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Chapter 73 ARM Processes and Their Modeling and ForecastingMethodology

The class of ARM (Autoregressive Modular) processes is a class of stochastic processes, defined by a nonlinear autoregressive scheme with modulo-1 reduction and additional transformations. ARM processes constitute a versatile class designed to produce high-fidelity models from stationary empirical time series by fitting a strong statistical signature consisting of the empirical marginal distribution (histogram) and the empirical autocorrelation function. More specifically, fitted ARM processes guarantee the matching of arbitrary empirical distributions, and simultaneously permit the approximation of the leading empirical autocorrelations. Additionally, simulated sample paths of ARM models often resemble the data to which they were fitted. Thus, ARM processes aim to provide both a quantitative and qualitative fit to empirical data. Fitted ARM models can be used in two ways: (1) to generate realistic-looking Monte Carlo sample paths (e.g., financial scenarios), and (2) to forecast via point estimates as well as confidence intervals. This chapter starts with a high-level discussion of stochastic model fitting and explains the fitting approach that motivates the introduction of ARM processes. It then continues with a review of ARM processes and their fundamental properties, including their construction, transition structure, and autocorrelation structure. Next, the chapter proceeds to outline the ARM modeling methodology by describing the key steps in fitting an ARM model to empirical data. It then describes in some detail the ARM forecasting methodology, and the computation of the conditional expectations that serve as point estimates (forecasts of future values) and their underlying conditional distributions from which confidence intervals are constructed for the point estimates. Finally, the chapter concludes with an illustration of the efficacy of the ARM modeling and forecasting methodologies through an example utilizing a sample from the S&P 500 Index. B. Melamed ( ) Department of Supply Chain Management and Marketing Sciences, Rutgers Business School – Newark and New Brunswick, Rutgers University, New Brunswick, NJ, USA e-mail: melamed@rbs.rutgers.edu