Robust And Mean Targeting Estimators For Integer-valued GARCH Models

Traditional time series models deal with continuous data(such as monthly rainfall in a certain area,the daily trading price of a stock,etc.).Results of modeling integervalued data with these models are less effective,which is especially true when the value is less than 20.The common pitfalls are: larger deviations when approximated by continuous data,results of prediction are not integer-valued.These data are widely encountered in the practice of actuarial,economics,finance,epidemiology,hydrology,meteorology,etc.For example,the daily number of outpatient visits to a hospital due to a specific disease,the monthly number of insurance policies for a given insurance plan,the hourly number of defective items produced on a production line,the daily number of certain products marketed by shopping malls,the hourly number of customers arriving at a barber shop,etc.These dependent data are quite different from the traditional time series of real numbers and are more difficult to study.Common time series models are autoregressive models and GARCH models,which respectively model the conditional mean and the conditional variance,and they all have corresponding integer-valued forms.Research about integer-valued time series models originated from first-order integer-valued autoregressive models based on thinning operators in the 1980 s,later a Poisson integer-valued GARCH model was proposed.This model assumes that the conditional distribution of the data is Poisson distribution,and the conditional mean(called the intensity process)is a linear function of historical observations and historical intensities,which can handle the volatility of integer-valued data,especially the heteroscedasticity.It is well known that using GARCH models to handle financial data is very popular,and its inventor Robert Engle thus won the 2003 Nobel Prize in economics.The integer-valued GARCH model is a generalization of the GARCH model with an integer-valued version,which attracts much attention once it appears.The most commonly used estimation method for unknown parameters of an integervalued GARCH model is maximum likelihood,but its numerical results are very sensitive to the choice of initial values and algorithm strategies,which is the same as classical GARCH models.To eliminate this numerical dilemma,closed-form estimates without using a numerical optimization process is a good alternative.Another idea is analogy to the idea of variance targeting estimation by introducing a mean targeting estimation(MTE)to reduce dependence on the initial value by reducing the number of parameters optimized numerically.When outliers exist in the data,a common idea is using robust estimation,we consider robust estimates based on closed-form estimates and -estimates,respectively.The main content of this thesis is divided into three parts,details as follows:1.Robust closed-form estimate.As an alternative to using numerically optimized estimators,one may want to consider closed-form moment estimators,which may reduce effectiveness,but have the advantage of being easy to compute and can be used as initial values for some estimates that required numerical optimization.We give the closed-form estimate for the integer-valued GARCH model,and establish the consistency and asymptotic normality of the estimator.When there are outliers in the considered problem,it is very important using robust estimate and filtering outliers in some situations.We consider the robust form of closed-form estimators by using the sample mean and autocorrelation functions respectively with their robust estimators based on weights,ranks and signs.The robust estimator gives a more reasonable result in the case of additive or transient outliers.The performance of the new estimator is demonstrated by five practical examples from the stock market using in-sample prediction,out-of-sample prediction and scoring rules,and recommend methods that should be used in practice,as well as discuss other possible methods for defining robust estimates.2.Data-adaptive robust estimate based on modified Tukey's biweight loss. estimation is a relatively common robust estimation method and two papers have studied two types of -estimate for integer-valued GARCH models,but there are some gaps in establishing large sample properties of parameters' estimators.To establish the existence,uniqueness,consistency and asymptotic normality of estimators,we assume that the loss function is three-times-continuously-differentiable or the corresponding estimation equation to be twice-continuously-differentiable,but existing Huber's or Tukey's loss functions do not satisfy this requirement.First,we propose a loss function called tri-weight,which is a modification of Tukey's biweight function and is three-timescontinuously-differentiable.Furthermore,we propose a new hybrid loss function,with relatively small errors using the tri-weight and relatively large ones using exponential squared loss.Mallows' quasi-likelihood estimate is defined,which can be regarded as an -estimate.The existence,uniqueness,consistency and asymptotic normality of the estimator are established,and an algorithm to calculate the estimator is also given,in which the selection of two tuning parameters of the loss function is data-driven.Finally,the performance of new estimators was examined through simulation and three real-data examples were analyzed to demonstrate the superiority of the new estimators.3.Mean targeting estimate.As mentioned earlier,numerical results of maximum likelihood estimator(MLE)for parameters of integer-valued GARCH models are sensitive to the choice of initial values and algorithm strategies.To eliminate this numerical dilemma,we propose an alternative for MLE and name it as MTE,which is an analog of the method of variance targeting estimation in the GARCH model.MTE relies on a re-parametrization of the model such that the expression of the conditional mean explicitly contains an unconditional mean.The first step introduces an estimator of the unconditional mean and in the second step MLE is used to estimate the remaining parameters.The two-step estimation process may undermine the accuracy of the asymptotic distribution for MLE,but the loss of accuracy cannot be known by perception.In addition,for non-Poisson models,we cannot take for granted that MLE is superior to MTE.The advantages of MTE may not be limited to numerical simplicity,which also ensures that the estimated unconditional mean of the model is equal to the sample mean.When the true model is not Poisson,MTE provides a better fit than MLE,which needs to be further examined by prediction.We establish the consistency and asymptotic normality of MTE,and give a comparison with MLE and discuss the advantages of the new method.Simulation and real-data examples further validate the superiority of the MTE over MLE.