus, D. K. and Zin, S. E. 1993 Long-memory in¯ ation
(
)
Huizinga, J. and Mishkin, F. 1984 In¯ ation and real interest
(
)
uncertainty: evidence from the term structure of interest
rates on assets with di€ erent risk characteristics, Journal of
25
rates, Journal of Money, Credit, and Banking, , 681±700.
39
Finance, , 699±712.
Baillie, R. T. 1996 Long memory processes and fractional inte-
(
)
Jonas, A. B. 1983 Persistent memory random processes, PhD
(
)
73
gration in econometrics, Journal of Econometrics, , 6±59.
Dissertation, Harvard University.
Li, W. K. and McLeod, A. I. 1986 Fractional time series mod-
Baillie, R. T., Chung, C.-F. and Tieslau, M. 1996 Analyzing
(
)
(
)
in¯ ation by the fractionally integrated ARFIMA-GARCH
73
eling, Biometrika, , 217±21.
11
model, Journal of Applied Econometrics, , 23±40.
Mandelbrot, B. and Van Ness, J. W. 1968 Fractional Brownian
(
)
Bollerslev, T. 1986 Generalized autoregressive conditional het-
(
)
motions, fractional noises and applications, SIAM Review,
31
eroskedasticity, Journal of Econometrics, , 307±327.
10
, 422±37.
Bollerslev, T. 1987 A conditionally heteroskedastic time series
(
)
Mankiw, N. G. and Miron, J. A. 1986 The changing behavior of
(
)
model for speculative prices and rates of return, The Review
the term structure of interest rates, Quarterly Journal of
69
of Economics and Statistics, , 542±7.
101
Economics,
, 211±28.
Bollerslev, T., Engle, R. F. and Nelson D. B. 1994 ARCH mod-
(
)
Mishkin, F. S. 1990 What does the term structure tell us about
(
)
els, in Handbook of Econometrics, Vol. 4, Chapter 49, Eds
(
)
25
future in¯ ation? Journal of Monetary Economics, , 77±95.
R. F. Engle and D. L. McFadden, North-Holland,
Amsterdam.
Mishkin, F. S. 1992 Is the Fisher e€ ect for real? A reexamina-
(
)
tion of the relationship between in¯ ation and interest rates,
Bollerslev, T. and Wooldridge, J. M. 1992 Quasi-maximum like-
(
)
30
Journal of Monetary Economics, , 195±215.
lihood estimation and inference in dynamic models with
éksendal, B. 1992 Stochastic Di€ erential Equations, third edi-
(
)
11
time-varying covariances, Econometric Reviews, , 143±72.
tion New York: Springer-Verlag .
Brennan, M. and Schwartz, E. 1977 Savings bonds, retractable
(
)
(
)
3
Vasicek, O. 1977 An equilibrium characterization of the term
( )
bonds, and callable bonds, Journal of Financial Economics, ,
5
231±62.
structure, Journal of Financial Economics, , 177±88.