Generalized Method of Moments and Generalized Estimating Functions Using Characteristic Function

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.


Introduction
In many applied fields, data analysts often have to use distributions with density 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 [1] and we shall use it to motivate inferences procedures based on characteristic functions instead of densities. Both the GNL and GAL distributions provide better fit to log returns data in finance. The density of the GNL distribution is more complicated than the density of the GAL distribution. The book by Kotz et al. [2] gives a very comprehensive account of the GAL distribution. Obviously, if these distributions with properties just mentioned are used for modelling, we still want to be able to estimate the parameters and perform tests for validating the models used.
Often maximum likelihood (ML) procedures are difficult to implement due to the lack of closed form for the density functions for the models being used and even the ML estimators are available, when they are used with the Pearson chi-square statistics in general do not lead to distribution free statistics further complicate ML procedures. Therefore, it is natural that we aim at a unified approach to estimation and testing. Inferences developed in this paper will be unified using GMM approach but with the use of estimating function theory to select sample moments using moment conditions extracted from model characteristic function or more precisely the square of its modulus for constructing the GMM objective function. Subsequently, estimation and testing can be carried out. Before giving more details of the developed GMM procedures of this paper and how they differ from GMM procedures in the literature and the advantages of the new procedures, we shall give more details about the GNL distribution where the use of characteristic function appears to be more natural than the use of the model density. The GMM methods developed are also less simulation intensive than simulated methods which appear in the paper by Luong and Bilodeau [3] and faster in computing time for implementing.
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. [2] (page 189), Luong [4] and since the GAL distribution can also be obtained from the distribution of the difference of two gamma random variables, we shall consider first the characteristic function of the gamma distribution in example 1 and subsequently in example 2 and example 3, we shall consider respectively the characteristic function of the GAL and GNL distributions.
First, recall that the characteristic function ( ) It is well known than the characteristic function of the Gamma distribution in algebraical form is given by with β being the scale parameter and ρ being the shape parameter, 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 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 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 [1] (p 475), 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

Empirical Characteristic Function and GMM Procedures in the Literature
For inferences, we assume that we have a random sample of size n which con- is the vector of the true parameters with 0 ∈ Ω γ , 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 [5].
and define S to be the limit covariance matrix of the vector ( ) ng γ under the true parameter 0 γ when n → ∞ and let Ŝ be a preliminary consistent estimate of S and from which we can obtain a preliminary consistent estimate The following expectation properties are quite obvious and the elements of the covariance matrix for ( ) g γ can be found explicitly using the following The above identities are results of Proposition 3.1 given by Groparu-Cojocaru and Doray [7] (p 1992) or results from Koutrouvelis [8] (p 919).
Observe that for the K-L procedures or GMM procedures based on the above 2k sample moments, we need to fix the points 1 , , k s s  where we can make use of the real and imaginary part of the model characteristic function ( ) s φ γ and there is still a lack of general criteria on how to choose these points, see discussions by Tran [9] but it is recommended that these points are equally spaced, i.e., 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 [7] (p 1996) have shown that in practice we need at least 10 k ≥ which means that at least 20 sample moments are needed for the procedures to have good efficiency and in these situations, the matrices S and Ŝ are often nearly singular and inverting such large matrix often create difficulties and we shall see that GMM procedures proposed in this paper with the use of theory of estimation function to select sample moments will only need a number of sample moments which is less than 10 in general instead of at least 20. In addition, the number of points from the model characteristic function used to construct sample moments also goes to infinity as the sample size n → ∞ .
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 [10] have introduced GMM methods with a continuum of moment conditions and Carrasco and Kotchoni [11] have used the empirical characteristic function and developed GMM procedures based on objective functions which match the empirical characteristic function with the model counterpart using points which belong to a continuum interval. Using a continuum of moment conditions is a solution for the arbitrariness choice of selecting points of the characteristic functions to extract information but the procedures might be difficult to implement for practitioners meanwhile our procedures remain simple and closer to the classical GMM procedures with a finite number of moments but we shall use estimation theory to select sample moments and the number of points will be selected equally spaced in the interval ( ) 0, π and the number of points will go to infinity as n → ∞ .
In fact, the points of our procedures are selected with and observe that the spacing used is 0 n π → , as n → ∞ and also observe that 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 ( ) 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.

Estimation
The theory of GMM procedures are well established in the literature, see Martin  [12], Hayashi [13], Hamilton [14] but based on the assumption that sample moments are already selected. In this paper, we focus on how to select moments for model with characteristic function being simple and has closed form but the model density is complicated and we do not make use of the classical empirical characteristic function as other GMM procedures being proposed in the literature. Here, we focus on the square of the modulus of the model characteristic function to build sample moments and since the modulus might not include all the parameters of the model such as the case when there is a location parameter, we shall also include two moments which focus on the model distribution mean and variance to complete the set of sample moments and for practical applications, we do not need more than ten sample moments for the use of the proposed GMM procedures.
We shall define the sample moments focusing on where we partition the vector γ into two components, These variances and covariance terms can also be obtained using results given by The vector sample moments for the developed GMM procedures is given by , ; ; , , ; , ; ; ;

Testing Moment Restrictions
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 de-

Numerical Illustrations and Simulations
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 [1] (p 47).
The characteristic function of the symmetric GNL distribution only has 4 parameters as α β = , it is easy to see that its characteristic function is reduced to ( )  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.