7/27/2019 SSRN-id1467184 2

1/43

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: trasapp@iastate.edu.

7/27/2019 SSRN-id1467184 2

2/43

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.

7/27/2019 SSRN-id1467184 2

3/43

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

1

7/27/2019 SSRN-id1467184 2

4/43

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

2

7/27/2019 SSRN-id1467184 2

5/43

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.

3

7/27/2019 SSRN-id1467184 2

6/43

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

4

7/27/2019 SSRN-id1467184 2

7/43

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).

5

7/27/2019 SSRN-id1467184 2

8/43

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

6

7/27/2019 SSRN-id1467184 2

9/43

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.

7

7/27/2019 SSRN-id1467184 2

10/43

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 +=

8

7/27/2019 SSRN-id1467184 2

11/43

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

9

7/27/2019 SSRN-id1467184 2

12/43

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

10

7/27/2019 SSRN-id1467184 2

13/43

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,

11

7/27/2019 SSRN-id1467184 2

14/43

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

12

7/27/2019 SSRN-id1467184 2

15/43

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

13

7/27/2019 SSRN-id1467184 2

16/43

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.

14

7/27/2019 SSRN-id1467184 2

17/43

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

15

7/27/2019 SSRN-id1467184 2

18/43

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.

16

7/27/2019 SSRN-id1467184 2

19/43

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

17

7/27/2019 SSRN-id1467184 2

20/43

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.

18

7/27/2019 SSRN-id1467184 2

21/43

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

19

7/27/2019 SSRN-id1467184 2

22/43

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 ,

20

7/27/2019 SSRN-id1467184 2

23/43

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

21

7/27/2019 SSRN-id1467184 2

24/43

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

22

7/27/2019 SSRN-id1467184 2

25/43

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.

23

7/27/2019 SSRN-id1467184 2

26/43

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

24

7/27/2019 SSRN-id1467184 2

27/43

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.

25

7/27/2019 SSRN-id1467184 2

28/43

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

24

2

22

5

4

2

1

2

22

2

22

3

2

2

1

2

29

2

2

10

9

1

2

27

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

7/27/2019 SSRN-id1467184 2

29/43

References

At-Sahalia, Y. (1996), Testing Continuous-Time Models of the Spot Interest Rate, Review ofFinancial Studies 9, 385-426.

Andersen, T., and J. Lund (1997), Estimating Continuous-Time Stochastic Volatility Models ofthe Short-Term Interest Rate,Journal of Econometrics 77, 343-377.

Andersen, T.and B. Sorensen (1996), GMM estimation of a Stochastic Volatility Model: AMonte Carlo Study,Journal of Business and Economic Statistics 14, 328-352.

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.

Brenner, R., R. Harjes, and K. Kroner (1996), Another Look at Alternative Models of the Short-Term Interest Rate,Journal of Financial and Quantitative Analysis 31, 85-107.

Chan, K., G. Karolyi, F. Longstaff, and A. Sanders (1992), An Empirical Comparison ofAlternative Models of the Short-Term Interest Rate,Journal of Finance 47, 1209-1227.

Chapman, D., J. Long, and N. Pearson (1999), Using Proxies for the Short Rate: When are Three

Months Like an Instant?Review of Financial Studies

12, 763-806.

Cox, J., J. Ingersoll, and S. Ross (1985), A Theory of the Term Structure of Interest Rates,Econometrica, 385-407.

Dunsmuir W. (1979), A Central Limit Theorem for Parameter Estimation in StationaryVectorTime Series and Its Application to Models for a Signal Observed with Noise,Annals of Statistics 7, 490-506.

Engle, R., and G. Lee (1996), Estimating Diffusion Models of Stochastic Volatility, in P. Rossi,ed., Modelling Stock Market VolatilityBridging the Gap to Continuous Time,Academic Press, San Diego, 333-355.

Gallant, R., D. Hsieh, and G. Tauchen (1995), Estimation of Stochastic Volatility Models WithDiagnostics, Working Paper, Duke University.

Hamilton, J. (1994). Time Series Analysis. Princeton University Press, New Jersey.

27

7/27/2019 SSRN-id1467184 2

30/43

Harvey, A., E. Ruiz, and N. Shephard (1994), Multivariate Stochastic Variance Models, Reviewof Economic Studies 61, 247-264.

Harvey, A., and N. Shephard (1996), Estimation of an Asymmetric Stochastic Volatility Modelfor Asset Returns,Journal of Business and Economic Statistics 14, 429-434.

Jacquier, E., N. Polson, and P. Rossi (1994), Bayesian Analysis of Stochastic Volatility Models(with critical commentary),Journal of Business and Economic Statistics 12, 371-417.

Longstaff, F. and E. Schwartz (1992), Interest Rate Volatility and the Term Structure: A TwoFactor General Equilibrium Model,Journal of Finance 47, 1259-1282.

Melino, A. and S. Turnbull (1990), Pricing Foreign Currency Options with Stochastic Volatility,Journal of Econometrics 45, 239-265.

Merton, R. (1973), Theory of Rational Option Pricing, Bell Journal of Economics and

Management Science 4, 141-183.

Nelson, D. (1991), Conditional Heteroskedasticity in Asset Returns: A New Approach,Econometrica 59, 347-370.

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

5, 177-188.

Vetzal, K. (1997), Stochastic Volatility, Movements in Short Term Interest Rates, and BondOption Values,Journal of Banking and Finance 21, 169-196.

Watson, M. (1989), Recursive Solution Methods for Dynamic Linear Rational ExpectationsModels,Journal of Econometrics 41, 65-89.

28

7/27/2019 SSRN-id1467184 2

31/43

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.

7/27/2019 SSRN-id1467184 2

32/43

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.

30

7/27/2019 SSRN-id1467184 2

33/43

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.

31

7/27/2019 SSRN-id1467184 2

34/43

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%

32

7/27/2019 SSRN-id1467184 2

35/43

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%

33

7/27/2019 SSRN-id1467184 2

36/43

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%

34

7/27/2019 SSRN-id1467184 2

37/43

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(%)

35

7/27/2019 SSRN-id1467184 2

38/43

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

7/27/2019 SSRN-id1467184 2

39/43

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

37

7/27/2019 SSRN-id1467184 2

40/43

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(%)

38

7/27/2019 SSRN-id1467184 2

41/43

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)

39

7/27/2019 SSRN-id1467184 2

42/43

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)

40

7/27/2019 SSRN-id1467184 2

43/43

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