Non-Linearity in Econometric Modeling, Vol. 1

Table of Contents

1. Frontmatter

2. 1. Importance of Filters in Data Processing Pipeline

   Sarit Maitra

   Abstract

   Linearity refers to a mathematical relationship within a system or model where outputs are directly proportional to inputs and adhere to the principle of superposition. In contrast, non-linearity describes relationships between inputs and outputs that are more complex, involving feedback loops, thresholds, or interactions that cannot be represented by a simple straight line. In nonlinear dynamic systems, noise can interact with the underlying dynamics to produce complex behaviors such as chaos, bifurcations, or stochastic resonance. It can obscure the distinction between genuine nonlinear patterns and randomness, and in some cases, may even induce or amplify nonlinear phenomena.

3. 2. Volatility Modeling

   Sarit Maitra

   Abstract

   Volatility is a statistical measure of dispersion of returns for a given security or market index. Understanding volatility in time series is crucial for several reasons, especially in fields like finance, economics, control systems, and signal processing. Understanding volatility in finance is crucial because it acts as a bridge between uncertainty and decision-making. It helps investors quantify, price, and manage risk, making it central to everything from portfolio construction and option pricing to financial regulation and macroeconomic policy. In this chapter, we will discuss volatility and how to model it using both traditional statistical approaches and modern machine learning techniques (with an emphasis on traditional statistical approaches), exploring methods that capture its dynamic nature and improve forecasting accuracy.

4. 3. Artificial Neural Networks and Hybrid Volatility Modeling

   Sarit Maitra

   Abstract

   In Chap. 2  we introduced a hybrid volatility modeling framework based on the ARIMA–GARCH approach, which integrates traditional time-series and econometric techniques. While GARCH models have been the cornerstone of volatility modeling due to their ability to account for volatility clustering and persistence, they rely on parametric assumptions that might limit their flexibility. Recent advances in Machine Learning (ML) and Artificial Intelligence (AI), particularly Artificial Neural Networks (ANNs), offer powerful, data-driven methods that can capture complex, nonlinear patterns and dependencies often present in financial time series but difficult to model with traditional methods. This chapter provides a brief introduction to ANNs and their applications, focusing on volatility modeling, to explore how these models can complement or enhance classical econometric frameworks like ARIMA–GARCH by potentially improving predictive accuracy and capturing nonlinearities without strict distributional assumptions.

5. Chapter 4. Dynamic Volatility and Option Valuation

   Sarit Maitra

   Abstract

    This chapter explores both the theoretical and practical aspects of option pricing, focusing on the Black-Scholes model, Binomial Tree method, and the critical role of the yield curve in determining discount rates. It emphasizes advanced volatility modeling techniques, including GARCH, to capture the nonlinear and dynamic nature of financial markets. While traditional models like Black-Scholes assume constant volatility, this chapter demonstrates how integrating volatility forecasts and alternative pricing methods leads to more accurate valuations and better risk assessment. Through detailed analysis of option prices, Greeks, and sensitivity to market factors, the chapter provides a comprehensive framework for understanding and applying modern option pricing tools.

6. Chapter 5. Markov Switching Models, Threshold Auto Regressive Models, and Smooth Transition Models

   Sarit Maitra

   Abstract

   This chapter explores nonlinear time-series models, focusing on Markov Switching Autoregressive (MSAR), Threshold Autoregressive (TAR), and Smooth Transition Autoregressive (STAR) models. MSAR models capture regime-dependent dynamics with probabilistic transitions governed by a Markov process, making them suitable for modeling phenomena such as economic cycles or stock market volatility. TAR models extend this framework by allowing deterministic regime shifts when a threshold variable crosses a certain level, enabling the modeling of systems with abrupt behavioral changes. STAR models further generalize TARs by introducing smooth transitions between regimes, capturing gradual changes in time-series behavior. Together, these models provide a flexible toolkit for analyzing complex, nonlinear dynamics in economics, finance, and other applied fields.