Long-Maturity Forward Rates
1
Charlotte Christiansen2
The Aarhus School of Business, Denmark
First Version: August 13, 2001
This version: April 24, 2002
1 Comments and suggestions are solicited.
2 Charlotte Christiansen, Department of Finance, The Aarhus School of Busi-
ness, Fuglesangs Alle 4, 8210 Aarhus V, Denmark. Phone: +45 8948 6691, fax:
+45 8615 1943, email: [email protected], homepage: www.asb.dk/~cha.
Long-Maturity Forward Rates
Abstract: The forward-rate curve is almost always downward sloping at the long-maturity
end. This paper contains an empirical analysis of the causal relation between the slope
of the forward-rate curve at long maturities and the volatility of the long-term spot rate.
The term-structure theory predicts a negative correlation. A bivariate regime-switching
model is applied. In the low-variance state (the usual state), a negative correlation between the long-maturity forward-rate slope and the long-rate volatility is recovered, i.e.
the greater the volatility, the steeper the forward-rate curve. The volatility dependence is
both statistically and economically important. In the high-variance state, the relation is
reversed.
Keywords:
Volatility
Forward-Rate Curve; Forward-Rate Slope; Long Maturity; Regime Switching;
JEL Classifications:
G12
1
1 Introduction
The long end of the term structure of interest rates has not been subject
to extensive investigation, which is perhaps surprising given its perceived
importance, cf. for instance Dybvig and Marshall (1996). It has been established empirically that the forward-rate curve is downward sloping for long
maturities. Here we add to the understanding of why this is so. We investigate the relationship between the long-term forward-rate slope and the
term-structure volatility using a financial econometrics approach.
The recent paper by Brown and Schaefer (2000) underscores that new
insights can be gained from analyzing the long-maturity end of the term
structure of interest rates. They discuss why the long end of the forward-rate
curve is almost always downward sloping. A dynamic term-structure model
is used to show that the slope of the forward-rate curve at long maturities is
proportional to minus the term-structure variance. An extended description
hereof is included in Section 3 below.
The previous literature has established the causal relation between interestrate volatility and the shape of the term structure. The larger part of the
empirical studies of the shape of the term structure of interest rates are
based upon spot rates rather than forward rates. Litterman, Scheinkman
and Weiss (1991) were among the first to suggest that interest-rate volatility
and the shape of the yield curve are correlated. Litterman and Scheinkman
(1991) show that the majority of the variation in the yield curve is driven
by 3 factors, namely level, slope, and curvature, and that the curvature is
closely connected to the term-structure volatility. An increase in the level of
the short-rate volatility implies a decrease of the individual yields (i.e. the
yields of different maturities decrease by different amounts), cf. Longstaff
and Schwartz (1993). When the volatility is relatively high, the yield curve
is likely to be upward sloping, cf. Engle and Ng (1993). Christiansen and
Lund (2001) show empirically that the yield curve is steeper and more concave, the greater the short-rate volatility.
In contrast to the cross section analysis in Brown and Schaefer (2000), Dy2
bvig, Ingersoll and Ross (1996) deal with the time series perspective: Dybvig
et al. (1996) show that absence of arbitrage restricts the permissible evolution
of the shape of the forward-rate curve: The long-term forward rate cannot
fall over time. Within a one-factor term-structure model Brown and Schaefer (1994) document a relationship between forward rates and the short-rate
volatility. Yet, it is not a simple one-to-one relationship. Carverhill (2001)
calibrates long-maturity (above 13 years) forward rates to the Vasicek (1977)
term-structure model which has been refined to account for downward sloping forward rates and non-attenuating volatility. Carverhill (2001) argues
that these features are empirical regularities of long-term forward rates.
In this paper, the slope of the long-term forward-rate curve and the longterm spot rate are analyzed in a bivariate regime-switching framework where
the conditional long-rate volatility is included in the mean equation. The
level of the long-rate variance defines the two states. The economy spends
most of its time in the low-variance state. In the low-variance regime, the
long-maturity forward-rate slope depends negatively on the long-yield volatility, and the dependence is significant in both a statistical and economic
sense. Thus the long-maturity region of the forward-rate curve is steeper
(downward sloping), the larger the volatility. This corresponds to the presumptions formed on the basis of Brown and Schaefer (2000). In contrast, in
the high-variance state, the long-maturity forward-rate slope depends positively on the long-rate volatility and the forward-rate curve is flatter, the
higher the volatility.
The remaining part of the paper is organized as follows. Section 2 introduces the forward-rate data. The theoretical relation between the slope
of the long-maturity forward-rate curve and the term-structure volatility is
laid out in Section 3, and the following section contains the empirical model
framework. The empirical results are analyzed in Section 5. Finally, Section
6 concludes.
3
2 The Term-Structure Data
The Bliss (1998) term-structure data are applied: The McCulloch and Kwon
(1993) zero-coupon bonds. In particular, the extracted zero-coupon bond
prices are transformed into 1-month forward rates:
+1 )
f = ln(b ) − ln(b
(1)
where b is the price of a zero-coupon bond at time t that matures in n month.
In this way 1-month forward rates starting at 10, 15, 20, and 25 years are
calculated. The McCulloch and Kwon (1993) data are constructed such that
extrapolation outside the range of maturities of the underlying bonds is not
permitted. The available sample period is from 1987 through 1998: A total
of 628 weekly observations.
We apply discrete - as opposed to instantaneous - forward rates, because
discrete forward rates are in use by the market. The same kind of reasoning
has led the term-structure literature from the Heath, Jarrow and Morton
(1992) model for instantaneous forward rates to the market model of Miltersen, Sandmann and Sondermann (1997) and Brace, Gatarek and Musiela
(1997) for discrete forward rates.
It appears from Figure 1 and Figure 2 that the forward-rate curve is
downward sloping from around 15 years. This can also be illustrated by
looking at the frequency of downward sloping forward curves at different
maturity segments. Four different line segments are considered, namely 1015 years, 15-20 years, 20-25 years, and 15-25 years, cf. Table 1. At the very
long end of the forward-rate curve, it is almost always downward sloping,
e.g. from 15 to 25 years more than 99% of the observations represent such
downward sloping curves. This closely resembles the findings of Brown and
Schaefer (2000). It is also evident that it is not till at the very long end
that the slope of the forward-rate curve turns negative. We follow Brown
and Schaefer (2000) and pay particular attention to the slope of the forward
curve at the segment between 15 and 25 years to maturity:
25
15
(2)
S ≡ f −f .
n
n
n
t
t
t
n
t
t
t
4
t
Admittedly, this maturity combination is more or less arbitrary.
In the following, the 25-year zero-coupon yield (the long rate) as well as
the slope of the long-maturity forward-rate curve (i.e. the difference between
the 25-year 1-month forward rate and the 15-year 1-month forward rate) are
analyzed. These variables are described by their summary statistics in Table
2.
The long rate has a mean of 7.69% and a standard deviation of 0.97%.
The long rate is strongly serially correlated and so are its squares. Thus, a
model for the long rate should include heteroscedasticity. It is safe working
in levels for the long rate: Testing the hypothesis of a unit root where the
alternative includes five lags, a trend, and an intercept results in a p-value
below 5%.
The average 25-15-year slope is negative and equals -1.48% and has a
standard deviation of 0.87%. The distribution of the slope is skewed to the
left and leptokurtic. The slope as well as the squared slope series are strongly
serially correlated. Again, the empirical model should include heteroscedasticity. Not surprisingly the null hypothesis of a unit root is rejected at any
usual level of significance.
3 The Slope of the Long-Maturity ForwardRate Curve
In order to illustrate the relation between the slope of the forward-rate curve
and interest-rate variance, Brown and Schaefer (2000) apply the Gaussian
two-factor time-homogeneous affine term-structure model, cf Duffie and Kan
(1996). We do not consider the choice of the dynamic term-structure model to
be crucial, rather, a model is chosen for illustrative purposes. The advantage
of this specification is the simplicity it implies.
The forward rate at time t with maturity τ is denoted f (t, τ ), and the
variance of the τ -period zero-coupon yield is denoted σ2.1 The forward rate
i
i
1
i
i
Notice that the notation for instantaneous and discrete forward rates are almost iden-
5
maturing at time τ depends linearly on the variance of the τ -period yield.
The spread between the forward rates at maturity τ1 and τ2 can be expressed
as:
1 σ2τ 2 − σ2τ 2 for τ > τ .
f (t, τ2) − f (t, τ1 ) = · · · −
(3)
2
1
2 22 11
A simplifying assumption about the size of the two mean-reversion parameters is introduced: One of them is close to zero, while the other is not.
Then the terms not explicitly included in equation (3) are negligible:
1 σ2τ 2 − σ2τ 2 for τ > τ .
f (t, τ2 ) − f (t, τ1 ) ≈ −
(4)
2
1
2 22 11
The assumption lies in continuation of the usual parameter estimates.
We restrict our attention to long maturities, i.e. large values of τ . The
volatility of the zero-coupon yields is assumed to be constant for the longmaturity segment of the yield curve which corresponds well with the findings
of Carverhill (2001). Let the variance of the long-term zero-coupon yield be
denoted σ2, whereby (4) is reduced to:
1
(5)
f (t, τ2 ) − f (t, τ1) ≈ − σ2 (τ22 − τ12 ) for τ2 > τ1 .
2
(5) implies that the slope of the forward-rate curve at the long-maturity end
depends negatively on the long-rate variance. When the forward curve is
downward sloping, this means that the greater the volatility, the steeper the
forward-rate curve is.
The manner in which Brown and Schaefer (2000) conduct the empirical analysis is as follows. They assume that equation (5) holds exactly, i.e.
the functional form of the relationship between the slope and the long-rate
volatility is taken as given. From equation (5) an estimate of the long-rate
volatility is obtained from the long-term forward-rate slope. It is subsequently investigated how well the thus obtained long-rate volatility forecasts
and approximates the time series volatility and reversely, the predictability of the slope of the long forward-rate curve using long-yield volatility is
analyzed.
i
i
i
l
l
tical: f (t, τ ) and ftτ , respectively.
6
Here we investigate whether there is empirical support for the hypothesis
that the slope of the long end of the forward-rate curve depends negatively on
the long-rate volatility. In other words, we do not make use of the functional
form expressed in equation (5). Yet, the hypothesis that we investigate is
based hereon. The analysis is conducted using an empirical model.
4 The Empirical Model Framework
To investigate the volatility dependence described in the previous section, we
apply a bivariate model for the long-term (25-year) zero-coupon yield and the
slope of the long end of the forward-rate curve, namely the 25-year minus
the 15-year 1-month forward rates. We let Y denote the long-term zerocoupon yield at time t and S the slope of the long-term forward-rate curve.
Furthermore, the analysis is extended to accommodate regime switching.
Regime switching has been applied successfully in empirical term-structure
analysis by amongst others Hamilton (1988), Cai (1994), and Gray (1996).
Thus, we investigate whether the relationship between the long-rate and the
long-maturity forward-rate slope varies across economic regimes.
A first-order 2-state Markov chain is assumed, cf. Hamilton (1994, chapter 22). The state of the economy at time t is denoted s . “First-order” means
that the probability that s equals j only depends on the most recent value
of s −1: prob[s = j |s −1 = i, s −2 = k, s −3 = l, . . .] = prob[s = j |s −1 = i].
At each t the economy is in one out of two possible states, i.e. the state
variable equals 0 or 1. However, the state of the economy is unobserved by
the econometrician. The transition probabilities between the two states are
assumed constant:
t
t
t
t
t
t
t
t
[
prob[s
prob[s
prob[s
prob st
t
t
t
t
= 0|s −1 = 0]
= 1|s −1 = 0]
= 1|s −1 = 1]
= 0|s −1 = 1]
t
t
t
t
7
t
= p
= 1−p
= q
= 1 − q.
t
(6)
The transition probabilities are estimated at the same time as the remaining
parameters of the model. The above assumptions imply that the unconditional probability of being in a particular state at time t is also constant,
e.g.:
1−q
(7)
prob[s = 0] =
2−p−q
The level of the long-rate variance defines the two states. In addition, the
level of the variance of the long-maturity forward-rate slope is also regime
switching. s = 0: “low” variance for the long rate and “low” variance for
the slope. s = 1: “high” variance for the long rate and “high” variance for
the slope. In order to reduce the number of states - and thereby the number
of parameters - the variance of the long-rate and the variance of the slope
are assumed to be of the same relative size.
The conditional mean vector evolves according to a VAR specification in
order to accommodate for the serial correlation in the data. The order of the
VAR is determined by minimizing the Schwartz criterion in a homoscedastic single-regime setting; here leading to a second order lag specification.
The conditional mean equation also contains volatility-in-mean terms (more
hereon follows shortly).
Thus, the mean equation is as follows:
t
t
t
Yt
= Φ0 + Φ1
St
+γ
st
h1t
Yt−1
St−1
+δ
+ Φ2
st
h2t
+ .
Yt−2
St−2
(8)
t
Φ0 is a vector of constants and Φ1 and Φ2 are constant matrixes. The
error vector, , has mean zero and conditional covariance matrix H ≡
t
h1t
h12t
h12t
h2t
t
. The volatility-in-mean parameter vectors are constant in
for s = 0
δ for s = 0
each regime: γ =
and δ = 0
, respectively.
γ1 for s = 1
δ1 for s = 1
The conditional means depend on the conditional volatility of the long
rate as well as the volatility of the long-maturity forward-rate slope. Thus, it
st
γ0
t
t
st
t
t
8
is possible to test statistically if the slope depends negatively on the long-yield
variance as predicted by the theory in Section 3. Moreover, other interesting
insights can be gained from studying the other volatility-in-mean parameters
in equation (8).
Due to the heteroscedasticity in the data uncovered in Section 2, the conditional variances are described by autoregressive conditional heteroskedastic
(ARCH) processes. The appealing feature of applying the ARCH framework
is that it so easily lends itself to incorporating (a function of) the volatility in the mean specification, cf. Engle, Lilien and Robins (1987). Here we
have formulated a volatility -in-mean specification, the most commonly used
approach in the literature. Naturally, the specification of the function of
the volatility that enters into the mean specification does not influence the
qualitative conclusions.
Laumoreux and Lastrapes (1990) have shown that if there is an unaccounted structural change in the data, the volatility process will be perceived
to be highly persistent even if this is not the case. This potential problem
is overcome by using the switching-ARCH (SWARCH) model, cf. Hamilton and Susmel (1994) and Cai (1994). Here, the conditional variances are
assumed to evolve according to Cai (1994) SWARCH(1) processes:
hjt
= ω + α (
j,s t
j
2
−1)
j,t
(9)
for s = 0
for j = 1, 2. The ω-parameters are conω 1 for s = 1
stant in each regime. To ensure strictly positive variances and covariancestationarity the following must be fulfilled; ω 0, ω 1 > 0, and α ≤ 1 for
j = 1, 2.
To identify state 0 as the low-variance regime the following restrictions
are imposed:
where
ωj,st
=
ωj,0
j,
t
t
j,
ω1,0
j,
≤ ω1 1 and ω2 0 ≤ ω2 1.
,
,
,
j
(10)
The conditional covariance process is assumed to be described by the
constant conditional correlation (CCORR) specification of Bollerslev (1990)
9
which has been extended to the regime switching framework by Ramchand
and Susmel (1998). The conditional covariance is proportional to the product
of the conditional volatilities:
h12t
=ρ
h1t
st
(11)
h2t ,
for s = 0
.
ρ1 for s = 1
Apart from the conditional covariance specification, the bivariate model
in Ramchand and Susmel (1998) is somewhat different from ours: Volatilityin-mean effects are not accommodated and the Hamilton and Susmel (1994)
SWARCH specification is applied. The bivariate regime switching model in
Kugler (1996) is homoskedastic and thereby also quite different from our
specification.
The parameters of the model are estimated simultaneously using the
Quasi Maximum Likelihood (QML) technique with a Gaussian likelihood
function and applying Bollerslev and Wooldridge (1992) robust standard errors. The estimation is conducted in GAUSS using the GAUSS module
Constrained Maximum Likelihood. A combination of the Berndt-Hall-HallHausman and the Newton-Raphson numerical maximization algorithm is applied. Starting values are set to their unconditional values.
and the correlation is constant in each regime:
ρst
=
ρ0
t
t
5 The Empirical Effect of the Long-Rate
Volatility
The model introduced in equations (8)-(11) allows us to investigate the influence of the long-rate volatility on the slope of the long end of the forward-rate
curve. Table 3 shows the results from the estimation; the upper part of the
table contains the estimates of the mean parameters, the middle part the
covariance parameters, and the bottom part the transition-probability parameters.
Before we scrutinize the impact of long-rate volatility on the slope of the
10
long end of the forward-rate curve, it is convenient to introduce a couple of
simplifications. Some of the volatility-in-mean parameters, γ0[1], γ1[1], and
δ0 [1], are not individually significant, nor are they jointly significant: The
robust Wald test for H0: γ0[1] = γ1[1] = δ0[1] = 0 gives rise to a p-value
of 14%. This means that in the low-variance regime the long rate does
not depend on volatility at all. In the high-variance regime, the long rate
only depends on the slope volatility. The correlation between the two data
series is insignificantly different across the two regimes: H0: ρ0 − ρ1 = 0
has associated a p-value of 69%. The two hypotheses combined result in a
p-value of 18%. We apply the general-to-specific modeling approach to go
from the general model to the restricted specification where γ0[1] ≡ γ1[1] ≡
δ0 [1] ≡ ρ0 − ρ1 ≡ 0. The thus resulting model is denoted the “restricted”
model, and the parameter estimates are available in Table 4. The remaining
analysis is based on the results in Table 4, which is organized as the previous
table.
When the economy is in state 0 today, the probability of also being in
state 0 next week is 75% (p̂), whereas, when the economy is in state 1, the
probability of also being in state 1 next week is 11% (q̂). The unconditional
probability of being in state 0 is 78%, cf. equation (7). The economy spends
most of its time in the low-volatility state, so, high volatility is the exception
rather than the rule. From the ordinary t-test, q̂ appears not to be significantly greater than zero. However, this test is not applicable here, because
the state-1 parameters are not identified under the null hypothesis. Judging from the usual t-statistic, p̂ is significantly positive and also significantly
smaller than 1. The same caveat with respect to unidentified parameters
under the null hypothesis applies. Instead, we interprete the transition probabilities as providing evidence that the economy is in state 0 most of the
time, whereas, it is only sporadically in state 1.
To investigate this preposition further, the “ex ante” and the “smoothed”
probabilities are calculated, cf. Hamilton (1994, Chapter 22). The information available at time t is denoted Φ , and T is the most recent observation
t
11
in the data set. The ex ante probability describes the probability of being
in state 0 (say) today based on the information from the previous week;
prob[s = 0|Φ −1 ]. In contrast, the smoothed probability of being in state 0 is
based on all the available information in the data set: prob[s = 0|Φ ]. When
the smoothed probability of being in state 0 at time t was above 50%, i.e.
prob[s = 0|Φ ] > .5, we say that the economy was in state 0 at that time.
The smoothed probability of being in the high-variance state is illustrated in
Figure 3. The figure supports our presumption that the economy is usually
in the low-variance state and once in a while it transfers to the high-variance
state.
Fortunately, the conditional variances are statistically smaller in the lowvariance state than in the high-variance state, because otherwise the definition of the two regimes would be meaningless. The Wald test of no switching
in the variances processes, ω1 0 − ω1 1 = ω2 0 −ω2 1 = 0, has a p-value far below
1%. It is also noticeable that the variance processes are not very persistent.
The ARCH-parameters, α1 and α2, are equal to .11 and .17, respectively,
which is far lower than what is usually seen in single regime models.
The slope of the long-maturity end of the forward-rate curve and the
long rate are negatively correlated. The correlation coefficient equals -18%
(in both regimes).
The VAR(2) specification of the mean equation appears to be reasonable.
Firstly, the parameter estimates are hardly different in the unrestricted and
the restricted specifications. Secondly, the Wald test for the null hypothesis
of a lower lag order, i.e. VAR(1), is rejected at any usual level of significance.
Moreover, the hypothesis of an AR(2) specification is also rejected, (the pvalue equals 4%).
The theory in Section 3 predicts a negative relation between the long-rate
volatility and the slope of the forward-rate curve at the long-maturity end. As
the economy is usually in state 0, we hypothesize that the volatility-in-mean
parameter for the long-forward slope in state 0 is negative, i.e. γ0[1] < 0.
The volatility-in-mean parameter is in fact negative and significant, (the onet
t
t
t
T
,
,
,
12
,
T
sided p-value is far below 1%). Moreover, the parameter value is numerically
large, γ̂0[1] = −4.8, which indicates that not only is the long-yield volatility
statistically significant in explaining the slope, it is also of vast economic
importance. Thus, in this state, the higher the interest-rate variance, the
smaller the slope of the forward-rate curve at the long-maturity region. As
the forward-rate curve is downward sloping (i.e. negative) at the long end,
this means, more steeply downward sloping.
In the high-variance state, the relationship between the long-maturity
forward-rate slope and the long-rate volatility is reversed, it is significantly
positive, (the one-sided p-value equals 4%). So, in the high-variance regime,
the long-maturity region of the forward-rate curve is greater - and thereby
flatter - the greater the interest-rate volatility.
The long-rate volatility dependence of the long-maturity forward-rate
slope is not statistically identical in the two regimes, the p-value for δ0[1] −
δ1 [1] = 0 takes on a value below 1%. Overall, the negative relationship dominates the positive relationship, and the slope tends to become steeper as the
interest-rate volatility increases.
The long-rate volatility does not enter significantly into the long-rate
mean equation in either of the two states.
The mean equation also includes the volatility of the long-maturity forwardrate slope. The long rate only depends on the slope volatility in the highvariance state. This dependence is significantly positive, however numerically
small, δ̂1[1] = .13. In the high-variance state, the slope depends negatively
on its own volatility, and the dependence is significant and numerically large
δ̂1 [2] = −2.7. Still, the long-rate volatility is much more important for the
slope than its own volatility.
In conclusion, our empirical findings support the ex ante hypotheses
formed on the basis of the dynamic term-structure model.
13
6 Conclusion
The forward-rate curve is almost always downward sloping at the longmaturity end, i.e. for maturities above 15 years. This paper has applied
the results in Brown and Schaefer (2000) to form a testable hypothesis that
can possibly explain this phenomenon. Brown and Schaefer (2000) use a
two-factor affine term-structure model. By introducing a couple of additional assumptions, they have shown that the slope of the forward-rate curve
at long maturities is proportional to minus the volatility of the long-maturity
spot rate. Therefore, we have conjectured that the long-maturity forwardrate slope depends negatively on the long-rate volatility.
In this paper, the slope of the long-term forward-rate curve and the longterm spot rate have been analyzed in a bivariate regime-switching framework
where the conditional long-rate volatility has been included in the mean
equation. The level of the long-rate variance has defined the two states. The
economy has been found to spend most of its time in the low-variance state.
In the low-variance regime, the long-maturity forward-rate slope has been
shown to depend negatively on the long-yield volatility, and the dependence
has been found to be significant in both a statistical and economic sense.
Thus the long-maturity region of the forward-rate curve has been seen to be
steeper (downward sloping), the larger the volatility. Thus, we have found
empirical support for the conjecture stated above. In contrast, in the highvariance state, the long-maturity forward-rate slope has been found to depend
positively on the long-rate volatility and the forward-rate curve is flatter, the
higher the volatility.
The expectation hypothesis (EH) states that the forward rate for a future period equals the short rate expected to prevail at that time (ignoring
term premia). Thus, within the EH understanding of the term structure of
interest rates, downward sloping forward-rate curves are evidence that the
market expects the short rates to fall in the long run. Interestingly, when
the spot-rate curve is upward sloping (by far the most usual shape), the EH
implies that the short rates are expected to increase in the future. The styl14
ized facts of the shape of the spot-rate and the forward-rate curves provide
contradicting information regarding the expectation of the short rate for the
near and more distant future. In future research it would be interesting to
look further into this apparent contradiction.
15
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18
10-15 years 15-20 years 20-25 years 15-25 years
4.8%
72.6%
98.6%
99.7%
The table reports the percentage of observations where the forward rate curve is downward
sloping at different segments of the forward rate curve, i.e. when ftn2 − ftn1 < 0 for n2 > n1 .
The weekly data cover the period 1987-1998.
Table 1: Negative Slopes
19
Y-25 f-15 f-25 S-2515
Mean
7.69 8.20 6.74 -1.46
0.96
Std.dev. 0.97 1.01 0.84
Skewness -0.33 -0.23 -0.51 -1.61
6.41
Kurtosis 2.44 2.31 3.72
AC, 1
.988 .985 .950
.936
AC, 5
.943 .941 .819
.812
.900
AC, sq, 1 .987 .983 .954
.700
AC, sp, 5 .941 .938 .833
The table reports the summary statistics for the following time series: 25-year zero-coupon
yield (Y-25), 15-year forward rate (f-15), 25-year forward rate (f-25), and the difference
between the 25-year and 15-year forward rates (S-2515y). The time series of interest rates
are measured in per cent per annum. For each time series the following statistics are
reported: mean, standard deviation, skewness, kurtosis, autocorrelation (order 1 and 5),
and autocorrelation of the squared time series (order 1 and 5). The weekly data cover the
period 1987-1998.
Table 2: Summary Statistics
20
Φ0
Φ1
Φ2
γ0
γ1
δ0
δ1
ω0
ω1
α
-0.039
0.882*
0.185§
0.111*
-0.192*
0.318
-0.709
0.154
0.527§
0.009*
0.023*
0.117§
ρ0
ρ1
p
q
Yt
(0.059)
(0.041)
(0.073)
(0.041)
(0.073)
(0.479)
(0.548)
(0.098)
(0.211)
(0.001)
(0.006)
(0.046)
-0.195*
-0.148
0.734*
0.081
0.487*
-0.014
0.667*
0.006
0.252*
-4.583*
3.962§
-0.383§
-2.911*
0.019*
0.167*
0.173*
(0.054)
(0.091)
(0.085)
(0.091)
St
(0.167)
(0.026)
(0.051)
(0.024)
(0.050)
(1.689)
(1.592)
(0.155)
(0.574)
(0.004)
(0.037)
(0.049)
QML estimates of the 2-state regime switching model for the long yield, Yt, and the slope
of the long end of the forward rate curve, St , where Yt is the 25 year zero-coupon
yield
Yt
and St is the difference between 25 year and 15 year 1 month forward rates:
=
Φ0 +Φ1
Yt−1
St−1
+Φ2
Yt−2
St−2
St
√h + δ √h + . has mean 0 and conditional
1t
s
2t
t
t
√ √
)2 , for j = 1, 2, and h = ρ h h . p and q are
+ γst
t
covariance: hjt = ωj,st + αj (j,t−1
12 t
st
1t
2t
the transition probabilities of the 2-state Markov switching process. The weekly data
cover the period 1987-1998. Bollerslev and Wooldridge (1992) robust standard errors in
parenthesis. ¶(§) [*] indicates that the parameter is significantly different from zero at a
10% (5%) [1%] level.
Table 3: 2-State Regime Switching Model
21
Φ0
Φ1
Φ2
γ0
γ1
δ0
δ1
ω0
ω1
α
0.004
0.888*
0.196*
0.108§
-0.201*
0.132§
0.009*
0.023*
0.113§
ρ
p
q
Yt
(0.042)
(0.047)
(0.074)
(0.047)
(0.073)
(0.058)
(0.001)
(0.006)
(0.048)
-0.179*
0.753*
0.109
0.511§
-0.032
0.670*
0.022
0.253*
-4.807*
3.420¶
-0.369
-2.716*
0.019*
0.174*
0.168*
(0.041)
(0.085)
(0.186)
St
(0.209)
(0.042)
(0.068)
(0.039)
(0.065)
(1.791)
(1.903)
(0.787)
(0.627)
(0.004)
(0.036)
(0.058)
QML estimates of the 2-state regime switching model for the long yield, Yt, and the slope
of the long end of the forward rate curve, St , where Yt is the 25 year zero-coupon
yield
and St is the difference between 25 year and 15 year 1 month forward rates:
Φ0 + Φ1
Yt− 1
St−1
+ Φ2
Yt−2
St−2
+ γst
Yt
St
=
√h + δ √h + . γ [1] = γ [1] = δ [1] = 0.
1t
s
2t
t
0
1
0
t
has mean 0 and conditional covariance: hjt = ωj,st + αj (j,t−1 )2 , for j = 1, 2, and
√ √
h12t = ρ h1t h2t. p and q are the transition probabilities of the 2-state Markov switching
process. The weekly data cover the period 1987-1998. Bollerslev and Wooldridge (1992)
robust standard errors in parenthesis. ¶(§) [*] indicates that the parameter is significantly
different from zero at a 10% (5%) [1%] level.
t
Table 4: Restricted 2-State Regime Switching Model
22
Figure 1: Forward Rate Curve Surface Plot
23
Figure 2: Yearly Average Forward Rate Curves
24
Figure 3: Smoothed State 1 Probability
25