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Estimating Continuous-Time Stochastic Volatility
Models of the Short-Term Interest Rate:
A Comparison of the Generalized Method of Moments and the Kalman Filter
TRAVIS R. A. SAPP
Forthcoming in the November 2009Review of Quantitative Finance and Accounting
JEL Classifications: G12, C51
Keywords: Stochastic volatility, short interest rate, generalized method of moments; GMM;Kalman filter; quasi-maximum likelihood
______________* Travis Sapp may be reached at College of Business, 3362 Gerdin Business Bldg., Iowa State University, Ames, IA50011-1350, Phone: (515) 294-2717, Fax: (515) 294-3525, email: [email protected].
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Estimating Continuous-Time Stochastic Volatility Models of the Short-Term Interest Rate:
A Comparison of the Generalized Method of Moments and the Kalman Filter
Abstract
This paper examines a model of short-term interest rates that incorporates stochastic volatility asan independent latent factor into the popular continuous-time mean-reverting model of Chan etal. (1992). I demonstrate that this two-factor specification can be efficiently estimated within ageneralized method of moments (GMM) framework using a judicious choice of momentconditions. The GMM procedure is compared to a Kalman filter estimation approach. Empiricalestimation is implemented on US Treasury bill yields using both techniques. A Monte Carlostudy of the finite sample performance of the estimators shows that GMM produces more heavilybiased estimates than does the Kalman filter, and with generally larger mean squared errors.
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I. Introduction
The short-term risk-free interest rate is an important and fundamental economic variable.
It is a key determinant in pricing fixed income securities and derivatives, as well as equity
derivatives, and it serves as a reference asset in various asset pricing models such as the Capital
Asset Pricing Model and the Arbitrage Pricing Theory. Because of its importance numerous
studies have attempted to model and estimate the stochastic behavior of the short rate, but to date
such efforts have met with only limited success.
There are two common approaches to modeling the short rate that have been taken in the
literature. The first specifies an underlying stochastic process, usually in continuous time, which
is hypothesized to generate yields and then attempts to estimate the model based on a
discretization, and employing the statistical distribution implied by the model. For instance, a
model based on Brownian motion would imply a Gaussian distribution for estimation purposes.
Specifying models in continuous time has certain theoretical appeal, especially for derivatives
pricing, but can also lead to intractable transitional densities for empirical estimation.
The second approach specifies a discrete-time model and is then more flexible in
selecting a statistical distribution which captures the salient features of the empirical data. For
example, ARMA models can be used to capture the structure in the conditional mean, and
ARCH models have been found to provide a good description of the conditional second moment.
These models have been estimated under Gaussian as well as other distributional assumptions.
Although the discrete-time model may have a structural interpretation which could also be
modeled in continuous time, or, as Nelson (1991) shows, the discrete-time model may under
certain regularity conditions converge to a particular continuous-time model in the limit, the
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discrete-time parameters and distribution may not necessarily correspond to a particular
continuous-time specification.
Originating with the arithmetic Brownian motion model in Merton (1973), researchers
have tended to favor the first approach outlined above, where a theoretical model for the data
generating process is specified in a continuous-time framework. Subsequent work by Vasicek
(1977) modified the Brownian motion specification to allow for mean reversion in a continuous-
time framework, and the Cox, Ingersoll, and Ross (CIR) (1985) square-root process introduced
the first heteroskedastic theoretical model for the short rate, also in continuous time. As these
particular continuous-time models admit closed-form solutions, they are popular specifications
for empirical work, but they are not necessarily the most accurate models in describing the
empirical features of the data.
Chan, Karolyi, Longstaff, and Sanders (CKLS) (1992) show that at least eight of the most
popular continuous-time specifications can be nested within the stochastic differential equation
( ) dZrdtrdr ++= (1)
For instance, setting = yields the CIR square root process, while setting = 1 and= 0
gives geometric Brownian motion. CKLS use this specification to compare the various models
and show that a level effect, where higher interest rates are associated with higher volatility, is
an important characteristic of the short rate dynamics. Specifically, models which do not allow
for a level effect (i.e., = 0) are shown to perform especially poorly.
All of the models represented by the CKLS process are one-factor models in that they
allow the drift and diffusion functions to depend only on the short rate. The CKLS model does
allow for one important source of conditional heteroskedasticity in the short rate, the so-called
level effect. However, a problem with one-factor models is that they are typically rejected in
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empirical work, as in Ait-Sahalia (1996) for instance, who tests and rejects all of the most
popular one-factor specifications. The overwhelming amount of evidence documenting time-
varying volatility in financial data in general, and in interest rates in particular (see, e.g.,
Bollerslev, Chou, and Kroner (1992)), suggests that a second factor corresponding to conditional
volatility is required to describe the short rate dynamics.
Studies which have attempted to incorporate time-varying volatility include Longstaff
and Schwartz (1992) and Brenner et al. (1996). These papers use discrete Euler approximations
to continuous-time models and incorporate time-varying volatility through GARCH
specifications. Longstaff and Schwartz use a Gaussian distribution for maximum likelihood
estimation, whereas Brenner et al. employ a Students tdistribution in order to better account for
the widely documented leptokurtosis found in interest rate data. The Brenner et al. paper is of
particular interest because that study finds statistically significant estimates of a conditional
variance specification that is the product of a GARCH(1,1) process and a levels effect, thus
showing that both level effects and stochastic volatility are key characteristics of the short rate
dynamics.
Simulation methods can also be used in order to estimate the parameters of a continuous-
time stochastic volatility process. The approach of Andersen and Lund (1997) is to specify a
continuous-time stochastic volatility process and estimate the continuous-time parameters using
a semi-nonparametric simulation technique known as the Efficient Method of Moments (EMM).
However, while EMM represents an interesting avenue of current research, simulation-based
estimation methods tend to be rather complex to implement and out of the reach of mainstream
users. The difficulty of EMM is compounded by the sensitivity of the parameter estimates to the
choice of the auxiliary model used in the estimation procedure.
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Due to the difficulties of estimating continuous-time models, many studies have relied on
GARCH specifications to capture stochastic volatility. While GARCH provides a good empirical
description of the data, it is a deterministic volatility specification. Specifically, GARCH is a
predictable function of past squared shocks to the mean of the process and does not allow for a
second independent source of risk. In contrast, a competing class of models for time-varying
volatility which has gained much recent interest in the literature is the stochastic volatility
specification. Stochastic volatility offers an attractive alternative to GARCH, and has theoretical
appeal due to its consistency with continuous-time modeling specifications.
Vetzal (1997) adopts a two-factor specification for the short-rate, modeling volatility as a
separate latent factor, and finds that this outperforms a one-factor model. Similar to Vetzal
(1997), this paper also models interest rate volatility as a separate latent factor while allowing for
level effects. Specifically, a second stochastic factor for the volatility process takes the form
( ) vdZdtd ++= 22 lnln (2)
where is a second independent Wiener process.dZ
Following CKLS, a minimal amount of structure is placed on the interest rate process by
retaining their GMM framework, while generalizing the CKLS model to a stochastic volatility
setting using Taylors (1986) stochastic autoregressive volatility (SARV) specification, which is
a discrete approximation of the process in (2). In the literature, applications of the log-normal
stochastic volatility model have typically been limited to the foreign exchange and stock markets
due to their straightforward application under these circumstances.1 Applications to interest rate
series, however, present an added degree of complexity due to the presence of mean reversion
1 Examples of the stochastic volatility model applied to foreign exchange rates include Melino and Turnbull (1990);Harvey, Ruiz, and Shephard (1994); and Jacquier, Polson, and Rossi (1994). Examples of the stochastic volatility
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and volatility level effects, both of which are represented in the specification of the CKLS model
in equation (1).
Vetzal (1997) casts the two-factor estimation problem in a GMM framework, but
unfortunately is unable to estimate all of the parameters of the model. In particular, he finds it
necessary to arbitrarily fix the level parameter in order to estimate the other model parameters.
As I argue below, in all likelihood, this is due to his reliance on an excessive number of moment
conditions. Thus, a contribution of this paper will be to show that a more judicious choice of
moment conditions allows the full stochastic volatility model, including the level parameter, to
be estimated within a unified GMM framework.
Another method for estimating a latent variable specification which has recently appeared
in the literature is a quasi-maximum likelihood (QML) procedure that casts the stochastic
volatility (SV) model in state-space form and uses a Kalman filter to estimate the parameters.
This procedure may also be adapted to the more complex two-factor interest rate model which
incorporates level effects. I implement the Kalman filter approach for both models and the
results are examined for comparison with the GMM estimation technique.
Using a sample of 2,769 weekly observations on the three-month U.S. T-bill rate, the
original CKLS model is first estimated using the same GMM specification as in the CKLS paper.
Then the SV model is estimated by GMM and also by QML using the Kalman filter. The two
procedures give similar results for the stand-alone SV model, though the Kalman filter estimates
a higher degree of volatility persistence. Finally, the joint CKLS-SV model is estimated by both
GMM and QML. Results show that the level effect remains significant in the presence of
stochastic volatility, with a value for the level parameter, , of 0.38 when estimated via GMM
model being estimated on stock returns include Jacqier, Polson and Rossi (1994); Gallant, Hsieh, and Tauchen(1995); Harvey and Shephard (1996); and Engle and Lee (1996).
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and 0.43 via QML, both of which are significantly less than the value of 1.5 found by CKLS.
This result is roughly consistent with that of Brenner et al. who use a GARCH specification and
find that the level effect drops to around 0.5. Furthermore, the results found are similar to those
of Andersen and Lund (1997), who use the EMM simulation procedure and get a value for the
level parameter of 0.54, though their model is rejected. The GMM specification is not rejected by
Hansens test of overidentifying restrictions.
A Monte Carlo study of the stochastic volatility interest rate model reveals that both
GMM and QML give robust parameter estimates for the approximate sample size used in the
empirical study, though GMM has a larger bias for the level parameter and produces larger mean
square errors. Hansens test of over-identifying restrictions for GMM is shown to reject the
model slightly more often than theory would predict, though it is not rejected in the empirical
implementation on Treasury bills. Overall, results from Monte Carlo simulation indicate that the
Kalman filter estimation procedure is generally more robust than that of GMM.
The remainder of the paper is organized as follows. In Section 2, the stochastic volatility
model is first described, and then the joint CKLS-SV specification is derived. Section 3 describes
the data used in the empirical estimation of the model. Section 4 presents and discusses the
results of the estimation procedures. Section 5 examines and compares the performance of the
GMM and Kalman filter estimators in a Monte Carlo setting. Section 6 makes some concluding
remarks.
II. Estimating the Stochastic Volatility Interest Rate Model
A. Generalized Method of Moments Estimation
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The one-factor CKLS model was given in equation (1) as ( ) dZrdtrdr ++= . As
noted earlier, this equation nests at least eight popular one-factor interest rate models. A
difficulty that arises in going from the CKLS continuous-time model to empirical estimation is
that, depending upon the value of, a different distribution for interest rate changes is required.
For instance, the Vasicek and Merton models specify a normal density, whereas the CIR model
implies a non-central chi-square density. CKLS avoid this apparent difficulty by adopting a
GMM framework, where specification of a density is not required for estimation. Specifically,
CKLS use the following Euler approximation of the continuous-time process specified in (1):
tttt rrr ++= 11 (3)
[ ] [ ] 2122,0 == ttt rEE
CKLS estimate the parameter vector with elements , , 2,andusing the following four
moment conditions h(, xt) :
(4)( ){ }
( )
0xh =
=
12
122
21
221,
ttt
tt
tt
t
t
rr
rrEE
While this specification allows for one important form of heteroskedasticity, the level effect, it
does not incorporate general stochastic volatility, which Brenner et al. show to be a significant
characteristic of the short rate dynamics using a GARCH specification under an assumed Student
terror density. I next introduce a simple stochastic volatility model and describe how the CKLS
model can be generalized to include true stochastic volatility, while retaining the GMM
estimation framework.
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GMM has been used in several papers such as Melino and Turnbull (1990), and Jacquier,
Polson, and Rossi (JPR, 1994) to estimate stochastic volatility models. For an observed series of
data yt, the simple lognormal stochastic volatility model that approximates the continuous-time
process given in equation (2) is specified as
(5)tutt
ttt
u
Zy
++=
=
2
12 lnln
where Zt andut are mutually independent i.i.d. N(0, 1) random variables. Note that this differs
from a GARCH-type specification in that volatility is not a deterministic function of past squared
shocks to the mean equation (i.e., the variance equation has its own error term), and hence
volatility is latent, or unobservable. The assumption of lognormality of the volatility process is a
convenient parameterization because it allows closed-form solutions for the moments and
precludes negative variances. This model specification also relies upon the assumption of
normality in the mean equation, thus imposing more structure than in the CKLS model.
In their paper JPR use 24 different moment conditions derived from the statistical
properties of the lognormal specification of the stochastic volatility model. Letting ( )= 1/m
and ( )222 1/ = us , the analytic expressions are as follows:
( )
( ) ( )( )
( ) ( )
( ) ( )( )( ) ( )( )10,,1
10,,1/2
3
/22
/2
2222
44
33
22
==
==
=
=
=
=
jEyyE
jEyyE
EyE
EyE
EyE
EyE
jttjtt
jttjtt
tt
tt
tt
tt
(6)
where, for any positive integerj and positive constantsp, q,
( ) ( )8/2/exp 22sppmE pt +=
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and ( ) ( ) ( ) ( )4/exp 2spqEEE jqtptq jtpt = .
The three parameters , , andu are then estimated from any three or more of the above 24
moment conditions using GMM. Common specifications include the subsets
3 moments: tyE , ( )2tyE , and 1ttyyE 5 moments: tyE , ( )2tyE , ( )4tyE , 1ttyyE , and ( )2 12 ttyyE 9 moments: tyE , ( )2tyE , 3tyE , ( )4tyE , 1ttyyE , 3ttyyE , 5ttyyE , ( )2 22 ttyyE , and
( )2 4t2tyyE 14 moments: 9 moments + 7ttyyE , 9ttyyE , ( )2 62 ttyyE , ( )2 82 ttyyE , and ( )2 102 ttyyE
Andersen and Sorensen (1996) conduct a Monte Carlo study of GMM estimation of the
lognormal stochastic volatility model and show that the just-identified model (three moment
conditions) is especially unstable. They recommend using at least nine moment conditions, but
note that using too many moment conditions also leads to inefficiency due to the
correspondingly large number of elements in the GMM covariance matrix which must be
estimated from the data. I note that Melino and Turnbull (1990) and Vetzal (1997) each use 35
moment conditions to estimate a stochastic volatility model, and both studies are unable to
estimate the level parameter. Instead they estimate the model while arbitrarily fixing the value of
the level parameter, which is quite unfortunate as this is a parameter of central interest. Based on
the caution to avoid an overabundance of moment conditions, I adopt the more parsimonious
fourteen-moment specification as prescribed in Andersen and Sorensen.
The stochastic volatility model may be combined with the CKLS one-factor interest rate
specification to give a two-factor model as follows. The discretized CKLS model specification
was given in equation (3) as
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tttt rrr ++= 11
Factoring the error term t into three separate components of interest, stochastic volatility t , a
level effect , and a standard normal shockZt , we can write . The combined,
discretized CKLS-SV model that we wish to estimate is then given by
1tr
tttt
Zr1
=
tttttt Zrrrr 111 ++=
(7)tutt u ++= 2
12 lnln
where [ ] [ ] 2121221211 |,0| == ttttttttt rIZrEIZrE
andItis the information set available at time t. The expectations in the last line follow as a result
of the independence ofZtandut. Specifically,
[ ] [ ] [ ] [ ](0,1)i.i.d.~since0
|||| 111111
NZ
IZEIrEIEIZrE
t
tttttttttt
=
=
and
[ ] [ ] [ ] [ ]
[ ] [ ][ ] ( )22
1
21
221
1211
21
221
12
1211
21
221
2
1since|
constantis|since||
||||
=
==
=
=
t
tttt
ttttttt
tttttttttt
r
ZEIEr
IrIZEIEr
IZEIrEIEIZrE
If< 1, then the variance process is stationary and the unconditional variance, 2, will exist.
Since ( )1/ is the unconditional mean of the logarithmic variance process and ( )2
is its variance, the unconditional variance of the interest rate process is given by
2 1/ u
2
( ) ( )
+
=
2
22
121exp
u (8)
2 Note that because is stochastic, in making the transformation out of logarithms we use2 = exp[m +s2/2].2t
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GMM exploits the convergence of selected sample moments to their unconditional expected
values, so the CKLS model and the stochastic volatility process can be effectively linked by
substituting the expression for the unconditional variance into the second moment of the CKLS
model. Also, the errors from the mean equation must be normalized by the level term, ,
before entering each of the other moment conditions. The actual analytic moment expressions
used in estimating the combined CKLS stochastic volatility model are listed in the Appendix.
1tr
Since the CKLS model and the SV model share the same analytic expression for the
variance, we are left with a vectorh(; xt) ofr= 4 + 14 - 1 = 17 total moment conditions for
GMM estimation. The GMM estimate is the value ofT ( )u ,,,,,= that minimizes
the quadratic form
J= [g(; xT)]' [g(; xT)],1 TS(1 x r) (rx r) (rx 1)
where the vector of moment conditions h(; xt) are replaced by the sample estimates given by
g(; xT) (1/T) h(, xt)=
T
t 1(rx 1) (rx 1)
and is an estimate ofTS
S = E{[ h(0, xt)][ h(0, xt-v)]'}.( ) =
=
T
t vT
T1
/1lim
(rx r) (rx 1) (1 x r)
Due to the presence of heteroskedasticity and serial correlation in the yield data, the weighting
matrix is estimated using a Bartlett kernel with a fixed bandwidth of 26 (corresponding to six
months of weekly observations).
TS
B. Quasi-Maximum Likelihood Estimation via the Kalman Filter
Harvey, Ruiz, and Shephard (1994) show how the lognormal stochastic volatility model,
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which was given in equation (5) as
tutt
ttt
u
Zy
++=
=
2
12 lnln
can be represented in linear state-space form and then estimated by maximum likelihood using a
Kalman filter algorithm. Specifically, if we square the first equation and then take logarithms,
letting ht= ln , we get2t
(9)22 lnln ttt Zhy +=
Since Zt is a standard normal random variate, the mean and variance of are known to be
-1.27 and2/2, respectively. If we think of the log variance, ht, as representing an unobservable
state, then we can cast the stochastic volatility model in state-space form as
2ln tZ
(10a)ttt hy ++=2ln
tutt uhh += 1 (10b)
where t = and where the constant term, besides representing the actual mean of the
variance process, will also absorb the mean of and so is denoted as *.
2ln tZ
2ln tZ3 The formulation
given in (10) is a standard time-varying parameter state-space model where the first equation is
the measurement equation and the second equation is the transition equation. Assuming
normality of the disturbances and the initial state vector, this type of problem is very amenable to
estimation via the Kalman filter. However, since the error in the measurement equation is no
longer normal due to the log-square transformation, the exact likelihood function is no longer
known to be normal. Using a result from Watson (1989), it can be shown that treating t as if it
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were NID(0, 2/2) will still yield consistent and asymptotically normal parameter estimates.4
Also, asymptotic standard errors which are robust to the specific form of non-normality in tcan
be computed using the methods outlined in Dunsmuir (1979)5.
In addition to handling the basic stochastic volatility model as described above, the
Kalman filter may also be adapted to the broader CKLS-SV specification which incorporates
level effects. This estimation approach has been used, for example, by Ball and Torous (1999).
Recall the model which was given in equation (7) as:
tttt rrr ++= 11
tutt u ++= 2 12 lnln
This can also be represented in linear state-space form and then estimated by maximum
likelihood using a Kalman filter algorithm. Specifically, recalling our factorization of the
residual t into , if we write the first equation in terms of the residualttt Zr 1 t and square it, we
get
221
22tttt Zr
=
and then taking logarithms and letting ht= ln , we have2t
(11)212 lnln2ln tttt Zrh ++=
Note that due to the log transformation the level term is no longer in the exponent and this
equation is thus linear in the parameters. Therefore, the Kalman filter, which requires linearity in
the parameters, may still be employed for estimation. Again, treating the log variance, ht, as
( )
3 For purposes of comparison with the constant found elsewhere in the paper, * is converted to the constant
by using
=
127.1 . Standard errors are obtained for via the delta method.
4 The result is summarized in Hamilton (1994), p.389.5 The formulas are rather lengthy and so are not reproduced here. The interested reader can consult Dunsmuir
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representing an unobservable state, we can cast the stochastic volatility model, including level
parameter, in state-space form as
tttt rrr ++= 11 (12a)
(12b)tttt rh +++=
12 ln2ln
tutt uhh += 1 (12c)
The first two equations are the measurement equations in this specification, and the third
equation is the transition equation. Estimation is implemented in two stages, where the first stage
employs ordinary least squares for equation (12a) to fit and, and the second stage applies the
Kalman filter to the resulting t.
III. Data
The data for the empirical part of this paper consist of 2,769 weekly observations on
three-month U.S. Treasury bill yields spanning the period from January 1954 through December
2006. The data were obtained from the Federal Reserve Bank of St. Louis in bank discount yield
form, and were converted to effective annual yields before beginning the analysis. A graph of the
yield data appears in Figure 1, and the differenced series is displayed in Figure 2. Both graphs
show strong evidence of heteroskedasticity. Evidence for a level effect can clearly be seen in the
graph of squared yield changes versus lagged yield levels, which is shown in Figure 3.
Summary statistics for the data are given in Table 1. From the table we see that the
unconditional mean level of the three-month yield is 5.47% with a weekly standard deviation of
3.11%. The first-order autocorrelation of weekly observations on the three-month yield is
slightly less than one at 0.997 and further lags are seen to die off very slowly, indicating that the
(1979), or Ruiz (1994), where a convenient summary of the formulas may be found.
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weekly short rate series is a near-integrated process. The results of an augmented Dickey-Fuller
test indicate that we are unable to reject the null hypothesis of a unit root in the interest rate
series at the 5% level, although explosive interest rates should be ruled out on theoretical
grounds. The differenced series and squared yield changes both exhibit positive serial
correlation, which in the squared changes is seen to be especially persistent.
IV. Estimation and Results
In this section of the paper I fit the CKLS-SV model to actual short rate data. Before
estimating the two-factor model, however, I first estimate the CKLS model and the stochastic
volatility model separately for purposes of comparison with the results of the joint CKLS-SV
model. Estimation of the SV model is carried out by GMM and then also by the Kalman filter-
QML procedure in order to compare the results from the simpler model which does not include a
level parameter.
A. Estimating CKLS and SV Separately
The standard CKLS one-factor model represented in equation (1) is estimated by GMM
using the four moment conditions listed in (4), and the results appear in Column 1 of Table 2.
The level effect, as represented by the parameter, has a value of 1.59 and is similar to that
found by CKLS who obtain = 1.5 on their shorter sample of monthly T-bill yield data. The
constant parameter is not statistically significant, and the mean reversion parameter, though
slightly negative, is also not statistically different from zero. Specifically, since the dependent
variable is differenced yields, the first-order autocorrelation of the short rate process is given by
1 +, and failing to reject= 0 implies the presence of a unit root in Treasury yields.This result
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is also consistent with that of CKLS and reflects the high degree of serial correlation present in
the interest rate process.
The stochastic volatility model was estimated by GMM using the fourteen-moment
specification, and results are presented in Column 2 of Table 2. The volatility autoregressive
coefficient is 0.976, indicating a high degree of persistence. Since there are r= 14 moment
conditions which were used to estimate a = 3 parameters, the model has 11 overidentifying
restrictions in that more orthogonality conditions were used than are needed to estimate . In thiscase, Hansen (1982) suggested a test statistic for the joint hypothesis that all of the sample
moments are equal to zero. This test statistic is calculated as the sample size TtimesJ, the value
obtained for the objective function at the GMM estimate :T
J-statistic = TJ~a2 (ra) (11)
For the stochastic volatility model, Hansens test of the overidentifying restrictions gives a p-
value of 0.659. Thus, a chi-square test of the overall model fails to reject the overidentifying
restrictions, indicating a reasonably good fit.
The QML-Kalman filter estimation technique is also applied to the SV model, and results
are presented in Column 3 of Table 2. The volatility persistence parameter= 0.996 estimated
from QML is higher than that obtained from GMM, suggesting a nearly integrated volatility
process. The higher estimate ofunder QML also leads to a smaller estimate of the standard
deviation parameter u = 0.11 since the two are related.6 Overall, the QML-Kalman filter
procedure gives results similar to the GMM approach, though with greater volatility persistence.
6 Specifically, this can be seen by looking at the coefficient of variation for the stochastic volatility process, which is
given by 11
exp2
2
u . Hence, as the estimate ofbecomes large, that ofu tends to decline.
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B. Estimating the Joint CKLS-SV Model
The generalized method of moments parameter estimates for the combined CKLS-
stochastic volatility model are provided in the fourth column of Table 2. As in the case of the
standard CKLS one-factor model, the constant and mean-reversion parameter in the mean
equation are not statistically significant. The coefficient of mean-reversion in the variance
equation, , has dropped slightly to 0.972. In Vetzal (1997) the level parameter, , was
artificially fixed to values of 0, 0.50, and 1, and he reports that the model with = 1seems to
provide the best overall fit. Here, using less than half the number of moment conditions as
Vetzal, is freely estimated to have a value of 0.38, and is highly significant. Note that the
magnitude of the level parameter is significantly less than the value of = 1.59 estimated from
the one-factor CKLS specification. By introducing a second stochastic factor, we are now largely
able to account for time-varying volatility through the stochastic volatility process rather than
completely relying upon the level term to capture volatility changes. The volatility of volatility is
estimated as 0.27, which is relatively unchanged from its GMM estimate in the SV model.
Since there are 17 moment conditions which were used to estimate six parameters, the
model has 11 overidentifying restrictions. Hansens chi-square test of the overidentifying
restrictions with 11 degrees of freedom results in a p-value of 0.522, indicating that we are
unable to reject the joint CKLS-SV model.
The fifth column of Table 2 shows estimates of the joint CKLS-SV model using the
QML-Kalman filter approach. Two results in particular stand out. First, the level parameter has a
value of 0.43 which is very similar to the value of 0.38 obtained from GMM. This tends to give
reassurance that the estimate ofobtained from GMM is reasonable. These estimates are much
lower than the one-factor CKLS estimate of 1.59. Both are also less than the assumed value of
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0.50 in the commonly used Cox, Ingersoll, Ross (1985) square-root process, with the GMM
estimate being significantly lower. Second, the volatility persistence parameter is 0.99, which
is larger than the value of 0.97 obtained from GMM, and again suggests that volatility shocks
tend to be extremely long-lived.
An interesting by-product of the Kalman filter estimation procedure, which is not
available with estimation by GMM, is the ability to compute an estimate of the state vector. A
graph of this smoothed Kalman conditional standard deviation versus absolute yield changes is
provided in Figure 4. The estimated volatility appears to track actual yield changes fairly closely.
When volatility is driven by an unobservable latent factor, as posited in this paper, it can be
shown that this estimate of the volatility path is superior to that provided by GARCH models
because the Kalman filter uses not only past information but subsequent observations to infer the
value of the state vector.
C. Allowing for a Structural Break
Due to the change in the Federal Reserves monetary policy, the period extending from
October 1979 to October 1982 is characterized by both unusually high levels of interest rates and
increased interest rate volatility. Some researchers (e.g. Bliss and Smith (1998)) have suggested
that this period corresponds to a fundamental change in the interest rate data generating process
and so constitutes a structural break. CKLS test for a structural break in the interest rate process
and fail to find any evidence that their model parameters are significantly different after this
period, whereas Brenner, Harjes, and Kroner (1996) and Bliss and Smith (1998) do find evidence
of a structural break. I next examine whether the model is still significant if we allow for the
possibility of a fundamental change in the interest rate process by re-estimating the model on the
subset of data covering only the period 1983-2006. The parameter estimates are given in Table 3.
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A few results from the estimation on post-1982 data emerge which are worth noting. The
level parameter from the CKLS one-factor model has dropped to 0.73 for this period. The GMM
estimate of the mean reversion parameter, , for the pure stochastic volatility model is equal to
0.88. But when we estimate the combined CKLS-SV model, the level parameter, , actually goes
up to 0.94, while the estimate of increases to 0.92. Although this behavior differs from the
earlier results over the entire sample period where the level effect dropped significantly in the
presence of a stochastic volatility factor, the results over this sub-sample still show that both
level effects and stochastic volatility are significant aspects of the short rate dynamics. Hansens
J-test of overidentifying restrictions has a p-value of 0.598, indicating that the model
specification is not rejected over the sub-sample period. The debate over whether a structural
break in the data generating process actually occurred during the 1979-82 period is likely to
continue indefinitely, but most studies nevertheless include this period in the data sample for
empirical estimation.
V. Parameter Estimation in a Monte Carlo Setting
The finite sample properties of GMM estimation of the basic lognormal stochastic
volatility model have been well-documented by Andersen and Sorensen (1996), and Ruiz (1994)
and Jacqier, Polson, and Rossi (1994) have examined the finite sample performance of the
Kalman filter approach. However, in the present model specification we have introduced two
parameters for the conditional mean of the short rate process as well as a volatility level
parameter in the conditional variance. It would be useful to have some idea of how well these
estimation techniques of the proposed model specification are able to recover the true values of
the parameters given a limited number of observations. Therefore, having applied both the GMM
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and QML estimation procedures to actual treasury yield data, I next examine the finite sample
performance of the two-factor model under each estimation approach within the context of
Monte Carlo simulation.
The vector of parameter values used for the data simulation is = (, , , , , u) =(0.12, -0.02, 0.50, -0.25, 0.95, 0.36). These parameter values were chosen so as to approximate
the values encountered in the actual data set, and thus reflect a high degree of autocorrelation in
both the mean and volatility processes. Since is of central interest,the Monte Carlo simulation
is also performed under two alternative values of this parameter, = 0 and = 1. The number of
observations in each simulated series was set equal to 2,800 in order to match the length of the
Treasury yield series that was used earlier in the paper for estimation. Data was simulated 1,000
times using the combined CKLS-stochastic volatility model given in equation (7) and the
parameters were estimated for each resulting series by GMM and by QML as described in the
previous section.
A. Finite Sample Performance of the GMM Estimator
The first set of simulations is conducted with = 0.5. This value is of primary interest
since it most closely approximates the value of gamma found in the empirical estimation. GMM
estimation on each simulated series was allowed up to 200 iterations to converge, and results
from the first 1,000 simulations for which convergence was achieved are presented in Panel A of
Table 4. First, I note that 34 of the estimation attempts failed to achieve convergence. Second,
the results show some amount of bias in all of the parameter estimates, but the bias is particularly
pronounced for the variance constant, , which has a 46% upward bias, and for the level
parameter, , which has an upward bias of 18%. Third, the root mean square error (RMSE) for
most of the parameter estimates is reasonably low, with the exception of the variance constant ,
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and the level parameter , which both have RMSEs roughly half the magnitude of their
respective mean estimates.
The distributions of each of the GMM parameter estimates are displayed as histograms in
Figure 5. From these graphs, we see that the estimates of the parameters ,,,,andall
have roughly symmetric distributions. However, the distribution of the level parameter, , is seen
to be peaked and highly skewed to the right. GMM seems to struggle with identifying and we
should therefore bear in mind that values for this parameter are estimated with some degree of
imprecision.
A natural by-product of each Monte Carlo GMM estimation is Hansens J-test statistic
for overidentifying restrictions. This allows for a straightforward investigation of the rejection
rates of the J-test in finite samples under the proposed fourteen-moment specification. A
histogram of the fractiles of thep-values, whose expected distribution is asymptotically uniform,
is provided in Panel A of Figure 7. Of particular interest is the left 5% tail of the distribution
where model rejection occurs. The distribution is slightly skewed to the left, with Hansens test
over-rejecting the model a little more often than it should, but not to an alarming extent. Even so,
the model estimated by GMM on T-bill returns in this paper is not rejected.
B. Finite Sample Performance of the QML Estimator
Estimation using the Kalman filter was performed on the same 1,000 simulated series that
were used for GMM, and none of these estimations failed to converge. Results are presented in
Panel B of Table 4. For the central parameter of interest, , it is interesting to note that the
Kalman filter delivers estimates which are downwardbiased by 9%. This is in stark contrast to
the 18% average upward bias of the GMM estimates for the level parameter. The root mean
square error of is slightly larger under the Kalman filter at 0.214 versus 0.192 for GMM. Both
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estimation techniques do a reasonably good job of identifying the volatility persistence parameter
and the volatility of volatility u, though the Kalman filter does so with smaller root mean
square error in each case. The Kalman estimate of the volatility constant has a bias of 6% and
is identified far more precisely than under GMM, where the bias is 46%. The root mean square
error of this parameter is much smaller under the Kalman filter, roughly one-third the value
produced by the GMM procedure.
Histograms of the distribution of each of the parameter estimates from the simulation are
displayed in Figure 6. A comparison with the distributions produced from the GMM procedure
in Figure 5 highlight the generally lower bias and smaller error of the Kalman filter estimation.
The distributions of parameter estimates from the Kalman filter are also generally less skewed.
Overall, the quasi-maximum likelihood Kalman filter estimation technique appears to be more
robust than the GMM approach, though both procedures produced similar results in the empirical
analysis using Treasury yield data.
C. Simulation Under Alternative Values of the Level Parameter
Monte Carlo simulation is next performed for two alternative values of the level
parameter, = 0 and= 1. Results for GMM estimation when = 0 are reported in Panel A of
Table 5. In this case, GMM estimation failed to converge for 20 of the simulation trials. The
amount of bias in the parameter estimates is comparable to that seen for simulations with =
0.5. The bias is again particularly pronounced for the variance constant, , which has a 42%
upward bias. The RMSE of 0.075 for is relatively large compared to the mean estimate of
0.059. In Panel B of Table 5, results from the QML estimation show substantially less bias in the
variance constant, where has a 22% upward bias. The mean estimate of gamma under QML is
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0.188, which is considerably higher than that of 0.059 under GMM. However, both techniques
again show negligible bias for the volatility persistence parameter.
A histogram of the fractiles of the p-values for HansensJ-test of GMM overidentifying
restrictions is provided in Panel B of Figure 7. This figure shows that the J-test again performs
reasonably well for this model when = 0. There is some concentration of probability to the left
half of the distribution, but theJ-test only slightly over-rejects at the 5% level.
Table 6 reports estimation results for simulations with = 1. The number of trials for
which GMM failed to converge is now 190, which is significantly higher than either of the other
two simulations. This shows that GMM struggles more to identify parameters when the volatility
level parameter is relatively high. Panel A shows results from GMM estimation, where we see a
substantial increase in the bias of the parameters and, which are at 23% and 22%,
respectively. GMM shows an upward bias of 15% in the level parameter and 38% in the
volatility constant . In Panel B, results from QML estimation show an even greater bias for the
parameters and, which are at 47% and 49% respectively. Evidently, both GMM and QML
struggle to correctly identify the conditional mean parameters when the volatility level effect is
relatively strong, as reflected in the value of = 1. The bias of the other four parameters is
noticeably smaller under QML than under GMM.
Finally, Panel C of Figure 7 shows the distribution ofp-values for Hansens J-test of
GMM overidentifying restrictions. From the figure it is apparent that the J-test performs rather
poorly for this model. There is a heavy concentration of outcomes in the left half of the
distribution, and the test rejects the model much too often at the 5% and 10% levels.
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VI. Concluding Remarks
Both level effects and time-varying volatility which is independent of the interest rate
have been documented as important features of the short rate dynamics. At least two previous
studies which have examined a model incorporating stochastic volatility as an independent latent
factor into the popular CKLS mean-reverting model have failed to estimate the level parameter
using GMM. This paper estimates the entire two-factor specification, including level parameter,
within a unified GMM framework by using a judicious choice of moment conditions. Estimates
from GMM are shown to be very similar to those produced by Kalman filter estimation. The
empirical results for the two-factor model on Treasury yield data show that the level effect drops
significantly when an independent volatility factor is introduced, but that both are highly
significant characteristics of the interest rate volatility dynamics. The increased generality of the
model analyzed in this paper over popular one-factor models better captures the volatility
dynamics of the short rate process, and this implies it is a better choice for the pricing of interest
rate derivatives.
Simulation evidence on the performance of the specific model proposed in this paper for
the sample size used in empirical estimation is also analyzed. Although the results of the Monte
Carlo study indicate that the GMM model specification does a reasonably good job of identifying
the true underlying parameters, the level parameter is estimated with some degree of
imprecision and with substantial bias. Although both estimation procedures provide nearly
identical empirical estimates for weekly T-bill data, simulation results indicate that the Kalman
filter is generally more robust for the two-factor model than GMM. The Kalman filter approach
appears to overcome the estimation difficulties faced by GMM in the presence of a near-
integrated variance process, while having the added benefit of providing an estimate of the
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sample path of conditional volatility. However, both approaches remain easily implementable
alternatives to more complex and computer-intensive techniques such as Efficient Method of
Moments.
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Appendix
Moment Conditions Used in GMM Estimation of the CKLS-Stochastic Volatility ModelNote:
11 = tttt rrr
26
( ){ }=tE xh ,
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
2
210
2
22
11
10
2
1
2
28
2
22
9
8
2
1
2
26
2
22
7
6
2
1
2
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2
22
5
4
2
1
2
22
2
22
3
2
2
1
2
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2
2
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9
1
2
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2
2
8
7
1
2
25
2
2
6
5
1
2
23
2
2
4
3
1
2
2
2
2
2
1
1
2
24
1
2
23
1
2
2
1
12
22
1
2
22
1
1
1exp
11
2exp
1exp
11
2exp
1exp11
2exp
1exp
11
2exp
1exp
11
2exp
14exp
141exp
2
14exp
141exp
2
14exp
141exp
2
14exp
141exp
2
14exp
141exp
2
1
2
1
2exp3
189
123exp22
1812exp
2
121exp
121exp
uu
t
t
t
t
uu
t
t
t
t
uu
t
t
t
t
uu
t
t
t
t
uu
t
t
t
t
uu
t
t
t
t
uu
t
t
t
t
uu
t
t
t
t
uu
t
t
t
t
uu
t
t
t
t
u
t
t
u
t
t
u
t
t
t
u
t
t
u
t
t
tt
t
rr
rr
rr
rr
rr
rrabs
rrabs
rrabs
rrabs
rrabs
r
rabs
rabs
rr
r
r
= 0E
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References
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Ball, C., and W. Torous (1999), The Stochastic Volatility of Short-Term Interest Rates: SomeInternational Evidence,Journal of Finance 54, 2339-2359.
Bliss, R., and D. Smith (1998), The Elasticity of Interest Rate Volatility: Chan, Karolyi,Longstaff, and Sanders Revisited, Federal Reserve Bank of Atlanta working paper.
Bollerslev, T., R. Chou, and K. Kroner (1992), ARCH Modeling in Finance, Journal ofEconometrics 52, 5-59.
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Gallant, R., D. Hsieh, and G. Tauchen (1995), Estimation of Stochastic Volatility Models WithDiagnostics, Working Paper, Duke University.
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Harvey, A., E. Ruiz, and N. Shephard (1994), Multivariate Stochastic Variance Models, Reviewof Economic Studies 61, 247-264.
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Ruiz, E. (1994), Quasi-Maximum Likelihood Estimation of Stochastic Volatility Models,Journal of Econometrics 63, 289-306.
Taylor, S. (1986), Modelling Financial Time Series, Wiley, New York.
Vasicek, O. (1977) An Equilibrium Characterization of the Term Structure,Journal of FinancialEconomics
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Watson, M. (1989), Recursive Solution Methods for Dynamic Linear Rational ExpectationsModels,Journal of Econometrics 41, 65-89.
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29
Table 1
Summary Statistics
Means, standard deviations, and autocorrelations of weekly observations on three-month maturity U.S. Treasury billyields are computed from January 1954 to December 2006. The variable r denotes the yield on Treasury billsmaturing in three months, andris the change in three-month yields. Autocorrelations for thej-th lag are denoted
byj,and the number of observations is denoted byN.
Variable N MeanStandardDeviation 1 2 3 4 5 6 7 8
r 2769 0.05473 0.03113 0.997 0.992 0.987 0.982 0.976 0.970 0.964 0.959
r 2768 0.00001 0.00231 0.270 0.067 0.049 0.085 0.053 0.003 -0.086 -0.038
(r)2 2768 0.00054 0.00285 0.253 0.279 0.231 0.207 0.237 0.199 0.258 0.226
* Based on the results of an augmented Dickey-Fuller test using MacKinnon critical values, we are only able toreject the null hypothesis of a unit root in the interest rate series with ap-value of 0.06.
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Table 2
Estimation Results 1954-2006
The column labeled CKLS gives GMM parameter estimates for the model
ttttt Zrrrr
111 ++=
where rt is the three-month T-bill yield. The column labeled SV gives GMM parameter estimates for the stochasticvolatility model
tutt
ttt
u
Zy
++=
=
2
12 lnln
where yt is the first difference of T-bill yields, minus the mean. The third column presents quasi-maximumlikelihood estimates for the stochastic volatility model. The fourth column presents GMM parameter estimates forthe combined CKLS-stochastic volatility model. The last column presents quasi-maximum likelihood estimates forthe combined CKLS-stochastic volatility model. T-statistics are given in parentheses.
CKLS SV SV (QML) CKLS-SVCKLS-SV
(QML)
0.0182
(1.27)
0.0078
(0.72)
0.0182
(1.04)
-0.0031
(-0.94)-0.0010
(-0.43)-0.0031
(-0.77)
2*
0.0001(1.84)
0.0188(7.61)
0.0307(1.44)
0.0051(7.97)
0.0066(1.61)
1.5920(13.54)
0.3869(71.42)
0.4292(2.86)
-0.1133(-0.85)
-0.0166(-1.71)
-0.1681(-1.06)
-0.0696(-3.06)
0.9763(34.98)
0.9962(434.23)
0.9716(36.36)
0.9881(268.03)
u0.2754
(1.75)0.1135(5.90)
0.2691(2.07)
0.2002(11.28)
ObjectiveFunction[p-value]
J 0 J = 0.00312[0.659] = 5374.9 J= 0.00366[0.522] = 6346.5
* With the exception of the CKLS model, these are impliedunconditional variances. The unconditional variance is
computed from( ) ( )
+
=
2
22
121exp
u . Standard errors are computed via the delta method.
Jdenotes the minimized value of the GMM criterion function, and is the value of the maximized log likelihoodfunction. Where applicable,p-values are reported for Hansens chi-square test of overidentifying restrictions.
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Table 3
Estimation Results 1983-2006
Results presented in this table are based on the sub-sample of T-bill yield data extending from January 1983 toDecember 2006, and comprising 1,256 weekly observations. The column labeled CKLS gives GMM parameterestimates for the model
ttttt Zrrrr 111 ++=
where rt is the three-month T-bill yield. The column labeled SV gives GMM parameter estimates for the stochasticvolatility model
tutt
ttt
u
Zy
++=
=
2
12 lnln
where yt is the first difference of T-bill yields, minus the mean. The third column presents quasi-maximumlikelihood estimates for the stochastic volatility model. The fourth column presents GMM parameter estimates forthe combined CKLS-stochastic volatility model. The last column presents quasi-maximum likelihood estimates forthe combined CKLS-stochastic volatility model. T-statistics are given in parentheses.
CKLS SV SV (QML) CKLS-SVCKLS-SV
(QML)
0.0071(0.61)
0.0198(2.10)
0.0071(0.89)
-0.0018(-0.80)
-0.0038(-1.82)
-0.0018(-1.36)
2*
0.0011(2.00)
0.0061(6.53)
0.0004(5.95)
0.0009(2.42)
0.7338(5.72)
0.9387(17.90)
0.7602(6.13)
-0.7075(-2.37)
-0.0093(-0.81)
-0.6113(-1.07)
-0.5523(-2.76)
0.8776(17.05)
0.9979(369.42)
0.9240(13.02)
0.9248(34.21)
u0.5584(4.13)
0.0617(2.59)
0.3070(1.72)
0.3114(4.71)
ObjectiveFunction
[p-value]
J 0 J= 0.01318[0.126] = 2361.7 J= 0.00742[0.598] = 2822.6
* These are implied unconditional variances computed from( ) ( )
+
=
2
22
121exp
u .
Jdenotes the minimized value of the GMM criterion function, and is the value of the maximized log likelihoodfunction. Where applicable, p-values are reported for Hansens chi-square test of overidentifying restrictions.
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Table 4
Estimation Results from Monte Carlo Simulation: Gamma = .5
Estimation of the CKLS-SV model was performed on 1,000 simulated return series. Each series of length 2,800 wasgenerated using parameter values = (,, , , , u) = (0.12, -0.02, 0.50, -0.25, 0.95, 0.36). Panel A reports theresults based on GMM estimation, and Panel B reports the results based on quasi-maximum likelihood estimation.
Reported statistics are based on the first 1,000 simulated samples for which convergence was achieved. Sinceconvergence using GMM was not obtained in 34 of the simulated samples, a total of 1,034 simulations were actually
performed.
Panel A: GMM Estimation
True Value Mean Median RMSE Bias % Bias
0.12 0.131 0.130 0.029 0.011 9%
-0.02 -0.022 -0.022 0.005 -0.002 10%
0.50 0.590 0.518 0.192 0.090 18%
-0.25 -0.365 -0.352 0.172 -0.115 46%
0.95 0.933 0.934 0.028 -0.017 -2%
u 0.36 0.345 0.346 0.064 -0.015 -4%
Panel B: QML Estimation
True Value Mean Median RMSE Bias % Bias
0.12 0.132 0.129 0.034 0.012 10%
-0.02 -0.022 -0.022 0.006 -0.002 10%
0.50 0.453 0.451 0.214 -0.047 -9%
-0.25 -0.265 -0.256 0.069 -0.015 6%
0.95 0.945 0.946 0.013 -0.005 -1%
u 0.36 0.367 0.366 0.046 0.007 2%
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Table 5
Estimation Results from Monte Carlo Simulation: Gamma = 0
Estimation of the CKLS-SV model was performed on 1,000 simulated return series. Each series of length 2,800 wasgenerated using parameter values = (, , , , , u) = (0.12, -0.02, 0, -0.25, 0.95, 0.36). Panel A reports theresults based on GMM estimation, and Panel B reports the results based on quasi-maximum likelihood estimation.
Reported statistics are based on the first 1,000 simulated samples for which convergence was achieved. Sinceconvergence using GMM was not obtained in 20 of the simulated samples, a total of 1,020 simulations were actually
performed.
Panel A: GMM Estimation
True Value Mean Median RMSE Bias % Bias
0.12 0.130 0.127 0.028 0.010 9%
-0.02 -0.022 -0.021 0.005 -0.002 9%
0.00 0.059 0.056 0.075 0.059 -
-0.25 -0.354 -0.345 0.156 -0.104 42%
0.95 0.934 0.936 0.027 -0.016 -2%
u 0.36 0.341 0.343 0.059 -0.019 -5%
Panel B: QML Estimation
True Value Mean Median RMSE Bias % Bias
0.12 0.132 0.128 0.033 0.012 10%
-0.02 -0.022 -0.021 0.005 -0.002 10%
0.00 0.188 0.024 0.346 0.188 -
-0.25 -0.305 -0.282 0.097 -0.055 22%
0.95 0.946 0.946 0.010 -0.004 0%
u 0.36 0.365 0.364 0.038 0.005 1%
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Table 6
Estimation Results from Monte Carlo Simulation: Gamma = 1
Estimation of the CKLS-SV model was performed on 1,000 simulated return series. Each series of length 2,800 wasgenerated using parameter values = (,, , , , u) = (0.12, -0.02, 1.00, -0.25, 0.95, 0.36). Panel A reports theresults based on GMM estimation, and Panel B reports the results based on quasi-maximum likelihood estimation.
Reported statistics are based on the first 1,000 simulated samples for which convergence was achieved. Sinceconvergence using GMM was not obtained in 190 of the simulated samples, a total of 1,190 simulations wereactually performed.
Panel A: GMM Estimation
True Value Mean Median RMSE Bias % Bias
0.12 0.148 0.148 0.056 0.028 23%
-0.02 -0.024 -0.024 0.010 -0.004 22%
1.00 1.147 1.145 0.214 0.147 15%
-0.25 -0.346 -0.330 0.191 -0.096 38%
0.95 0.939 0.941 0.051 -0.011 -1%
u 0.36 0.316 0.320 0.089 -0.044 -12%
Panel B: QML Estimation
True Value Mean Median RMSE Bias % Bias
0.12 0.177 0.168 0.083 0.057 47%
-0.02 -0.030 -0.028 0.015 -0.010 49%
1.00 0.939 0.939 0.115 -0.061 -6%
-0.25 -0.262 -0.257 0.060 -0.012 5%
0.95 0.945 0.947 0.013 -0.005 0%
u 0.36 0.362 0.359 0.042 0.002 0%
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Figure 1
Weekly U.S. T-Bill Rate, Three-Month Maturity, January 1954 to December 2006
0
2
4
6
8
10
12
14
16
18
20
1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006
InterestRate(%)
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Figure 2
First Difference of T-Bill Rate
-3
-2
-1
0
1
2
3
1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006
YieldChange(%)
36
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Figure 3
Plot of Squared Weekly Changes in Three Month Yield Against Lagged Yield
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14 16 18 20
Interest Rate Level (Percent)
SquaredYieldChange
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Figure 4
Kalman Filter Smoothed Standard Deviation vs. Absolute Yield Changes
0.0
0.5
1.0
1.5
2.0
2.5
1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006
StandardDeviation(%)
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Figure 5
Distribution of GMM Parameter Estimates from Monte Carlo SimulationTrue parameter values are given in parentheses. Frequency refers to the percentage of observations out of 1,000.
.00
.10
.20
.30
.40
.50
.04 .08 .12 .16 .20 .24 .28
Frequency
Alpha (.12)
.00
.10
.20
.30
.40
.50
-.05 -.04 -.03 -.02 -.01 .00
Frequency
Beta (-.02)
.00
.10
.20
.30
.40
.50
0.0 0.4 0.8 1.2 1.6
Frequency
Gamma (.50)
.00
.10
.20
.30
.40
.50
-2.0 -1.5 -1.0 -0.5 0.0
Frequency
Omega (-.25)
.00
.10
.20
.30
.40
.50
0.75 0.80 0.85 0.90 0.95 1.00
Frequency
Phi (.95)
.00
.10
.20
.30
.40
.50
.0 .1 .2 .3 .4 .5 .6 .7
Frequency
Sigma (.36)
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Figure 6
Distribution of QML Parameter Estimates from Monte Carlo SimulationTrue parameter values are given in parentheses. Frequency refers to the percentage of observations out of 1,000.
.00
.10
.20
.30
.40
.50
.04 .08 .12 .16 .20 .24 .28
Frequency
Alpha (.12)
.00
.10
.20
.30
.40
.50
-.05 -.04 -.03 -.02 -.01 .00
Frequency
Beta (-.02)
.00
.10
.20
.30
.40
.50
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Frequency
Gamma (.50)
.00
.10
.20
.30
.40
.50
-.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 .0
Frequency
Omega (-.25)
.00
.10
.20
.30
.40
.50
0.88 0.90 0.92 0.94 0.96 0.98 1.00
Frequency
Phi (.95)
.00
.10
.20
.30
.40
.50
.2 .3 .4 .5 .6
Frequency
Sigma (.36)
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Figure 7
Distribution ofP-values from the GMM Omnibus TestThe figure shows the distribution ofp-values for the test of overidentifying restrictions based on each GMMobjective function from the Monte Carlo simulation. The theoretically expected percentage of outcomes within each5% fractile is marked on the graph with a horizontal line. Frequency refers to the percentage of observations out of1,000 falling within each fractile.
Panel A: Gamma = .5
0.00
0.05
0.10
0.15
0.20
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95P-value Frac tiles from [0, .05] to [.95, 1]
Frequency
Panel B: Gamma = 0
0.00
0.05
0.10
0.15
0.20
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
P-value Fractiles from [0, .05] to [.95, 1]
Frequency
Panel C: Gamma = 1
0.00
0.05
0.10
0.15
0.20
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Frequency