Estimating Function Method’s Generalization For Multi-parameter Model

This paper first overview the development about the estimating function, introduce some known theories and application to one-parameter time series models.’We also introduce the concept of information.Recently there has been growing interest in using nonlinear time series model-s in finance and economics,Inference for nonlinear time series had been studied in Thavaneswaran and Abraham (1988) and in Thavaneswaran and Heyde(1999) using estimating function theory.An onlinear model was proposed in Abraham and Tha-vaneswaran(1991) and using nonlinear statespace formulation,filtering and smoothing had been studied(see Granger,1998 for moredetails).Many financial series such as re-turns on stocks and for exhibit leptokurtosis and time-varying volatility.These two fea-tures have been the subjec to fextensive studies ever since reported them.The random coefficient autoregressive(RCA) models,the autoregressive conditional heteroscedas-tic(ARCH) models(see Engle,1982;Engleand Gonz a lez-Rivera,1991) and the general-ized autoregressive conditional heteroscedastic (GARCH) models of Bollerslev (1986), provide a convenient framework to study time-varying volatility in financial markets. In Thavaneswaran etal.(2005) and in Leipus andSurgallis(2003), the correlation proper ties for RCA-GARCH models have been studied in detail. In a ddition to GARCH models,estimates of volatility can also be obtained using historical volatility, implied volatility derived from option pricing formulas(see Anh andlnoue,2005, for example)and stochastic volatility models (for example,see Taylor,1994). For continuous-time s-tochastic volatility models that allow for long-range dependence.see Anh et al.(2002). In this paper,discussion centers one stimating the model parameter for the class of GARCH and RCA volatility models using combined estimating function theory.Estimating function theory is well suited to financial data(see Bera etal.,2006 and Pandher,2001, for example). The combined estimating function method has been stud-ied by Naik-Nimbalkar and Rajarshi(1995) and alsoby Thompson and Thavaneswaran (1999) in the filtering context(seealso McLeish and Small,1988;Heyde,1997). Another application of the combined estimating function method for hypothesis testingjbased on ARM A model swith GARCH errors is given in Ghahramani andThavaneswaran(2006). Recently, Ghahramani and Thavaneswaran(2009) have studied GARCH model identi-fication by combining leasts quares and least absoluted eviation estimating functions and the method has been applied to identification problems with several real finan-cial datasets. Combined estimating functions had also been studied in Godambe and-Heyde(1987).Random Coefficient AutoRegressive (RCAR) models are obtained by introducing random coefficients to an AR or more generally ARMA model. These models have second order properties similar to that of ARCH and GARCH models. In this article, a Bayesian approach to estimate the first order RCAR models is considered. A couple of Bayesian testing criteria for the unit-root hypothesis are proposed:one is based on the Posterior Interval,and the other one is based on Bayes Factor. In the end, two real life examples involving the daily stock volume transaction data are presented to show the applicability of the proposed methods.In this paper,the combined estimating function method is applied too btain opti-mal recursive estimates of the parameter in autoregressive model swith GARCH errors and of the parameter in RCA models. Combinations of least squares and quadratic es-timating functions, as well as combinations of leasts quares and LAD estimating func-tions,are considered. The following example motivates the use of estimating function theory for recursive estimation of the parameter in certain time series models.For some time series,the conditional moments,which depend on the parameter are of interest. The combined estimating function method has been applied to estima-tion for some volatility models(See Ghahramani and Thavaneswaran [Combining Esti-mating Function for Volatility, Journal of Statistical Planning and Inference,2009,139, 1449-1461] for details). Compared with some other methods,the estimating function method is based on looser conditions.We tend to use nonlinear (quadratic,for exam-ple) combined estimating function. It turns out that the combined estimating function for the parameter in autoregressive with GARCH errors and RCA models contains maximum information. Estimating function method has been applied to one-parameter time series models, for example,recursive estimation for parameter. In this paper,the estimating function method is generalized to multi-parameter models.To demonstrate the theoretical results,we give two numerical examples.Lastly,we list some further work.