Zero-Modified Skellam Integer-Valued GARCH Model

Integer-valued time series data can be the number of events observed or integer values of a particular pattern.Many scholars have attempted to deal with integer-valued time series,mainly focusing on the discussion of nonnegative integer values,but they cannot fit time series data with negative observations and negative correlations.Subsequently,scholars gradually develop models on the Z-valued sample space,which can handle all integers Z = {...,-1,0,1,...} data and generalize the discrete time series models on non-negative integer values to Z values.In recent years,many studies of time series have been based on integervalued autoregression（INAR）models and integer-valued generalized autoregressive conditional heteroscedasticity（GARCH）models.For the study of Zvalued time series,some INAR models based on signed sparse operators and the rounding operator have been proposed.On the assumption of conditional distribution of integer-valued GARCH models,scholars have proposed the integer-valued GARCH model based on Skellam distribution and the GlostenJagannathan-Runkle GARCH（GJR-GARCH）model based on shifted geometric distribution by considering discrete distributions on Z values.We propose an improved Skellam model based on the zero-modified Skellam distribution for a particular integer 0 in this paper,namely the zeromodified Skellam integer-valued GARCH model.This model takes into account an additional parameter which is detailed for integer 0 by considering datasets with different frequencies of zeros in order to fit the data better,match data characteristics and capture volatility in non-zero-mean time series.In addition,the model has a simple conditional distribution form,which makes our model more flexible.In this paper,we introduce the standard Skellam distribution and its corresponding integer-valued GARCH model,several generalizations of the Skellam distribution and some theory properties,plot the bars of these distributions for different combinations of parameters and give the definition and statistical properties of the zero-modified Skellam integer-valued GARCH model,including the mean,variance,covariance and partial autocorrelation coefficients.Secondly,we estimate the parameters of the model by the conditional maximum likelihood method and conditional least squares method.We make numerical simulation by generating samples from the zero-modified Skellam distribution model.We test the modified model by giving the log-likelihood ratio test statistic.Finally,we illustrate the good performance of the newly proposed model by analyzing two instances from different stock exchange markets.