Moment Properties and Quadratic Estimating Functions for Integer-valued Time Series Models

Recently, there has been a growing interest in integer-valued time series models. In this paper, using a martingale difference, we prove a general theorem on the moment properties of a class of integer-valued time series models. This theorem not only contains results in the recent literature as special cases but also has the advantage of a simpler proof. In addition, we derive the closed form expressions for the kurtosis and skewness of the models. The results are very useful in understanding the behaviour of the processes involved and in estimating the parameters of the models using quadratic estimating functions (QEF). Specifically, we derive the optimal function for the integer-valued GARCH (p, q) known as INGARCH (p, q) model. Simulation study is carried out to compare the performance of QEF estimates with corresponding maximum likelihood (ML) and least squares (LS) estimates for the INGARCH (1,1) model with different sets of parameters. Results show that the QEF estimates produce smaller standard errors than the ML and LS estimates for small sample size and are comparable to the ML estimates for larger sample size. For illustration, we fit the 108 monthly strike data to INGARCH (1, 1) models via QEF, ML and LS methods, and show the applicability of QEF method in practice. Keyword: Skewness, kurtosis, martingale difference, quadratic estimating functions, integer-valued.


Introduction
An increasing number of studies that involve integer-valued time series data can be found in the literature.Zeger (1988) extensively studied the monthly cases of Polio infection in the U.S. from 1970 to 1983.Johansson (1996) considered the effect of lowering speed limits on the number of accidents while Li et al. (2014) investigated the implication of crime cases over time.As a result, there is a need for integer-valued time series models extended to include autoregressive moving average models, the first of which were introduced by Brockwell and Davis (1991) and Emad and Nadjib (1994).
Later, Ferland et al. (2006) extended the classical generalized autoregressive conditional heteroskedastic model with Poisson deviate.To account for overdispersion, Zhu (2011) introduced a new version of Ferland's model with negative binomial deviate.For cases of data with excess zeroes, Zhu (2012) proposed the zero-inflated models with both Poisson and negative binomial deviates.Here, we re-examine some of these models and present simpler derivations of their moment properties using martingale difference.Such martingale difference have been successfully applied to various time series processes, see for example, Thavaneswaran and Abraham (1988) and Ghahramani and Thavaneswaran (2009).These results are very significant for the development of simpler theories on integer-valued time series models, in particular, for estimating the paramaters of the models using the estimating functions method.
The paper is divided as follows: In Section 2, we propose a general class of integervalued time series models including important models given in Ferland et al. (2006), Zhu (2011) and Zhu (2012).We also derive the basic properties of the model, namely, formulae for the mean, variance, autocovariance and autocorrelation using a new approach, i.e by employing martingale differences.In Section 3, we present the higher order moment properties of the model up to order 4 by using martingale difference.In Section 4, we derive the optimal function for INGARCH (p,q) model via quadratic estimating functions (QEF).Simulation study is conducted to compare the performance of QEF, ML and LS estimates for INGARCH (1,1) model.We illustrate the QEF method in practice to the monthly strike data set given in Jung et al. (2005).Concluding remarks are given in Section 5.

Moments of Integer-Valued Time Series Models
Following the notation in Ferland et al. (2006), we consider four types of integer-valued time series model: Poisson (INGARCH), negative binomial (NBINGARCH), zeroinflated Poisson (ZIPINGARCH) and zero-inflated negative binomial (ZINBINGARCH) with conditional mean   is of the form:  t is the  −field generated by We now apply the martingale difference, TP , u is a martingale difference sequence, the equation can be rewritten in using backward operator, B, as If all the roots of   0 , the equation ( 4) can be written as and the variance of t X is given by and since for large t,  can be easily expressed in terms of  .For the processes INGARCH, NBINGARCH and ZIPINGARCH, it can be easily shown that the corresponding values following the index c = 0, 1 appearing in the probability mass function of the zeroinflated negative binomial distribution (see Zhu, 2012).However, we note that the strict stationary properties have been studied only for the INGARCH (p, q) model by Ferland et al. (2006).As highlighted by Zhu (2011Zhu ( , 2012)), different approaches are required to exhibit the properties for the other three models and are of interest in future work.
The first aim here is to derive the general formula for the first two moments, the autocovariance and the autocorrelation of the integer-valued process   t X of the form in equations (1-2).The result is given in Theorem 1.
Theorem 1: Under the first and second order stationarity assumptions, (a) Proof: The mean of t X can be obtained by taking the expectation of equation (3).Since 0 ) ( , 1(a) follows.From equation (5), we notice that the process can be represented as a general form of a time series process (see Abraham and Ledolter, 2009), therefore, the variance, autocovariance and correlation of t X are obtained.

Skewness and Kurtosis
In the literature, only the first two moments and the autocovariance are given for integervalued time series models.In this section, following respectively.The proof is given in Appendix 1.
Example: Using Theorems 1 and 2, we derive the following results for four distributions with p = 1 and q = 1.From equations ( 1) and ( 2), the process   t X is such that It can be shown that the weight j  is given by where Therefore, the summations of the weights j  are given in the following form: From Theorem 1(c), the autocovariance of the   t X process with order (1,1) can be written as where a is as defined earlier for the different models.
Using the same arguments as in Section 2, we can find the skewness and kurtosis of .
respectively.On the other hand, However, the similar corresponding expression for ZINBINGARCH is complicated but can still be solved using standard mathematical software.

General Theory of Quadratic Estimating Functions
Godambe (1960) was the first to introduce regular estimating functions (EF) that satisfy certain conditions and procedures for choosing an optimal EF.The requirement for a regular EF,   θ ; t X g are: where  is the parameter space, (iii) is differentiable under the sign of integration, (iv) .
According to Godambe (1960), to find the optimal EF, say two criteria should be satisfied.First, the estimated parameter should be as close as possible to the true value.This means that the variance  should be minimized and therefore The second criterion is that the expected values of the derivatives of the function   . By following both criteria, the optimal EF,   θ ; * t X g can be defined as follows:

Definition 1
Let G denote the class of all regular EFs.The   Godambe (1985) studied the inference of discrete time series processes using estimating functions.He considered a class : θ of EF which is linear combination of t h 's where the expected value . The theorem below is the result in Godambe (1985) on optimal EFs for the dependent case.
: θ be the class of all EFs where t h and .Then, the optimal estimating function ; The EF method was later extended by Liang et We estimate the parameter θ using two classes of martingale differences and such that The following theorem is obtained from Liang et al. (2011).
estimating functions, the optimal estimating functions is given by

The INGARCH (p, q) Model
In this subsection, we focus on INGARCH (p, q) model given by From equations ( 6) and ( 7), the martingale differences considered are Therefore, one can conclude that, if the conditional mean and conditional variance are the same, the QEF method can be reduced to the EF method.Since we have 1   q p parameters, then . Hence, the optimal QEF for each parameter are:     The formulation of optimal equations for the other three models which are NBINGARCH (p,q), ZIPINGARCH (p,q) and ZINBINGARCH ) , ( q p models are the same as INGARCH (p,q) model.The optimal equation(s) for the additional extra parameter (s) in the above three models can be derived using optimal estimating functions in Theorem 4.An optimal estimate of θ can be obtained by solving the equation(s)

Simulation Study
Let N and n be the number of simulations and the sample size generated respectively from the INGARCH (1, 1) models given by Model 1: Here, we demonstrate how to estimate the parameters using the QEF method: • Step 1-Generate the data: We first generate the data from given true values.Then, we choose the observations numbering from 100 to 100 + n.
• • Step 3-Estimate the parameters: Using nleqslv, we solve the simultaneous optimal equations ( 11) to ( 13) in R-cran programming language in order to obtain the QEF estimates of  , 1  and 1  for the INGARCH (1,1) model. .The performance of each parameter estimates is measured using bias, standard error (SE) and root mean squared error (RMSE).The results are shown in Tables 1 to 3.   the QEF estimates give the smaller values of SE and RMSE compared to other two methods.However, as n increases, the SE and RMSE values for the QEF estimates are always marginal smaller than ML and LS estimates.Secondly, as expected, as n increases from 100 to 1500, all the SE and RMSE of QEF, ML and LS estimates are consistently decreases.Lastly, it is important to point out the computational times for the QEF method is four times shorter than the ML method and three times shorter than the LS method when the simulation is done using R-cran programming.R codes are available upon request.Therefore, we can conclude that the QEF method provided consistently accurate estimates and computation effective than the ML and LS methods in the parameter estimation of INGARCH models.

Real data Example
We apply the proposed methodology to analyze the 108 monthly strike data from January 1994 to December 2002 given by Jung et al. (2005).The data are available at the U.S. Bureau of Labor Statistics (http://www.bls.gov/wsp/)(see Weiβ, 2010).It describes the number of work stoppages leading to 1000 workers or more effectively idle during the period.The time series is given in Figure 1.We fit the data using the INGARCH (1,1) model via the QEF, ML and LS methods.Then we obtain the parameter estimates together with their respective standard errors in parenthesis are shown in Table 4.We observe that, the QEF and ML methods give the same values of estimates.As expected, the standard errors of the QEF estimates are the smallest as compared to other two methods. .According to Kedeem and Fakianos (2002), for the specified model, the sequence t z should have mean and variance close to 0 and 1 respectively and the sequence does not have serial correlation.We found that in our case, the mean and variance of the Pearson residuals are 0.032 and 1.009 respectively and are thus close to zero and unity as desired.Moreover, by using Ljung-Box (LB) statistics, the results from Table 5 indicate that there is no significant serial correlation in the residual.

Concluding Remarks
This paper studied the moments of four integer-valued time series models, namely, the Poisson, negative binomial, zero-inflated Poisson and zero-inflated negative binomial models.We used the martingale difference to derive the higher order moments of all four models.The results for the first two moments are similar to those found in Zhu (2011) and Zhu (2012), but the derivation was much simpler.In addition, we derived the higher order moments of integer-valued time series up to order 4.However, the results hold for only the INGARCH (p, q) model.Further investigations on the stationarity of the other three models are required.Furthermore, we developed the quadratic estimating functions method mainly focusing on the INGARCH (p, q) model.
To investigate the performance of the QEF method compared to those of the LS and ML methods, simulation was carried out to obtain the estimated parameters together with their standard errors.The results showed that the QEF estimates give smaller standard errors and computation effective compared to the ML and LS estimates.Lastly, we model the monthly strike data using the INGARCH (1,1) model via QEF method.The adequacy of fit was investigated using diagnostic tools based on the Pearson residuals.
For the future research, other estimation methods such as Kalman filter can be considered.
. Similarly, we can show that Therefore, the covariance is given by

Theorem 4 :
In the class

Step 2 -
Initialize the parameters: We set the initial values for 1 other hand, we take the value of  to be the mean X  , of the generated data in Step 1, namely,Ferland et al., 2006).

Figure 1 :
Figure 1: The monthly strike data from January 1994 to December 2002.
b) Using the third moment and Theorem 1(b), the skewness of t X is Using the fourth moment and Theorem 1(c), the kurtosis of

XIV No.1 2018 pp157-175 159
r is the number of successful trials, t p is the probability of successful trials and  is the inflation parameter.Pak.j.stat.oper.res.Vol.
al. (2011) to the case where the first four conditional moments are known.The functions used are called quadratic estimating functions (QEF).

Table 5 : Diagnostics for INGARCH (1,1) model using QEF method.
Jung et al. (2005)amine the model adequacy, from Figure2(a), there is no trend observed indicating the randomness of the residuals and in Figure2(b), the plot does not exceed the dotted line.Therefore, the INGARCH (1, 1) model is a good fit for the monthly strike data given inJung et al. (2005).