Local lagged adapted generalized method of moments and applications

In this work, an attempt is made for developing the local lagged adapted generalized method of moments (LLGMM). This proposed method is composed of: 1) development of the stochastic model for continuous-time dynamic process; 2) development of the discrete-time interconnected dynamic model for statistic process; 3) utilization of Euler-type discretized scheme for nonlinear and nonstationary system of stochastic differential equations; 4) development of generalized method of moment/observation equations by employing lagged adaptive expectation process; 5) introduction of the conceptual and computational parameter estimation problem; 6) formulation of the conceptual and computational state estimation scheme; and 7) definition of the conditional mean square epsilon-best sub-optimal procedure. The development of LLGMM is motivated by parameter and state estimation problems in continuous-time nonlinear and nonstationary stochastic dynamic model validation problems in biological, chemical, engineering, financial, medical, physical, and social sciences. The byproducts of LLGMM are the balance between model specification and model prescription of continuous-time dynamic process and the development of discrete-time interconnected dynamic model of local sample mean and variance statistic process (DTIDMLSMVSP). DTIDMLSMVSP is the generalization of statistic (sample mean and variance) drawn from the static dynamic population problems. Moreover, it is also an alternative approach to the GARCH (1,1) model and its many related variant models (e.g., EGARCH model, GJR GARCH model). It provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equations. Furthermore, application of the LLGMM method to stochastic differential dynamic models for energy commodity price, U.S. Treasury bill yield interest rate U.S.-U.K. foreign exchange rate exhibits its unique role and scope.