_{1}

GMM inference procedures based on the square of the modulus of the model characteristic function are developed using sample moments selected using estimating function theory and bypassing the use of empirical characteristic function of other GMM procedures in the literature. The procedures are relatively simple to implement and are less simulation-oriented than simulated methods of inferences yet have the potential of good efficiencies for models with densities without closed form. The procedures also yield better estimators than method of moment estimators for models with more than three parameters as higher order sample moments tend to be unstable.

In many applied fields, data analysts often have to use distributions with density functions having complicated forms. They are often expressed using mean of series representations but model characteristic functions are simpler and have closed form expressions. For actuarial sciences, the compound Poisson distributions are classical examples and for finance, the stable distributions fall into the same category. These are infinitely divisible and many infinitely divisible distributions share the same property of having much simpler characteristic functions than density functions. We shall examine in more details using the Generalized Normal Laplace (GNL) distribution which is obtained by adding a normal component to the GAL random variable hence can be viewed as created by a convolution operation. The GNL distribution was introduced by Reed [

The GNL generalizes the GAL distribution, the density of the GAL can be obtained in closed form but depend on Bessel functions, see Kotz et al. [

First, recall that the characteristic function ϕ ( s ) of a random variable X is a complex function defined as

ϕ ( s ) = E ( e i s X )

and it can be expressed as

ϕ ( s ) = R e ϕ ( s ) + i I m ϕ ( s )

with the real and imaginary parts of ϕ ( s ) given respectively by R e ϕ ( s ) and I m ϕ ( s ) . We can also use polar forms instead of algebraical forms to express complex numbers or functions.

The modulus of ϕ ( s ) is defined as

| ϕ ( s ) | = [ ( R e ϕ ( s ) ) 2 + ( I m ϕ ( s ) ) 2 ] 1 / 2

and the argument ω ( s ) of ϕ ( s ) is defined as ω ( s ) = arctan I m ϕ ( s ) R e ϕ ( s ) . This

allows to express ϕ ( s ) = | ϕ ( s ) | e i ω ( s ) and depending on situations, the polar form of ϕ ( s ) can be simpler to handle than its algebraical form as illustrated in example 1 which gives the characteristic function of the gamma random variable using both representations for the gamma distribution.

Example 1 (characteristic function of the gamma distribution)

It is well known than the characteristic function of the Gamma distribution in algebraical form is given by

ϕ γ ( s ) = ( 1 − i β s ) − ρ , γ = ( ρ , β ) ′

with β being the scale parameter and ρ being the shape parameter, β > 0 and ρ > 0 . Now before giving the polar form of ϕ ( s ) , we give the polar form of z ( s ) = 1 − i β s first and using properties of the modulus as well as properties of the argument of a complex number we then give the polar form for ϕ γ ( s ) . The modulus of z ( s ) is denoted by

| z ( s ) | = [ 1 + β 2 s 2 ] 1 / 2

and the argument of arg ( z ( s ) ) = arctan ( − β s ) and since the function arctan ( x ) is odd, arg ( z ( s ) ) = − arctan ( β s ) . This allows the representation in polar form for z ( s ) .

Using properties of the modulus, since ϕ ( s ) = ( z ( s ) ) − ρ , so | ϕ ( s ) | = | z ( s ) | − ρ . With z ( s ) = [ 1 + β 2 s 2 ] 1 / 2 e i ( − arctan ( β s ) ) and since ϕ ( s ) = ( z ( s ) ) − ρ , it is easy to see that the characteristic function of the gamma distribution is given by ϕ γ ( s ) = [ 1 + β 2 s 2 ] − ρ / 2 e i ( ρ arctan ( β s ) ) , γ = ( β , ρ ) ′ using polar form.

Using the characteristic function of the gamma distribution in polar form, we can find the characteristic function of the GAL distribution which can be considered as the difference of two independent gamma random variables.

Example 2 (characteristic function of the GAL distribution)

Among many representations in distribution of the GAL distribution, the one which makes use of two independent random gamma random variables allows the following representation for the GAL random variable X, see proposition

4.1.3 given by Kotz et al. [

an equality in distribution, G 1 and G 2 are independent and identically distributed as G which follows a gamma distribution with scale parameter equals to one and shape parameter being ρ , the parameter θ is a location parameter with − ∞ < θ < ∞ . The parameter κ controls the skewness of the GAL distribution, κ > 0 and if κ = 1 , the distribution is symmetric. The parameter σ is a scale parameter with σ > 0 . Using the representation with gamma random variables it is easy to see that by letting

1 α = σ κ 2 , 1 β = σ κ 2 and X = d θ + 1 α G 1 − 1 β G 2 , κ = 1 if α = β .

The characteristic function ϕ ( s ) for the GAL distribution in polar form is given by

ϕ γ ( s ) = ( 1 + s 2 α 2 ) − ρ / 2 ( 1 + s 2 β 2 ) − ρ / 2 e i ( θ + ρ ω 1 ( s ) + ρ ω 2 ( s ) ) , γ = ( θ , α , β , ρ ) ′

ω 1 ( s ) = arctan ( s α ) , ω 2 ( s ) = arctan ( − s β ) , using the characteristic function of

the gamma distribution in polar form as given by example 1. Instead of using, replace it by ρ μ then

ϕ γ ( s ) = ( 1 + s 2 α 2 ) − ρ / 2 ( 1 + s 2 β 2 ) − ρ / 2 e i ρ ( μ + ω 1 ( s ) + ρ ω 2 ( s ) ) , γ = ( μ , α , β , ρ ) ′

Using this parametrization, it is easier to connect with the GNL distribution with the representation of the GNL random variable as the convolution of a normal random variable with a GAL random variable. The GAL is symmetric if α = β and its characteristic function can be further simplified and reduced to

ϕ γ ( s ) = ( 1 + s 2 α 2 ) − ρ e i ρ μ or ϕ γ ( s ) = ( 1 + s 2 α 2 ) − ρ e i θ .

Observe that often it is relatively simple to find characteristic function of a distribution of a convolution of two independent random variables using characteristic functions of the component independent random variables and also the characteristic function of the GAL distribution does not depend on the Bessel functions and is much simpler than its density despite the density of the GAL density has closed form expression. The GNL random variable X can be created by adding an independent normal random variable to a GAL random variable and allows the following representation as introduced by Reed [

X = d σ ρ Z + ρ μ + 1 α G 1 − 1 β G 2

or equivalently,

X = d σ ρ Z + θ + 1 α G 1 − 1 β G 2 .

Z is a standard normal random variable and independent of G 1 and G 2 with G 1 and G 2 are as defined as in example 2. Since the characteristic function for

the standard normal random variable is e 1 2 s 2 and the characteristic function of

the GAL distribution is already obtained, the polar form of the GNL distribution can be also obtained and it is given in the following example.

Example 3 (characteristic function of the GNL distribution)

From the representation of the GNL random variable it is easy to see that the characteristic function of the GNL distribution in algebraical form is

ϕ γ ( s ) = e ρ i μ s − ρ σ 2 s 2 2 ( 1 1 − i s / α ) ρ ( 1 1 + i s / β ) ρ

which is given by Reed [

ϕ γ ( s ) = e − ρ σ 2 s 2 2 ( 1 + s 2 α 2 ) − ρ / 2 ( 1 + s 2 β 2 ) − ρ / 2 e i ρ ( μ + ω 1 ( s ) + ρ ω 2 ( s ) )

with ω 1 ( s ) and ω 2 ( s ) are as defined in example 2.

Using the modulus of ϕ γ ( s ) ,

| ϕ γ ( s ) | = e − ρ σ 2 s 2 2 ( 1 + s 2 α 2 ) − ρ / 2 ( 1 + s 2 β 2 ) − ρ / 2 ,

we also have

ϕ γ ( s ) = | ϕ γ ( s ) | e i ρ ( μ + ω 1 ( s ) + ρ ω 2 ( s ) ) , γ = ( μ , α , β , σ 2 , ρ ) ′

As for the GAL distribution, if α = β , the GNL distribution is symmetric and its characteristic function is further simplified and is given by

ϕ γ ( s ) = e − ρ σ 2 s 2 2 ( 1 + s 2 α 2 ) − ρ e i θ with θ = ρ μ

Reed [

c 1 = ρ ( μ + 1 α − 1 β ) , c 2 = ρ ( σ 2 + 1 α 2 + 1 β 2 ) ,

c r = ρ ( r − 1 ) ! ( 1 α 2 + ( − 1 ) r 1 β r ) , r > 3

The GNL distribution provides a better fit to log returns data than the GAL distribution and both these distributions provide much better fit to log returns data than the normal distribution. In addition, all integer moments exist for these distributions and they are also infinitely divisible like the normal distribution which makes them being good alternatives to the normal distribution. From the characteristic function of the GNL distribution, it is easy to see that the real and imaginary part of the characteristic function are given respectively as

R e ϕ γ ( s ) = | ϕ γ ( s ) | cos ( ρ ( μ + ω 1 ( s ) + ρ ω 2 ( s ) ) )

and

I m ϕ γ ( s ) = | ϕ γ ( s ) | sin ( ρ ( μ + ω 1 ( s ) + ρ ω 2 ( s ) ) ) .

For inferences, we assume that we have a random sample of size n which consists of X 1 , ⋯ , X n of independent and identically distributed continuous random variables and they are distributed as X, with common characteristic function ϕ γ ( s ) , γ is a p by 1 vector of parameters of interests with γ = ( γ 1 , ⋯ , γ p ) ′ , γ 0 is the vector of the true parameters with γ 0 ∈ Ω , the parameter space is assumed to be compact. The number of parameters in the model is p. In fact, most inferences procedures based on characteristic function proposed in the literature are still valid if X has a discontinuity point with mass attributed at the origin such as in the cases of the compound distributions. If X is discrete, it is often preferred to work with probability generating function rather characteristic function and for related procedures using probability generating function, see Luong [

Commonly proposed GMM procedures in the literature are based on the empirical characteristic function which is the counterpart of the theoretical one and it is defined as

ϕ n ( s ) = R e ϕ n ( s ) + i I m ϕ n ( s )

with the real and imaginary parts given respectively by

R e ϕ n ( s ) = 1 n ∑ i = 1 n cos ( s X i ) and I m ϕ n ( s ) = 1 n ∑ i = 1 n cos ( s X i ) .

For example, the K-L procedures proposed by Feuerverger and McDunnough [

g 1 ( γ ) = ( 1 n ∑ i = 1 n cos ( s 1 X i ) − R e ϕ γ ( s 1 ) , ⋯ , 1 n ∑ i = 1 n cos ( s k X i ) − R e ϕ γ ( s k ) ) ′

and the rest of moments are similarly formed but based on the imaginary part of the empirical characteristic function and the imaginary part of the model characteristic function,

g 2 ( γ ) = ( 1 n ∑ i = 1 n sin ( s 1 X i ) − I m ϕ γ ( s 1 ) , ⋯ , 1 n ∑ i = 1 n sin ( s k X i ) − I m ϕ γ ( s k ) ) ′ .

By letting

g ( γ ) = ( g 1 ( γ ) g 2 ( γ ) )

and define S to be the limit covariance matrix of the vector n g ( γ ) under the true parameter γ 0 when n → ∞ and let S ^ be a preliminary consistent estimate of S and from which we can obtain a preliminary consistent estimate S ^ − 1 for the inverse of S then the related GMM objective function Q ( γ ) can be formed, i.e., Q ( γ ) = g ′ ( γ ) S ^ − 1 g ( γ ) and minimizing Q ( γ ) will give the vector of K-L estimators.

The following expectation properties are quite obvious and the elements of the covariance matrix for g ( γ ) can be found explicitly using the following identities which are established using properties of trigonometric functions, we have:

E ( R e ϕ n ( s ) ) = R e ϕ γ ( s ) , E ( I m ϕ n ( s ) ) = I m ϕ γ ( s ) ,

C o v ( R e ϕ n ( s ) , R e ϕ n ( t ) ) = ( R e ϕ γ ( s + t ) + R e ϕ γ ( t − s ) − 2 R e ϕ γ ( s ) R e ϕ γ ( t ) ) / 2 n ,

C o v ( I m ϕ n ( s ) , I m ϕ n ( t ) ) = ( R e ϕ γ ( t − s ) + R e ϕ γ ( t + s ) − 2 I m ϕ γ ( s ) I m ϕ γ ( t ) ) / 2 n ,

C o v ( R e ϕ n ( t ) , I m ϕ n ( s ) ) = ( I m ϕ γ ( t + s ) − I m ϕ γ ( t − s ) − 2 I m ϕ γ ( s ) R e ϕ γ ( t ) ) / 2 n (1)

The above identities are results of Proposition 3.1 given by Groparu-Cojocaru and Doray [

Observe that for the K-L procedures or GMM procedures based on the above 2k sample moments, we need to fix the points s 1 , ⋯ , s k where we can make use of the real and imaginary part of the model characteristic function ϕ γ ( s ) and there is still a lack of general criteria on how to choose these points, see discussions by Tran [

Koutrouvelis [

V ( cos ( s X ) ) = n V ( R e ϕ n ( s ) ) → 0 and V ( sin ( s X ) ) = n V ( I m ϕ n ( s ) ) → 0

as s → 0 .

and argued that we should select points in the range of ( 0 , π ) as points near 0 that we need to focus when extracting information from the model characteristic functions. Despite that the K-L procedures have good potentials for generating good efficiencies for estimators but it is often numerical difficult to implement, as the studies of Groparu-Cojocaru and Doray [

The proposed GMM procedures with the selection of the sample moments based on estimating function theory will be developed in the next section. With the original sample, we also transform it to a sample of n observations which are still independent and we work with the original sample and the transformed sample to construct moment conditions.

Carrasco and Florens [

In fact, the points of our procedures are selected with

s i = π n ( i − 1 2 ) , i = 1 , ⋯ , n

and observe that the spacing used is π n → 0 , as n → ∞ and also observe that

the spacing mimics the behavior of the optimum spacing and numerically it bypasses the difficulties of having to find explicitly the value of the optimum spacing by minimizing the determinant of the asymptotic covariance of the K-L estimators if the K-L procedures are used.

For the proposed methods, we need the additional assumption that the first four integer moments of the model distribution exist but in practical situations, this assumption is often met.

The proposed procedures make use of sample moments which focus on extracting information from the square of the modulus of the characteristic function

| ϕ γ ( s ) | 2 using the points s i = π n ( i − 1 2 ) , i = 1 , ⋯ , n and clearly there will be as many points as the sample size.

For model, with a location parameter μ , the modulus | ϕ γ ( s ) | and consequently | ϕ γ ( s ) | 2 will not depend on the location parameter μ and we need another two sample moments beside the sample moments which make use of | ϕ γ ( s ) | 2 to take care of this situation. The example given below will help to clarify the problem that we might encounter when the modulus | ϕ γ ( s ) | or the square of the modulus | ϕ γ ( s ) | 2 is used for inferences.

For the normal distribution with the vector of parameters γ = ( μ , σ 2 ) ′ , the characteristic function is

ϕ γ ( s ) = e i μ t − σ 2 t 2 2

and its modulus is

| ϕ γ ( s ) | = e − σ 2 t 2 2

and the square of the modulus is

| ϕ γ ( s ) | 2 = e − σ 2 t 2 ,

The location parameter μ is missing in | ϕ γ ( s ) | and consequently, in | ϕ γ ( s ) | 2 . This is to illustrate that there might be one parameter being left out if inferences procedures are solely based on | ϕ γ ( s ) | or | ϕ γ ( s ) | 2 . This also means that for GMM procedures which make use of sample moments formed using | ϕ γ ( s ) | or | ϕ γ ( s ) | 2 , there should also be other sample moments to take into account the parameters being left out and if there are parameters being left out, it only affects the location parameter of the model in general, so if we use two additional sample moments which take care of the parameters being left out beside the sample moments which make use of | ϕ γ ( s ) | 2 , the GMM procedures will be viable. As mentioned often at most there is one parameter in the model which is not included in | ϕ γ ( s ) | 2 so the proposed will make use of additional moments which are based on the mean and variance of the model distribution beside the moments based on

| ϕ γ ( s ) | 2 using s i = π n ( i − 1 2 ) , i = 1 , ⋯ , n

We hope to achieve good efficiency yet preserve simplicity by not using more than ten sample moments, this achieved by using the theory of estimating function for building sample moments which make use of

| ϕ γ ( s ) | 2 using s i = π n ( i − 1 2 ) , i = 1 , ⋯ , n

Therefore, it is relatively simple to implement and all can be done within the classical context of GMM procedures without having to rely on a continuum of moment conditions which the practitioners might find difficult to implement. The use of theory of estimating function appears to be new and not included in proposed GMM procedures in the literature which focused on the use of the empirical characteristic function. The new procedures also make use of transformed observations besides the original observations.

The paper is organized as follows. Section 1 introduces the commonly used GMM procedures which are based on empirical characteristic function, the approach taken here does not use the empirical characteristic function and relies on estimating function theory to select sample moments based on the square of the modulus of the model characteristic function. The new GMM procedures are introduced in Section 2.1 with the choice of selected sample moments aiming to provide efficiency for GMM estimation. In Section 2.2 the chi-square test for moment restrictions which can be interpreted as goodness-of-fit is presented. In Section 3, illustrations for implementing the methods using the GNL distribution and normal distribution, the methods appear to be relatively simple to implement yet being very efficient based on the limited studies and appear to be better alternatives the method of moments (MOM) in general.

The theory of GMM procedures are well established in the literature, see Martin et al. [

We shall define the sample moments focusing on | ϕ γ ( s ) | 2 . Let us consider the basic estimating functions

h ( x i , s i ; γ ) = R e ϕ γ ( s i ) ( cos ( s i X i ) − R e ϕ γ ( s i ) ) + I m ϕ γ ( s i ) ( sin ( s i X i ) − I m ϕ γ ( s i ) ) , i = 1 , ⋯ , n

Clearly, the basic estimating functions are unbiased, i.e.,

E γ ( h ( x i , s i ; γ ) ) = 0 , i = 1 , ⋯ , n

Using | ϕ γ ( s ) | 2 = ( R e | ϕ γ ( s ) | 2 ) 2 + ( I m ϕ γ ( s ) ) 2 , we can also express

h ( x i , s i ; γ ) = R e ϕ γ ( s i ) ( cos ( s i X i ) ) + I m ϕ γ ( s i ) ( sin ( s i X i ) ) − | ϕ γ ( s i ) | 2 , i = 1 , ⋯ , n (2)

Now we can construct the optimum estimating functions for estimating γ or more precisely for parameters which appear in | ϕ γ ( s ) | 2 using results of Godambe and Thompson [

1 n ∑ i = 1 n h ( x i , s i ; γ ) E γ ( ∂ h ( x i , s i ; γ ) ∂ γ j ) v γ ( h ( x i , s i ; γ ) ) , j = 1 , ⋯ , p (3)

and v γ ( h ( x i , s i ; γ ) ) denote the variance of h ( x i , s i ; γ ) and can be obtained explicitly, see expression (1).

We would like to make a few remarks here. First note that it is easy to show that

E γ ( ∂ h ( x i , s i ; γ ) ∂ γ j ) = − 1 2 ∂ | ϕ γ ( s ) | 2 ∂ γ j

using E γ ( cos ( s X ) ) = R e ϕ γ ( s ) and E γ ( sin ( s X ) ) = I m ϕ γ ( s ) and clearly if there is one parameter of the model says γ l which does not appear in | ϕ γ ( s ) | 2 then

E γ ( ∂ h ( x i , s i ; γ ) ∂ γ j ) = 0 and there is no optimum estimating function for this parameter and if we want to estimate all the parameters, we need an extra estimating function. We use the following notation for the vector of optimum estimating function discarding the ones with E γ ( ∂ h ( x i , s i ; γ ) ∂ γ j ) = 0 . The vector of optimum estimating functions for parameters included in | ϕ γ ( s ) | 2 adopting the convention discarding those with E γ ( ∂ h ( x i , s i ; γ ) ∂ γ j ) = 0 is given by

1 n ∑ i = 1 n h ( x i , s i ; γ ) E γ ( ∂ h ( x i , s i ; γ ) ∂ γ 1 ) v γ ( h ( x i , s i ; γ ) )

where we partition the vector γ into two components, γ = ( γ 1 γ 2 ) with the property that all the parameters appear in | ϕ γ ( s ) | 2 form the vector γ 1 and all the

remaining parameters are included in the vector γ 2 . In general, if γ ≠ γ 1 in then γ 2 is reduced to a scalar. Therefore, the vector of optimum estimating function in general is either a vector of p elements or p − 1 elements and consequently when these estimating functions are converted to sample moments, we shall have either p or p − 1 sample moments.

We shall let g 1 ( γ ) be vector the sample moments which make use of points of | ϕ γ ( s ) | 2 as

g 1 ( γ ) = 1 n ∑ i = 1 n h ( x i , s i ; γ ) E γ ( ∂ h ( x i , s i ; γ ) ∂ γ 1 ) v γ ( h ( x i , s i ; γ ) ) ,

v γ ( h ( x i , s i ; γ ) ) can be obtained using the real and imaginary parts of the model characteristic function and from the definition of h ( x i , s i ; γ ) , it then follows that

v γ ( h ( x i , s i ; γ ) ) = ( R e ϕ γ ( s i ) ) 2 v a r ( cos ( s i X i ) ) + ( I m ϕ γ ( s i ) ) 2 v a r ( sin ( s i X i ) ) + 2 ( R e ϕ γ ( s i ) ( I m ϕ γ ( s i ) ) ) c o v ( cos ( s i X i ) , sin ( s i X i ) ) (4)

Using the identities as given by expression (1), the variance of cos ( s i X i ) is

v a r ( cos ( s i X i ) ) = ( R e ϕ γ ( 2 s i ) + 1 − 2 ( R e ϕ γ ( s i ) ) 2 ) / 2 (5)

and the variance of sin ( s i X i ) and the covariance c o v ( cos ( s i X i ) , sin ( s i X i ) ) are given respectively by

v a r ( sin ( s i X i ) ) = ( 1 − R e ϕ γ ( 2 s i ) − 2 ( I m ϕ γ ( s i ) ) 2 ) 2 , (6)

c o v ( cos ( s i X i ) , sin ( s i X i ) ) = ( I m ϕ γ ( 2 s i ) − 2 ( R e ϕ γ ( s i ) ) ( I m ϕ γ ( s i ) ) ) / 2 . (7)

These variances and covariance terms can also be obtained using results given by Groparu-Cojocaru and Doray [

Now define two additional sample moments g 2 ( γ ) and g 3 ( γ )

g 2 ( γ ) = 1 n ∑ i = 1 n ( X i − E γ ( X ) ) ,

The location parameter if it belongs to the model but is not included in γ 1 , it will appear in E γ ( X ) which is the mean of the model distribution and it can be obtained by differentiating the model characteristic function and for GMM procedures we prefer to have the number of sample moments exceeding the number of parameters in the model, so we also consider the following sample moment which makes use of the variance of the model distribution V γ ( X ) and it can also be obtained by differentiating twice the model characteristic function, i.e.,

g 3 ( γ ) = 1 n ∑ i = 1 n ( X i − V γ ( X ) ) .

The vector sample moments for the developed GMM procedures is given by

g ( γ ) = ( g 1 ( γ ) g 2 ( γ ) g 3 ( γ ) )

and notice g 1 ( γ ) makes use of the transformed observations s i X i , i = 1 , ⋯ , n but g 2 ( γ ) and g 3 ( γ ) make use of the original sample observations X i , i = 1 , ⋯ , n .

The vector of the proposed GMM estimators γ ^ is obtained by minimizing the criterion function,

Q ( γ ) = g ′ ( γ ) S ^ − 1 g ( γ )

with S ^ − 1 being a positive definite matrix with probability one and it will be defined subsequently after the definition of S and its inverse S − 1 , S ^ − 1 is a consistent estimate of S − 1 .

For finding elements of the matrix, we can first express the components of the vector of sample moments as

g ( γ ) = ( g 1 ( γ ) g 2 ( γ ) g 3 ( γ ) )

with

g 1 ( γ ) = 1 n ∑ i = 1 n m 1 ( x i ; s i , γ ) , m 1 ( x i ; s i , γ ) = h ( x i , s i ; γ ) E γ ( ∂ h ( x i , s i ; γ ) ∂ γ 1 ) v γ ( h ( x i , s i ; γ ) ) ,

g 2 ( γ ) = 1 n ∑ i = 1 n m 2 ( x i ; γ ) ,

and let

The matrix

Now if we have a preliminary consistent estimate

and its inverse which is

The vector of the GMM estimators

which is a r by p matrix, r is the number of sample moments used or equivalently the number of elements of the vector

and

and consequently, the asymptotic variance of

One of the advantages of GMM procedures is that it can lead to distribution free chi-square test. The asymptotic null distribution of the statistic no longer depends on the parameters using statistic based on the same objective function to obtain the vector of GMM estimators

For testing the moment restrictions

Assuming we have already minimized

For illustrations of the newly developed methods, we shall examine the symmetric GNL distribution and compare the efficiencies of GMM estimators vs the efficiencies of method of moment estimators (MOM) as given by Reed [

The location parameter instead of being

It is not difficult to see that

The vector of parameters of the symmetric GNL distribution is

which gives the following derivatives

and the variance of the basic estimating equation

The first two cumulants which give the mean and the variance of the distribution

are

Reed [

By fixing

Using simulated samples, we can estimate the individual mean square error for each parameter for each estimator. The vector of GMM estimators is denoted by

with these quantities estimated, we can estimate the asymptotic overall relative of efficiency of these two methods as

Since GMM procedures perform better than MOM procedures for estimating

In the second limited study, we consider the normal distribution which is also used for modelling log-returns data. The normal distribution is

We only have 3 sample moments for GMM estimation for the normal model and we also use

the estimate ARE is close to 1 for the parameters being considered in TableA1 & TableA2.

Based on theoretical results and numerical results, it appears that:

1) The new GMM procedures are relatively simple to implement. The number of sample moments can be kept below the number ten yet the methods appear to have good efficiencies and offer good alternatives to MOM procedures which in general are not efficient for models with more than three parameters.

2) The proposed procedures are simpler to implement than GMM procedures based on a continuum of moment conditions and consequently might be of interests for practitioner who want to use these methods to analyze date where the model characteristic function is simple and have closed form but the density function is complicated, these situations often occur in practice.

3) The methods are less simulation oriented and consequently faster in computing time for implementations.

4) The estimators obtained have good efficiencies for some models being considered but more numerical and simulation works are needed to confirm the efficiencies using different parametric models and larger scale of simulations. In addition, further studies are needed for the topic on adding sample moments to make the chi-square goodness-of-fit test consistent without creating extensive numerical difficulties when it comes to obtaining the efficient matrix which is used for the quadratic form of the GMM objective function.

The helpful and constructive comments of a referee which lead to an improvement of the presentation of the paper and support from the editorial staff of Open Journal of Statistics to process the paper are all gratefully acknowledged.

The author declares no conflict of interest regarding the publication of the paper.

Luong, A. (2020) Generalized Method of Moments and Generalized Estimating Functions Using Characteristic Function. Open Journal of Statistics, 10, 581-599. https://doi.org/10.4236/ojs.2020.103035

−0.04 | −0.03 | −0.02 | −0.01 | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | |
---|---|---|---|---|---|---|---|---|---|

0.10 | 0.3709 | 0.0084 | 0.0104 | 0.2435 | 0.0419 | 0.5732 | 0.5725 | 0.0006 | 0.5008 |

0.11 | 0.0000 | 0.0834 | 0.0000 | 0.7649 | 0.0002 | 0.4282 | 0.0025 | 0.2395 | 0.0859 |

0.12 | 0.1362 | 0.2370 | 0.2177 | 0.0021 | 0.3731 | 0.5520 | 0.1105 | 0.0000 | 0.6634 |

0.13 | 0.0034 | 0.0880 | 0.0000 | 0.1197 | 0.0002 | 0.0670 | 0.2390 | 0.0188 | 0.1897 |

0.14 | 0.0530 | 0.0000 | 0.0004 | 0.4699 | 0.0702 | 0.0220 | 0.0001 | 0.0725 | 0.2501 |

Simulation studies for symmetric GNL distributions with

−0.04 | −0.03 | −0.02 | −0.01 | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | |
---|---|---|---|---|---|---|---|---|---|

0.10 | 1.0005 | 1.0009 | 1.0008 | 1.0005 | 1.0012 | 1.0014 | 1.0001 | 1.0011 | 1.0004 |

0.11 | 1.0005 | 1.0020 | 1.0036 | 1.0021 | 1.0026 | 1.0035 | 1.0020 | 1.0022 | 1.0038 |

0.12 | 1.0009 | 1.0020 | 1.0015 | 1.0018 | 1.0008 | 1.0032 | 1.0018 | 1.0013 | 1.0031 |

0.13 | 1.0022 | 1.0021 | 1.0040 | 1.0020 | 1.0040 | 1.0012 | 1.0023 | 1.0020 | 1.0012 |

0.14 | 1.0017 | 1.0022 | 1.0023 | 1.0025 | 1.0050 | 1.0157 | 1.0008 | 1.0019 | 1.0036 |

Simulation studies for normal distributions