Inference for Interest Rate Models Using Milstein’s Approximation

Inference for Interest Rate Models Using Milstein’s

Approximation

Theodoro Koulis, Aera Thavaneswaran

Department of Statistics, University of Manitoba, Winnipeg, Canada Email: [email protected]

Received October 9, 2012; revised November 24, 2012; accepted December 8, 2012

ABSTRACT

A class of martingale estimating functions based on the first two moments of the observed process provides a conven- ient framework for estimating the parameters of diffusion processes [1]. In the Bayesian set up, combined estimating functions had been studied for diffusion processes in [2] with filtering applications. However, when the conditional mean and the conditional variance are functions of parameters of interest in a diffusion process model, the basic mar- tingales generating components of quadratic estimating functions are such that one is an absolute continuous function with respect to the other [3, p. 94]. Hence, the combined martingale estimating functions cannot be constructed for con-

tinuous-time diffusion processes. In this paper, a general framework for parameter estimation of discretely observed

interest rate models is developed by using the Milstein approximation and closed form expressions for the information gain are also obtained. The method is used to study the estimates of the parameters for an extended version of the Cox- Ingersoll-Ross interest rate model.

Keywords: Interest Rate Models; Combined Estimating Functions; Information; Diffusion Processes; Milstein Approximation

1. Introduction

Inference for discrete-time stochastic processes using estimating functions was discussed in [4]. In [1] and [5], estimation for semimartingales was studied using esti-mating functions. In addition, filtering and prediction problems were studied in [6] and [2] using estimating functions in the Bayesian context.

The standard method of estimation for parameters in the drift coefficient of interest rate models [7] involves the calculation of a likelihood ratio (Radon-Nikodym derivative) and hence the maximum likelihood estima- tor(s). This is less than straightforward for complicated

models, and indeed it is not available at all because of

the non-existence of the Radon-Nikodym derivative. The estimating function method, however, allows estimators to be obtained straightforwardly under very general condi- tions on the first two conditional moments [3, p. 131]. They can deal, in particular, with the situation in which the Brownian motion in a diffusion is replaced by a gene- ral square-integrable martingale as in [1]. The combined estimating function approach used in this paper, based on selection of an optimal estimating function from within a specified class of martingale estimating functions, in- volves assumptions on the first four conditional moments of the underlying process. In most realistic situations the

diffusions cannot be observed continuously, so discrete time approximation to stochastic integrals or a direct approach using the discrete time observations is required.

Recently in [8], among others, the estimating functions approach was used to study the estimation problems for some discretely observed interest rate models. However, these methods involve the closed form expressions for the first four conditional moments, obtained by Ito’s ap-

proximations, and these are not available for general

time-homogeneous diffusion process models and in par- ticular for an extended CIR interest rate model.

In [9], the asymptotic theory of the maximum likeli- hood estimator for diffusion models was studied, first by using Milstein’s approximation of diffusion processes [10], and further by approximating the conditional transi- tion density by a normal density by ignoring the skew-ness and kurtosis.

In this paper, we study combined martingale estimat-

ing functions for interest rate models and show that the combined estimating functions are more informative when the conditional mean and variance of the observed

process depend on the parameter of interest. This paper

tions for discretely observed continuous-time diffusion processes based on closed form expressions for the first four conditional moments via Itô’s formula. In Section 3, the theory of combined estimating functions is applied to general diffusion processes.

Suppose that is a realization of a dis-

crete-time stochastic process and its distribution depends on a vector parameter



y t_t, 1, , n



 belonging to an open subset

the p-dimensional Euclidean space. Let

of



, , P_



y t

F

denote the underlying probability space, and let be

the σ-field generated by . Let

1

t t t



y1, , y t_t, 1





, ,y,



h h y 

M

 , be specified q-dimen-

sional vectors that are martingales. We consider the class

of zero mean and square integrable p-dimensional

martingale estimating functions of the form

1 t n

 

 

1

1

: n

n n t

t t

g  g  a

 

_ 







M h_

where are matrices depending on

, . The estimating functions

1

t

a

1 t 1

y y

p q t n  

, , 1 gn

 



are further assumed to be almost surely differentiable with respect to the components of θ and such that

 

1

n y

n

g

E 

 



 

 _ 

 F  and

   

1

y

n n n

E g_  g   F _ are non-

singular for all   and for each . The

expecta-tions are always taken with respect to

1

n

P_. Estimators of

θ can be obtained by solving the estimating equation

 

0

n

g   . Furthermore, the p p matrix

   

y1

n n n

E g_  g   F _is assumed to be positive definite

for all  . Then in the class of all zero mean and

square integrable martingale estimating functions M,

the optimal estimating function gn

 

 which

maxi-mizes, in the partial order of nonnegative definite matri-ces, the information matrix

 







1 1

1

1

1 1 1 1 1

1 1

n

y t

g t t

t

n n

y t y

t t t t t t t

t t

h

I a E

h

E a h a h a E







 





    

 



  

  _ 

 

 

  

 _{  }

  

 _{ } _

    







F

F F

is given by

 





1 1

1

1 1

1

n

n t t

t

n

y y

t

t t t t

t

g a h

h

E E h h





 

 



 







   _{  }

  _  _{ }

 

 





F F h_t

and the corresponding optimal information reduces to

   

1

y

n n n

E g_   g   F _. The function gn

 



 _{is also called the optimal esti-}

mating function and has properties similar to those of a score function in the sense that E g_{ n}

 

 _{ }0

  and

   

n

 

n n

g

E g  g  E 

 

   

 _{   } _

  _

  _  _.This is a more general

result in the sense that for its validity we do not need to assume that the true underlying distribution belongs to the exponential family of distributions. Moreover, it fol-lows from [11, p. 916] that if we solve an unbiased esti-mating equation g_n

 

 0 to get an estimator, then the asymptotic variance of theresulting estimator is the

in-verse of the information g_n. Hence the estimator

ob-tained from a more informative estimating equation is asymptotically more efficient.

I

2. Combined Estimating Functions for

Discretely Observed Diffusions

In this section, we discuss the discrete time results on combining estimating functions and obtain the closed form expression for the gain in information. Assume the

real-valued continuous time process

 

yt is recorded

discretely at the time points where h is the dis-

crete interval of observations of , 2 , ,

h h

 

yt . Now we consider

the observable discrete-time process



y t_th, 1, 2,



with conditional moments

 

_{ }1

y

t E yth t h

  _{ } F _ _ (2.1)

 

_{ }

2

1

Var y

t yth t h

  _  _

 F  (2.2)

 



 



 

3 1

y

t E yth t t h

   _   F _ _ (2.3)

 



 



  4

1

y

t th t t h

к  E_ y   _ _

 F  (2.4)

where _{ }y₁ is the σ-field generated by t h

F

 



1h t1h



. That is, we assume that the third

and the forth moments of t do not contain any

addi-tional parameters. In order to estimate the parameter θ

based on the observations 1h n , we consider two

classes of martingale differences

,t1



, ,

y y

y

, ,

y  y_h

 

 



m_t  y_th _t ,t1, , n



and

 

 

 



_{Mt  mt}2 _{  t}2 _,t1,,_n



_{, where the quad-}

ratic variation and covariation of m_t and M_t are

 

2 2

1 ,

y

t t h t

t

m E m_ F _ _

 

2 4

1 ,

y

t t t h t

M E M_ F  _к t

and

 1

, y ,

t t t h t

t

m M E m M_ F _ _

respectively. The optimal estimating functions based on

sponding information are given by

 

 

1

1

,

1 , m

n

t t m

t _t

n

t t g

t _t

m g

m I

m

 

   

 









  

   

  





(2.5)

 

 

2 1

2 2 1

,

1 .

M

n

t t

M

t _t

n

t t

g

t _t

M g

M I

M

 



 



 







   



 





 





(2.6)

The following theorem provides optimality of the combined estimating function based on martingales

and M_t for the multi-parameter case. mt

Theorem 1. For a discretely observed process, in the

class of all combined estimating functions of the form

 

 



1 1

1

: n

C C C t t t t

t

g g  g  a m b M

 

_   _







,



t

(a) The optimal estimating function is given by

 

1



1 1



n

C t t t t

g  



_ a m_ b M_ , where 1

2 ₂

1

, 1 ,

1 t t t

t

t t t t t

m M m M

a

m M m m M

 

 

 



   _{ } 

 

  _  _

  _   _

 

t

and

1

2 ₂

1

, , ₁

1 _;



t t t t

t

t t t t t

m M m M

b

m M m M m

 

 

 



  _{ } 

 

  _ 

  _ 

 

(b) the information

 

C

g

I   is given by

 

1 2

1

2 2

2 2

, 1

1 1

, ; C

n

t g

t _{t t}

t t t t

t t

t t t t t

t t

m M I

m M

m M

m M m M



   

   

   

   







 

 

 

 

 

   



 

   



 

  

   

_{  } _{  }  

  



(c) the gain in information

 

 

C m

g g

I   I   is given by

1 2

1

2 _{2 2}

2 2

, 1

, 1

, .

n

t

t _{t t}

t t t t t

t t

t t t t t

t t

m M m M

m M

m M m

m M m M

   

   

   

   





 

  

 

 

   



 

   

 

  



   

_{  }  _{  }  

  



t

Example 1 (Combined Estimating Functions for Cox-

Ingersoll-Ross Model). Recently there has been a grow-

ing interest in studying inference for interest rate models. In most realistic situations, the diffusion cannot be ob- served continuously, so discrete time approximations to stochastic integrals or a direct approach using discrete time observations is required. As a concrete illustration of the methodology, we shall discuss the estimation for the Cox, Ingersoll and Ross [12] short-term interest rate model of the form





,

t t t t

dy  k y  dt y dW (2.7)

with y00, k0, 0, 0 _and Wt is the

standard Brownian motion. The unknown parameters of

interest are _



_{k, , }2



_{. Let}

 



, ,





 1



e

kh

t t k yt h



       



   . It is of interest

to note that in this example we can obtain the closed form expressions for the first four conditional moments by using Itô’s formula for

itional moments of yth d u

t y ,

are

2,3, 4

u . calculate

The first four

as

p-cond (see A

pendix A for the details):

 

 

1 ,

t t

     (2.8)

 

2



 

 



2 _{2 1 2 ,}

2

t t t

k



      (2.9)

 

₂42



3

 

1 3

 

2

 

3 ,



t t t t

k



        (2.10)

and

 

63



 

 

 

 



3 _{4 1 6 2 4 3 4}

4

t t

k 

         

 

4

3 .

t t t

t   

Then based on the discretely observed observations the martingale differences are

(2.11)

0, ,h 2h, , nh,

y y y  y

 

t th t

m  y   , and 2 2

 

_.

t t t

M m   Also,

 

 

 

2 _, 4 _,

t t Mt t t

m       and

 

, t t .

m M   The derivatives are given by

 

_

_{ }

_

1 ,1 e ,0 ,

t kh

t h

   



 _

 

 (2.12)

 



   

  









 

 





1

2 t 1 t 2

  

2 2

2

2 1 hk _t 1 1 2hk _t 2

   

 

    

 

2 ₂

2

1 e 2

2

kh t

k

k k

  _





 

 

 _

  

 _{ }

 

 

 

 

Hence, the optimal estimating functions bas

martingale differences and the corresponding infor-

(2.13)

ed on the t

mation matrix are given by

 

 





 





_{ }

 

1 2 1

2 1

11 12

12 22 e

, 1 e

0 0 0 ,

0 0 0

m

t t h n

kh t

t

m _{kh n t}

t t

m m

m m

g

y m

h

g _m

I I I _ I I

   

 



 





 

 _ 

_ 

 

 

  _{ } 

 

 

 

 

 

 

  

 

 





where

 

 

2 2

11 2

1

1 , n

t m

t t

I h 

  









 

_{ }

12 2

1

1

1 e kh ,

t _t

n t m

I h 

  



 



(2.14)





2

_{ }

22 2

1

1

1 e .

m kh

t t

n I

 





 



Similarly, the optimal estimating functions based on the martingale differences Mt and the corresponding information matrix are given by

 



   

  









 

 





 

2 1

2 2

1

1 11 12 13 12 22 23 13 13 33

1 1 1 2 2

2

1 e

, 2

2 1 2

1

,

M

t t t

t

t

kh

t

M _t

t

t t t

t

M M M

m m M

g

M

n

M M

n

hk hk M

M k

M g

k M

M M

I I I

I I I I

I I I



 

 

  





 







  



 

 

  

_{  }

 

 

 

 

 

 

 

 

  

 

 



(2.15)

where

2 ₂

n  

 





2_k t







   

  





2

4 11 4

1

2 1 1 1 2 2

, 4

t t

M

t n

t

hk hk

I

M k

  

 

   









2



   

  

4

12 3

1

1 e _{2 1 1 1 2 2}

, 4

kh

t t

M

t n

t

hk hk

I

M k

    



    

 





   

  







 

 



2 13 3

1

2 1 1 1 2 2 2 1 2

, 4

t t t

M

t _t

n hk hk

I

M k

     

 

     

 



t





4 4

22 2

1

1 e ₁

, 4

kh n M

t _t

I

M k

 













 

 

2 2

23 2

1

1 e 2 1 2

, 4

kh

t t

M

t _t

n

I

M k

  _{  }



  







 

 



2

33 2

1

2 1 2

1

. 4

t t

t _t

n M I

M k

  



 





The optimal combined estimating function using mt and Mt is given by

t (2.16) where

 



1 1



1

,

C t t t

t n

g  a m b M 







 





   

  







 

 





2

2

1 ₂

2 ₂

1

2 1 1 1 2 2 ,

1

2

, _{1 e} 1 e ,

1

2

2 1 2 ,

2

t t

t t

t t t

kh kh

t t

t

t t _{t t t}

t t t

t t

hk hk m M

h

m k m M

m M m M

a

m M _{m k m M}

m M

k m M

   





  



 

 

     

 

 

 

 

 _{    }

 

  _{   }

 _{  }

 

 

 

 

 

and

 





   

  







 

 





2

2

1 1 2 2

2

t t

hk hk

k M

   

1 ₂

2 ₂

* 1

2 1 1 ,

, 1 e , 1 e

1 .

2

2 1 2

2

t t

t t t

kh kh

t _t

t

t t _{t t t}

t t

t

h m M m M

m M _{m M}

b

m M _{m M k M}

k M

  



  



 



   



 

 

 

 _{    }

 

  _{  }

 _{  }

 

 

 

 _ 

 

 

Further, let Wt

 

 be the matrix

 

 

2

 

2

 

 

11 12 13 12 22 23 13 23 33

,

t t t t

t

W

w w w w w w w w w

       



   

    

_  _

 

   

 

 

 

  

 

 

(2.17)

where

 

_

_{   }

_{  }

2

11 2

1

2 1 1 1 2 2 ,

t

t t

h

w hk

k  

  

  _     hk _





 





_

_{   }

_{  }

2 2

12

2

1 e 1

2

2 1 1 1 2 2 ,

2

kh t

t t

h k

hk hk

k

 

  



 

 _     _

2 _{1 e} kh

 _ 

w





 





3

22 1

1 e

,

kh

t h h

w y

k 



 

 

 

 

23

1 e

2 1 2 ,

2

kh

t t

w

k   

 

 _   _

33 0

w  .

The information associated with the optimal combined estimating function is

 

 

 

 

 

_{ }

1 2 1

2 2

, 1

1

,

1 _.

C

t g

t _{t t}

t t

t

t t t

t

t t

n _{m M}

I

m M

m

m M W

M m M



   

 

   



 







 

 

 

 

 

  

   



 

 



  _t

  



Note: If we allow η in (2.7) to be a function of k, then the estimating function gM

 

 and the combined esti- mating function gC

 

 become intractable.

3. Combined Estimating Functions for

General Models

For extended versions of the CIR model, closed form

expressions for the first four conditional moments cannot be obtained easily by using Itô’s formula, as was done for the CIR model. Recently, the Milstein’s approxima- tion was used in [9] to obtain the first two conditional moments of the diffusion. In this section, we use Mil- stein’s approximation to obtain the first four conditional moments and construct the optimal estimating functions.

Consider the diffusion process given by the time-ho- mogeneous stochastic differential equation of the form









dy_ta ,y_t dt b ,y_t dW_t. (3.1) Where a and b are the drift and diffusion functions,

re-spectively, and is the standard Brownian motion.

A special case of (3.1) is the Brownian motion with constant drift and diffusion:

,

t W

dy_t dtdW_t

where  0

ven 0

. In this case, the conditional distribution of gi

t

y y y is a normal with mean yt and

n

varia ce _2_{t. If we consider the geometric n}

,

t

Brownia motion given by

dyty ttd y Wtd , then log

 

y_t

with  0 becomes a Brownian motion

with drift with _{   } 2 _{2 and   . In this case,}

the conditional distribution of log



y_t



given

 

0

 

log y log y is also normal. The CIR pr

o the

ocess can followi

be re-parameterized t ng form:



1



dyt   2yt dt y Wtd .t

In this case, we have computed the first four condi-tional moments of the process to use in an estimating function framework. Extended versions of the CIR pro- cess model have been proposed for modeling interest rate processes. For example, some consider the constant elas-ticity of variance process of the form





2

1 2

dyt    yt dtyt dWt

or the nonlinear drift diffusion process [13] given by





4

2 1

dy _  _ y _ y _ y dt

1 2 3 4

1 2 3 d .

t t t t

t t t

y y W

  

For more general extended models, the diffusion is a

function of the observation yt and hence, closed form

ex

r the conditional moments cannot be easily obtained by solving differential equa- tions obtained by repeated application Itô’s formula.

the first four cond

pressions of the conditional distributions, as well as closed form expressions fo

However, Milstein’s approximation can be used to obtain itional moments.

Milstein’s approximation applied to (3.1) produces

 



 





 



 







 

1 1 1

1

, ,

1

th t h t h h t

y 1





2



, , 1 ,

2 t h t h t

t

y y a y h b y h b

 

 





 



(3.2)

e

b  y  y _  h

wher



y b



_

b y 

 and t ~N 0,1



, i.i.d. Unlike the Euler

approximation for diffusion processes, the Milstein ap-

proximation does not yield a conditional nor al distribu-

tion for The distribution implied by the

M

m

 1

th t h

ilstein approximation is a mixture of a normal and chi- square distributions. By using (3.2), the first four condi- tional moments of yth given y t1h are approximated

by

 

   

.

y y





1 , ,

t yt h a yt h h

   _   _1 (3.3)

 



 



 





 

2 2

1 ,

2

t b yt h h

    _

.4)





2 2 2

1 1

1

, t h y , t h ,

b  y  b  y  h

 (3

 

3



 1





 1



2

3 3

3 , ,

t b yt h by yt h h

    _  _

 





 

(3.5)





3

1 1

, _{t h y} , _{t h} ,

b  y _ b  y _ h 

 



 





 



 





 

bserved observations





 





4 2 3

1 1

4 4 4

1 1

4 2

1

15 , ,

15 _{, ,}

4

3 , .

t t h y t h

y

t h t h

t h

b y b y h

b y b y h

b y h

   

 



 

 









(3.6)

Then based on the discretely o

0, ,y yh 2h, , ynh, the martingale differences are

y

 

t th t

m y   , and 2 2

 

t t t

M m   . In this case,

we have

 

2 _,

t t

m   (3.8)

 

 

 









 





 

4

4 1 ,

t t

t

t h

M

y

   

 _

 





 





4 4 4

1 1

4 2

1 7

, ,

2

2 ,

y

t h t h

t h

b y b y h

b y h

 



 







(3.9)

2 3

14b by ,y h



 

, .

m M

2 2 m M, t t

t t

m M

  , then

t

t   (3.10)

In addition, if we let

 





 





 







 



 







 



2 1 2

2 1

2 4

1 1

2 4

1 1

4 ,

2 ,

9 6 , ,

.

4 28 , 7 ,

y t h

t

y t h

y t h y t h

y t h y t h

hb y

b y h

b y h b y h

b y h b y h

 



 

 





 

 



 _ 

 

2 2

 _{ } 

 



 _{ } 

 

(3.11)

The optimal estimating functions based on the mar-

tingale differences m_t and M_t, and the corresponding

information are given by

 

 

1

,

1 t t m

t _t

n

n

m g

m

 

   



  

  



(3.12)

1

, m

t

t t g

t

I

m



 







  



 

 

2 1

2 2

t _(3.13)

1

,

1 ,

M

t t

M

t

t t

g

t _t

n

n

M g

M I

M 





 



 





   



 





 





The combined estimating function and the corre- sponding information follow from Theorem 1 by



taking



,



   .

Example 2 (NLD Process).The nonlinear drift (NLD)

diffusion process for modeling interest rates was intro-duced in [13]. Here we consider the following NLD



1



1 2 1 2

dy_t    y_t dt   y W_td _t (3.14)

where 10, 20, 012 2, and 21 2. These ranges are chosen to guarantee a posi-tive recurrent solution to the SDE. For this process,

parameter



,

 

1 2



a  y    y ,B



,y



  1 2y, and









1 2

1 2

, 1 2

y

b  y   y 

  . The Milstein ap

tion gives the following discretized version of the pro- cess:

proxima-

 



 



 



2



1 2 1 2

1

1 .

4 t

t h

1

1 2

1 1

th t h t h

t

y y y h

y h h

  _ 

    

(3.15)

  

 

  



In this case, we have the following for the first four conditional moments of the discretized process:

 

 1



1 2  11

t yt h yt h

    

 

  



h, (3.16)

 





2 2

1 2 1 2

1 , 8

t h t

 

 

  2 3 3 3

2 1 2 1

4 4 2 2 2

2 1 1 2 1

2 2 2 2 1

7 7

2 2

7

2 4

32

2 ,

t h t

t h

t h

M h h y

h h h y

h y

  

   









 

  



(3.18)

 

 





2 3 3

1 2 1 2 2 .

2   yt h  h 8 h

  

The estimating function and corresponding informa- tion based on are given by

, t t

m M  

3 1 (3.19)

t

m

 

 

 

 

 

2 1

1 11 12

t t

t

t h

n m

h

I I I

 



 

 _ 

 

 

  



2 1

, m

t t

t n

g

m h

y



  



  



 

 





 (3.20)

12 22

,

m m m

g _{I I}

 

 

where

m m

 

2

11 2

1

1 ,

m

t t

n

I h

 







 

 

2

12 2

1 ₁

t yt h t 1

,

m n

I h

 

 _





 

 

2

22 2 2

1 1

1

m n _.

t t h t

I h

y  

 





Moreover, the estimating function and corresponding information based on Mt are given by

 

 





 

1 2 1

1 11 12 12 22

, 1 4

M

t t

t M

t t h

t

t

M M

M

n

n

M g

M h

M g

y h M

h

M

I I

I

I I









 

 

 _ 

 

 

  _ 

_ 

 

 

 

  

 





(3.21)

where

2 11

1

t M _t

1 _,

n M I h



 



1 2



2 12

1

1 4 ,

t h M

t _t

I h

M  









n y h

 





2

2 1

2 22

1

1 4

t

n _h

M

t _t

y h

I h

M  









In this case,  



    1, 2, ,1 2



and the optimal

combined estimating function using mt and Mt is given by _C

 

_tn₁



_{t 1 t 1}



t t

g  



_ a m_ b M_{ , where}









2 1

2 1

2 1

2 1

1

1 ,

, ₁

1 ,

t t

t t

t t t

t t t

t t

t t t

m a

m m M

m M b

m M m

 



 

 



 

 



 



 _{ } 

  _  _

 

 

_{ } 

  _  _

 

 

,

t

M

with

 



1



1

, ,0,0

t

t h h hy

 

 

 _ 

 and

 





2

2 2 1

0,0, , 1 4 .

t

t h

h hy h

 _

 

 _{ } 



The information for the combined estimating function

 

C

g

I   is given by

 



₂



1 2 2

1

2 2

1 1

1

, . C

t t t t

t g

t _{t t}

t t t t t

t n

t

I

m M

m M m M

   

 

   

   

   







   

  

 

   

    

_  _

 

    

  

  



4. Conclusion

For discretely observed general interest rate models, the ed estimating function method allows estimators to be obtained straightforwardly unde very general con- ditions on the first four conditional m ments. In this pa- per, we have studied inference for interest rate models, first by using the Milstein approximation, and then com-

bining estimating functions using marting differences

and have obtained the closed form expression for the information gain.

REFERENCES

[1] aran and M. E. Thom

0.2307/3214183

combin

r o

ale

A. Thavanesw pson, “Optimal Esti-mation for Semimartingales,” Journal of Applied

Prob-y, Vol. 23, No. 2, 1986, pp. 409-417. abilit

doi:1

[2] A. Thavaneswaran and M. E. Thompson, “A Criterion for Filtering in Semimartingale Models,” Stochastic Proc-esses and Their Applications, Vol. 28, No. 2, 1988, pp. 259-265. doi:10.1016/0304-4149(88)90099-3

[3] C. Heyde, “Quasi-Likelihood and Its Application: A Ge- neral Approach to Optimal Parameter Estimation,” Springer Series in Statistics, Springer, 1997. doi:10.1007/b98823

he Foundations of Finite Sample Es-timation in Stochastic Processes,” Biometrika, Vol. 72, No. 2, 1985, pp. 419-428. doi:10.1093/biomet/72.2.419 [4] V. P. Godambe, “T

for Semimartingales,” Stochastic Processes and Their Ap-plications, Vol. 22, No. 2, 1986, pp. 245-257.

doi:10.1016/0304-4149(86)90004-9

[6] U. V. Naik-Nimbalkar and M. B. Rajarshi, “Filtering and

pp 0.1080/01621459.1995.10476513 Smoothing via Estimating Functions,” Journal of the American Statistical Association, Vol. 90, No. 429, 1995,

. 301-306. doi:1

[7] A. Paseka, T. Koulis and A. Thavaneswaran, “Interes Rate Models,” Journal of Mathematical Finance, Vol. 2 No. 2, 2012, pp. 141-158. doi:10.4236/jmf.2012.22016

t ,

. [8] A. Thavaneswaran, Y. Liang and N. Ravishanker, “Infe

ence for Diffusion Processes Using Combined Estimating Functions,” Sri Lankan Journal of Applied Statistics, Vol 12, 2011, pp. 145-160.

, “Asymptotic Theory of Maxi-d Estimator for Diffusion MoMaxi-del,” Work-

r-[9] M. Jeong and J. Y. Park mum Likelihoo

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Conditional Transition Density for the Milstein Scheme,” Economics Discussion Paper 1998-W18, Nuffield Col-lege, Oxford, 1998.

[11] B. G. Lindsay, “Using Empirical Partially Bayes Infer-ence for Increased Efficiency,” The Annals of Statistics, Vol. 13, No. 3, 1985, pp. 914-931.

doi:10.1214/aos/1176349646

[12] J. C. Cox, J. E. Ingersoll Jr. and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, Vol. 53, No. 2, 1985, pp. 385-407. doi:10.2307/1911242 [13] Y. Ait-Sahalia, “Testing Continuous-Time Models of The

Moments of the CIR Process

The first four conditional moments of the CIR process





dyt k yt dt y Wtd ,t

may be obtained by using Itô’s formula on successive powers of the process. This gives





dyt k yt dt y Wtd ,t

2 1 2

d 2 d 2 d

2

t t t t

y  y y   y t,

3 2 2 2

dy_t 3 dy y_{t t} 3 y t_td ,

4 3 2 3

dyt 4 dy yt t 6 y ttd ,

with y

 

0 x as an initial condition. We let

 



j 0



j t

m t E y y x , so that the first four conditional

moments of yt satisfy the following differential

equa-tions:

 



 



1 1

d

, dtm t  k m t 

 

 



2



 

2 2 1

d _{2 2}

dtm t   km t  k  m t ,

 

 



2



 

3 3

d _{3 3 3}

dtm t   km t  k  m t2 ,

 

 



2



 

4 4 3

d _{4 4 6}

dtm t   km t  k  m t ,

with initial conditionsm1

 

0 x,m2

 

0 x , 2

 

₀ 3

m₃ x , and _m

 

_{0 x}4

4  . Solving the differential

equations in turn yields:

 





1 e ,

kt m t _{ }  _x

 



2

 

2









2







2 2

2

2 2 2 2 2

e e

2 2

kt kt

k k x k x kx

m t

k k k

      _     _

   ,

 





 











 

















 











2 3 2 3

3 2 2

2 2 2 2 2 2

2 3

2 2

2 3 3 2 3 3

e

6 2

3 3 2 2 2 3 3 2 3 6

e e

2 6

kt

kt kt kt

k k k k x

m t

k k

k k x kx k k x kx

x

k k

         

         



 

    

 

       

  _ 3_e3 _,

 







 















 















 













 



 













2 2 2 2 2 2

4 3 3

2 2 2 2

2 3

2 2 2 2 2 3

3 3

2 2 2 2 2 3

4 4 4 3

2 3 3 4 6 2 3 3 4 6

24 6

3 3 4 6 2 2 2

4

3 3 4 6 2 3 6 6

6

4 6 3 3 2 4 12 24

. 24

kt

kt

kt

kt kt

k k k k k k x

m t e

k k

k k k x kx

e k

k k k x kx k x

e k

k k k x kx k x

e x e

k

             

      

      

      







 

      

 

    



     



     

 

Hence, the first four conditional centered moments are given as





 

 

 













2 2

2

2 1

2

2

2 e 2 e

2

t t

kt kt

E y

m t m t

x x

k

  

     

 _ _

 

 

     ,





 

 

   

 

















3 ₂

3

3 1 2 1

4

2 2

3

3 2

3 e 3 2 e

2

3 e ,

t t

kt kt

kt

E y

m t m t m t m t

x x

k x

  

   



 



 _ _

 

  

    

 





 

 

   

   

 





















 

4 ₂

2 4

4 1 3 1 2 1

6

2 3

3 4 4

4 6 3

3

4 e 6 2 e

4

4 3 e 4 e 3 .

t t

kt kt

kt kt

t

E y

m t m t m t m t m t m t

x x

k

x x

  

   

   

 

 

 _ _

 

   

    