The provided text is a comprehensive overview of the development and application of nonlinear time series analysis, focusing on both parametric and nonparametric methods. It begins by introducing the canonical model of nonlinear time series, which describes the evolution of a process over time with a nonlinear function and stochastic noise. The text then delves into various approximation techniques, including Volterra expansions and local polynomial approximations, which are used to estimate the nonlinear function \( f \) in the model. Nonparametric approaches, such as kernel regression and local linear approximation, are discussed in detail, along with their asymptotic properties and conditions for consistency. The text also explores recurrent neural networks and wavelet transforms as alternative methods for analyzing nonlinear time series. The ergodic properties of nonlinear time series, including Lyapunov exponents and Kolmogorov–Sinai entropy, are examined to understand the long-term behavior and chaotic dynamics of the system. Piecewise linear models and threshold autoregressive models are introduced as popular parametric specifications, with a focus on their estimation and inference. The text also covers models that introduce nonlinearity through the error term, such as the GARCH and stochastic volatility models, which are widely used in financial econometrics. Testing for linearity and Gaussianity, both nonparametric and parametric, is discussed, along with their applications in model selection and hypothesis testing. Finally, the text addresses the forecasting capabilities of nonlinear models, comparing their performance with linear models and discussing the challenges and benefits of using nonlinear models for forecasting. The article concludes with a discussion of non-nested hypothesis testing, which is crucial when comparing models that do not share a common null hypothesis, and provides an overview of the most widely used non-nested hypothesis tests in econometrics.The provided text is a comprehensive overview of the development and application of nonlinear time series analysis, focusing on both parametric and nonparametric methods. It begins by introducing the canonical model of nonlinear time series, which describes the evolution of a process over time with a nonlinear function and stochastic noise. The text then delves into various approximation techniques, including Volterra expansions and local polynomial approximations, which are used to estimate the nonlinear function \( f \) in the model. Nonparametric approaches, such as kernel regression and local linear approximation, are discussed in detail, along with their asymptotic properties and conditions for consistency. The text also explores recurrent neural networks and wavelet transforms as alternative methods for analyzing nonlinear time series. The ergodic properties of nonlinear time series, including Lyapunov exponents and Kolmogorov–Sinai entropy, are examined to understand the long-term behavior and chaotic dynamics of the system. Piecewise linear models and threshold autoregressive models are introduced as popular parametric specifications, with a focus on their estimation and inference. The text also covers models that introduce nonlinearity through the error term, such as the GARCH and stochastic volatility models, which are widely used in financial econometrics. Testing for linearity and Gaussianity, both nonparametric and parametric, is discussed, along with their applications in model selection and hypothesis testing. Finally, the text addresses the forecasting capabilities of nonlinear models, comparing their performance with linear models and discussing the challenges and benefits of using nonlinear models for forecasting. The article concludes with a discussion of non-nested hypothesis testing, which is crucial when comparing models that do not share a common null hypothesis, and provides an overview of the most widely used non-nested hypothesis tests in econometrics.